Basic methods of soliton theory

64

I. CONSERVATION LAWS & ALGEBRAIC-GEOMETRIC SOLUTIONS

A fl A{ n Aj (1 ^ ii ^ m). Then for some divisor Q on T, the sheaf Q is isomorphic to the sheaf 0(Q). The sections of the latter sheaf over A are, by definition, meromorphic functions f on A with the condition ( / ) +Q\A = 0, where Q\A is obtained from the divisor Q by neglecting all points not included in the set A. Since q can be deformed continuously to zero (the zero distribution corresponds, by definition, to the divisor Q — 0), the degree of Q is equal to zero. We denote the class of the divisor Q by q. If q coincides with the principal part of some rational function on r , then q = 0 (check this as an exercise). T h e o r e m 3 . 1 . a) The divisor (ip) of a Baker function ip corresponding linearly equivalent to Q in the class q. b) dimH°(V- q) = dimff°(Z> + Q). Proof. Represent

to q is

(1 ^ i ^ m) for

- 9.) on Ai D Aj. Therefore, a Baker function

; q) to the classical problem of the computation of dim H° (T> + Q). In particular, if deg T> ^ g — 1, then V+ Q is not special for q in an open dense everywhere set (with respect to the natural coefncientwise topology on the space of sets of polynomials Mk-1)}). Hence dimH°(V;q) = degT> - g + 1, for such q (see above). This equality will be the fundamental theoretical result in the sequel. Various problems of construction of algebraic-geometric solutions of differential equations will be reduced to it. The reader can find the Riemann-Roch theorem and the above cited results on algebraic curves, for example, in [La], [Fay], and [Mu]. To conclude this section, we turn to the more analytic (and constructive) point of view, connecting Baker functions to Riemann theta functions (see [Ba], [BC], [IM], [Kri3], [Mat3]). C o m p u t a t i o n s of Baker functions via t h e t a functions. Let us choose a basis a i , . . . , ag, 6 j , . . . , bg of 1-cycles on the Riemann surface T with the following

§3. ALGEBRAIC-GEOMETRIC SOLUTIONS OF BASIC EQUATIONS

65

intersection indices: (ai,aj)

(au bj) = SJ,

= (bi, bj) = 0,

where S1 is the Kronecker symbol. We normalize the basis of differentials of the first kind (holomorphic differentials) {«,-} by the condition

L

Uj=6i

(l£i,j£g).

The matrix B — (b{), given by b3 = jh.Wj is called the period matrix. The theta function is defined by the following power series: 0(p) - ^ exp{m(Bk,k) fc€Z» where p G Cg, (x,y) = X ^ = 1 xjVj f ° r relations are easily checked:

x =

+

Kxj)i

a) 0(-p) = 9(p), b) 6(p + k)= 6(p) for k € Z», c) 9(p + b>) = exp(—nib3, — 2mpj)9{p),

an<

2iri(k,p)},

^ V ~ XVJ) ^ ^ ' • ^

e

following

where b3 is the j - t h column of B.

Let us fix a certain point PQ € F. Then the Jacobi map

r 3 P H->. W ( P )

def

gives the vector w(P) which is defined up to elements of the period lattice A = Z9+ 2 ? = i ^fr7 C O and depends on the homotopy class of the path of integration from Po to P. With the help of w, we can relate the (/-dimensional complex torus C ' / A and the abelian variety Jac T. If functions p? = 0(u(P) —p — r)/6(u(P) — r) defined for p € C and some r = r i , r 2 € C are not identically zero as functions of P € T, then their ratio p* /p?2 is a rational function o n T j P because of the properties b) and c) of theta functions. Consequently, under the assumption p£ ^ 0, oo the class of the divisor (p£) of the (multivalued) functions p*. on T does not depend on the choice of r. The corresponding r form an open subset of C depending on p - see below. The mapping a : C / A 9 p •"•(!>?)€ JacT

66

I. CONSERVATION LAWS &. ALGEBRAIC-GEOMETRIC SOLUTIONS

induces an isomorphism between C ff /A and J a c T , which obviously does not depend on the choice of the point P0. Using a we can describe u purely algebraically: a(u>(P)) is the class P - P0 of the divisor P - P0. By the Riemann theorem (see, for example, [Fay] T h e o r e m l . l ) , there exists a class of divisors A 6 J9'1, for which 2A = K (K is the canonical class of V) with the following properties. We define w(T>) G Cg mod A for a divisor V of degree g by the relation a(w(V)) = V - A - P0. Then a) every nonspecial V ^ 0 is zero or the divisor of the (multivalued) function 0(w(P) - w{V)) of P G T; b) V is special =*• 9{u{P) - w(V)) = 0 for all P; c) V-P0 is special =*• 0(t»(2?)) = 0. C o r o l l a r y 3 . 1 . If a divisor S of degree g— 1 with property 2S = K is not special, then

w(s+p0) = c+J2bbi> J'=l

where ( = %),

£ = \(j)

€ \Z« C C.

(see e.g. [Fay] Corollay 1.5).

Proof. Because of the property c) of the mapping w, it is enough to show that 0(z) ± 0 =* %C G | Z for any 2 = C + £ ' = ! & * ' » £ £ € ^ Z 9 , where *££ = 53?=i Cj£j G | Z . But by properties a), b), c) of theta functions, we have: 0{z) = 6{-z) = 0{z - 2z) = e x p ( 4 « *£C)0(*)• The definition of 6 and w and the proof of Corollary3.1 can be given in a completely algebraic way (D. Mumford). In any case, it is easy to establish the following. E x e r c i s e 3 . 1 * . Without resorting to theta functions, and using the Riemaim -Roch theorem, show that a) for arbitrary {Pj} C T, 1 ^ j ^ j - 1, the divisor K — £)?=i Pj J S linearly equivalent to a divisor of the form ]C?=i Pj, Pj G T, where K is the canonical divisor ofF; b) for every x G Jac T the intersection of the divisor i%2jZi Pj ~ (ff~ l)Po} C Jac Y shifted by x and V = a(u>(T)) C J a c T either coincides with T or is represented in the form {Qj — Po, 1 = j = 9} for some nonspecial divisor Q — 5 ^ ' = 1 Qj, Qj € T fuse a)). • For a distribution q we can make a differential w ? on T of the second kind (with zero residues) that has principal part dqj at the points Pj (1 ^ j ^ m), is holomorphic outside suppg = {Pj}, and is normalized by the conditions / o . u)q = 0. Set vq = j ^ \Sb ui> • • • ' fb w ?) € C s . If q is the principal part of a rational function, then vtt — 0.

§3. ALGEBRAIC-GEOMETRIC SOLUTIONS OF BASIC EQUATIONS

67

P r o p o s i t i o n s . 1 . IfV ^ 0 is not special of degree g then the function

belongs to H°(T>; q). Also a(vq) = q (see above). Here JPo u)q is calculated for the same path from Po to P as w(P) for Po € suppg. Proof. Add to the path from Po to P a 1-cycle homotopic to 2 > - i ( a j a . ; + Pjbj) for some aj,fSj e Z . Then exp(fp

u>q) is multiplied by exp{—2iri(v q ,/3)}, where

(3 = Xftj). It follows from the properties b), c) of theta functions that the ratio of theta functions in the formula for ip is multiplied by exp{27ri(vq,P)}. Hence ip is a single-valued function on T with exponential singularity of the form required in Definition3.1. Calculating the divisor of
any Baker

function

3.2. T h e main construction. Let A be a rational function on T of degree n. This means that the divisor of poles of A is represented in the form (A)oo = S i l l ef^R-Vi where S i l l e^° = n {ef° € Z+ is the multiplicity of i?f°). The function A defines a map T \ {i2f°} —»• C, which can be extended to a regular covering A : T —¥ P 1 of degree n. Here P 1 = C U oo is the projective line over C with A considered as the coordinate. The fiber of A over a point z 6 P 1 defined with the multiplicities is denoted by (A) z : (A) z = S i l ' i ef-^f» S i l ' i el — ni where {iZf} = A - 1 (z) and ef is the ramification index of A at the point R*. Points iZf are taken pairwise distinct. We fix a certain indexation of those points and choose a local parameter ki
q = q0 + xq1+tq~\

q = q0 + xq1 + tq'c2,

where x, t £ K, c 6 C*, go is an arbitrary distribution on I \

68

I. CONSERVATION LAWS & ALGEBRAIC-GEOMETRIC SOLUTIONS

Let us fix a divisor P on T of degree g + n — 1, the support of which suppZ> does not intersect A _1 (oo) = {iZ? 0 }. Further we always assume that x and t are taken from the subset in R 2 , for which the divisor V + Q — (X)z of degree g — 1 is not special, where Q 6 g i n the notations of the previous section (see Theorem3.1). This subset is open and dense everywhere. The concrete choice of z is not essential here, because all fibers of A are linearly equivalent. We use the abbriviations k{ = fci)00, e* = ef°, m = m°°, iZ; = R?°. Let us introduce the functions fr, ( l ^ r ^ m , l ^ s ^ e r ) i n H°{T); q) normalized by the conditions

(*J-V.-*r e 'W*>) = 0

(3.2)

for 1 ^ j ' ^ m. Because of the nonspeciality of the divisors V + Q — (A)oo of the Baker functions, frs are uniquely determined by (3.2) and form a basis in H°{T>; q). We change the indices as follows: fi = frs for i = s + e r _i H and put (p = \f\,...

1- ex + e 0

(1 ^ i ^ n, e 0 = 0),

,fn).

T h e o r e m 3 . 2 . There exist unique g( n -vaiued functions U,V(x,t) and U,W(x,t) (q of (3.1B)) such that

(q of (3.1A))

U

(3.3) V

(3.3A,B)

<*

=

I+A*'

(

Vt =

c2U

W

\(i=w

+

l

T=\JV-

Proof. It results from the nonspeciality of V + Q — (A)oo that the functions '(^ _1 )v«)> where not all c,- € C are zero and r* are some polynomials. But then £ " = i ayi € H°(V - (A)^,; q) = {0}. Using the nonspeciality of V+Q+l{\)z) when / ^ — 1 and the linear independence of {fi}, we obtain that (z — X)'fj for 1 ^ i ^ I, 1 ^ j ' ^ n form a basis of the space H°(D + l(\)z); q). Functions (fi)x and (fi)t in the case A, B belong respectively to fl°(X> + (A) i ; g) and #°(2>-|-(A)_i;g) (A), #°(2> + 2(A) + 1 ;g) (B) (additional poles arise only because of the differentiating the exp factor). • Note that if we replace the divisor T> with an equivalent W, the matrix functions corresponding to V, U', V, W are obtained from U, V, W by conjugating by a

§3. ALGEBRAIC-GEOMETRIC SOLUTIONS OF BASIC EQUATIONS

69

constant invertible matrix (check this as an exercise). In particular, we can also relate the matrices U, V, W, defined up to a conjugation, to a divisor V containing points of A _1 (oo) = {Ri}. In order to do this, it is necessary to replace V by equivalent V, the support of which does not intersect A - 1 (oo), and to apply the costruction of Theorem3.2. It gives the following. Exercise3.3. Let I he a distribution on T such that vj € A (under t i e notations at the end of §3.1), U = U(qo) be that of Theorem3.2. Here we show explicitly the dependence of qo from the definition of q (see (3.1 A,B)). Then for a constant invertible matrix £ = £j a) U(qo + I) = £U(qo)£~1 (analogous statements hold for V, W with the same

£); b) Sp(UV)(qo+l) = SP(UV), Sp(UZ)(q0 + l) = Sp(U2x)(q0), i.e., Sp(UV), are quasi periodic functions of x, t. •

Sp(f/J)

Jordan form of U, V. We suppose that for « £ C U o o the fiber (A) z does not intersect s u p p P . We make an n x n matrix $( r ) of "values" of

w e * a ' £ e the coefiicient of kf.~* in the expansion of the function

*{Z) =

(
For arbitrary z the matrix $( z ) is invertible, since V + Q — (A)2 is not special and supp V n A - 1 (z) = 0. For z = oo, $(<») = I because of the normalization condition. Note that the columns of $( z ) are permuted when we change the indices of the points in X 1(z). Until the end of this section we suppose that the local parameters fcr± satisfy the relation fcr± = 1 =(= A for 1 ^ r ^ m * = m * 1 . In a neighbourhood of each point R* = R*1 € A _ 1 ( ± l ) we write the distribution q*1 in the form

(3-4+)

g

+ i = ^ ^ ( f c r - + 1 ) ^ - + 1 + ..., 3=1

( 3 - 4_ )

9"1 = E "rAKlY7-s+1

+ •••,

70

I. CONSERVATION LAWS & ALGEBRAIC-GEOMETRIC SOLUTIONS

modulo holomorphic part for appropriate /i r > J , ur>, € C. The latter expansion is only for the case (3.1 A). We make a n x n matrix from m + square blocks U£,... , UQ = ( m [ ' J ) , . . . , U™ located along the diagonal constructed as follows. For i > j set m*'1 = 0, for i ^ j set ml'3 = fj,rj-i+i, where 1 ^ », j ^ e+ are the indices of rows and columns. Analogously we make V0r and Vo replacing + 1 , (M by —1, v. P r o p o s i t i o n s . 2 . The (constant) matrices Uo, VQ defined for t i e distributions q±x (see (3.1), (3.4)) are equivalent to the matrix functions U, V in Theorem3.2 with respect to conjugation by invertible matrices. Proof. P u t %/;(*) = ( $ ( ± i ) ) - 1 V - Then V (± ^exp(—q) is normalized in a neighbourhood of the fiber (A)±i by the relation (3.2) for fcJ± instead of kj, and (3.5)

( V ( + ) ) * = (1 - A)(*(-11)CT*(+1)),fr<+> - $ ( + 1 1 ) ( $ ( + 1 ) ) ^ ( + ) .

Differentiating the exp-factor of the expansion of •t/>'-+) in a neighbourhood of the points of A _ 1 ( ± l ) , we can easily show that ( V ^ + 0 * IS written in the form (1 — X)Uoip^ modulo regular terms. Consequently $7? 1 jC/'$( +1 ) = UQ. Analogously, replacing + with — we can check the relation $7^ 1 jV$(_i) = Vo (for the case (3.1A)). • E x e r c i s e 3 . 4 . P u t y,T = p r > 1 (1 ^ r ^ m = m = 1 j . We denote the order of zero of(l — X)q+1—fxratthepointR^1byK,r(l^Kr^er = e* 1 ). Let [x] be the integer part of x £ R and Ma(s) be the Jordan block of order (s + 1) x (s + 1) with a € C on the principal diagonal and with the unity (if s > 0) on the (right) neighbouring diagonal. Then the Jordan standard form ofUo and, hence, ofU consists of K\ blocks Af / . i ( [ c i - 1 / ' « i ] ) » - - - ,M„1{[ei-K1/Ki]),K2 blocksMll2([e2-l/K2]),... ,Af„a([e2K2/K2]),..., Km blocks Mpm([em - l / « m ] ) , . . . ,AfMm([e /Km])- • T h e o r e m 3 . 3 . a) The functions U, V considered in Theorem.3.2 satisfy system (0.2) in the case (A); U, V are equivaient respectively to the matrices Uo, Vo in Proposition3.2. If supp2?D(A)o = 0, then for arbitrary invertible constant matrices G\, G2 the function g = G\^(o)G2 is a solution of equation (0.1). b) In the case of (3.IB), let q1 = q]. = (1 — A) -1 Ci in a neighbourhood of the points i2+ € A _ 1 ( + l ) for 1 ^ r ^ r<j, q1 = qj. = (1 — X)~1c2 in a neibourhood of the Rf for ro < r ^ m+ and ef H \-e+ = p, where c\,c2 € C, c = c\ — c2. Then W = [U, Ux] and U is a solution of the GHM equation (0.9) with the condition (0.10). Proof. For an open set 0 ^ A C C the fibers (X)z z € A are simple and X x(z) H s u p p P = 0. We can assume that the points R\ depend on z € A continuously for a

§3. ALGEBRAIC-GEOMETRIC SOLUTIONS OF BASIC EQUATIONS

71

fixed index 1 = i = n. It results from (3.3) and (3.4) that U, V, and $(A) = f $ ( A ) satisfy relation (0.3) (see Introduction), where A € A. Therefore U,V(x,t) are solutions of (0.2) because A € A is arbitrary and $(A) is invertible. (We use $(*) for the sake of simpUcity. One can obtain (0.2) directly from (3.3,3.3A)). In a neighbourhood of the points B° (1 _ r _ m°) we have: 1 — A = 1 + 0(fc*°0). Hence, (3.3, 3.3A) imply that ($(o))* = *7$(o), ($(o))t = ^$(o) andgr = G i $ ( 0 ) G 2 satisfies (0.1) (see Introduction). Let us turn to statement b) and consider the functions
=

(p(plkr+)

in a sufficiently small punctured neighbourhood of each point i2+ £ A _ 1 ( + l ) for p — exp(27n'/e+), 0 _ / ^ e+ — 1. Then the functions of 1 — A,

? " =f 1+Kl-V '£ p*'"1 V ° , er

(1 = ^ e+)

i=o

satisfy (3.3, 3.3B) in a suitable punctured neighbourhood of + 1 € P 1 . After the obvious change of indices (cf. the definition of
(1-A)™'

wiere m _ 1, {fMr,6r} C C, 5r = 6, if fir = p.a. Construct the vector Baker functions

72

I. CONSERVATION LAWS & ALGEBRAIC-GEOMETRIC SOLUTIONS

proof of Theorem3.3, b) and (3.5)), check that Q = S T ^ f y + i ) ) * is a solution of equation (2.5), §2 (cf. [Du2], [Kri3]). 3 . 3 . Reality conditions. Let T be a ramification divisor of the covering A : T - • P 1 of §3.2. By definition T = £ p e r ( e ( - f ) — 1)-P> where e(P) is the index of the branch of A at the point P . If P = R'T for z € C U oo, 1 ^ r ^ mz (see above), then e(P) = ez. The divisor (dA) of the differential (the differential 1-form) dX on T is easily calculated via T: (dX) = T — 2(A)oo. Since (dX) belongs to the canonical class, and, hence, its degree is equal to 1g — 2, then d e g T = 2(g + n — 1). The last equation is a special case of the Hurwitz formula (see, for example, [La]). For a rational function / on T we denote the trace of / with respect to A by Tr / . It is a rational function of A defined as the sum of the images of / by n distinct embeddings of the field C ( r ) of rational functions on T into the algebraic closure of the field C(P J ) = C(A) of rational functions of A. If the fiber (X)z = Y^i e*Rzr does not intersect the poles of / ,

Trf(z)=J2
*€P J .

An important (and denning) property of T is that for any function / € H°(T) the function T r / does not depend on A (i.e., is constant). Let us suppose further that T is equipped with a real structure introduced as an anti-involution a of the field C(r) (an automorphism of order two, acting on z G C C C ( r ) by the usual conjugation z •-»• z). the action of
(3.6)

Kp) = r{Pa).

The point P" is uniquely determined by (3.6), since we can compute the value of every rational function in C(T) at P" by this relation. The set T R = { P € r , P" = P } of real points of T is either empty or a disjoint union of the components diffeomorphic to the circle S 1 (called the real ovals of T). The action of cr is naturally transferred to the divisors (their linear equivalence classes) and distributions by extending the pointwise action of a on their supports. If r is supplied with projective coordinates (i.e., T is detrmined by polynomial equations in an appropriate projective space), then the complex conjugation of the coordinates gives an anti-involution on T, if the ideal of the defining equations of r is invariant under the coefficientwise conjugation. In this case real points of T are simply points with real coordinates. This more traditional definition is less

§3. ALGEBRAIC-GEOMETRIC SOLUTIONS OF BASIC EQUATIONS

73

convenient for us because even in the simplest cases (for example, for hypereUiptic curves) it is easier to introduce a directly on rational functions generating C ( r ) , without projective embeddings. Let us assume that A" = A hereafter. Then we can choose the system of local coordinates fc;iZ at points Rz € (A) z , z 6 C U oo in such a way that kt 'x = ±(z — A) or ± A _ 1 for z 6 C, z = oo and that kfz — kjtj for all pairs (Rz)" = Rj. Let us check it. In fact, fcI]Z = (±(2 — A)) 1 /*' for : £ C and k{ = A - 1 / 6 ' for z = 00 are defined up to a root of unity of order 2ez. Let us denote the sign of (z — A) or A - 1 by £i,z. We can put £i:Z = + either for (Rz)" ^ Rz or for odd e\. Then it is easy to find the desired it. If (Rz)a = R\, e\ = 21, I € Z+, and ifc?z = pkilX for p' ^ 1 and some kitZ = (z — A ) 1 ' 6 ' , A - 1 ' " , then it is necessary to take £j )2 = —. In this case ki}Z = C^t.z f ° r C2 = P w > u be a real parameter at Rf of the desired type. Let us take the distribution q and the divisor T> of Theorem3.2, satisfying the relations (3.7a) (3.7b)

q" = -q, V

+ V ~ T,

which are consistent in the following sense. If q and V are subject to (3.7), then Q" + Q ~ 0 for Q € q and T> + Q has proberty (3.7b). Here (3.7a) consists of three relations of the same kind for q>±1 and qo (recall that x, t are real parameters). The equivalence of 2?" +2? and T means that TD" + T>—T = (w) for a rational function w on T. Being the ramification divisor, T is obviously (T-invariant (though the points of its support can be non-real). Hence we can choose real w (with the property w" = w). Let us suppose that for some z £ P ^ = R U 00 the fiber (A)z does not intersect supp T> (hence supp T>" either). The permutation of the points of A - 1 (2) = {Rz, 1 ^ r 5= mz} under the action of a will be denoted by c = cz : (Rz)" = Rzciry Define the permutation d of indices from 1 to n by the relation c(r)-l

d{ez0 + • • • + e'T_x + s) = J2

el + s,

t=0

where e£ = 0, 1 ^ r ^ mz, 1 <| 5 ^ e*, e* = e(.R*). Put

In a neighbourhood of Rz (1 ^ r ^ m) we expand ez(k-i)w = j2
74

I. CONSERVATION LAWS & ALGEBRAIC-GEOMETRIC SOLUTIONS

modulo holomorphic part. For 1 ^ r ^ mz consider a e* x e% matrix Wr, setting W = (w^), where W? = w*^^ (1 <| i,j g ez) if * + j - 1 ^ e* and iu,r,i = 0 z if i + j > e + 1. Let us recall that £',* = £r,z(z — A),e r t jA _ 1 for z ^ oo, z = oo, e r ,z = ± 1 . Arranging W 1 , . . . , W,... , Wm along the principal diagonal, we get def

a matrix Wz of order n x n. Finally, let flz = WZCZ. Since wze ^ 0 for e = e*, 1 SI r ^ m r , ft* is invertible. The matrix flz is hermitian: ftj" = flz, where the sign " + " means hereafter the hermitian conjugation (composition of the transposition and the complex conjugation). In fact, (Wr)+ = W c ( r ) because of the reality of w and the relation k%x = fcc(r),z for 1 ^ r ^ m r . The conjugation Wz -> CzWzCj1 induces the permutation c = cz of the blocks {W,-}. We set fi = Sloo- Let us define $( z ) by the above system of "real" parameters {fcj)Z}. T h e o r e m 3 . 4 . For
gn;

b) ^ ( z j f t , * ^ ) = f i , z € R U o o , s u p p D n A _ 1 ( i ) = 0; c) c/fi + QC/+ = o = vo. + nv+. Proof. First of all, the trace of u>' = Tr(ipiW
Res,(Tr(/)
For z ^ oo we have u{ = Resz(Tr(ipiW(p')(\

- z) _ 1 d(A - z)) =

m'

= J2er ^R'r (rfcr,rz in a neighbourhood of Rz. Strictly speaking, in this computation we can take arbitrary constant er,z instead of e r)Z = ± 1 . Expressing e^.k~\w via wz 3 (see above), we find that the matrix (u>l) coincides with $ ( z ) f t z $ j \

§3. ALGEBRAIC-GEOMETRIC SOLUTIONS OF BASIC EQUATIONS

75

(this is also true for z — oo). In particular, the latter matrix does not depend on the choice of z (and on the parameters x, t € R either) and (u>l) = $(oo)^$(oo) = ° " To prove b) we use the relation ($( z ))i = (1 — z ) - 1 ( 7 $ ( z ) for a simple fiber (\)z, z ^ ± 1 which does not intersect suppl? (see Theorem3.3). The relation $ + = ftj^T^ft gives c) for U. Analogously we can check c) for V. D The matrix ft-1 can be regarded as an hermitian form: ft_1(a,6) = ( f t _ 1 a , 6 ) , where (a,b) = b+a is the standard hermitian form, a,b € C". With respect to the anti-involution defined by this form (sending A to A* = ftj4+ft-1) the matrices U, V are *-anti-hermitian because of Theorem3.4,b). If supp2? ("1 (A)o = 0, then g — $(o)$/"^ is a *-unitary solution of (0.1) for $( 0 ) = $o(x,i), where x, i are some fixed (admissible) values of x, t. This is a result of claim a) and Theorem3.3a). Note that LJ and ft-1 are defined up to multiplication by real constants. We will show that linear equivalent divisors V lead to the equivalent forms ft-1 (with respect to base changes in C" and multiplication by constants from R*). We can examine the form ft(a, 6) = (fta, b) instead of ft-1, since they are equivalent (ft+ft-^ft = ft). If V = V + ( / ) , where / € QT), then the fuction w' such that (to') = V + (Vy - T, (w'y = w' is of the form w' = pfwf for suitable p € R*. Here, as for T>, suppl? 7 n A - 1 (oo) = 0. Hence we can introduce an invertible matrix ft' = ft^ for w'. Set / = J ^ i U,,K + ° ( f c r r ) in a neighbourhood of Rr € A _ 1 (oo) and define the matrix F following the construction {w*a} i-+ Wz with /,-,, instead of {w*a}. Then the matrix C00F+C00 corresponds to {/",} and ft' = W^CX = pFW^CcoF+CooCco = pFSlF+. The equivalence of ft and ft' can be established in a different way. Let us take a fiber (A) z not intersecting suppX>,2?/. Then ft i s equivalent to ftz and ft' is equivalent to Q'z by Theorem3.4b). If (A)z is simple, then ftz and ft^ are diagonal and il'z = pFQzF+ forF = diag(/(i?f),...,/(^)). Let J$+n~x be the variety of linear equivalence classes of divisors of degree g + n — 1 satisfying condition (3.7b). In the notation of §3.1, J9+n-l

=

^ g jg+n-1^

+ x =

f}.

The above construction makes it possible to assign an equivalence class of nondegenerate hermitian forms ft (up to proportionality) to each class V € l 7 / + " ~ 1 . For this purpose one can either: a) choose V € ~D not intersecting (A),*, and construct ft' from V or b) given any Z> € £>, define ftz at a general enough fiber

76

I. CONSERVATION LAWS & ALGEBRAIC-GEOMETRIC SOLUTIONS

(A) z . The equivalence class of Q' or Qz is the class of hermitian forms corresponding to V. If (p+,p_) is the signature (type) of the form fl, the map T> t-t K = \p+ —p~\ is extended to a continuous map /c:J'+"-1->{0,...,n}. Really, any continuous deformation of V in J9+n~1 can be lifted to a pointwise continuous deformation of divisors T) € V not intersecting (A)^,. Theorem3.4b) gives that « is constant on connected components of J9+n~1 • Exercise3.6*. J9+n~x / 0. (see [Fay]; a purely algebraic proof is in the author's paper "On reality conditions in finite zoned integration", Doklady of the Academy of Sciences USSR, 1980, 252-5, pp.1104-1108). • In the particular case of K = n, which will be mostly considered further, the statement of the exercise is checked below. P r o p o s i t i o n s . 3 . a) The variety J'9+n~1 is a principal homogeneous space over a commutative (real) Lie group J® = {x £ J° = S&cY,x" = — x}. In particular, j-g+n-i an^ j - o are j s o m o r p f l i c as smooth manifolds. b) The connected component of zero {J®)o of the group J® is isomorphic to the torus (S1)9, S1 d= {z € C, \z\ = 1}. The factor group J°/{JS)o is isomorphic g d to Z2~ , where 2 is the number of real (a-invariant) points of order 2 in J°. In particular, j° is isomorphic to the disjoint union of2d~9 copies of(S1)9. Proof. lfV € J/" 1 "" - 1 , then the map J? B x i-> V + x € J9+n~l is an isomorphism (2?i,X>2 € 3%+n~X =^ T>i -V2 e J°). But the existence of at least one point D £ J$+n~^ is guaranteed by the statement of Exercise3.6 (see also Proposition3.4,b)). Let us denote the kernel of the map e : J° B x >-> x—xa € J® by J® . Since every connected Lie group is divisible and e(y) = 2y for y 6 J°, then e(J°) D {Jo)oMoreover, s.(J°) is connected because J° is, and e{J°) = («7
§3. ALGEBRAIC-GEOMETRIC SOLUTIONS OF BASIC EQUATIONS

77

is isomorphic to Z*. Since S C J„ D J® and d ^ g (see above), the factor group {JS n J ? ) / 5 is isomorphic to J°/{J°)o• A n t i - h e r m i t i c i t y . Here we discuss the condition which implies the (positive or negative) definiteness of the form fi considered above (i.e., the equality K = \P+-P-\

= «)•

P r o p o s i t i o n 3 . 4 . a) If there exists T> € J " / + n _ 1 for some K = n, then all fibers (A) z , z g i U o o are simple and real. b) Conversely, the reality and simplicity of all (X)z, where z g K U o o , iead to a representation T = VO + T>Q for some Vo ^ 0. Such a divisor Vo corresponds to the form Q.0 with K — n. Proof. First, let us suppose that a certain fiber (A) z , where z 6 R U o o = P R , has non-real points. Changing the point z a little (in the topology of P R ), we may assume that (A) z is a simple fiber. Take a divisor V € V such that X~1(z) ("I suppZ> = 0. Then the form fl* = diag(u>(iZ*))Cr for the permutation matrix Cz of a acting on {R*}, should be definite. However Cz ^ I and {Cza, o) = 0 for a vector a with only one nonzero component not stable under the action of Cz. This contradiction means that Cz = I. Second, Qz is equal to diag( W ) because of the "absence" of Cz at an arbitrary (not necessarily simple) fiber (see the definition of flz). If some index e* = e is not equal to the unity, then (Wa,a) = 0 for o = \0,... ,0,1) € C 6 . Hence ezT = 1 and the fiber is simple. Conversely, if T does not contain real points then supp T is represented as follows: T = T>o + T>Q. For such Vo we can take Wo = 1- The reality of the fibers (A) z , z € P i , results in Q° = I. • Till the end of this section we assume that V, cr, A satisfy the conditions of Proposition3.4. Then T is separable: T \ TR consists of two connected components (by definition). Indeed, P 1 is separated by the circle P K into two components. Therefore the image (with respect to A) of any path connecting two points P\, P2 € T with projections A(Pj), A(P2) in different components of P 1 \ P R intersects Pjj and, hence, the path itself intersects A J (P R ) = TR. The number of components does not exceed 2. This is a general fact. In the given situation it is easily derived from the existence of A. T h e o r e m 3 . 5 . a) If K = n for V € J§+n~1, then there exists a divisor V € V such that V - (>)«, ^ 0. b) In the case K = n, the function

78

I. CONSERVATION LAWS & ALGEBRAIC-GEOMETRIC SOLUTIONS

Proof. Constructing the function w by V, put u = wdX. Since fiz is definite for any z € Pjj, w has a constant sign over any oval of TK (the poles of w belong to s u p p T C T \ T R ) . The form u is regular on V. Choosing the orientation on Pjg and using A, we can pull it back to an orientation on ovals of TK. With respect to this orientation, w does not change its sign on FK, which contradicts its regularity because of the Cauchy theorem (fr w is zero for regular LJ). Let us recall that tp, U, V were constructed under the condition that "D+Q—(A)oo is non-special for t>, q satisfying (3.7), Q € q. However, because of statement a), any divisor of type (3.7b) with K = n (in particular, 27 + Q) becomes non-special after (A)oo is subtracted. • Exercise3.7. Prove directly that any solution g of equation (0.1) constructed in Theorem3.3 is regular for all x,t € R, if it is unitary. (Show that g can have only poles with respect to i or t. Such a singularity necessarily causes a singular point of the function gg+ = I). D T h e o r e m 3 . 6 . The form Q is definite only for one component ofJ£+n~1 . Every other component contains at ieast one class with a representative T> such that T2 — (^)oo = 0> which leads to the existence of singular solutions U,V(x,t) for a suitable choice of the initial data (2?, qo). Proof. The set of classes of divisors S € J9+"-1 for which 2 5 ~ T is identified with Z 2 9 in the following way (see Corollary3.1 §3.1). Introduce a quadratic form

onZ 2 s = {(x;y), x,y e z^},

9

{(x; y)) = tyx = Y^l *tyi mod 2, i=i

where x = \xi), such that

y = \yi).

Then the class S corresponds to an element 5 € Z 2 S

H°(S-(X)oo)

=

{0}^(s)=0.

As follows from Proposition3.3, the classes 5 belonging to different connected components {J§+n~1)a C Jl+n~x correspond to different affine Z2-subspaces £ a C Z 2 * by the mapping S •->• s. These affine subspaces E a do not intersect and are translations of a certain linear Z2-subspace Z 2 = E C Z29. By virtue of Theorem3.5a), the component ( J r / + n - 1 ) o containing the divisor VQ of Proposition3.4,b) corresponds to an isotropic affine subspace Eo (3 G Eo =£• (s) = 0). The dimension £0 over Z2 ( = dimzj E) is equal to g, i.e., the half of the dimension of Z 2 S . Therefore Eo is a maximal isotropic affine subspace in Z 2 * (check this as an exercise or see [Fay]). Any other connected component {J$+n~1)\ not containing divisors S such that

§3. ALGEBRAIC-GEOMETRIC SOLUTIONS OF BASIC EQUATIONS

79

S — (A)oo ^ 0 corresponds t o K = n and is mapped to S i ^ So- But then So U S i becomes an isotropic affine subspace of dimension g + 1, which is impossible. • We can assign not only «, but also a sharper discrete invariant II = ±{sgnj w} to each component of j % + n ~ 1 . Here w is a function constructed by V from the corresponding component, sgn • w = w/\w\Tj is the sign of w on the j - t h real oval r R of the curve T (1 ^ j ^ p, p ^ n). The set ±{sgn • w} does not depend on the choice of concrete T> and w and is a function of the component. The sign of w on each oval T^ is constant, since (w) — V + V = T and, hence, the poles and zeros of w on Tin have even multiplicities. Note that K = | Yl*j=i h sSaj w\i where lj is the degree of the covering S1 = F}R -*• P 1 3* S 1 induced by A. In a more general situation of an arbitrary real curve T that is a union of p real ovals r R (TR = (J>=i F R ) w e c a n construct an analogous invariant II' as well. Instead of JS+n~1 we take Jg'1 - {x 6 J9~x,x + x" = K}, where K is the canonical class of I \ There exists a real differential form w on T with divisor (w) = V + V" for each V € x £ Jj>~1. The zeros and poles of u on T R have even multiplicities. Therefore, fixing the orientation on one of components of T \ TR and the corresponding orientation on TR, we obtain that w is of constant sign on each oval r R . Hence we can set II' = ± {sgn^ u>}. In the case considered above the mapping V i-> V — (A)oo results in an isomorphism J%+n~1 —> J%~x and the coincidence II' = II, since w = wdX. Theorem3.5 can be generalized as follows: Exercise3.8*. T i e invariant II' establishes an isomorphism of nected components of » 7 / - 1 and the set {(±1) 7 } mod (±1). In number of connected components is equal to 2P~1. The component sponding to II' = { + 1 , . . . , + 1 } mod (±1) does not contain x 9 D [Fay], Proposition 6.8). D

the set of conparticular, the of J$~x correfor V ^ 0 (see

3.4. Curves w i t h an involution. The previous discussion gave us algebraic-geometric solutions with values in fl[„ and its real forms (u n etc.). In order to obtain solutions with values in other Lie algebras (or Lie groups) or in homogeneous spaces of type Sn_1, we need more special curves. We will describe below methods of constructing of algebraic-geometric Sp2m(C)-fields, 0 2 m(C)-fields, and S 2 m _ 1 -field. All notations from §3.2 remain the same. Let us assume that there is an involution r on T (its action is written on the left) such that r A = A. By definition, r is a C-automorphism of the second order of the field C ( r ) extended to an action on points P € T by the formula / ( TP) (Tf)(P) for arbitrary / 6 C(r) (see (3.6)). We impose the following conditions on a divisor

80

I. CONSERVATION LAWS & ALGEBRAIC-GEOMETRIC SOLUTIONS

T> of degree g + n — 1 on T and a distribution q from (3.1A): q + Tq = 0,

(3.8a,b)

V + TV ~ T,

where the ramification divisor T is defined in §3.3. These relations are self-consistent (see (3.7a,b)): T(D + Q) + (V + Q) ~ T, if Q e q. For a suitable function v € C ( r ) we can set (v) = V + TV — T. Moreover Tv = ±v, since Tvv~1 € C Let us determine which data lead to the sign + . Let us denote the smooth curve with the function field C ( r ' ) = {/ € C ( r ) , rf = / } by T' . Then the covering A : T -> P 1 is factorized as A = A/ o n, where T —>• T' is induced by the embedding C ( r ' ) C C ( r ) and A' : I" ->• P 1 corresponds to the function A restricted to T'. In particular, n = 2m for m = deg A' and T = -K*T'+Tn, where w*T' C T is the pullback (the inverse image) of the ramification divisor T" of the covering A', and TV is the ramification divisor of the covering 7r. If T„ ^ 0, then v cannot belong to C ( r ' ) , since the divisors T> + TV — T and T„ are obtained from some divisors on T'. When TK = 0, the covering n is non-ramified (by definition) and the divisor (v) can be regarded as a divisor on V. However v does not necessarily belong to C ( r ' ) (it holds only when the sign is + ) . P r o p o s i t i o n s . 5 . T i e variety J$+n~1 of the classes of divisors V with properties (3.8b) is connected in the case T„ ^ 0 (in this case Tvv~* = — 1). IfTw = 0 then it consists of two isomorphic components ( j 7 / + n _ 1 ) ± for T* ^ 0 where the signs are the signs of Tvv~1. The connected components of J?+n~1 are isomorphic to the abelian variety P = { Tx — x, x € JT0} (called Prym variety of the covering w) of dimension g — g' = g' + d e * r — 1, where g' is the genus ofT'. Proof, (see [Fay] Ch4; cf. Proposition3.3). The variety J^+n~x is isomorphic to the algebraic group J° = {x S J - 0 , Tx + x = 0}. Let (»7°)o be the connected component of zero of ,7° and (jT°)o be the component of zero of the group J° = {x £ J°, Tx = x}. Then (J r °) 0 + {J°)o D 2J° and, consequently, (J°)0 + {J?)o = J° because of the connectivity of J°. In particular, each component of J° contains at least one point of the group J® n {J?)o, which consists of the points of the second order. Therefore the components of j° are in one-to-one correspondence with the points of the second order in (jT°)o (all of them he in J®) modulo (J°)a O (J?)o d=IT = {rx + x=ry-y,x,ye J0}. Since (g) = TV + V + TV T

- V* «- ( Tgg) =

2n(V),

and C ( r ' ) * = { gg} for g € C[T)*, V,V* € T (see [Se2] Ch.2 §3), then the points of the second order of J'° = Jac V generate JT with respect to the natural map

*• : J'° -»• J° .

§3. ALGEBRAIC-GEOMETRIC SOLUTIONS OF BASIC EQUATIONS

81

When T* ^ 0, n* does not have the kernel (divisors non-equivalent to zero remain non-equivalent after being lifted to T). Indeed, if (/) = TT*V for a divisor V on V, then Tf = ±f, p € C(r"), and ( / 2 ) = W. But then, adding the function / to C(r"), we can construct a curve between T and V which is an unramified covering of r". Hence, / € C(r"). An analogous argument shows that the kernel of ir* is isomorphic to Z 2 when Tw = 0. Really, if / g C ( r ' ) and (/) = ir*V for some 7>', then / is uniquely determined up to the multiplication by elements of C(r")*. Since the dimension of J' is g, the number of points of second order in J' is equal to 22g . Therefore the group Ir contains either Z 2 9 (T T ^ 0) or Z 2 9 ~~ (when T*. = 0). The dimension of (J°)o = n*J' is equal to g'. Consequently, the group of points of the second order (J°)o is isomorphic to Z 2 9 . This proves the proposition. • Following §3.2, let us define the vector Baker function (p = % + Q lies in the same connected component as T>. Theorems 3.2 and 3.3 allow us to construct a solution U, V of system (0.2). Let us discuss the restriction on U, V following from condition (3.3). We define a matrix A z for z £ P 1 = C U oo exactly in the same way as ftz in §3.3 for the function v instead of w. Here we assume that suppl? fl A - 1 (z) = 0 and take the matrix of the permutation r of the points of the fiber (A)z instead of the matrix Cz of the permutation a in the same fiber. However (in contrast with §3.3) *AZ = eA z , where e = 1 only if 7V = 0 and V 6 ( t 7 / + n _ 1 ) + ; otherwise e = —1. Just as in Theorem3.4, we can check Proposition3.6. a.) A = Ax = (Ij), where I? — Ti((piVT
J, where A = diag(t;(.R J )), 1 fS j
(A)cc = E J m = 1 ^ + r (E7= 1 ^)Replace V with an equivalent divisor V = V+(f), f e QT)* (which conjugates U and V by a constant matrix). Then v is multiplied by Tff and the matrix A is replaced by A d i a g ( / ( ^ > ) / ( T i i i ) ) . Changing the function / , we can make A equal to the unit matrix J m .

82

I. CONSERVATION LAWS & ALGEBRAIC-GEOMETRIC SOLUTIONS

02m(R)-fields. In addition to (3.8), now let the assumptions of §3.4 be fulfilled, T>, q satisfy the relations (3.7), and r
J, A = diag(v(i?j)), 1 ^ j ^ m (see above), ft = diag(io(iZi), 1 ^

i ^ 2m. Replacing V by a linear equivalent divisor, we may assume that ft = Iim. Since ( Tw w) = (u v"), then Tww = pvv" for p € R* (here the commutativity of a and r and the reality of w were used). Considering the latter equation at the points of the fiber (A)oo, we obtain that p > 0. Substituting v for p~1l2v, we come to the equality rww = vv", which gives the relation v(Rj)v(Rj) = 1, 1 ^ j S; m. There exists a diagonal matrix A of order m such that AA = Im and A = A2. Let F be (A XA\ the following matrix of order 2m: F = -4= I . . . j . Then we have FF+=I2m.

F'F = A,

Consequently, the pair F~lUF, F~1VF is a solution of (0.2) with values in 02m(R) (due to Proposition3.6c), Theorem3.4c)). The variety J"/"^" - 1 of the classes of divisors V with conditions (3.7, 8b) is isomorphic to the disjoint union of the components (S1)9 , where g = Ig1 — 1. We recommend calculating the number of the connected components of J^n~1 as an exercise. Anyway it is easy to see that J%%n~x does not contain the classes of divisors V for which V — (A)oo ^ 0 (Theorem3.5). Hence U, V are well defined for all x, t (and bounded). Analogous statements hold for s p 2 m ( K ) (instead of 02m(R)) in the case of e = —1. 5 2 m - 1 - f l e l d s . First, we are not suppose n to be even and adjust the construction of §3.2 in order to get a solution g of the P C F equation (0.1) with the condition g2 = I. Let us assume that there exists an involution 7 on T such that 7A = A - 1 . Then 7 permutes (A)o and (A)oo and preserves (A)±i. We numerate the points {R°j} = A-^O), {Rf} = A - ^ o o ) with respect to 7: ^ i J f = RO. i n §3.2 W e constructed a matrix g = $( 0 ) of "values" of the vector function tp at the points of (A) 0 . By virtue of Therem3.3, g is a solution of (0.1). We also assume that V, q (see above and §3.2) satisfy the relations (3.9)

7 9

= 9,

y

V = V.

