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This development led to the formulation of the extremely important
method of inverse scattering theory.
4
In general outline it reduces to tracing the evolution
in t of the space of solutions of the linear problem Lty = l\£ ( A a constant) in terms of the "scattering data." The scattering data consist of the discrete part of the spectrum of L , the normalization constant of the eigenfunctions, and also the scattering matrix for the (rapidly decreasing) potential u from the continuous part of the spectrum.
For further
details see the survey of Faddeev [23], the extensive literature, and also Sec. 4 of Chap. 4. 0.7. The Variational Formalism and the Hamiltonian Property. x
equation can also be written in the form „ _J?
The Korteweg-rle Vrles
where T T = 7 (~^)'^!T17T
is the
Euler—Lagrange operator or the variational derivative. This is a somewhat unusual Hamiltonian form: the standard form of the equation of a Hamiltonian evolution for a vector-valued function a of dimension In is /i, = i Mi
aw
.
.
&H
Pn !——
•'fiW
. where E is the (ftX«) identity matrix, // = fiff
''
n (u)") is the Hamiltonian, and — — = ,-?—,-•.,;—.. Various aspects of the Hamiltonian property were investigated in the important work of Gardner [36], Lax [43], and Zakharov and Faddeev [12]; it was shown, in particular, that the conservation laws commute in the sense of the Hamiltonian formalism. M, = - J ^ - —
Equations of the form
, where T,-{-Xx — 0 is some conservation law for ut = 6uux — uxxx,
have received the
name of higher Korteweg—de Vries equations. The solutions described in part 0.4 were characterized invariantly as flows induced by the Korteweg-de Vries equation on the stationary manifolds of conserved quantities defined by the ordinary differential equation -4—=0. 6u There are other equations. Practically all the effects described above for the Korteweg— de Vries equation were subsequently found for a large number of physically interesting equations including the sine-Gordon equation, the nonlinear Schrbdinger equation, the equations for self-induced transparency, etc. For other "suspect" equations part of the properties have been verified sometimes by numerical experiment. 0.8.
The Plan of the Article.
In the vast majority of studies pertaining to the
Korteweg—de Vries equation and its analogues, a substantial role is played by a system of purely algebraic structures connected with these equations which do not depend on assumptions of analytic character, the choice of function spaces, existence and uniqueness theorems, etc. The principal aim of this article, as previously mentioned, consists in displaying and systematically expounding the origins of the theory of these structures. This objective has determined the choice of the material as well as the order in which it is introduced. The first chapter is devoted to the foundations of the variational calculus with higher order derivatives which is necessary for the natural introduction of the conservation laws and the Hamiltonian structure. Here an attempt is made to follow the invariant interpretation of the variational calculus in terms of differential forms and vector fields on spaces of jets withoutwhich the formulas, which become more complex with increasing order of the derivatives, are hard to interpret and work with in a practical manner. Special attention is focussed on the basic facts of the Hamiltonian and Lagrangian formalisms. 5
In the second chapter a detailed study i s made of the structure of general Lax equations as well as of an enigmatic system of wave equations of Benney which displays many of the features of the systems described above but so far does not f i t into the general theory.
We
consider i t an i n t e r e s t i n g object for future i n v e s t i g a t i o n s . The t h i r d chapter i s devoted to the Lax equations of multisoliton and quasiperiodic type.
Here we have also strived to display in the clearest possible way the mechanism of the
appearance of the algebrogeometric structures in the theory of the equations without writing out e x p l i c i t formulas for t h e i r solutions which i s done in a number of other surveys and i s b r i e f l y considered in Sec. 6 of Chap. 4.
Exceptions are the "solitons of higher rank" which
are here obtained by algebrogeometric methods for the f i r s t time.
One common feature of a l l
problems solved should be emphasized: the introduction of an auxiliary fiber bundle over an algebraic manifold and the i n t e r p r e t a t i o n of the equation as the problem of finding a connection in t h i s bundle with certain additional p r o p e r t i e s .
Recently the problem of "instanton"
solutions of the Yang-Mills equations (more p r e c i s e l y , the duality equations) in Euclidean field theory with the group SU(2) has been reformulated and advanced in t h i s manner (Atiyah following preliminary work of Penrose, t ' H o o f t , A. S. Shvarts, Polyakov, and o t h e r s ) .
It
3
reduces to the c l a s s i f i c a t i o n of two-dimensional complex vector bundles over ^ (C) , which are t r i v i a l on a certain class of lines in P3 . F i n a l l y , facts regarding p a r t i c u l a r i n t e r e s t i n g constructions and methods which have not yet been s u f f i c i e n t l y thought out or subjected to systematization are collected in the fourth chapter.
The exposition here follows the sources cited in the corresponding s e c t i o n s ;
proofs for the most part are omitted.
The only exception which deserves mention i s thebeginning
of Sec. 3 of Chap. 4 where an attempt i s made (not e n t i r e l y successful) to invariantly define a very i n t e r e s t i n g Lie algebra introduced by Estabrook and Wahlquist in connection with t h e i r theory of "prolongation s t r u c t u r e s " and generalized conservation laws. The reader should not take the scattered references and credit of authorship for various r e s u l t s too s e r i o u s l y . Many similar works were done almost simultaneously and almost i n dependently; many approaches revealed a parallelism unknown to t h e i r authors; many ideas hung in the a i r and continued to hang in the a i r some time after formal f i r s t publications.
The
history of our question, i f i t deserves such, remains to be w r i t t e n . In s p i t e of the length of the a r t i c l e , many i n t e r e s t i n g facts have remained beyond i t s scope. F i r s t of a l l , the analytic theory of the method of inverse s c a t t e r i n g has been omitted completely in s p i t e of i t s importance and the fact that i t motivated the inception of many of the purely algebraic constructs described here. Regarding t h i s question, the reader may find abundant information in the l i t e r a t u r e c i t e d . Secondly, very l i t t l e attention i s devoted to specific solutions of p a r t i c u l a r equations or to t h e i r physical i n t e r p r e t a t i o n . Third, we have l e f t untouched the i n t e r e s t i n g p a r a l l e l theory of discrete systems such as the "Toda l a t t i c e " and such of i t s principal applications as the explanation of the Fermi—Pasta-Ulam paradox. Fourth, i n t e r e s t i n g investigations of flows of Lax type in finite-dimensional Lie algebras and the many-particle problems related to such algebras have been omitted. The 6
informed r e a d e r w i l l p r o b a b l y d i s c o v e r s t i l l more omissions v o l u n t a r y or i n v o l u n t a r y . Among t h e r e s u l t s n o t c o n t a i n e d i n t h i s work b u t n a t u r a l l y r e l a t e d t o i t mention should be made of the i n v e s t i g a t i o n of B o g o y a v l e n s k i i and Novikov [1] and of G e l ' f a n d and D i k i i [3] on r e s t r i c t i n g Hamiltonian flows t o s t a t i o n a r y manifolds of c o n s e r v a t i o n l a w s .
They a r e of b a s i c im-
p o r t a n c e for u n d e r s t a n d i n g t h e r e l a t i o n between s o l i t o n s and c o n s e r v a t i o n laws and m e r i t generalization to the multidimensional case. We have n o t endeavored t o compile a complete b i b l i o g r a p h y .
In p l a c e of t h i s the b i b l i o g -
raphy i n c l u d e s surveys w i t h l a r g e b i b l i o g r a p h i e s and c o l l e c t i o n s devoted to s p e c i f i c of the t h e o r y [ 2 , 8 , 2 3 , 2 6 , 3 3 , 4 4 , 4 7 , 5 0 , 5 3 ] .
aspects
Beyond t h i s , p a p e r s having a d i r e c t
rela-
t i o n to the q u e s t i o n touched on h e r e have been s e l e c t e d as have s e v e r a l works to which we do n o t r e f e r b u t which, i n our o p i n i o n , deserve s p e c i a l 0.9.
attention.
I t i s i m p o s s i b l e t o o v e r e s t i m a t e t h e r o l e t h a t t h e a u t h o r ' s many c o n v e r s a t i o n s
w i t h I . M. G e l ' f a n d and a l s o the work of G e l ' f a n d and D i k i i [2-5] p l a y e d i n the design and plan of t h i s p a p e r . The s e l e c t i o n of m a t e r i a l f o r the p a p e r was made d u r i n g a s p e c i a l course which t h e a u t h o r gave i n the mechanics and mathematics department of Moscow S t a t e U n i v e r s i t y i n 1975/ 1976 and from t h e i n t r o d u c t i o n t o a seminar i n 1976/1977.
The p a r t i c i p a n t s i n t h e c o u r s e and
seminar p r o v i d e d t h e a u t h o r w i t h a g r e a t d e a l of m a t e r i a l which d i r e c t l y o r i n d i r e c t l y
af-
f e c t e d the c o n t e n t of t h e p a p e r . In p a r t i c u l a r , a l a r g e p a r t of t h e new r e s u l t s of Chap. 1 belong t o B. A. Kupershmidt; t h e i r p r e s e n t a t i o n i s based on h i s p u b l i s h e d p a p e r s and n o t e s which were k i n d l y given t o the author before t h e i r p u b l i c a t i o n . c a r r i e d out j o i n t l y by B. A.
The i n v e s t i g a t i o n of Benney's e q u a t i o n s i n Chap. 2 was
Kupershmidt and t h e a u t h o r .
The a l g e b r a i c reworking of the
Gel'fand—Dikii t h e o r y i n Chap. 2 d e r i v e s from a r e p o r t of M. S. Shubin i n t h e seminar i n which t h e s i m p l i c i t y of t h e formalism of p s e u d o d i f f e r e n t i a l o p e r a t o r s over a one-dimensional b a s e was r e v e a l e d .
The r o l e of bimodules and c o n n e c t i o n s to which Chap. 3 i s devoted was
c l a r i f i e d by V. G. D r i n f e l ' d .
From t h e r e p o r t of S. I . G e l ' f a n d t h e a u t h o r f i r s t
understood
t h e t e c h n i q u e of Estabrook and W a h l q u i s t , w h i l e t h e e x p o s i t i o n of the r e s u l t s of Lax in Sec. 5 , Chap. 4 i s based on t h e n o t e s of I . Ya. Dorfman.
F i n a l l y , c o n v e r s a t i o n s w i t h B. A. Kuper-
s h m i d t , M. A. Shubin, V. G. D r i n f e l ' d , V. E. Zakharov, I . Ya. Dorfman, and D. R. Lebedev were v e r y u s e f u l t o the a u t h o r .
I am happy t o e x p r e s s my d e e p e s t g r a t i t u d e t o them a l l . CHAPTER I THE VARIATIONAL FORMALISM
1.
Differential Equations: Three Languages 1.1.
The Classical Language.
In this language we first of all choose a notation for
the independent variables, say, xlt...,xm
, and for the unknown functions, say itt, ...,#„. m
Let £ = (&i,..., £m) where kt>0 KJ*> the derivative
are integers, and let \k\=£kl.
We denote by the symbol
^ — r — . A system of differential equations relative to {«J is a 7
collection of relations of the form Fj(*i
«<*>,..., a<s»)=0,
*«: tiu...,un;
(1)
where the Fj are some functions. It is sometimes convenient to distinguish one of the variables, say, t (the "time" as opposed to the spatial coordinates xx
xm), and to consider a system of evolution equa-
tions of the form
-gL^tij.t^Fjixu
...,xm,t;
a,
an; «<*>
«(*>).
(2)
We point out that Fj does not depend on the derivatives of ut with respect to t\ j = l, ...,
n.
1.2. The Language of Differential Algebra. Let ^ be a ring, and let M be a left A module. We recall that a differentiation of A into M is any additive mapping d:A^>-M with the property d(ab)=adb + bda for all a, b£A .
The algebraic analogue of the system of equation (1) is a structure consisting of a ring A , some Lie algebra D with a differentiation into itself, and an ideal Id A , for which Dial
. More precisely, we suppose that Ft
in (1) is infinitely dlfferentiable in all its
00
arguments and set A = U C~(jd, ... , xm\ «<*> 11 < / < « , \k\ are formal indepenm
dent v a r i a b l e s .
Let f u r t h e r
D=
'^Adj,
where
3; t a k e s
X; i n t o
8,7,
and
u<*> i n t o
ullk+ei) ,
l-i
s; = (0.. .1.. .0) n
d*>...da Fj
(1 sits at the j-th place), and let / be the ideal in A , generated by all
. The structure (A, D, I) corresponds to (1). Any smooth solution of (1) corre-
sponds to a homomorphism A ^ C 0 0 ^ , ,.. , xm) = K, which is the identity on K, contains / in its kernel, and takes d,
into T— .
'
axj
I f t h e r i g h t s i d e s of (2) do n o t depend on t
e x p l i c i t l y , then the a l g e b r a i c analogue
of (2) i n t h i s c o n t e x t can be c o n s t r u c t e d by n o t i n t r o d u c i n g (A, D)
and t h e a d d i t i o n a l d i f f e r e n t i a t i o n
[X,
for a l l
t e x p l i c i t l y : i t c o n s i s t s of
of " e v o l u t i o n " X : A-*-A , d e f i n e d by t h e c o n d i t i o n s
/, and XUJ = FJ ( t h e r i g h t s i d e s of (2) for
/= 1
n).
of (2) corresponds t o a /(-homomorphism A-+C™(Xi,..., xm, t), which t a k e s X into
dj i n t o
d/dx,,
and
d/dt.
V a r i a t i o n s a r e p o s s i b l e i n the d e f i n i t i o n of t h e r i n g A . (1) do n o t depend e x p l i c i t l y on /4 = R[«,-""] is
Any smooth s o l u t i o n
x,
For example, i f the Fj
in
and a r e polynomials i n u, , i t i s p o s s i b l e t o s e t
and c o n s i d e r t h e a l g e b r a i c o b j e c t modeling (1) t o be a minimal i d e a l i n A, which
£>-closed and c o n t a i n s
Fj .
I t i s a l s o p o s s i b l e t o t a k e A t o c o n s i s t of a n a l y t i c
func-
t i o n s , meromorphic f u n c t i o n s , germs of f u n c t i o n s , e t c . , depending on t h e p r o p e r t i e s of Fj and the s o l u t i o n s of i n t e r e s t t o u s . 1.3. class
The Geometric Language.
C°° ) f i b r a t i o n
played by a p o i n t on
n : N-+M ,
Here we s t a r t w i t h some l o c a l l y t r i v i a l , smooth ( i . e . ,
The r o l e of t h e independent v a r i a b l e s
M , and t h e r o l e of t h e unknown f u n c t i o n s
smooth s e c t i o n s : M-+-/V of t h e f i b r a t i o n 8
it .
uu...,un
xx
xm
i n (1) i s
i s assumed by a
I n o r d e r t o d e f i n e a geometric o b j e c t
correspond-
ing to the system (1) i t i s necessary to introduce the tower of j e t spaces of the fibration n .
We r e c a l l the corresponding d e f i n i t i o n s . Let s,
at a point
and
s2 be local sections of the fibration
n. .
They are tangent to one another
|eiV to order fe^sO, i f they pass through t h i s point and t h e i r Taylor s e r i e s in
any local coordinate system for n(£) coincide through order k .
A k-jet
at the point | i s
the equivalence class of local sections which are tangent to one another at the point \ h
order k .
Let J n be the set of a l l
k -jets.
to
I t i s equipped with a n a t u r a l smooth manifold
s t r u c t u r e and there i s a tower of smooth fibrations
• •. /^JI-*/ 6 - 1 !!-*-.. .-*-/°jt = JV--»-M .
analogue of system (1) i s then a closed subset in a s u i t a b l e story of the tower
An
/*n , p r e -
scribed by the vanishing of the r i g h t sides of (1). More p r e c i s e l y , the connection of the c l a s s i c a l notation (1) with geometry i s realized by means o-f a choice of consistent local coordinate systems on a l l the spaces J"~ . with a choice of a pair of neighborhoods *&VcN tion
and ir(;)6(/ = 7:(V)cM, such that the r e s t r i c -
~:V-+U i s diffeomorphically equivalent to the projection
diffeomorphism i s a local chart for
We begin
R mn ->R m .
The choice of t h i s
r.\ prescribing i t i s equivalent to prescribing a local
coordinate system in N of the form («.,...,«,,; x,, . . . , xm) such that the r e s t r i c t i o n s of
(«/)
to each fiber of ~ give a local chart in t h i s f i b e r , and the (•*/) are l i f t s from the base. With respect to t h i s local chart a local chart i s canonically constructed in each J"x with a coordinate system which i s denoted by (u(/\ manner.
Let s be some local section of s
by the functions coordinate
!
u], ... , u neC' °(xu • • • , •*«)•
.x,| |/| < fc) and i s uniquely defined in the following it|y , which i s represented in a local chart of it I t s fe-j.et i s a point of
«(," at t h i s point i s by definition
'—r .
of the section s at a l l i t s points i s a smooth section of to
J"- , and we denote i t by the same l e t t e r 5 .
J"r..
The value of the
Obviously the collection of
fe-jets
Jkr. ; i t i s called the l i f t of s
The l i f t s of sections are compatible with
the projections of the j e t spaces. I t i s now clear that the system (1) defines some subset <1» in
Jkr. , and i t s solutions
are those sections of IT, whose l i f t s l i e in <1>. The connection of the geometric language with the language of d i f f e r e n t i a l algebra i s as 00
follows: A = U
C°°(y*it) (imbeddings with respect to the natural projections), D = DC
is the
fc-0
Lie algebra of "horizontal" differentiations of A (for the precise definition see the following section),
I cA
is the ideal of functions which vanish on the "lifts $ " (the geometric
definition of a lift is most simply introduced by the condition of differential closedness of the corresponding ideal). 1.4.