Replacing V by a linear equivalent divisor, we may assume that 7Z> = V (for the sake of simplicity). This is a special case of the Hilbert Theorem 90. We will sketch

§3. ALGEBRAIC-GEOMETRIC SOLUTIONS OF BASIC EQUATIONS

83

the proof here. Put (/) = "'V - V. Then if f = c € C* and (dividing / by take / such that 7 / / = 1. Consequently, / _ 1 = 7 2/3/ _ 1 , where for a sufficiently general x 6 C{T). The divisor V + (y) becomes 7invariant. Since we replaced V by an equivalent 7-invariant divisor, the new matrix g is conjugated with the old one by a constant matrix. The invariance of V, q gives that 7 acts on the space H°{V; q). Since 7 is an involution, we can choose a basis {£,-, 1 ^ j ^ n} of H°(T>; q) such that ~!, q satisfy (3.7), (3.9). We also assume that Q is definite, in particular, all fibers (A) z , z € R U o o are purely real and simple. As above, we take T> such that 7Z> = T>. Then 7u> = ±tw. However w does not change the sign on c-i/2) w e c a n y = (x + 7 x / )

def

TR. Hence yw = w. In particular, fio = floo = ^- Replacing 3? by an equivalent 7-invariant divisor, we may get tto = fi = ^ Then (see Theorem3.4) grgr+ = / for # constructed above. We need one more involution r on T which commutes with and q satisfy (3.8). Since 7X> = V, then ^w = ±v. Let ^w = u. Then A 0 = A and gh V = A (Proposition3.6). Hence, F _ 1 g F is an S 2 m _ 1 -field. The analysis of the variety of the classes of divisors V = 7X> with conditions (3.7-3.8b) for which A is definite and yv = v is left to the reader as an exercise. In the following section we will discuss examples which illustrate the constructions of this section. Note that the above results are directly related to the Euler equation on 04 and other Lie algebras (see [Al], [Du3], [Manal], and [Chll]). 3.5. Application: discrete P C F equation. In this section, following [Ch9], we briefly describe a technique of integrating certain difference analogues of equation (0.1) and system (0.2). This discretization is based on the process of constructing algebraic-geometric solutions. A more systematic approach to difference analogues of integrable soliton equations makes use of the r-matrix Hamiltonian formalism and the general theory of the r-functions and Hirota equations (cf., for example, [TF2], [DJM], [UT]). We do not discuss these directions here. The integration of difference analogues of GHM and other equations studied above can be done by similar technique and is left to the reader as an exercise (for the HM and the NS, see [TF2], [DJM]; we also refer to the papers of M. Ablowitz with coauthors). As to the theory of the Toda lattice and related topics, see the book of M. Toda "Theory of nonlinear lattices" (Springer), and

84

I. CONSERVATION LAWS & ALGEBRAIC-GEOMETRIC SOLUTIONS

books [CD], [TF2] (the last book contains systematic exposition of various problems connected with models on lattices). The discrete PCF system is the system of equations (3.10a) (3.10b)

^ = -{VkUk at a Vk - Vjt-! = -(VkUk a

-

UkVk^),

- £/*V fc -i),

on the functions {Uk, Vk, VM~I,M ^ k ^ N} of t € K with values in n x n-matrices. Let M ^ N (EZ, K, a € C be certain fixed parameters, K ^ a, K / 0 ^ a. If we take a = 2 and formally replace £/*, V* by Z7(x), V(x) and K _1 (Vt — Vjt+1) by dV/dx we will arrive at (0.2). Similar to (0.2), system (3.10) is the compatibility condition of the equaitons d

**

1

at

a —A

$* - $ t - i =

T/*

r{Uk - l ) $ t - i , K —A

for arbitrary A, M — 1 ^ fc ^ N (I is identified with 1 hereafter). It follows from (3.10b) that ^ ( l - ^ )

=

( l - ^ ) ^

for the powers Vfcs, s = 1,2, Let us assume further that Uk, Vk are diagonizable and that their eigenvalues are in a general position. Then the traces Sp(Vfcs) and (equivalently) eigenvalues {v'(t)} of the matrices V* for 1 ^ i ^ n do not depend on k. Using (3.11a), we obtain that SpJ7jt, Sp{/£, etc. do not depend on t. Hence the eigenvalues {n'k} of the matrices Uk for M ^ fc ^ N do not depend on t either ( cf. the analogous statement for the PCF). Our goal is to integrate system (3.10) under the boundary condition (3.10c)

VM-!C

= CVN

for a diagonal constant invertible matrix C and fixed {v'(t)}, 0 4 } m t n e c a s e °f a general position. Let T, A, (A)2 = JZ"=i Rf denote the same as in §3.3. We take a divisor D o n T of degree g + n — 1 (g = genus(r), n = deg A) and N — M+l divisors Sk = X)£=i Q'k of degree n. Let us suppose that 2 t = A f ( 5 t — (A)oo) = (h) for a certain rational function h on T. Put Vk = V + £ ? = M ( ( A ) « - 5,), M - 1 ^ k ^ iV. We will assume

§3. ALGEBRAIC-GEOMETRIC SOLUTIONS OF BASIC EQUATIONS

85

that neither all Sk nor ( A ) ^ , nor (A) a have points of multiplicities more than 1. Moreover let the support of £>* not intersect the fibers A _ 1 (oo), A x ( a ) . We can introduce the set of Baker functions {k ^ 0. They have essential singularities in the form exp((a — A ) - 1 J0 i/'(r)dr) at the points Rf, and axe normalized by the condition
HI) T h e o r e m 3 . 7 . a) The matrix functions Uk, V* oft are uniquely determined (3.11a) (3.11b)

= - L - VkVk, at a —A
by

M-l^k^N, M^k^N.

K — A

b) The functions {UkyVk} satisfy system (3.10) with the boundary

matrix

c = $W))Moreover (x\ — K~1X(Q'k) and the eigenvalues of {Vj} coincide with v'(t) from the definition of ipk. Proof. It is parallel to the proofs of Theorems 3.2, 3.3. (and goes even when the supports of the divisors and fi, v are not generic). The formulas for fi result from relation (3.11b) at the points Q\ as tpk tend to zero. The expression for C follows from the equation k, let us consider the matrices
~n-9k \9k + gk-i) = (gk-i +

gktekii

86

I. CONSERVATION LAWS & ALGEBRAIC-GEOMETRIC SOLUTIONS

C = (8ih(R°/2)).

where M ^ k ^ N, CgN = gM-iC, imply that:

*-§*.

Relations (3.11), (3.10)

to+„_.*£,-j-i.^-!).

We will also describe the procedure of integration of the "twice discrete" analogue of the P C F equation in the generic case. Given K ^ K, O^K^CC, M ^ N £Z, and a set of divisors Si = J^JLi Q\ °f degree n (M ^ / ^ N), let us define the divisors Vk,i =Vk + Z ) | = M ( ( ^ ) « _ Si), where T>t were introduced above, M - 1 fs I ^ N, and ^2l=jg{Si — (A)s) = (h) for a certain rational function h on T. We define a set of rational functions ^pi' and a vector function (pk'1 by the relations (ipi' ) + £>*,; ^ 0 and the same normalization condition as above. E x e r c i s e 3 . 9 . a) Matrices Uk,i, Uk,t that axe uniquely defined by the relations K — A

K — A

satisfy the equations Uk,,Uk-iJ

- UkjUk,i-i

= Uk,i - Uk,i-i - -(Uk,i -

Uk-i,i),

*{Uk,i - Ut,i-i) + K{Uk,, - Uk-i,i) = 0, with the boundary

conditions CUN,I

= UM-I,IC,

C^jb,jv =

for C, C de&aed with respect to h, h as in

Uk^_1C,

Theorem3.7.

b) The function gk,i = *{$ = (
M-l

^k^N,

M — 1 ^ I ^ TV satisfies the equation (3-12)

9k,l(9k}-i

with the boundary (S'h(R^)),

conditions

C = (5^h(R^).

depend respectively

+9k-i,i)

= (St.'-i + 0 * - i , / K " - i , l - i

CgN,i = giu-ijC,

Cgk^

= 9kM-\C

f°r C —

The eigenvalues fi\, Ji\ of the matrices Uk,i, Uk,i do not

on I and on k and are equal to K~1\(Qtk),

K~1X(Q'k).

•

The equation (3.12) is an analogue of the one suggested by A. M. Polyakov [Pol]. Another variant of the Polyakov lattice (the periodic nonabelian Toda lattice) was

§3. ALGEBRAIC-GEOMETRIC SOLUTIONS OF BASIC EQUATIONS

87

integrated by I. M. Krichever (see [Du2]). In conclusion we mention that there are quite a few papers devoted to the two-dimensional nonabelian Toda lattice including [LeS], [MOP], [DrS], [KW], [UT]. There are also a lot of results on integrating various concrete difference equations including those due to H. Flashka, D. MacLaughlin, S. V. Manakov, 0 . I. Bogoyavlensky, V. B. Matveev, M. A. Sail. E. Date. Our aim here is to demonstrate the fundamental principle of integrating periodic lattices, which is the passage from the continuous parameters x, t connected with distributions on T to discrete parameters k, I corresponding to translations in the group Jac r by divisors of degree zero. 3.6. C o m m e n t s . The formula for the functions on T with exponential singularities and g poles from Proposition3.1 is, in fact, from [Ba] (similar functions also appeared in ealiear works). Those functions were intoroduced by G. Baker to simplify the results by J. Burchnall, T. Chaundy [BC], who studied pairs of commuting differential operators of one variable x of coprime orders. Later this problem appeared to be very important for the so-called "finite zoned integration". If one of these operator is of order 2 (and the order of the second is odd) the description of such pairs is equivalent to integrating the stationary (d/dt = 0) higher KdV equations (§2.5). S. P. Novikov reduced the construction of the finite zoned solutions of the KdV equation to the last problem (see [Nl], [Dul], [IM], [Mar], [Lax] and [McKvM]). In the context of the KdV and similar equations, the Baker (Baker-Akhiezer) functions played the main role in the works by S. P. Novikov, B. A. Dubrovin. A. R. Its, V. B. Matveev (see [DuM]). I. M. Krichever developped the technique of Baker functions and made it more universal. He applied it (see [Kri3]) to obtain algebraic-geometric solutions of the matrix Zakharov-Shabat equations (see §2.4 above). The results of J. Burchnall, T. Chaundy were generalized to differential operators whose orders are not coprime with each other ([Dr], [Kri4]), which gave new types of solutions of soliton equations (see [KN], [Mani], and the papers by S. P. Novikov and P. G. Grinevich). In paper [Ch7] the construction of algebraic-geometric ("finite zoned") solutions of the P C F equations and of general equations with rational zero curvature representation (the Zakharov-Mikhailov systems) was made (including anti-hermitian solutions). The variety of anti-hermitian algebraic-geometric solutions of the P C F (and the general Zakharov-Mikhailov systems) was proved to be connected in [ChlO]. These solutions are always regular (which was demonstrated in the same paper). Concerning the results of §3.4, we note that the curves with complex involution were studied by E. Date, S. P. Novikov, I. M. Krichever, B. A. Dubrovin and other authors for various problems. One can find typical examples applications of the "finite zoned integration" in works [Du3], [Kri6], [N3]. The present exposition is based mostly on works [Chi,7]. To study the real-

88

I. CONSERVATION LAWS & ALGEBRAIC-GEOMETRIC SOLUTIONS

ity conditions we reproved several results from the book by J. Fay [Fay] (to make the book self-contained and because Baker functions are very convenient for many problems concerning real curves and their Jacobians). The reality and regularity of algebraic-geometric solutions are connected with rather delicate (and sometimes insufficiently studied) problems of real algebraic geometry. In particular, it is important to describe the varieties of special divisors T> of type (3.7-9) in the most general situation (when the forms are not definite). In this section we do not touch upon the "completeness" of the constructed families of algebraic-geometric solutions. For this purpose one should define the latter as solutions of higher stationary equations of Lax type (i.e., as generalized "finite zoned" solutions). Actually this "abstract approach" is directly related to the classification of maximal subtori of proper linear groups over projective line P 1 (see [Chll]). There are quite a few more specific questions. For example, we can show that the construction of §3.2 gives anti-hermitian U, V in §3.2 if and only if there exists a certain anti-involution a on the corresponding curve and T> and q satisfy property (3.7). Checking such statements is beyond the framework of this book.

§4. Algebraic-geometric solutions of Sin-Gordon, N S , etc. In §4.1 we construct algebraic-geometric solutions of the Sin-Gordon equation, based on the results of §2.5 (that is, without using the results of §3). These solutions are then lifted to S 2 -fields. In §4.2, without consulting the results of §3, we describe algebraic-geometric solutions of the VNS equation, and analyze the NS equation in detail. §4.3 is devoted to comparing the constructions in §3 and §§4.1,2. In §4.4, as an application, we present a method of constructing algebraic-geometric solutions of the four-dimensional duality equation (the self dual Yang-Mills fields). In the first two sections we use the definition of the Baker function and Proposition3.3 from the material of §3. The rest of the section is independent of §3 except for the proof of Theorem4.4, b) which is based on the results of §3.3. 4 . 1 . Sin-Gordon equation, 5 2 -fields. Let r be a hyperelliptic curve given by the equation •Zg

(4.1)

S = \H(\-aj),

for pairwise distinct 0 ^ otj S C

More precisely, the curve T is the totality of

points (A,/x) (E C 2 satisfying (4.1) as a set, completed by the point Ooo = (A = oo,fi/X = oo). The embedding T \ Ooo C C 2 gives the structure of a smooth (check it) submanifold on r\Ooo- The analytic structure at Ooo is introduced, given a local parameter koo in a (punctured) neighbourhood of Ooo, by the relations A - 1 = fc2^, / i - 1 = k^+1 +0(k^+2). Here 0(fc£f +2 ) is a power series in k^, j S> 2fl- + 2 and the second relation is necessary only to fix the sign &<» = A - 1 ' 2 . The field of rational functions C ( r ) on T is generated by A, fx. By the formulae r A = A, Ty. = — (i we can define an involutive automorphism r on C ( r ) , which extends to the involution of points of F by the relations (3.6). (The coordinate A of an arbitrary point P € T does not change under the action of T, while the /i-coordinate, p{P), is multiplied by — 1.) The action of r on functions, points, divisors and distributions of T is denoted by r (-). The branching points of the mapping A : T —> P 1 induced by A (i.e., fixed points of r ) are the points Ooo, 0\,... , 02g, Oo, defined by A(0 7 ) = ctj, A(0o) = 0 . In a neighbourhood of Oo we choose the local coordinate fco = /u/A. The involution r acts on the parameters fcoo, fc0 by multiplication of — 1. Following the notation of §3.3, let Tr be the trace of A and let T = Ooo + OQ + J Z j i i Oj be the branching divisor of A. By the Hurwitz formula the genus of Y is equal to d e g T / 2 — deg A + 1, and thus equal to g. Let us assume that the coefficients of the polynomial FJ ?=l (A—ay) are real. Then we can introduce an anti-involution a on T, putting A" = A,

fj." = u.

89

90

I. CONSERVATION LAWS & ALGEBRAIC-GEOMETRIC SOLUTIONS

Under the action of a each point P € T with the coordinates (A, /i) maps to the point with the coordinates (A,p). The points Ooo, 0o are real (i.e., er-invariant) and k^a = fcoo, fcg = fc0. As a consequence of A being real, the set of points {Oj} is mapped to itself under the action of a. We call divisors 2? ^ 0 on T of degree g admissible if they satisfy the condition

Va + V~YdOj.

(4.2)

7=1

For each such V we can set V + V - Yf-Lx Oj = (v), where v is a suitable real (v" = v) rational function on T uniquely written in the form g

-1

v = Coo + A* XI ciXi'

(4.3)

7=1

where Coo = v(Ooo) € R, Cj € R. T h e o r e m 4 . 1 . a) A necessary and sufficient condition for the existence of admissible divisors on T is that all real numbers among { a i , . . . , <*2j} are negative. b) Let us suppose that there exist exactly 1m real numbers among {ctj}, 1 ^ j ^ g (their number should be even, since remaining a.j must be pairwise conjugate), and all of them are negative. Then the variety of classes of admissible divisors T> is isomorphic to a disjoint union of 2 m copies of^S1)9. Proof. We suppose that there exists an admissible divisor T>. Then for a function v corresponding to V, applying r to (4.3), we obtain the relation

(4.4)

c L -Tvv

=A^-A'-1) j=\

2

I I ( A - a,-)" 1 . >=i

From Proposition3.3b) it follows that classes of admissible divisors form a (7-dimensional real variety. There exist only a finite number ( ^ 29~m) of admissible divisors with the same function v. Thus, there exists an admissible divisor V for which Coo 7^ 0 and zeros of v do not have multiplicities (in particular, otj 6 R =>• Oj £ suppD,X> ff ). Define Pj by V = Y,'=i pi ^ d P u t Kpj) = Si- T h e n

(4.5)

*v v = *4 ]I(A - Sj)(X - 5j)^2X. 7=1

§4. ALGEBRAIC-GEOMETRIC SOLUTIONS OF SIN-GORDON, NS, ETC.

91

Comparing this with (4.4), we have the identity fj(A >=1

aj)

- Xij^cjX'-1)2^

= ^ ( A - *,-)(A - Sj),

i=l

>=1

and calculation of the constant term of both sides gives us the relation

(4.6)

Uoi = f[SiSi. i=\

>=i

In particular I L i i ai > 0, and consequently in a small enough neighbourhood of A = 0, the right hand side of (4.4) is positive under the condition A > 0. If a\ — m i n { a j , R 9 a, > 0}, 1 ^ j' ^ g, then in the interval 0 < A < a\ the right hand side of (4.4) takes all values from 0 to oo and, among others, c ^ . Then Tv v(X') = 0 for some 0 < A' < a i and both points on T with coordinate A = A' must be real (A' I l j i i C ^ ' ~ ai) > 0)- One of the points is necessarily in the zero divisor of v and since this has the form 2>+T> a , this point must have multiplicity ^ 2. But we chose V such that all zeros of v are simple. This is a contradiction. Now, conversely, let us suppose that, if ay € R, then ay < 0, 1 ^ j ; ^ 2g. In order to prove the existence of an admissible T> it is enough to check that the class of the divisor 0^ — Oo belongs to the connected component of zero (J°)o of the group J° of real ( is equivalent to an admissible divisor. The fact that the class 0<x, — Oo belongs to the group («7°)o follows from the fact that the points Oao and Oo are contained in one real oval {(A,/x) € Tu,0 ^ A ^ +00} of the curve r . To calculate the number of components of J® (cf. Proposition3.3) we count the number of real points of second order of Jac Y. That ay is negative is not used here. An arbitrary point of second order of Jac V is uniquely represented by a class of divisors oL = Y.)=i °h ~ s°°°i where L = { Z i , . . . , / , } C { 1 , . . . , 2g}, 1 5S s ^ 2#, U ^ Ij for i ^ j (check this as an exercise). Conversely, the classes of divisors OL of the sets L — {l\,... , /,} containing Zy< and lj such that Of. = 0\., correspond to the real points. Hence, the number of such points is equal to 2 2 m + ( ' _ m ) = 2 m + J , where 1m is the number of real points Oj, 1 ^ j ^ 2g. Consequently, J° consists of 2 m connected components. • C o r o l I a r y 4 . 1 . a) For an arbitrary admissible divisor T>, divisors Z> — Oo and V - Ooo are not special, i.e., V = £ ' = 1 Pj, where all \{Pj) are pairwise distinct and not equal to 0, 00.

92

I. CONSERVATION LAWS & ALGEBRAIC-GEOMETRIC SOLUTIONS

b) Let T> be admissible, A(P) = z € R for some point P 6 T and {ctj} be numbered so that {otj, 1 ^ j 5; 2m} = {a^} PI R and 0 > a j > 0:2 > . -. > a 2 m . Then for 0 ^ i < m, a2i+i >z>

a 2 i + 2 =* P * = P, r

z < a 2 m or a2i > z > a 2 ,- +1 => P * = P , z >

a i

=>• P " = P,

v(P)v(

T

P) ^ 0,

t;(P)u( r P ) ^ 0, w(P)w( r P ) ^ 0.

Proof. It is necessary to prove that u(Ooo) = Coo ^ 0 for an admissible divisor 2? (v(O 0 ) = "(Ooo) by virtue of (4.3)). If c ^ = - , then, as follows from (4.4), V+V'T + T T> + TV contains the point Ooo with multiplicity 2(2/ + 1) for a certain integer I ^ 0. This is, however, impossible, since the point Ooo appears in each divisor V, V, rV, Tiy with the same multiplicity. Statement b) is obtained by direct analysis of formula (4.4). As A € R varies from +00 to —00, the functions Tvv(P) and (J?(P) change sign at the moment when A goes over each point 0 1 , . . . ,at2m (fx2 changes sign also when A goes over Oo and Ooo). The zeros of Tvv for real A may have only even multiplicities. Therefore, for z < a 2 m , z > a j , and also in an arbitrary interval cij+i < z < aj, the function Tvv(P) does not change sign. • Solutions of t h e Sin-Gordon equation. Let V ^ 0 be admissible, v the rational function on V corresponding to the divisor 2> (cf. above). Turning to the construction of algebraic-geometric solutions, we fix a distribution q = ikoaX + ikot, x,t € R. As q" = —q, T>+Q for Q € exp(—q)(Ooo) = 1. In addition, necessarily, ^>exp(—q)(Oo) ^ 0. P r o p o s i t i o n 4 . 1 . For P i {Oj, 1 ^ j< ' f* 2g} (4.7)

vW{P)

+ vW{

(4.8)

T

P) = v(P) + v( rP) =

H>'{00)

2v(0oo),

= 1.

Prxwf. Since q" — —q, the function il}^" is rational on V and

vMr

€ H°(Y,Oj) C H°(T). i=i

By virtue of the fundamental property of the branching divisor T, the functions Trw, Tr(vt/>ij>'') do not depend on A. This implies (4.7) (moreover, the equality

§4. ALGEBRAIC-GEOMETRIC SOLUTIONS OF SIN-GORDON, NS, ETC.

93

v+ Tv = 2v(Ooo) is an obvious consequence of (4.3)). Substituting P = OQ in (4.7), we obtain (4.8). • Put, as above, V = £ ? = 1 Ph A(F,) = Sj. Set n ' i j <*> = c 2 , c € R, n j = i *i = A. Then
(-iyArW{O0),

a(x,t) = where r^tp{00) = exp(-2q)i>2(00). From (4.8) it follows that aa = c 2 .

T h e o r e m 4 . 2 . T i e quasi-periodic real-valued (4.9)

function

ot(x,t) = i l o g ^ - 1 ^ ! , ^ - 1 ^ ) * ) )

deSned for all x,t € R (modulo 2n) satisfies the Sin-Gordon equation (0.8): ctxt + sin a = 0. Proof. If we put qfr exp(—9) = 1 + ^>ifcoo + (^(fc2^) in a neighbourhood of Ox, then V>„ = (« - A)V> for u = 2»(V»i),. In fact, ^ x : c - (u - A)V> € #°(X> - O ^ ; q) = {0}. The function Til>iptw for w = A | i _ 1 n y = 1 ( A - Sj) belongs to # ° ( T ) . Thus, the trace Tr(Tipiptw) does not depend on A. Computing the last function at A = 00 and A = 0, we obtain the relation (V>i )t = if- Eliminating ipi, we get the equation ut + 2ax = 0. On the other hand, substituting A = 0 in the formula for i/>xx, we find that « = ( l o g ^ ) „ + ((logV)x) 2 = i(loga)« + i((loga),)2 = —2axx

~

1\a'i

for a = z log a. Rewriting u and a via a(x,t) (2.25)), we have

(4-10)

= —ilogc + a(x, (c _ 1 /4)<) (cf. §2

(§«« + j(««)2)« = !(«-*%

Using the fact that a is real, we sum (4.10) and its complex conjugate. Canceling ax on both sides of the resulting equation, we arrive at the statement of the theorem. •

94

I. CONSERVATION LAWS & ALGEBRAIC-GEOMETRIC SOLUTIONS

E x e r c i s e 4 . 1 . Let 2?, V ^ 0 be divisors of degree g which do not contain Oo, Ooo on the curve T given by equation (4.1) (this time, without the reality condition on the coefficients). Under the assumption of non-speciality of T>+Q — Ootoo, construct Baker functions ip, ifr for V and V ~ T) + Ooo — Oo, as above. Replacing tl>" by T^>, prove equalities (4.7), (4.8). Check that functions a, a corresponding to T>, T? (cf. (4.9)) are related by the relation a = —a and each satisfy each (4.10). Summing (4.10) for a and the analogous equation for 5 = —a, show that a is a solution of the (complex) Sin-Gordon equation. • E x e r c i s e 4 . 2 . Under the assumptions [Ch4]) that 9

of Theorem4.2

9

(or Exercise4.1) show (cf.

\

X-S where fj = \(Qj)

and {Qj} are zeros ofip (i.e., (V>) = J^j=i Qj ~ ®)<

b) _ K

'

re(w(V)+vq-e)9(w(V))] l6(w(V)-e)0{w(V)+vq)\

* '

where.PQ = Ooo, e = w(Oo), and the cycles {a>} are chosen r-invariant (cf. §3.1). (Note that substitution of such a into (4.9) results in the formula in [KoK].) O Lifting a t o S 2 -flelds. We begin with the computation of two Wronskians. Define {/, g}x = fxg - fgx, {/, g}t = ftg - fgt for functions / , g of x, t. We keep all the previous conventions (though the reality condition is not used in the following proposition). P r o p o s i t i o n ^ , a) {ip, Ti>}x = 2 ^ n ? = i ( ^ - ^ ) b) {*, T4>h = 2iMA- 1 an j 9 = 1 (A Sj)-1;

- 1

;

Proof. The principal part of {ij>, Tip}x at the point Ooo is equal to 2ik^. For generic x, t, the poles of 0*J/> - 1 at the point Ooo coincide with the zeros of ij). Therefore, the poles of / = V ' I V ' - 1 — T(1l}x'~1) w e contained in the set of zeros of Ttfjr/j. Since {Oj} are fixed points of r , all Oj, I ^ j ^ g, must be contained in the set of zeros of / . There are no other zeros of / (we already know that the number of poles of / does not exceed 2g + 1). Therefore for general enough x, t the divisor ({V>, Ti/>}x) of the function {if>, Tx/>}x = Tij>ii>f is equal to YljLo Oj — V — T V — 000. The condition that x, t are general enough can be omitted, since the last divisor is independent of x, t. Thus we get formula a). Analogously we obtain that

§4. ALGEBRAIC-GEOMETRIC SOLUTIONS OF SIN-GORDON, NS, ETC.

95

T {{ij>, Ti>}t) = Y?jLi Oj + Ooo -VV-Oo and at the point O0 the principal part T 1T of {0, V'}t is equal to 2ik0~ ^{0o)Recalling the definition of ko, we obtain

b).

•

Now for points P G T« (<3> P" = P), assuming that v(P) Tv(P) > 0 (cf. Corollary4.1), we put

H ^ M c r 1 = #,

H 'PMOoo)-1 = ft,

where px,(32 € Rand/??+/?f = 1. This is possible since v(P)+v(TP) v(P), v{TP), ^(Oooj^iust have the same sign. Set & = f M(P), Then e, = M°{P), i2 = M{ rP) and (4-11)

= 2v(0oo) and & =f M{TP)-

6fi+«a = l

by (4.7). P u t

a = I s 2 I € R3.

Hereafter (,) denotes the Euclidean scalar product, c, A the same as in Theorem4.2. T h e o r e m 4 . 3 . a) (s, s) = 1, b)(sx,sx) = 4\(P), c)(sust)=4\(P)-1c2. Proof. Statement a) is derived from (4.11) by direct calculation (which, by the way, establishes the local isomorphism of 0\ and O3 x O3). Differentiating (4.11), we obtain the identities (4.12a)

(»«,««) = 4 { 6 » f 2 } * { f i » 6 } * .

(4.12b)

(*,»«) = 4 { 6 , & } , & , & } « .

For example, for the proof of (4.12), we use ((S!)x)2 +((S2)X)2

=4(^2)^(2)*, 2

((s3)x) =

-MteMtotih-

96

I. CONSERVATION LAWS & ALGEBRAIC-GEOMETRIC SOLUTIONS

Summing these identities and making simple transformations, we arrive at (4.12a). T Replacing ip(P) = P^ti, ip{P) = P^ti m t n e formulae of Proposition4.2, we have a

(4.13a) (4.13b)

{6,?2}* = KfrMP) II( A ( P ) - 'i)" 1 ' tti,f2}f

= A(P)a{6,f2},.

Substituting the above relations into (4.12), we obtain (4.14a)

f[(HP)-Sj)-1WP)-sir1,

(»„-,) = leuhfoffiiP)* i=i

(4.14b) (*t,st) = A(P)-2c2(sx,sx). For the derivation of (4.14b) we used the formulae (4.8) and (4.6). Prom (4.5) follows the relation

(Aft)2 = MPMrPMoo)-2 = = f[(X(P)

- 5j)(X(P) - J 7 ) M ( P ) - 2 A ( P ) ,

>=1

which gives statements b) and c) from (4.14).

•

The vector function s satisfies the equations of the 5 2 -field (0.5) (cf. Introduction). For the S 2 -field s(x,t) = a(x, ( c - 1 / 4 ) t ) the more symmetric relations ( » „ sx) = 4A(P),

(at, st) = (4A(P))" J

hold. By the coordinate change x i-» (4A(P)) - 1 a;, t <->• ( 4 A ( P ) ) - 1 t the field I turns into an 5 2 -field in the normalized coordinates (cf. (0.6)). Thus, the function arccos(5 I ,5 x ) satisfies the Sin-Gordon equation. It is easy to check (cf. (4.12)) the formula

(««,•«) = 2{fc,? 2 M?i.6}t+ 2{€i,&M6.?2}*Replacing { f r , ^ } * by means of (4.13b) and using Theorem4.3, we obtain the relation (sx, 8t) = 2(a + a), from which it follows that ( • „ « , ) = (c-V4)2(o + = cos a.

a)(x,{c-1/4)t)

§4. ALGEBRAIC-GEOMETRIC SOLUTIONS OF SIN-GORDON, NS, ETC.

97

In this way, s(x,£) turns out to be a lifting of a(x,t) to an S2-field, and A(P) > 0 is a parameter reflecting the non-uniqueness of this lifting. In Ch.II (§1.2) we construct a family of transformations of the P C F , which allow us, in particular, to calculate the S2-field s for arbitrary (admissible) points P in terms of the field s constructed for one such A(P) by a certain general method which works for any S^-fields. Exercise4.3. In the notation

of Theorem4.3,

w(P)w(Ooo) > 0, s3 ^ 6 ^ i + 6 ? 2 , H

r

for A(P) < 0 (P € TR) and for

P ) f (Ooo)- 1 d= - /?f (Pi 6 R;, show that

(a,*), = 1 , ( • « , « « ) . = 4A(P), ( • „ - , ) , = 4(A(P))- 1 c 2 , where (a, «)« = s2 — s\ — s\.

D

Theorem4.3 and Exercise4.3 make it possible to construct algebraic-geometric Chebyshev nets (cf. Introduction and [H]) on the sphere S2 and on the Lobachevskii plane. The Chebyshev nets for the latter arise as asymptotic lines for isometric regular embeddings of parts of the Lobachevskii plane into the Euclidean space E3. The impossibility of such an embedding of all the Lobachevskii plane was proven by D. Hilbert (see [Tc]) as a corollary to the absence of solutions for all i , / 6 R o f the equation axt = sin a for the net angle 5 with the natural restriction 0 < 5 < 7r (which reflects the regularity). The result of D. Hilbert (after the change t H-> — i) leads to the following claim: for an arbitrary divisor V of type (4.2), there are i,i £ R such that the divisor ]T)?=i Qj (*ke z e r o divisor of i/>(x,t)) which is equivalent to V + Q (Q € q(x,i)) satisfies the relation ( n ? = i 7 i ) 2 = c2> where 7y = A(Qj) (cf. Exercise4.2, a)). Apparently, this is not the only geometric fact possessing an interesting algebraic interpretation for solutions of equations (0.8) and (0.5) constructed in Theorem4.2, 4.3 (see also Exercise4.3). 4.2. V N S equation. Let r be a curve with an anti-involution a, A a real function on F of degree n. We keep the notation of §§3.2,3. For simplicity we assume that (A)oo = Y^=i Rj is a simple fiber with real Rj. Pick out the point Rn and put q = ifc~xx + ik~2t, where k„ = A - 1 is the local parameter at Rn. Let T> ^ 0 be a divisor of degree g on T not containing points of {Rj} which satisfies the condition n

(4.15)

V +V

~T-2^i?,.

98

I. CONSERVATION LAWS & ALGEBRAIC-GEOMETRIC SOLUTIONS

As in the previous section, (4.15) is consistent with the obvious relation q" + q = 0. Put V + V — T + 2 Ylij=i Rj = (w) f ° r a rational function w normalized by the relation w(R„) = 1. Define a real matrix ft = diag(A 2 w(i2i),... ,X2w(R„-i)) of order n — 1. H V + Q — f£„ is not special (as usual, Q € q), then H°(T>; q) is generated by the Baker function normalized by exp(—q)rj>(R„) = 1. T h e o r e m 4 . 4 . a) Put rj = i/>(Rj), 1 ^ i ^ n - 1, r = V j ) (4.16)

Then

t r t = r « + 2(ftr, r ) r ,

wiere (,) is the Euclidean scalar product. b) There exists a divisor V on T with condition ft > 0, if and only if X does not have real branching points. The variety of classes of divisors D with ft > 0 is then connected and isomorphic to (S1)9. The functions r(x,t) corresponding to such V are regular and bounded for all x, t € R. Proof. As in the proof of Theorem4.2, we find that ipxx — (u — k~2)i/> for u = 2i(i>i)x tends to zero in a neighbourhood of .Rn once divided by exp(g), where exp(q)»/> d= l + V>ifcn + 0(fc*). Consequently, xpxx-uip-ii>t € H°(V-R„; q) = {0} and

(4.17)

tyt

= i>xx - ui>.

Now we express u in terms of the coefficients of the expansion of the rational function ipip" in a neighbourhood of iZ n . The differential form wipxil>erd\ has a pole only at the point R„. Hence ResRn(wi/?xil>''d\) = 0. Computing the residue, we get the relation Resfl,, {wiX^'dX) — (ipi ) x = 0, which implies u = —2HesR7l(\wiptl)''dX). {Rj}. Therefore, by the residue formula,

T h e form Xwij>tj}'rdX has poles only at

n-l

u = ^Resfli(Aw>VV''7<*-M = - 2 ( f t r , r ) . i=i Substituting the obtained expression for u and values of i> at points R\,... ,iZ„_i (instead of ij?) into (4.17), we have (4.16). Statement b) is directly derived from Proposition3.4 and Theorem3.5,6, because relation (4.15) for "D is equivalent to condition (3.7) for V + Y^Zi Rj- D It is easy to formulate and prove the theorem for divisors V ^ 0 containing Rj, 1 = 3 = n — 1 ( D u t n ° t R„). Note that Theorem4.4 can be derived (for (A)i instead °f (^)oo) from Theorem3.3 by the reduction procedure from the GHM to the VNS (cf. Introduction, Exercise2.1 in §2.1 and below §4.3). The direct proof mentioned becomes conceptually more transparent if we use the dual Jost functions (§2):

§4. ALGEBRAIC-GEOMETRIC SOLUTIONS OF SIN-GORDON, NS, ETC.

99

Exercise4.4. a) For T with a marked point PQ and a divisor T> of degree g we put V — K — T>+2Po, where K, belongs to the canonical class ofT. Let q — k^1! (ko is a locai parameter at PoJ> i> € H°(V;q) be normalized by x[> exp(—q){Po) = 1 under t i e assumption of the non-speciality ofV + Q. Then V — Q is not special (Exercise3.1) and we can introduce ij> 6 H°(i>; —q) by the condition exp(l)ip(Po) = 1. For a differential form u with the divisor (w) = T> + V — 2Po the formal series — ^jjfcoincides with the function ip* defined in §§2.2, 2.5. (Verify that

for any m — cf. [Ch5]). b) If there is an anti-involution a on T, PQ = PQ and T> + V we can take V, ip" asV,%j>. • Exercise4.5*. in t i e notation of Theorem4.4, but for q = ik^x that r satisSes the modi£ed vector KdV equation

~ K. + 2Po, then

+ ik~3t,

show

rt + rxxx + 3 ( f i r , r ) r I + 3(Qrx, r)r = 0. (Follow the proof of Theorem4.4, or use Exercise2.5 (§2) in scalar variant, or the identity (2.19) — cf. [Ch5].) D One advantage of using divisors T> of degree g (not of g+n — 1, as for the GHM) in the construction of solutions of the VNS is the possibility of obtaining the formula directly in terms of theta functions: Exercise4.6. a) In the notation of §3.1 for PQ = R„, show that

r 3 - cxnf f^ u \ VH„

9{u;iRj)

~ W{V) ~ "'W^P*) J9{u,iRi)-u>('D))0(w(V)+v,)

t

(for n = 2 t i i s formula was obtained in [I]). b) Analyzing possible singularities of Tj (cf. a)), give a direct proof of the regularity of any solutions of (4.16) in Theorem4.4 with ft > 0. (If 0(w(V) +vq) has zero for x, then u = -2&2\og0(w(V) + vq)/dx2+ const, takes arbitrarily large positive value. But u = - ( f t r , r) •£ 0. For n = 2 this argument is due to A. R. Its, the first proof of the regularity of r for n = 2 was given by V. P. Kotlyarov.) D

100

I. CONSERVATION LAWS &; ALGEBRAIC-GEOMETRIC SOLUTIONS

C a s e n = 2 ( N S ) . It is convenient to give the curve T for the NS by the equation 2j+2

(4.18)

e2 = H (A - pj),

ft

^ , 6 C for j ± I,

>=i

where there are two points Ri, R2 (with a local parameter A - 1 ) over a point A = 00 on T, which are distinguished by the value of the function A - 9 - 1 e . If the polynomial YijLi (^ — Pi) has r e a l coefficients, then an anti-involution a can be introduced, putting A" = A, e" = e. We assume further that r A = A, Te = —e. Branching points of A are points Ej g T for which X(Ej) = Pj, 1 ^ j ^ 2g + 2. In this way, A does not branch at infinity in contrast to the curve (4.1). Note that both points Ri, R2 are real. The matrix ft now reduces to one number v = w\2(Ri) equation (4.16) becomes the scalar NS: irt = rxx +

6 R, and

2u\r\2r.

v can be positive only when there is no real number among {Pj, 1 ^ j ^ 2g + 2}. Let us suppose the last condition on {Pj} is attained. Real points of second order of Jac T = J° are generated by divisors Ej — EJ for 1 ^ j ^ g + 1 and when g = 2m + 1 is odd, by one more divisor E ^ 1 ^ + EJ) ~ E ^ + 2 ( ^ + Ef), where EJ = E2g+2-j- The only relation among them is Y."jt\{Ei - EJ) ~ 0. By virtue of Proposition3.3 the number of connected components of the variety J<j{R2) of classes of divisors V with condition (4.15): V + V ~ Y ^ 2 Ej - 2Rt is equal to 1 for even g = 2m and 2 for odd g = 2m + 1. This is also verified in another way: tracing the signs of X~9~1e on each of two lcomponents of T \ {Ri, R2} when A € R changes from - c o to +00, and checking that for g = 2m, 2m + 1, the set TR consists respectively of one and two ovals (cf. Exercise3.8). In the case g = 2m + 1, as we showed above, v > 0 on one of the components of J%(R-2) — 3**"a~x\ on the other component necessarily v ^ 0 and there exists a non-regular solution r (Theorem3.6 and 4.4). If g = 2m, then always v > 0. Thus the case of imaginary {Pj} is completely examined. Now let us assume until the end of this section that at least one of Pj is real and, consequently v ^ 0. Then there exists an isomorphism of the curves (4.1) and (4.18) which sends r to r and a to a. In addition, a pair of real points in {Ej} corresponds to the points Go and Ox (if there exists one real Pj, then there exists a second one since their total number 2g + 2 is even). In particular, using the proof of Theorem4.1b) we can describe the group S of points of second order of J° = {x € i7°, x + x" = 0 } , which coincides with the group of points of second order of J° = {x £ J°, x = x"}, in the following way:

§4. ALGEBRAIC-GEOMETRIC SOLUTIONS OF SIN-GORDON, NS, ETC.

101

First we arrange {/?,}: let ft < ft < ... /?2»+l =02(i+l),---

>02g+l = ^2(j+l)-

Then £ is generated by Z2-linearly independent classes ei, ej of the divisors e\= E2 — Ez,...,a= t\ = E3 — E4,...

En — E2i+i,...

,e, = E2i+i — E2(i+i),...

,

e p _i = .E 2 (p-i) , eg= E2g+i

—

E2p-i,

— E2(g+1y

Let us show that {e?, 1 ^ i ^ g) generate the intersection £Q = €C\ {j„)o, where (J%)o is the connected component of zero of J%. Analyzing the sign of the right hand side of equation (4.18), we find that the ovals

rj, d= {P e r./fc+i ^ \(P) ^ ft(l+1) for 0 ^ i ^ p - 1 exhaust all the set f R = { P € T, P" = TP} of "imaginary'' points o f r . I f l ^ i ^ p — 1 , then the zero divisors 0 and e° are connected in J° by a connected variety of divisors Pi — P2, P\t2 G Tfc, the classes of which belong to J®. For g ^ i ^ p connect fti+i and ft(j+i) = fti+i by sa axc 7 C C which is invariant under complex conjugation and intersects R c C a t the point ft. Then 0, e° € 7 = {Pz — P f , z € 7 } . The image of 7 in J° is a connected curve in ,7°. Therefore e ? , . . . ,2} € SoAs we found g independent elements e ^ , . . . ,e£ in £0, we obtain (cf. Proposition3.3) that precisely one element of £ = ©?=i ^ e i ** contained in each component of J®. This corresponds to Exercise3.8, since p is equal to the number of real (or imaginary) ovals of V. We will not use this statement. It is given to help the reader to understand the structure of J° clearly. We now turn to the description of the connected components of J9{R2). It is more convenient to examine the variety J<> of classes of divisors with the condition (4.19)

V + V° ~T-P0-P£,

P0ef&.

The map S : V i-> V - R2 + P0 establishes the isomorphism of Jg{R2) and Jjf. Further we will assume that V = YZZl T>i+W ~^0, s u p p l y € T'R, s u p p P 7 C\ (fK \

f ° ) = 0. Put £(£>) = {£Ti=degZ>,-

mod 2,

lgzgp-1}.

102

I. CONSERVATION LAWS & ALGEBRAIC-GEOMETRIC SOLUTIONS

P r o p o s i t i o n 4 . 3 . If there exist real numbers among {/?,}, then the vaiue of £(2?) depends only on the index of the connected component of J% containing the class ofD ^ 0. The invariant £ gives an isomorphism of the set of connected components of J<> and Z ? " 1 = {; q') with the divisor {
T

P) = 0 = v(P) + v(

T

P).

Consequently, "(P) = tpy>'T(TP) and ip(P)v?(P) ^ 0 for such values of z. But then y>y)'r(z) may have only an even number of zeros and poles (counted with multiplicities) in the interval /?2«+i ^ z ^ /?2(i+i) for arbitrary indices 0 ^ i ^ p - 1. This means that E(Z>) = T,{V). As a by-product, we have shown the existence of the representation 2> + V = T T>1 + V1 for a certain divisor V1 ^ 0 (using the fact that ipifi' € C(A)). Hence, if P € s u p p P , then either TP € s u p p P (when P = r P , the multiplicity of P in T> should be even) or TP € suppP"* and TP" € s u p p P . This makes it possible to represent the divisor V in the form (4.20)

V = V + Ax + r A i + A 2 + TAJ,

supp 2? e

fR,

where T>, A j , A2 ^ 0. Conversely, it is easy to see that any divisor T> ^ 0 of degree g of the form (4.20) satisfies (4.19). In particular, as g ^ p — 1, any value of the invariant £ € Z£~ is attained by some divisor V = T>. We already know that the total number of components of J% is precisely 2 P _ 1 . Therefore E is one-to-one on components of J%. • C o r o l l a r y 4 . 2 . a) Ifp = g + 1 (i.e., all {/3j} are real), then the support of any divisor V ^ 0 the class V of which belongs to the component {Jg)i C J% where the invariant £ takes the value {o~i = 1,1 5: i ^ g} is contained in the set U^Z i r R .

§4. ALGEBRAIC-GEOMETRIC SOLUTIONS OF SIN-GORDON, NS, ETC.