A Comparison of the Languages. The three languages briefly described are not
equivalent either mathematically or esthetically; each has its advantages and shortcomings.
9
The classical language makes the fewest explicit assumptions on the form of the functions Fj,
the sense of the derivatives «/*', the domains of existence of solutions, etc. It
affords the freedom of interpreting, e.g., the uj
as various types of generalized solutions
(and therefore, possibly, not elements of any natural ring).
It also permits when necessary
accentuating other algebraic structures which are essential for investigating the system and its solutions, for example, linear topological structures. However, this language is not suitable for investigating global properties of solutions related to the topology of the manifold of independent variables and fibrations of the unknown functions.
Further, this
language may poorly express the invariant properties of Eqs. (1) and (2) and their consequences . The language of differential algebra is better suited for expressing such properties and puts at the disposal of the investigator the extensive apparatus of commutative algebra, differential algebra, and algebraic geometry; this is especially true if the Fj in (1) and (2) are polynomials, and we are interested in special classes of solutions. The numerous "explicit formulas" for the solutions of the classical and newest differential equations have good interpretations in this language; the same may be said for conservation laws. However, the language of differential algebra which has been traditional since the work of Ritt does not contain the means for describing changes of the functions «j and the variables xt
and for clarifying properties which are invariant under such changes. This is one of the
main reasons for the embryonic state of the theory of so-called "Backlund transformations" in which there has been a recent surge of interest. The geometric language is especially well suited for formulating and clarifying global and invariant properties of general systems of equations and for applying to them the theory of differential-geometric constructs and ideas.
Its main drawback is its generality which,
on the one hand, requires a rather lengthy development of foundations without concrete applications and, on the other hand, creates the risk of overlooking interesting properties of special classes of equations connected with fortuitous additional structures. For these reasons the present paper is written in a broken and somewhat eclectic jargon in which modes of expression from all three languages are mixed in those proportions which to the author seemed most suitable for the object of study.
Equivalent or comparable formula-
tions of the same facts and constructions in different languages are often presented. 1.5.
The Lagrange and Hamiltonian Equations.
The classes of equations which will be
of main interest to us in this work are engendered either by a Lagrangian or Hamiltonian formalism.
Their choice and investigation is strongly motivated by the finite-dimensional
case which corresponds to the projection n: N-*- (point).
In this case the analogue of our
"Lagrange" problem consists in choosing a smooth function L : N-*-R and finding its stationary points grad
L = 0 , i.e., from a general point of view it is not intrinsically a problem in
differential equations.
The "Hamiltonian" problem is obtained if a Hamiltonian structure is
given on N which makes it possible for each Hamiltonian H : N-t-R to construct the appropriate vector field XH
10
on N', it is required to investigate the properties of its trajectories.
If the base M i s not a point then the role of the finite-dimensional configuration (or phase) i s taken over by the infinite-dimensional space of sections s:M-+N
of the fibration
ic , and the f i r s t problem i s to choose a s u i t a b l e class of functionals on the sections from which the Lagrangian and Hamiltonian can be selected.
Here we adopt the point of view of
the c l a s s i c a l v a r i a t i o n a l calculus according to which the i n i t i a l functionals have the form u)(s)=Jco*
, where » in a local chart has the form L(xt; uf^dxiA-
• • A'dxm- and ms i s the
M
r e s t r i c t i o n of to to the corresponding l i f t all
s .
(If M i s not compact
<«(s) are defined on
to — on s u f f i c i e n t l y rapidly decreasing
sections.) In the Lagrange problem to i s called a Lagrangian density (or simply the Lagrangian) ,
8\to* = 0
in c l a s s i c a l notation.
M
In the Hamiltonian problem i t i s , in addition, necessary to give a Hamiltonian s t r u c t u r e : o>*+X~ , where X~ i s the d i f f e r e n t i a t i o n of evolution corresponding to the Hamiltonian Not every such (linear) mapping defines a
Hamiltonian s t r u c t u r e .
to .
In analogy with the
finite-dimensional case, we introduce on {«>} the Poisson bracket by the formula {to,, u>2}=X-2» and we require t h a t the mapping ^Xs
be a Lie-algebra morphism: -^{o,, 5,} = 1-^,1 -X~l (the
commutator of the evolution f i e l d s ) . I t i s well known that by means of integration by p a r t s the condition
8\
to a system of d i f f e r e n t i a l equations (the vanishing of the v a r i a t i o n a l derivatives or the Euler—Lagrange operators of the form to). bracket
X^2
.
This same mechanism works in studying the Poisson
The formalism of the v a r i a t i o n a l calculus i s based on the fact that e s s e n t i a l -
ly a l l computations are carried out with the integrands, i . e . , in the algebra of certain operators on forms on the space of j e t s . derivatives in xt
Since for us Lagrangians and Hamiltonians with
of a r b i t r a r i l y high orders are e s s e n t i a l , i t i s important to clarify the
invariant meaning of the c l a s s i c a l constructions and formulas;, dealing with these objects becomes complicated as the order of the derivatives increase, while invariance under change of coordinates i s almost not amenable to v e r i f i c a t i o n . f i r s t chapter.
This i s the basic objective of the
In Sec. 8 we r e s t r i c t our consideration to the case of a one-dimensional base
and introduce the class of Hamiltonian structures which are important in the sequel. In conclusion we remark that the choice of the class of functionals i s not the only possible one nor even the most important.
{to} on the sections
In field theory, for example, the
class of functionals of basic importance are those generated by i n t e g r a l transformations of polynomials of the form P{us(xM)
«*(*<">)) , where (*<",..., *<">) €MX • •. X M=-MN , and us
are the coordinates of the section.
They include, say, Fourier transformations of the co-
o r d i n a t e s , and, i f generalized kernels are admitted, also a l l functionals of the form
Moreover, they form a r i n g , while functionals of the form «i constitute only a l i n e a r space. A systematic development of the v a r i a t i o n a l formalism in t h i s class has apparently not been carried out in s p i t e of the importance of t h i s problem. 11
§2.
Fields and Forms on the Space of Jets The Basic Model. We consider a fibration N-*M, locally diffeomorphic to the
2.1.
projection Un+m-+Rm , and we describe a number of concepts and constructions related to differential forms and vector fields on various stories of the tower of jets.
The defini-
tions and formulations of results are given in invariant terms which automatically ensures their consistency in local models (an exception is the Legendre operator for which see Sec. 4).
For this reason computations may be carried out in local coordinates if desired which
clearly indicates the connection with the classical formalism.
We thus introduce a number
of bundles and homomorphisms between them, but we carry out almost all computations in a and a diffeomorphism with projection R n+m ->R" 1 ), and we assume
single local chart ( KV'-V-+U
as in Sec. 1 that A'=C0O(i/), At = C°°{J'r.v), i>0, and A_l=^K(zA(j'^Al(z s
we identify with the induced homomorphism A-+-K': P>-+P . C
*«; up\\k\
°°<*>
dxk°
2.2.
Vector Fields on Jets.
In a local chart we have AL =
«; for any s.
dxkn
1
s:U-+V
A section
m
If I is a smooth manifold, B = C~(L) , we denote by D(B)
the Lie B-algebra of vector fields on L, considered as differentiations of B into itself. If C-*-B
is a homomorphism induced by a smooth mapping we denote by D(B/C) the subalgebra
of fields of D(B), which are trivial on the image of C. k>l
For
a
Xk\A,=X'[ D(Ak)
a field Xk£D(Ak)
. Since J*KV = J'*vXR i
is called an extension or lift of a field X,eD(At), , any field Xt extends to D(Ak) .
denote by D(A) the set of all differentiations X:A-*A a k>i,
for which X\AfiD
tion
_d_
|/|
, such that for each i there exists
^-module freely generated by the partial derivatives
Any element of D(A) can we w r i t t e n u n i q u e l y a s an i n f i n i t e l i n e a r combina-
X= ZJP, J— +2jQk,i—TTT!
T
'' tt
sively restricting
Pn Qi.&A.
The c o e f f i c i e n t s
P-Qk.i
a r e determined by s u c c e s -
du
k
X t o A, w i t h
i-+oo.
Obviously,
Pj = XXj, Qk,i=Xuk'K
v e r i f i e d t h a t D(A) forms a Lie A-algebra. 2 . 3 . LEMMA. L e t PQA b e such t h a t P'=0 f o r a l l s e c t i o n s Proof.
I f P(Xj, uk»)^0
for functions
Finally, we
(A(, Ak).
In a local chart D(At) is a free I d_
D(AltAk)(z
We denote by
the submodule generated over Ak by all elements of D(At) in D(Ak) .
if
, then t h e r e e x i s t
s.
lp v«'>€R , such t h a t 1
uycK , such t h a t a t t h e p o i n t (£,) we have {d^ldx ^..
I t i s easily
Then P = 0 . P(Zj, v<'>)=£0.
l
s
.dx jn)u t(Zj)=vk'K
We look They d e f i n e
s
a s e c t i o n s, f o r which P ^f=0. 2.4. s:U-+V
LEMMA.
There e x i s t s a unique mapping D(K)->D(A):
and f o r a l l
Proof.
XeD(K),
PeA
we have (XPy = XP> .
I t i s an imbedding of Lie
D(K) i s f r e e l y g e n e r a t e d o v e r Af by the f i e l d s
2 «<*+e')dlduf*£D(A), where ei=(0... 1.. .0) (one sits at the i,*
12
X<-*X such t h a t f o r a l l
dldxt
.
sections /f-algebras.
We s e t dldxl = di =' ^-+dxt
i-th place) and we extend this
definition by /C-linearity to a l l of D{K) .
Obviously, dt\K = dldxlt (<J,aJ*>)5 = ( a ^ ' ) J , = ^ («}*>*) •
From the fact that P^*PS i s the i d e n t i t y on K and from formal properties of derivatives i t follows that (XPy = XP* for a l l The uniqueness of
X
The fact that X^-X
XqD(K)
and
P£A.
follows from Lemma 2 . 3 . i s a Lie-algebra homomorphism follows from the computation [X, Y\ P° = XYPS - YXPS = X {YPy - Y iXPf = =(XYpy - (Yxpy=([X,
and the uniqueness of Obviously,
Y] py
[X, Y] .
dj i s the " t o t a l derivative" with respect to Xj in the c l a s s i c a l terminology.
We denote by BcczD(A)
the image of D{K) under the imbedding
forms a Lie subalgebra in D(A).
X+-+X .
I t i s clear that ADC
In a local chart we s h a l l write P<" = (?{<.. .6'mP.
This i s
s
consistent with the notation tttp and the homorphisms P*+P . 2.5. [F, X]=0
LEMMA.. There e x i s t s a unique imbedding of AD(AQlK) into D(A):Y*+Y such that for a l l XeD,. and YP =YP for a l l P ^ V
fields which commute with Proof.
Dc and are t r i v i a l on K .
The action of
Y on xt
and
uk coincides with the action of
for | £ | > 1 i t i s defined by the con dition s i b l e formula for
The required properties of
l)
{,)
[K, £>c]=0: Yai = (Yuk) . i)
7 has the form
hence valid everywhere.
I t s image i s the Lie subalgebra of a l l
7P = ^l{Yuk) ' ^T
Y, and on aj,°
Therefore, the only pos-
.
Y axe. obvious on the generators of Dc and A0 and are
Any field which commutes with De and i s t r i v i a l on K, can be r e -
presented in the form 2 Q*° d (/> » and therefore has the form of
Y .
Finally, such fields
obviously generate a Lie algebra. We denote by
DevdD(A)
the image of AD(A0jK) in D(A).
the system of evolution equations t i v e " by v i r t u e of t h i s system.
Assigning to each Y&Dev
Uj,t — YUj , we see that Y:A-+A i s the " t o t a l time derivaThe condition [Dc, P] = 0 means that the t o t a l time and space
derivatives commute. 2.6.
LEMMA,
algebra in D{A). with
a) [Dev, ADc\dADc
. b) Dev + ADc = Dey®ADc
(sum of spaces) i s a Lie sub-
The r e s t r i c t i o n of Dev + ADK to A0 defines an isomorphism of- t h i s space
AD(A0). Proof, b)
a) For 7&D„, X&)e,
The r e s t r i c t i o n of
P$A we have {?, PX\ = (YP) J?GADC, since
J Q*1"^" + 2
#/dfcD.y + ADe to >t0 coincides with 2 Q * 5 5 T
S S i ^ + S " ! ' ' ^ ) - ^ uniquely determined for any f i e l d of AD(A0),
[7,JC]=0.
Therefore,
Q.and
+
/?, are
whence b ) .
13
For any element Dtv-\-ADc . .
X&AD(A0) we denote finally by X&D(A) the corresponding element of
As i s evident from the proof of Lemma 2 . 6 , on D ev and Dc t h i s canonical l i f t
operator coincides with those defined in Lemmas 2.4 and 2.5 respectively. Although Dev-\-ADc far from exhausts D(A),
these d i f f e r e n t i a t i o n s completely suffice in
order that i n t e r i o r products with them should uniquely determine the d i f f e r e n t i a l forms on the space of j e t s (see below). 2.7.
Indeed, we have the following r e s u l t .
LEMMA. Fields of the form X + Y , where X&D(A0lK),
YeD(K\ form a basis of the
k
tangent space at any point of J %v • Fields X\Ak have the form ^ ( Q i ) u •- ^
Proof.
used in Lemma 2.3 shows that for any point prescribed values
;>
Qi
at t h i s p o i n t .
, and an argument analogous to that
ZGJknv i t i s possible to find
Q&A with any
Thus, X generates the spaces of v e r t i c a l tangent
v e c t o r s , while the spaces generated by Y for Y^DIK) project onto the e n t i r e tangent space O at it(c) and therefore give the lacking horizontal complement. ( I t should be mentioned that Y takes Ak onto Ak+l, so t h a t , s t r i c t l y speaking, Y does not define an ordinary tangent However, we s h a l l use t h i s lemma to verify that each d i f f e r e n t i a l form on Jkr. i s
vector.
determined by i t s values on f i e l d s of the form X-\-Y, and for t h i s purpose our argument suffices.) 2.8. of
The de Rham Complex.
For i>— 1 we denote by QAt= ©2M, the e x t e r i o r algebra
C°° d i f f e r e n t i a l forms on J'IC with d i f f e r e n t i a l rf:2M,-> 2*+M,-.
At-*-Ak (k>i)
The canonical imbedding
define a system of imbeddings QAi-^QAk consistent with d .
We set
QA =
lim2^ = u QAh 2 M = Z&A • —•
i—i
i—t
In a local chart Q»A i s freely generated over A by elements of the form atttl Adu^Adx^A w&JAt
••• Adxib,
i, < . . . < * „ t a + 1 < . . .
For any vector field XeD(At)
A---
and forms
u
the standard compositions ixuigS^Mj ( i n t e r i o r product) and Lx > = (ixd+dix)u>dQJAi
(Lie derivative) are defined.
If k>i,
then tjfio = fx« and L^—Lx^.
Therefore,
possess the usual p r o p e r t i e s .
In p a r t i c u l a r , the Lx (respectively,
X, and are d i f f e r e n t i a t i o n s
X&D(Ak), i s an extension of XQD(At), and w&i(A,)czQ(Ak), ix and Lx are defined for X(*D(A),
(respectively, a n t i d i f f e r e n t i a t i o n s ) of the algebra QA .
over, \Lx,d\ = 0, \LX,LY\=L[X,Y\;
Lpx--=PLx+dPAix\
[Lx, iy] = iix,ri', ipx = Pix-
More-
We s h a l l sometimes
write Xm in place of Lx® . We now concern ourselves with the r e s t r i c t i o n of forms of
QA to sections.
Any section
s defines an algebra homomorphism QA-+QK- «>*-*•«>*. 2.9. Proposition, a) There e x i s t s a unique operator -z-.QA^-QA with TQA
i s the ideal in 2^4, generated by KertnS'-A.
d) t/.jf=*TijfT for a l l
14
XeDev + ADc; tZ.]f=Z.jFT for a l l
XeDeV.
that
e)
xrfx = xd
Proof,
and
xdxd=0 .
a), b).
We d e f i n e an .A-homomorphism of a l g e b r a s
QA->AQK on t h e g e n e r a t o r s
m
^{du^)=^u^+ti)dxi.
by the formulas *(dXj)=dxt,
It is clear from the definitions that
xQAczAQK, (T(I>)*=U)*, and x2 = x. Uniqueness follows from the fact that AQK module with generatorsrfjCj.A-• • /\dxik mined by its
is a free
A-
, and therefore any of its elements is uniquely deter-
s-images according to Lemma 2.3. m
We remark t h a t
tdP^^djPdx,
for a l l
x2 = x , we have
c) Since
and i n d u c t i o n on
t show t h a t
follows from t h e f a c t t h a t
PeA
Ker x = Im(l —x).
.
The i d e n t i t y
Im(l — t)n2 i -AcQ i _ 1 A(Im(l — x)n2M) .
Kerx i s an i d e a l c o n t a i n i n g
d) We s h a l l show f i r s t
o>Av — T(«)A V ) =
that
t i o n and Ker x i s an i d e a l , i t s u f f i c e s t o v e r i f y t h i s i n c l u s i o n i . e . , on
«<*' — 2 tlf^^dxj
The r e v e r s e i n c l u s i o n
Im(l — r)f\Q1A
L-^KeiiCzKeT x for X&Dey -\-ADc .
.
Since
Lj
is a
differentia-
on t h e g e n e r a t o r s of
. Because of t h e a d d i t i v i t y of L^, w i t h r e s p e c t to
t o c o n s i d e r ^f s e p a r a t e l y for
(U) — •M>)AV + ' W ) A C ' — X *)
Ker t,
X , i t is possible
and A'g.AD,.; we have
XGAD(AQ!K)
= <%«{*> - ^ (f.jU^dXj
+
U^dLjXj).
i )
the
For A'=2.Q{* -T»)
r i g h t
s i d e
i s
equal
t0
0—T)<*Q{*)eKerT.
For X=Pdt
the right side
i s e q u a l to P(\ — x) rfw^+VgKer x. Now xZ.^x — xLjf = x£^(x — 1) and -X'g.AD (,40/A') , then Ljf(xrfa<*))=xrfQ(*> .
i.e.,
rfKerxcKerx
If,
moreover,
^ = 2Q{*'—J$\
Ij f x=xZ.^x=xIj f .
xtf —tdx = x<* (1—x) . and hence
If
u>* = 0
xtf(l—x)=0.
for a l l
s , then (rf
Multiplying the equation
s,
xd = xdt on t h e r i g h t by
2
d , we f i n d t h a t 3.
Xjflmxclmx , s i n c e by t h e p r e c e d i n g c a l c u l a t i o n ( f o r Therefore
e) Obviously,
ImZ.y(x — l)clm(x — l ) = K e r x , whence T-Lj = tL-gt.
(xrf) =0 .
Integration by Parts 3.1.
Probably a l l t h e i n v a r i a n t i n f o r m a t i o n c o n t a i n e d i n the c l a s s i c a l p r o c e d u r e of
i n t e g r a t i n g the v a r i e d Lagrangian d e n s i t y by p a r t s i s c o n t a i n e d i n the n e x t p r o p o s i t i o n . Here we do n o t even assume t h a t the "form i n t h e v a r i a t i o n s " u>(*QlAQmK (du\k) i s t h e " v a r i a t i o n " of
«<*>) i s o b t a i n e d from a L a g r a n g i a n ; t h i s c a s e w i l l be d e a l t w i t h i n the n e x t s e c t i o n . 3.2.
to3^Q1AQm~1K a)
u) =
Proposition.
Let
<»&ilAQmKc:QA.
There e x i s t forms
a>l£AQlA(fimK, ^1&QxAQmK,
w i t h the f o l l o w i n g p r o p e r t i e s : o>j - | -
b) For a l l
X£Dev
we have
ix
The forms ml, w2 a r e determined u n i q u e l y .
The 15
form to3 can be n o r m a l i z e d by the following a d d i t i o n a l r e q u i r e m e n t : we choose
vQA£m/(
arbitrar-
i l y and impose on w3 t h e c o n d i t i o n : c)
tio 3 = v.
Then u)3 e x i s t s and i s u n i q u e l y determined i n t h e case m = l and a l s o i n t h e case a>($ZlAiQmK, i f we r e q u i r e , i n a d d i t i o n , t h e i n c l u s i o n Proof.
Existence.
I t suffices
IO&A&AQQ^K-
t o v e r i f y t h e e x i s t e n c e of
<»i, u>2 w i t h p r o p e r t i e s a ) , b)
on t h e a d d i t i v e g e n e r a t o r s of t h e group Q1AQnK which we t a k e t o be t h e forms dX\/\...f\dxM t h e forms
in a local chart.
We s e t
k(.i) = \k\ and c a r r y out i n d u c t i o n on £(<»). For A(u>)=0
«>,=u), u> 2 =u) 3 =0 s a t i s f y a) and b ) .
Suppose t h a t £ ( « ) > l a n d l e t £ / > l . -d,Prf« J ( *- , <» ArfJC,A • •.
o ) = p a « ( * ) A r f x , A . . . ,\dxm= + d, (Pduf-'S
)/\dxxA
- . . Arf.*m=u>' + v.
and t h e s e l i e i n t h e d e s c r i b e d subgroups of ix*-=h(dipdu?~'l)
Ad.*:A• • • /\dxm) e
= df ( X ^ f - ' ' dXl/\...
Adxm+P
(Xud^dx,
= dt(P(XUif- ' e )
=(-l)'- xrf/ 7 (Pda ( !*- /
2J4 .
Further,
+i^{Pdu\*>Adx1A.../\dxm)
e}
1
Then
Adxm+
By t h e i n d u c t i o n h y p o t h e s i s i t i s p o s s i b l e t o choose w[, u^ and u>3 w i t h for a l l XQD^,
Pdu^A
A ...
=
Adxm=
)dXlA-..Adxn=
Adx'tA...
Adx,_lAdxl+lA---Adxn
=trfty.
I t remains t o s e t o>1=2 = a>j-j-v, a>3 = 3-f v\ Without d e s t r o y i n g p r o p e r t i e s a ) and b) we may add t o o>3 any element the p r o p e r t y
*jf v "=0
p l a c e of (i>3 f o r any
for a l l
X^Dev.
v^AQmfC, s i n c e
In p a r t i c u l a r , i t i s p o s s i b l e t o t a k e a>3 —f(o3-|-v i n i-^(AQmK)=[0}.
Then
TO>3
i s r e p l a c e d by TV-=V, which gives
condition c ) . Uniqueness.
Let io = |-}-^
and i-^ = zdi-^a>'3 f o r a l l X , i . e . ,
of forms which s a t i s f i e s a ) , b ) , c ) .
Then f o r a l l
X^Dev
^J, «>2, <«3 i s a n o t h e r
triplet
we h a v e , on s e t t i n g o)j=(oi—OJ,',
i-£»\ — — i jf<»2 = — ~di~x<j>3. From t h e proof of Lemma 2 . 7 i t follows t h a t any form v a l u e s of
t^
f o r a l l X(*D{A0lK),
i t suffices
of
x :
QlAQmK i s u n i q u e l y determined by t h e
s i n c e such ^ g e n e r a t e a b a s i s f o r t h e v e r t i c a l
v e c t o r s a t any p o i n t of t h e s p a c e of j e t s . 2
(3 )
t o v e r i f y t h a t i^u>j = 0.
tangent
T h e r e f o r e , t o e s t a b l i s h t h e uniqueness of <»! and
For a l l s e c t i o n s
s we have by (3) and t h e d e f i n i t i o n
(«V»i)s=— dVx^zYWe choose any point xf*M. , a small open neighborhood U of it with boundary dll, and we consider the fields X for AT with support strictly inside rc_1(£/). By Stokes' formula J M m rf t
( jf">3)*=
I (fjea>3),=0> dCJ
16
f o r
a11 such
~X-
I f
%
2iPtctiliAdxlA~-Adxm
^>i = (=1
and Xu, = Qi, then
(2PiQi)i|c/=0
for a l l
Q.GAi > whence Psi\u=0
s and
and t h e r e f o r e
t h i s c o n c l u s i o n remains i n f o r c e i f i t i s assumed t h a t 0^=0)2=0 , and a c c o r d i n g t o (3) xcf/yio3=0 for a l l Even t h e n o r m a l i z a t i o n of n e s s of
u>3 .
u>3 depends on
X.
We remark t h a t Finally,
X(*Dev.
u>3 by c o n d i t i o n c ) , however, does n o t g u a r a n t e e t h e u n i q u e -
We cannot d e s c r i b e t h e complete k e r n e l _fl
Kerxrfj^, b u t i t c o n t a i n s ,
e.g.,
*6D, ev
m 2
dxdQ ~ K . I n d e e d , •zd [ijd\d)=i
(dij) dzd =
= t (Ljf — i-^d) did =
-X-^l-cd=iL-gidid=0,
a c c o r d i n g t o P r o p o s i t i o n 2.9 d) and e ) . I t remains t o c o n s i d e r c a s e s i n which we a r e a b l e t o g u a r a n t e e t h e uniqueness of The c a s e
m= \ .
Up t o a term i n AQmK, which i s u n i q u e l y n o r m a l i z e d by c o n d i t i o n c)
and l i e s i n t h e k e r n e l of a l l t h e «i3=2/>A*dtt/*)"PA*^-
w3.
If
fdi^,
t h e element
then
T<%C73=0,
a>3 can be r e p r e s e n t e d i n t h e form
xd^Pj&fj^dAP/kQj^dx^O
for a l l
Q} =
XufiA,
k
i . e . , V P^QJ** = const for any The c a s e
w^A^K,
Qj. .
m£AQlA0Qm-lK.
i s obvious from t h e c o n s t r u c t i o n of t h e element u>3(
rfa
a)
From t h i s i t
e a s i l y follows t h a t
The e x i s t e n c e of
u>3 i n t h i s subgroup f o r
o>3 a t t h e b e g i n n i n g of t h e proof.
t h e d i f f e r e n c e of two c h o i c e s of
m l
&AQ - K.
Pjk=0.
")3
(a^A^K
Up to a term of
AQmK
can be r e p r e s e n t e d in t h e
Further,
i j ^ 3 = 2 A a i < » / = 2 XUi ( ^ P^^XiA<
• • A<£xt A • • • A ^ ) ,
i
x d i ^ 3 = 2 i-lY-% (%*iPt,i) dxi A • • • Ad*mi.t
If the last expression is identically zero, then, since Xut =QiQA may be chosen arbitrarily and independently, we find that for each i
the sum V ( — l)'-1Xdj(QPi,;)does not depend on Q .
It is easy to see that this is possible only for Pf ,=0,
which completes the proof.
We denote the forms o^ and a>3, constructed in Proposition 3.2 on the basis of the form respectively.
We shall indicate their explicit forms in co-
ordinates. We begin with the following classical lemma. Let B be a (not necessarily commutative) ring, and let
For any sequence of numbers tl§ i2, i3,..., l
...dt :
d(0) = id . 3.3. LEMMA.
For any x, y£B
we have
17
d{h,.. .,ik)xy-(-\)kxd{ik,..
.,h)y=
k
= 2 < - xrldt
(dv**' ••••'*)*w»*' • • • • 'i)y)fl
0-1
The proof is obvious.
3.4.
COROLLARY. We assume t h a t m
C (/;*, / ) > 0 , l < / < m ; k,l$N ,
[d„dj] = Q for a l l 1 < i , y
Then there exist numbers
such t h a t for a l l seN™,
w
d**«/-(-l) jed^=
2
(- 1 )" I + 1 C(/;*, l)dj(d*xdly).
Proof. We write out the formula of Lemma 3.3 for all sequences {h, • •., ijS\) with
w 2 e ' = s = s » take their arithmetic mean, and collect like terms.
We remark that for m = l we have C{J, k,l) — l.
3.5.
We can now write down formulas for 8a> and §vw.
Let u>= ^ ^ j sdtl\s)dx\A-..
Adxn .
We set */»* =rfjc,A •.. Adxm, dfx=afJC, A • • • Arf-«/_i Arf-*7+i A • • • Adxm. Then by 3.4 we find that l»=^i(-\fd*Pljlu1Ad'"x, s
• _ fc. -
2
•( -
1
) w + l c
@'Pi,Mk))^dmx-
For any ~X&Dtv we have i^ (d^d'P, /f^*'Adm.*)) = *<% X(—1) /-1 (&Pl%sdu^)Adfx.
Replacing here du\k) by (rf — •td)aj*) » w e do
not
change the r i g h t s i d e .
i t i s also possible to take du\k) — u\kJre^dXj in the term zero.
After t h i s change the form under the sign idij
Therefore
In place of (d —•zd)u\k'>
d"*x ', the remainder terms give
i s in the kernel of
x.
Finally,
adding v to i t in order to s a t i s f y condition c) of Proposition 3.2, we obtain f i n a l l y
$v<°=* +
2
M) , " + 'C-(/.-Mtf , / > , 1 ,W
-u^)dxj)AdTx.
It+l+tj-S
4. The Euler-Lagrange Operators and the Legendre Transformation 4.1. THEOREM. Let
form which belongs locally to AQ Kl
m
m l
TZ:N^>-M
, i.e., a
Then there exist forms 8m and Su> , locally belonging
to AQ AQQ K and Q^AQ ~ K , respectively, such that for all Xf*DeY + ADc we have (locally)
18
xl^u) = x (ijd + di Y) o) = x (i jf 8iu 4- dijS*) and, moreover,
(4)
xSu> = u>.
The form 8«> i s always u n i q u e l y d e f i n e d by t h e s e c o n d i t i o n s , w h i l e the form Sco i s u n i q u e o r i n t h e case ^€AlQmK i.f i t i s a d d i t i o n a l l y r e q u i r e d t h a t
l y defined i n t h e case m=\ SueA&A&^K Here
(locally'). 8 i s c a l l e d t h e Euler—Lagrange o p e r a t o r , and S i s t h e Legendre t r a n s f o r m .
Proof,
a) The l o c a l c a s e .
We s e t 5u>=8*flfu>, Sio = Sadw .where 8 and ^ a r e defined as i n
P r o p o s i t i o n 3 . 2 , where v i n c o n d i t i o n c) i s
w.
We then o b t a i n a c c o r d i n g t o 3 . 2 :
rfu)==8a)-)-u)2; i^tOo = xrf/^S'j) for X^Dev\
From t h i s we deduce t h e i d e n t i t y (4) for X^Dev If
X(*DeV,
then
and
XQADC
xSu> = co.
(5)
individually.
whence a c c o r d i n g t o (5)
£^J>=0,
xL^io = x/-^cfu) = •zi-^ (Sin -\- co2) = xt^8a) -f- idijSin. The uniqueness properties of l
m
is noted that do, b^AQ AlQ K, i^du>, ij^Sa) for
8J> and St> follow from those proved in Proposition 3.2 if it so that these forms are uniquely determined by all values of
X(*Dev , and these values lie in AQmK ; therefore, x is the identity on them,
and hence the decompositionrfio= Sw-f-u>2 with the properties postulated in Proposition 3.2 follows from (4) for X&Dev . It thus remain to verify (4) for X(
We shall first establish that x(iy2m+1i4) = 0,
i.e., (i-^Qm+1A)s — 0 for all sections s . Since (QmnA)s={0}, s
(ijfv) = tA-v*
for Xi*D(K)-
Since
ij
it suffices to verify that
is a differentiation and restriction to s is a homomor-
phism of the algebraof forms, it suffices to verify it on the generators. coincide.
On A both sides
Further, (ijdXjY = {XxjY = (**,)* = ix (dXjY,
(M«J«)-(^J*') , -(S»J* 4 * , ) ^«)'-2^ + * , , , ^I.
Using this, we see that it is necessary to verify the formula idij^a = -zdi-^Sio. We have xd ipSui = -zL-^S'-a—i(ijdS)=xZ.j-.Su>. From P r o p o s i t i o n 2 . 9 we have 19
But TSU> = CO, which gives the required result. b) The Global Case. Since izing it are consistent
8co is locally uniquely defined and the properties character-
with the operation of restricting to a subdomain, all local con-
structions automatically fit together. where uniqueness is guaranteed.
For So an analogous argument goes through in cases
Aside from these cases, as has been shown, it does not hold;
e.g., it is possible to add locally to Sco any element of dxd(AQ™-2K) without changing the values of idixSa Sa
for all XeDev+ADc
. Therefore, to prove the existence of a global form
a special treatment is necessary in order to establish the possibility of consistent
local constructions. This has been done by B. A. Kupershmidt, but we shall not reproduce the proof, since it will not be needed below.
It is essentially a question of the vanishing of
a certain cohomological obstacle, and the problems which arise here merit special attention. The question as to what addition conditions (functorial property, for example) it is necessary to impose on So> in order that S© may be uniquely chosen has not been solved. 4.2. Classical Formulas. We shall apply the formulas of 3.5 to the case a>=PdmX, dm = 2^) r f »j"Atf m JC
. We obtain "«£.A. *./«. 6P
V/
iv>,W dP \(s)
Further, S«_.+ V
^(rf«|«-U^Adx^dfX,
where
_« i7r _2(-.)««c ( y, 4 .o( si ^ w )«» (it can be shown that C(J;fc,l)
depends only on A + e y ,/).
5. The Variational Complex 5.1.
Definition of the Complex. The beginning of the complex is locally as follows: A^AQtK*...
'*AOi*Kt*AQlAfir"K.
The fact that (trf)2==0, is proved in Proposition 2.9 d).
The equality *wf=0 is established
as follows. The operator *Lxd on AQ^K can be transformed in two ways. xljfrf=T (ijd -f- di y) d=id
20
(6)
(%rf).
First of all,
Secondly, by formula (4) and P r o p o s i t i o n 2.9 d) for XeDew + ADc xLJd = xL^xd = xi^xd + xd{i-RSxd). From t h e u n i q u e n e s s of t h e E u l e r - L a g r a n g e o p e r a t o r we t h e r e f o r e f i n d t h a t We s h a l l now i n d i c a t e how t o extend complex (6) to t h e r i g h t .
8tfif=(X
The e x t e n s i o n o p e r a t o r
w i l l a l s o be e s s e n t i a l l y an Euler—Lagrange o p e r a t o r but i n a l a r g e r bundle « X l d
:NXR-*-MXR-
L o c a l l y we d e n o t e by xm+l
Km
C°°(Xi, ...,xm+l),
t h e c o o r d i n a t e on t h e a d d i t i o n a l f a c t o r
AI^UC"^,
n;k£Nmn,
a<*>|t = l
|ft|
R , and we s e t
=
The c a n o n i c a l inbedding
AaAW,
c o r r e s p o n d i n g t o t h e p r o j e c t i o n of « X i d onto *, i n l o c a l c o o r d i n a t e s can be i d e n t i f i e d w i t h t h e imbedding a<*)*-»fcttj*,0\ which t h u s h a s an i n v a r i a n t meaning: t h e c o o r d i n a t e s "along t h e fiber"
ut
i n t h e e x t e n s i o n t o NxR
The imbedding lift
from it t o
a r e assumed t o be independent of t h e new v a r i a b l e .