103

On the other components for p = g + 1 and on any components of J§ for p < g + 1 there are classes T> with a representative V ^ 0 which contains the point PQ. b) Only in the case p = g + 1 and for a unique connected component of J${R2), namely for 6_1{{J§)\), is it true that arbitrary representatives V ^ 0 do not contain i?2- The solutions of the NS (with v < 0) corresponding to this component are defined and bounded for all x,t € K. Proof. For p = g+1 an arbitrary divisor V of the form (4.20) with E(X>) = {crj = 1} must have at least one point on each oval r R (1 ^ i ^ g). On the other hand degl> = g and V — V in (4.20), as s u p p P C U f = i r R . In all the other cases for arbitrary values of £ we can find V of type (4.20), one point of which lies in r j j . If suppX> 9 J? 2 for X> € Ji{Ri), then the divisor 5{V) =V-R2 + P0is effective and contains PQ. The converse is also true. D E x e r c i s e 4 . 7 . Show that T is divided by T R , if and only if p = g + 1, i.e., {/?y} C R. (In the case /?i ^ R, any points z\,z2 € C \ R can be connected by a smooth p a t h not intersecting with P H \ A ( F R ) . For z\ = z2 this path (cycle) can be chosen so that only the point /?i among all {/3j} is inside it.) • 4 . 3 . R e l a t i o n s w i t h t h e c o n s t r u c t i o n s in §3. The use of divisors of degree g and a scalar Baker function ij> for constructing solutions of the VNS, in principle, does not supplement new features to the construction in §§3.2-3.3. Following the general recipe of the reduction from the GHM to the VNS and Proposition3.2, we can construct a vector Baker function i\> = tl>+ = ($( + i)) - 1 y> from q in §4.2 and V of type (3.7b). Then i/>x, jj)t are expressed in terms of iff and ( $ ( + 1 ) ) - 1 ( $ ( + 1 ) ) j : . The latter matrix has the form 5^7=1 (r3^i ~ rj-'i+i) f ° r appropriate TJ (cf. Introduction). The vector r = \rj) satisfies the VNS and is uniquely determined from the first component of i}> which is the function tj) also introduced in §4.2. Hence, essentially, Theorem4.4 is a specialized form of the theorems in §3. The above construction of solutions of the Sin-Gordon equation and the 5 2 -fields provides a more interesting example. The formal mechanism which we actually made use of in the proof of Theorem4.2 is the interpretation of the Sin-Gordon equation as a higher KdV at a "finite" point k = 0 (§2.5). Let us recall that scalar Lax equations (in paticular, higher KdV's) can be written in the form (2.5) (cf. ibid.). Consequently, the Sin-Gordon equation is represented in the form (2.5a), which allows us to obtain the matrix zero curvature representation for it. The corresponding matrix formal Jost function ^(k) is introduced by the formula

*

w -

^ i

-i)

\-ikti>x(k)

-ik^x(-k)j'

104

I. CONSERVATION LAWS & ALGEBRAIC-GEOMETRIC SOLUTIONS

where ip(k) is the formal solution of the equation - ^ „ + utj) =fc2^>and u is related to a as in Theorem4.2. Let il>(k) be the expansion of the Baker function i/>(P) in §4.1 in a neighbourhood of Oco with respect to fc_1 = kx = A" 1 / 2 . Then $(fc) can be expressed as an expansion of the matrix of the "values" of the vector Baker function,

in a neighbourhood of A = oo according to the definitions in §3.2. However, ij) is defined not on T but on the two-fold covering T' of the curve T, for which C[T) 3 Al/2,

r(Al/2)

=

Al/2

Next, we discuss in detail how to get tp from rp. Beginning with the construction of P , we establish the connection of the constructions in §3.3 and §4.1, and, in partcular, lift a of (4.9) to an Sl/2-field. In order to study specific characters of the Sin-Gordon equation more flexibly, we make a slight change in the general procedure of §2.5. Define a hyperelliptic curve T' by the equation 2g 2

(4.21)

e

= n(A'-a;-)(A' + a'j), >=i

where a'- = a •' for 1 ^ j ^ 2g and {otj} are the same as in (4.1). In the notation of §4.2, the curve T is given by equation (4.18) for 1 % j ^ 2g', g' = 2g — 1, {Pj] = {±atj}. Its genus is equal to g' and there is an involution r ( r A = A, r e = —e) on it. If (cf. §4.1) T satisfies the conditions of Theorem4.1, V with the anti-involution
fi = e\'.

This covering is extended t o a mapping from (P 1 )' with coordinate A' t o P 1 with coordinate A = (A') 2 ; we also denote this by the letter n. The latter covering is ramified exactly over the points A = 0,00, over which A : V —>• P 1 also ramifies. Therefore, n : T' -> T is not ramified (this is also seen from the Hurwitz formula, g' — 1 = 2(g — 1). Let us bring together all the mappings in one diagram. On the

right, we show what is occurring over the point 00 = (A = 00), 00' = (A' = 00).

(P 1 )'

> P1

oo'

>• 00

§4. ALGEBRAIC-GEOMETRIC SOLUTIONS OF SIN-GORDON, NS, ETC.

105

The covering n commutes with r on F" and on T. All coverings commute with a. Let T>, q on T be the same as in §4.1. Put

V1 = 7r -1 D, q' = ik^x + ik^t + ik\x + ik£t (q' is a lift of q on V). The parameters k\£ are obtained, by definition, from the local paxameter koo at the point Ooo by lifting to the points Ri, R2, respectively. If we choose a point R[ by the condition that k^ = ( A ' ) - 1 in a neighbourhood of R[, then fc^, = — Tkl0 = — ( A ' ) - 1 in a neighbourhood of iZ^ ( T koo = — &«>)• Analogously the local parameters in a neighbourhood of points (A') _ 1 (0') = {R% , R\ }, k0' are defined by fcoj namely, fc0' = A e _ 1 . The degree of V is equal to 2 deg I? = g' + 1. Because of (4.2), 2g'+2

Tf + {Vy ~ T = J2 Ej, i=i

where X(Ej) = flj € { ± a y } , T' is the branching divisor of A'. Therefore, the divisor V and the distribution q' satisfy (3.7) and we can apply Theorem3.4 to T', a, V, q', Xz> = 1 — y* zi taken as the parameter A (z' € R). The condition of non-speciality of the divisors "D — Oo, T> — 0^ is automatically satisfied. This condition is equivalent to non-speciality of the divisors V = (R[ + i%) ~ V - (i??' + R%'). The matrices U, V in Theorem3.4 corresponding to V are diagonalized (cf. §3.2):

^

v

~^(o

-,)•

v

~i(°

-•)• •'-n-i-

This formula can be checked by rewriting (the principal part of) k]£ and fc0' via 1 — \zi and 1 + Xzi. If the value of Az< is equal to 00, then A' = —z' and A = (z1)2 = z > 0. Analogously, the point A2< = 0 = A' — z' is mapped by TT to the same point A = z on P 1 as A z ' = 00 does. Put A _ 1 (z) = { P i , P 2 } , where /u(Pi) > 0. The points of the fibers of A' over A' = —z, A' = z' are denoted respectively by {R^z ,R^Z }, {R{ ,-Rf } and ordered by the condition s(Ri ) > 0. The genus of I"" is odd, hence the pairs z {Ri } and {Rf } belong to different components of T'R (there are exactly two of them). Note that n(Rf') = Pt, K(R^') = P2, but n(RYz') = P2, ^(R^z>) = PiWe can take the function v lifted from T (§4.1) with the divisor (v) = V + Vs = E>=i Oj as the function w' on V (cf. §3.3) with the divisor (u/) = V + (Vy -

106

I. CONSERVATION LAWS & ALGEBRAIC-GEOMETRIC SOLUTIONS

£ i = i Oj. Hereafter we will identify functions on T with their lifts onto I"" with respect to IT. Applying Corollary3.2, we obtain that Q = Q00 = diag(v(P 2 ),f(Pi)),

fio

= diag(»(A),i;(ft))

(see §3.3) are definite quadratic forms. As in §4.1, put 0} =

1

2 2v(P1)v(0oo)-\f3 2

=

^v(P2)v(000)-\

To conclude the inspection, it remains to relate the vector Baker function

with the Baker function i> in §4.1. If we set V' = {TV)f, where / = Tve]lgi=1{X - ^ ) _ 1 . then V, */>' form a basis in H°(V; q'). Constructing the vector Baker function normalized at the fiber (Ar/ )i or (Xzi )oo with the aid of ip, i\>', we easily obtain two types of zero curvature representations for a in (4.9). We do not do this here, but use ifr, if)' to lift a to an SE/2-field, following Theorem3.3 and 3.4. As in the proof of Theorem4.3, put T h e o r e m 4 . 5 . The matrix function

ofx,t£R

is an SU2-6eld (satisfies (0.1) with values in

Sfo)-

Proof. Put

*°

/ ftPy) \ev(P2)4>(P2)

^(ft) -ev(P{W{Pi))'

\ °°

/ HP2) \ev(P1)i;(P1)

V(Pl) -ev{P2)j>{P2))

\ '

where e = n ? = i ( z — ^») - 1 (II>=i( z ~ ai)) > a n ( i w e t a ^ e the positive sign of the root. The matrices $0, ^00 represent the matrices of "values" of %4>,il>') at points Az< = 0, Xz- = 00 in the definition in §3.2. From Theorem3.3 it follows that " P ^ o satisfies (0.1). Putting ft = B^B*,, fi0 = B0B$, where Bx = diag(/? 2 ,/3i), 5 0 = diag(/?i,/? 2 ), we obtain by Theorem3.4 that G = B^V^$0B0 is an S I V field. The rest of the proof is the direct calculation of G using the relations

det * „ = -e(v(P2W(P2)iflV) + »(PiMW(A)) = —2ei)(oo)

(Proposition4.1).

D

§4. ALGEBRAIC-GEOMETRIC SOLUTIONS OF SIN-GORDON, NS, ETC.

Exercise4.8. Reiate the U2-6eld G = Gy

107

J for G of Theorem4.5 with the

property G2 = I to the vector function s of Theorem4.3. Show that

«-(? ih-o 7)+-(-.' :)• Substituting

G in (0.1), reduce Theorem4.3 to Theorem4.5.

•

In the construction of Theorem4.3 we could use not only A(P) — z > 0 but also z < 0. For such P the function * is a field on a hyperboloid (Exercise4.3). An analogous extension is also possible for Theorem4.5: Exercise4.9. Examine t i e anti-invoJution
= R}<* {ir{R)° = x(R),\(ir(R))

< 0}.

For z < 0 and a' instead of a, trace the construction of Tbeorem.4.5 and show that the corresponding matrix SI is not definite (cf. Corollary4.1). • We will showed that the concrete methods of constructing algebraic-geometric solutions of §§4.1,2 can be deduced from the general method in §§3.2,3, based on vector Baker functions normalized at points of a certain real ( ^ 0 of degree g + 1 with the standard reality property (3.7a). Replacing it with an equivalent one, we may assume that V = TV ^ 0 (cf. (4.20)). Put V + V° - T = (w) for w with conditions w" = w = — Tw. We fix a real point P on V. We assume further that V, P, x, t are in a generic position. The basis {ipi,^} C H°(V;q) is normalized by the conditions exp(-9)vi,2(0oo) = 0,1,

exp(-q)
= 1,0.

Computing Tr(w we obtain the relation w(P)Vl{P)V2{ where wx

T

P) + » ( T P ) 9 i (

T

P)V2{P)

= (A;00u;)(Ooo), f 0 = (fcou>)(0o) fT

= exp(-g)fc^Vi(Ooo),

€

= 2u/ooV>J° = -2wop3,

*i v>° = exp(-?)*o"V2(Oo).

108

I. CONSERVATION LAWS & ALGEBRAIC-GEOMETRIC SOLUTIONS

Taking the normalization of tp\, fV2»

ifijt = «v3vi-

Since V = TV, then (p° = Ttpu
•w)« = —r—*vi, •IWQ

S =^1^2 +^1^2,

where we set tt/>i = Vi(-f)i *^2 = ¥J2(-P)- Removing the real factors -(l/2)w(P)fw00, -(l/2)w{P)/w0 by a suitable substitution of x, t (they are easily calculated explicitly in terms of V), we get the solution of the system of equations in [NP] with the symplectic group of chiral symmetry. 4.4. Application: the duality equation. Let (XJ) be coordinates of C 4 , {Aj} be functions on C* with values in Q[„ (j = 1,2,3,4). Put

Fik = [Vj,Vk] = {Ak)Xi - (Aj)Xk + [Aj,Ak]. The system of differential equations, (4.23)

i*12 = F34,

P13 = F43,

-Fl4 = i*23,

on matrix elements of functions Aj is called the complexified Euclidean duality equation. It is convenient to introduce vector fields Vi = A - iZ>2,

V 2 = Z>3 - iZ>4,

Va=-Vi-W2,

V 4 = -2>3 - *P 4 .

Following A. A. Belavin and V. E. Zakharov [BZ], we rewrite (4.23) in the form of the relation (4.24)

[Vi + AV2, V 3 - A"1 V4] = 0,

§4. ALGEBRAIC-GEOMETRIC SOLUTIONS OF SIN-GORDON, NS, ETC.

109

which is fulfilled for all values of the parameter A (check the equivalence of (4.23) and (4.24)). P u t further V , = d/dzj + Bj (1 ^ j ^ 4) for the new variables z\ = 5(^1 + ix2), Z3 = \{~Xl

+ ix2),

z2 = \(x3 + ix4), 2 4 = | ( - X 3 + *Xi).

System (4.23) for real x i , x2, x3,14 and anti-hennitian {Aj} is called the duality equation. This equation is a reduction of more general Yang-Mills equations. If we extend the hermitian conjugation " + " to T>j, V7-, setting (d/dxj)+ = —d/dxj, the anti-hermitian condition of {Aj} can be written in the form V 3 = V * , V2 = V4" briefly. Correspondingly, Aj are expressed in terms of Bi, B2 in this case by the formulae: Ai = (1/2)(B1 A3=

- B+), A2 = (i/2)(Br

(1/2)(B 2 - Bt),

M = (i/2)(B2

+ B+), + B+).

Below we sketch the construction of anti-hermitian solutions of system (4.23). Modification of this construction for obtaining complex solutions of (4.23) or Aj with values in other real Lie algebras (definite and indefinite - see §3.3,4) is not complicated and left to the reader as an exercise. Let r be an algebraic curve with an anti-involution cr, for which A" = —A -1 (in contrast to the situation of §3.3). This, in particular, means that there do not exist real points on T, since the values of A at such points should satisfy the unsolvable equation A = — A - 1 . As in §3, we denote by T and Tr the branching divisor and the trace of the covering A : T —> P 1 corresponding to A. Let qi, q2 be two distributions with support in the set of zeros and poles of A and V a divisor of degree g + n — 1 where g is the genus of T and n is the degree of A. We suppose that the distribution <7i + A<j2 is regular at zero, and the distribution A - 1 ?i + q2 is regular at poles of A. We also impose the condition of reality on V: V+V" — T = (w) for a real {wa = w) rational function w on I \ Repeating the steps of the argument of §3.3, we consider the vector function ip = \ipi,... , (pn), where {jU)<^J). Then U does not depend on A and is hermitian and invertible (Theorem3.4). Let us suppose that U is definite, i.e., VUV+ = ± 7 for a suitable matrix-valued function V of 21, 22-

110

I. CONSERVATION LAWS & ALGEBRAIC-GEOMETRIC SOLUTIONS

T h e o r e m 4 . 6 . a) There exist unique g[„-valued functions B\, B2 of z\, z2 such that

(Vi+AV2)p=0,

Vi,2=5

+ Bi,2.

OZl,2

b) For functions Bx = VB1V~1

+ ^V~\

B2 = K 5 2 V - 1 +

^V~\

"gauge equivalent" to Bii2, the corresponding u„-valued Ai, A2, A3, A 4 of x i , x 2 , X3, 14 (cf. (4.25)) satisfy system (4.23). Proof. Statement a) is proved completely analogously to the proof of Theorem3.2 and based directly on the regularity of 91 + Xq2, A - 1 ^ + q2. Turning to b), we use the notation and the relations in Theorem3.4. For generic c € R, we have

* (e) n«*^ c - 1) = u, where $( c ), ftc are the matrices of values of ip, w at points of (A)c (the fibers of A over c and — c - 1 are ordered consistently with the action of a). Consequently, together with the operator V i + CV2 the operator V3 — c _ 1 V 4 for V3 = Vj", V4 = V j must also be zero on V$( c ). From the invertibility of the latter matrix it follows that relation (4.24) and statement b) hold. D In spite of the obvious resemblance between the present construction and the discussion in §§3,4, there are also some fundamental differences. The higher dimensionality of the duality equation becomes apparent in the fact that the distributions 92 axe not uniquely defined, from the equation nor from its natural reduction. Let us recall that in the case of the general algebraic-geometric P C F (cf. Theorems. 3) corresponding distributions are recovered from the eigenvalues of functions U, V which are invariants of system (0.2). The reality condition (A* = —A -1 ) apparently cannot be combined with any natural normalization of

§4. ALGEBRAIC-GEOMETRIC SOLUTIONS OF SIN-GORDON, NS, ETC.

Ill

Exercise4.10. For the curve s2 = F(X) of the type (4.18), where F is a polynomial of degree 2g + 2 for odd g with the property F ( - A - 1 ) = X~2g~2F(X), put X' = -X'1, s" = eX-"-1. Then for a divisor V satisfying V + V - T, and, zu z2 close enough to zero, the 2 x 2-matrix U (cf. above) is definite (corresponding {Aj} take values in \x2). • Exercise4.11. (cf. Exercise3.3).

The Lagrangian

density

for self-dual Yang-Mills £elds of Theorem4.6 is quasi-periodic function of x\, x2, X3, X4, i.e., it can be written in terms of certain rational functions on the Jacobian variety of T . D 4.5. C o m m e n t s . "Finite-zone" integration of the Sin-Gordon equation was done for the first time by V. 0 . Kozel and V. P. Kotlyarov [KoK] (cf. the formula of Exercise4.2 b) and (4.9)). They also considered the corresponding real solutions and showed their regularity. The method of [KoK] was simplified (in the complex version) by A. R. Its (cf. [Mat3]), and then generalized and refined in the works [Ch4], [Ch5] and [Ch2]. In [Ch5] on the basis of the general definition of the duality of Baker functions, the reality condition was interpreted algebraic-geometrically. In [Ch2] the number of connected components of the variety of real solutions of this equation was calculated (cf. also the author's paper in the Doklady of the Academy of Sciences, USSR, 1980, 252-5, pp.1104-1108). Concerning the duality of Baker functions, some results can be found in [BC]. The lifting of algebraic-geometric real solutions of the SinGordon equation to S2 -fields was done in [Ch5] using one idea of [NP]. Note that the construction of Kozel-Kotlyarov is connected with the problem of eigenfunctions of certain two-dimensional Schrodinger operators which was solved in [DuKN] (cf. also [DuNo], [N3]). Concrete formulae in theta functions for curves of small genus are given in [BBM], [BE], [DuNa], [ZJ] and others. Algebraic-geometric solutions of the NS equation (cf. formula of Exercise4.6a) for n = 2) were constructed in [I], [IK]. This construction was made more invariant and generalized to the VNS in the work [Ch5]. After that in [ChlO], the description of the variety of such solutions of the VNS and NS equations was obtained with the aid of [Fay] for various fi, v and their regularity was studied in the development of the result by V. P. Kotlyarov (see Theorem4.4, Corollary4.2, and Exercise4.6). The correspondence of "finite-zone" solutions of the Sin-Gordon equation to suitable algebraic-geometric SC^-fields (§4.3), induced by a certain unramified covering of the spectral curve was established (in the complex variant) by E. Date (cf. [Dat2]).

112

I. CONSERVATION LAWS & ALGEBRAIC-GEOMETRIC SOLUTIONS

He also constructed "finite-zone" solutions of the so-called massive Thirring model (cf. [Datl]) which is similar to the equations studied in this section, and found an explicit formula for invariant finite-zone SZ<2-fields (e.g., for finite-zone solutions of the Pohlmeyer-Lund-Regge-Getmanov system). Note that statement b) of Corollary4.2 (in the case p ^ g) can be derived from the result of [DuNo], where the reader can also find a discussion of reality properties of finite-zone solutions of the Sin-Gordon equation in the language of the period matrices and arguments of corresponding theta functions. The method of constructing algebraic-geometric Yang-Mills fields was proposed in [Ch7] (cf. also [Chi]). We mention before concluding this section that (complete or partial) trigonometric degenerations of solutions in theta functions and their relation with multi-soliton solutions are discussed in considerably many articles. One of the first works in this direction was done by A. R. Its, I. M. Krichever, V. B. Matveev (cf. [Mat3]). Among applications of non-degenerate and degenerate theta solutions of soliton equations, we note their use in the framework of the WKB method (Whitham method) in the works of V. P. Maslov, S. Yu. Dobrokhotov, H. Flashka, J. Forest, D. V. MacLaughlin, M. V. Karasyev and others. Theta solutions of the Sin-Gordon equation (Exercise4.2,b)) can be seen in C h . l l of the book by G. Baker, "Abel's Theorem and the allied theory including the theory of the Theta functions", Cambridge, 1897 (cf. also the book [Mu]).

CHAPTER II

BACKLUND TRANSFORMS & INVERSE PROBLEM

The inverse scattering problem method occupies a central place in soliton theory. For the study of a soliton equation, it is not enough to know some of its exact solutions, rather we must describe classes of solutions with certain analytic properties. The inverse problem in principle allows us to do this for solitons, i.e. for all solutions rapidly decreasing to zero (or stabilizing) as x —• ±00. There are versions of the inverse problem method which allow us to study families of solutions of more general types, but here we do not discuss those methods. For instance, the results in §§3,4 of the previous chapter lead to an analogue of the inverse problem method for the class of almost-periodic solutions of the basic equations. See [ZMaNP], in which the KdV is discussed in detail, and [TF2]. The Backlund-Darboux transformations are closely related to the inverse problem method. By means of these transformations we can construct new solutions of a soliton equation from an arbitrary solution by solving an auxiliary system of linear differential equations. This system is in fact the associated linear problem. Therefore it is not surprising that Backlund-Darboux transformations are directly related to the inverse problem method. The so-called multi-soliton solutions turn out to be a result of the application of these transformations to the trivial (zero or constant) solutions. Generally speaking, they are superpositions of the simplest one-soliton solutions (of constant velocity and form) as t —> 00 and can be written in terms of trigonometric functions of x and t. These solutions can also be obtained as the degeneration of the algebraic-geometric solutions of the previous chapter. The Backlund-Darboux transformations transform solitons into solitons and preserve multi-soliton solutions. In §1 we examine the action of the Backlund-Darboux transformations on arbitrary solutions of the basic equations ( P C F , GHM) and their reductions ( S n _ 1 fields, solutions of the Sin-Gordon and the NS) from various viewpoints. We demonstrate two different fundamental approaches to these transformations, extending the classical results of Backlund and Darboux and making them as explicit as possible. The group theoretic aspects of them are studied in [Chll]. In the inverse problem the Backlund-Darboux transformations ("dressing transformations'') control the discrete spectrum (§2, §3).

113

114

II. BACKLUND TRANSFORMS & INVERSE PROBLEM

In §2 elementary facts about the inverse scattering problem are presented systematically up to the reduction to a suitable Riemann problem. The latter is based on the following. Solutions $ ± of the equation

which are analytic with respect to a in the upper and lower half planes respectively are constructed by the method of M. G. Krein (via the triangular factorization of Jost solutions). Here Q(x) is a matrix function integrable over the x-axis and U0 is a diagonal anti-hermitian matrix J70 = diag(/i!,... ,/z„) in which it is not necessary to suppose that the numbers {/**} are pairwise distinct. Then the ratio S = ( $ _ ) _ 1 $ + for a € R does not depend on x. Moreover, the map Q(x) \-t S(a) turns out to be one-to-one modulo "discrete spectral data" and is infinitesimally invertible. This theorem (together with the construction of $ ± ) is fundamental in §2General theorems of §2 are applied to the basic equations and their reductions in §3. Here Q and $ ± depend on t. We can find a simple transformation law of the entries of S and the discrete spectral data with respect to t, which then allows implementation of the corresponding integration procedure. For pairwise distinct {Hk) the dependence on t results directly from the definition. When some of the fik coincide, we need a more detailed discussion. We consider S and the scattering data for 0„ and S" - 1 -fields in more detail. As an application, the trace formulae axe derived which allows us to express the local integrals of motion of Ch.I in terms of the scattering data and then clarify and interpret some of the statements obtained in §1 (ibid). In particular, we solve the problem of their functional independence. We note that the examples of the scattering data of concrete solutions of the NS equation and the so-called derivative NS equation (DNS) considered at the end of §3 are closely related to the numerical study of multi-soliton solutions of those equations as well as to the variational formulae established in §3. On one hand, these examples show how to get and study certain solutions with the scattering data expressed in terms of elementary functions. On the other hand, they illustrate how to use the inverse problem to describe nonlinear effects in soliton equations theoretically and numerically. The numerical (computer) simulation of these equations is very interesting, but we do not discuss this here. The general goal of this chapter is to give a systematic algebraic exposition of the direct scattering problem oriented to applications. Our more concrete purpose is to connect the algebraic theory of the Backlund-Darboux transformations and the methods in Chapter I with the scattering theory.

§1. Backlund transformations The first §1.1 contains the definition of the Backlund-Darboux transformations, "dressing" transformations of solutions of the basic equations (first without imposing reduction constraints). In §1.2 differential relations between the initial and transformed fields are calculated for transformations of the simplest type and specialized in t h e case of chiral Un, On, S " - 1 - fields. Then in §1.3 we demonstrate the connection of the transformations of §§1.1,2 with the classical Backlund transformation of solutions of the Sin-Gordon equation. §1.4 consists of applications and a study of local conservation laws. In §1.5 we show the relationship of our constructions to the classical Darboux transformations. As examples, the KdV and the NS equations are investigated. The material is self-contained and does not use any special facts about differential algebras or the theory of differential equations. It is enough for the reader to know basic theorems on linear differential equations (cf., for example, [A2]). 1.1. Transformations of basic equations. In this section g(x,t) is a solution of the P C F equation (0.1). The pair U(x,t) and V(x,t) is a solution of system (0.2) corresponding to g. We also consider 'simultaneously the case when U(x, i) satisfies the GHM equation (0.9) with conditions (0.10). Let us recall that in both cases the corresponding equations are the consistency conditions of the equations

(1.1)

*,=(l-7)tf* = j^tf*,

(1.1A)

$ t = ( 1 - -y-^V*

(LIB)

*

f =

(_£_

E

r

+ r

= j - j - ^ V*. L

f P

7,I7,])#,

where, as in Ch.I, the letters A,B correspond to the P C F and the GHM. Using the conservation laws Sp(U')t = 0 = S p ( V s ) r (s = 1,2,...) for the P C F , we perform the following reduction. Fix the eigenvalues (fii(x),... ,fin(x)), (vi(<),... ,t/n(t)) of the matrices U and V which do not depend on t and on x respectively. Let us assume that U and V are diagonalizable. For the GHM, the corresponding requirement is condition (0.10):

U-c^I'+ci

£

i=i

If,

;=p+i

where ci t 2 € C, c = c t — c 2 ^ 0, 1 ^ p < n. We do not specify the domains of U, V in this section. One may assume that the variables x, t belong to an appropriate open connected dense everywhere domain in R 2 .

115

116

II. BACKLUND TRANSFORMS & INVERSE PROBLEM

A conjugation by a constant invertible matrix F : U *-¥ F~1UF and V i-4 F~ VF is the simplest transformation of (0.1), (0.2) and (0.9) which preserves the eigenvalues of U and V in an obvious way. If U and V are *-anti-hermitian (cf. Introduction), then F should be chosen *-unitary (like g). This conjugation, called the chiral symmetry, lifts to a transformation g *-¥ F~1gV for an arbitrary invertible constant matrix T>. Our first goal is to construct a one-parameter family of transformations which is a generalization of the chiral symmetry and an analogue of the Pohlmeyer transformation [Poh] for S n _ 1 -fields. We construct a solution $o = $(z!*;7o) of system (1.1,2) for a fixed value of 7 = 7 o € C * = C \ {0} which is uniquely determined by the value $(xo, t 0 ; 70) at a certain point (xo,*o) in the domain of definition of U and V (or of only U for the GHM). In order to do this, we can first find $(xoi*;7o) by (1-2A or B) and then (1.1) determines all $0. It only remains to check that this $0 satisfies (1.2) for all x, t. This follows from the uniqueness theorem of differential equations, since the difference of the left and right hand sides of (1.2) must be a solution of (1.1) with zero boundary value at t = to (by the construction of $0) v i a the zero curvature representation. 1

Proposition!.. 1. a) Let $(x,t;*f) (1.1,2A). Then the function $(*,*; 7) =

be an arbitrary matrix

solution of

system

$o1$(x,t;77o)

satisfies, in the case of the PCF, the same system for the pair of functions U(x,t)

= 7 o$o~ 1 tf$o,

V(x,t)

=

7o-

1

$0-1V$o,

corresponding to the principal chiral Geld 7j = QQ1 $(a;, t; —70). The Geld g is related toU andV by QxT1 = U, gtg-1 = V. The eigenvalues ofU and V are obtained from {fij}, {VJ} by means of by 7Q 1 respectively. b) For system (1.1,2B) the function U of a) satisfies 10Ut

= [U, U„] + 22 2 ( 7 o - 1 - \)UX

and relation (1-10) with c = 70c. Proof. We have M "

1

= $0 J ( l - «nti)U9o ~ *o _ 1 (*o)* = (7o-77o)$o"lt;'$o-

multiplication

117

§1. BACKLUND TRANSFORMATIONS 1

Analogously we can compute $ t $

and check the relation for g.

D

2

Note that for real 70 (and real c in case B) it is easy to get a solution of (0.2) and (0.9) with the same eigenvalues as those of U and V from U and V by means of a constant linear transformation of the variables x and t. Exercise 1.1. Let us denote g of Propositionl.l by'g = g(J\ and write g ~ g' if g' is obtained from g by the chiral symmetry (see above). Then a) g(-yi-n) ~

b) c)

((5(7i))(72);

g^-g, 1 1 ff(- )~ff" .

D

Now we generalize the above construction, by multiplying $ on the left by matrices depending on 7. Let K\ and Ki be two subspaces of C" of complementary dimension, that is, dim-Ki + dini-K^ = n, and let Ai ^ A2 € C For a given pair U and V fix two invertible solutions of system (1.1,2) at A = Aj,A2 respectively. We denote the projection onto the space $1 K\ parallel to *&iKi by P. We assume further that A lj2 ^ ± 1 (PCF), ^ 1 (GHM) and $ ! # ! n $ 2 ^ 2 = 0. Put B =

/

_ ^ 1 A L p A2 — A

T h e o r e m l . l . a) If $ satisfies (1.1,2) in some neighbourhood of the point A' € C, then $ = J5$ is a solution of (1.1,2) in a neighbourhood of X' ^ Ai and A2,±l for (1.3)

U = U + Pr(\i-\2),

V = V +

Pt(\2-\1).

b) Matrices U and V are equivalent to U and V respectively jugation by matrices B(±l)). The function

(by means of con-

>-('-'&*)> is a solution of (0.1) corresponding to U and V in the case of the PCF. Proof. The definition of $ implies the formulae (1.4)

l^-

1

= BXB-X

+ -J—

BUB,

(1.5A) $ , $ - ' = BtB'1

+

—^-BVB, 1 -(- A

(1.5B) Qt*-1

= BtB'1

+

°

BUB-1

+

-L-BIUIUJB-1.

118

II. BACKLUND TRANSFORMS & INVERSE PROBLEM

We will show that the right hand side of (1.4) is rational with respect to A and does not have poles at A = Aj, A2. In a neighbourhood of A = Ai we can represent it in the form $ x $ _ 1 where B - 1 $ is a certain analytic continuation of $1 in a neighbourhood of the point A = Ax. As B'1 = I - X£~J\P, Q^O?

= /TiOi(A - A i ) " 1 + Ol

d

=K.u

where 0\ = 0\^ is the ring of formal power series in (A — Ai) with coefficients depending on x, O" = 0\ ®c C"; $ _ 1 C " is the image of O" under multiplication of the matrix $ _ 1 by all vector functions from O". Differentiating $ £ 1 = O" with respect to x (here (0")x = O" and {JCi)x = K.\ since K\ does not depend on x), we obtain the relation

It follows from the last inclusion that $ T $ _ 1 is analytic at the point A = Ai. An analogous argument holds for A = A2 and for (1.5) instead of (1.4). In this way we see that the right hand sides of (1.4) and (1.5) which depend rationally on A do not have singularities at the points A = Ai,A2 (at the poles of the matrix elements of B, B_1) and may have poles only at A = ± 1 of first order (or less) for (1.4,5A), and at A = 1 of order ^ 2 for (1.5B). Taking the equality, B(\ = 00) = I into account we obtain the relations

Sff.

1-A

rV( in the case of A ), ; 1+ A v '

^

+ r

i_Sr

( a a e

B)

for U = J5(1){7B(1) _1 and V = B ( - l ) y S ( - l ) _ 1 and a suitable matrix function W. In the case of (B) in a neighbourhood of A = 1 we can express the right hand side of (1.5B) in the form ( 5 $ ) t $ _ 1 B _ 1 where $ is the formal solution of (1.1,2B) from Propositionl.2, Ch.I for U. Since B$ is a solution of the same type for U and W = [{/, Ux] (cf. §1.4, Ch.I). This argument proves all the claims of the theorem except for (1.3) since the relation of g with U and V is trivial, as g~ = B(0)g. The remaining formulae are obtained from the expansion of (1.4) and (1.5A) with respect to A - 1 in a neighbourhood of A = 00. D

§1. BACKLUND TRANSFORMATIONS

119

E x e r c i s e l . 2 . If we determine U and V byU and V and get 70 by the construction of Propositionl.l, apply the transformation in Theoreml.l to U and V for X\ and A2 corresponding to 71 and 72 and again let the transformation in Propositionl.l for^Q1 act on the result, then we obtain matrix functions which come from U and V by the transformation of the type described in Theoreml.l with -y'l — 7i7o and 72 = 7270- All transformations from the therorem for X'12 = (7i,2 + l)/(7i,2 ~ 1) are given by the above construction for suitable initial K\ and K2 corresponding to Aj and A2. • Developing the proof of Theoreml.l (cf. also [Chll]), we arrive at the following proposition. P r o p o s i t i o n ! . 2 . Let $ be a solution of system (1-1,2) which is defined and analytic in a neighbourhood of the points A i , . . . ,Ajy ^ ± 1 and B(X) a rational matrix function of A for which a) B(oo) = I, b) matrix elements of B and B~Y do not have poles at A ^ A i , . . . , XN, c) $ - 1 B - 1 0 | l = K.i do not depend on x, t, i.e., are generated as Oi-modules by a vector-valued series in the parameters A — A; which are constant with respect to x and t. Here Oi = 0\t is defined for the local parameter A — A j ^ i ^ J V a s for 0\ (cf. above). Then $ = B $ is a solution of (1.1,2) which is analytic for A in some domain and corresponds to certain solutions U and V of (0.2) or (0.9) with the same eigenvalues as U and V. • In the next chapter we discuss the restrictions on {fCi} and $ which ensure the existence of B. For the moment we turn to an example which illustrates Propositionl.2 and shows what happens with Theoreml.l when the points Aj and A2 are equal. Let $ = $1 + (A — Ai)$j + o(A — Ai) be the expansion of an invertible solution $ of system (1.1,2) in a neighbourhood of the point A = Ai ^ ± 1 . For cr £ C* and a constant matrix Q with the property Q2 = 0, put F = $1 + ^[Q and denote a rational matrix function of A (depending also on x, t) which transforms a vector of the form Fz, z € C to Fz —
= (F + a(A - A i ) - 1 ^ ) ^ "

= $(J
120

II.

BACKLUND TRANSFORMS & INVERSE PROBLEM

C o r o l l a r y l . l . For
V =

above, the functions V-<j{FQF-1)t

are solutions of (0.2) and (0.9) which are equivalent to U and V. Proof. B=I-cr(\-\1)-1FQF-1.

D

E x e r c i s e l . 3 . Carry over Theoreml.l and Proposition 1.2 to equations (2.3) and (2.5) of Ch.I, constructing B from solutions at points A i , . . . , XN of the proper analogues of system (1.1,2B) with the normalization B(k = 0) = I and B(k = oo) = I for (2.3) and (2.5) respectively, where k = (1 - A ) - 1 (cf. §2, Ch.I). • E x e r c i s e l . 4 . ([BZ], [CGW], [Taka], [UN]). In the notation of §4.4, Ch.I, let {Aj} be a solution of the complex duality equation (4.23) and $ a solution of the system ( V ! + A V 2 ) $ = 0, (V3-A-1V4)$ = 0 for Vi constructed from {Aj}. If a rational function B of A satisfies conditions b) and c) of Propositionl.2, then $ = B$ satisfies the same condition as $ with an appropriate {A'j} which form, a solution of (4.23). O 1.2. Backlund transformations of U„, On, and S n _ 1 - f l e l d s . In this section we describe a more traditional approach to Backlund transformations via a differential relation between initial and transformed solutions of the equations under consideration. Hereafter we denote the complex conjugate and the hermitian conjugate of a matrix A by A and A+, respectively. For the unitary P C F it is necessary in the construction of Theoreml.l to choose conjugate parameters Ai = Ao, A2 = Ao for Ao ^ Ao, Ao j= ± 1 , and then take one (constant) space K C C" and construct the hermitian projection PK onto $\K = 9{\o)K. Then P% = PK =» B(0)+B(0) = I=>g+g = I^U+ + U = 0 = V+ + V in the notation of Theoreml.l if g+g — I. However, the projection PK admits another definition as a solution of a certain system of differential equations. Proposition!. . 3 . The projection PK is a solution of the system of equations (1.6a)

PPX = —L=-PU(P 1 — Ao

- J),

(1.6b)

PPt = —^PV(P 1 + Ao

- I).

Conversely, any hermitian projection P satisfying (1.6) for anti-hermitian solutions U and V of the PCF system (0.2) can be uniquely expressed in the form PK for Ao and a fixed invertible solution $(Ao) of system (1.1,2A).

§1. BACKLUND TRANSFORMATIONS

121

Proof. Relations (1.6) together with their hermitian conjugates are equivalent to the absence of poles in the right hand sides of (1.4,5A) at A = AO,AQ and, consequently, they are satisfied by PK- Using hermitian conjugation and the identity P2 = P, we can rewrite (1.6) into an equivalent form (1.7a)

Px = -^PU(P 1 — Ao

- / ) - (1 - A 0 )(P -

I)UP,

(1.7b)

Pt = —L^PViP 1 + Ao

- I) - (1 + A 0 )(P -

I)VP.

An arbitrary solution P of the above system is uniquely determined by the value P(xo, to) at an arbitrary point (xo, to) in the domain of P. Therefore P is uniquely expressed in the form PK- d T h e o r e m l . 2 . Let g(x,t) be a unitary solution of the PCF equation (0.1), Ao (E C with Ao ^ ± 1 . The system of equations (l-8a)

<7i<7_1 - gxg_1

= Ao(<7ff-1)i,

(1.8b)

gtg'1

= -A 0 (flS _ 1 ) t ,

(1.8c) has a unique for any initial satisfies (0.1). ofU = ^xg~l

- gtg'1

\ogg~1 + Aoffff-1 = (Ao + Ao)/ unitary solution (gg+ = I) in a neighbourhood of a point (xo,to) value g(xo,to) which is unitary and satisfies (1.8c). The function g The eigenvalues of the matrices U = gxg_1 and V = gtg~1 and those and V = gt^)*1 coincide.

Proof. For a certain matrix function P(x, t) we can put g = Bog, where Bo = j _ A a p ^ p If y is a unitary solution of (1.8), then by (1.8c)

#o

= 9 9 - 1

P. Ao

Consequently, P2 = P. Since g is unitary, Bo is unitary and the projection P is hermitian. Substituting g = Bog into (1.8a,b), we get formulae (1.7) which are equivalent to (1.6). Hence P = PK for a certain subspace K (Propositionl.3). On the other hand, any unitary matrix <7(xo,
122

II.

BACKLUND TRANSFORMS & INVERSE PROBLEM

0 n -flelds. Now let us assume that g(x,t) € On, i.e., g~ = g. The simplest example of the Backlund transformation of O n -fields is constructed from the initial subspace K C C (cf. above) for which (K,K) = 0, where (,) is the standard hermitian form on C" (which is anti-linear for the second argument). Fixing $(Ao), —

= _

def

we can put $(A 0 ) = $(Ao), since $(A) = $(A) together with $(A) is a solution of system (1.1,2A) for real U and V. We denote the hermitian projections onto $(\Q)K and $(\o)K by PK and Pjf. Then Pg=- = Pic- Now we take $(Ao) with its values in the complex orthogonal group (i.e. $(A 0 )*$(A 0 ) = I ) . This is possible, since *U + U = Q- V + V. In this case

= ('$(Ao)$(A0)tf,F) = (K,K) Hence, PKPji

= P^PK

= 0.

= 0.

Proposition]..4. The

function

S ( A ) = J _i^P_^Lz2jop Ao — A

Ao — A

for P — PK above satisfies the condition of Propositionl.2 and is orthogonal for A € R. The orthogonal Geld g and the anti-hermitian currents corresponding to B $ have the form (1.9a)

~99->

=I - ^ f ^ P

- ^ f - ^ P = 5(0), A

Ao

(1.9b)

U = U + (Ao - A 0 )(P - P)x,

o

V = V +

(X0-X0)(P-P)t.

Proof. From the equation PP = 0 follows the formula

B-\X)

= I - h ^ - P - ^_Z^£p. Ao — A

AQ — A

In particular, B(X)B(X)+ = I. Taking the trivial relation B(X) = B(X) into account, we see that B(X) is orthogonal for A € R. Properties b) and c) of Propositionl.2 should be checked for A = Ao, Ao. Since $(Ao) = $(Ao) and B(Xo) = B(Xo), it is enough to check it only for the point AoWe have: B-10Xo = $(A 0 )((A - A o ) " 1 * + KX + (A - X0)OnXo),

§1. BACKLUND TRANSFORMATIONS

where K

is the orthogonal complement of K in C n .

123

D

Here we will not analyze the differential equations defining P (cf. Proposition]..3) and express the transformation g •-• 7j in the form of a differential relation (as in Theoreml.2) in the case of On-fields. 5" _ 1 -fields. We now impose one more reduction condition on an 0n-field g. Let g2 = JT. If we introduce the parameter 7 = ^ y instead of A, it follows that def

g(x,t)$(x,t;— 7) satisfies (1.1,2A) together with $ , where $ ( 7 ) = $(A). The converse is also true up to the chiral symmetry of g. Put 70 = (Ao + l)/(Ao — 1). We keep all previous notation, in particular, $(T~O) = $(70)Proposition!..5. Let us assume that 70 € «R (•& |Ao| = I) and g$(io)K — $(70)AT at some point (xo,to) in a (connected) domain of definition of $(70) for x, t. Then g2 = 1. Conversely, the last relation (the involutivity ofg) imphes the above restriction for 70 and K, if the matrices U and V corresponding to g are not non-degenerate (do not preserve any proper subspace of C which is constant with respect to x and t). Proof. Let us suppose that g2 = I and that U and V are non-degenerate. Choose one solution $ ° ( x ; 7 ) of equation (1.1) normalized by the condition $°(a:o;7) = I (temporarily we omit t and do not consider equation (1.2A)). We may assume that this function $ ° is an analytic function of 7 in a certain domain which is symmetric with respect to the reflection 7 i-t —7 and contains 70 and j 0 as a result of the well-known theorem regarding analytic dependence on parameters of solutions of differential equations. Then the function $ ° = $(x; 7)B(xg; 7)"" 1 , constructed from the function $ = 5 $ ° , is a solution of (1.1) for U with normalization $°(xo; 7) = J , which singles out $ ° uniquely among all solutions. The function $ ° is analytic and invertible in the same domain as $ ° except at the points 70 and 7 0 . Due to the uniqueness of solutions of differential equations, $ ° = g(z)*(x;

-7)*(*o; ^ r ^ o )

- 1

.

The above function is expressed in terms of $(x; —7), B(x; —7) and hence is analytic and invertible everywhere except at —70 and —7 0 . Consequently, if 7 0 7^ 70, $ ° must be invertible at 70, which contradicts the assumption that U and V are nondegenerate. Moreover, if g2 — I, then / - (Ao - Ao)(P/Ao - P/Ao) = / - (Ao - \0)(gPg/Xo

-

gPg/X).

124

II. BACKLUND TRANSFORMS & INVERSE PROBLEM

Therefore gPg = P, since the pairs of projections P, P and gPg, gPg are orthogonal. This proves the necessity. Reversing the last discussion, we obtain that

g*(jo)K = $(j0)K

&gPg =

P^g2=I.

We can prove the remaining part, noting that $(7o) _1 <7$(7o) = $( — 7o) _1 <7$(7o) does not depend on x, t, if 70 € JR. • In this section we apply the last reduction, restricting ourselves to involutive g for which the corresponding projection Q = (I — g)/2 is an orthogonal projection onto a Revalued function q(x,t) with the condition that (q,q) = 1. As we checked in the Introduction, equation (0.1) then follows from the equation of S n - 1 -fields (0.5). Let us recall that U = 2q A qx and V = 2q A qt for an 5 n _ 1 - field q. Also recall the notation:

(pAq)z =

(z,p)q-(z,q)p

for p, q, z e C. Following Propositionl.4, we select the projection P onto the one-dimensional subspace Op for the solution
= 0,

gImr

= - I m r * > I m r = ±q.