A-+AM d e t e r m i n e s t h e imbedding QA-+QAW which a l s o c o r r e s p o n d s t o t h e
*Xid .
QlAQmK w i t h i t s image i n 21.A(1>QmA'(I,, we may c o n s i d e r
Identifying
t h e composition of t h e o p e r a t o r s 1
QlAQmK^QiAWQmK^-^AO)Q^K^)-1lQiA^Qm+1Kil),which
1
d e n o t e simply by BCt* ' (B* ' i s t h e E u l e r - L a g r a n g e o p e r a t o r , and fibration
i:Xid ) .
t
(1)
we
is the operator
x
for
the
This c o n s t r u c t i o n can o b v i o u s l y be i t e r a t e d which l e a d s t o t h e sequence (7) L^(2)01 J 4(2)gm+2/^-(2)_ > _
5.2.
THEOREM.
#
_
Sequence (7) is a complex which we shall call the (local) variational
complex of the fibration T..
The global complex is a bundle complex the local sections of
which have the form (7). Proof. We first of all introduce a new operator x+:Qm*tA-+ Q^A^K for all Xu . ..,XkeD(AlK)
by the condition:
m+
and t»eQ *A i*,. • •f'xft-+U) = t (**,.. -fXftW).
The e x i s t e n c e and u n i q u e n e s s of t h e o p e r a t o r
t + a r e o b v i o u s , s i n c e a form i n QkAQmK can be
c o n s i d e r e d a skew-symmetric, m u l t i l i n e a r f u n c t i o n of k m
Q K
.
I t is clear that
5.3. Proof.
LEMMA.
x* i s
(8)
f i e l d s i n D(AlK)
with values in
-A-linear. o>£AQmK we have 8 .
For any Lagrangian
I t s u f f i c e s to v e r i f y t h a t £^8u> = xi-^dS® for a l l X^DeV,
because of Lemma 2 . 7 .
According to formula (4) and P r o p o s i t i o n 2.9 d ) , we have / , - « ) = i L - a=xdi-^Sm + i % 8u>=x (Lj — i^d) Sio + i jj-B J> = =xL-g xSu> — xi-^dSo + ix&M=iLj a — xijdS®+i-j^-oTherefore, 5.4. Proof.
xi^dSia = i^o, LEMMA.
as required.
t0> =*<>>•:+ on
I n view of t h e
QtAQ^K-
A - l i n e a r i t y of both o p e r a t o r s , i t s u f f i c e s to v e r i f y k)
m
e q u a l i t y on g e n e r a t o r s of t h e form du\ f\d x kind t h e o p e r a t o r
x
+
is the identity.
(
and /\da f>/\d™x.
this
On forms of t h e
first
F u r t h e r , a s i s n o t hard to check, 21
z+ (rfu<*> Aduf
Adfx)
whence i t f o l l o w s t h a t t h e v a l u e of
= ( - l ) ' " 1 (u^du**
af+t')duip)f\d«x,
-
x(')t + on t h i s form i s e q u a l to (even i n
(-ly-^u^u^^-a^u^^dx^Adx.A... On t h e o t h e r hand, t h e v a l u e of
QAm)
Adxm.
t(') on our form i s equal to
The l a s t two e x p r e s s i o n s o b v i o u s l y c o i n c i d e . 5.5.
Completion of t h e Proof of Theorem 5 . 2 .
According t o Lemmas 5 . 3 and 5.4
8 < I V 1 > 8 = 8 ( 1 W r f S = 8<1>,(,)dS. But
8 (1) -: (I, rf=0,
A(l).
by t h e argument a t t h e b e g i n n i n g of t h i s s e c t i o n a p p l i e d to t h e r i n g
Thus, (7) i s a complex a t t h e term QlAiimK and hence a t a l l remaining new t e r m s : 3(a+I>,:j<«»)__ 0
for a l l
verified
The f a c t t h a t i t i s a complex i n t h e terms t o t h e l e f t of QlAQmK was
a>0 .
earlier.
The e x a c t n e s s of complex (7) i n t h e terms to t h e l e f t of
AQmK
Rham complex t o t h e j e t s ) was r e c e n t l y proved by A. M. Vinogradov.
( t h e " l i f t " of t h e de The e x a c t n e s s of t h e
g l o b a l complex (7) i n t h e remaining terms was e s t a b l i s h e d by B. V. Kupershmidt.
We s h a l l
r e s t r i c t o u r s e l v e s to t h e proof of t h e c l a s s i c a l p a r t ( " i f t h e v a r i a t i o n a l d e r i v a t i v e of t h e Lagrangian i s equal to z e r o , then i t i s a d i v e r g e n c e " ) . 5.6.
THEOREM.
Proof.
Ker6=ImTd. .
We c o n s i d e r a c o n t r a c t i v e homotopy
which t h e complex (6) i s
0 < * < 1 , tpo X, that for
= 1
d and
-t)~2\).
(point).
be i t s c a n o n i c a l l i f t
t o D(A) m
tZ.jjf<»=I]f(fl f o r a l l * < 1 , and hence
<»eA2 K.
o*Lj—-j
t o formula ( 4 ) , we find
, over
defined: *,:(.*„ Uj)~(tXi, uj exp[l - ( 1
I t i s obvious t h a t
Vc.N
Let XfiD(AQ)
where
with
be t h e c o r r e s p o n d i n g v e c t o r f i e l d , and l e t
a c c o r d i n g t o Lemma 2 . 6 . Since Xt
onto i t s e l f
Since Xt(*D(A0) , i t i s obvious
i s t h e d e r i v a t i v e of
Applying
9*
that
^zdilljSv+rfljt*, since, as i s e a s i l y seen, the field
Xt
i s not defined, i t s l i m i t
t e g r a t i n g (9) on t. from 1
9*T = T9* ( t h e mapping
1
0 to
?}£x<» e x i s t s a s
<->l
for a l l
and
a\k)) .
m.
Therefore,
Although in-
1, we f i n d 1 _
_ a , = \ A-y*dt = trfC 9 J i 7 S a d t + x\
22
1 _
' _
def
= xrft\ y*ijSmdt + %jj w*i^Udt = — «%(w) — ij>2(8
since ~d = tdx.
Thus, 1
o
and we find that if
8u = 0,
'
then
We remark t h a t if the form
«<*> ( r e s p e c t i v e l y , in «<*>, JC ; ), then
so are also Sw, hm polynomials by v i r t u e of the formulas of 4.2 and with these also
k
t|i2(8u))
, since
The Formula n
Fur ther
*<»**,=2 u^dx,
8
a>=Pdmx .
We s e t
Then 8a>= 2 ,_,
^ldul/\dnx.
+ «{em+«' dxm+u whence
i-i
n 6P ( W dx mviAdxiA T(»> 8
.• • A^»-
i-i
Therefore, 8(0,(08(0 = 2 gF/(S "i*" 41 ')
4*iAdxm+lAdax=
rt
'
/An \^rn+v
,|t W <S
= 2 ( 2 m ( - W (^r>5T, i
] ' JujAdx^Adnx ~ 2 IBSJ)
dalAdxnAAdxlA ... A ^ J .
Further,
Therefore, after division by dxMlrXf\dmx
the formula
8(1)T<1)8O)==0
assumes finally the following
form:
We recall now that over A the variables dkufm+l)
are free.
Therefore, for all i, j
we ob-
tain a family of equalities among differential operators which is equivalent to the formula 8(OtO)8=0 . 5.8. LEMMA. For any P&A and l
We shall need this lemma and its corollary in Sec. 7 to investigate the Hamiltonian structure. 5.9. COROLLARY. For any
XeDer
23
Proof. 6.
We apply the i d e n t i t y of Lemma 5.8 to
Xuj and sum on /'.
Nother's Theorem and Lagrangian Conservation Laws Let (a=PdmxeAQmK be some Lagrangian.
6.1.
t h a t the equation
s
(6co) =0
I t i s evident from the formulas of 4.2
r e l a t i v e to unknown s e r i e s
system of Euler-Lagrange equations for the functions
s s
coincides with the c l a s s i c a l un$ with Lagrangian
ux
P .
Its
solutions are called extremals of the Lagrangian w. 6.2.
I n t e g r a l s , Flows, and Symmetries; Analytic Version. m 1
integral for the form u> i s a form M^AQ ~ K such t h a t
tt(t')=0
A conservation law or an
for any extremal s of the Lagrangian
t h i s means t h a t vgi4 i s a functional of the space of j e t s which i s constant along
any extremal.
In the general case for
I 2(—1)'-1
v = £ Ptdfx
the r e l a t i o n
rf(vJ)=0
has the form
along any extremal. m
A flow for
a) i s a f i e l d
Y(
If <s>=Pdmx, F = 2 Q A
iym i s an i n t e g r a l .
2—1
m
then ty(i)=^| (— \)l~lPQldfx
. It is evident from this that if P does not vanish on its ex-
i-i
tremals, then for any conservation law v there is a unique current / such that v = i-pio. The classical Nother theorem affords the construction of conservation laws on the basis of a Lie group O, which acts on K:N-*-M If X
action s^co*.
and preserves the Lagrangian
io or even just the
is an element of the Lie algebra of the group G, considered as a field
fur
on N , then the G-invariance of o> implies that 1 ^ = 0 . More generally, let X(*AD(A0)
be
s
a field on the jets such that (L^
Integrals, Flows, and Symmetries: Algebraic Version.
The definitions of the
preceding section appeal to the set of all extremals of the Lagrangian u>, which may be empty or very complicated.
In practice integrals, currents, and symmetries always satisfy the
following algebraic version of the definition whose application does not require vanishing Dc -closed ideal in ^4, generated by expressions
on the extremals. Let /(8a) be the minimal of the form ixh,.. minimal
•*>„>>, where X£D(AlK),
£>e-closed ideal in the ring
K„ .. .,YmeD(K).
QA,
Let J(h^aQ'AQ^K
generated by /(&«>) .
We call v an algebraic conservation law iffi?vg./(8u>). We call if iyia
is an algebraic conservation law.
be the
Finally, X&AD(A0)
Y
an algebraic current
is an algebraic symmetry if
L-jfwQj (8
We point out that the use of this definition makes it possible to obtain additional information. dv
For example, if v is a conservation law, then the explicit representation of
as an element of /(8a>) determines important invariants; the theory of characteristics of
Gel'fand and Dikii [2, 3] is based on this. 6.4.
THEOREM,
NSther theorem).
24
a) If A" is a symmetry, then ti^Sa
is a conservation law (the formal
,
Y
b) For any flow Proof.
the field Y is a formal symmetry.
We write the proof in the analytic version; the algebraic version may be con-
sidered analogously. a) Since X is a symmetry, for any extremal s we have (Lj
s
from (4) that (i^w) -\-d(i^Sj>) =0
It therefore follows
on the extremals, whence d{ijSo>Y<=d(iixSa>Y= _(/x8
The last inequality follows from the fact that locally 8o> = 2 So" ^^mjc
»
so
that
'x*""
is a
linear combination of the g^-r which are zero on the extremals. b) Since Y
is a flow, we havetf(t7co)I=Oonthe extremals, so that (L7u>Y=(diy» +
iyd*»y=(i7du>Y.
But t h e l a s t e x p r e s s i o n i s zero a s v e r i f i e d i n t h e proof of Theorem 4 . 1 a ) . 7.
The H a m i l t o n i a n S t r u c t u r e 7.1.
We s h a l l g i v e a l o c a l d e f i n i t i o n of t h e Hamiltonian s t r u c t u r e on
TZ;N-+M\
global-
i z a t i o n goes through a u t o m a t i c a l l y . 7.2.
Definition.
Let
Y
b e an R - l i n e a r o p e r a t o r
T:AQmK-+bev
.
We d e f i n e a
m
b i l i n e a r c o m p o s i t i o n law on AQ K, by s e t t i n g {
(10)
The operator T gives a Hamiltonian structure if it takes { . }r into the commutator in the Lie algebra DeY ,
r{u)„
Comments.
(11)
Imxrf=Ker8.
The m o t i v a t i o n for t h i s d e f i n i t i o n was given i n Sec. 1; i t i s p a r a l l e l -
t o one of t h e c h a r a c t e r i z a t i o n s of a f i n i t e - d i m e n s i o n a l Hamiltonian s t r u c t u r e . here
m
^AQ K
i s c o n s i d e r e d a s a r e p r e s e n t a t i v e of t h e f u n c t i o n a l
u)(s) = W
In p a r t i c u l a r ,
on s e c t i o n s .
M
Since
tdv = 0 ( f o r a form
we r e q u i r e t h a t 7.4.
vg.i4Qm-1/f w i t h compact s u p p o r t o r on r a p i d l y d e c r e a s i n g s e c t i o n s ) ,
KerTiDlm'tt/ .
The f o l l o w i n g r e s u l t i s obvious from t h e d e f i n i t i o n s .
(We sometimes omit V i n
t h e n o t a t i o n {«>i, «>2}r) • 7.5.
THEOREM.
If
r:AQ m K-*• D eT d e f i n e s a Hamiltonian s t r u c t u r e , then for any
we have {•"I.^ + H .
•"ljGKerr,
U)
{">!. {<%, ^ i ) + K . {<°i. 2>} + K . K - <°i}}GKer r .
25
In p a r t i c u l a r , the operation { . }r induces on AQmK/KerY
the s t r u c t u r e of a Lie algebra,
m
and T induces an imbedding of the Lie algebra AQ K/Ker T->Dev . A
n
7.6. Since KerficKerT , we can always represent T as a composition AQmK->Im8-*• D,ev Indeed, lm^CzAQ1A(^mK , and our operator B in a l l concrete cases w i l l be a d i f f e r e n t i a l operator defined on the whole module AQ1A(j2mK. We c a l l such an operator a Hamiltonian if r = 5 o 8 defines a Hamiltonian s t r u c t u r e . 7.7.
LEMMA. For any a d d i t i v e operator 5:Im8-»-Z)ev we have £8 {«!, m2}B6 = B8 (/Btol8u>2).
Proof-.
For any X^Dtv,
(12)
by formula (4) and item 5.1 we have Ijf<»2 == ijf8
Setting here JT^B^ 7.8.
and applying (10) with r = 5 8 , we obtain (12).
We rewrite the Hamiltonian condition (11) of the operator T = 5 8 in terms of
structures related to the choice of a local coordinate system (Xj, at). Let us f i r s t agree on the following notation. the module of column vectors of height
Let 5s be an s
n with elements in 3 - .
A-module. 8
If
3*" denotes
P6S " , then
P'
denotes
the transposed row vector ( t w i l l generally denote the transpose of a matrix of any s i z e ) . If QeAn, P£$>n, then Q'P^
•7=:A-*-An takes
The operator /6P
^=|
W 5u /.
we wrxte
denotes the scalar product
OH
P&A into the vector
2^p<•?= = (£)
8tf
\otti;
(cf. 4 . 2 ) .
In place of
-=-.
6a'
Further, l e t A[DC\ be the ring of d i f f e r e n t i a l operators over A, generated by t o t a l d i f f e r e n t i a t i o n s along the base.
For any P&A we denote by D 0/ (P)(
Frechet derivative
^ < p > = 2 £ p 7 ^ *=aj....a!-,
as the
(nXtl)
matrix (not the determinant) with the operator Du.(Pi) at the site (//) . Its invariant interpretation will become clear in the next section. In order to write the Hamiltonian condition (11) in this notation, we identify the m
m
modules AQ K, A2M 0 2 A' and £>ev, respectively with A, A", A" m
mappings: PdxlA—Adxm^-P,^Pidutdx1A-Ad x^(Pt),X<^(Xai) l
n
amples lies in M„(A[DC\) 7.9. we have
26
Proposition.
by means of the following
for XeAD(AolK). n
operator B:AQ A0Q K-*-Dev
A-
will be represented by an operator B:A -*-A",
Then the
which in all ex-
. The operator B:An-i-An
is Hamiltonian if and only if for any P, Q(*A
Proof.
We s h a l l verify that the l e f t and r i g h t sides of (13) are i d e n t i c a l with the oh = Qdmx, <»2 = Pdmx
l e f t and r i g h t sides of (11), r e s p e c t i v e l y , for T=Bl,
According to (12), the l e f t side of (11) i s £3(iB^fi^) 8u>, i s represented by the vector
-=; ; the field
.
• According to 4 . 2 , the form
i s therefore represented by the vector
B&IDJ
n
B^S: . 6u
by
M8«>2)=2 AXuidmx
From the formula
^B~
.
i t follows t h a t the form
fZTi °"<
F i n a l l y , the operator fio = r
J'B&D.H
i s represented
i s represented by the operator B-= .
This
reduces the l e f t side of (11) to the l e f t side of (13). In order to compute the r i g h t side of (11), we note f i r s t of a l l that the field Bow2 8P
i s represented by the vector
—
—
Br= .
Further, for any R&A and X^D&V
we have XR —
n
2 — % {Xut)W = ^ Du(R)XUil i.fc
du
<-l
i
this explains the meaning of the Frechet derivatives. Thus,
'
_ _ for any vector R^A" we have XR = D(R)Xu
.
Applying t h i s to the case X=Bhml
we write BIUUBOM^U in the form £ ) [ £ — ) B - S .