Here we use the identities (Re r, Re r) = 1 = (Im r, Im r) and (Re r, Im r) = 0, which are equivalent to the constraints, (r, r) = 2 and (r, r) = 0. Thus we have shown that we can find the solution

125

§1. BACKLUND TRANSFORMATIONS

the function p analogous to that given in Propositionl.3 and to calculate if in terms of p. P u t 270 = K G R. We apply the relation (q,qx) = 0 = (q,qt) derived from the equation (q,q) = 1. Differentiating r = y/2tp(tp, y ) - 1 / ' 2 , we have the following equations: (1.10a)

rx = - ( 1 + iK)Ur -

%

-j{Ur, r)r, JK-1

1

(1.10b)

-1

rt = - ( 1 - I K ) Vr + —— (Vr, r)r,

derived from the formula {
+

iK

)(U

f)

+ g (l ~

i/c

)(P> U) =

iK

{U
and the analogous formula for (tp, ip)t. Substituting r = p + iq into (1.10), we get the equations (1.11a)

(p + Kq)x = (p, qx){Kp - q),

(1.11b)

(/ep - g) t = - ( p , ?/)(p + «g),

(1.11c)

(p,p) = l,

(p,q)=0.

Let us derive, for instance, (1.11a) from (1.10a). We have 2px = Up — nUq + K,(Uq,p)p = 2(9 A g x )(p - nq) + «(2(g A g x )g,p)p = -2(9«,p)« - 2«9 X +

2K(qx,p)p

= 2nqx + 2(p, qx)(np - q). E x e r c i s e 1.5. Show that any solution p of system (1.11) can be obtained by the above procedure from the solution
> 7Q + 1 in- 1 . . Ao = = = — cos p + 1 sin p, 70 — 1 */s + 1 2

/i« + l\ _

\IK — 1 / _ /i«:-l\2 gr = ( . . . ) r. VlK + 1 /

126

II. BACKLUND TRANSFORMS & INVERSE PROBLEM

Consequently, gr' = r,

gr' = r'

-1

for r' = (iK — l)(i« + l ) r = (is'm/3 — cos/?)r. Since the image of the projection Q = (I — q)/2 is included in the two dimensional space spanned by the vectors r and r (or p and q), Q is the projection onto q = I m r ' = psin/? — qcos(3. By Propositionl.5 the function q must satisfy (0.5). The points U = 2q A qx and V = 2q A qt corresponding to J e a n be easily expressed in terms of p, using (1.9b): U = U + 2sinP(q A p ) x ,

V = V - 2sin^(9 A p)t.

T h e o r e m l . 3 . a) Ifq is a soJution, the system of equations (1.12a)

(« + *)«

(1.12b)

(q-q)t

(1.12c)

(f,5)

«2 + l (q,qx)(q-q), 2 «~2 + l

{q,qt)(q + q), *

(q,q)

2

-i

K2 + 1

in q(x, t) € K" i a s a unique solution for K £ K, K ^ 0 and an arbitrary initial value q(xo,to) satisfying (1.12c) at a certain point in a (connected) domain where q is defined. b) Each solution q of system (1.12) satisfies the equation of the Sn~1-Geld. Moreover, \U = qAqx

= qAqx

l y = qAqt

=qhqt

i ^ A ? ) ^ , ^ ) + {qKq)t,

=

(qt,qt) =

(qx,qx), {qt,qt),

Proof. The existence of solutions of (1.12) for an arbitrary initial condition of type (1.12c) is proven above. The uniqueness reduces to standard properties of differential equations. Therefore it follows from Propositionl.5 that any solution q of system (1.12) satisfies (0.5). The formulae for U and V were obtained before and the norms of derivatives of q and q coincide since, by Proposition 1.2, Sp U2 = —2(qx,qx) and SpV 2 = -2{qt,qt). • E x e r c i s e l . 6 . Check directly that solutions of system (1.12) satisfy (<jx,qx) = («*,?*)

(0.5),

(qt, qt) = (qt, qt)-

(Using A, multiply (1.12a) by qx and qx, sum these equations and then compare the result with (0.2). Multiplying (1.12a) scalarly by qx and qx, derive the statement on the norms. The proof is analogous for t.) •

127

§1. BACKLUND TRANSFORMATIONS E x e r c i s e l . 7 . Modify Theoreml.2 and 1.3 for the case of the *-unitary i.e., for the case of indefinite hermitian forms.

fields,

E x e r c i s e l . 8 . Show that, in contrast to the constructions in §1.1, in the present section transformations do not diminish the defining domain of the initial field for x and t due to the hermitian condition. (Singularities of the transformed solutions U and V of Theoreml.l might appear only if $\K\ fl $2-^2 7^ {0} for some x, t. In the hermitian case, $i.Ki and $2-^2 are always orthogonal and, consequently, do not intersect.) D In conclusion, we describe the action of the transformation of Proposition 1.1 on an 5 n - 1 -field. It is easy to show that for 70 € R (in the notation of that proposition) these transformations are compatible with the restriction to an arbitrary real subgroup of the group GL„ of complex matrices. More exactly, if g is a G-field and $ ( x , ( ; ±70) has its values in G, then g is also a G-field. def

Proposition]..6. Let K 6 R, K ^ 0 and Jet $(*) = $(x,t;«;) € On be an arbitrary solution of system (1.1,2A) for currents U = 2q A qx and V = 2q A qt corresponding to an Sn~1-6eld q. Then qW = $/]A<7 is an Sn~1-&eld; currents U^ and V^ constructed from qW are obtained from U and V by the transformation of Proposition 1.1 for 70 = K, 9**0 = « * ( - } « , ,

q(tK) =

K-'^qt.

Proof. First, we check the last formulae:

^-*(-«)& = -*r« l ,(*(.)).* ( > = 1(« - l ) * ^ = | ( K - l)* ( - 1 ) (2fc).

This implies qx

= K$,K-.qx.

An analogous calculation proves the formula for t.

Using the orthogonality of $ and the above formulae for qx compute U(«\ V("> and {q(K\q(K)). D

and q\ , we can

E x e r c i s e l . 9 . Denote the function q for 7 = —i by q+. Derive from Exercisel.2 that ((q(-K))+yi/K) satisfies system (1.12), where the transformation q i-¥ q^ is as defined in Propositionl.6. Any solution of (1.12) can be thus obtained for a suitable initial value of q+. D E x e r c i s e l . 1 0 * . In the notation of Theorem4.3, Ch.l, take two algebraic-geometric S2-fields constructed for two points P = Pi and P = P2 (for the same curve T, the function X and corresponding divisors). Then the second solution is obtained by applying the transformation q t-> ql") for K = (A(P 2 )A(Pi)~ 1 ) 1 / 2 to the first solution. D

128

II. BACKLUND TRANSFORMS & INVERSE PROBLEM

1.3. Sin-Gordon equation. We specialize Theorem 1.3 to the case of 5 2 -fields in coordinates x, t normalized by the conditions (qx,qx) = 1 = (qt,qt) (cf. Introduction). Then (qx,qx) = 1 = (qt,qt) (Theoreml.3, b)). Putting {qx,qt) = cos a, {qx,q~t) — cos a we have that a and a satisfy the Sin-Gordon equation (0.8). Our goal here is to derive differential relations on a and a. This can be done by direct calculation using (1.12) but we take another approach which is not direct but more "geometric.'' Put q(K) = q and q^K^ = p. Then q and p satisfy the following system derived from (1.12) or (1.11) (see Exercise 1.9): (1.13a)

(p + q)x = ( P , ? i ) ( p - q),

(1.13b)

(p-q)t

(1.13c)

= (p,qt)(p + ql,

(?,p) = l,

(P,9)=0.

By Proposition 1.6, (qx,qx)

= « 2 = (px,px), 2

(qt,qt) = «~ = (Pt,Pt),

(qx,qt) = c o s a =

(qx,qt),

{px,Pt) = c o s a =

(qx,qt).

As (qx,q) = (qt,q) = 0 = (px,p) = (pt,p), the vector q belongs to the plane px spanned by px and pt by definition, and p g q1- is expressed as a linear combination of qx and q\. Here we use the condition n = 3 and the equation (p, q) = 0 (which, by the way, characterize the cases K = ± 1 in Theoreml.3 and is the main reason why we turn to §"and p from q and q). Let us denote by {a, 6} the angle from the vector a to the vector 6 with respect to a certain orientation on the plane spanned by o, 6 € R 3 . We choose an orientation in R 3 and then define the orientation of frames (q, qx, qt) and (p,px,pt) consistently with this orientation, and finally define the orientation of planes px and g x accordingly. Changing signs of a and /?, if necessary (these angles are defined up to signs by their cosines), we can put, mod 2ir, (1.14a)

a = {p, qt} - {p, qx},

(l-14b)

a = {q,pt} -

{q,px}.

Differentiating the equation (p, q) = 0, we obtain the relations cos{p, qt} = - cos{§",pi}, cos{p,qx}

=

-cos{q,px}.

§1. BACKLUND TRANSFORMATIONS

129

If {?>Pt} — {?)?<} = "•) t Q e n inevitably {p,qx} + {q,px} = w. The converse case (mod 2x) is also possible. We suggest that the reader draw the figures for this and check this geometric statement. The last equation makes it possible to determine the sum and the difference of formulae (1.14a) and (1.14b) resulting in the formulae a + a = 2{p,qt}, Recalling that (qx,qx)

a-a

=

2{p,qx}.

= K2 and (qt,qt) = K~2, we finally get the relations KCOS(—— ) V

(1.15) «

2

={p,qx),

J cos^—— J =(p,qt).

In the last formulae, in accordance with the sign of K and the choice of the signs of a, a and a ± a may be interchanged and K may be multiplied by —1. We use (1-15) and the fact that qxt and pxt are parallel to §"and p respectively (equation (0.5)): - i • fa + a\(a

r~~\ (pt,q*) = ( £ £ ) « =

+

a

\

- * s i n ( ^ ) ( ^ )

We substitute the above relations into the scalar product of formula (1.13a) and qt, and into the scalar product of formula (1.13b) and qx. As a result we obtain the classical Backlund transformation (1.16a)

/a + a\

=

. /a — a\

(-2-), '""'(-r)-

(1.16b) Strictly speaking, we deduce (1.16) from (1.12) for a certain choice of the angles a and a and for a suitable location of the frames (q,qx, qt) and (p,px,Pt) (cf. (1.15)). Again we suggest that the reader check that in other cases a + a and a — a may be interchanged and the sign of K may also be different. E x e r c i s e l . i l . By generalizing the previous observation or by direct analysis, show that for any solution a of the Sin-Gordon equation (0.8) with arbitrary initial value a(xo, to) there exists a solution a of system (1-16) and that it satisfies equation (0.8). •

130

II. BACKLUND TRANSFORMS & INVERSE PROBLEM

Relation w i t h t h e Lax formalism. Above, we obtained the Backhand transformation (1.16) as a corollary to the existence of the zero curvature representation for the P C F equation, performing five (or rather six) auxiliary reductions. It is natural to expect that (1.16) can be obtained directly from the analogous representation of the Sin-Gordon equation itself. Now we turn to the description of such representations (exactly speaking, to the description of one of them). Put

n-( H

l

d

0\

-{o

-iJ&

ax fQ

+

yy. _ 1 / cos a 2 \ sin a

l\

TVi oj' sin a \ — cos a J '

The function a(x,t) € C satisfies equation (0.8) if and only if the differential operator [d/dt + K~1W, H] is divisible by the operator H — | « J from the right. In fact, this commutator is equal to

* ~ l s i n a ( ( ! ~o)lt

+

axK))+i(ail+Silia)(l

l(-K

where the operator in the parentheses coincides with I

_ ) (H — \KI).

corollary of this statement we get the zero curvature representation Gordon equation: d_ dx

1 / —K

ax\

2 \ — ax

K

5 K

J ' dt

1

2

/cos a \ sin a

o)'

sin a

\

— cos a J

AS a

of the Sin-

0.

Equation (0.8) is also interpreted through the above representation as the compatibility condition of the following system of equations:

da*)

*—**iU :";)•

(1.17b)

i>t =

-K^Wijf.

Using these calculations, we can rewrite the Sin-Gordon equation in the Lax form (cf. §2, Ch.I): (0.8) «• {H2)x = -\[WH-\H2]. The operator WH~X here is an integro-differential operator. Therefore it is more convenient to use (and check) the last relation multiplied by the "denominator": (H2)XH

=

[W,H2}.

§1. BACKLUND TRANSFORMATIONS

131

Another way to construct the Lax pair for the Sin-Gordon equation uses only differential operators, but in 4 x 4 matrices. Note that the problem of searching for the Lax pair is somewhat abstract, since for any concrete purposes system (1.17) or the zero curvature representation of the Sin-Gordon equation are enough. Thus, let a satisfy (0.8) and let tfr = '(V'l) W be a solution of system (1.17) for K € R, K ^ 0, where ^1,2(^11) € R. Below we may regard a and K not only as real numbers, but also as complex numbers. Then 1/) is taken from C 2 . Put

2) = — K_1ip cos a.

(1.18b)

b) The function a = ct + 2(3, where

l)x + y ^ 2 = ^ L (^2)1 + y V ' i = 2^2Multiplying these equations crosswise and subtracting the result multiplied by 2 K - 1 from the first equation of the system multiplied by 2^2* we obtain the equality 02(V>i)* - $i(ih)x

+ y - ^ i + ^2) = Klfolfe-

Dividing this by ij>\, we get (1.18a). Rewrite equation (1.17a) in the following way: K-i

K-i

r-cosaV-i

— s i n 0 ^ 2 = (ipi)t,

K-i

K-i

2-Sma^'1

+

~2~~cosa^2

=

(fa)t-

Multiplying the first equation by fa and subtracting the second multiplied by ^ , we obtain the formula: K-i

(i/>i)f2)ti>i + -x-sina(il>l

-tp\)

= - K _ 1 cos 0:^1^2.

132

II. BACKLUND TRANSFORMS & INVERSE PROBLEM

As above, we divide this by ip\ and get (1.18b). After substituting ip = tan(/?/2) and u = a + (5, system (1.18) has the following form: ux = resin/3, (!-19)

_ . - -1i a /3 t — —re sinu.

It is easy to see that the transformations u = u, ^ — —j3 and re = —re bring (1.19) to itself. Lifting conversely from (1.19) to (1.18), we obtain that 5 = u — (3 = o + 2/3, ip = —ip and re = —re satisfy system (1.18). Consequently, 5 = a + 2/3 is a solution of (0.8). In order to find the relation connecting a and a, we substitute tp = t a n ( ( 5 — a ) / 4 ) into (1.18) and come exactly to (1.16). • To systematize the construction in Theoreml.14, we find a by the method of §1.1 (using the B-factors), starting from system (1.17) (cf. Exercisel.3). Put ip1 = \4>2,ipi) and $ ' = I

I \I> I

l for vectors ip and matrices $> of rank 2.

Then both matrix functions "£(«) and $'(—«) are solutions of system (1.17) (check this). We follow the construction in §1.2, but with the involution defined above, rather than complex conjugation. All functions and parameters may be considered to be real, or, if necessary, complex numbers. For K ^ — «o, $0 is an invertible solution of (1.17) for re = «o, ipo = Xipiifa) is the first column of $ o , and P is the projection onto i/>o parallel to V'o- Explicitly,

Note that P + Pl = I. Put (cf. Theoreml.l) B(K)

= I -

- ^ - P .

K 0 + re

Then B satisfies the conditions of Propositionl.2 for system (1.17) and «i = «o and K2 = —«o (instead of (1.1,2), and the values Aj and A2 of the parameter A). Consequently, for \t(re) = B(K)9(K) functions \ t r \ t - 1 and * t $ _ 1 are analytically continued to rational functions of re with poles (of first order) only at the points re = 0,re = oo, if $ is an invertible solution of (1.17) on some domain for re.

§1. BACKLUND TRANSFORMATIONS

133

As B(oo) = I, the principal part of " i ^ ^ - 1 at x = oo has the same form as M and $ t \ t - 1 = — K~1Wii, where W is a matrix which does not depend on K. It remains to check the consistency of B with the involution:

i-^-pi

B(-Ky =

Ko — K KQ

—K

K—

KQ

The appearance of the scalar factor (K + KO)/(K — «o) for B does not affect the formulae of the type (1.4,5). Hence, Wl = — W and M(—K)1 — M(K) for a matrix M which is M for the function $ instead of if) in the right hand side of (1.17a). Let us compute M and W. P r o p o s i t i o n l . 7 . The matrix functions M and W defined fay the construction in §1.1 from the solution a of equation (0.8) (and from ip = 4>i/ip2 for the solution ip of system (1-17) at the point KQ) have the same form as M and W for a = a + 2/3, where if = tan(/?/2) as in Theoreml.4. Proof. We check the statements of the proposition only for W since the case of M is analogous. By definition W coincides with B(0)W^B(0) - 1 (cf. proof of Theoreml.l). We have: B(0) = 1 - 2 P = B(O)- 1 1 /y-i-y >p-\-if~l \ —2

-2 \ = / cos/? -sin/A (p —
The rest of the proof is a direct calculation using the fact that 5 ( 0 ) and W are matrices of orthogonal reflection in in the lines of R 2 . • 1.4. Application: local conservation laws for t h e Sin-Gordon equation and 5" _ 1 -flelds. Let a(x,t) be a solution of the Sin-Gordon equation (0.8). An equation Q = ffe is called a local conservation law for (0.8), where £ and n are, generally speaking, certain "local'' functions of a (for instance, elementary functions of a and derivatives of a with respect to x). We call the term £ the local density of the conservation law. Note the asymmetric role of x and t in this definition. If the function £ is absolutely integrable in x from —oo to oo and n has limits n(±oo, t) as x —> ±oo and T}(—oo,t) — n(oo,t), then

d f°° —

f°° ((x,t)dx=

nx(x,t)dx = i7(+oo,*)-ij(-oo,0 = 0

134

II. BACKLUND TRANSFORMS & INVERSE PROBLEM

(cf. Ch.I). Consequently, ^^Cdx is an integral of equation (0.8). It is always so for an exact differential ^ £ of an arbitrary function ( of the above type. Therefore densities of local conservation laws are interesting only modulo exact derivatives of absolutely integrable functions of x. The procedure of constructing local conservation laws from the Backlund transformation of the Sin-Gordon equation reduces to the following two steps. First, we construct a solution S of equation (1.16) in the form of the formal power series S = S0 + a x * - 1 -|

h5,K-1 H

.

It is easy to show by expanding (1.16a) in the series of K - 1 and comparing its coefficients that the coefficients Ss are uniquely determined from this equation. Moreover, the a , are expressed as differential polynomials in a (i.e., polynomials in a and derivatives of a with respect to x). Particularly, So = a and 5 i = 2ax. Secondly, we substitute the above series S into the conservation law /, „„•, (1.18)

/a — S\ Kcosf——J

+K

_•, /a + S\ 1 cosl j =0,

which is easily derived from (1.16). Then we equate the coefficients of the expansion of (1.18) with respect to K~X. AS a result we obtain infinitely many series of local conservation laws. We list the formulae for the first three of them: (1.18a)

(«i/2)< = (cost*)*,

(1.18b) (1.18c)

(axaxx)t 2

( a * / 8 + a „/2 + axaxx)t

+ (ax s i n a ) * = 0, + {(a2x/2) cos a + axx sin a ) * = 0.

Law (1.18b) is trivial, since axaxx = (a2.)x/2 is an exact derivative. The densities of all these laws are differential polynomials (without constant terms) not only in a but also in ax. Therefore, if we assume that the values of a mod 2w for x = ±oo coincide, then we can construct integrals of equation (0.8) from conservation laws (1.18) under the assumption of the absolute integrability of ax, axx, E x e r c i s e l . 1 2 . a) Using the uniqueness theorem for differential equations, show that the formal power series for 5 in terms ofn'1 defined by (1.16) satisfies (1.16b). b) Check that the density £* for the coefficient of K~k (k = 1 , 2 , . . . ) in conservation law (1.18) is written as a differential polynomial in ax, and that rjk is a sum of (two) differential polynomials in ax multiplied by sin a or by cos a. c)* Show that the densities ft for even k are exact derivatives, as for (1.18b). O Further we show that the construction above and the statement of Exercisel.12 reduce to results in Ch.I. Now we discuss other methods of enumerating local conservation laws of equation (0.8). For example, we can make use of the zero curvature

135

§1. BACKLUND TRANSFORMATIONS

representation (1.17) and directly apply the methods of §§1,2 of Ch.I for equation (1.17a) or for the operator H. For the computation of the densities £ it is more convenient to use the interpretation of the Sin-Gordon equation as a "higher KdV" equation (cf. §2.5, Ch.I (2.25)). Let us recall that (0.8) is equivalent to the pair of equations (1.19)

Lt = [A,L],

where L = —^ + u,u — a%/4: + iaxx/2 c

Lct =

[Ac,L%

and A = — ( a j + f a * )

s

- j - is anintegro-

c

differential operator and L and A are formal complex conjugation (ic = —i and ac — a). Independent of the form of the operator A (for the KdV it is a third order differential operator), the densities h of conservation laws (2.21a) of Ch.I are given by the same formulae as for the KdV. We can also take the formal complex conjugate of (2.21a). Thus we obtain the following recurrence algorithm for computing the densities hk of local conservation laws for the Sin-Gordon equation: hi = u, m-1

hm+i — —(hm)x + 2 J hkhm-k, Jfc=i

where u = —U2 is expressed via ax by the above formula. S n _ 1 - f l e l d s . Regarding (0.8) as a reduction of the equation of an 5" - 1 -field q (0.5), we construct an analogue of conservation law (1.18). In the notation of §1.2, by differentiating the function log(^J, tp) with respect to x and t (cf. formulae (1.10)) we get the relation K{Ur,r)t+K-l{Vr,r)x =0. Rewriting it in terms of p and q (r = y/2tp(tp, ip)*1?2 conservation law (1-20)

K{p,qx)t + K-1(p,qt)x

= p + iq), we derive a

= 0,

where p satisfies system (1.11). We are going to show that conservation law (1.20) corresponds to (1.18) after the reduction. In the notation of §1.3 the functions p and q satisfy (1.20) for /e = 1. On the other hand, (cf. (1.15))

(P, 9r) = « cos(—"^—J. Hence we indeed obtain (1.18).

(?> %) = « - 1

cos —

( o—)'

136

II. BACKLUND TRANSFORMS & INVERSE PROBLEM

Thus the construction for the Sin-Gordon equation described above reduces to Theoreml.4, §1, Ch.I (§1.5). We called the corresponding conservation laws local Pohlmeyer laws. Moreover, (1.20) is a direct consequence of formula (1.20) of Ch.I (surprising coincidence of the numbers!). In particular, Corollaryl.2 and 1.3 of Ch.I generalize the statements of Exercise 1.12 (we will discuss it in detail later). Let us examine the procedure for computing local conservation laws for 5" _1 -fields which generalizes the computations for the Sin-Gordon equation. P r o p o s i t i o n l . 8 . For an Sn~1-£eld q there exists the unique solution p of the equation (1.11a) of the form p = X)tLoP* K_ *> where pk(x,t) € R n . Such a p satisfies equations (1.11b) and (1.11c). Proof. Not using the results of Ch.I we will prove this and give an algorithm for constructing {pit}. Rewrite equation (1.11a) in the form (1.21)

« _ 1 ( p , qx) + « ~ V * + qx = (p, q*)p-

Then it is easy to see that po = qx/\qx\, where we set hereafter \z\ = (z,zy/2 ^ 0 for z € K". Substituting qx = Po\qx\ into (1.21) and collecting the terms for K~k, we obtain the following relation: (1-22) \q*\((Pk>Po)po + p t ) = { expression of q, po,... , p * - i and their derivatives by x }. Assuming that p i , . . . , p * - i are already found, we can define p* by this relation. Taking the scalar product of both sides of (1.22) with po, we get the equation l<7i|(p*)Po) = { expression of q, p o , . . . ,Pk-i

and their derivatives by x }.

Express (p*,Po) in terms of q and the previous {pj} then substitute it into (1.22). Then equation (1.22) transformed in such a way can be solved with respect to p*. It is easy to check that the obtained pk also satisfies the original equation (1.22) (i.e. before the substitution of (pt,po))- Thus we proved the uniqueness and found an algorithm for computing {pt}. Let us denote the series generated by {pjt} by p. Let us check that (p, q) = 0. Taking the scalar product of (1.21) with q, we get the formula K~l{p,q)x = (p,?)(p,9x)As the constant term of (p,q x ) is equal to \qx\ =/= 0, the series (p, q) must be a zero series in « - 1 . Analogously, multiplying (1-21) by p and using the orthogonality of p and q just proven, we have the identity «-1|p||. = 2 ( p , f c ) ( | p | 2 - l ) ,

137

§1. BACKLUND TRANSFORMATIONS

from which it follows that |p| 2 = 1. By a similar method we can check that this p satisfies (1.11b). Note that the first equation of system (1.11) is enough to define p as a formal power series (cf. Theoreml.3). D Substituting the above p into (1.20) and expanding this relation into power series, we obtain an infinite series of local conservation laws for (0'.5). Here are the first two nontrivial laws (cf. [Poh]): f 1 I d qx |2-| _ f ( g x , g t h I2|fc|lax|fc|l h I \qx\ J , ' f 1 I d t 1 d qx \ | » 5 I d qx |*> \2\q • 2\q \\qx\x\J\dx \qx\J\ 88\q | a ,x\l3s l aI idx ; | o\q . | xl\ I it x\\dx\\q x\\dx x\dx\q {qx,qt)\ ad qx |-"i ((qx,qt)i 3 \dx\qx x\ "*" I 2\q 2|«J» \dx\QA\ h

n

In the coordinates x, t normalized by the conditions \qx\2 = 1 = \qt\2, they are written in the particularly simple form: {|««| 2 /2}« = { ( « , , * ) } « ,

(1.23a) (1.23b)

{\qxxx\2/2

~ %x*| 4 /8}< + {(qx,qt)\qxx\2/2}x

= 0.

E x e r c i s e l . 1 3 . a) Following the proof of Propositionl.8, write out the recurrence relations for pk (cf. §1, Ch.I, or [OPSW]). b) Show that in the normalized coordinates the densities £ of the local conservation laws constructed above are polynomials of scalar products of qxx, qxxx and so on (cf. Exercisel.6, Ch.I), and that the rj are also this kind of polynomial multiplied by [-¥ f(x) dx. Otherwise the integrals / ^ (,dx would be useless. Before ending this section, we briefly explain how to extract the local conservation laws of the Sin-Gordon equation from those of S2-fields. Let q be an S2-field in the normalized coordinates. Take an orthogonal basis of R 3 , q, qx and q + qxx ({q,qxx) = — \qx\2 = —1). Starting from (0.7) (Introduction), we see that l««*|2 = o£ + 1,

\q + qxX\2 = a2x.

Differentiating the first relation, we get the formula (qxxx, qxx) = ctxaxx. Further: (qxxx,qx) = ~(qxx,qxx) = -ot2x - 1, (qxxx,q) = -(qxx,qx) = 0 (|g x | 2 = 1 =*• {qxx,qx) = 0). Consequently, \qxxx\2

= (qxxx,q)2 2

= a +ai

+ (qxxx,qx)2 + 2al.

+ a~2(qxxx,

q + qxx)2

138

II. BACKLUND TRANSFORMS & INVERSE PROBLEM

As (qx,qt) = cos a, (1.23) implies relation (1.18a): (cosa)* = ( a £ / 2 ) t , and we get a conservation law which is a linear combination of (1.18a), (1.18c) and the derivative of (1.18c) with respect to x. E x e r c i s e l . 1 4 . Show that the orthogonality of the basis q, qx and (q + qXx)a~l in R 3 allows us to express any scalar products of derivatives ofq with respect to x in terms of differential polynomials of ax. • 1.5. D a r b o u x transformation; Nonlinear Schrodinger equation. In the notation of §2.5, Ch.I, let tj) = (1 + X ^ i k~'ips)ekx be an abstract formal power series in the parameter fc_1 with indeterminate scalar coefficients il>j{x). In §2.5, we constructed (differential) fractional powers of Lr from ^>, for which Lr%l> =fcrV>+ 0{k-1)ekx. Here

^ = fe)r+E«-fe) for r ^ 1, and the uI|T. are expressed as differential polynomials of ^>i, ^ 2 , With the help of Lr we can introduce pairwise commutative differentiations of the ring of differential polynomials of {ip3} by (2.17) of Ch.I: (1-24)

^

= Lrxl>-

krrp.

These differentiations commute, by definition, with d/dx; and dekx/dtr = 0. Note that the differentiation d/dt\ acts on {i/>a} as d/dx. Let if)1, i/>2,... ,^> m be a set of functions of x,ti,t2,... that are solutions of (1.24) at k = fci,&2,... ,km € C, where (1.24) is regarded as an (infinite) system of equations with functions u; ) r as coefficients. For concreteness we could restrict ourselves to a finite number of indices r. There may be repeats among hi, but {V"'} should be in generic position (see below). Put

TJ>(k) =

j>(k)exp(*Tkrtr)

and analogously define */>' (here, the exponential factor plays its role only in the differentiations with respect to tr). We call the function (1.25)

*'(*) = y

'

t - i ^ ^ . - - ^ ^ ) )

w.(f

^-)

139

§1. BACKLUND TRANSFORMATIONS

the Darboux transformation

of the function ij>{k) for m ^ 1. Here

Wl ,..,/o

d f

= det((0-y)

is the Wronskian of the set of functions {fi}, 1 ^ i,j ^ I. Just like ipe~kx, we have x[>'e~kx = ij)' exp(—kx — X)%=i ^ r *r) is a formal power series of fc_1 with constant term equal to 1, where V>' is defined in terms of %j) by the same formula (1.25). T h e o r e m l . 5 . Define t i e fractional powers L'T and the differentiations using rp'. Then for r ^ 1 dip' df dtr ~ dt'r'

d/dt'r

that is, the differentiations d/dtr and d/dt'r coincide on the ring of differential polynomials in the coefficients ip'a of the series ip'e~kz. Here we assume that Proof. Set / = Wm+iffi,... we have

y-=rw^L.M dt r

, ip"1,^).

By the commutativity of d/dx and

iu^,f#

d/dtr

*-&

1=1

m

.

=^Wm+1(^1,...)Lr^,...^m,^)+^m+1(^,...,^m,Lr^).

Let us fomulate one simple property of Wronskians. Let tp, o i , . . . , a m be functions of x, Wm(a\,... ,am) ^ 0. Then each of the derivatives d"

1=1

where L is a differential operator of order r in x, and the coefficients of L and the functions {ci} are differential polynomials in iji1,... ,xj>m divided by powers of

140

II. BACKLUND TRANSFORMS & INVERSE PROBLEM

Wm(i>1,... ,ipm)- Let us show that c,- = 0. First, substitute ij> = ip1,... by one in the above formula. Then / = 0 and consequently

, ^ m one

for 1 ^ j ^ m. As W m ( ^ 1 , . . . ,t/>m) ^ 0> it is necessary that c* = 0 for 1 ^ i; ^ m. Thus ctr

otr

where L' is a differential operator of order r which does not depend on k (with the highest coefficient equal to unity). Since di/>'/dtr = 0(k~1)ekx, it follows from the above relation for V that this operator must coincide with the operator L'r constructed from tp'. • C o r o l l a r y l . 2 . Let us assume that functions «; )S (s = p, r) satisfy the ZakharovShabat equation which is the consistency condition of the pair of equations

L

~ZT =

P$I

dt

oy for some p,r, or that they satisfy the Lax equation which is the consistency of equations

condition

= L j> - fcpV,

p dt Lrtp = krip.

(1.26)

T i e n t i e same holds for the coefficients u\ s corresponding to the function ij>' constructed above. D K d V equation. As is well-known (cf. §2.5, Ch.I), up to normalization this equation appears as the consistency condition of equations (1.26) for p = 3, r = 2. Let ( d\2 Then

/d\3 La =

KTJ

3 +

d

2UYx

3 +

4"-

§1. BACKLUND TRANSFORMATIONS

141

It follows directly from formula (1.25) that the Darboux transformation of the function u for the function V"1 which satisfies equation (1.26) at a point k = k\ is the function u' = u + 2vx, where v = ( l o g ^ 1 ) I . Indeed, V'' = V>i - i>v, + (v2 +k2-

r/>'x = -$xv i>xz + (u-k2W

k2)j>,

(2v2-2(k2-u))4,'.

=

Here we use the formulae ipxx = (k2 — u)ij>, vx = (kf — u) — v2. Actually we are now repeating the classical argument by Darboux (and partly reproving Corollaryl.2). Note that for arbitrary m the transformed function u is equal to u' = u + 2(logWm(i>\...,4,m))xx, which is a result of Crum (cf., for example, [DT]). Now we connect the Darboux transformation with the Backlund transformation in §1.1. For this purpose, following the method in §2.5, Ch.I, it is necessary to turn to the first order system from the equation tj>xx = (k2 — u)ip and the corresponding relation for */>*. We have:

1.27)

WrV

^k2

QJ

^

QJ,

*-^i(Ar)

r M

) ,

where ip(k), tp*(k) are two non-proportional solutions of system (1.26) for p = 3, r = 2 at the point k. For A = k2, Aj = k2 let B

V(A-A 1 ) + u'!

-vj'

We check that B satisfies the conditions of Proposition 1.2, i.e., $ 1B XG\ does not depend on x, t in a neighbourhood of any point A = I € C U oo. Indeed, B

A - A ! V ( A - A i ) + t>2

B-1C?? = C(A-A 1 )- 1

v)'

(l)+o2

= *(*t)((5)C + 0?)'

142

II. BACKLUND TRANSFORMS & INVERSE PROBLEM

Oi = OAI in a neighbourhood of the point A = Ai, where the module in the parentheses does not depend on x and t. It is necessary to examine the point A = co since B has a pole there. We have

B-'Ol = ( J ) C + X-'Ol = ViX-'Ol), because the expansion of $ diag(e - * x , ekx) into a series in a neighbourhood of k = oo is easily computed and begins with I ,

, J. Strictly speaking, in Propositionl.2

"critical" points of B (points A i , . . . , Aw) must not coincide with singular points of $ (points A = ±1), as is the case here. However the method of the proof of the proposition does not change in the given situation (check this). Thus, by Proposition 1.2 ii'x(^')~1 is a rational function of A with a unique pole of first order at the point A = oo. (The estimate of the order of the pole at oo follows from the independence of ii!~lB~10\0 of x.) Moreover it is easy to prove that $ ^ ( $ ' ) _ 1 coincides with the right hand side of (1.27) for u' = u + 2vx instead of u. An analogous computation works also for $ J ( $ ' ) - 1 . Of course, the last statement can be checked by a direct method. We presented the above reasononing in order to illustrate the coincidence of the transformations given in §1.1 and the Darboux transformations in the case of Lax equations. The following example is more interesting. Apply the result of Corollary 1.1 for $ instead of $ and Q = I

) with a = 1. Set as before A = k2, Ai = k2. Then

prtp-i _ ( -V>(*i)i>x{ki) v2(fci) "\ .-i v 2 * * - { -(^(fci)) y>(fci)v>x(fci)y ' where V-A = ' d^/dX, and d = d e t F = W2{i/?(k1),^*(Jfci)) + W 2 (V>(fci),iMM). that W2(ip(ki),ip*(ki)) does not depend on x. It follows from Corollary 1.1 that $ I * - 1 = BXB~X

Note

+5(*I*-1)S-1

(for B = I - (A - X^FQF'1, * = BV) is a rational function of A with a unique pole at A = oo. In addition, the principal part of the above expression is equal to

(i:> \u

The constant term of — ^x^

oy + [*w*

can be easily calculated. It is equal to

' V i o y j - V " Q) + d\2wi{ki)

-Hh)J-

143

§1. BACKLUND TRANSFORMATIONS

Let us denote the first column of the matrix * by \ipi,varphi2).

{tp2)s = (A - u)tpi + ^{hfyidT1 (Vi)xx = (A - u)yi -

-

Then

2(iP(ki))xTl>(k1)
2(dx/d)xip1.

In the derivation of the last formula we use the relation dx = ^ ( ^ l ) 2 which follows from the equation {ij>\)xx = ip + (A — M)V>AIn this way, starting with the transformation of §1.1 with "confluent zeros" Aj = A2, we construct from a pair of functions ij> and u connected by the relation xpxx = (A — M)V> the functions • k\. Analogously we can generalize Crum's formulae with arbitrary numbers k\,... , km via suitable matrix Backlund transformations (but only for Lax type equations). N S equation. As a conclusion of this section, we deduce the Darboux type formulae for the NS equation. We omit the complete derivation in general cases, referring the reader to appropriate literature. Continuing in the line of Propositionl.7, we apply the reduction of Theoreml.l to this equation. The NS equation irt = rxx + 2ur\r\2 (u> = ± 1 correspond to the "attractive" and "repulsive" cases) turns out to be (cf. [ZMaNP], [TF2]) the consistency condition of the system (1.28)

- 5 - = Af¥,

-=- =

ox

N*

at

for M

_ ( —iA - w r \

N —(

—

2*A2 + * w r r

—2wAi— iu>rx \

which is called the zero curvature representation of the NS. Together with i/>(\) = \4>\, $2), the function^(A)* = \ipx(X), —w^i(A)) is also a solution of system (1.28) with the parameter A instead of A. Let P be the projection onto the direction of V^Ao) = t ( 0 i , V§) parallel to ^>(A0)*. Such a projection P satisfies the conditions of Theoreml.l. We can write an explicit formula for it:

144

II. BACKLUND TRANSFORMS & INVERSE PROBLEM

Following Theoreml.l, let us introduce B = I - *f~^P and * ' = B^>. Analysis of the relation * ; ( $ ' ) _ 1 = BxB-1 + BMB~l (cf. (1.4,5)) in a neighbourhood of the point A the same form as M: iu(r' — 0 0 -i(r' — r)

r)

oo shows that M' = * i ( * ' ) _ 1 has

'•(» J).

) = ( A o - A 0 ) P, 2(A0

Ao)

0

^^1^2

tffft+unfifi

0

As a result we construct a new solution of the NS equation from r: (1.29)

,

4f Im A0 1 + uxpip'

where y> = tl^/xj^. It is easy to write the differential equation for y> (the Riccati equation) and to prove the analogue of Th'eoreml.4. Note that in the case of ui = + 1 the function r' is defined on the same domain of x and t, as r, since 1 + ipTp ^ 0 (cf. Exercisel.8). We can also apply Propositionl.2 (instead of Theoreml.l) with appropriate generalization for a set of points Ai, A i , . . . , Aw, Ajv, and vector functions t^(Xi), V»(A,)* as the images of principal parts of matrices B, B_1 at the points A i , . . . , A# and A j , . . . , A/v respectively. We search for B of the form

B = I-Y =Qi t?Ai-A Calculating B~1, we obtain a system of linear equations determining matrices Qi via {V'(Ai)}- After that we get an iV-fold iteration of formula (1.29). In the following exercise, we formulate the final result, leaving the proof to the reader (cf. Crum's formula (1.25)). E x e r c i s e l . 1 5 * . Let V'(Ai) = '(V'iiV'i) D e solutions of system (1.28) for a solution r of the NS at pairwise distinct points A = Ai,. ,AJV ( A i ^ A j j . Put

-Mfii

A,- = A,-,

^N+i = A,',

§1. BACKLUND TRANSFORMATIONS

for 1 ^ i ^ N. Consider a 2N x 2N-matrix,

145

the k-th row of which has the form

(1>"?«:, A*, A A ^ , A | , A ^ f c , . . . ,Xk

_1

, A f c ),

and a matrix of the same size with k-th row

(l.^.At.At^.Al.A^,,..,^-1,^-1^). Let us denote the determinant of the first by Ao and of the second by A i . r = ro — 2iwAo/Ai is a solution of the NS equation. D

Then

E x e r c i s e l . 1 6 . Following Propositionl.7, show that formula (1.29) can be obtained as a corollary of Theoreml.l specialized for the Heisenberg magnet st = sx sxx,

(cf. Introduction

§0.3).

(s,s) = 1

•

1.6. C o m m e n t s . Backlund and Darboux transformations of integrable equations are a brilliant part of the soliton theory. They are closely connected with local conservation laws, the inverse problem method and the algebraic-geometric technique. In more algebraic versions of the soliton technique, the infinite dimensional groups of BacklundDarboux transformations (actually, they are loop groups -see [Chll]) appear even before introducing soliton equations. It is remarkable that only after the appearance of soliton theory did the equivalence (in principle) of classical Backlund transformations [Ba] for the Sin-Gordon equation (1.16) and Darboux transformations (see [Dar]) for the equation i\>xx = (fc2 — u)ip (§1-5) became completely clear, though they were invented more than one hundred years ago. The work by K. Pohlmeyer [Poh] plays an important role in the matrix generalization of Backlund transformations. He considers transformations of 5 n - 1 -fields in the form of an overdetermined system of differential relations on initial and transformed fields. In the article [Ch6] this result was generalized to orthogonal fields and moreover was attached to the solutions $ of the corresponding linear system (1.4,5). We mostly follow this paper and [Ch2] in this section. Compared to [Ch2,6] or [Poh], however, we make the constructions more transparent, because today we better understand the invariant meaning of Backlund transformations. Theoreml.l and Proposition 1.2 are taken from [Ch3] (results close to Theoreml.l can also be found in works by other authors, for instance, in one of the papers by Chudonovsky). Theoreml.2 is contained in [OPSW], in which results of [Ch6] and [Poh] are transfered to Un-fields, though the approach is different from ours (it is less constructive and more in the style of Pohlmeyer). In the same paper the

146

II. BACKLUND TRANSFORMS & INVERSE PROBLEM

statement of Theoreml.2 is formulated (which is in fact a complete version of the results by K. Pohlmeyer). Note also the work [U] which is devoted to Backlund transformations for principal chiral fields (in this article, particularly, the relation of infinitesimal Backlund transformations to the Lie algebra of formal currents is established). In the works [CGW], [UN] (and others) the duality equation is examined by an analogous technique (cf. also [Taka]). It is important to note the close relation of Backlund transformations to the "dressing" transformations of Zakharov-Shabat [ZSl,2] which correspond to adding discrete scattering data of the associated linear problem (see the next section). This relation was not immediately clear, although it was rather well-known that the construction of multi-soliton solutions by the inverse problem method coincides with the application of Backlund transformations to trivial solutions of integrable equations. In articles from [Mi] (and other papers) the examples that the operation of adding "zero" to the coefficient a of the corresponding linear problem induces a certain Backlund transformation were considered. Now all these important pioneering observations are included in the general scheme and the connection is obvious. The constructive form of the classical Backlund transformation for the SinGordon equation (Theoreml.4) is from [Ch2]. (I used the analysis of this transformation performed by K. Pohlmeyer). Similar arguments are contained in [AKNS2], [Lu] and in the paper by H. Flashka, D. McLaughlin [Mi]. It might be that such an interpretation is contained in some classical works on the Sin-Gordon equation. The result of Theoreml.4 was reproven by V. B. Matveev and M. A. Sail who generalized the Darboux transformation (cf. Proposition 1.7). As to the zero curvature representation of the Sin-Gordon equation, we refer to [Takh], [AKNSl] and [ZTF], [TF1]. For local conservation laws of the Sin-Gordon equation, see [DB], [ZMaNP], [TF2] and [Poh], [Ch2]. The connection of Backlund transformations and the Darboux-Crum transformations was established (in principle) by F. Estabrook and H. Wahlquist, in particular, in the paper by H. Wahlquist in [Mi]. Note also Hirota's formula (cf. [Mi]) which reflects this connection, and a series of other close observations (cf. also [DT]). B. V. Matveev extended the technique of Darboux transformations to the ZakharovShabat equations (Corollaryl.2) and several similar soliton equations. Theoreml.5 is a generalization of the result by B. V. Matveev (see, for example, [Matl,2]). At the present time there are many papers devoted to Backlund-Darboux transformations of concrete equations, which we do not review here (there are also works on the "invariant" theory of such transformations -see [Chll]). We also mention a series of results by M. A. Sail [Sa], results by A. R. Its, D. Levi, A. I. Bobenko on the Backlund transformation. The statement of Exercisel.5 is from the work by M. A. Sail [Sa] (cf. also the paper by H. Wahlquist in [Mi]). It seems that the

§1. BACKLUND TRANSFORMATIONS

147

Backhand transformation for the NS equation was considered for the first time by G. Lamb. We do not touch upon the conceptually important papers by R. Hirota, works by M. Sato and the group of Japanese mathematicians, in which Backlund-Darboux transformations appear as a result of the action of a central extension of the group GLoo on the r-functions (cf. for example, [DKM] or [Chll]). We also do not mention the beautiful differential-geometric interpretation of Backlund transformations and their applications to soliton surfaces (started with the classical results by E. Beltrami, L. Bianki and others). See, for instance, [LR] and [Ba], [Ei], [H], [Tc]. We also refer the reader to more recent works by R. Sasaki and A. Sym (cf. [Sy]) on this subject. The results by F. Griffiths generalizes this direction. Works by H. Wahlquist, F. Estabrook and others on pseudo-potentials and the "theory of continuation" for soliton equations have direct relation to Backlund transformations (cf. [EsW], [C]). Certain geometric aspects of these transformations are connected with the work by J.-L. Verdier on the classification and description of instanton chiral fields on spheres and with a series of results by A. A. Belavin, A. M. Polyakov, V. L. Golo, A. M. Perelomov, H. Eichenherr and others on instantons in chiral models. Ending this far-from-complete list of references, we mention works by R. Anderson, I. H. Ibragimov, G. Neugebauer, S. Orfanadis, M. A. Semenov-Tian-Shansky and A. B. Shabat. Recently, transformations of solutions of integrable equations connected with the action of the Virasoro algebra on compact and non-compact Riemann surfaces have been studied. In contrast to classical "pointwise" Backlund transformations it is necessary here to use the $ in a neighbourhood or on certain contours in the spectral parameter.

§2. Introduction t o t h e scattering t h e o r y In §§2.1 and 2.2 we introduce the monodromy matrix T for equation (1.5) of Ch.I in the case that the diagonal matrix UQ has purely imaginary entries and that the matrix function Q has absolutely integrable entries. (Note that in the literature, the matrix T is also called the scattering matrix, though "monodromy" dominates.) Herein we construct solutions of (1.5) Ch.I which are analytic on the upper and lower half-planes of the spectral parameter and study analytic continuations of the principal minors of T. In §2.3 we briefly discuss variations of the afforementioned construction in the case when some of eigenvalues of UQ coincide and in the case of a general (not necessarily purely imaginary) complex matrix UQ. §2.4 contains the "discrete scattering data," a formulation of the inverse scattering problem for (1.5) (a Riemann problem on the real axis), and certain results on its solvability. Then in §2.5 we calculate the variational derivatives of the entries of T, allowing us to prove the "infinitesimal" solvability of the inverse problem and to compute the Poisson brackets of the entries of T. The reader does not need any preliminary knowledge of scattering theory. We use only the theory of linear differential equations (cf., for example, [Al]) and fundamental facts from the theory of complex functions of one variable (cf., for example, [Ca]). 2.1. M o n o d r o m y matrix. In this section, Q = (q°) is a matrix function of x € R with values in the set of n x ra-matrices whose entries q' = ?'(a;) (located at the crossing of the r-th row and the 5-th column) are absolutely integrable functions of x from —oo to oo. Put \q{x)\ = max\q'r{x)\,

CQ =

r 3

\q(x)\dx. J-oc

-

Throughout this section except at the end of §2.3 we assume that the constant diagonal matrix UQ = d i a g ( / / i , . . . ,//„) (cf. Ch.I) has purely imaginary entries \ij which are ordered in the following way: def

(2.1)

HJ

= ia.j,

aj g R,

j > s

=•

aj ^ a3.

Note that there can be repeats among the /ij. To simplify formulae we denote the matrix-valued function exp(aUox) by Rg — RQ(X; a). Finally, following the notation in Ch.I, we put Q' = (q'r), where q'"r = q* if fj,, yt ^ j r , and q'r = 0 otherwise, i.e., q1^ = q°(l — 5£'), or in other words,

Q'(z)e[cro,fl[B],

[Q-Q',u0] = o. 148

§2. INTRODUCTION TO THE SCATTERING THEORY P r o p o s i t i o n 2 . 1 . There exist unique invertiblematrix-valued and E-(x; o) (where x, a € R) of the equation (2.2)

EX + QE =

which satisfy the normalization

149

solutions E+(x; a)

aU0E,

conditions

E± -¥ Ro^E±(x;

a ) - Ro(x; a) = o(l)

as x —)• oo (for E+) and as x -¥ —oo (for E-).

These are called the Jost

functions.

Proof. Arbitrary solutions of (2.2) can be written in the form EC via an invertible solution of this equation, where C is a matrix which is constant with respect to x. The existence of E± is checked constructively in a simple way, by explicitly writing the function in the form of "ordered exponentials". Put (formally) for p ^ 0 /•±oo r±oo

K%+1\x;a)=

/ Jxi

Jx

(2.3)

0)

tf£

/-±oo P

~[[l%1QR0(x,+1;a)dxp+1...dx1,.

•••/ Jxv

„

n

= J. OO

E±(x; a) = Ro(x; a) £

#£>(*; a).

As usual, by the definition of integrals, i i runs in the interval [x, ±oo) from x to ±oo; x 2 € [xi,±oo) and so on; fx °° f(y)dy

= —f*xf(y)dy.

The existence of each

of the integrals in the formulae for K± follows immediately from the absolute integrability of the entries of Q and the fact that Ro (for | exp(anjx)\ = 1) is unitary. Moreover, it is easy to prove the following estimate for the entries ( f c ^ 1 )> o r " t n e matrix K{^+1):

(2.4)

1 £,>£»,

|A4^)|^|^L1,

where we abbreviated |(fc2' + 1 ) );| to Indeed, |fc^

\k^+1)\rj.

'|J does not exceed

• def

C±oo y±oo

/ Jx

/ JXl

y±oo P

'•'/ Jxp

J\\+i)\dxP+i---dxi s=0

150

II. BACKLUND TRANSFORMS & INVERSE PROBLEM

The factor np appears here due to the matrix multiplication; it is equal to the number of products of matrix entries of RQQR^1 in the sum for (&2' +1) )J- The integrand of I is invariant under an arbitrary permutation of the arguments x j , . . . , Xp+i. Consequently we can change the integration over the domain {x s +i € [x s , ±00), 0 ^ s = p} C R p + 1 , where we put xo = x for consistency, to be integration over all the space R'+1, and

/=

p

^OO

yOO

P

dx\ (?+i)iy--y-.nw*^)i^i... (P + I)-' 7-00 7-oo £ '"' ' ny

,+in

P+1

(P+ 1)!'

Using (2.4), we obtain that series (2.3) for E± converge absolutely and uniformly for x, a € R. Due to the uniform convergence, we can differentiate these series term by term with respect to x. Thus we get ( f f £ + 1 ) ) , = -^QRoK^, since RQ = exp(aUox).

(flb), = <*UoRo,

Therefore functions E± satisfy equation (2.2)

Integrating with respect to Xi iiin the formulae for K± following variant of estimate (2.4)

(2.4a)

, it is easy to obtain the

|fcip+1)(x)|^n"c±(x)^.

Hereafter we denote the absolute value of the integral fx °° \q(x)\ dx by CQ(X). Obviously CQ(X) ^ CQ and CQ(X) -*• 0 when x -> ±00. Consequently, (2.4a) imphes E±(x; a) -¥ RQ(X; a)

as x -> ±00.

To complete the proof we must check the invertibility of E±. Considering the exterior power / \ n C instead of C", we get a differential equation for the determinant, det E, where E is a solution of equation (2.2): (2.5)

( d e t £ ) x + S p Q d e t £ = aSpZ7 0 det.E.

Since det E± —> det Ro ^ 0 as x —>• ±00, it follows from (2.5) that det E± do not vanish for any x, t 6 R. •

§2. INTRODUCTION TO THE SCATTERING THEORY

151

Deflnition2.1. The invertible matrix-valued function T(a) for a € R, deterT mined uniquely by the relation E+T = E- and expressed in terms of K± of (2.3) by OO

= +oo) = Yi, K~\x = +°°)

T = ^E-ix

p=0

is called the monodromy matrix of equation

(2.2).

Set Q° = Q — Q' and denote solutions of equation (2.2) by E"± normalized as in Proposition2.1 for Q° at a = 0. L e m m a 2 . 1 . a) Let E± = f E^R^1. The entries of E± - E^ and T - E™(x = +oo) as functions of a are Fourier transformations of certain absolutely integrable functions on the real axis. b) The functions E± and T are continuous and invertibJe on R U oo and E±(a ->oo) = E%' = E±(a -» - o o ) , T{a -+ oo) = E™(x = +oo) = T(a ->• - o o ) . In particular, when Q = Q' we have E"± = I and E±,T

-+ I as | a | —>• oo.

Proof. We set £ i p + 1 ) ( z ; a ) = RoK^^^ix;

a)

for p ^ 0 and, use (2.3), to write an explicit formula for the entry (A± )* of iCjf . For notational ease let i = io and j = iP+i, x = xo- Then the entry above is equal to the sum of the following terms over all sets 1 ^ i'i, (2> • - - i ip ^ n: /-oo

oo

/

/

P

expfta £ >

-ooJ-oo

v

i m

-a,p+1)(xm - x m + 1 ) j -

m=0

p

p

m=0

m=0

• 1 7 9.'m+1(a;"»+i) J J 0 ( ± ( x m + i

-xm))dxp+i...dxx,

where 8(x) = 1 for x ^ 0 and 0(x) = 0 for x < 0. The integrand in (2.6) is an absolutely integrable function on Rp+1 (see estimate (2.4)). If Oim, ^ °t m / + 1 f ° r some m', then we can make the non-degenerate coordinate transformation Xm = xm,

for m ^ m ,

p

*m' = 2 j ( a , - » ~ aip+l)(a;m - Sm+l)m=0

152

II. BACKLUND TRANSFORMS & INVERSE PROBLEM

Integrating with respect to all xm, m > 0, without xmi, we can write term (2.6) in the form

eiaxf(x)dx

/ «/ — C

for a certain absolutely integrable function / of x = xmi which also depends on the indices {im}, P and XQ = x. We can estimate $™ 1/(501 dx as in (2.4). As a result, we see that this integral does not exceed Cg + 1 /(p + 1)!. Thus {k±

')' is represented as the sum of the Fourier transformation

f

exp(iax)g(x)

dx

J — oo

of a function g, for which

J— <

and the "anomalous'' terms of (2.6) such that aim = a,i0 for a l l O ^ m ^ p + 1 (recall that {k^+1))i is a sum of np terms of the form (2.6)). Returning to the summation over p (E± = I + Y^,tLi K± )•> we obtain the first statements of the lemma. Indeed, the sum of the "anomalous" terms over all sets {im} and indices p is precisely equal to E™. The invertibility of E™ follows immediately from equation (2.5) for det.E|° (E± are invertible for a € R by Proposition2.1). If Q' = Q, then E£ = I. This is obvious, since a,„ = aim+1 => g £ + 1 = 0 =>• (2.6) = 0. In order to finish the proof, we use a well-known property of Fourier transformations of absolutely integrable functions. • Before ending this section we construct and study analytic continuations of the boundary columns of E± (corresponding to eigenvalues y.\ and fi„) by direct analysis of formula (3.3). In the next section we complete these results using purely algebraic methods. Set E± = E±R^1 as before. Let us denote the fc-th column of E± by e ± . Define analogously e± = exp(—iaa.kx)e±. Set T _ 1 = ( r ^ ) and, for consistency of notation, T = (t}+i). We will assume that fj.„ = fj,n and /J.r = p„ for the column e'+ and the entry f_T, respectively, H„ = m = p r when e'_, t+r are considered. We call such indices r and s boundary

153

§2. INTRODUCTION TO THE SCATTERING THEORY

indices. Let e ± " denote the s-th column of the matrix E± of Lemma2.1 and

e^'

denote the corresponding entry of E™. We remind that e

±

=

1

»

e

±r -

d

r:

when Q' = Q, where

l1

0

1' =

...,1» =

w

0

W

L e m m a 2 . 2 . a) The vectors e + , e l , and t*_T, t"+r for boundary indices r, s may be continued to analytic functions of a on the upper half-plane I m a > 0. b) For boundary indices r and s, and for I m o ^ O we have: | a | - • oo =• ea± -> e^",t'±T

-» £ ? ; ,

where we denote the analytic continuations by the same letters. c) For boundary indices r and s, and for Im a > 0, x -+ +oo =*• e + ->• l ' , e l r -> t+ P ) x -> - o o =*• e l ->• l*,e+ r -»• r l r . I f / i j ^ /ii, then limx_,+<x> e l j = 0; if fij ^ /i„ then lim;c_>_00 e ^ = 0, where I m a > 0. d) Statements a), b), c) hold if we replace the condition that Im a ^ 0 (Im a > 0 for fij 7^ /^I in cjj by Im a ^ 0 (respectively Im a < 0) and exchange the places of fix and fin in the definition of boundary indices (i.e., regard that fir = fi\ = fia for e+, t*_r; and fir = fin = p, for e l , f + r ) . Proof. For simplicity, let s = n = r and sign = + . Any other combinations of the sign ± , boundary indices r, s, and the sign of I m a are considered in the same way. We follow the proof of Proposition2.1 and Lemma2.1. First of all, the absolute value of the exponential factor e x p ( i a £ ^ _ 0 ( a i m — <*i„)(xm — xm+i))

in formula (2.6) for j = ip+i = n does not exceed 1 for I m a ^ 0

(recall that a* ^ a n , i m + i ^ xm for the index sign + ) . Consequently, estimate (2.4) is valid and the n-th column of the series J + Y.%\ K±* ( s e e ( 2 - 3 )) converges

154

II. BACKLUND TRANSFORMS & INVERSE PROBLEM

uniformly for x and a to a continuation of e!f. in the upper half-plane I m a > 0. Moreover, this vector function is represented in the form eiaxf(x)dx

+ ^ J—oo

for I m a ^ 0 (see the proof of Lemma2.1) for a suitable vector function / which is absolutely integrable. Here, f(x) — Oif x < 0, since X)m=o( a 'm —a «r.)( a: m —x "»+i) ^ 0 for an arbitrary set { i x , . . . , ip}. If Re a = a, Im a ^ 0, then 00

/

_

_

eiaxe-ilmaxf(x)dx.

Thus elf. is analytic for I m a > 0 and e+ —• e^? n as |a| —>oo, I m a ^ 0 (statements a) and b) for el}.). Now let us use estimate (2.4a) for I m a ^ 0 and r = n and an analogous estimate for x < 0 taking the exponential factor into account: (2.4b)

\k%+1)\]

^ e x p ( - I m a(a„ +

QK

' °,

a j

) f ) ^ ^

(lma^0,x<0)

p\

Its derivation does not differ at all from those of (2.4), (2.4a) for a € R. We find that elf. converges uniformly to 1" as i —• +<x> and that for h > 0 and I m a ^ h, the function e j • converges uniformly to zero if y,j ^ /i„ (•$> aj ^ o „ ) . This proves statement c) for elf.. Analogous estimates also hold for the entry £"„ of the matrix T - 1 , but we do not need this. By the definition of T for a € R the entry e+ n converges uniformly to t " n when x -> — oo. This is also true for a = oo (Lemma2.1). As the function e!J.n is analytic on the upper half-plane and continuous (including at the point a = oo), the limit of elf„ exists as x —> —oo and for Im a ^ 0 by a standard theorem of complex analysis. Moreover, this limit is the analytic continuation of tZ„, for which statement b) follows from the property of e!f proven above. • L e m m a 2 . 3 . For boundary indices s, the vector functions e"+ and e l have no zeros (componentwise) and are linearly independent for arbitrary x € R and a in the upper half-plane, I m a ^ 0 including the point a — oo. Proof. Let us suppose that for a certain sign, e±(x; ao) = 0 = ea±{xo\ ao) where a 0 ^ 0, xQ € R. The vector function e(x) = e3.(x;a 0 ) defined for all x € R

§2. INTRODUCTION TO THE SCATTERING THEORY

155

satisfies equation (2.2) for a = OQ. Hence by the uniqueness theorem for differential equations, e(xo) = 0 implies e(x) = 0. Therefore CIQ is a zero of e± and es± for all a; (not only for x = XQ). But this is impossible since e±(x; a) -¥ 1* as x -¥ ±oo and for any a by Lemma2.2. The Unear independence of {e+} or { e l } is established analogously. By Lemma 2.1, e± are linearly independent also at a = oo. D 2.2. A n a l y t i c c o n t i n u a t i o n s . For ease of notation we assume that Q' = Q, in particular Ej° = I (see §2.1). We keep all previous assumptions and notation. For a G R construct from T(a) = (t*) the principal "upper" minors to = 1, ti

=t1,

t2 = t\t\ - t\t\,

... t„ = detT.

Here and futher, abusing the standard terminology, by minors we mean the determinants of the minors. Let us show that t„ = 1. Really, it follows from differential equation (2.5) that det.E = c d e t i i o for a suitable c € C, where E is a certain solution of (2.2). Here we use the equation SpQ = 0 which is derived from the condition Q' = Q. Consequently, deti?_ = det Ho by the normalization of E-. Hence, d e t T = det(.R0~1.E_(a: = +oo)) = 1. Moreover, Yl7=i *i( a ) 7^ 0 for the remaining minors which results from Lemma2.1: ti(a) —y 1 as \a\ -» oo. Define the principal "lower" minors of T, changing the order of rows and columns: tn

=

•!•> tn—\

=

*ni

tn-2 = O S ! } ~ t^K-l,

... To = det T = 1

Similarly, t-i =tl1,

... t-„ = d e t T _ 1 = 1,

?_„ = t0 = 1, r _ „ + 1 = tln,

... To = det T - 1 = 1

for T'1 = {tli). The relations among minors are given by the following formulae from linear algebra: (2-7)

T_m det T = tm,

tm det T'1

= *_ m ,

where 0 ^ m ^ n and the determinant of T may be any arbitrary invertible element (not necessarily the unity). It is probably helpful to give a sketch of the proof of these beautiful formulae.

156

II. BACKLUND TRANSFORMS & INVERSE PROBLEM

First note that (detT)V„

=

(T®n)Vn,

where T®" = T ® • • • ® T is the n-th tensor power of T and Vn is the antisymmetrizer (the canonical projection of (C")®" onto the exterior power / \ n C ) . Therefore T-1 det T = S P n _ 1 ( ( I ®

T^'-^Vn),

where S p n _ x is the trace over the last n — 1 matrix indices of the tensor products. Here we apply the relation Sp n _i Vn = 1. Analogously, using the identity SPn-m ^n = T>m-I we can check the formula ( T - i ) ® m detTVm

=

Sp„_ m ((7® m ®

T®(n-m))V„),

where S p n _ m is taken over the last (n — m) matrix indices. By calculating the trace of the last formula (for all m matrix indices), we get relation (2.7). We call the decomposition T = T°To the triangular factorization of T, where T° is a lower triangular matrix and To is an upper triangular matrix. In order for this decomposition to be unique, we need to fix the diagonal of T° or To. Let diagT 0 = diag( since the minors t\,... , tn of the matrix T must equal the products of the corresponding minors of the matrices T° and To. This factorization might not be defined for all a € R, since in general, the entries of T° and To have denominators that are products of the minors ti,... , tn-i (as we shall see later, the entries of T° are polynomials in tJt). In a neighbourhood of the point a = oo, such a decomposition is well-defined (see above). Analogously, set T = ToT° , where To and T° are upper and lower triangular respectively, diagTo = diag(f 0 ,*i,.-. Jn-i) and d i a g f 0 = d i a g ^ " 1 , . . . J'l^T'1). Such a decomposition, called the conjugate triangular factorization, exists in a neighbourhood of a = oo. Let us construct two new solutions of (2.2) for a € R from E±: $ = E+T° =

E-(T0)-\

* = E+T° = E_(fo) -1 . Set $ = QR^1, * = $.Ro~\ $k = Vfc exp(—ajifca:), and ipk = ipk e x p ( - a ^ * : r ) , where 0.

§2. INTRODUCTION TO THE SCATTERING THEORY

157

Minors tp and t-p of ty are continuous functions of a € R and admit analytic continuations to the lower half-plane Im a < 0. b) If we denote the analytic continuations by the same letters, then for 1 ^ p ^ n, as \a\ —• oo tp->l, tp -»• 1,

$-»I,

(Ima^O),

tf->-I,

(Ima^O).

Proof. By construction, ipn = e " , ip1 — e l , i/)1 = e\, ijin = e " and t\ = fj,
(2.8)

f\ A{zx A . . . A zp) = Azx A Az2 A . . . A 4 z p .

We number the coordinates of /\p C" in the natural lexicographic order by the sets 1 ^ ii < i2 < ... < ip ^ n. Then at the crossing of row (»'i,... , i p ) and column Uii •• • iip) of matrix /\p A stands the corresponding minor which is equal to the determinant of the matrix (a[") = (aj m ) of order p x p (we remind that minors in this book mean the determinants of the minors). If (ij,... ,ip) = (ji,... ,jp), then the corresponding diagonal element of XPA is the trace of the matrix (a{") = (aj™). It is easy to check that XpUo is again diagonal. Its (I'I, . . . , i p )-th entry is equal to p^ H Mi,,. There is another obvious relation: /\p Ro = exp(ax(XpUo)). If the first coefficient Hi H (- \xp of matrix XPUQ is equal to the coefficient mx -\ 1- /*,-, then i'i = 1, »2 = 2, . . . , and i, = s for s = max{jf, fj,j ^ p.p,j < p} and p s + i = . . . nP = Hi,+1 — ... = Hip. We assume the condition Q' = Q in this section. This implies that the entry of XPQ at the crossing of the first column with the index ( 1 , 2 , . . . ,p) and row ( i ' i , . . . ,ip) for which p ^ + h m = fi\ + 1- \ip

158

II. BACKLUND TRANSFORMS & INVERSE PROBLEM

is equal to zero. An analogous statement holds for the first row. We can check by the same method that for the ( i i , . . . ,ip)-th element of XpUo which coincides with its last element fin + /j,n-i + • • • + (*n-P+i, the entries of XPQ with indices {(n - p + 1,... , n ) , ( » i , . . . ,ip)) and ((t'i,... ,ip),(n-p + 1 , . . . ,n)) are zero. Differentiating /\p E for an arbitrary solution E of equation (2.2), we obtain the relation

(f\ E)x + {XpQ)(/\ E) = a(A'CM(A E )Indeed, for any invertible matrix function A(x), differentiating the second formula of (2.8), we get the equation

The equation for /\p E has the same structure as (2.2). For p = 1 we have equation (2.2) and for p = n we get (2.5). The entries of XPQ are sums of entries of Q and, consequently, are absolutely integrable from x = — co to x = +oo. Note that /\P(E±) = (/\"E)±, i.e., tf(E±) -» A P Ao as x -+ ±oo, and ft" E+ = (ft" E-)(ftp T). This follows from the multiphcativity of /\p. Thus we have seen that we can apply Lemma2.2 to XPQ, XPUQ, /\P E± and / \ p T instead of Q, Uo, E±, and T. The above vanishing conditions for the entries of /\p Q is, of course, weaker than the relation (/\ p Q)' = /\p Q. But even this statement is enough to show that the limits of /\p e± and (/\p T)5- r in statement b) of the lemma are equal to unity for s = 1 = r or s = n = r. This is obvious from the proof of Lemma2.2. We want to use Lemma2.2 exactly for such indices. Set s*.=e*+1A...Ae£, ^=et+1A...Ae!l. Let -elf. = e°_ = 1. Then for 0 5: p ^ n the vector ructions etj. and the function tp (the first coefficient of the matrix /\p T) are analytic for Im a > 0. Moreover, «£ -* lp+1 A . . . A 1",

et ^•l1A...Alp,

t,-H

as \a\ -> oo, I m o ^ 0. In particular, we obtain the proof of the statements concerning the tp. Note the formula (2.9)

tp = ep_t\ep+,

l^P^n.

Now we turn to $ . For a £R,1 < p
s T 1 A
p

tP-iet, 1

§2. INTRODUCTION TO THE SCATTERING THEORY

159

which follow from the definition of $ . We show that for each x and for 1 < p < n, the function
for a large enough \a\ and x = x(a). But then e*T Ae+ = 0 (z can be expressed as a linear combination of either e[_,... , eL _ , or e+ > • • • i £+)• Consequently, for an appropriate sequence \aj\ -» oo, Xj = x(a.j), the vectors e"L, • • • , &~ , e+ , • • • , e+ are linearly dependent and tP = ffL A ?+ = 0. As the minor tp does not depend on x, we see that tp —>• 0 when | a | —»• 0. This is a contradiction since we know that tp —> 1 when | a | —¥ 0. Now let us replace t and e± in (2.10) by their analytic continuations in the upper half-plane. We will show that this inhomogeneous system of linear equations on 0. We will now prove the continuity of (pv at a € R and analyticity on the upper half-plane. L e m m a 2 . 4 . Let us suppose that for certain analytic vector functions e±(ct) € / \ P C" (1 ^ p ^ n) of parameter a 6 C, system (2.10) with tp from (2.9) is uniquely solvable in a punctured neighbourhood of a point oro and ^ . ( o o ) ^ 0 for all p (e°± ^ 1). Then a) there exist vector functions which are holomorphic at c*o, / ± ( a ) , • • • , / ± ( « ) such that

SL=/iA...A/£, p

e +-1=fpA...Af^,

160

II. BACKLUND TRANSFORMS & INVERSE PROBLEM

(l^p^n); b) the solution
tp-iet = / i A / £ A . - . A / r ' A y " (see (2.10a)), where
(2.11)

ffp

= £ > / ! +(a-a0)mg, «=i

where g(a) and functions ct(a) for 1 ^ i ^ p — 1 are holomorphic at ao- Really, the exterior product / i A . . . A flT A gp = e^T A g1" has a zero of order m at ao (equation (2.10a)), and / I , . . . ,/f. - 1 are linearly independent at ao, since / i A . . . A/£~ = e^T is not zero at that point. Therefore g(ao) is linearly expressed in terms of / I ( a o ) :

(2.11')

g"(a) = £ > ; / ! ( « ) + (a - a 0 ) g ' W ,

where c\ e C, <j' is holomorphic in a neighbourhood of ao. The order of the zero of / I A . . . A flT A g' at ao is m — 1. Analogously we can represent g' in the form (2.11'), constructing g" and so on. Finally we come to g^m^ for which /lA...A/r

1

A^(a

o

)^0.

If we set g = g^m\ then relation (2.11) is fulfilled for suitable functions Ci(a). Hence we can take g as / £ . This proves statement a). Keeping the same notation, we use (2.11) for g^ = (a — ao)m(fipt~i1, where m p is the order of pole of £ t~i.j at ao (i.e., the maximum order of the pole of the

§2. INTRODUCTION TO THE SCATTERING THEORY

161

components of
Multiply by / £ A / I A . . . A / 1 ° _ 1 A / 1 0 + 1 A . . . A fl~l on both sides of this equation. Then the right hand side is identically zero as a varies ( / ' A e+~ = 0 by statement a)), and the left hand side can be rewritten as

±cioF_-1A?p1+(Q-ao)V, where g' is holomorphic at ao. Thus t p _ i = e^T A e+~ has a zero of order not less than m at ao (see (2.9)). Therefore (pv is holomorphic at ao. • L e m m a 2 . 5 . For an arbitrary invertibJe matrix T = (if), the entries of the matrices T°, To1, To a n d ( T ° ) - 1 constructed above and of the matrices VT0, X ^ T 0 ) - 1 , VT° and VT0~l are polynomials in t\, where T> = d i a g ( i 0 f i , " . ,U-iU,...

,t„_it„),

V = diag(t 0 *i, • • • ,*i—i*i, - - • ,*n-i*n)Proof. Set E+ = I and E- = T in Lemma2.4 and define e?± from E± by the same formulae as above. Then $ = T° is a solution of system (2.10). Let a € C, T = (t*), where t{ = t\ + a fij, where the fij are linear combinations of entries of T. We can always take fij such that the entries of T° are analytic in a certain punctured neighbourhood of a = 0. Applying Lemma2.4 to $ = T ° , we obtain that all entries of T° are analytically continued to a = 0 for suitable {fij}- Thus entries of T° (which are a priori rational functions of t\) are regular for all concrete T and must be polynomials of t\. Indeed, singularities of T° (if any) form a hypersurface V (of codimension 1) in the space of all invertible T. Therefore there exists a matrix T € T> and an entry of T° with a pole at a = 0 for any choice of {fij}, which contradicts the fact proved above. We can apply an analogous argument to TQ - 1 (set E- = J, E+ = T _ 1 ) . Transposing and inverting the decomposition T = T°TQ, we show that the entries of the other six matrices are polynomial. • Lemma2.4 for ^{x; I m a > 0.

a) above assures that
162

II. BACKLUND TRANSFORMS & INVERSE PROBLEM

The continuity of tpp for I m a ^ O follows from Lemma2.5. To complete the proof of the theorem concerning $ , we have only to check that oo. The statements of the theorem for $ are proved in exactly the same way. We leave the proof of the analogues of formulae (2.9), (2.10) and Lemma2.4 to the reader as an exercise (see formula (2.13) below). We can also use the following method. Set II = (ir{), where TT{ = S"^ and A{a)u = f I L 4 ( - a ) I I . Then E% and Tu correspond to the matrices IIQII and — HUQII in the sense of §2.1. Moreover, the involution u> transforms the decomposition T = T°TQ into the conjugate triangular decomposition of Tu. Consequently, $ " (which is analytic on I m a < 0) is 9 constructed for IIQII and — TiUoIL and all statements for \t follow formally from the properties of $ . • Corollary2.1. a) The multiphcity of each zero ao of vector functions ipp (I ^ p 5i n), i.e., the minimum order of the zero among all the components of (fP at a 0 (im a 0 ^ 0), does not exceed the multiplicity of the zero of the minor t p _ i and t i e minor tp at that point. The zeros of (p1 A . . . A • tp*, -¥ tp and ipp -¥ 0 x -¥ —oo =>•
and ipp -¥ 0

p

I m a > 0,x -» ±oo => (p q -¥ 0

for q < p, for q > p, if nq ^ (ip,

where ip? is the q-th component of (pp. c) At points in the lower half-plane the multiplicities of the zeros ofipp (1 ^ p ^ n) do not exceed the multipHcities of the zeros oftp-i and tp (cf. a)). The zeros of ifr1 A . . . A il>p (with multiphcities) coincide with the zeros oftoh • • • tp-i and the zeros ofr/>p A . . . t/)n coincide with the zeros of tptp+i ---tn- For Im a ^ 0 we have: x -¥ +oo =*• ij)p -t tp-i p

and ipp -¥ 0 p

x -+ - o o =>• j> -> tp and ip -> 0 p

Im a < 0, x ->• ±oo =4- i/? -> 0

for q> p, for q < p, if fiq ^ nP.

§2. INTRODUCTION TO THE SCATTERING THEORY

163

Proof. By Lemma2.3 the vector functions e^. do not vanish for Im a ^ 0. Thus the zeros of (p9 are contained (with multiplicities) in the set of zeros of tp-\ and tp (see formula (2.10)). Applying (2.10) step by step, we obtain the relations ^ A . . . A ^ = (
=

(tp...tn)£p+-1,

which prove statement a). Prom the definition of e± (and Lemma2.2) it follows that for I m a ^ O , ?+^lp+1A...Mn, 2

£*_ - > 1 A . . . A 1 " ,

(*-•+«>) (a:-4--oo).

The first component of lim I _ ) .+ 00 e^. (the coefficient of l 1 A . . . A lp) is equal to tp, while the last component of limx-f-oo PL (the coefficient of l p + 1 A . . . A 1") is equal to t-p = tp. For I m a > 0 and [iq ^ fip, q > p the coefficient of lq A (•) in the expansion of lim I - + + 0 0 e?_ is equal to zero, where (•) is an exterior product of certain basis vectors of the form V (j ^ q). Correspondingly, for q < p, the coefficients of 1 ? A (•) of lim^-^-oo e^. are zero (see Lemma2.2) if I m a > 0, fj.q ^ nP. Substituting the limits limx.+ioo e^. into relation (2.10) we get the asymptotics of 0 is analogous. Let us assume, for example, that Im a > 0, y.q j= nP and q> p. Then epT1(x

= oo) A
= co).

Vectors of the form lq A (•) (see above) do not appear in the expansion of e^T (x = oo) ^ 0 with respect to the standard basis vectors of /\p~ C . Consequently, if • v as x —¥ ±oo are calculated analogously. We can also use the involution w from the proof of Theorem2.1. D

164

II. BACKLUND TRANSFORMS & INVERSE PROBLEM

Exercise2.1. Using Lemma2.2, compute the limit of(pp for Im a > 0 and ofij)v for I m o < 0 as x —> ±oo. (Corollary2.1 allows us to do this computation in the case when the eigenvalues pp have multiplicity 1). • Exercise2.2. a) Using estimate (2.4), show that the minors ti,... , * n - i and ti,... , t„-i do not have zeros for Im a ^ 0 a n d l m a ^ 0 respectively, ifexp(ncQ) < 2 (cf. [ZMaNP], Ch.I §10). b) Check that for compactly supported Q (zero outside of a finite interval), the functions E± are analytic for all a € C except at a = oo. D 2.3. Variants. When some of the eigenvalues {/i,} coincide, the entries of T behave better analytically than in the case of pairwise distinct {/x,}. For instance, if p i = /i2, then not only t\ but also the entries t\, t\ and t\ are analytically continued to the upper half-plane (Lemma2.2). Theorem2.1 only guarantees that t\ and ti = t\t\ — t\t\ are analytic. It is exactly the same for E± — more entries admit analytic continuation to the upper and lower half-plane. This means that the procedure of constructing analytic solutions of equation (2.2) from E± can be refined and, in fact, simplified in the case when {/i,} have multiplicities. Our first goal in this section is to modify Theorem2.1 in the case when some of the fij coincide. The reader can skip this section ignoring all dots in the indices when reading the next sections. First of all, we modify the decomposition T = T°To (a variant of the Bruhat decomposition with respect to the Borel subgroup of lower triangular matrices). Keeping the notation and conventions of the previous section, set i* = m a x { j | pj = m },

t'» = min{j| y.t; = m } - 1

for 1 ^ i ^ n; the number of distinct values which i* and i» might take is the same as the number of pairwise distinct Hi among {/*>}• If m / Hj for all j ' ^ i, then i* = i, it — i — 1. Let us introduce the factorization T = T*°T£, where Tm0 d = (r? J ) and T0* = f ( r ^ ) and

* > 3 =*• r oi = 0, Pi = Hj,i^

j =^r^ = 0 .

This decomposition is uniquely determined if we fix the diagonal of T£. Let

diag T0* = diag(l, t£,...

, tr1,... , t~]),

§2. INTRODUCTION TO THE SCATTERING THEORY

165

where the ti are the "upper" principal minors (see §2.2). Then the entry r°j for which (Mi = /ij (•$=> it = j») is equal to the determinant of the matrix of order i* + 1 consisting of the entries tJ" of T for I = 1,2,... ,i*,i and m = 1,2,... , t», j . In particular, ri1 =t}i for £ij = //i = /i,-. Similarly, we can express T4 •* (for yn — fij) in terms of the matrices A T. Note that for p = p*, the minor det(r; },i* =j*—p) of the matrix T'° is < , , # " ' * _ 1 ) and V + t * = *p+iIn an analogous way we define a decomposition T — TQT*°, where zero entries of TQ and T°° are arranged as the zeros in the transposed matrices T"° and *TQ respectively. We fix diag T # 0 = diag(f- 1 , q.1,...

, t-.1,... , 1),

where ti are "lower" minors of T (see §2.2). The minor d e ^ r ^ , i* = j * = p*) of the matrix T£ is equal to tpjy1, ~p'~ '. The minors {i,-., £;< } cannot be zero for a € R with big enough absolute values. This ensures the existence of the above decomposition "almost everywhere" for a G R. T h e o r e m 2 . 2 . a) Functions $* d= E+T'° = E^Tf)-1 and tf* = E+fj = E-{T*°)~l are continuous functions of a € R and can be analytically continued for a in the upper and the lower half-plane, respectively. This is also true with respect to the functions rt} and r ^ for m = p.j, where T*° = (T{ }) and T£ = (?&). b) If we denote the analytic continuations by the same letters and set $* =

S ' J ^ 1 and $ • = VR^1,

then T? -> s>;,

$• -+ / ,

? j "• Sj,

$ • -+ I

as \ot\ -> oo, I m o ^ 0 or I m a ^ 0, /i,- = p.j. c) For p — p*, p < pi < ... < pk g (p + 1)* = p', set $•{*)

= £ • ! A . . . A £** A £* P l A . . . A £•"",

^ " > = £ » n A . . . A £ • » A $*+1 and analogously define e_''

A ... A £*n,

and £ + ' via e_ and e + respectively. Then,

£•_<"> = d _ e l " \

166

II. BACKLUND TRANSFORMS & INVERSE PROBLEM

where

d- = (tp)kf[ti., «=1

4"} = det(r° «; 1 ^ i, j ^ k)tj

f[ U,, i=p'+l

{qi} runs over the sets p < qi < ... < qt ^ p'. The zeros of 0, lim $ ' = d i a g ( l , . . . ,*,-.,... ,*„.), x—•—oo

Urn $' = [T"°]. Analogously for Im a < 0, Urn $ * = diag(f 1 .,.. . , £ . , . . . ,1), x—y—oo

hm $ ' = [27]. X-++OO

The same relations also hold for [$*] and [\£#] instead of$'

and \P* when Im a ^ 0.

Proof. The only difference in the proofs here from those of Theorem2.1 and Corollary2.1 is the extended use of Lemma2.2. Lemma2.4 and Lemma2.5 are transferred to the case of analytic functions with dots without difficulty. Set P.+I

elp(x;a) = e L A . . . A e f A ? . £

/\

C\

n-p.+l

e7-1(x;a) = e^.Ae^+1A...Ae!le

f\

C,

keeping the notation ejj. of the previous section. Then for a £ R the following generalizations of (2.10) hold

Mp'p=tPXI,

(2.10a')

e?

(2.10b')

£*'A«£=

P*

£) »=n-p*+l

T

f«+_1»

§2. INTRODUCTION TO THE SCATTERING THEORY

167

where for 1 ^ p ^ n, m = ftj) are analytically continued to the upper half-plane. Hence 0 (respectively, r = 0). Let V± = exp(u± © Uo), U± = exp(u±). Represent T in the form T = r , 0 r 0 * = T Q ' T , 0 for "almost all" a € R and suitable T*°(a) 6 V-, where TS{a) € U+, f0'(a) € V+, f'°(a) € U— Then the functions $ ' and $ * defined by the formulae of Theorem2.2 are meromorphically continued to the upper and lower half-plane of a respectively. IfQ(x) 6 u+ © U-, then $* —>• / and \t* —• J as \a\ -+ oo ( I m a ^ 0 or I m a ^ Oj. • Unitarity conditions. Let ft be a diagonal matrix consisting of ± 1 . Set A* = ft.A+ft, where A+ = %A) is the hermitian conjugate of the matrix A. In the most important applications, Q* + Q = 0 for a suitable matrix ft. We assume here that this condition, called the *-anti-hermitian condition, is fulfilled. All the previous conventions and notation remain valid. Then (E±(a)y

= E±(a),

( T ( a ) ) ' = T " 1 (a)

for a € R, i.e., E±, T axe *-unitary matrix functions. This is a direct consequence of the identity R£ = R^1 and the definition of E±. Using (2.7), the following relations are easily obtained:

% = ((T0)')-1©*,

T° = (^^((To)*)- 1 ,

168

II. BACKLUND TRANSFORMS & INVERSE PROBLEM def

where V = diag(t 0 *i,M2,- • • ,*n-i*n), a € R. If a G C, then we set A*(a) = (.A(a))* for any matrix function A(a). In this notation * ( a ) = ($*(a)) - 1 2>*(a) for Im(a) ^ 0. Indeed, the last equation is true for a € R and by the Maximum Principle, is also true for all a ( I m a ^ 0). An analogous relation for $* also holds as well (see Theorem2.2): ¥•(<*) = ( $ " ( a ) ) ~ 1 I > " ( a ) , (2.12)

def

V

= d i a g ( * i . , . . . ,t,-.
The c a s e of an arbitrary matrix UQ. We will briefly sketch the necessary changes in the construction of analytic solutions of (2.2) for matrices Uo (diagonal) which are not purely imaginary. Though the approach is almost the same, the construction becomes complicated. First of all, we no longer have the concept of a (total) monodromy matrix. The entries of Q{x) are assumed to be absolutely integrable for x from —oo to co as before. For the sake of simplicity, let Q' = Q. We associate to Uo = d i a g ( ^ i , . . . ,fin), Hj € C two sectors M± = M±{Uo) C C: M± = ±{a € C| k < I => Re(nka)

^

Re(tna)}.

For {HJ} C JR with the standard ordering (see above), M ± are the upper and lower half-planes. Sectors M± may consist of only zero or of half-lines of the form R ± a , where R± = { ± r 2 , r € R } , a € C. Let us assume that the interiors M± of sectors M± are not void. Then MJ. are (connected) simply connected domains (intersections of half-planes). For a £ M+, we define vector functions e l ( : c ; a ) , e " ( x ; a ) as the first and last column of series (2.3) respectively. Estimate (2.4) for |fc+ p+1) |" and |fci p+1) |j remains valid without change. Thus these series converge. Moreover, estimates (2.4a) and (2.4b) also hold. In an analogous way we can construct vector functions e ^ (a € M+) for fjt, = y.n or fx, = fix respectively, and eL (a € M _ ) for fj., = //i or l*s = A^n respectively. With all these we can rewrite Lemma2.2 and 2.3 for vector functions e± defined by series (2.3) and boundary coefficients T defined by series (2.5) for a € M+ (instead of the upper half-plane) and a € M _ (instead of the lower half-plane) respectively. Note that in §2.1 the series which define E± and T were meaningful for all a € R. This allowed us to pose a question about analytic continuations of the entries of these matrices (or a linear combination of them). If such continuations are found, then they are uniquely determined. Now the situation is different. We need to define the entries of analytic solutions of (2.2) constructively. We construct the vector functions e ± ( i ; « ) € /\p C from the boundary columns of formulae (2.3) for \PQ and XpUo instead of Q and Uo- These functions are

§2. INTRODUCTION T O THE SCATTERING THEORY

169

continuous for a € M+ and analytic for a € M+. As in the proof of Theorem2.1, e+ is constructed from the last column of the UE+ -formula" and ev_ from the first column of the UE--formula", 1 ^ p ^ n. Analogously we can introduce tp(ct) (the first coeflicient of the /\ p -analogue of the formula for T). In particular, e i = e " , eL = e l , and
and so on). Analogously for a € Af_ (starting with 7+ = e i ,

7 ! = e " and t„-i = tJJ), let us construct 7±(x; a) G / \ P C , tp(a) and 7 J , where 1 ^ p ^ n. Note that tn = 1 = f0 and e£ = e!l = 1 = 7"! = 7!!. We also set i 0 = 1 = tn and e^. = 1 = 7±P r o p o s i t i o n 2 . 2 . For any x <E R and for each 1 ^ p ^ n, there exists a unique soiution £ p ( x ; a ) € C" of system (2.10) which is continuous on M+ and analytic on M+. Analogously there exists a unique analytic (continuous) solution if>° of the system of equations

for a G M I (a € M-). We construct the matrices $ and "P from { £ p } and {i/>p} as the columns and set $ = $iio and * = $iZo, #0 = exp(aUox). Then $ —• I and $ —>• J when | a | —¥ 00 (a € M±), and $ and $ satisfy equation (2.2). Proof. The solvability of (2.10) (or (2.13)) is an equation of the form TTP = 0 (or 7TP = 0 ) , where np (or 5rp) is a polynomial of the components of e^., ejjT

} depend analytically on Hi,... , / i n in the domains M\ = {Re/** > Re^tjt+i,A: = 1 , . . . ,n — 1} and M°_ = — M\ respectively and continuously in their closures M.±. Now let us prove, for example, that system (2.10) is solvable for p = 2 for (/*i,... ,nn) € M+. General p and (2.13) are completely analogous. Set T = M+ fl {Re^i = Re^2}- For (m) € T, more terms in series (2.3) and in the series for T converge than in all the domain M+. Actually we can handle those series after restricting to T as we did in the case m — fj.2 (see §2.1, Theorem2.2). This allows us to apply Lemma2.2 to its full extent. In addition to eL = eL and eL we can define a continuous C"-valued vector functions eL of (fij) € T, such that eL A eL = eL. Analogously we define a continuous function t\ of (fij) 6 T. Then
170

II. BACKLUND TRANSFORMS & INVERSE PROBLEM

Thus we have proven the solvability of system (2.10) for p = 2 at (/*,) G T. Hence, iti\j: = 0. The real codimension of T in M.+ is one. Hence iti — 0 on M+ (n-2 is analytic). Similarly, TTP = 0 and np = 0 for all p on A4+ and Af _ respectively. Therefore systems (2.10) and (2.13) are solvable for any p, x G R and for arbitrary (Hj) € M+. Of course, this also holds for functions of a € M± at fixed /Ui,... , /UnMoreover, using the solvability of (2.10), (2.13) proven above, it is easy to show the uniqueness for these systems in a neighbourhood of the point a = oo (i.e., for big enough |a|, a G M±), to derive relation (2.2), and to find the asymptotics of $ and # when \a\ —> oo. Finally Lemma2.4 implies that $ and $ have unique analytic continuations from a neighbourhood of a = oo to corresponding sectors a G M+,M2. which are continuous for all a G M ± . • It remains to cover the complex plane C by sectors, where the above construction can be applied. For an arbitrary permutation a of n elements we denote the corresponding matrix by Pa:

i^hW

/ 2,7(1) \

:

,

Pa(a\)P^=(a^).