Similarly, the second term of the commutator
gives the second term of the r i g h t side of (13). 7.10.
— gp and R = B^ ,
This completes the proof.
We s h a l l now concern ourselves with transformation of the c r i t e r i o n (13). We
r e c a l l f i r s t the formalism of the adjoints of d i f f e r e n t i a l o p e r a t o r s . have A[Dc] = A[du...,dn],
[dhdj]=0.
Let L£Mn {A [Dc]) 5
which act in the natural way on S ", where 5
s
(the ring of
i s any
In a local chart we
(nX n) matrix operators
A [D c ]-module).
We define an additive
+
mapping Z.—Z.+, by s e t t i n g for any matrix aeM„(j4):(ad*) = ( — l)l*id*°a', where
a' i s the transpose
of the matrix a . The following r e s u l t i s c l a s s i c a l (and can be deduced without d i f f i c u l t y from 3.3 and 3.4). LEMMA, a) For any L, M we have (Z.+)+ = L, (LM)+=M*L+. b) For any PeA",
7.11. where
5
S
Q&\
i s an A [Dc\ -module we have P(LQ-
{L+pyQedlg;>+... +dng>.
The operator L i s called formally symmetric (respectively, skew-symmetric) if (respectively,
IS=L
L*=—L ) .
The next r e s u l t i s obtained immediately from Lemma 5.8 and the d e f i n i t i o n of the Frechet derivative. 7.12.
LEMMA. For any element P£A the operator D (^=\ i s symmetric.
I t i s moreover clear from the considerations of 5.7 that the part of the v a r i a t i o n a l complex AQmK^-AQ1A0QnK-*AWQlAWQm+1Kil) with the i d e n t i f i c a t i o n s of 7.8 may be replaced by
27
6/4a D(-)-D+()
A+A* + M„(A[Dt]), and the theorem of B. A. Kupershmidt on the exactness of t h i s complex means that D+(Q) = D(Q), if and only if there e x i s t s a PQA, such that Q=?=T=- There i s thus an effective c r i t e r i o n _ 6u for determining if a vector QcA" i s a vector of p a r t i a l derivatives of some element of A. Suppose now that BQM„(A[DC]) i s some d i f f e r e n t i a l operator. 7.13.
THEOREM,
a) If B i s skew-symmetric and BeM„(K[Dc\) , then B i s Hamiltonian.
b) If B i s skew-symmetric and Bf<Mn(A0[Dc]) , then i n order that B be Hamiltonian i t i s necessary and s u f f i c i e n t that for a l l P, Q(*A the following i d e n t i t y hold:
[D,B}^B^-[D,B]^B6Z=B[^-d-i^\
(14)
We begin with t h e following lemma. 7.14.
LEMMA. Suppose that B&Mn(A>[Dc]) i s skew-symmetric.
±mB^)=D(^)Bb4-OmB^
Then
+8Zd-*%
(15)
where the last term is a column vector with coordinate - = 3 — = at the i-th row, and 3— is flu d"/6«
<*"<
the r e s u l t of applying ^— to the coefficients of the operator B. Proof.
By t h e d e f i n i t i o n s
Further,
'©*3*-)-['w*s+^«'§+ff'ttg-* m
2 ^ iditias u
C7ZJ
an
d 2i4 i s considered as an A-bimodule with m u l t i p l i c a t i o n
u>P=Pa> for
any PGA, a> Therefore, for any X(
Using the formula XR=D(R)Xu
, we rewrite the right side of (17) in the form
[(*SiO (I) *">S>(1)«+ (SIf)'*}
<«>
We now note that for any R&A we have djRdmx = ( — l)'-ITflf (/?#"*). Therefore, transposing the operators i n (18) while replacing them by t h e i r a d j o i n t s , as i n Lemma 7.11 a ) , we do not 28
change the value of the r i g h t side modulo Imzd. In the f i r s t term of (18) we transposed the operator operator 5£>(^£) .
D ^ = l a n d in the second term the
Recalling t h a t B i s skew-symmetric while D ^ = ) and £> ^ £ ) are symmetric,
we obtain f i n a l l y
From the uniqueness of the Euler-Lagrange operator i t follows that the form following on the r i g h t side of (19) i s id (*£t B^Ld^x} 7.15.
Proof of Theorem 7.13.
If BeMn(K[Dc]) , then to the vector Xu£A\ sions coincide, since
^f=0XeDeV,
.
%
This and (16) give the assertion of the lemma.
By formula (15) the l e f t side of (13) i s equal to
Moreover, in t h i s case they give BX^
BD
[%)=D(B^)*
and XB ^ p ,
K[DC] commutes with Z>ev .
s i n c e when
a
PPlied
respectively, and these expres-
Thus, the c r i t e r i o n (13) i s s a t i s f i e d in
t h i s case. Using the information obtained, for B^AQ\DC\ formula (14) obviously coincides with c r i t e r i o n (13). In Sec. 8 we s h a l l apply the c r i t e r i o n (14) to prove that two special operators B over a one-dimensional base are Hamiltonian: the Gel'fand-ilikii operator and the Benney operator. 7.16. r.:N->M .
Hamiltonian Conservation Laws.
Suppose V defines a Hamiltonian structure on
For any form m^QrfjdA . . . /\dx„£AQmK
system of evolution equations u,*=T(
io or Q
i t s Hamiltonian and B
The form ux = Pdxl/\
to the field
I »
there corresponds the
in the notation of the preceding s e c t i o n s .
the corresponding Hamiltonian operator.
. . . /\dxm , or the coefficient P, i s called an integral or a conserva-
tion law for t h i s system if
{
t h i s means that for an evolution
In the corresponding analytic formulation
s, because of our system,
dt
M
M
M
i.e., \<»l is a quantity conserved in time.
In the notation of 7.8-7.9 P is an integral
M
for the Hamiltonian Q, if
8u \ 6a'
Su J
29
The integrals Pi and P 2 are said to commute if {<«i,<»2}r€Ker8, i.e., ou V, ou1
ou I
We note in conclusion that if {<«>, ..., {u>,«>,}..;}gKer 8 (k>2
factors
sections s, which are uniquely included in the flow with Hamiltonian co, it is possible to \
find k— 1 quantities conserved in time as follows. We have \-§r) '
{(o.u),}.. .}=0 . Therefore W
M
M
is a polynomial in t of degree k — l in general.
M
Normalizing
M
the initial time on a given trajectory of the flow so that the (k — 2)-th coefficient of the polynomial is zero, we find that the remaining k — 1 coefficients are invariantly defined, conserved quantities. 7.17.
Stationary Manifolds of Integrals. Let wt be a conservation law for a Hamilton-
ian system of evolution corresponding to a field
r(i») . By definition Zr^.i^Glm"^ or
ir(»)8«)iGlnitrf . The latter condition means that if s is an extremal of the Lagrangian
ID, ,
then within a short time in the linear approximation the evolution of s, under the field r(u>), will also be an extremal or that the "field extremals u>1 ."
T(m) is tangent to the manifolds of
It is possible to give a precise meaning to the last assertion and to prove
it at least for /re=l and polynomial integrals «>i, when the manifold of extremals W can be identified with a finite algebraic manifold by assigning to each extremal its value at x — 0. Then T (co) defines on it a flow which is also Hamiltonian with Hamiltonian computed on the basis of «) and m1. W
This assertion is nontrival because the natural class of functions on
is not obtained by restricting functionals W to
functions on
W in the natural structure of W.
T I it consists, for example, of smooth For a precise formulation and proof see
the papers of Bogoyavlenskii and Novikov [1] and also Gel'fand and Dikii [3], We shall make use of this remark in Chap. 3 where in place of the equations we will solve jointly the system ut=T(
and 8'J>1=0 , where wt
is an integral.
ut=T(m)u It is
just this procedure which distinguishes in an invariant way the class of multisolution and finite-zone solutions of the Korteweg—de Vries equation as already mentioned in the introduction. 7.18. Examples, a) Let
n = 2r , and B = \ E ~ Q \ * where E is the identity matrix of —•
go
order r . The system of evolution at = B—=- is traditionally called a Hamiltonian system 6u with Hamiltonian P in "canonical coordinates". It is also Hamiltonian in our sense according to Theorem 7.13 a ) , since B is a differential operator of order zero and B*=—B . Below we shall show how it is possible to define a "cotangent fibration" n:Ar->Af to any fibration ir:Af->./W, a canonical Hamiltonian structure on it, and to each local coordinate system (xu uj) on N a system of coordinates (xit Uj, Vj) on N, such that in these coordinates the operator B corresponding to this structure has the form ( E J.
30
b) Let
m=l
and B — d = di .
That equations of the form ut = d-s—, which include the
usual and higher Korteweg—de Vries equations, are Hamiltonian was established by Lax in the functional version and in terms of Fourier coefficients by Gardner.
Our proof of Theorem
7.13 i s a generalization of the proof of Lax as reworked by M. A. Shubin.
A formalized
version of Gardner's arguments w i l l be given at the end of t h i s section. 7.19. The Cotangent F i b r a t i o n . The cotangent fib r a t ion to TZ:N^-M we c a l l the fib r a t i o n lz:N-+M, where N = T*{N!M)®Amit*T* (M). Here T(N/M) i s the tangent bundle to N along the fiber
"*, T (M)
i s the tangent bundle to M , and the a s t e r i s k denotes dualization; _
_ a
it
the tensor product i s taken over vV ; v i s the composition N-+N-+M, where a:N-+N
i s the
natural projection. Sections of a a r e n a t u r a l l y identified (locally) with forms in Q1A0QmKLet [Xi, Uj) be coordinates on N. On the basis of these i t is natural to define coordinates (xt, Uj, vj) on tf (/ = 1> • • •.») as follows. If x£N l i e s on a section of o, c o r r e al sponding to the form ^lPjduJ/\d'nx , then vj{x)=P1{a(x)). i 1 _ " Let A) be the ring of smooth functions on ic-1((/), where U i s a neighborhood with coordinates
(x).
to the projections We consider section
Then A0 = C°° (xt; u}, vj),
and the natural imbeddings KdA0c:A0
correspond
i: and o . P = 2 VjdU]Admx(*AQQ1(A0)QmK. I t possesses the following property: if the
s of the f i b r a t i o n
a corresponds to the form
to see that t h i s property determines
p uniquely.
uigQ1 (Aa)QnK » then p*=
Thus, f
I t i s easy
i s defined canonically and
globally. Let
Dev be the evolution d i f f e r e n t i a t i o n s for
it" .
XQDey , then i-^d^lAaQmK
If
and
the formula i jd p = 2 (Xvjduj—Xttjdvj) d'x /-i
shows that t h i s mapping Dev-+QlAQQ"K i s an isomorphism of l
m
inverse operator, B:Q A0Q K-+Dev
A0-modules.
We denote by B the
, and we set r=-58:I0Qm/r-vDeV.
7.20.
THEOREM.
In the coordinates (xh a,, vj) on N with the conventions of 7.8 (as
z
applied to w ) the operator —B
i s represented by the matrix
/0 - £ \ I» n ) »
the
°Perator
^ is
Hamiltonian, and the system of evolution corresponding to the Hamiltonian Pdmx(
6P
«
*P
31
All t h i s follows readily from the definitions and Theorem 7.13 a ) . We note further that the c l a s s i c a l expression !-2fL.-Q--- PJL\dmx i s equal to iBta,B^m>2 where v\=Pdmx, Qdnx . 7.21.
The Hamiltonian Property in Formal Fourier Coefficients.
In t h i s and the follow-
ing sections we b r i e f l y describe the formalism of v a r i a t i o n a l calculus in terms of Fourier coefficients in the simplest case: M i s the unit c i r c l e , Af = vWXR .
We r e s t r i c t ourselves
to Hamiltonians which a r e polynomials in u and do not depend e x p l i c i t l y on s t a r t from the algebra A = C[u, u',...].
x , i . e . , we
We s h a l l f i r s t introduce formal Fourier coefficients oo
(•n„|ftgZ)
"Vne2Kinx. As the Fourier
, assuming that the Fourier s e r i e s of u i s F(u) = 2 n^—oo
s e r i e s for uu) i t i s then natural to take F(uU))=^(2un.yvne2lti',x, n U)
The Fourier coefficients of the polynomials in u i n f i n i t e l y many v a r i a b l e s B = C[v„] .
vn .
w i l l , however, be special s e r i e s of
In order to introduce the corresponding r i n g , we f i r s t s e t
In t h i s ring we introduce the order function
OTd('^cl.v±\=mla^otdv±\c±^'0) then obviously
.
B = © BN
i . e . , F(a (y) ) = (2ic/By©n .
If
BN
ord l©^'.. .ti°*) = |«i|+• • •+|«*| and
i s the space ot polynomials purely of order
( d i r e c t sum). We s e t £? = I I BN •
N,
The elements of the space
OO
B
are i n f i n i t e s e r i e s
2 fft> /N^BN •
Since
ord(/£) = o r d / + o r d g
for f£BM, g§BN, B
N=0
has a natural ring structure: (2/v)(2 ^ ) = 2 A * '
K^/K-MSM
'
We call the ring
5((^iJr)) = ( 2 / ^„2xinx
fn£B, zc>0,
n0>0,v\n\>nQ,
ord/„>c|«|} the ring of formal Fourier s e r i e s ; i t has the natural m u l t i p l i c a t i o n \^ifme2,tlm \^hpe2"'px
, where hp = y
fmgn
.
The convergence of the s e r i e s for hp in B
x
) (2w2"iw)= i s ensured
m+n=p
by the fact that OTd(fmgn)>c(\m\ 2 ix
B((e " ))
+ \n\)-+co together with
, because ord/t„> min c(\m\ +
\m\, \n\ .
The product l i e s in
\n\)>c\p\.
m+n=p
We extend to B((e2mx))
the differentiation -^-
by continuity: we extend from B to
B and then set -^-(e2ni"x) = 0 . This is clearly possible, since OTd[~-)>otd/~n.. We similarly define a differentiation d, which on B has the form dvn = 2itivn. We introduce a ring homomorphism fM-*^/ 1 8 ')), by defining F(a) = 2 " / ' " . It is not hard to see that F
is an imbedding which commutes with d. Let
for any P g A 7.22. LEMMA. For any P&4
32
we have
F
n[-^-)=-^-F0(P).
F(ua))=dJF(u)
F(P)=^Fm(P)e'2Mmx
Proof. We consider the two mappings of A-+B({en
They are both differentiations of (;)
Further, on the generators «
A
into the
)):
A -module B((e2Mx))
they coincide, taking
(relative to F ) .
a<'> into (2i:in.y'e2ainx'. They therefore
coincide everywhere, whence
We note now that F0°d = 0,
so that we may "integrate by parts" under the sign of
F0 , and
the last expression is equal to
'•(§'--<-•>'*«'(&))-M& which proves the lemma. 7.23.
Corollary.
6P The equation «/ = 1.
Thus, s e t t i n g pn=v_n,
4n = %^* H(p, q) = iF0(P), we find that on the "hyperplanes" z>0=const
our equation formally has the Hamiltonian form dp!L=_dH_ dt dqn'
dt
dqndH dPn
Further, the Poisson bracket may be w r i t t e n
^ S
= (2al^o(Q)e^
whence 00
FoiQ, -P} = 2(-2w«) a -^-F 0 (Q) 5 | x F 0 (P) = n—oo
in agreement with the c l a s s i c a l formalism. 8.
Special Hamiltonian Operators Over a One-Dimensional Base
8 . 1 . In t h i s section we s e t tn — l, Xi—x, d=dl=dldx, K=Caa{x), and we work with the sequence of rings -An = U C°°(JC; a<*> | 0 < / < r t , 0 < & < i ) , and also with t h e i r union A=\jA. . We point out t h a t the enumeration of the variables Uj begins with zero in contrast to the preceding conventions. In correspondence with t h i s the enumeration of the elements in the rows and columns and also the enumeration of rows and columns in different matrices also begins with zero. 33
We define the operator
Bn^Mn^ (An [
B « = 2 [Bn.id'-(-iyd'oB'Hil],
(Bn,i)ab
0 for a+b + i + l>n, (' + ") for a+b + i +
BnjeM^M,
a+b + i + l=n+2.
These operators were introduced by Gel'fand and Dikii [ 4 ] .
l^n+2,
Following their, work, we s h a l l
show in the next chapter t h a t the Lax equations Lt = [P,L\ , where I = d n + 2 + 2 utdl are _ «-o Hamiltonian in the s t r u c t u r e defined by the operator Bn, with a s u i t a b l e Hamiltonian depending on P. We further introduce the operator B^Maa(A[d]) by the formulas B=Bd + doB', BeM„(A), fl
(fiu={;0"a+6-i for Here Ma,{A)
(21)
for
a+b>l, a+b=0.
denotes the group of matrices with an infinite number of rows and columns down-
ward and to the right beginning with the zero (index) columns and rows. MociAld])
The elements of
are considered as left operators on the A -module Af of finitely supported
infinite columns with elements in A (finitely supported means that only a finite number of the coordinates are nonzero).
The range of such an operator lies in the A -module A°° of
all infinite columns. The scalar product PlQ is defined if at least one of the two vectors 6P
— —
P, Q
is finitely supported.
All vectors T = for P(*A are finitely supported, and there-
fore the B-= are defined but are not necessarily finitely supported. Frechet Jacobian D(P) ly large j
Each row of the
is finitely supported for any P£A°° , since DUi(Pi) = 0 for sufficient-
(depending on i ; see the definition in 7.8). Therefore, D(P) 00
to any vector of A
.
may be applied
The left and right sides of equalities (13) and (14) of Sec. 7 are
thus defined, and we take them as the definition that B
be Hamiltonian in the ring A
of
functions on the space of jets of the corresponding "infinite-dimensional" fibration (the projective limit of the finite-dimensional fibrations).