W(r>)/ Set Ug = PcUoP-1 and Af| = M±(U£). We call permutation a and sector M± admissible, if (M±)° ^ 0. Admissible sectors {A/+} cover the whole complex plane and any two of them can either coincide, or intersect in a half-line, or intersect at the origin (and analogously for {M"}). When pi,... ,fj.„ are in a generic position (i.e., lines /u7R are pairwise distinct), the number of admissible sectors is maximal and equal to n(n — 1). In each admissible sector M " we construct a solution of equation (2.2) for US and Q" = f P^QP'1 by Propostion2.2. Set $<"> = p - 1 * ^ . Then $(") is a solution of (2.2) for Uo and Q which is analytic for a G M" and continuous up to the boundary. When some of {fij} coincide, there is one deficiency in the above construction. It is easy to see that

for admissible a and a'. Therefore, generally speaking, one admissible sector M+ might correspond to several different permutations a' and functions $ ( " ) (their number is equal to Ili'C 1 * — **)'» * n e o r < i e r °f * n e centralizer of UQ in the permutation group). The best way to define $
§2. INTRODUCTION TO THE SCATTERING THEORY

171

Exercise2.4. a) Following the method of Proposition2.2, construct analogues of$" and <J>* of Theorem2.2 in sectors M±(U0). (Rewrite system (2.13) for * ' and then generalize it and (2.10*) to the case of an arbitrary matrix J70). b) Show that the function $•(") defined in the same manner as $('> depends only on its admissible sector M+, i.e.,

In particular, ** = $•(''*') for w0(l,... proof of Theorem2.1). •

,n) = (n,n - 1 , . . . ,1) (cf. the end of the

2.4. Riemann-Hilbert problem. We begin with a "spectral" interpretation of the degenerations of the functions $ " and \f* of Theoreml.1,2, i.e., points a in the upper or the lower half-plane at which these functions are not invertible. We keep all assumptions and notation of those theorems. We can naturally regard the zeros of det $* = IJiLi *>. an< ^ det $ * = Yli=i *<• = III=i *»• a s P°J n * s °f * n e discrete spectrum of equation (2.2) in the upper and the lower half-plane respectively. Let us recall the usual definition of a point of the discrete spectrum ao. For such points, equation (2.2) (or any other similar differential equation with parameter) has vector solution ^ which is normally integrable from —oo to +oo. The concept of normal integrability depends on concrete problems. Note that { t , } U n = 0 U {?'*}, P r o p o s i t i o n 2 . 3 . a) For each zero ao & R of functions ]\{ r,-, or fj^ ti., we can hnd a vector solution ip(x;ao) = X'Pk) ^ 0 of equation (2.2) and two numbers a, 6 € R, 6 > 0 for which \e"• oo,

1 ^ h ^ n.

b) ^ n " = i *>.( a o) ^ 0 ^ Iir=i *>.( a o) for I m a 0 > 0 or, respectively, I m a o < 0, then for any a € R and an arbitrary solution • +oo or \x\ —• —oo. Proof. Let r p .(ao) = 0, I m a o > 0 for a certain p. We use the notation of Theorem2.2. In a neighbourhood of ao, set §q = (a — ao)**
172

If tp,(ao)

II.

BACKLUND TRANSFORMS & INVERSE PROBLEM

= 0, then (see equation (2.10a*)) (a - ao)-*-«n* A = ( a - a0)-k<tpX-

= &- A 3 * ± 0

for all q with the condition q* = p* (<=> p* < q ^ p*). Thus tp, has a zero of order kq at ao since e*_* does not vanish by Lemma2.3. Moreover, all kq coincide with each other (and with fcp»). Using (2.10a*) directly or statement c) of Theorem2.2, we get that

( A £•*)A*+ = v # ' ~ p - - 1 ) £ + >

*+ («•>)*o.

The order of the zero of the left hand side at point ao is greater than or equal to (p* — p»)fcp.. Therefore tp,(ao) = 0 =$• tp*(a0) = 0. Applying this argument to tp(instead of tp,) and so on, we finally get a contradiction since t„ = 1. Let p be the minimal index for which eL* Agp(ao) = 0 (p > 1, since e_" A g1 = eL ^ 0). It is easy to check by using induction and (2.10a*) that fl1A...Ajp*(a»>)

= ce!*(ao)

for c ^ 0. We can also use formulae c) of Theorem2.2. As a result, g1,... ,gp* p are linearly independent at ao and g (ao) is expressed linearly in terms of them. Finally we obtain the identity: p«

gp(a0) =

Y,Ci9"(a0), ?=i

where c,- G C and g* = exp(afiix)g'. The coefficients c; do not depend on x, since gp, {g9} are solutions of equation (2.2). Let ap > a' > ap, (recall that /x, — ia,, q > r => aq > ar), 0 < b' < ap — a' and b' < a' — ap< for a', 6' € R. Set a = Imaoa', b = Imao&' and y> = gp. Letting |x| —¥ oo, we obtain the estimates: \
\
Im

"»

as x -> +oo, a s i - + -oo.

Here we use the definition of ip — gp and the relations obtained above (together with formulae d) of Theorem2.2). Thus statement a) is proven for J\ ti,. For f [ t,-, the proof is the same.

§2. INTRODUCTION TO THE SCATTERING THEORY

173

Now let us suppose that J\ ti. does not have a zero in the upper half-plane. Then at any point a = cto with Imao > 0, the matrix $* is invertible and (p'1,... ,
v(«o) = ] T Ciip'^ao),

a € C

i=i

by the uniqueness theorem of differential equations. Let IQ = min{i,Cj ^ 0}, i° = max{i,Cj ^ 0}. Finally we obtain the following estimates with the help of Theorem2.2 (cf. above): |v» t l (ao)| > | C i | e - | x | a i ° I n , a ° |VP i 3 (ao)|>|C 2 | e l*l°.-°

I,na

°

as x - • +oo, asx^-oo,

where C\ ^ 0 ^ Ci, for certain constants fci.fo. Analogous estimates for SP* hold when I m a o < 0. Applying the inequality i° ^ i 0 , we get the proof of statement

b).

•

Now let $ be an arbitrary analytic solution of equation (2.2) in a punctured neighbourhood of the point a = ao for — co < x < +oo. We assume that det $ ( a ) ^ 0 for a ^ ao and the entries of 4 ( a ) are meromorphic at ao. For scalar (C-valued) analytic functions of a, the most important invariant of zeros and poles is their order. For matrix functions, the degenerate points (the poles of the entries of $ or $ - 1 ) are described by a set of vector spaces which we now construct for $ . Actually we have already encountered an analogous problem in §1 (see Proposition!..2). Let us recall the notation we use there. Let Oo = Oao be the ring of formal power series of ( a — ao) whose coefficients depend on x, and let Ofi = O ®c C" be the space of vector series. Set Co = {Ei|fc c,(x)(a - a 0 ) S k € Z } and 0% = O0 c C" (then O0 is the field of fractions of the ring Co)- We call an (9o-module AC C Ofi a lattice when it generates GQ as an Oo-module (i.e., O0fC = AC and O0K. = OS). It may happen that $ ( a o ) _ 1 cannot be defined. However we can always define $ ( a ) _ 1 as an element of GL„(OQ) (as an invertible matrix with entries in Oo). Set ACo = $ ( a ) - 1 0 o . Then ACo is a lattice since d e t $ ^ 0. We will show that ACo is generated as an C?o-module by vectors which are constant with respect to x. This independence from x can also be written in the form (ACo)x C ACo, which can be easily checked: (ACo)* = (Q- aUo^OS C AC0.

174

II. BACKLUND TRANSFORMS & INVERSE PROBLEM

With the aid of Co, we can formulate a trivial but useful criterion for $ to be analytic and invertible at ag: (2.14)

* e G £ B ( O 0 ) <» iCo d = * _ 1 O f f =
Obviously Co plays the role of the "vector multiplicity" of ceo. We will apply the construction $ t-t K.Q to functions $ = $ , $*, >J> and **. Let us recall that in the case of pairwise distinct {/*,-}, the functions $ and & coincide with $ # and * * (analogously for To, T°, T 0 and T°) and i* = £, z*. = i — 1. L e m m a 2 . 6 . Define (see Lemma2.5 and (2.12)) V = diag(t^ i _i),2> # = diag(t,-.*i.), l g i g n , and ana/ogous diagonal matrices T> and V for ti instead of ti. Then the functions X>$ - 1 ,1?*** - 1 , £ > § - 1 , a n d X ) * * * - 1 are analyticforIm a > 0, I m a < 0 respectively and continuous up to the boundary a £ R. When | a | —> oo, all these functions converge to I. Proof. These statements are easily derived from Theorem2.1, 2.2. We introduce an involution AT(a) — (*.A(—a)) -1 . Functions J5J and TT are the E±— and T-functions corresponding to the same matrix UQ and — *Q instead of Q; the decomposition TT(a) = ( T 0 r ( a ) I > ( - a ) ) ( D ( - a ) ~ 1 T 0 r ( a ) ) turns out to be the conjugate factorization of TT. Really, this decomposition has the necessary type. Moreover, the i-th diagonal element of T°T(a)T?(—a) is equal to t-^-a^ii-a^i^i-a)

= *<_,(-<*)

and, consequently, coincides with the ij-minor (lower (i— l)-th minor) of the matrix ^^(—a) or (*T(—a)) - 1 in view of formula (2.7) (the transposition does not change the principal minors). Therefore *(Z>$ -1 (—a)) becomes the ^-function for — *Q and has the desired analyticity and continuity by Theorem2.1. The statements of the lemma for the other functions can be checked similarly (see Lemma2.5). • In the notation of Lemma2.6, set $ _ = $ P _ 1 and $!_ = §•£>— l . For consistency of notation we write $ + and $ + instead of $ and $*. $ ± = $±RQ (2.15) (2.15')

and $ ± = *±-Kb> where Ro = exp(aUox). $:*$+ = RoSR^1, ^ l - 1 ^ ; = RoS'R^1,

Analogously,

When a € K we have

P _1 S =f f^T° = T°To\ V'^S'

=f JJ -1 !"* =

f'0^1.

175

§2. INTRODUCTION T O THE SCATTERING THEORY

According to Lemma2.5, 5 and S* are polynomials of the entries of T and, in particular, are continuous functions of a € R. Strictly speaking, this lemma can be applied only to 5 , but its formulation and proof hold in the case with *. By Lemma2.6 the functions &Z1 and $ 1 - 1 have analytic continuations in the lower half-plane. Let a J , . . . ,oc°N, be the pairwise distinct zeros of det $ + = Iliz=i ti. an<^ le* ot°N.+1,.-. ,<x*N. be the zeros of d e t $ l _ 1 = n " = i *«. "* * n e upper ( I m a > 0) and the lower ( I m a < 0) half-planes respectively. Furthermore for the sake of simplicity we assume that the minors ti,, ti, do not have zeros on the real axis a € R. Set (see (2.14))

q = *;_1(o")

(i^i^;),

where Oj = N± respectively. We also define the analogous lattices for $ ± (without *). All lattices K. introduced here do not depend on x. We call {atj,lCj} the discrete scattering data. E x e r c i s e 2 . 5 . Show that we can recover every zero of each minor ti, (1 ^ j- is N$) and the lattices [ T , 0 ] _ 1 O ^ (in the notation of Theorem2.2) from the set { a * , £ J } (1 ^ j ^ IV* j . Similar statements hold for { a ' . J C J . j > 1V+}, f and for a, K, without *. (Use statement c) of Theorem2.2, its analogue for $ , and Corollary2.1.) D T h e o r e m 2 . 3 . a) Let us suppose that a continuous function S(ot) € GLn, a € R, certain sets of pairwise distinct points {aj, 1 ^ j ^ N+,hnacj > 0}, {ay, N+ < j ^ N,Imctj < 0}, and a set of Oj-lattices {K.j} for Oj = Oaj are given. Also we suppose that there exist matrix-valued functions $ + ( x ; a ) and $Z {x;ct) which are continuous for I m a ^ 0 (respectively, I m a ^ Oj and analytic in I m a > 0 (respectively, Im a < 0), and satisfy the following properties: 0) i) ii) iii)

{ctj,j ^ N+} are zeros of det $ + and {<*j,j > N+} are poles of det $ _ ; $Z1*+ = R0SI%1; $ + , $ _ -» J for | a | -> oo; K.j = * ^ ( 0 ? ) (j g N+), Kj = *z\0*) (j > N+).

Then $ + , $ _ are uniquely determined by relations i), ii), and iii). exp(aU0x),

$± =

(Here RQ =

$±RQ.)

b) The functions $ ± satisfy relation (2.2) for a suitable matrix function Q(x). Let us suppose that functions $ ± g are invertible for all a (1m a^Qorhna^Q)

176

II. BACKLUND TRANSFORMS k INVERSE PROBLEM

and that they fulBll relations i) and ii). Then $ ± = i ? $ ± g for a matrix function B(x; a). This function B(x; a) is rational in a and corresponds to the functions $ + or $ _ (depending on the sign ofhsxaj) in the sense of Propositionl.2 of§l. The function B is uniquely determined from the relations: i) B(a = oo) = I; ii) B-^Onj = $ ± % . Proof. The proof of this theorem is shorter than its formulation. If $'± satisfy i) - iii) with the same 5 , and {ctj,Kj}, then $ + ( $ ! ) . ) - 1 = QZ1®'- for a € R. The left hand side of this equation is analytically continued in a to the upper half-plane and the right hand side to the lower half-plane (use the criterion (2.14)). As both sides tend to I when \a\ —> oo, Liouville's theorem gives that $ ± ( $ ± ) _ 1 = I. We can check the statement for $ ± ( $ ™ 8 ) - 1 analogously. By the same properties of analytic continuations we get

(s+M^1 = (*-)**:\ which is obtained by differentiating the relation $ I 1 $ + = S. In a neighbourhood of a = oo, both sides are equal to aUo — Q for a suitable Q up to 0(a~1) and, again by Liouville's theorem they coincide with alio — Q for all a. • According to Theorem2.3, the composition of arrows

CH+E±h+{#,*}h+SI{aj,Ki} (or, Q ^ S " , {«•,£•}) is a one-to-one correspondence. Indeed, $ + and $ _ are uniquely recovered from S and {oLj,Kj} and satisfy differential equation (2.2), which makes it possible to reconstruct Q{x) from $ + or $ _ . As is obvious from the theorem, the central part of this procedure is the solution of the regular Riemann problem, i.e., finding the functions $ ± g for ± Im a > 0 satisfying relations i) and ii) of the theorem. Then $ i (or, completely analogously, $5:) a r e given by the purely algebraic operation of "dressing" (see §1) in terms of $ " g and {aj,fCj}. Note that, in general, there exists exactly one relation for the dimensions of {ICj} and a sequence of inequalities for $ " * ( £ ; ) which guarantee the dressing procedure. Therefore, if we do not take any symmetries into account, Q could have singularities. Here we do not touch upon the natural (and difficult) problem of describing all S(a), a € R, which correspond to the matrix functions Q(x) with absolutely integrable entries. We only mention that the entries of S — I must be Fourier transformations of absolutely integrable functions (which is a simple consequence

§2. INTRODUCTION TO THE SCATTERING THEORY

177

of Lemma2.1). We refer the reader to the paper [DS] or to classical works (see [GK], [Kre]) for methods of solving this kind of problem, generalizing the Wiener-Hopf method. There is a huge amount of literature on this problem. We can take an arbitrary closed contour homeomorphic to a circle instead of the real line in Theorem2.3 (on which the function S was denned). The formulation and proof of the theorem do not change. The inner and outer domains bounded by this contour play the role of the upper and the lower half-plane. This contour may have a more complicated topology. For instance, it is easy to carry over Theorem2.3 to the case of an arbitrary matrix Uo (see the previous section) where 5 is defined on a collection of half-lines bounding sectors {M+}. Analogues of $ ± are defined in each of the sectors. They are the functions $ ^ . It is worth noting that the level of difficulty of the Wiener-Hopf method increases radically when the class of contours in the problem changes. In particular, the Riemann-Hilbert problem for the interior and the exterior of only one sector in C came to be studied only recently. For the readers who are not familiar with the inverse scattering problem and the Riemann-Hilbert problem (which is the recovery of Q and the corresponding $ ± from S and {ctj, Kj}), the following comments might be useful. When we turn from $ and $ to S, implicit dependence of $ and $ o n i (defined by equation (2.2)) is replaced by a simple evolutionary relation

(S-r^+foa) =

R0(x;a)S(a)R^1(x-a)

without loss of information. If we know values of $ + ( x o ; a ) and $ _ ( x o ; a ) for a point x = XQ, then after conjugation by the matrix RQ{X — x0;a) we obtain functions depending on x which satisfy relation (2.15) like $ ± . They are, of course, analytically continued to a suitable half-plane for all x. However, in contrast to the "true" functions $ + ( x ) and $ _ ( x ) , the functions Ro(x — xo)$±{x0)Ro(x — xo)-1 have essential singularities when | Q | —>• oo, a £ R. Hence such a naive attempt to recover directly $ + ( x ) and $ _ ( x ) from $ + ( x 0 ) and $ _ ( x 0 ) turns out to be inconsistent and, actually, we have to solve the Riemann problem for each x and the corresponding matrix Ro(x)S(a)R^1 (x) again and again. This makes the problem of describing the dependence of Q on x highly complicated even for a simple matrix S. For a € R, conjugation by the matrix Ro(x) is harmless and no singularities arise when a —> ±oo. As the function Q(x) can be recovered from S and 5* and discrete scattering data, the matrix T is uniquely determined by S and {<XJ,ICJ} (similarly for 5* and {<**,£*})• Let us rewrite the corresponding procedure without passing through Q. We will examine only the arrow S* t-¥ T and leave the case S ^ T to the reader (omit the index * and set i* = i, it = i - 1). We keep the notation of Theorem2.2. We assume that the minors {*,-.,£.} do not have zeros on the real axis, as before.

178

II. BACKLUND TRANSFORMS & INVERSE PROBLEM

P r o p o s i t i o n 2 . 4 . Let us assume that for the function T(a), a € R, the entries r{} and Tg, of the matrices T'° and T£ are analytically continued to the upper and the lower half-plane for fii = /J.J (<& i* = j * ) , respectively. Let rfJ, TJ^ —• 8? as \a\ -> oo. Then T is uniquely recovered from the matrix S* = P ' T Q - 1 T*° = 2>*T*°ro*_1 (see (2.15')), the zeros of the minors U., U. (1 5; i ^ n) on I m a > 0, I m a < 0 (with multiphcities) and corresponding lattices [ 7 " 0 ] - 1 C ^ and [ T o ] _ 1 0 ? for the matrices [T*°] and [T£] defined in Theorem2.2. Proof. We recall that diagT0* = diagfo" 1 ), d i a g f ' 0 = d i a g f o 1 ) , and V* = 6iag(ti,ti-) (1 ^ i ^ n). Consequently,

(2.16)

* * = ' d e t ( ^ , l ^ p , ^ . 7 ) = ]J£.*,.,

lSj^n.

The minors fy. and tj, are analytically continued to the corresponding half-planes and have zeros a3j, atj with multiplicities « Sj ., is,. (which are constant for j t < j ^ j*) as prescribed by the conditions of the proposition. Moreover tj.,tj, —y 1 when \a\ —• oo. Hence we come to the scalar Riemann problem and can solve it explicitly: 1

, ~. ,

f+^log^Z^))

1 /•+°°logK^Z11(^)) ,,

jjia-a.^-'

n t ^ ^ f l

where I m a > 0 for t, Ima < 0 for t, j ^ j * and log is a suitable branch of the natural logarithm. Note that given only the lattice [ T * 0 ] - 1 ^ - at the point aj, we can determine the minors t{, whose zero is aj and calculate the multiplicities (cf. Exercise2.5). T h e same is true for i j . and

[TQ]_1OJ.

As we have recovered tj,, tj,, we can express the matrix V' via the minors s'j. Thus the matrix TQ~1T*°

= T*°TQ~1

can be expressed via 5 * and, consequently,

this holds for T"° and TJ separately. Indeed, the triangular matrices T*° and TQ _ 1 have their diagonals expressed in terms of the minors of S' and using the triangular factorization of V'~1S', we can determine T J - 1 uniquely. Let us show that T£ and T'° are also expressed in terms of T^*~1S* (they are not triangular, and the algebraic arguments are not enough).

§2. INTRODUCTION TO THE SCATTERING THEORY

179

Let f 0 , - 1 T * ° = f • 0 _1 Ti'° for the matrices T*0 and T*° which satisfy the same conditions as f0* and T*°. Then the matrix T'oTJp 1 = A = f T*0T^°_1 commutes with Uo (in order to check this, examine the left hand side and the right hand side of the above equality). Using the notation of Theorem2.2, we have: [T 1 - 0 ][f 0 -]- 1 =A=[T- 0 ][T 1 - 0 ]- 1 As the lattices for [f'0] and [T*°] coincide with the lattices for [f0#] and [T*°] at the corresponding zeros • I when \a\ -> oo. Thus, A = I. D Before ending this section, let us show the constraints on the scattering data which should be imposed in the case of the *-anti-hermitian matrix Q (see §2.3). Equation (2.12) implies that ( $ _ ) _ 1 = $*, ( * ! _ ) - 1 = ($*)*• Therefore (2.15) is rewritten in the form (2-17)

~

$*$ = ~

R0SR0-1,

where S = (T°)*T° = (T 0 - 1 )*T 0 - 1 and S' = (T*°)*T'° = (T* - 1 )*T 0 — J . The matrices S and S' are *-anti-hermitian. We need only aci,... , <XN+ and corresponding K.j (or a j , . . . , a'N, , { £ ! } ) from the discrete scattering data. The remaining points { a w + + i , . . . , ajv} are complex conjugate to { a i , . . . , ajv} (N = 2N+). It is also easy to connect the sets {fCj, j ^ N+} and {ICj, j > N+} by means of the involution *. The matrix B of Theorem2.3 satisfies the relation B*(a) a € C (analogously for B').

= B(a)* = B~1{a)

for

2.5. Variational derivatives o f entries of T (Poisson brackets). Let us recall the notation: JJ = (S^Sj,) is the matrix with the unity as the (q,p) element and zeros as the other entries, A' = ( a ' ( l — Sfc)) for a matrix A = (a*), *A is the transpose of A. Set (see §0.5 of Introduction): [J], = £

JJ,

[/]' =

7=1

[A], = [I]PA[I]P,

£

ij,

[J]o = 0 = [/]",

?=P+1

[AY = [ J ] M [ / ] " .

We keep the notation of the previous section. For an arbitrary functional / of entries of Q we define a matrix Sf/SQ = {Sf/Sq?) whose entry at the crossing of the j - t h column and the i-th row is Sf/Sqf. When Q' = Q, the entries Sf/Sq{ are identically zero for m = y,j. Unless otherwise stipulated, we assume that Q' = Q.

180

II. BACKLUND TRANSFORMS & INVERSE PROBLEM

Proposition2.5. a) For E±(x;a)

of§2.i, we have

5
(l^p^ n). c) In the case of Q' ^ Q, the above formulas remain valid if the primes are dropped. Proof. Let us suppose that E_ and T* correspond to another matrix Qs(x). Then recalling the definition of T-matrices and using equation (2.2), we obtain the formula

oo

/

{E^Es_)xdx

-OO / O O

E+1{AQ)Es_dx,

-OO

where AT = Ts - T, AQ = Qs - Q. Therefore

A*£ = t6/ - t\ = Sp(ATJ')

-I

Sp(AQEs_nE^)dx.

By definition of the variational derivation of / = / ( « ) , we have as Au —> 0: OO

/

r y-

^ ( y ) A « ( 2 / ) d j / + o(Au).

Hence, changing the difference A to the variation 5, we estabhsh statement a) and its formulation without primes (when Q' ^ Q).

§2. INTRODUCTION TO THE SCATTERING THEORY

181

We use the formula 6 log det A = Sp(SAA *) which follows directly from the definition of the determinant of the matrix A to get:

|^(x) = -(*rom; 1 T 0 *- 1 )'t, (we expressed E± via $ ) . Here [T]*1 is the inverse of [T]p as a p x p-matrix. Since [T]p = [T°]P[T0]P, TQ[I]P = [T0]P and [I]PT° = [T°]p, we get to relation b) for tp, a € R. Both sides of the above formula are defined on the real axis and can be continued analytically to meromorphic functions on Im a > 0. Hence they coincide (where defined), by the uniqueness of analytic continuations. This argument can be also applied to the "lower" principal minors tp. • Exercise2.6. Show that {Inn*

6*Q(x)

= -($'[/]p$-1)',

for p — p* (in the notation of §2.3). Change statement include rf}, T^ when /J,- = fij (see Theorem2.2). •

b) of Proposition2.5

to

We will apply the results of Proposition2.5 to prove the infinitesimal invertibility of the mapping Q(x) >-¥ S*(a) defined in the previous section. First of all, completing the analysis started in §2.4, we shall find algebraic relations among the entries of S'. According to formula (2.16), we can calculate the "upper" principal minors s^ of the matrix 5*. We deduce an analogous formula for the "lower" principal minors under the assumption j * = j . We get

3*=f det(S;",j
n

JJ

tiX,

i=j+2

To prove this, we use the representation S* = T>TQ~1T*°. The lower minors of the matrices TQ _ 1 and T*° are easily calculated with the help of the formulae in §2.3 for det(r° y ,i* = p* = j*) and det(r^,z* = p* = j * ) ; V* = d i a g ^ . f ; . ) - Using this, we can define s^ which are equal to products of the corresponding lower minors of V',fo~1 a n d T * 0 .

182

II. BACKLUND TRANSFORMS & INVERSE PROBLEM

Comparing with formula (2.16), we come to the relations (2.19)

aj

= l,

S«.5;.+1=5; = det5',

(lgj^n).

In the case of 5 (see (2.15)) and the minors Sj and 7j defined by S instead of S', or for pairwise distinct eigenvalues /ij (then j * = j — 1, S = S°), we obtain the equations Jj-iSj = sn, 1 ^ j ' ^ n. Relations on 5 are exhausted by these formulae. If there is coincidence among {f*j}, then there are additional analytic constraints on the entries of S, as we already know. These constraints reflect the existence of the analytic continuations of proper polynomial combination of them into the upper and the lower half-planes of a. The matrix 5* is, in principle, free from analytic constraints since all the relations on its entries are algebraic. This statement can be easily made strict, if 5* is "near enough" to the unit matrix. First we check purely algebraically, not considering the dependence of T and S* on a. P r o p o s i t i o n 2 . 6 . a) Besides relation (2.19), the matrix S* defined by formula (2.15) via T satisfies the following additional constraints. Ifi* = j t , we denote by o~\, the determinant of the matrix of order i* + 1 formed by the s*m where I runs over the indices 1,2,... , i * , i and m over 1,2,... ,i*,j. Then o\ = Sjs* + 1 (see (2-16)). b) If S* satisfies all of the constraints of a), 11?= l s*# 7^ 0 and IliLi s*. ^ 0> then there is an invertible matrix T (for which fJiLi *«. ^ 0 ^ IliLi *>'.) which is related to S* by (2.15). c) Let T = 1 + SA and S' = 1 + SB for A = (cr?) and B = (&*). Then b{ = sgn(i* - j*)a{ + 0(8) for i* = j * (here, sgn means the sign and sgnO = 0). Proof. The decomposition S' = {V'T*0)^'1 is of the same type as the decomposition T = T'°TJ. In the beginning of §2.3 we calculate the entries of [T*°] (the diagonal of TQ is fixed) via the minors of T. We apply this result here noting that, in contrast to T'°, the matrix V'f*0 is more special: [V'f'°] = (Sjdi.) for certain di, € C This implies a). The proof of b) is similar to that of Proposition2.4. Let t^ and £;. satisfy (2.16) and (2.18). We represent S' = S^S^ = S^S*0 where S # 0 , S£, SjJ and S*° are of the same type as T*°, TQ, TQ and T*°, respectively. Such a representation exists since s,-. ^ 0 ^ 3 " ^ (1 ^ i ^ n ) . We may assume that diagSg = diag(
where 5*° = ( ? / ) . Then the upper and the

lower principal minors (with the index j = j*) of the matrix Ti = 5 , 0 5 o _ 1 are

183

§2. INTRODUCTION TO THE SCATTERING THEORY

equal to tj and tj respectively. The upper minors of Ti are computed directly from the constraints on diagSJ and [S'°]. To determine the lower minors, we apply the decomposition T\ = 5o _ 1 S*° together with formulae which express diagS* 0 , the minors of S* and [S*°] via {tj.Jj,}. Furthermore, S' = WTfT^1 for a suitable diagonal matrix V = diag(di.) and the matrices T*° and T'0 constructed from the matrix Ti defined above. Using the formulae for the minors of T\, we find that V coincides with the matrix T>\ corresponding to T\. Thus T = Ti is the desired matrix. Let us check statement c). The triangular factorization of T = I + SA gives the decomposition of A (up to o(5)) into a sum (instead of a product) of the matrices of the same type. In particular, T*° = I + SA° and T£ = I + 5Ao, where A°-Ao

= B = (sgn(i* - j*)a{) + A + o(S)

for a suitable diagonal matrix A (recall that S* = D * T , 0 T Q _ 1 ) . Hence to prove statement c) it is enough to show that diag B = 0. We have: diag S* = diag(t,-. U,) = diag(t,-. t-i,) = diag(£;. — ti,) + I = I mod o(6). Here we use formula (2.7) and the trivial relation ( / + -A) - 1 = I — 6A + o(S).

•

To complete the analysis of the algebraic structure of 5*, let us return to the functions T(a) and S*(a) for a e R denned from Q(x). We remind (Lemma2.1) that the entries of T—I belong to the space F(C\) of Fourier transformations of functions from the space C\ of absolutely integrable complex valued functions of x from — oo to +oo. This is also true for entries of 5* — / (which are expressed as polynomials of tJ{). Indeed, the space F(Ci) is a ring with respect to the multiplication of functions (which corresponds to the convolution in Ci). Let us introduce a norm in F(Ci) which is induced from C\. Each variational derivation Ss'1 /6q' is a linear operator from C\ to F(L\). We will compute them at the "point" Q(x) = 0. We have: E± = Ro(x; a) = exp(aU0x) and T(a) = I = S'{a) for Q = 0. Combining Proposition2.5, a) and Proposition2.6, c), we obtain Corollary2.2. For Q(x) = 0 t i e variational derivation Ssjm/6q'

maps f € Cx

to ,oo

sgn(m* — I*) I

exp(ia(a m — ai)x)f{x)dx

€ -F(£i)

J—oo

for I = r and m = s, and to zero if either I ^ r orm ^ s (recall that fim = iam € iR, m* = I* # a m = a j , sgnO = 0). D We obtain that the map Q(x) t-> S'(a) — / is an isomorphism of the topological vector spaces of the matrices with entries in C\ or F(Ci) in a neighbourhood of

184

II. BACKLUND TRANSFORMS & INVERSE PROBLEM

Q = 0, S° = I. Note that for a matrix Q(x) sufficiently close to zero, the minors U, ti do not have zeros in the corresponding half-planes and those discrete scattering data are trivial (see Exercise2.2). Hence the inverse scattering problem for the matrices S'{a) close enough to J admits a unique solution, if Q is taken from a neighbourhood of zero even if we take into consideration the discrete spectrum. Poisson bracket. Another application of the formulae in Proposition2.5 is the computation of Poisson brackets for the entries of T. In order to avoid the problem of convergence, we assume that Q(x) = 0 if x ^ a or x ^ b (a < b). Let us call the following integral (assuming the existence) the Poisson bracket of two given functionals / , g of the entries of Q:

(2.20)

{/, g} = jf Sp ( (j£(*)) [Uo, J|(*)] ) dx

(Sp is the trace). In particular, for a < z and j / < 6 w e obtain the relations

{«?'(*),ir(y)} = I Sp(i/*(* - x)[J70> my - x)]) dx Ja

Here S(x) is the delta function ( / f(x)8(x—y) dx = f(y) and 8(x) = S(—x)) and the variational derivatives and Poisson brackets are considered as generalized functions. The original definition (2.20) follows from the last formula since by the chain rule

= E

/

/

^T<*)<«*(*),rf(v)}||(v)

dxdy

for any functionals / , g, where the summations are taken over all indices. We also have the following trivial identities:

Wi(zWM} =-{qar(yU(z)h {{
+ {{9,h},f}

+

= —{g,f})

{{h,f},g}=0.

and satisfies

§2. INTRODUCTION TO THE SCATTERING THEORY

185

Exercise2.7. Check these two properties of {,}, using (2.20). Assume that functionals f, g and h are twice continuously differentiate with respect to Q (see [Chll]). • Proposition2.7. a) For a, ft € R (a ^ j3), 1 ^ i,j,k,l ^ n, {ti(a),t'km

= ^ ( e ^ - " > < " - « V f c ( a M / 3 ) -e<«-«><-«'t|(a)*i(/?)),

{*?'(a), £(/?)} = _ L ^ ( a $ ( / ? ) - *?(«*)*£(/?)), where $ ) = f d= JJo^TJIbfa)- 1 . b) For finite or infinite a, /? the brackets {t*(a),t/(/?)} are well-defined and identically zero for Ima,/? ^ 0. Proof. We use (2.20) and Proposition2.5 and get W(ot),
t

Ft(m)dx

Ja

= f

Sp([FtW,U0}F;(aY)dx

Ja

= f

Sv([Ft(P),Uo]FHa))dx,

Ja

where we set Fj(a) = E-(x; a)IjE+(x; a)*1, applied the transposition before taking the trace and used the identity Sp(A'B) = Sp(A'B'). As a consequence of the fundamental equation (2.2) (2.21)

Sp(F}(a)F,k(P))x = (a - 0) Sp([U0, F;(a)]i?(/?)).

Really, Sp{[Fj(a), Q]F,k(P)+FJ(a)[F,k(/3),Q]) = 0, since Sp(AB) = Sp(BA). Using (2.21), we get (a -/?){
- Jfc(a)*|lJV(a)&( a )t* J / J ^ a ) ) .

186

II. BACKLUND TRANSFORMS & INVERSE PROBLEM

Conjugating this by RQ and RQ, we get the formulae of a). Analogously we can establish the relation (a - 0){tk(a),tk(0)}

=

Sp(Fk(a)F,(/3))\r=ba

for the functions Fj = $[I]j$~l which satisfy the same equations (2.21) (see Proposition2.5). At x = a or x — b, the functions $ , Fk and Fi are simultaneously upper or lower triangular matrices. Therefore, Sp(Fk(a)Ft(P))\r=bi

= Sp(diag^(a)diagF,(^))l^a-

As diag Fj = [I]j for x = a,b and j = k,l, we get statement b).

•

At the end of this section we give three examples which illustrate the coincidence of the number of entries of Q = Q' and which is independent of the number of entries of S*. The most popular case in the inverse problem method is n = 2, /^i 7^ M2- In this case: T = T°T0

~ \c d)~\c

= T0T°

iy \o a-1 ) ~ \o djycd-1 \)'

since det T = £2 = to = 1 = to = *2, h = a. and t\ = d. Hence (see (2.15)), 5 = diag(F^_1)T°T-1

d 0 \ f d-1 0 d) ycd-1

0 \ (I l)\Q

~b\ a )

(\ \c

-b\ 1 )"

Relations (2.19) are reduced to the equations s\ = 3"i = 1. The case n = 3, pi = ^2 ^ H3- One has: T = T * 0 T Q = 7 J T *°

'a c

6 d

0\ / l 0 } ( 0

0 1 -1

(de - fi/)^1 \ ( a / - ce)*" 1 0

0'

187

§2. INTRODUCTION TO THE SCATTERING THEORY t2 = ad - be, * = d i a g ^ . f ; . ) = /l 5* = [ 0

\h

0 1

0\ / l 0 ( 0

k g)

0 1

\0

0

6/-de\ /l ce - a / J = I 0

t2

)

\h

0 1

dia.g(g,g,g),

6/-
k

1

/

Let us assume now that T is *-unitary and n = 3; (ii, fi2, and H3 will be arbitrary purely imaginary numbers. We have:

fa T = T°T0 =

0 <2

c \h

ak-hb

0\ 0J 1/ \ 0

0

in the notation above, and S = (r°)*T° = 1

(

ft(T°)*ftT° —6

—W1W26 a a + W1W266 /l

—(J2CL>3f

hui\u2 \

—/ 1

J/

The entries of S satisfy the unique (real) relation s^si = det S. We can express aa in terms of the entries b, f and h which are independent. The quadratic equation for aa has one or two non-negative solutions for W2W3 = ± 1 respectively, if it has real solutions at all. The number of real parameters determining S is equal to 6 and coincides with the number of real (functional) parameters for the *-anti-hermitian matrix Q with zeros on the diagonal. 2.6. C o m m e n t s . The aim of this section is to give a systematic introduction to the inverse scattering problem method in the formulation by A. B. Shabat (see [Shl,2]). Actually we are concerned only with the direct scattering problem. This includes the construction of the scattering matrix and solutions of the original linear problem (in our case (2.2)) which are analytic in the spectral parameter. Since we always consider a one-dimensional variable x, we prefer the term "monodromy matrix" to the term "scattering matrix" and use the notation "T". We understand the inverse problem (from T to Q) as the Riemann problem. In a more traditional approach, one usually includes in it the reformulation of the Riemann problem in terms of integral equations and the proper existence theorems. For instance, the inverse problem for the KdV equation can be reduced to the Gelfand-Levitan-Marchenko equations. In the case of matrices of order 2, a systematic exposition of the inverse scattering problem can be found in [TF2] (see also [F2], [DT]). As to arbitrary matrices,

188

II. BACKLUND TRANSFORMS & INVERSE PROBLEM

the corresponding methods have not yet been worked out completely. There are a number of theoretical problems. For instance, the space of T-matrices corresponding to the matrices Q with absolutely integrable entries have not yet been fully described. The construction of the analytic solutions $ and 4" of equation (2.2) by means of the triangular factorization is due to M. G. Krein (see, for example, [GK], [Kre]). For n = 2 the triangular factorization is trivial. We can construct analytic continuations for the columns of matrices E± directly. In this case the study of the discrete spectrum is essentially simple. Note that we do not examine real points where $ and $ are degenerate (i.e., the zeros of the minors U, ti on the real axis). For n = 2 such points are studied in detail in [TF2]. We include this "analytic" section because of the following reasons. First of all, its content is closely related to more algebraic material in other sections. In general, the inverse problem method is used in the soliton theory more algebraically than analytically (though, of course, such a statement should be understood with reservations). Another reason is that, in spite of the popularity of the inverse problem method, the direct scattering problem for n > 2 has not been discussed in the literature in a sufficiently complete and rigorous way. Usually the analyticity of $ and <]> is explained by the limiting procedure from comactly supported Q(x) (equal to zero apart from a finite interval) to absolutely integrable functions (see, for example, [ZMaNP], [Shi]). This procedure is not simple and still uses a kind of reduction as in §2.2 to prove the analyticity of $ and $ . This way or that way, we need exterior powers of the matrices E± which play a central role in the present section. Our approach sits next to the methods in other sections and has explicit invariant meanings from the viewpoint of the representation theory. The analytic part (§2.1) consists of elementary direct estimates. We follow paper [Ch2] and drop the usual condition of pairwise distinctness of numbers fij (1 ^ j ^ n). Our "algebraized" approach to the direct problem makes the necessary refinement easy. We need to take possible coincidence of certain fij into account for concrete equations ( 5 n _ 1 fields, the GHM and the VNS for n > 2). From the theoretical viewpoint this refinement is necessary to handle the case of an arbitrary reductive group instead of GL„ (Exercise2.3). Another demonstration of the universality of our procedure is the construction of the system of analytic solutions of equation (2.2) for arbitrary complex fj,j (see Proposition2.2). We give this construction as an illustration only without going into detail. However, one can easily complete the story and prove the counterparts of Theorem2.3, Proposition2.6 and Corollary2.2. The procedure of dividing the complex plane into sectors and constructing solutions of (2.2) in each sector is not new. For a series of soliton equations, such an approach was applied by R. Beals, R. Coifman, P. Deift, D. Kaup, A. V. Mikhailov, E. Trubowitz and others (see e.g. [BDZ]). As we noted above, there are difficulties in working out the analytic

§2. INTRODUCTION TO THE SCATTERING THEORY

189

methods of solving the associated Riemann problem. We hope that our constructions simplify the technique of the inverse scattering problem especially the study of the analytic continuations. Even in the well-known case of the purely imaginary matrix Uo with pairwise distinct /ij, when the results in this section are mostly classical, it seems to be useful. As is obvious from the proof of Theorem2.3, we can apply the Riemann problem in order to solve equation (2.2) for functions Q(x) of other types (not only for normally integrable ones) if we choose suitable contours and matrix functions S(a) with appropriate properties. This was pointed out in [ZMaNP], [Kri5] (see also [TF2]). The corresponding analytic technique can be rather involved (except for the case we studied here and for algebraic-geometric Q). The formulae for the variational derivatives of the entries of T is universal, too. Note that the results of Proposition2.5 are rather well-known (see, for example, [ZMaNP], [Sb.2], [TF2], and also [Ch2] for the formulae for principal minors) and often used in soliton theory. The proof of the infinitesimal solvability of the direct problem by means of the computation of variational derivatives (for pairwise distinct py) can be found in [Sh2]. The Poisson bracket (2.20) turns out to be a special case of the Gelfand-Dickey bracket (see [GD3]), if fij are in a generic position. Such brackets play an important role in mathematical physics and the soliton theory. For instance, for n = 2 bracket (2.20) is one of the fundamental objects studied in the book [TF2]. The results of Proposition2.7 (and the variants) are also known (see [ZMaNP], [TF2]). Analysis of examples of the Riemann problems connected to soliton equations is the subject of the next section. The reader can also refer to books [ZMaNP], [TF2], [AS], [CD]. Note that Ch.III of [ZMaNP] contains main formulations related to the matrix Riemann problem for pairwise distinct {JJ,J}. See also papers [BC1, BC2, BDZ] on more exact and systematic theory. We also mention the paper of A. Fordy (J. Phys. A, 17, (1984)) devoted to an analogous problem.

§3. Applications of t h e inverse problem m e t h o d In §3.1 we construct "stabilizing" *-unitary solutions of the basic equations by means of the Riemann problem (and the results of §2). In §3.2 we calculate the asymptotic expansions of the principal minors of the T-matrix. We also deduce the trace formulae, making it possible to compute the integrals of the basic equations (§1) from the scattering data. The next §3.3 is devoted to the study of special cases. We consider 0„-fields, 5" _ 1 -fields, and equations with n — 2. In §3.4 we examine the nonlinear Schrodinger equation (NS) in more detail, and compute the entries of T (scattering coefficients) for several solutions. In §3.5 we study the derivative nonlinear Schrodinger equation (DNS), which is an analogue of the NS and an example of a more general associated inverse problem. In §§3.2, 3.4 and 3.5 we use the formulae for variational derivatives from §2 proving the involutivity of local conservation laws and computing the variations of the discrete scattering data in the case of the NS and DNS. We remind the reader of all necessary constructions as needed. When reading §§3.4 and 3.5 it is useful (but not necessary) to be familiar with basic facts about the inverse problem method for the NS equation presented in the books [ZMaNP] and [TF2]. 3 . 1 . Inverse problem for basic equations. Let U(x,t), V(x,t) be a solution of the P C F system (0.2) (Introduction) or let U be a solution of the GHM equation (0.9). As in Ch.I, (3.1)

U = F0UFo\

Q d= F^iFoh

= Q',

where UQ = diag(/xi,... ,/j.n) for constant {ftj} C C and a suitable invertible matrix function Fo(x,t). Let us recall that

For the GHM we impose restriction (0.10) on UoMl = • • • = Up = cl> Mp+l = • • • = Mn = C2, c = cj — C2 ^ 0, 1 ^ p < n. The functions FQ and Q are not uniquely determined by U. By Corollaryl.l of Ch.I we can choose the function Fo so that it is subject to the relations (3.2A)

(3.2B)

2(Fo)tF0-1

=V&

2F^ (F0)t = F'1 VF0 = f W,

K\F*)t = [VM-\\Q,\V*M 190

§3. APPLICATIONS OF THE INVERSE PROBLEM METHOD

191

where we denote the formulae corresponding to the P C F by the letter "A" and to the GHM by the letter " B " , as in Ch.I. If $ satisfies system (1.1), (1.2A,B) of Ch.I for k — a (see also (1.1,2) of the present chapter and (0.3) and (0.11) of Introduction), then E = F 0 - 1 $ is a solution of the system of equations (3.3)

EX + QE = aU0E,

^

E E l

(3.4B)

EtE-1

< ~ = 2(2^1)* = c2a2UQ - c2aQ + [Qx, U0] - \\Q, [Q, U0}},

where W is defined in (3.2A) and c = Ci — c% (see §1.2, Ch.I). It is proved using constraints (3.2), the relation [U0,[Q,U0]] = -c2Q for the GHM (see (0.11)) and the following differential equation which may be checked directly: EtE-'= EtE-1

^-^W

-

F^{F0)U

= c2a2U0 + a[UQ, [Q, U0}] - ^ ( F o ) * ,

in the cases A and B respectively. Conversely, suppose that an invertible matrix function E(x, t; a) together with the functions Q and W satisfy (3.3) and (3.4) for the P C F (A) or the GHM (B). Then Fo is determined from Q and W by (3.1) and (3.2) up to left multiplication by a constant matrix, and U and V are uniquely determined by (3.1) and (3.2A) def

modulo conjugation by constant matrices. As $ = FoE, U, V are connected by formulae (1.1) and (1.2) of Ch.I for k = a, it is easily checked that the U and V constructed above are solutions of system (0.2) or U is a solution of (0.9) together with (0.10) respectively. Let us recall that E = F^1^ is analogous to the formal series 'i'.FJj -1 ^ from Ch.I (see Corollaryl.l and the proof of Propositionl.2). Using the results obtained there, or by direct computation, it is easy to show that the compatibility condition of equations (3.3) and (3.4A) is given by the following system: , Wx = ±[Uo,W]-[Q,W), Qt =

\\U0,W].