It is not hard to verify that the
remaining constructs of Sec. 7 with appropriate modifications carry over to A. The operator (21) was introduced in the work of Kupershmidt and the author [19] to study the system of equations for long waves with a free surface suggested by Benney. sequence of unknown functions
The infinite
u„, n > 0 , is, in fact, the sequence of moments of the horizon-
tal component of the velocity, and the system of Benney's evolution equations for it is found to be Hamiltonian with operator B
and a corresponding Hamiltonian.
Details may be found in
the next chapter. The following theorem is the main result of this section. 8.2.
THEOREM.
The operators
Bn and B are Hamiltonian.
The proof follows by means
of lengthy computations. Among the reasons for this is probably the fact that we do not know 34
an invariant (coordinate-free) characterization of these operators. We note that according to formula (20) there is also a formal "limit" Bx operators B n , which is a matrix operator of infinite order. Boo , by setting « n+2 =l, Uj=0 for j=n+l,.j>n+2.
from
The operator
of the
B„ is obtained
Although we are not in a position to
assign Boo a substantial meaning, in the computations to verify the Hamiltonian property it is possible to work directly with
Boo, since the identities of interest to us will
obviously be preserved under the substitution
«j = 0 for j>n, j^n+2,
These identities are obtained in the following manner. fiO and -a we consider the columns AT=(X0, Xu ...)'
u n+2 =l.
In place of the columns
and Y = (Y0, Y\, • • •)' with formal variables
as coordinates; the action of d* on them is interpreted as the conversion of Xt, Yk i
i
(
the formal variables d Xi = X i^, d'Yk=Y l>),
— 6u
into
independently of one another and of a<" . Each
element of- the left and right sides of (14) will then be a formal infinite sum of monomials in a, X, Y and their derivatives which is trilinear in u, X, Y ; we simply verify that the coefficients of each such monomial at corresponding places on the left and right coincide. (Here X and Y are not to be confused with the previous notation for fields on jets!) 8.3.
For the reformulation of the Hamiltonian criterion (14) which we shall use below,
we introduce some further notation. Let P,QdA DUj(PQ)=PDUi(Q)
+ QDu.(P)
. Since DUj is a differentiation, we have
, or in vector form D-t (PQ)=PD-t
(Q)+QD-t
(P), where D-,=(DB., D a „ ...).
Now let P, Q be columns. From the last equality it then follows immediately that D-,(P'Q) = P~'D(Q)+~QtD(P) .
Finally, let C
be a matrix and Z
a column.
Then CZ is a column, and
D(CZ)=CD(Z)+Z'D(C),
(22)
where the expression ZlD(
which contains at the
infinite column containing at the finite matrix with
t - t h row 2
th element the operator
d.
Gi={i-th place C)', D(C) i s an
i - t h place the matrix **
D{Ct), and Z'D(C)
i s an in-
( k-tb. row of -D(C,))=2 Z„D-t(Clk) and hence with
ij
d , the identity
d(Pdx) =^iDU[(P)dui/\dx
implies that DUl and
From (22) we therefore find that D(CdlZ) = CD(d'Z) + (dlZyD(C)=:Cdl°D(Z)+Zw'D(C). D (
where
ZW = d'Z and the circle following
example, the notation
-
^ ZkDUj(Cik) . *>o
Since d commutes with D commute with
z
f-th place
C is a transfinite object —
dl on the right is inserted in order that, for
dl°CD(Z) not be interpreted as (d'C)D(Z).
In this notation the criterion (14) after obvious calculations may be rewritten in the following manner.
35
8.4. (—lydloBj)
The Hamiltonian property of the skew-symmetric operator -^{Bfl1 —
Proposition.
is equivalent to the following identities between the infinite columns:
B (y,X'd-i/YU)-X'U)dlLy)
= (y XU)td-p--(-\)ldioXt%\BY
-(M same with X~Y).
(23)
Here the r i g h t and l e f t sides of (23) correspond to the l e f t and r i g h t sides of ( 1 4 ) , dBjldue i s the column of i n -
respectively, and the following additional notation i s used: f i n i t e matrices containing a t the i - t h place the matrix _ , _ the matrix with (t*)-element 2^ Xt ^ (Bj,u); B'j={B))~. 8.5.
4= ( i - t h row of B)) ; XU)t-=L da du'
is
We begin with the verification that the operator (21) is Hamiltonian. The
identity (23) to be verified assumes the form {X'—dX, etc.) (Bd+d°B')(X'¥Y'-X''dJ?v\=(xt'
§t+doX'
<24)
~)(Bd+d°B<)Y-(iiie*mevirhX~Y).
We expand here all the brackets. On both sides there are sums of monomials each of which contains A""' and YW (possibly transposed).
The ordered pair
ij we call the type of the
corresponding monomial. By collecting all terms of the same type, we bring (24) to the form BrX<M-Y'+(B+B')X'J^Y"-Bt'Xt'^Y+(B+B^Xi'-^Y,-{B+B<)Xi'i¥!rY'du
du
du
du
-(B+B')Xr^-Y=X'^rBrY+X'^r(B'+2Bl')Y' + Xr
+ X'^-(B+B')Y" du'
du
da'
+BI)
B''Y+Xr
dlB
du'
du'
d B
( ±B') (B + BWdu'
du
(25) + (the same with X~Y).
We shall prove that this is an identity by showing that the terms of each type on the left and right cancel. All calculations making use of the concrete form of the matrix B we reduce to the following lemma. 8.6. LEMMA. For any i>0, and any X and Y the following assertions hold: a) The expression X'-^-Bi(i)Y
is symmetric in X and Y .
du*
b)
(B+B')WX'-^Y=X'^r(B+B'Yl,Y.
c)
W
da
da' Ht)
B X'-¥LY=Y'£rB X. du
da'
d) The expression X'~(B+BlYl)
Y is symmetric in X and Y .
du*
Proof, a) and d ) . •S-(Btab)'=^iXl>bh,a+b-\
The matrix X'-^z-
du'
has at the ad -th place the element
, where 8 is the Kronecker symbol. Further, (B'^)dc = cu^lc_1 ADt
fore, the a-th element of the column
X'-^=r-Y
is equal to
ou*
2
b,c>0 d=a+b-l
36
bC *»*"»iVl K «=2 ^U+c-2X>Yc b,c>0
2 Xb
£5,
. There-
It is obvious that this expression is symmetric in X , Xl-^=-
Similarly, the matrix
Y.
has at the ad -th place 2 Xb-g— (Bfl(,)=2 Xba&d,a-y>-i ,
°~a
d
b>0
0 a
while the matrix (B+B')' t the dc-th. s i t e has the element element of the column X'-^-iB+B'Y^Y is du' »,cX>
6X>
(c + d) a ' ^ - i »
so
that
the
fl
~th
»,c>0
and this expression is symmetric in X, Y. b) The element at the d -th place of the column X'—=-Y Further, (B+B')$=(a+ d )"£V-r is equal to 2
b
is X'-*—Y
= ^, Xbbhd,b+c-iYc
.
Therefore, the a-th element of the column (B+£<) ( l ) X'-^LY
(«+ * + c ~ ! ) " i ' V * - ^ ' '
»,c>0
In analogy with the computation at the start of the proof, the a-th element of the column X'^-(B + B')WY i s *£-K da' 2
Xbb(c+d)tt^d_xYe=
»,cX> d—a+b-l
2b(a+b
+ c-l)u«J>b+c_2XbYc.
»,cX>
c) As above, the a - t h element of the column BWX'^M-Y
is
S abUaU+c-2XbYe . t>,c>0
On the other hand, the matrix y^ald,a+c-iYc 2
, and (B*)lJl = bui%d-i •
Kjafrtta+j+e-z^Tj .
Y'-^=j has a t the Therefore, the
ad - t h place the element a-th element of Y*~BmX
^.Ycr~{Bacy= i s equal to
The proof of the lemma i s complete.
b,c>0
8.7. Verification of the Identity (25). Type 00. On the left there are no such terms, while on the right there is X'-^-B'Y du' Type 01. We must verify that
— (the same with X**Y
B'X'6AY-^X'-^riB' du du1
+ 2BnY'-Y1'
d
) , i.e., zero by Lemma 8.6 a ) .
< g + B '> out
B'X.
On the r i g h t there i s the monomial X'—zr-B'Y' and a similar term with opposite sign and du' X ++K\ which cancel by Lemma 8.6a). The remaining terms we transform in accordance with Lemma 8.6 b) and c ) : X*2j!r(B,+Bt')Y'-(B,+Bl')X'-£-r\ du' du , i -Y' -^-B X=-B'X'M-Y'. du' du Their sum is obviously equal to the monomial on the left side. Type 02. We have the equality at once and it follows from Lemma 8.6 b) .
37
Type 10. We must verify that d
(B±~B">
-B'X'^Y^X'' du
By Lemma 8.6 a) the term X
BtY-Y'^-(B'+2Bt')X'.
du'
du'
y^-5 Y on the right cancels with the term symmetric to it. We
transform the remaining terms according to Lemma 8.6 c) and b ) :
X' '2i-BtY=B'Y' du'
-?2-X', du
-Y'-^rdu' (B'+B')=-(B'+B')Yt^-X'. du The sum of the right sides is equal to — B'Y'-Jl-X'. that Y'-j=-X'=X
The proof is completed by the remark
-jF^Yi which is immediately evident from the definition of these expressions.
Type 11. We must verify that
(B+B^X'-^-Y'-iB+B^X'-^Y'^X'^^-iB+B^Y'-Y'-
*&+»"> (B+B')X>.
By transforming the first term on the left according to Lemma 8.6 b) and cancelling with the corresponding term on the right, we obtain the equivalent identity:
-(B+B')X''d^LY'=Xt' du
*£-(B+B<)Y' -Yr du*
On the left we interchange X and Y, replacing B'
by
*@±]*l(BJr.B<)X\ du' * B,
again apply Lemma 8.6 b ) , and
cancel by Lemma 8.6 d ) : l 0=X''-¥L(B+B<)Y'-Y du'
'du' J!L(B+B<)X:
Type 20. We must verify that
-(B+Bf)X''^Y^ du On the left we make the change X"**Y
-Y' du' — iB+B^X".
and B'^-B ; applying Lemma 8.6 b ) , we obtain the re-
quired result. This completes the proof that the operator (21) is Hamiltonian. We now proceed to verify that the operator (20) is Hamiltonian. put B=Bca = llmBn
(formal limit), B,=llmB„,j
It
;X
In criterion (23) we
and Y are infinite columns.
n
The next lemma is verified by straightforward computations using (20), and we limit ourselves to formulating it.
38
8.8.
LEMMA,
a) I n t h e
a - t h element of t h e column ^.BXt^L-YW )
of a monomial of t h e form
only t h e
coefficient
8)
X
to + p + 6-/)! S r(a L aipi(6—/)!
(« + /)! (a + p + 6 - / ) ! (? + /)'1 at/! ~~ a! pi (6—/)! y!/I J*
Here and below a l l i n d i c e s a r e i n t e g e r s c o n s i d e r e d e q u a l to zero for y > 8 b) I n t h e
> 0 ; terms w i t h
(respectively,
-
(respectively,
-^=-K only t h e c o e f f i c i e n t of a monomial
)
of t h e form A tf>K
1VH.A+<+.yrr
K
iw
^LV
'
c) In t h e
(a + P + V + 6+Ti + l)!
a - t h element of t h e column
XWF<, 8) aW ry+ p +v+fl+n+2
(g + /)|
M « + T + /+1)1 PI («-/)«»ll
can be n o n z e r o .
For
, , l
«!/!
\^XW TJ = 0
(f5 —/)l ) a r e
7>P ).
^^BX(^J)
O-th element of t h e column
(8—J)[
(26)
(cc + p + v + 6 + ti + l)!
(? + /)! 1
, U / ;
M a + T + /+l)!8!(p-/)!tl! " W T J -
-rJj-BY
only t h e c o e f f i c i e n t of
this coefficient
(q+p)!r(a + ct + p + 6+l)! , a\f>\ L(B + a + P+l)!6! " ^
i s equal to
i s e q u a l to
n f t (T
+ 6)'l V! 6! J'
'
,00. ^Z8'
For r(^=0 this coefficient is equal to /
n„4*(T
v
'
+ 6 + r))!(q + P)! V!8!T)!
a!p! '
(29)
d) In the a-th element of the column ZJ(~iy+ldJ°X<-?J-BY X4e)^(6)Ba+a+B+V+6+T.+2' c a n V,
^6V
1Vu.fi+/+„+i
'
De
only the coefficient of ou'
•*••
nonzero, and this coefficient is equal to
(g + P + a-Z + iDICa + a + P + e + tn-l)! (a + a + p + 6 - / + r i + l ) ! a ! p ! ( 6 - / ) ! t ) ! / !
y J ^ ^
1tf+
'
^ ( « + P +fe+ Ql(V + 6 + Tl-fe-0l a! pi Y! ( 6 - 0 ! ( l - * ) l *l "
*
,^n.
K
* '
We remark t h a t p a r t s a) and b) of t h i s lemma r e f e r to t h e e n t i r e l e f t s i d e of ( 2 3 ) , w h i l e p a r t s c) and d) r e f e r only to h a l f of t h e r i g h t s i d e , t h e remaining terms b e i n g cons i d e r e d by permuting
X++Y , i . e . , the p a i r of i n d i c e s (a, B) **~(y, 6 ) .
We now proceed to d e s c r i b e t h e c a n c e l l a t i o n s of s i m i l a r t e r m s . 8.9.
LEMMA. + P + 6-/)! 2. (qa!p!(6—/)! 6
Proof.
A f t e r d i v i s i o n by
(a
g^
)!
(« + /)! _(a+P)! (a + a + P + 6 + 1)! etl/1 — a!?! (« + a + p + l)!6! *
t h e sum on t h e l e f t i s t h e c o e f f i c i e n t of xa^YaTb
in
t h e polynomial 2 {X + 7 T + P + 1 W {Y + T)a+t
=(X + T)a+*(K+Tf
(X + T)^-(Y
+ Tf+^
j<6
39
Separating from the two first factors the coefficient of XaJ^-rYa-s, s
l
and (Y + T?+l
the coefficient of X ' Y T i (after the binary expansion of (X + Tf+ by
X — Y) , w e obtain
{l+T^{\+TY+*(l
8.10.
2
and division
( t )l ,)'?)» which is the coefficient of T6 in the polynomial
+ Tr , i.e., just COROLLARY.
and from the third
{
i ; « ^ +l .
All terms of the form (26) cancel with the first term of (28) and its
transform under the substitution (a, P)*-*-(T. 8) . 8.11.
LEMMA.
2/
iw
(« + T + 6 + l)I
(a + /)l
.
1v>(Y +
8)!_
Proof. W e make use of the identity (q + y + 6+1)! (6-/)!(a + T + /+I)!
(tt + y + 6)! . (g+ Y+ 6)! (6-/-l)!(a + v + /+I)! "•"(6-/)! (a + y + /)! '
8.12.
COROLLARY.
The expression (27) is equal to
8.13.
COROLLARY.
The first sum in (30) is equal to fv _ 1V)+n+i (« + P + H)» (g + 6)l ' aipit)! «16t *
8.14.
COROLLARY.
The second sum in (30) is equal to , _ itf+j-H,(« + P + y + 6 + T| + l)l(« + p)l ' aipiaiT)l(« + p + 7+l)!
1
Proof. We first sum o n / for fixed k with the help of Lemma 8.9 (with different values of the parameters) and then sum on k using the same lemma. 8.15.
According to 8.12 and 8.14, the terms of (27) cancel in the second sum of (30)
and its transform under the substitution (a, B)-"-^, 6). The remaining term of (28) and the coefficient of (29) cancel with the transform of the first sum in (30) under the substitution (a, &)*-(y,
6) according to 8.13.
Finally, the last remaining terms cancel by symmetry also
by 8.13. CHAPTER II THE STRUCTURE OF THE BASIC EQUATIONS 1.
Introduction 1.1.
40
The principal purpose of this chapter is to describe the construction, algebraic
s t r u c t u r e , and conservation laws of some classes of nonlinear p a r t i a l d i f f e r e n t i a l equations and also ordinary d i f f e r e n t i a l equations r e l a t e d to them.
The introduction i s devoted to a
brief description of these c l a s s e s . 1.2. istic
Lax Equations.
Let
33 be some a s s o c i a t i v e algebra over a f i e l d of character-
k, which l i e s in the center of 53; l e t dr53->53 be a d i f f e r e n t i a t i o n of $ which i s
t r i v i a l on k .
The algebra of d i f f e r e n t i a l operators 53 [d] consists of a l l expressions of
the form 2 btdl, b0H , with the commutation r u l e d°b =db-irbd
.
If L=2Jbldl,
(—0
number N
bN=j=0 , the
1-0
i s called the order of the operator L and i s denoted by
known that the l e f t
S3-module of operators of order
ordi .
I t i s well
i s freely generated by l,d,...,d
•
Basic examples: a) variable
53 i s the ring (of germs) of smooth, analytic or meromorphic functions of the x, of the v a r i a b l e s
x, t , or of the variables
b) 53 = /fe[«<»|i=0,...,/t; j>0\;
d:u\'^a\'+^,
x, y, t; A=R or C; d = d/dx.
* = Q , R or C.
c) S = AJZ(5B0) i s the ring of (ZXO matrices over a ring 530 of the type described in a) or b ) ; d i s the unique extension of d from 530 to iW, (33Q) . We suppose further t h a t in $ there are defined two additional d i f f e r e n t i a t i o n s d2
such t h a t
d, dt, d2 are pairwise commutative.
b)
du d2 a r e some d i f f e r e n t i a t i o n s commuting with
t i o n f i e l d s of Chap. I .
dt and
In example a) di = d/dt, d2 = d/dy ; in example d , i . e . , formal analogues of the evolu-
We s h a l l usually write dt, dy in place of
For any L = ^btdl&^[d\ we s e t
d lt d2 in case b) as well. d2 .
The symbol
[P, L]
A pair of operators P, L&-3) [d] i s called a solution of the s t a t i o n a r y Lax equation (respectively, the Lax equation, the equation of Zakharov—Shabat) if [P,L]=0 diP=[P, L], diP+d2L=[P,
L]).
(respectively,
For b r e v i t y , we s h a l l often refer to a l l these equations as
Lax equations. We remark that in case b) (and similarly in the matrix case) the "unknowns" in the Lax equations, in addition to
P, L ± include the fields dlt d2, and solutions of these equations
in such rings correspond to what in algebraic geometry are called "generic points" of the algebraic manifolds represented by systems of algebraic equations.
From the more t r a d i t i o n a l
point of view, when working with these r i n g s , we shall investigate the s t r u c t u r e of the Lax equations themselves rather than t h e i r solutions in the sense of t r a d i t i o n a l a n a l y s i s .
The
l a t t e r case corresponds to rings of type a) and i s treated in Chap. I I I . The main r e s u l t s pertaining to Lax equations are proved in Sees. 2-5. These include the following: an e x p l i c i t description of operators P with the property ord[P, LJ^ord L—1 or ordi—2 ; establishing that the equations Lt=[P, L] are Hamiltonian over commutative rings of type b ) ; formalization of the method of Zakharov—Shabat. We make use of the
41
technique of fractional powers of Gel'fand and Dikii in a considerably simplified and formalized version. 1.3.
Another version based on the study of the resolvent may be found in [ 5 ] .
The Benney Equations.
This name we give to the following system of equations
for the two-dimensional motion of a nonviscous, incompressible fluid in a g r a v i t a t i o n a l field in the long-wave approximation: u(+uux—Uy^ux\
ht+\\udy\
d^ + hx=0,
=0.
Here — O O < J C < O O i s the horizontal coordinate;
the time;
u=u(x,
0<#
i s the v e r t i c a l coordinate;
t
is
//, t) is the horizontal component of the velocity a t the point (x, y)
at
time / ; h(x, t) i s the height of the free surface above the point notation
tit i s an abbreviation for
-gr u(x, y, t) , e t c .
(x, 0) a t time
The
The system of units is chosen so
that the g r a v i t a t i o n a l acceleration and the density are equal to one. \Ujcdfi a r i s e from the equation of continuity ux-\-vy=0
t.
Integrals of the type
where v i s the v e r t i c a l component
of the velocity and from the boundary condition v—0 a t y=0 ; these r e l a t i o n s make i t possible to eliminate v, by expressing i t in terms of «; •v(x,y,t) =—\uxdn o
•
For the r e -
maining details of the derivation see the work of Benney [27] who first discovered an unexpected property of the system: the existence of an infinite sequence of conservation laws for it. The Benney equations display a number of unusual properties. "We do now know of a Lax pair for them; the conservation laws in the interpretation of the present work (in contrast to the formal derivation of Benney) are obtained from a nonlinear integral equation with a parameter.
This equation enables us to obtain the conservation laws of Miura [46] for the
Benney system. We further establish the Hamiltonian character of the "reduced system" (with the additional condition
« y =0 ) and the commutativity of the reduced integrals. These
results are then generalized to the full system. Our exposition is based on the work of Kupershmidt and the author [19]. The main results pertaining to the Benney equations are formulated in more detail in Sec. 6; Sees. 7-13 are devoted to their proofs. 2.
The Commutator and Fractional Powers of Differential Operators *
2.1.
In t h i s section we begin the study of the Lax equations. If dtL=Lt = \P, L\ or
[P, £ ] = 0 , then in any case ord [P, L]
We therefore f i r s t investigate the conditions
under which the commutator of two d i f f e r e n t i a l operators has lower order. ff
We s e t
L=^undn, n=0
42
M
P= 2 m—0
v
mdm-
2.2.
LEMMA. Setting drtw = wia)
for any w(*S3, a > 0 , we have
i*4- 2
\[«)^)-(l)w%)}dm+a-a-
a,m,n>0L^
'
\ i
I
J
Proof. According to Leibnitz's rule d'(«zi)=2 (! a=0 *
\uwvu~a). '
2 . 3 . COROLLARY. The coefficient of dM+ff in [P, L] i s equal to [vM, uN] . ord[P, £ ] < M+N — 1, if «^6 i s contained in t h e center of 33 . 2.4.
[v, L] = [v, uN] d" +(Vu'N —NuNv)d"~l+
COROLLARY. For v£3d we have
order
In p a r t i c u l a r ,
(terms of
In p a r t i c u l a r , if the center of 33 i s infinite-dimensional over
k, then
the l i n e a r space of those P, for which ord [P, L\
COROLLARY.
on those coefficients
The coefficient of
[P, L] depends only
Vj of the operator P and t h e i r derivatives for which
j>k . The
terms depending on ?>*. Vk+i and t h e i r derivatives in t h i s coefficient have the form [**. «Ar]+ ['*+!• uN-i] + Mv^xUN-NaNv'k+v Below we shall always assume that duN—0).
(1)
ttN i s i n v e r t i b l e in 33 and i s a d-constant
(i.e.,
The operator L i s fixed, while P may vary.
2.6.
THEOREM.
If a v , a^_, l i e in the center of 33 and d i s the dimension over k of
the space of ^-constants i n 33 , then the dimension of the space of operator P&B\d\, for which
ordfP, L\
does not exceed
(M + \)d .
I t i s equal to {M + l)d, if d:33-+33 i s a
s u r j e c t i v e mapping ( i t i s possible to " i n t e g r a t e on x" in the ring 33 ) . Proof.
For k = M, M— 1, .. .,0, —1 we c a l l the "k-th equation" ( r e l a t i v e to N+k
condition t h a t the coefficient of d Af-th
in [P, L] vanish.
equation i s t r i v i a l l y s a t i s f i e d .
v;) the
I t i s evident from (1) that the
Further, for fixed vM,...,vk+2
the k-th equation,
according to ( 1 ) , has the form v'k+l = ii> , where w i s a polynomial in u^1, «*.a), vf), M>l>k
.
If i t i s solvable a t a l l , then any solution i s obtained from one solution by addition of any d -constant
in 33 .
I t i s c l e a r l y solvable if d:33-+33 i s s u r j e c t i v e .
This completes the
proof. 2.7.
COROLLARY.
If the d -constants in 33 coincide with
k ( e . g . , 33 i s the ring
;)
of germs of functions of x or 53 = fe[?ij ] ) , then the dimension of the space described in Theorem 2.6 does not exceed 7W+1 .
( I t w i l l be shown below that in the commutative case
i t i s equal to M-\-\, even if d:33-*33 i s not s u r j e c t i v e . ) For matrix rings 33 i t i s possible to obtain an analogous estimate under l e s s r e s t r i c t i v e assumptions which we now axiomatize. 2.8.
Let adu^:33-^33 be the operator
&&UN(p) = [u,N, b\ .
We s e t 33+=Ker adttAf, 33~—lm ad«j\r
and assume that the following conditions a r e s a t i s f i e d : 43
a) $ = $ + @ 3 r
(as a
b) The o p e r a t o r
^ + S _ c ^ " , S"S+CS".
k -space);
adUff-33~-*3B~ i s b i j e c t i v e .
c)
If for
WG33 we d e n o t e by
w + , w~ t h e components of •© i n
UN t h e c o n d i t i o n s a ) - c ) a r e s a t i s f i e d then we c a l l
commutative r i n g
SB every element i s s e m i s i m p l e :
+
$ = $ J , 98~={0} .
S + and 53", r e s p e c -
UN s e m i s i m p l e .
In a
i n t h e g e n e r a l c a s e our
terminology i s m o t i v a t e d by t h e f o l l o w i n g example. 2.9.
Example.
nonzero elements sites
Let
cfck
S3Q be commutative,
on t h e d i a g o n a l .
for which c^Cj,
and
$B=Ml(3B0),
Then
and
UN be a d i a g o n a l m a t r i x w i t h
SB* c o n s i s t s of m a t r i c e s w i t h zeros a t t h o s e
SBT of m a t r i c e s a t t h e s i t e s
ij,
for which
easy to v e r i f y t h a t a l l c o n d i t i o n s 2.8 a ) , b ) , and c) a r e s a t i s f i e d . semisimple m a t r i x of
Mt(k)
ct = Cj.
It
is
This i m p l i e s t h a t any
i s a l s o semisimple i n our s e n s e of t h e word.
As i s known, t h e
converse i s also t r u e . I n c o n d i t i o n s 2.8 we d e n o t e by d t h e dimension of
KetdnSB*
over k .
In Example 2.9
i t i s equal to t h e sum of t h e s q u a r e s of t h e m u l t i p l i c i t i e s of t h e e i g e n v a l u e s of 2.10.
THEOREM.
If
UN
i s i n v e r t i b l e , c o n s t a n t , and semisimple and
dimension of t h e space of o p e r a t o r s i s p r e c i s e l y equal to d(M+\) Proof.
P
with +
if
+
d:33 -+33
otd[P, L\<.N—2 is
d(MJr\)
.
It
The proof i s analogous t o t h e proof of Theorem 2 . 6 , b u t i t i s somewhat more
We do i n d u c t i o n k downwards assuming t h a t v~, w i t h i n d i c e s g r e a t e r t h a n o r equal to
&-fl.
vk+i
as w e l l a s
vk:
= w.
vk+2, .. .,VM a r e a l r e a d y defined from e q u a t i o n s Then t h e c o n d i t i o n for s o l v a b i l i t y of t h e
c o n s i s t s i n (by 2 . 8 b ) ) UN-I]+-(NUNV"II+1)+
[vM,
If i t i s s a t i s f i e d , Uff-iB^T
, then the
surjective.
\vk, uN\ + K + 1 , uN-i] - NuNv'k+l
vk
UN-X£3B~
does n o t exceed
complicated due to t h e f a c t t h a t t h e k - t h e q u a t i o n now c o n t a i n s
k for
uN .
then
. Therefore,
hypothesis.
Hence
v^
[vk+u v£+l
i t necessarily e x i s t s if
= W+.
i s uniquely determined.
UN-I]* = [V£+V
UN-I]+
and
v~+l
On t h e o t h e r hand,
is surjective.
Ujv-i]+ = 0, s i n c e
i s a l r e a d y d e f i n e d by t h e i n d u c t i o n
i s determined up to an element of d:ffi^ffl
[v£+v
K e r d f | 5 3 \ i f i t e x i s t s a t a l l , and
This completes t h e i n d u c t i o n s t e p and
the proof. 2.11.
Using t h e method of G e l ' f a n d and D i k i i , below we c o n s t r u c t e x p l i c i t l y for each
o r d e r an o p e r a t o r
P w i t h t h e c o n d i t i o n ord[P, L]
p r o p r i a t e f r a c t i o n a l power of We d e n o t e by with
3B) .
not of r i n g s .
44
L .
a s t h e d i f f e r e n t i a l p a r t of an a p -
To t h i s end we i n t r o d u c e t h e formal r i n g of symbols.
£ t h e f r e e v a r i a b l e and c o n s i d e r t h e polynomial r i n g
3B[Z] (;
The mapping L*+-L:£ bid'>-*£ bfi- induces an isomorphism of l e f t The t r a n s f e r a l of m u l t i p l i c a t i o n of o p e r a t o r s t o
commutes
3B -modules b u t
33[l] we s h a l l c a l l composi-
t i o n and denote i t by °: PL—P°L
.
In order to describe composition in
the inner s t r u c t u r e s , we introduce two d i f f e r e n t i a t i o n s in dl'^i toft t-*£ibfil~l 2.12.
.
53 [S] in terms of dbfi1
53[S]: d:2b^'-*^
and
The next lemma follows from the Leibniz formula.
LEMMA. PL = 2 •% d\PdaL. 0L>0
The idea i s now to use the well-known construct of extensions of commutative rings and extensions of d i f f e r e n t i a t i o n s to them in order to learn how to extend composition by the formula of Lemma 2.12 and thus to construct useful ring extensions for d i f f e r e n t i a l operators. ( * = ®((f1)) = J 2
For our purposes a single extension suffices:
*iE'|*ie«| (the ring of
U—oo
formal Laurent s e r i e s ) . formulas. . We s e t
Obviously d and
j
dj extends to 53((H"1)) by continuity with the same
o r d ( 2 ^ ' ) = max{/fe|&fc=£0} and further |j
ordO= — oo) . The norm [| || i s non-Archimedean: ||a+6||<max(||a||, || 6||) . Moreover, ||afc||< IIaIIII* II. Ilflll = 1 for ae53\{0} and | | a | | = 0 if and only if a=0. The algebra ^((r 1 )) i s complete in t h i s norm. For any additive operator F:®-+@ we s e t ||Z71| = sup|J-Fa||/|i«II-J<1
.
Obviously, ||
33((E'1)) the composition
In analogy with Lemma 2.12, we introduce in
«•»•=!! i <£«*•*. aX>
o, by s e t t i n g (2)
2.13. LEMMA. Series (2) converges in norm for any a, 6e53((S-1)) and defines on 53((S-1)) the s t r u c t u r e of an a s s o c i a t e k-algebra. Proof. The convergence of (2) follows from N[j-d£a-0 as a-*oo . B i l i n e a r i t y in a and b i s obvious. The i d e n t i t y i s 1653 . Associativity i s verified as follows: {a°b)oc = \2±dZad°b}c= L<x>0
A>0
J
Lflx) "'
2 a.p,v<&
J
ZW-^tPM^bdtc, F
* "'
A,B,T
" '
I t i s not hard to see that the s u b s t i t u t i o n o = r, $ = A + B — T, f = ^ _ r defines a b i s e c t i o n of the terms of both s e r i e s and corresponding terms coincide. The proof of the lemma i s complete. We now suppose that M i s a l e f t 53-module to which the action d: d(bm) = db-m+ bdm has been extended for £653, m£M . We denote by M (<£-*)) the l e f t 53((S-1)) -module {^mfil\ m&M, 3i0 Vi > i0, nii = o} with the obvious action. We extend the action of d to M^V1)) coe f f i c i e n t - w i s e . I t i s not hard to see that formula (2) enables us to define on M((i~1)) a new s t r u c t u r e of a l e f t 53((?-1)) -module r e l a t i v e to composition if we take a653((t"1)), bqM((£-1)). 45
Right composition M((S-1))X33((;"'))-+M((<--% i s defined similarly if with action
d .
M is a right
33-module
-1
F i n a l l y , if
M i s a 33 -bimodule, then M((' )) with l e f t and r i g h t composi-
-1
tion becomes a 53((? ))-bimodule.
The only nonobvious a s s e r t i o n s here concern a s s o c i a t i v i t y ,
and i t s v e r i f i c a t i o n in the proof of Lemma 2.13 c a r r i e s over automatically in a l l the required cases.
We s h a l l need t h i s construction in Sec. 3.
An i n t e r p r e t a t i o n of the elements of the ring
33 ((S-1)) with negative powers of \
as
symbols of i n t e g r o d i f f e r e n t i a l operators i s given in Sec. 5. 2.14.
^ bfil = b i s i n v e r t i b l e in
LEMMA.. An element
m u l t i p l i c a t i o n and composition if and only if Proof.
b,03
33((I'1)) both with respect to
i s i n v e r t i b l e in 33, where ft=ord ( 2 btV).
I n v e r t i b i l i t y with respect to multiplication i s well known.
Further,
CO
(^b,ll)ob^lVn
= l + c, o r d c < —1, as i s evident from ( 2 ) .
^(—i)lc°l
Therefore,
e x i s t s in
oo
®((S-1))i
and i s inverse to \-\-c
c°' = Co...oc, i
times).
in the sense of composition:
ft„^',=(l+c)0(~1,
Then
inverse i s established s i m i l a r l y . term of X must be equal to
on
i s inverse to 2 * ' "
F i n a l l y , if
b„ $"*, so that
b°x=\
tne
right.
write
A left
or x°b = \, then by (2) the leading
bn i s i n v e r t i b l e .
We proceed to the extraction of r o o t s . xfi
(1 + c ) ° ' 2 ( ~ " " ) ' c ° ' ~ * ^ we
If the element c = ^ t t « S
N
and W = UN .
i>0
s e r t i o n , we introduce the following notation.
is
N-th
an
power of
In order to e s t a b l i s h the converse a s -
We c a l l an element
W03 Af-admissible (A r >0) ,
S3*+-33: X*+2J wlxwN~l~l i s a b i j e c t i o n . In a commutative ring 33 with unique <-o division by N precisely the i n v e r t i b l e elements are admissible. In a matrix ring the if the mapping
N-l
element
o> = diag(Ci, . . . , c,) i s admissible if and only if a l l elements
clrc^~l~l (r, s = l , . . . , /)
^ (-0
are different from zero. 2.15. LEMMA. Let
c = '^lu„ln, N>0.