Analogously, the compatibility of (3.3) and (3.4B) leads to the equations (3.6a) (3.6b)

i Q « = Q W + [[, Q< + >], = Q£

+ [[, Q(->],
192

II. BACKLUND TRANSFORMS & INVERSE PROBLEM

where Q = Q<+> + Q<"> and [i/o,Q ( ± ) ] = ±cQ ( ± > (see equation (0.13) of Introduction). Note that \{Q,[Q,U0]} = c[0< + ) , Q^] (see (3.4B)). Thus equation (0.2) and (0.9) together with (0.10) can be transformed into (3.5) or (S.6), and vice versa. Further in this section we assume that U* + U = 0 = V + V for an anti-involution A* = £IA+Q v.'ith ft = diag(w 7 ) and uij = ± 1 (where A+ = \A) is the hermitian conjugation). Then we can assume for such U and V that Fo is a *-unitary matrix (that is, FoF0* = I) and, consequently, Q* + Q = 0 = W* + W. Note that the matrix UQ in this case must be purely imaginary: {fj.j} C iR, c, ci, c-i S z'K. For the GHM equation, the second equation of (3.6) is obtained from the first by applying *, since Q^ — —{Q^)*We leave the general complex case t o the reader. Almost always, the most interesting applications are for soliton equations with certain reality conditions (for instance, for *-anti-hermitian or *-unitary solutions). We further assume that the derivatives Wx, Qt (PCF), Qx, Qxx and Qt (GHM) exist and that the entries of Q, [Uo, W] and of Q, Qx and Qxx for the P C F and GHM, respectively, are absolutely integrable with respect t o x on the real axis. Such *-anti-hermitian W and Q are called stabilizing (as always, Q' = Q). In this case there exist the limits W+ and W- of the function W(x, t) for x —• ±oo (which possibly depend on t). By equation (3.5),

[U0,W±] = (Q(±oO,t))t=0. We may assume that the variable t belongs to a n interval containing the initial point t'. The above conditions on the solutions Q and W are equivalent to the existence of the derivative Ut and the existence and the absolute integrability of the entries of the matrix functions Ux and Vx (PCF) and of U, Uxx and Uxxx (GHM) which are constructed from Q and W by the above procedure. Now we are going to check this statement. By (3.1), Fo depends continuously on x and there exist limits F ( ± o o , t). Therefore the entries of Fo and F 0 - 1 are bounded functions of - c o ^ x ^ +oo. Since Ux = Fa[Q, Uotfo1,

V=

FoWFo1,

Wx = F 0 ([Q, W] + WX)F^ = ±F0[Uo,

W)F0-\

we get the equivalence of absolute integrability of Q and Ux, and that of [Uo, W] and Wx. Analogous reasoning proves the same equivalence for Uxx, Uxxx and Qx, Qxx in the case of the GHM. Note that the absolute integrability of the entries of Ux and Vx implies the existence of the limits U(±oo,t) and V(±oo,t). We call functions U and V satisfying all the above conditions stabilizing.

§3. APPLICATIONS OF THE INVERSE PROBLEM METHOD

193

Scattering data. Let the numbers \i, (where VQ = diag(pj)) be ordered as in §2, i.e., ifij ^ ifik if j < k, and assume that ici > ici and ic > 0. We construct from *-antihermitian stabilizing matrix functions Q and W (satisfying (3.5) or (3.6)) solutions E±(x,t;a) of equation (3.3) and the monodromy matrix T(t;a) (in the notation of §2) for a € K. By the conditions on Q and W, the entries of Qt are absolutely integrable (see (3.5) and (3.6)). Therefore we can differentiate the series for E± and T term-wise with respect to t since they converge uniformly and absolutely. The functions (E±)t and Tt obtained in this way have the same continuity properties as E± and T (see Lemma2.1, b)). Moreover, (E±)t,Tt -¥ 0 when \a\ —¥ oo. Recall from §2.3 that E±(t; a ) ' = El1 (t; a),

T(t; a)* = T " 1 (t; a).

In §2 we introduced matrices S and 5* (see formulae (2.15) and (2.15*)). These matrices are *-hermitian: S* = S and (5*)* = S* (see (2.17)). We also constructed solutions $ and $* of equation (3.3) for I m a ^ 0 which depend on x and a (and now also on t) and are analytic with respect to a, as well as the discrete scattering data (which depend only on t): {ahKj

* * - * (
{ « • , * ; & * - * (
where 1 ^ j ^ N+,N+ and a.j,a*j are the zeros of det $ and d e t $ * (see §2.4) respectively. If all \ij are pairwise distinct, then S = S°, $ = $* and {otj,Kj} = {a',JC*} (and i* = i, i, = i - 1). As in §2, we assume, restricting the class of Q under consideration, that I m a j > 0 and I m a * > 0 (i.e., d e t $ and d e t $ * do not have real zeros). Recall that T = E+*E-

where RQ = exp(aUox). A(0;a) = l b y :

= R^lE-{x

= +oo)

= E+tRoix

= -oo),

Let us define a function R(t; a) with the normalization

5

'=2(2a^I)^


R = exp(c2a2U0t),

(GHM).

If W- = W(-oo,i) does not depend on t, then R = exp(*W_/(4a - 2)) (for the PCF).

194

II. BACKLUND TRANSFORMS & INVERSE PROBLEM

T h e o r e m 3 . 1 . The matrix T depends on the variable t as follows: (3.7)

T(t; a) = R{t - t'; a)T{t'; a ) ^ " 1 ^ - t'; a),

where the right hand side of (3.7) is denned for all a e R U oo, since [W-,T(a = 1/2)] = 0 (PCF) and [a2U0, T(a)] -» 0 as | a | ->• oo (GHM). The same formula holds for the matrix S* defined by formula (2.17) of $2.4 (see also (2.15*)). Analogously, the evolution of discrete scattering data is determined by the relations (ct')t = 0 and (3.8)

K'{t) = R(t -1'; <*')£•(?),

l^j^N'+.

Proof. Let us denote the right hand sides of formulae (3.4A) (PCF) and (3.4B) (GHM) by M. Now we temporarily assume that a ^ 1/2 (A), a ^ oo (B). The compatibility of (3.3) and (3.4) given by the equation [d/dx+Q-aU0,d/dt—M] = 0 implies the following general property: For an arbitrary invertible solution E of equation (3.3), the function Et—ME also satisfies the same equation and is obtained from E by multiplying on the right by a matrix which is constant with respect to x. Hence, in particular, we can set (E±)t

- ME =

(C±)x=0,

-E±C±,

Cl = CZ\

Letting x -¥ ± o o , we get C+ = M (+oo, <),

C_ = M ( - o o , t).

Here we use the definition of E± (E± —• Ro when x —¥ ±oo) and the equation [Uo, W±] = 0 derived above in the case of the P C F . As E- = E+T, (3.9)

Tt = C+T - TC-,

W+T(a = 1/2) = T(a =

1/2)W-

The latter formula is for the P C F only, and follows from the differentiability of T with respect t o t and the continuity of T with respect t o a for all a. Since T(a) —> I, when | a | —• oo, we have that [a2Uo,T(a)] -> 0 for the GHM when \a\ -> oo by the same reasoning. The differentiability of E± and of T with respect to t follows from the absolute integrability of Qt, as noted above. In the case of the P C F , the functions WE±(a = 1/2) satisfy (3.3) for a = 1/2. It is easy to deduce the second formula of (3.9) from this.

§3. APPLICATIONS OF THE INVERSE PROBLEM METHOD

195

Now we will remind the definition of S*. The factorization T = T'°TQ determined by the conditions

is uniquely

r?' = 0 = < £ r ^ ,

4

= 0 = ^ 4 ,

diagTS = d i a g ( l , . . . , tj*,...

for

i <j,

1

, t" ),

where T*° = (r° J ), T0* = ( T £ ) , »* = max{j | W = / / ; } , i, = min{j |/iy = /<,-} - 1 (see §2.3) and the fy are the upper principal minors of T. Then S* d£ (T*°)*T*° = ( T o - 1 ) * ^ - 1 , (see (2.17)). Note that, by definition, the minors tj, do not have real zeros (det $* =

n?=i*>.)L e m m a 3 . 1 . T i e minors tj,(a)

for I m a ^ O do not depend ont(l^j^

n).

Proof. Set c± 7 = £ i = i c!y, where C± = ( c ^ ) (see (3.9)). As [t7 0 ,C±] = 0, (3.10)

(tj)t = c+jfj - *>c_y,

where j = j * .

In order to check these formulae, we turn from C to the exterior power f\} C". Only the diagonal entry is non-zero among the entries in the first row and the first column of the matrix \}C±. By the principle of analytic continuations, the relation holds not only for a (E R but also for all a in the upper half-plane. It follows from (3.10) that the zeros of tj (and their multiplicities) for j = j * do not depend on t (by the uniqueness theorem for differential equations — cf. the proof of Lemma2.3). Since c±j(t) 6 iR and (tjij(a))t = 0 for a € R (here we differentiate tjij using (3.10)). But tj(a) for a such that I m a ^ 0 is uniquely determined from tjij(a), a € R and its zeros in the upper half-plane by the Maximum Principle, since tj —• 1 when \a\ —>• oo (Theorem2.2). • The matrices C± belong to the normalizers of the Lie subalgebras of g[n in which the matrices T*° and TQ take their values. Moreover, C± commute with diagTg. The above lemma implies that diagTg does not depend on t. Consequently, using the uniqueness of the factorization T = T°°TQ, we have

(r-°)t = (c+r-°-:r0c_), (

'

'

and (S')t

{TS)t = [C-X], = ((r*- 1 )*T 0 —*)« = [C-,S'}.

Thus the statement on S* holds.

196

II. BACKLUND TRANSFORMS & INVERSE PROBLEM

L e m m a 3 . 2 . aj [T'°] d= (S^T?') b) W- = W(-oo,t)

(see §2.3; satisfies [T*°]t = [C_, [T'°]].

= W(+oo,<) = W+.

Proof. From formula (3.11), we have [T ,0 (*)] = R+(t - i')[T , 0 (*')]^-(< - * ' ) - 1 . where we temporarily denoted R by R- and introduced t h e analogous matrix R+ for W+ instead of W-. This formula holds for I m a ^ 0 (recall that the entries of [T°°] are analytically continued to the upper half-plane — Theorem2.2). Set Tj = [Tt0]-l(O?) (in the notation of §2) for the same a] (1 ^ j ^ JVJ) as above. Then

Tj(t)=R-(t-t>;a')Tj(t'). Let us define a matrix

[T*°] = JL(* - t')[T'°{t')]R-{t -

t'Y\

and construct an analogous lattice

r_/=f[r:0]-1(c>;). Then we have [T1°]*[T1°] = [T ,0 ]*[T ,() ] for a € R and Tj = T-j, which follows directly from the definition of [T*°] and the formula for [T*°(t)]. As [T*°], [T*°] -»• J when | a | —>• oo (Theorem2.2), we have that T°° = T^° by the unique solvability of the Riemann problem of type (2.17) (cf. the proof of Proposition2.4). But then R- = R+ and W- = W+. D From this lemma and relation (3.9) we get formula (3.7). Note that for the GHM the statements of Lemma3.1 and 3.2 are trivial, since in this case C± = c2a2Uo. It remains to check the dependence of tC'ont. As in the case of E = E±, we get for E = $ * that (3.12)

$J - M$* = - $ * C ,

Cx = 0.

Computing the matrix C by means of Lemma3.1 and Theorem2.2 d), we obtain: * - • + « > = * • * • = f Q'R^1 -»• diag(< i# ). Thus, C = M- since [Uo,M-] = 0 = [diag( +oo, (3.12)tendstotherelationfor[T*°] ( derivedfrom(3.11). Set $ * = $*R. Then fy = Af$* as proved above. The standard argument in §2 or in §1 (see Propositionl.2) leads to the fact that the lattices $ , _ 1 ( C ? " ) = R^1^' do not depend on t. This is equivalent to relation (3.8). •

§3. APPLICATIONS OF THE INVERSE PROBLEM METHOD

197

E x e r c i s e 3 . 1 . Show that formulae (3.7) and (3.8) hold for S and {ctj, K.j}, if the matrix WL = W+ is diagonal. •

Exercise3.2. Rewrite Theorem3.1 for the Lax equations (2.3) and (2.5) ofCh.I, setting R = exp(a m f V) for V and m from §2 of Ch.I.

O

Inverse problem. Let us assume that the entries of stabilizing Q (PCF) or of Q, Qx and Qxx (GHM) are absolutely integrable and W(t) is continuous in a neighbourhood of a point t'. As above, Q and W are *-anti-hermitian, Q' = Q, and [W, Uo] = 0. Let us construct from Q the T-matrix and the scattering data S'(a) and {K*,a*}, where I m a * > 0 for 1 ^ j ' ^ iV+ (see above). We assume that the zeros {a*} of the product 11?= I *;. do not he on the real axis and [W,T(a

= $)) = 0,

2

[a U0, T(a)] -» 0,

(PCF), when \a\ -)• oo,

(GHM).

It follows from the previous analysis and Theorem2.3 (see also formula (2.17)), that there exists at most one stabilizing pair of solutions Q(x, t) and W(x, t) of equations (3.5) or (3.6) such that Q{x,t') = Q(x) and W(±oo,t) = W(t). Hence, the Cauchy problem of this type for differential equations (3.5), (3.6) is uniquely solvable. Note that the function W appears only in system (3.5) (PCF). We now put the results obtained above and those found in §2 together in order to construct the solutions Q(x, t) and W(x, t) of the corresponding equations in terms of given Q(x) and W(t). This procedure is called the inverse problem method. Below we omit * for the sake of simplicity of notation. The statements below are also true for "true" S and {aj,fCj} (defined in §2), if the matrix W is diagonal (see Exercise3.1). Let R be constructed by W(t) for the P C F (see above) and, as before, R - exp(c 2 a 2 l/o<) for the GHM. Set

Kj(t) = R(t - t'; ai)Kh

l^j^N+.

Note that S(t;a) is defined for all a including the points a — 1/2 (PCF) and a = oo (GHM) and is continuous with respect to a, since at these points (where R has essential singularities) S satisfies the same commutativity relation as T (see above). Let us assume that for all x and t the Riemann problem (3.13)

$** = RoS(t;a)R^\

^ ( C ? ) = JtoK>(t),

is solvable, where a 6 R, 1 ^ J' ^ N+, $ = $ ( x , t ; a ) is analytically continued to the upper half-plane ( I m o ^ 0), and $ —> I for I m a ^ 0, as \a\ —• 0 (see

198

II. BACKLUND TRANSFORMS & INVERSE PROBLEM

(2.17)). Set $ = $RQR. Then $ satisfies equations (3.3) and (3-4) for suitable functions Q and W. The first of these equations follows from Theorem2.3. $ * $ - 1 is calculated analogously. Really, $ * $ does not depend on t and, consequently, the function $ t $ _ 1 = —($*) _ 1 ^t is meromorphically continued to the upper and the lower half-planes. Due to the constraints on ICj{t), the entries of this function do not have singularities at a = aj,cij, 1 ^ j' ^ iV+ and may have poles only at a = 1/2 (PCF), a = oo (GHM). Furthermore, using the equation $ ( a = oo) = I and algebraic results in Ch.I (for the GHM), it is easy t o show that the function $ t $ _ 1 can be written in the form of (3.4) for a certain W(x,t) in the case of the P C F and (the same) function Q in the case of the GHM. Thus, from Q(x) and W{t) we have constructed solutions

Q{x,t) = aU0 - $A~\

W(x,t) = (4a - 2)$ f $~ 1

of equation (3.5) or equation (3.6). If the Riemann problem (3.13) is solvable and Q(x,t) is a stabilizing function, the pair of functions Q(x,t) and W(x,t) is a stabilizing solution of (3.5) and W(t) = W(±oo,t). Defining FQ, U and V by formulae (3.1) and (3.2) (recall that V and V are constructed from Q a n d W uniquely up to conjugation by constant matrices), we get a stabilizing solution U{x,t) and V(x,t) of the system P C F (0.2) or the GHM equation (0.9) of type (0.10). There is an explicit formula for Fo in terms of $ : Fo = C$(a = 0)-1, where Cx = Ct = 0. Recall that U = FoUoF^1 and V = F 0 W F 0 _ 1 . The function g(x,t) = C^~1(a — 0 ) $ ( a = 1)C" satisfies equation (0.1) for the case of the P C F . We see that g(x, t) is a *-unitary principal chiral field for arbitrary constant •-unitary matrices C and C". If there is no discrete spectrum (i.e., Yij=i tj, (a) vanishes nowhere), then (3.13) is called a regular Riemann problem. In the opposite case, when S = I, solving (3.13) reduces to a purely algebraic procedure and the corresponding Q, W and U, V are called multi-soliton solutions. We study such solutions from an algebraic viewpoint in [Chll]. Returning to the general case, we can use (3.13) t o construct solutions Q r e g and Wtes from the matrix S, dropping the data of the discrete spectrum. Then Q and W can be obtained from Qlee and Wtes by a Backlund-Darboux transformation introduced in §1 of this chapter (see Theorem2.3). Analogous statements hold for U and V. Properties of the action of Backlund-Darboux transformations on the class of stabilizing solutions strongly depend on the type of the matrix fl defining the anti-involution *.

§3. APPLICATIONS OF THE INVERSE PROBLEM METHOD

199

P r o p o s i t i o n s . 1. a) Let 0 = 7 (i.e., * = +). We suppose that for a given matrix S, the regular Riemann problem (3.13) is solvable and that Qtes(x,t) is a stabilizing function. Then for arbitrary ctj (imaj > 0) and ICj, the problem (3.13) is also solvable. The pair Q(x, t) and W(x, t) is deGned everywhere. Moreover, it is a stabilizing solution of (3.5) or (3.6) for sufficiently generic {ICj}. b) Let ifii = . . . = ifip = 1 = u>i = . . . = u>p and inP+\ = . . . = i/i„ = —1 = Wp+i = . . . = u>n, where ft = diag(wy) and Uo = diag(/uy), 1 ^ p < n. Then for an arbitrary function Q{x) with absolutely integrable entries, the data of the discrete spectrum are absent (i.e., the product 17?= i tj. does not vanish in the upper half-plane). Proof. In the case of an anti-hermitian matrix Q (or U), we can use the results of §1.2 on Backlund transformations. Adapting the result of Exercisel.8 to this situation, we get that for a stabilizing Q'e8(x,t), the functions Q and W (or U and V) do not have singularities for all x, t € R. As a result of a more detailed study of the asymptotics as |x| —>• oo, for sufficiently generic {ICj}, the entries of Q (or the entries of Q, Qx and QXx) are absolutely integrable with respect to x, if the entries of Q re * are. Now we suppose that assumption b) holds. Then Q = -Q* = - f t Q + f l = Q+, (Uo1Q)+

= -QUo1

= Uo1Q.

Suppose that 17?= I */'. aas a z e r o a * a o with Im a 0 > 0. By Proposition2.3 there exists a solution ip of equation (3.3) for a = ao and a, 6 6 R with b > 0 such that \eaxipk(x; a)\ < e~"x>

when \x\ —• oo,

k =

l,...,n.

Here we can set a = a' Im ao for a real number a' such that — i^ti = 1 > a' > iy,n = —1. Choose a' = a = 0. Then we may take any 6 = 5 ' I m a o with 0 < b' < 1 (cf. proof of Proposition2.3). Let (p,q) = q+p be the hermitian scalar product of the vectors p,q e C". The operator L = U^d/dx + U^Q is self-adjoint. Thus we get OO

/

yOO

(
(L
J — OO OO

/

i>00

(
((p, (f) dx.

J—oo

All of these integrals are finite and nonzero due to the conditions on (p. Therefore ao = 5b; this is a contradiction. •

200

II. BACKLUND TRANSFORMS & INVERSE PROBLEM

3.2. A s y m p t o t i c expansions and t h e trace formulae. The transformation Q t-+ T has many properties in common with the Fourier transformation. It is well known that a function which is the absolute value of the Fourier transform of an absolutely integrable function such that all its derivatives are also absolutely integrable decays more rapidly than any function \a\a as \a\ —• oo. We will get similar results for the entries of T. For a while we won't be taking the dependence of Q and T on t into account. As before, Q is *-anti-hermitian, Q' = Q. In this section we assume that all the entries of Q and their derivatives with respect to x of any order are absolutely integrable from —oo to oo. Let us introduce the following notation which generalizes (2.3): rrfcoo *±oo

K^\Q1,...,Q")=

/

*±oo P

J[B^1Q'Ro(x.)dx,...dx1,

•••/

where p ^ 1, and set K± = I and K± = Ro{x)K^'(x)R^1(x). Here we assume that the matrix functions Q1,... , Qp have entries which are absolutely integrable from x = —oo to +oo. The functions K± and K± depend on a € R (let us recall that RQ = exp(aUox)). In particular, in this notation, oo

1

E± = E^

= J2 * ± fa1. • • • .
OO

T=YJK(I\Q\...,Q") p=0

r=+oo

for the solutions E±(x; a) of equation (3.3) constructed in §2.1 and the matrix T(a). Provided that the entries of the derivatives Q\.,... Q? are also absolutely integrable, we define an operation A by the formula: AK^

= K^(Ql

Q\...,QP)

+ ... + K%\Q\...

, Q>-\
T h e l a s t s e r i e s is We set AK^ = 0 and A £ £ l 0 i < 4 p ) = T,™=oAK±}term-wise absolutely and uniformly convergent, if the absolute values of all the entries of Qp and Qp for p = 1,2,... are dominated by an absolutely integrable function independent of p.

P r o p o s i t i o n s . 2 . The entries of the function \UQ,T] decay more rapidly any function | a | ° , a € R when \a\ -»• oo (a € R). Proof. We will apply the following lemma.

than

§3. APPLICATIONS OF THE INVERSE PROBLEM METHOD

201

L e m m a 3 . 3 . a) AT = a[U0, T], b)AE± = a[U0,E±] - QE± = (E±)x. Proof. We check the lemma only for E+ (for E- and for T, the argument is similar). The second equality of b) follows immediately from (3.3). Let us prove the first one. Integration by parts for p ^ 1 gives the following formulae for A, d^f ak^HQ1,...

A, = Kip)(Q\... +

, Q'-1, [UQ, Q'],Q,+1

,Q,-\Q'x,Q'+\...

K{+p-1)(Q\Q2,...

,Q")-

K^iQ1,...

,Q'- ,Q")

An =

2

,Q>)

forKKn,

i4, = « 1 ^ _ 1 ) ( Q 2 , . . . ,Q*)-K(>-l){QxQ\... 1

,Q'Q'+\...

1

^'-'Q',...

1

,...,Q>):

,Q>),

1

R!r \Q ,...,Q'- ,Q>- Q>).

Summing the At for I = 1 , . . . , p , we get the equation (3.14)

AK{p)

= a[U0, K(+p)] - Q'K^iQ2,...,

Replacing Ql = Q in (3.14), we get b).

Q').

•

Estimates (2.4), (2.4a) and (2.4b) in §2 remain valid for KM and K(p) which depend on Q1,... ,QP. Denote the maximum of all the entries of the functions Q1,... ,QP by |g(a:)|. In this general situation, the statement of Lemma2.1 holds, when we substitute the corresponding entry of Qm+1 for g,'m+1 in formula (2.6) and define E%n = E±(Q\... ,QP,...), T« en and ( £ | e n ) ° ° in the same way. In m n m particular, the entries of A E^ and A T « e n for A m = A o • • • o A tend to zero as \a\ —»• oo (a € K) for arbitrary m ^ 1, since they are Fourier transformations of absolutely integrable functions. Applying formula b) repeatedly, we obtain the statement of the proposition. • Following the notation of §2, set n-p

K=

ep+1/\...Ael(x;a)e

f\ p

elA.../\ep_(x;a)

€ /\C\

C\

202

II. BACKLUND TRANSFORMS & INVERSE PROBLEM

where 1 _i p ^ n, e ± is the 5-th column of E±. Let us recall that tp is the p-th principal minor of the matrix T (see §2.2). By Theorem2.1, ejj. and tp are analytically continued to the upper half-plane ( I m a > 0) and tp —> 1, e+ —> F + 1 A . . . l n and e"_ -> l 1 A . . . A lp when \a\ -» oo. Set /P*=^-+1)A...A^*P', for the function $* defined in §2.3, i.e., fp is the exterior product of the columns of the matrix $* corresponding to nP,+i = . . . = nP*. We assume, as before, that the entries of Q and all their derivatives are absolutely integrable and Q' = Q. We also assume that the minors tp* for all p do not have real zeros. P r o p o s i t i o n s . 3 . Suppose that p* = p. Then tp, e^., and fp admit asymptotic expansions, i.e. can be represented by series in a - 1 when | a | —• oo and I m a ^ 0. Proof. It is sufficient to show the existence of asymptotic expansions of e^. for p = 1 and e^. for p = n. Indeed, e?_ is the first column of the matrix function /\p E-, and tp is the first entry of / \ p T (similarly for e+). Therefore the standard reduction argument of the previous section can be applied. Let p = 1 (the case p = n is analogous). If 1* = 1 then pi = Hs for s ^ 1. Hence by Proposition3.2, the absolute values of the entries tJ for 1 < s ^ n decay more rapidly than any | Q | " as \a\ —• oo. Then this also holds for |
T*T = i=»f;w>if{|2 = i, .7=1

where ft = diag(w,), UJ = ± 1 . Let g(a) be a meromorphic function on the upper half-plane which is continuous for I m a ^ 0 . Then g(a) has the form

if g(a) —• 1 when \a\ —> oo. Here we assume that the integral exists. We see that G is the product of the powers of the monomials ( a — a,-) over the zeros and poles aj of the function g. Let us recall that G(a) = G(a). If |gr(a)| = 1 + O^al-"-1) 7-1 as \a\ —• oo, then the integrals J ^ ^ |log|fli(«)||« ||d« exist for j ^ p. Expanding G(a)/G(a) into a power series and approximating (z — a ) - 1 by — a - 1 2 j = o ( z / a ) ' ' for I m a > 0 as \a\ —• oo, we obtain g(a) = J2cja-i

+

o(\a\-P-1),

§3. APPLICATIONS OF THE INVERSE PROBLEM METHOD

203

where ]T^_ 0 CjCt~i is a partial sum of the Taylor series for g in a neighbourhood of \a\ = oo. By continuity, the last formula is also valid for real a. Using the property of |
AeL = a(U0 - m)eL - Qel =

(el)x.

We will show the existence of an asymptotic expansion of eL by induction. Let us suppose that r

1

ei = I +]ry »~s+o(i«rr) 3=1

when a —• oo and I m a ^ 0 for c* = cs(x) € C". For r = 0 this follows from Lemma2.2. We will now construct c r + 1 . It follows from (3.16) (see also (3.14)) that A 2 eL = (a(U0 -m)-

Q)2Sl

-

Qxel,

where A 2 = A o A. Analogously, r

Ar+1eL = ar+1(^0 - m ) r + 1 e L + £>>(•) + (-l)r+1(Q)r+1eL, >=i

where the coefficients (•) of a 1 , . . . , a r depend linearly on the components of the vector function eL and polynomially on the entries of Q, Qx, As A r + 1 e l = o(l) t r+1 when \a\ —> oo (see above), (J7o — A i ) ^ - and, consequently, the components ~el_ k for k > 1 of the vector function e\_ admit expansions of the desired type up to a~T~1. Thus it remains to prove the statement for the first component eL iUsing the fact that E- is *-unitary, for a 6 R: r+1

^ l e i ^ l ^ l e L ^ + ^o;* fc=l

2

= l+0(|a|--2).

Jt=2

Therefore, | e l j / | = 1 + o(|a|~ T ' - 1 ) for a suitable polynomial / ( a - 1 ) . Applying (3.15) to g = eL 1f, we obtain that eL i admits an asymptotic expansion up to the term with a~r~1. Thus the induction step from r to r + 1 is completed. Now we turn to fp.

204

II. BACKLUND TRANSFORMS & INVERSE PROBLEM

The vector function fp{x;a) tions (3-17)

G /yp~p'

C" is uniquely determined by the rela-

~ ", fpAep+=tptpp-p'-1)ep+'.

This follows from formula (2.10*) and relation (2.9). We need the expression for fp in a "sufficiently small" neighbourhood of a = oo (where at least tp and tp, have no zeros). Replacing S T ' ' and tPiPt by their asymptotic expansions obtained above, we get the asymptotic expansion of fp (cf. proof of Theorem2.1, 2.2). • Let p = p* = p« + 1, i.e., fip is an eigenvalue of Uo of multiplicity one. Then by Proposition3.3 tp-\, tp, e^j., e^T and fp = (fi'p = Cp9 have asymptotic expansions as \a\ —> oo. Trace formulae. We keep the previous assumptions and notation. In particular, we assume that p = p*, 1 fs p ^ n. We rewrite relation (3.15) for g = tp in a more explicit form. Let atip,... , arp be the zeros of tp in the upper half-plane and let « i p , . . . , Krp be their multiplicities. Here r = r(p) (which depends on p), and, as above, tp does not have zeros on the real axis. Then in a suitable neighbourhood of a = oo *"* J-oo

z

~

a

f=i(a~

a

ip)JV

Expanding with respect to a - 1 as \a\ —> oo with I m a ^ 0, we get the following formulae

(3.18)

log*,, = ^2^ka~k, .

f00

* = ~Z /

as \a\ -> oo, with I m a ^ 0, , r (p)

° t _ 1 b g | t p ( a ) l d a - if E

Kjp{a

i» - "ip)-

The convergence of the integrals in the expression for c£ is guaranteed by the fact that \\tp\ — 1| < \a\a for any a G R when \a\ —• oo. The last inequality needs to be checked only for p = 1 = 1*, which has been done already (see proof of Proposition3.3). Now we will compute the expansion of log tp in another way. Let us denote the (p* + 1 , . . . ,p)-th component of the vector function fp by fp Recall that fp(x; a) G

§3. APPLICATIONS OF THE INVERSE PROBLEM METHOD

205

/ \ ( p p , ) C n and fp is equal to the minor d e t ( ^ ' , p * < i,j ^ p). Theorem2.2 implies that fp -> tp~p' when x -> - o o and fp ->• tptpp~p'~1) when x ->• +00, where I m a ^ O . Consequently, l o g ( V ; . 1 ) = [°° (log f>)xdx.

(3.19)

By the methods presented in Ch.I we can find an asymptotic expansion for (log / ? ) * . We will review the necessary results. Let * = ($0 + * i a _ 1 H h* 3 a - ' H ).Ro be an invertible formal solution 1 of equation (3.3), * = VR^ . In §1.2, Ch.I, we denoted the minor d e t ( ^ ; p « < hi = P) by m p (W), where $ = (fy), and introduced a formal power series £ p = ( l o g m p ( $ ) ) x = J ^ ^ l j Cjt a _ p - The coefficients f£ do not depend on the choice of a solution $ and are polynomials in the entries of Q, Qx,Qxx, • • • without constant terms (see Theoreml.2, Ch.I). In §1.4, Ch.I, we discussed methods of computing the coefficients of the series (p. In particular, for p = 1 = 1* and Cp = C1 w e have:

^l = y^ 1

iW,

=

_ y * g'g'^»^i

i^K-p-

tt^-^'

P r o p o s i t i o n 3 . 4 . The series C coincides with the asymptotic expansion of the function {logfp)x as \ct\ —• 00. The following relation (trace formula) holds:

f°°

(3.20)

<*-#=/

Cldx,

for 1 ^ p ^ n and cj = 0. Proof. For a fixed formal solution $ of equation (3.3) (see above), set 7?p = V> A . . . A V"* and 7?p = xj)v+1 A . . . A ^ " , where ^ is the j - t h column of
206

II. BACKLUND TRANSFORMS & INVERSE PROBLEM

Moreover, any series in a - 1 satisfying these equations up to scalar factors is proportional to gP. These claims are immediate consequences of the invertibiUty of $ . Thus, by relation (3.17) and the proportionality of the series £%*' and ir±p', the asymptotic series for fp differs from gT only by a factor which is constant with respect to x. Therefore Cp (the logarithmic derivative of the (p, + 1 , . . . ,p)-th component of the series g p ) coincides with the asymptotic expansion of ( l o g / £ ) x . Use (3.18) and (3.19) to complete the proof. • We remind that for the proof of the trace formulae we assumed that the entries of Q and all their derivatives are absolutely integrable with respect to x from x = —oo to +oo. In particular, all the functions ££ are absolutely integrable from —oo to oo. As can be seen from the proofs of Proposition3.3 and 3.4, we can weaken this condition, not requiring integrability of all derivatives of the entries of Q. If -one of the integrands in either side of (3.20) is integrable for certain p, k, then both sides of (3.20) are well-defined and coincide for such p, k. We now look at several applications of the trace formulae. If p = 1 = 1*, then for k = 1 r 1

oo

/

n

( )

log \U I da - 2i V

Kjl

Im an

= V"

«>oo

" *

/

|g?| 2 dr.

Obviously, both sides of (3.21) are purely imaginary (belong to —iR + for fl = / ) . In general, we have also f™ Re (P = 0, which follows directly from (3.20) and the definition of cj (see (3.18)). Compare this result with Corollaryl.2, Ch.I. In §1 of Ch.I in addition to C for Q, — I we studied the densities x (formal series in a or in 57) for the conservation laws of the Pohhneyer type. Note that in Ch.I the series x was defined only for p = 1 = 1* (exactly speaking, for p , + 1 = p = p*). The formulae, theorems obtained there can be extended easily to the general case. We will now compute the integrals corresponding to the series x of Ch.I, and to its variants for arbitrary p = p* using the scattering data. Let £1 = 1 and xp = (log(g p ,g p )) x , where gF is the series introduced in the proof of Proposition3.4 and (,) is the standard hermitian form on / \ ' p _ p * ) C™. In particular, xl = ( l o g ^ 1 * ^1))x for p = 1 = 1* coincides with x °f Ch.I. The components of the vector function fp defined above except f£ (i.e., with the multiindices not equal to the multi-index (p, + 1 , . . . ,p)) tend to zero when x —¥ ±oo and I m a > 0 (Theorem2.2). Therefore oo

/

xpdx = iog(tpt;i) •oo

+

log(tpt;.1),

§3. APPLICATIONS OF THE INVERSE PROBLEM METHOD

207

if we substitute the corresponding series in (3.18) for tp and
/

Cpkdx = 0,

fc

= l,2,...,

r=v °° since t„ = d e t T = 1. One can prove this fact purely algebraically. Using the technique in §1, Ch.I, we can check that £ ) - _ - . Cjt f ° r a n v ^ ' s a n e x a c t derivative of a certain differential polynomial of the entries of Q. This is the unique relation between integrals J_oo Ckdx in general. Corollary3.1. For an arbitrary finite set of indices k and any numbers Ip € iR (p = p*, 1 ^ p ^ n) with the condition 2 _ _ _ . 1% = 0, there exists a function Q(x) (of the above type) such that Ip = f_ (£dx. Proof. If the numbers Ip are sufficiently small, the statement follows directly from (3.20). Indeed, by Corollary2.2 (§2) we can find a matrix function Q(x) which corresponds to any prescribed *-anti-hermitian matrix S'(a) which is sufficiently close to I and satisfies the conditions of Proposition2.6 (§2.5). Multiplying Q(x) by suitable constants c € R+ and changing the variable x to c _ 1 x , we get arbitrary sets {1%} from those close to zero. These {1%} also satisfy the conditions of the corollary. Here we use the homogeneity of ££, which follows from results in §1, Ch.I (details are left to the reader). • Now let the pair Q(x,t), W{x,t) be a stabilizing solution of equation (3.5) or (3.6) (or let the pair U, V be a stabilizing solution of (0.2) or (0.9)). As before, the entries of Q (or, equivalently, the entries of Ux) and all their derivatives with respect to x are absolutely integrable. Then Lemma3.1 implies: Corollary3.2. For arbitrary p = p* and k, the integrals 1% = J^ Qdx do not depend on t and are integrals of motion of corresponding equations. The number of independent series among {Ip, k = 1 , 2 , . . . } is equal to the number of pairwise distinct fip minus one, i.e. all the integrals Ip are functionally independent of each other ifp > 1*. • Using the Hamiltonian structure introduced in §2.5, we can refine the statement of this corollary. Recall that

208

II. BACKLUND TRANSFORMS & INVERSE PROBLEM

for functional / and g of the entries of Q (formula (2.20) for o = - c o and b = oo). If / has the form J_00 Fdx, where F is a polynomial of {qf} and derivatives (<^)(*) = dk(q3i)/dxk of arbitrary orders (i.e., F is a differential polynomial of {
Therefore we can set {I%,I[} = f_00F£fdx, where p = p*, r = r*, k,l ^ 1 and F£'i is a suitable polynomial of the entries of Q. From Proposition2.4, b) it follows that {I%,I[} = 0 for any combination of indices. Consequently, the integrals 1% are not only independent (for p > 1*), but also pairwise involutive. This statement can be made purely algebraic, since the involutivity means that all the Fj^'* are exact derivatives of differential polynomials of {q?} with respect to x. In conclusion, note that relation (3.18) and the trace formulae (3.20) also hold without imposing *-anti-hermitian conditions on Q, W, U and V. The existence of integrals on both sides of (3.20) is sufficient. 3.3. E x a m p l e s of reduction ( 0 „ and S n _ 1 -flelds; n = 2 ) . We will discuss in this section the constraints imposed on the scattering data for orthogonal PCF's and S n - 1 -fields. We also study in more detail the above constructions in the special case n = 2. We keep all conventions from §3.1. Let U(x, t) and V(x, t) be *-anti-hermitian solutions of system (0.2) which are real-valued (i.e., U, V € g[„(R)). This means that U and V belong to a real Lie algebra o(r, n — r), where r is the number of pluses in the matrix Q, denning the anti-involution *. All the previous observations can be applied. Denote the matrix (JJ^-i) by II. The diagonal matrix UQ = diag(^ 7 ) which is equivalent to U is purely imaginary and hence satisfies an additional relation: UQ = IIE/oII = Uo- Indeed, the characteristic polynomial of U is real and its zeros (fij) are pairwise conjugate (with regard to multiplicities). Since —ifij G K are increasing, we get the desired relation for U0- Set A1 = TiSl( '^4)011 and assume that UQ = Q.U.. (This is equivalent to A*1 = A1* for an arbitrary matrix A). Taking the fact that U and V are real into account, let us take the complex conjugates of (3.1) and (3.2A). We can choose a matrix Fo of §3.1 such that F0 — FQU and, hence,

F0< = (3.23)

Vn,

Q = F^(F0)X W = 2F^\F0)t

= =

-Q\ -Wl.

§3. APPLICATIONS OF THE INVERSE PROBLEM METHOD

209

Here, as above, Fo is *-unitary and Q and W are *-anti-hermitian. Hence equations (3.23) are equivalent to the relations

Q = ngn,

w = uwu.

It follows from (3.23) that for any a € C and an arbitrary solution E of equation (3.3) (or systems (3.3), (3.4A)), the relation El = E~x holds identically, which is equivalent to E = HEH, if it is satisfied for a certain point x' (or a pair (x',t')). Here x' may be equal to ±oo. In particular, E± =• n j ? ± n , T — TITR .(recall that £ ( a ) d = £(??)) and El±=Eg1,

(3.24)

T = T~1

for E± and T of §2 (see also §3.1). Let us show that (3.25)

5" = VS'1!?,

(S'Y

=

V(S')-1:D°,

where S(t; a) and S'(t; a) are *-hermitian matrix functions defined above (see Theorems. 1 and its proof) and V = dutftjtj-!

),

V = diagfo. tj.),

l^j^n.

Note that t-j = tj = tj since T is *-unitary (see §2.2). The decomposition T = ( T ° ' ) - 1 ( T ( ) ) _ 1 has the same type as the triangular factorization T — T°To- Hence the calculation of the matrix (T<j) -1 reduces to the caluculation of its diagonal. We get diag(T0')-1=diag(n1roIir1 = diag(*„_i,... ,<„_,-,... , 1) = diag(<j). Here we used the relation t„-j = t-j and formulae t-j = tj derived from (3.24) (see (2.7) of §2), which imply the identities (3.26)

tn-j

= tj,

1 g j g n.

Consequently, (To')" 1 = VT0 and S ' = ((T^'To1)1 S* is proved exactly in the same way. Analogously we can prove the relations (3.27)

$' = P $ - J ,

= VS~XV.

(*•)* = 2 r ( $ * ) - \

The formula for

210

II. BACKLUND TRANSFORMS & INVERSE PROBLEM

where $ = ^ - ( T o ) " 1 and $* = E_(T 0 *) _ 1 (see above and §2). It is necessary to use (3.24) and the formulae for TQ and (T£ ) ' . These relations hold for analytic continuations of $ , $*, T> and T>* in the upper half-plane by the Maximum Principle. The corresponding conditions on the discrete scattering data Kj = $ - 1 ( 0 ! ? ) , where {otj} are the zeros of det $ (Imay > 0), have the following form: (3.28)

Kj = {z£

<9?| (z, VnSUCj).

C Oj } .

Here (p, q), = *pq is the standard scalar product of vectors p,q G C™,(!?^. Really,

where <$ = Z>nfi$ - 1 ftII. (Note that ILV = PET by (3.26).) Analogous statements hold for { £ ; , a ; } . E x e r c i s e 3 . 3 . a) Drop the *-(anti-)hermitian (unitrary) conditions. That is, for L-skew symmetric U and V, show that (3.23) implies relations (3.24) and (3.25) (for V and V' instead ofV and P * — see §2) and (3.26) - (3.28), where U0 is assumed to be purely imaginary as before. Rewrite equations (3.27) and (3.28) for the functions $ and 9* defined on the lower half-plane (see §2.2, 2.3). b) Show that the inverse problem (3.13) (or its variant for the pair $ and $ without the anti-involution * — see §2) reduces to solutions Q, W, U, and V of system (3.5), (0.2) with values in the Lie algebra o(r, n — r) (or its complexiGcation), if relations (3.27) and (3.28) are satisfied (together with their analogues for $ in the complex case) as well as equation (3.23). O Now we will discuss specific properties of the trace formulae of §3.2 for real (and *-anti-hermitian) Q. Prom relations (3.26) it follows that log(W.1)=1°g(<»-P-*n-P.) = log(< ( n _ p + 1 ) .< ( - 1 _ J ) + , ) .) = - l o g ( t ( n _ p + 1 ) . < ( - : l _ p + 1 ) # ) . Consequently, oo

*»oo

/

cin~P+1)'dx = 0.

$'dx+ -oo

J—oo

In particular, if p.p = 0, then p* = (n — p+ 1)* and f Cp dx = 0. We see that for such Q there are exactly [m/2] independent integrals of motion {/',& = 1,2,... }, where m = {p = p*} is the number of pairwise distinct eigenvalues fij, 1 5i j ' ^ n (here, [•] denotes the integer part). The proof is similar to the proof of Corollary3.1, 3.2. We leave the proof of the algebraic variant of formula (3.29) as an exercise for the reader.

§3. APPLICATIONS OF THE INVERSE PROBLEM METHOD

211

5 n _ 1 - f i e l d s . Now let U = 2q A qx, V = 2q A g, and g = 1 - 2Pq (where Pqz = {z,q)q and (a A b)z = (z,a)b - (z,b)a) for an 5 n _ 1 -field q(x,t) € Rn in the normalized coordinates (see Introduction (0.5) and (0.6)). Recall that (q, q) = 1 = (<7x>9x) and the eigenvalues of the matrix U are /xj = — 2i, fi, = 0 for 1 < s < n and / i n = 2i (they are ordered to make — ifij £ TSL increasing). The eigenvectors of U corresponding to the eigenvalues ±2i are proportional to qx ± iq respectively (see §1.5, Ch.I) and the matrices U and V are real skew-symmetric. Therefore we can apply all the previous results (for Q. = I, (•)* = (-) + ). Let us determine the first and the last columns f1 and / " of the matrix FQ defined by relations (3.1), (3.2A) and (3.23). Obviously, fn = u(qx+iq) for u(x,t) € C. As Q' = Q and, in particular, q" = 0 (see (3.1)), we have

(/»)+£ = 0 due to the fact that FQ is unitary. Thus 2ux + u(qx — iq, qxx + iqx) — 0 =>• 2uxu~1

= —2i ,x

=>• u = e~ c,

cx = 0.

Here we use the relations (qx,qxx)

= (q,qx)

=0,

(«,?**) = -(«*,?*) = - 1 where (,) denotes the standard Euclidean scalar product. Now we determine the constant c from (3.2A). Since Vf = u(2iqt — 2(qx,qt)q), we see that u-2(r)+Vr=4i(qx,qt) = 2(n+(F0)tF0-1fnu-2

= 2(/")+(/")4u-2

= {qxt + iqt,qx + iq) + 2ct = 4i(qx,qt)

+ 2ct

(note that we use equation (0.5) and the relation (qt, qt) = 1). Consequently, c< = 0. By the symmetry F0 = i^oll (see (3.23)), we obtain the equation f1 = fn. Thus up to a negligible unimodular constant factor c (cc = 1), fn = ^e-ix(qx f1 =

+ iq),

^=eix(qx-iq).

212

II. BACKLUND TRANSFORMS & INVERSE PROBLEM

It follows from the fact that Fo is unitary and that the other columns of this matrix are orthogonal to f1 and / " , and, hence, orthogonal to q. Therefore F^gFo

T2 = / ,

= exp(tf 0 *)r,

where T = (7/), 7 " = 7* = 1 and 7j = 1 for 1 < j < n. Indeed,

gr

= r-2ie-i*q = e2i*f1,

gf1 = e-2i*fn. We remarked in §1 (see also formula (1-21) of §1, Ch.I) that for an arbitrary solution $ ( a ) of equation (1.1) of §1, the function <7$(1 — a) also satisfies (1.1) for a — (1 — A ) - 1 . Consequently, F^1 gFoE{x\ 1 — a) becomes a solution of equation (3.3) together with E(x; a) (recall that E = F 0 - 1 $ ) . Using the above calculation of F^~1gFo, we obtain the relations (for a € R):

(3 30)

-

TE±(1

- a)T = e x p ( - t / o * ) £ ± ( a ) ,

rr(i-a)r = T(a).

As before, by the uniqueness theorem for differential equations, relation (3.30) can be checked by replacing solutions E± by their asymptotics Ro(x;ct) for x —>• ±00. We set Ay(a) = TA(1 — a)F for matrix-valued functions A of a. In §2 we defined the matrices T ' ° , T0*, f*° and T0* and the functions $* = E+T'° and \P* = E+TQ . The latter two functions are analytically continued to the upper and the lower half-plane respectively. Let us show that the involution 7 maps T*° and T0* to 7£ and T'° respectively, and $* to e x p ( - t / 0 z ) * * - The matrix (T0*)7 is of the same type as T*° (we assume now that / j 2 = . . . = fin-i = 0, 1* = 1, 2* = . . . = (n - 1)* = n - 1 and n* = n). Relations (3.30) and (3.21) imply that t~,(a) = tn-i{a)=

=h(a)

=

ti(l-a)

t „ _ 1 ( a ) = <„_i(l - a ) ,

which results in d i a g ( T 0 T = diag(?r 1 , • • • Jj.\

• • • , 1) = d i a g f ' ° .

Therefore

(T0*)7 = f , 0 =» (T*0)'1' = f0* ^ ( $ * ) 7 = exp(-i7ox)4'V

§3. APPLICATIONS OF THE INVERSE PROBLEM METHOD

213

Using these formulae, the definition of Sm (see (2.15*) of §2.4), and formula (2.12), we obtain the relations

(ST =

(3 32)

exp(0b*)(*T =

VS-1^,

_ (^'+)~11>.

Here V = diag(*,.*,-.) = d i a g ( t i , ( t i ) 2 , . - - , ( i i ) 2 , t i ) - Formula (3.31) and the second relation of (3.32) hold also for Im a 5; 0. All these functions are analytically continued to the lower half-plane (by definition, F(a) = F(a), F+(a) = (F(a))+ for scalar and matrix functions of a € C). Putting (3.32) together with (3.25) and (3.27), we establish the connection between the involution 7 and the anti-involution 1:

(ST = (ST, (3 33)

'

nrs*(i - a)rn = *s'(a) = T(a),

(3.34)

($T+=(*Texp(E/o*), _. I i r $ (1 - a ) r n = exp{-U0x)*'(a).

Applying (3.34) and the relation ti(l - a) = t^a) = h(a) (see (3.31)), we find that the points of the discrete spectrum a j (the zeros of det $* = Y\2=i *p* = (*i) n _ 1 ) and the lattices ICj = ^'~1(OJ) obey the following symmetries: Rea* = | , (3 35)

'

K) = {nrz(Q; - «)| «(« - a;) e JC; }.

Let us remind that K'j is generated by the meromorphic vector series z of the parameter (a — a'). The transformation z(a — a'A >->• z ( a j — a ) is the complex conjugation of the coefficients of z and the change of signs of the local parameter. We leave the converse statement as an exercise to the reader. Show that: by solving the Riemann problem (S.1S) for the hermitian matrix S* with the discrete spectral data { a * , £ J } satisfying (3.83), (3.35), (3.25) and (3.28) and the constraints of Proposition2.6 on S*, we can construct Sn~l-fields of stabilizing type (for which qx and qt tend to the common limits qx(oo,t) and qt(oo,t)). The number of independent local conservation laws considered in §1 reduces much more here than for the orthogonal P C F . Indeed, by (3.29) only the series {££} survives. Moreover, it follows from the formula t j ( l — a ) =
/

*00

?kdz = -OO

Cldx, J—OO

fc

= l,2,....

214

II. BACKLUND TRANSFORMS & INVERSE PROBLEM

This is consistent with the statement of Corollaryl.3 of Ch.I. It is easily shown that (3.36) and the identity J^ Re ('k dx = 0 exhaust all the relations among I'k = f ^

Cjt- Thus Imi^* are independent integrals of the equation of 5 n _ 1 -fields

(0.5) and linearly generate all the integrals {1% }. C a s e n = 2. Set n = 2 and Uo = dia.g(fti,fX2) where p = n\ — fi2 and, for the sake of uniformity, c\ = p i , C2 = //2 and c = fix — pi in the case of the Heisenberg magnet. Let u>i = 1 and u>2 = « , where ft = diag(u;i,u>2) defines the anti-involution *. Let us denote (see §2.5):

(3.37)

0=(_°r ? ) ,

T-(: - f ) ,

5-(J f ) .

Then

(3.38)

= f * = (aeX+be^el)

= (p1, p2) = (ei.wteL + a e i ) .

The functions a and 6 depend on a € R and $ depends on a and a;; a, tp1 and <^2 are analytically continued to the upper half-plane for a. We denote the pairwise distinct zeros of a(a) = d e t $ ( a ) by { a i , . . . , ajv} for Im a > 0. Recall that the aj are interpreted as points of the discrete spectrum of the operator d/dx — iUo<x + Q(x) (see Proposition2.3). As usual, we assume that a does not have real zeros. For the sake of simplicity we assume further that all the zeros of a are simple. Then the appearance of the lattice tCj = ^~1(02A in the discrete data indicates proportionality of the vector functions ip1 and
V1(<*j) = biV2(aj)

*

eL(a,) =

bje2+(aj).

Then Kj = \l,—bj)(a — a y ) _ 1 C + O2. In particular, the coefficients bj do not depend on x. We remind that a does not have zeros in the upper half-plane for w = —1. When u = 1, arbitrary values of { a j , Im a,- > 0} and any bj € C* = C\{0} r.-e allowed (Proposition3.1). If Q depends on t and is a stabilizing solution of system (3.5) with W(x, t) (for the PCF) or for system (3.6) (in particular, the function r(x, t) is absolutely integrable with respect to x on the real axis), then at = 0 and (^j)t = 0 for any j . The matrix W(+oo,t) = W(-oo,t) (for (3.6)) is then diagonal. Let W(±oo,t) =
§3. APPLICATIONS OF THE INVERSE PROBLEM METHOD

215

For simplicity, assume that w>i, 102 and w = W2 — wi do not depend on t. We know from Theorem3.1 that: (3.40A) fe(t; a) = exp(urf(4a - 2) _ 1 )6(0; a) b{t; a ) = exp(-/i 3 a 2 t)6(0; a)

(3.40B)

bj(t) = exp(w<(4a i - 2) _ 1 )6 i (0) exp(-n3a2t)bj(0)

bj(t) =

for (3.5), (3.6) respectively. The Riemann problem (3.13) becomes

(3.41) (3.41a)

l ^ ^ ( \ 3

J

(p =

_6W)=(^*'*'1*)' -¥ V

O G R

'

when I m a ^ 0, | a | —• 00,

and considered together with (3.39) uniquely recovers $(x,t;a), Q(x,t), W(x,t) from the scattering data 6(0; a ) a n d {aj,bj(0)}, 1 ^ j fz N.

and Here


(3.42)

(fi1 = e>- ( ( J ) + (-^- r ;; ( l ' ) < f X ') a"1 + o(«-)) .

In order t o check this formula, we substitute the asymptotic expansion of (fi1 in a neighbourhood of a = 00 up to a~2 (see §3.2) into the equation

<'•«)

*.-("o ; > + ( ° T >

and compare the constant terms and the coefficients of a - 1 ! 1 . We can also find the complete asymptotic expansion of (fi1 (provided that it exists) in this way. Moreover, using the method in Ch.I §1, we can compute
216

II. BACKLUND TRANSFORMS & INVERSE PROBLEM

such a solution r' we need to know (an arbitrary) ao with Im ao > 0 and a constant

68 e C*: a'(a) = a(a)

=-, a — ao

I m a ^ 0,

b'0(t = 0) = 6°. Here a; = 1, b'0 is constructed by r' using ip' (ao) and ip' (ao). We will write bo instead of b0. According to Theorem2.3, in order to find the function $ corresponding to r(x, t), it is necessary to find the hermitian projection P satisfying the relations B-'Ol

= $K0,

B = I-

^ ^ l p a0-a

(see the end of §2.4 and §1.2). Since K0 = %1, -&o)(« ~ <*o)C + 0$, P is the projection onto C ^ , where 10 =

($1)

=V(ao)-W2(ao),

&o =

MO'

(see above). Following the computations in §1.5 for the NS equation (see (1.29)), we obtain: to AA\ (3.44)

i ^ ( a 0 - a0)(fi r =r — = - ,

def/0/,0 fl-bofli \

Now we will check that it has the required discrete spectrum data. The analogous calculation of W from (3.5) is left to the reader. Formula (3.43) coincides with the first equation of (1.28) for a — A, where we substituted the values fij = =Fi for H\ and ^2- Strictly speaking, in accordance with (1.29), we should use the vector functions

= (p* exp(wjt(4a — 2 ) _ 1 ) for (A) (ipJ — (p' exp(fijfj.2a2t) in the case of (B)) and the constant &o(0) instead of (p* and 6o = bo{t). However, the exponential factors cancel. The function $ ' corresponding to r' has the form $ ' = B $ . Therefore a' = det $ ' =

detBdet*=(l-^!°spP)a

\

a0 - a

=

/

a2LZ^L, a-a0

§3. APPLICATIONS OF THE INVERSE PROBLEM METHOD

217

f>'X{aj) = bj b'j = bj (1 g j g N) and b' = 6 by the relations

S' = $' + $' Finally, from the construction of $ ' we have:

Thus r' from (3.44) indeed has the desired discrete spectrum. This solution is called a one-soliton solution in the background of the initial solution r (with the parameters ato and h®). 3.4. Scattering d a t a for certain solutions of t h e N S equation. We will apply the formula obtained above and describe the scattering data of certain solutions of the NS equation. The solutions in question are those which are obtained from multi-soliton solutions by restricting the domain of r(x, 0) to finite or semi-infinite intervals. This problem is of definite interest since in applied physics the models described by the NS equation always deal with functions r(x,t) on a finite interval. Our approach illustrates the inverse scattering method and makes it possible to trace the way the discrete spectrum appears (or vanishes) when r changes. The results below can be easily transfered to the P C F equation (n = 2) and the Sin-Gordon equation. We keep all notation from the end of the previous section and set w = 1. We turn to the associated linear problem (1.28) of §1 which is somewhat more traditional for the NS equation. Let fii — c\ = — i, \ii = c% = +i and p = —2i in equation (3.3) or equation (3.4B). Equation (3.3) remains unchanged, but (3.4B), which determines the ^-dependence, is replaced by the second equation of (1.28), differing a little by the coefficients. Exercise3.4. Check that for Q of (3.38) and ^1,2 = T-h * f l e z e r o curvature representation (1.28) for the NS equation is obtained by setting a = —2A in formulae (3.3) and (3.4B) and conjugating by the

*"-(!i>

matrix

We remind that the compatibility condition of system (1.28) is the NS equation (for ui = + 1 and in the standard normalization) (3.45)

irt = rxx + 2\r\2r.

The evolution of the coefficient b and the discrete scattering data bj (1 5; j ^ N) with respect to t is now given by formulae b(t;a)=eXp(4ia2t)b0(a), bj(t)=eXp(4ia2t)b0j,

218

II. BACKLUND TRANSFORMS & INVERSE PROBLEM

instead of (3.40B). It is convenient to introduce functions c,- = bj/a'(aj), where a' = da/da. The ^-dependence of c,- is given by the same relations (3.46) as for bj with c°j = Cj(t = 0) as the initial values. We begin with the formulae for 7V-soliton solutions of (3.45) and the corresponding ^-functions. They are rather well-known. Such solutions are specified by the following defining conditions: N

6(a) = 0 <** a(a) = TT ^ _ 5 > .

(3.47)

The equivalence is a consequence of the relations a(co) = 1 and \a\2 + \b\2 = 1 (fulfilled on the real axis). Since 6 = 0, the functions ip1 = ip1 exp(iax) and ip* = tp2 exp(—iax) are denned on the entire complex plane by formula (3.41) and are rational with respect to a. For any a € C, we set * > » = (i(a)) for a vector function

a-V(«)e-=(j)+£^> a - V f o V " = ( *1 +

(3.48)

where the vector functions x1 satisfy the following system of linear equations (1 ^ m <; N): N

(3.49)

X

+ 1,

_ { a m _« t ) ( « f c _ Q j .)

=

* -

U eXP(2tQmX)+ W£

i ^

Proof. The existence of expansion (3.48) for x' = &i *
§3. APPLICATIONS OF THE INVERSE PROBLEM METHOD

219

{a,}) and the equation ^ ( o o ) = e l ( o o ) = 1(1,0) (Lemma2.1). The functions %' are expressed by formulae (3.39) in terms of
-rfsMvW - (J) + £?2i!S±V fe ), eXP(

.exp(-^)^(at) = f ° ) + £ V

Cancelling {tp2{aj)*},

;— =1

'

we get (3.49).

-^)5V2(ajr-

act — otj

D

C o r o l l a r y 3 . 3 . The multi-soliton solution of equation (3.45) corresponding the set {aj, bj,l ^j ^ N} is given by the formula

(3-50)

to

Xj = ( *} } ,

r = -2iY,cJxi,

and satisfies the relation N «

x

/

rr{x')dx'

=2iY,Cjx{. _•_»

-OO

In particular, for N = 1, Ai = f + ir;, 6° = exp(2»7Xo + io 6 R and rj > 0,

v

;

v

(3.51') V

'

y

/ 7-00

'

cosh(2»7(x - xo + 4ft))

r r ( x ' ) d x ' = 47?V

^

-^

'

r.

' 1 + c 4i|(xo-«)

Proof. Formula (3.50) and the relation for rr are derived from (3.42) for fi = —2t and the limit of expression (3.48) for a —• 00. We also use the expansion iV

a(a) = 1 + a-1 £ ( " 7 - 3y) + ^ a " 1 ) .

220

II. BACKLUND TRANSFORMS & INVERSE PROBLEM

For the one-soliton solution r we have (see (3.49)): X\ = exp(2ia 1 x) l X\

+

^

^

= ic, e x p ( - 4 ^ ) ( 2 r / ) - 1 ^ + ^

X2 =

X + e 4,(x 0 -r)

^ ^

, ^

^

^

•

= 2tr/(a — a i ) 2 ) , we get the formulae

Substituting ct = 2irjbi (a' = da/da (3.52)

^

Xl =

'

l

+ e 4,( I O -x)

'

As r = -2iciX2, ^ r r ( s ' ) dz' = 4V + 2iclX\

=4^1 -

T

_ _ _

y

j .

It is easy to check that formula (3.51) coincides with formula (3.44) which was obtained using Backlund-Darboux transformations, when Vi = exp(—iax) and I rr(x) dx, "" J—ao

j

=

i

J—oo

which is obtained from (3.21) with n = 2, q\ = - r and ^ i | 2 = =F»- Compare (3.50) and the formula of Exercisel.15 as an exercise. Let r°(x) be a function which is absolutely integrable on the real axis and has the property that its derivative r r is absolutely integrable on R also. For a pair of points — O O 5 J 2 / < Z ^ O O we introduce a truncated function

r?(x)=e(x-y)e(z-x)r°(x), where 0 ( x ) = 0 for x < 0 and 0 ( x ) = 1 for x ^ 0. Using ry°, we will construct the monodromy matrix T*° of equation (3.43), the corresponding coefficients ay and by° (see (3.37)) and vector functions ip10 = eL° and
(3 53)

"



b?(x;a)eia*,



iai

a)e~

.

§3. APPLICATIONS OF THE INVERSE PROBLEM METHOD

221

From these relations we can write a j and by in terms of (fi1 and
(3.54a)

a*_(a) = a(a)+

(3.54b)

6*_(«) =

VC3 C L a-a/

^ -

j

X

l

N

a[a) e-^^i-^-M >=i

a

-

a

'

hold by (3.53) and (3.48). Hence for the calculation of a* and &*, one needs only the following property of T: TZ(a)T<(a)

=T » ,

y<(
where •'v

In the special case N = 1 we obtain: P r o p o s i t i o n 3 . 6 . Let N = 1, Aj = £ + ir), 6? = e2"Xo+i^o, lx = exp(4(x 0 - x)rj). Then for the function r° defined by (3.51) at t — 0, we get the formulae 2iv

a* = 1 &*- = a+ = 1 1

bi°

=

(1 \ I r a — 5i V

1

^ ( l + t.)-1, a — c*i 2i?7 (1 1 r 1 ) " 1 a —a x x 2 i(ai a)xb

-^ -

a — a\

°(i+ixr\

:_ ° ~ ^ ~

if) t a n h 2 ?

( ? ( g ~ ^0)) a — 57i

_ < * - £ + »7 tanh(277(a: a -ai

x0)) '

222

II. BACKLUND TRANSFORMS & INVERSE PROBLEM

a

»

1+

(i + z,)(i + z , ) ( " - 5 i ) ( 2lT)

'*

'

( e 2t-(or-Ol)(*-») _

1

)

j

(l + / f ) ( l + Z , ) ( a - a i ) 2^6°

"' (a-ajKl+Z.Kl+i,) 2"#

V

S

«-«J

•2,(„-al, A , , a - « i \

\1 + lza-aJ-

(a-«,)(!+ /,)(!+/*)

Proof. We apply formulae (3.54) for the \\ and Xi from (3.52) and the relations for T. For example, the identity T+^Tf.^ = diag(o,a) can be used to calculate at and b+ in terms of af. and fef.. • It is easy t o see that af. has a zero in the upper half-plane I m a ^ 0 only when x > xo- Analogously, a + has a zero in the upper half-plane only when x < XQ. This zero is f ± ir) tanh.(2r](x — xo)) respectively. The value x = xo is called the "center" of the soliton r(x,0). The discrete spectrum vanishes when we cut off more than one half of the one-soliton. Exercise3.5. Show that the coefficient a* calculated above for N = 1 has only zeros of the form £ + ipr}, p € R, if any. For z — xo = k = XQ — y (i.e. for the symmetric truncation), p satisfies the equation (« + K _ 1 -2)hp = 2 + h~XK + hit-1, where K = (p — l)/(p + 1) and h = exp(—2rik). In particular, a* has a zero in the upper half-plane only when e 4,) * > 3 + y/8. • Superposition o f "semi-solitons". Interesting effects arise for the sum of two sequential (nonintersecting) semi-soliton functions r. Let R?(x) = rH°(x) + r*,°(x) for a certain fixed x', where r° and r ° are denned by formula (3.31) at t = 0 with the same value of Ai, but with different coefficients b\ = exp(2»7Xo + *Vo) and 6° = exp(2r?xo + iipo)- Assume further that XQ ^ x' ^ xo- We set: I = exp(4(77(xo — x')),

I = exp(4T7(xo — x))

/o_/_ , ~ o_'\\ 12 (ll)\i/2 ' = exp(2j?(x 0 + x 0 - 2x')),

.. _ u = —2ir) oil — a

§3. APPLICATIONS OF THE INVERSE PROBLEM METHOD

223

The coefficients A and B° of the matrix T corresponding t o Ro are written as follows: 4=1 +

=r (u(l + 21 + 11) + u2(l - (iiy^e^0-^))

(i + 0(i + 0

v

,

'

where u = 2ir)/(a — a\). For the zeros of A we get a quadratic equation with respect to u. For simplicity, let us assume that x' — XQ = k — xo — x' and set
A=i+2«{i+*-ri+«\1+;)-£+(r-iy As a first illustration, let us consider the case when e = — 1, i.e., assume that initial one-soliton functions are in opposite phase. We can find two zeros of A by simple calculation: ai,2 = (£ ± rjcosh - (r]k)) + iT)tanh(r]k). The multi-soliton constituent of i?° (i.e., a solution of the NS with the same discrete scattering data as EP, but without continuous scattering data) is the superposition of two one-solitons with different velocities vi = —4£ — 4r;/ cosh^fc) and V2 = —4£ + 4?7/cosh(T/fc), and with the same amplitude. This means, qualitatively, that we obtain a pair of one-solitons of identical shape traveling apart asymptotically. We will study the case of identical phase in more detail. Let e = 1. Then the zeros ai,2 are written in the form: sinh^fc) ± 1 . . • cosn(277Kj For e 2 ''* > 1 + y/2, the coefficent A has two zeros in the upper half-plane. For e2i)k _: i -|- y/2 there is only one zero, a\. Note that the velocities of the resulting one-solitons (from E?) are equal to the velocity v = —4£ of the initial solitons, but they have different amplitudes. If we formally set u = 2ir}(a — Qi) for a € C, then we can consider the above formula for B° also for I m a > 0. Let us substitute a = ai_2 in this formula. We remind that the coefficient b is not analytic in the upper half-plane, generally speaking. However in the present case we can show that B°(0:1,2) = t ° / a + ' ( a i , 2 ) = B°<2- Here B^fi are ^-coefficients of HP defined at the zeros 0:1,2 of the coefficient A by the procedure of the previous section. This follows immediately from formulae (3.39) and (3.53) (substitute x = x'). By a simple calculation we get: <*l,2 = ? + *Aii2T7,

Ai>2 =

.3° 2 = ±exp(i 0).

224

II. BACKLUND TRANSFORMS & INVERSE PROBLEM

Exercise3.6. Show that ro

def B°la ^.'(01,2)

.(lie2**)2 2cosh(27;fc)

.

Construct t i e two-soiiton solution R,(x,t) from the scattering data {0:1,2, C?2} ^ o r Ima2 > 0, using formula (3.50). Check the relation

f

|E| 2 - |_R,|2dx = 4T?(1 - tanh(2»7fc)),

J —C

where i?(x,t) is a solution of (3.45) for which R(x,0) = R°(x). Show that when £ = 0 the function | i l s ( x , i ) | 2 is periodic with respect to t with the period r j = f ( a ! - a?) = g^r c o s h 2 ^ * ) / sinh(2r/fc). • Variations of t h e discrete spectral data (JV = 1). Let r(x,t) be the onesoliton solution given by formula (3.51). We compute the variations 8a\ =a\ — a\ and 5b\ = b\ — b\ in the case of an arbitrary (small) variation 5r° = Sr°(x) of the initial value r° = r(x, 0). Here Si is the zero of the coefficient a of the function r° = r° 4- Sr° and b° is given by formula (3.39) for r° (a = S i , t = 0). We assume that r° tends rapidly to zero when x —> ±00 as before, but maybe not to a pure one-soliton function. Combining the results of Proposition2.4 with formulae (3.53), we get the following general identities: (3.55a)

|

(3.55b)

| ^ |

-al(a)bt0(a)exp(-2iax),

^=

= -b*_°(a)a+(a) exp(2mx),

,T+0,

x def \

which hold for I m a ^ 0 (bx (a) = &J°(a)). We also take the relation Sa(ai) + da/6V*(ai )(k*i = 0 into account. Replacing af., r+o bx , a+ and &1° by their expressions for the one-soliton function r° (Proposition3.6), we obtain the formula (3-56)

8ai =

iv

f exp(-^o-2^-2,(x-x 0 )) fr0(x)dj J-ao 2cosh (2rj(x — XQ))

+ ir, r J-oo

exP(^o + p

+ 2??(x-xo))^^d:c

2 cosh (2r/(x — XQ))

§3. APPLICATIONS OF THE INVERSE PROBLEM METHOD

225

The value ai + Sai is consistent with exact relations for the zero of the coefficient azy in the limit when y = — k and z = k as k -¥ oo. Analogously, but after a longer computation we can establish the following formulae: (3.57a) y_oo2cosh(2r?(x-x0))

r

(3.57b) 7_00V2cosh(27?(x-xo))

' cosh2(2r7(x-x0))y

A * = exp(—2i£x — i<po)5r°(x)±exp(2i£x

r

+ i<po)5r°(x).

Here SXQ = xo — xo, Sfo = ipo — ¥o, xo and ipo are not derived from 6°, but from the coefficient c^, since c appears in formulae (3.50) and (3.51). More precisely, it is necessary to set c? = 2i»7exp(2?7Xo + i
6 ? (Si) 2j(ai)S'(a?i)'

, 95. a = da'

where the value of Si = a\ +6an computed above, the derivative of a at that point, &o and a t (defined by the suitable analogues of formulae (3.55)) are substituted. The point £ S R is chosen arbitrarily. Here we apply Proposition3.6 at full potential. Indeed, to compute the variation of at in (3.55), it is necessary to replace the index "—" by £ (analogously for b_ ). Therefore we must use the formula for a* and h* E x e r c i s e 3 . 7 . Deduce relations (3.57) by the above method. In particular, the following identities resulting from Proposition2.5: SJ"J

= a!.(a)a+(a)exp(-2iax),

| j ^

= -4i(a)&+(a)exp(2uM),

check

and similar formulae for b*° in which the indices "—" and "+" are changed by y and z respectively. (Use the fact that the expression for 3^ is independent of C and let £ tend to 00 in the final formula for Sc°.) •

226

II. BACKLUND TRANSFORMS & INVERSE PROBLEM

3.5. Application: D N S equation. Consider the following generalization of the NS equation which is of practical importance: irt = rxx — iui(rr2)x,

(3.58)

u) = ± 1 ,

called the derivative nonlinear Schrodinger equation (DNS). This is also a soliton equation and the associated Unear problem is close to (3.43). However it has new features and will illustrate better some of the general results of §2. In this section we apply the inverse problem method to (3.58). Before this, let us show how the NS arises as a limit of the DNS. Exercise3.8. Let r be a solution of (3.58). Then for (J > 0 t i e function r'(x,t)

J^eit}-2t-if>-1*r(x-2f3-1t,t)

=

satisfies t i e equation ir't = r'xx + W ( r ' ) 2 -

2iu>(r'(r')2),

which degenerates to the NS when /3 —>• oo. Equation (3.58) is the compatibility condition of the following two equations for the invertible matrix function $ .

**=(-;f ^ " C r '

(3.59a) (3.59b)

ii<eM - i -- (i tw

2

. ^ r 2+ C^- u , _C

2

r .r

^ ^

2iui(3r + u>Crx + i£rr2 \

-2C4 +u(2rr

J'

Though the dependence on the "spectral parameter'' £ is complicated, this greatly resembles the zero curvature representation (1.28) from §1 for the NS equation. In order to apply the results we already know (without developing the scattering theory with quadratic dependence on the parameter from scratch), we will use the parameter a = f2 instead of C,. Let us fix notation: H-(x) = - -

rr(x')dx',

Note that n does not depend on t.

n+(x) = - |

/

rf{x')dx',

n =

n-+/J.+.

§3. APPLICATIONS OF THE INVERSE PROBLEM METHOD

L e m m a 3 . 4 . Set$=(1Q

°_1 J $ ( * ° \

(3.60,

* , * - = (-•«

227

where $ is defined by (3.59).

Then

"£•).

Analogously, only a appears in the expression for $ < $ _ 1 . For this $ , if we introduce $ r e d from the relation

*<*->-=(; ? ) ( " * t W ) «p<-t-w))'

no.) then (3.60b)

$«d($redj-i

=

f

e x p ( - 2 i > _ ) ( r 2 r / 4 - »wr x /2) \

-*'a

n Following the procedure given in §2, we can construct solutions E±(x; £) of equation (3.59a) for a = f2 € R normalized as follows: E±(x\ () -> exp ( „ y U

. 1 iax J

when x —> ±oo.

We assume that the functions r and r x are absolutely integrable with respect to x on the real axis. Let us introduce the matrix T(£) by the usual relation E-

= E+T.

T = I _

It is more convenient to use E± = I _

._j I T I _

/._x

I E± I

. 1 and

. 1 rather than E± and T (see Lemma3.4). These ma-

trices E± and T are the S±-functions and the monodromy matrix for equation (3.60). These functions depend directly on a € R. The matrix T has the form T = \~

\b

I for suitable functions 5 ( a ) and &(£)

a J

(a £ R). Indeed, T is *-unitary and d e t T = 1, since $ x $ _ 1 of (3.59a) is *-antihermitian and S p $ x $ _ 1 = 0. Let us recall that the anti-involution * is given by the formula A* = QA+$l, £1 = diag(l,w). Set (3.61)

(J - f " ) ,

*W_,

S=e\,

* = (V\S),

228

II. BACKLUND TRANSFORMS & INVERSE PROBLEM

Here a = a and b = C-1& by the definition of T and T. The functions a, b and $ depend on o 6 R. We have the relations (3.61')

ip1 = ae\ +be\,

aa + uabb = l,


+

aet.

Let
for ip = ' (
as above. P r o p o s i t i o n s . 7 . a) The functions a, ip1 and (p2 are continuous and analyticallycontinued with respect to a in the upper half-plane I m a > 0. When \a\ -> oo and Ima ^ 0 ,

w ««> - «*• •-«• ( t -L) - (""t-' **5»3)/s) • b) For a € R the functions y 1 and y>2 are reiated by:

(3.63)

^ . ^ ( i _i-0 = (5 a-0(^.^)-

Proof. By considering $ r e d , we establish the analyticity and asymptotic behaviour of the functions a and y?1. Indeed, the functions (£red)_ =f(£L)red, (Ered)+

d

= (E+Yed diag(e*", c"'")

have the same normalizations as i?±, and satisfy (3.60b). Correspondingly, we have a = ated exp(i/z) and 6 = 6 red exp(—i/i). We then apply theorems from §2 to equation (3.60b) to prove statement a). Furthermore, the functions (p1 = e l and (p2 = e~+ satisfy relation (3.41), in which b is replaced by b = 6f and we substitute b — Q> for b. Thus in order to prove relation (3.63), it suffices to conjugate by

(1 ?)• ° Analogous to (3.39), we set (3.64)


b; € C,

l^j^N

§3. APPLICATIONS OF THE INVERSE PROBLEM METHOD

229

at the zeros o i , . . . , ON of the coefficient a in the upper half-plane (we assume that a ^ O o n the real axis). Then the function r is uniquely determined by the scattering data b(a), a € K, {ctj,bj}. Let us discuss this in detail. The right hand side of (3.59a) vanishes when £ = 0. Consequently, E±(£ = 0) = I and (3.65)

a(0) = 1,

$ ( a = 0) =

1

G ,; •

where (•) denotes a certain function of x. The asymptotics

when \a\ —>• oo (see (3.62)), the normalization condition (3.65), and relation (3.64) make the Riemann problem (3.63) uniquely solvable for known a, b and {ctj,/3j}. However, the coefficient a can be recovered from its zeros and the value of the function \a\2 = 1 — wa|&| 2 on the real axis because of the normalization a(0) = 1. Therefore a is uniquely determined by 6 and {a,-}. The function r can be found using $ from the corresponding differential equation. In the case of the DNS (as in the case of the NS equation), it is convenient to use (3.60) in a neighbourhood of a = oo. When \a\ —¥ oo and I m a ^ 0, we have the following relations: (3.66a)

(filexp(-iax)

->• iwre**'+,

(3.66b)

aip\ exp(iax)

—>

-re,ft~.

The first of these is included in formula (3.62), while the second is obtained by substituting (a part of) the asymptotic expansion of tp1 exp(iax) for \a\ —> oo into (3.60). We can also derive b) from a), using the equality d e t $ ( a = oo) = e1'1 and the fact that $ is *-unitary in a neighbourhood of a = oo. Note that for the first time in this chapter, it turned out that it is useful to normalize $ at the two points a = 0, oo. The DNS equation can be also regarded as a special case of soliton equations for which the associated linear problem depends on the spectral parameter polynomially (go back to $, f from $ , a). The ^-functions of such equations are analytic in sectors of the complex plane (not necessarily in half-planes) and have more complicated asymptotic expansions for \a\ —>• oo (see (3.62)). The dependence of the scattering data on t is determined in the standard way. It suffices to study the behaviour o f $ ( $ _ 1 at x -+ ±oo. Let us suppose that r(x,t)

230

II. BACKLUND TRANSFORMS & INVERSE PROBLEM

satisfies (3.56) and that r, rx and rxx are absolutely integrable with respect to x on the real axis. Then at = 0, («>)< = 0 and (3.67)

exp(4ia2t)b°{a),

b{t; a) =

bj(t) = exp(4ia**)*°,

i S i ^ ^ ,

i.e., they satisfy relation (3.46) as they do in the case of the NS. Analogously d

= bja'ictj)-1

Cj(t)

where a' =

= exp(4ia3t)cg,

da/da.

iV-soliton solutions. Let us construct solutions of (3.58) for which N

b(a) = 0 <=> a(a) = e x p ( i » TT

=*•, a —a

7=1

Ira a j > 0 (cf. (3.47)). As in the previous section, (p1 = ip1 exp(iax) and
(3.68)

a-V(«)e
J=1

where t i e vector functions x' — (

"

a

Xj =
7

j; ) = d i a g ^ w - 1 ) * ' 7

an

^ « = e*M+ are de-

fined by the following system of equations: N

h f i o l

(3.69a)

N

, V ^ ~ r™ J V^ c * e x p ( 2 t ( a m - a * ) s ) wxi + > Q m C; t Xi = n"m > \ c*exj = , ak - a m

N

(3.69b)

x

? +

w

£

^ ^ X J = exp(2ia m x),

*,7=1

where

of the form:

_ „ m def e t c , exp(2i(a m - a Q z ) kJ ( a m - ak)(ak - aj)

§3. APPLICATIONS OF THE INVERSE PROBLEM METHOD

231

(cf. (3.49)), and N

The corresponding solution r of the DNS is given by the formula (cf. (3.50))

(3.70)

Proof. This is similar to the proofs of Proposition3.5 and Corollary3.3. Using the expansion (3.68) and conjugating by * at the points Q i , . . . , ON, « I , • • • , O N , we get system (3.69). We use \> instead of x? t ° eliminate e , ''+, that is also an unknown function and is to be computed via {acj, CJ}. The expression for u in terms of { x i } can be found by substituting the value a = 0 in (3.68) and using formula (3.65). Finally, (3.70) follows immediately from relation (3.66b) (recall that a(oo) = e'*1, Now we write down the formtda for the one-soliton solution of the DNS. Let N = 1 and ai = £ + irj. We set hereafter ?7 = A 2 s i n 7 ,

f = u>A 2 cos7,

c° = 27jA _1 exp(27yzo + 2ia0),

where A € 'R+, and the angle 7 is chosen such that 0 < 7 < 7r. Note that ai = iA 2 exp(iw(7~7r/2)) = u;A 2 e , u ' 7 . As always in this section, u> = ± 1 is from equation (3.58). Since a{a) = -• ~ °* e'" and a(0) = 1, we have eifl = Q j / a j = e~2i^. a — ai 6 = T)(x-x0+

4wtA2 cos7),

Let

cr = £x + a0 + 2iA 4 cos(27).

Using Proposition3.8, we get (3-71)

X 2 = , 2. , , . ^ . ^ . 4^x . „ .

4r/ + uiaic1cie- i

'

Xi =

i x

(3.72)

= P'^

=

u4r)2 •+ "a i c i c i e - 4 ' 1 ' e4* + e ' ^ e49 + eiu^

u}-ra1e ^ °-^ w + Qie 4 "'^ 0 - 1 )

Substituting \\i " 2 and the values of all the parameters into (3.70) and taking the evolution relation (3.67) into account, we get the formula: (3.73)

r(x,t)

= -4iAsin7

e29+2ia

u

A9 + e' t

e40+

e

4S

e-iu-t

+ e""T

232

II. BACKLUND TRANSFORMS & INVERSE PROBLEM

We note that by (3.72), we have the relation 1 f°° —/ rr(x) dx = —ufi = 2~f,

(3.74)

*

J-oo

resulting in i

(3.74a)

/.oo

N

- / rr{x)dx=2Y,li J -°° ;=1

for an arbitrary iV-soliton solution r given by formula (3.70). Here fj = axgctj for w = 1 and ~fj = 7r — a r g a j when w = — 1 (0 < a r g a j < 7r). For Xj with pairwise distinct £,- = Rea,-, the last formula follows directly from (3.74). Indeed, asymptotically for large t, the function r is a sum of one-soliton functions r,- which correspond to each zero ctj for 1 ^ j ; ^ N. The distances between any two centers of Tj can be arbitrarily large. The point XQ = 4££ is called the center of the one-soliton function r(x, t) in the form of (3.73). Using (3.74) and the fact that /_f° rr(x) dx is independent of t, we get the desired identity. By continuity, we can check it for any {£j}. More rigorously, we can use the relation a(0) = e'M Ylj=i(aj/<*j) ~ 1> w hicb holds for any iV-sohton r. This implies (3.74a) modulo 2ir. It follows from identity (3.74) that the "energy" | J l ^ , »*r(s) dx of a one-soliton solution of the DNS does not exceed 2TT. This is not the case for the NS, as onesoliton solutions of this equation may have any energy by the formula

If

rr(x) dx = 2r]

2J-<

(see Corollary3.3). We can explain this qualitatively as follows. As the DNS equation has a higher order of non-linearity, its solutions which have a stable shape must be relatively narrower and steeper than such solutions of the NS equation. The other important difference between multi-soliton solutions of the DNS and those of the NS is the appearance of much more complicated phase modulations. The next exercise reflects this. Exercise3.9. a) Let r(x) = ®{x — y)Q(z — x)R, where R e C* is a certain constant, &(x) = 1 for x ^ 0 and Q(x) = 0 for x < 0. By integrating equation def

—

(3.60b), show that for such a function r and M = CJRR have the form a= 6=

ia(z y) e

-

(

ci

\ rrrr7T~\

the coefficients a and b

sm a 2

( ( - y)\A + M/a)

cos(a(z - y) y/l + Ma)

\ fte-ia(z+y) "—A , ,r/

icty/l + M/a

\

V1 + sln

(<*(z ~ »)V1 + M/a),

=

M a

l

'-

§3. APPLICATIONS OF THE INVERSE PROBLEM METHOD

233

b) For the same "step-like" r, but for the coefficients a and b given in §3.4 for the case of the NS, check the formulae a =

eia(z~

Vn + M

\

Re-ic(z+y)

iy/a2 + M

J

sin((z — y) v a 2 + M).

Show that in the upper half-plane, the number of zeros and the imaginary parts of the zeros of a for the NS (u> = 1) for rather small M grow almost uniformly relative to M. This is not true for the coefficient a of the step-like function in the case of the DNS (see a)). Check this by numerical experiments. O C o m p u t a t i o n of 8a\ for one-soliton. Using the method developed in the previous section let us find the variation Sa\ for the one-soliton function r°(x) of the DNS given by formula (3.74) at t = 0. We have the relations: (3.75a)

-P^l = or(x)

-cjaal{a)b^°{a)exp(-2iax),

(3.75b)

^ 4 = -wa6!°(a)a+(a)exp(2mx), or(x) Here, the coefficients a* and 6*° are defined as in the previous section for

rf

=Q(z-x)e(x-y)r°(x).

These identities follow from (3.55). Indeed, the linear problem (3.59a) is obtained from (3.43) for ^1,2 = T* by replacing r and r by r£ and fC, respectively. Consequently, for the DNS the right hand sides of (3.75) coincide with the right hand sides of formulae (3.55) after multiplying by w£ and substituting b = C,b for b. The appearance of the coefficient u> is connected with the fact that identities (3.55) were written under the assumption that w = 1. The coefficients ax_ and b1? are connected with (p1 by the usual relations (cf. (3.53)) ax_ exp(—iax) =
,3.76.) (3.76b)

*-*-»(•-(A) bx_ =

ei/i+ip+-2iax

(a-ai)

A,+£,^.). 2i T 29

2Ane ' ~

A2+ua1e-4e'

234

II. BACKLUND TRANSFORMS fc INVERSE PROBLEM

Using the multiplicative property of the T, we can derive the following formulae from (3.76): (3.77a)

V (a-a1)o;A2 + a1e-4V' _ e i/i+-2«o,r 2Arje2itr-29 b± = 2 ( a - a i ) A +wa1e-4*'

(3.77b)

Substituting (3.76) and (3.77) into (3.75), and then into the relation 2iA2sin7 Sai=

-eM-2i»l)

f

(ai)

'

we get the identities A 3 sin 7 e-W+i") e-2i»f (e29 + e -2«-<«7)2 '

Sea _ Sr(x) ~ fot Sr(x) ~

A 3 sin 7 eW+J°) e-2iu~t (e29 + e-29+iui)2

'

The following form is more convenient <JA2 _ Sr{x) ~

ujA 2 sin7(e-' u 'T f + 2 ( | i l -'' r ) - e '"-r- 2 ('+">) (e29 + e- 2 »e-'"T) 2 '

<5r(x) ~

m s m 7 e

2

<*A ...7*A»~ Jr(x) ~ (<Sr(x) >'

(e2» + e -2ff e -iu, 7 )2 '

*7

_7~^7~T

Sr{x) 7 ( <Jr(x)''

Exercise3.10. a) Show that the zeros of the coefficients at and a+ (from formulae (3.76a) and (3.77a)) are equal to f + ipr) and £ — ipr] respectively, where p = (A 2 - w a i e ~ 4 ' ) / ( A 2 + w a 1 e - 4 * ) . Check that, as in the case of the NS, at (or a+) has a zero in the upper half-plane, if and only if x > XQ (or x < xo for a+). b) Derive formulae for the coefficients a* and 6J° from (3.76) and (3.77) (cf. Proposition3.6) and formulae for the coefficients a and b in the case of the "superposition" of nonintersecting semi-solitons (see §3.4). •

§3. APPLICATIONS OF THE INVERSE PROBLEM METHOD

235

T h e trace formulae. Let us briefly describe the relationship between local integrals of motion (3.58) and the scattering data. We assume that all derivatives of r with respect to x are absolutely integrable on the real axis. Following the methods of §1, Ch.I, we construct a formal invertible solution, $ = $ exp ( a (

. J x J,

of equation (3.60a), where $ is a formal matrix power series in a - 1 . Set N

C = (io g ^) I = ^ o « - J i=i Then the series ( does not depend on the choice of $ and its coefficients are expressed as differential polynomials of r and r. Let us compute C by the method of Propositionl.3 of Ch.I (§1.4). Here n = 2. We set li = - e x p ( - 2 i / / _ ) ( r 2 r / 4 -

q\ = —rexp(2in-),

iwrx/2).

In particular, by formula (1.18), f

q\q\

111£

2 i V (rr) • /

-

c1 = ^3^ = 2-r

,

+

u>

— •—

2rri

(recall that m<2 = ±i for the DNS). The right hand side of equation (3.60b) is not *-anti-hermitian. However, the results of §3.3 (especially Lemma3.3 and Propsosition3.3) can be easily proved in this more general situation. For Lemma3.3, it is almost obvious. For the proof of Proposition3.3, we must use the relation |a r e d | 2 + au;|6 r e d | 2 = l and the analogous property of the matrix (i? r e d )± instead of the fact that E± and T are *-unitary (see the proof of Proposition3.7). These properties hold since E± are •-unitary. We remind that a red = aexp(—ifi) and 6 red = 6exp(i/j). In particular, we obtain that log \a\ decreases more rapidly than any power of \a\ when \a\ —> oo, and the vector function