Then for each
Af-admissible root of an
power w of UN there exists a unique element X£33((V~1)), for which
X
=d
N-th
and X = w\-\-
0(1) (O(S') denotes some series of order <£)• Proof. We apply the method of successive approximations. We set X_i = w\. N
N l
Xl.x=uN\ -\-0(\ ~ ) 1
Obviously,
. Suppose that for some r> — 1 we have already proven the existence of X?*=c+0(\N~r-2)
and its uniqueness up toO(r(,+1)) . We
X,e33((Z- )),
such that Xr = wl+0(\),
seek Xr+l
in the form X„x = Xr + xr+iZ~lr+1) . Using the distributivity of composition, we have Af-l
(X, + x r + 1 r ( r + I >)°" = X°N + 2 K'°x'r+t~(,+1)=Jc;<^-'-"
+ (remainder).
1=0 N.-l
We compute the terms on the r i g h t up to
N r 3)
0(\ ~ ~ ) .
The sum has the form 2
wx
' r+i'
w"-l-^N~r~'2 + 0(lN~r~3) . The remainder consists of a sum of products of j
46
of w it follows that xr+l 0(lN~r~")
X°r+i=c-\-
exists and is uniquely determined by the requirement
. This completes the proof. w
2.16.
L='2iundn£33[d]
We now choose an o p e r a t o r
power w of
UN ( i f i t e x i s t s ) .
X*=w\ + 0(\).
We c o n s t r u c t a r o o t X of d e g r e e S
We d e n o t e by
and an AT-admissible r o o t of an JV-th
1°
for any S=PN~1GQN
= ZAM
N of L, as i n Lemma 2 . 1 5 ;
t h e element XP£33{(\~1))
.
Let
i
Assuming that w and L are fixed, we shall often write below Vj{s;w, L)=Vj(s).
For s>0
we
set
/-o 2.17. c e n t e r of 2.18. and i s an
THEOREM.
S3 , t h e n even COROLLARY.
S^QN , we have ord[ < Ls > , L]
s>0,
ord [ < I* > ,
k .
Suppose t h a t
Then for any
— 2, o r d P < A f
UN, »JV-I l i e i n t h e c e n t e r of 38, t h a t
M>0
w
is invertible, a l l
w , t h a t t h e s e t of
t h e space of o p e r a t o r s
P£33[d]
of o r d e r <'AJ
is
invertible
with the property 0<s<MN~l.
i s f r e e l y g e n e r a t e d by t h e o p e r a t o r s < L > ,
s
ttN
d - c o n s t a n t s i n 33
s
Indeed, s i n c e t h e o r d e r of t h e o p e r a t o r sN
If a l l i e s i n t h e
L]
N - t h power of an a d m i s s i b l e element
coincides with o r d [ P , L]
For any
( L s > i s e x a c t l y sN, and i t s l e a d i n g g e n e r a t e a space of dimension
coefficient
M-\-l .
It re-
mains to u s e C o r o l l a r y 2 . 7 . For m a t r i x r i n g s , however, t h e o p e r a t o r s
{ Ls ) , i n g e n e r a l , g e n e r a t e o n l y a p a r t of t h e
space of i n t e r e s t to u s . 2.19.
Proof of Theorem 2 . 1 7 .
l a r , commute w i t h
N
L= X .
s
Z] = [f, L° — < L") ~\ .
A l l powers X from 2.17 commute p a i r w i s e and, i n p a r t i c u -
Therefore, in the ring s
But ord(L° —
ord[ < Ls > ~, Z]<jV — 1 and i s even
s
(L )
33((l~1)) w i t h composition we have
~ ) < — 1 by d e f i n i t i o n of < Z> > •
< A 7 —2, i f
a^ l i e s i n t h e c e n t e r of
Therefore, 33 (use i s made of
-1
C o r o l l a r y 2 . 3 which remains v a l i d for composition i n S((S )). by formula 3.
(2)).
The H a m i l t o n i a n P r o p e r t y for t h e N o n s t a t i o n a r y Lax Equations and Their 3.1.
I n t h i s s e c t i o n we s e t
33 = k\uj%
where
[<£*)",
i = 0 , . . . , J V — 2 (N>2),
Integrals j>0;
d:a<»-*a<>+»,
AT-2
are algebraically independent variables.
u l
Let further L=tf*-\-2J
$-
We
choose
w=\
<-o and
for
any
SGQAT
we
set,
as
in
2.16,
Z ° * = 2 Vj(s)V,
Vj(s)e33.
MAT*
47
According t o C o r o l l a r y 2 . 1 8 , a l l s o l u t i o n s of t h e Lax e q u a t i o n s
have t h e f o l l o w i n g form
P = '^lcl (L ')
f e r e n t i a t i o n ( i . e . , i t commutes w i t h
, cfck, sfcQN, Si>0\
diL==[P, L] i n t h e r i n g
33
5 1 = 9 ] , p : ^ - > ^ i s an e v o l u t i o n d i f -
d and i s t r i v i a l on k ) u n i q u e l y d e f i n e d by the con-
ditions , i\, £ = 0
N—2.
We shall here prove the following basic theorem. 3.2.
THEOREM,
l^tCjiL'1)
a) The e v o l u t i o n d i f f e r e n t i a t i o n
> £J» i s Hamiltonian w i t h o p e r a t o r
£JV_2
du
d e f i n e d by t h e c o n d i t i o n
dxL =
( c f . Sec. 8 , Chap. I ) and Hamiltonian
Jgc) »-*<*+'> Sj+l b) A l l Hamiltonians
V-i(r), r(
and a r e t h e r e f o r e c o n s e r v a t i o n laws f o r any of t h e Lax e q u a t i o n s d e s c r i b e d . We b e g i n w i t h a number of a u x i l i a r y a s s e r t i o n s . 3.3.
LEMMA.
Proof. (2)
Let
a, b£33((£-1)) .
We s e t
res ( ^ btl') = b_i.
Then res(aob — boa)£d93 .
I t suffices to verify t h i s for
a=Vim,
b = u\n .
I t f o l l o w s e a s i l y from formula
that (1B+B+1)
if
m+«-f-l>0
cases.
and e i t h e r
Let us suppose t h a t
/ra>0,
or
m>0, /i<0;
[m + n + lj Therefore,
rt>0,
but
mn<0
, and res[a, &]=0
i n t h e remaining
t h e second a l t e r n a t i v e i s t r e a t e d a n a l o g o u s l y . Then
(m + n + 1)!
res[a, b] i s p r o p o r t i o n a l t o valm+n+1)—(
l
'
\m + n + \}'
— l)m*a*1tt'Oi'a+'t+1> and i s a t o t a l
derivative
by a lemma of Chap. I . 3.4.
I n o r d e r to v e r i f y t h a t t h e Lax e q u a t i o n s a r e H a m i l t o n i a n , we a d a p t t h e r e s u l t s
of Chap. I on t h e c h a r a c t e r i z a t i o n of v a r i a t i o n a l d e r i v a t i v e s To t h i s end we d e n o t e by • 2 ' ( S )
lLyUSla\h-
We
dldXi
extend
to t h e j e t s .
hard t o d e m o n s t r a t e
48
8:;S->2'(53) be t h e u n i v e r s a l
Q'(S) i s f r e e l y g e n e r a t e d over S& by t h e elements •$ t o a d i f f e r e n t i a t i o n
This i s a formal a n a l o g u e of t h e o p e r a t o r of
to our formal c a s e .
t h e u n i v e r s a l module of d i f f e r e n t i a l s of t h e r i n g
i n t h e a l g e b r a i c s e n s e of t h e word, and we l e t I t i s w e l l known t h a t
^-
Obviously,
L^^
diS1^)-*"8'^).
of Chap. I where d/dxt
ffilk
differentiation. 8tt<'>
and
8/> =
by s e t t i n g d(8«(»)=8«(M-i>. i s the canonical
8d =
u\^ .
lift
I t i s not
t h e v a l i d i t y of t h e f o l l o w i n g formal v e r s i o n of Theorem 4 . 1 of Chap. I .
3.5. LEMMA.
P&3B there exists a unique representation of SP in the form
For any
N-i
2
a ^ S 1 ^ ) ; for t h i s r e p r e s e n t a t i o n
Q M + ^>
Qlr°=6A
. l
i-0
( A l l formulas of S e c . 4 , Chap. I p e r t a i n i n g t o t h e computation of v a r i a t i o n a l t i v e s a r e to be " d i v i d e d by 3.6.
w 1
2u £ '
0)
We d e f i n e t h e
r e p l a c e d by
S3((£"'))-module
deriva-
Sw^' ) . Q1 (33)((;-1)),
con-
1
j62 (^) w i t h t h e u s u a l r u l e of m u l t i p l i c a t i o n on t h e l e f t and
The d i f f e r e n t i a t i o n s
d and
t h e second by t h e u s u a l formula. on
and a r e to have dti\
53((c_1)).
We now pass to t h e r i n g
s i s t i n g of s e r i e s right.
dxi», (tn=l),
n
d|
extend to
2 1 (53) ((s -1 )): t h e f i r s t c o e f f i c i e n t - w i s e and
This e n a b l e s us t o i n t r o d u c e a n o t h e r a c t i o n of
53((£-1))
21~(53) ((;"')) — t h e composition 1 •*•*. a! a»0
I
According to t h e remark f o l l o w i n g Lemma 2 . 1 3 , 2 1 (53)((S-1))
i s converted i n t o a l e f t
($((S -1 )). °) -
The composition mob d e f i n e d by an analogous formula c o n v e r t s Q1 (53)((£""')) i n t o a r i g h t
module.
($J((; -1 )). °) -module.
I t i n f a c t becomes a bimodule, s i n c e t h e formula of
associativity
i s v e r i f i e d i n t h e same way a s f o r t h e a s s o c i a t i v i t y of m u l t i p l i c a t i o n i n
6O(COCC) = (6O(»)DC
53 ((;-)) . We extend t h e d i f f e r e n t i a l 1
8 : $-> 2 1 (33) to
8:53 ((?->)) ->• 0 1 (33) ((S-1)) c o e f f i c i e n t - w i s e .
2 1
mapping res: Q (®)(('"'))-* . (33) p i c k s o u t t h e c o e f f i c i e n t of
The
l
\~ .
The e a s i l y v e r i f i e d c o m p a t i b i l i t y p r o p e r t i e s of t h e s t r u c t u r e s i n t r o d u c e d a r e c o l l e c t e d i n t h e f o l l o w i n g lemma. 3.7.
LEMMA,
a) The o p e r a t o r s
res, d, 8 a r e p a i r w i s e commutative ( i n t h e sense of the
commutativity of t h e c o r r e s p o n d i n g d i a g r a m s ) . b)
o(aob) = Uob + aoU
for any
a, *633((?-1)) .
The second assertion follows from the fact that 8 also commutes with d% . We now apply this formalism to the computation of —^-^ . 3.8. Proof.
LEMMA. 8©,,(s) = sres 8ZoZ° ( * _1) modImd Let
A, = I c( * f ~ ,) 1
We wish to e s t a b l i s h
s^pN-1
.
Then
for
s$QN,
s>0
.
and v_x(s) = resX° p
Z=X°"
.
that 8 res A'°" =~N ' res 8 (X°N)°X°(P
m
mod Im d
and NS res AT 0 "=p8 (A'°")oA'°(''-Ar, mod Im d. According to Lemma 3 . 7 , p-i
JV8 res X°" = iV res 8 (AT0")=JV res 2
X°l o8A^oA"'(p~'-1,
i-0
Further, X°l o8AVXo(p-/_1> = 8A'oA'°(p-1,-[8A'oA'°(p-''_1), AT0'].
It is shown by the same argument as
in Lemma 3.3 that the residue of the commutator on the right lies in
Imd.
Therefore,
49
JVSres XePmNpteshXoX'(p-l)mod
Im
On t h e o t h e r hand, pl(X°N)oXp-N=p
J
Jc'oS^oX 0 ( p - i - 1 )
and s i n c e X'°hX°XP-i-1
= lX°Xr-i - [ZXoX'C-1-",
X°l],
t h e same argument shows t h a t p res 8 (X°N)oX°(p-w=Np
res hXoX" u-» mod
This completes t h e proof of t h e lemma. 3.9.
COROLLARY.
For
s£Q„, s>0 »
and 0 < i < A T —2
we have
6w_, (s)
Proof. According to the preceding lemma,
8xu(s)=sresn2 8^')°( 2
M*" 1 ) «'))•*
^sres(2(«)^< ) (*-0^ +/ - a Ws2( l - + f /+ i)^ +y+1) (*- 1 )8" i niodIin(?, whence t h e r e q u i r e d r e s u l t follows i n view of t h e c h a r a c t e r i z a t i o n of v a r i a t i o n a l
derivatives
i n Lemma 3 . 5 . 3.10.
Proposition.
For a l l
s>0,
S6Q„ and 0 < i < A r — 2
we have
<4>
^-M(5-I)=I2(;J(-^-5^^^ Proof.
R e l a t i o n s (3) can be c o n s i d e r e d a system of e q u a t i o n s f o r
which has t r i a n g u l a r form and can t h e r e f o r e be s o l v e d by i n d u c t i o n .
y_i(s—\),...,v_N+l(s—\),
The c l o s e d formulas
(4)
a r e most e a s i l y o b t a i n e d by w r i t i n g (3) i n o p e r a t o r form
tofe&^sQ
+ doiyv^is-l),
where T is the operator for increasing the index by one: T (v^)
(5) =v(*l+1 .
Equation (4) then
becomes
*>_,._, (5-i)=4(i- and the fact that (6) is the inversion of (5) is obtained by induction on /.
50
(6)
3.11.
Proof of Theorem 3.2 a ) . It obviously suffices to consider the equation
,/.] . We rewrite it in the algebra by (L*)~—L°s
(33 ((S-1)). o), replacing L by L and (Ls >
(it is recalled that [L°s, Z]=0) . We find on setting « Ar _ 1 =0, « w = l: AT-2
r JV
*-0
La-0
P-1
This implies the two identities AT-2
J" N
ff-l
ft-0
La-0
p-l
La-0
9-JV
1 J
J
(the second follows from the fact that the order of the commutator on the right is < — 1 ) The commutator in the first identity is equal to [(! Wv^{s)-{-^)v_^S)u^]^-r.
2
(7)
a
We sustitute in (7) the result of Proposition 3.10: B-l
We wish to represent the formula obtained in the form (cf. Sec. 8, Chap. I):
J.I
with appropriate
B.
kl
.
In order to compare (7) and ( 9 ) , we make i n (7) and (8) the change of
i n d i c e s : a — p — ~{ = k, f4-8 = y, P —1—8 = / .
We then find t h a t the sum of the f i r s t terms of
(7) leads t o the f i r s t terms of (9) i f we s e t
2(-»)6(*+/-./+1)n>^'-(/i;*)«/^ B
l.*i
=
do)
for j + k + l + \
for J + k + l + \>N
(we have used one of the identities for the binomial coefficients proved in Sec. 8, Chap. I), Similarly, the part of the second sum of (7) corresponding to given
J, k,I,
can be
written in the form (for k-\-l + j - \ - \
-£<-«v(!li)('V)*4r^-*^^M'n<-*(^£^> This agrees with (9) and (10) and completes the proof of Theorem 3.2 a ) . The fact t h a t the operator J£tBjdJ — ( — d)ioB,j i s Hamiltonian i s proved in Sec. 8 , Chap. I .
51
3.12.
Proof of Theorem 3.2 b ) . We f i x r
due to t h e system r e p l a c e d by
dx 1
dxL = [ < L > , L] .
a c t i o n duftiir ))-*®^ )),
5 and c o n s i d e r t h e e v o l u t i o n of
v_x(s),
Repeating word-for-word t h e proof of Lemma 3.8 w i t h 8
( i t i s only important t h a t 1
r and
[du d] = [du d g ] = 0 ) w i t h t h e
coefficient-wise
we find ) mod Im d.
Further, = \{Lr)
7Z]oZ°
= < If )~oZ°*-[Lo < Lr > TZ 0 ^- 1 '! +Z°*= < U >~ = [ < U ) ~ L°*\-[Zo < 2/ >~ S i n c e t h e r e s i d u e s of t h e commutators l i e i n
Imd
L°i-%
by Lemma 3 . 3 , we f i n a l l y o b t a i n _i(s)6
According t o t h e d e f i n i t i o n s of S e c . 7, Chap. I , t h i s means t h a t
XLi(r) and *Li(s) commute.
We shall now clarify the algebraic significance of commutativity. )
let B be some Hamiltonian operator, and let Q, PG^=fc[ay ].
Let H,=(a0, .. . ,tt/v-2)>
Returning to the conventions •.
of Chap. I, we denote by XQ the evolution, field XQU=B-= 6u
3.13. Proposition. If Q, P commute in the Hamiltonian structure with operator B , then XQ takes the minimal d -closed ideal JP, generated by the components of B-= into itself.
ou I n o t h e r words, t h e XQ flow i s t a n g e n t t o the f i n i t e - d i m e n s i o n a l manifold of 6P
solutions of the system of ordinary differential equations 5 - = = 0 . 6u Proof. Since [
But XQ^ = D (B^) 6a
\
B^i,
6« / 6u
and by the Hamiltonian criterion for B the latter expression is equal to B-=(-=: B~)-\6a \6a' Ou / D[B-=i-)B— . The first term is equal to zero, since the commutativity of P, Q implies \ bu' )
bu
that ~=-B-=G\md ou'
bu
cKer-=. bu
The second term (more precisely, its components) lies in Jp ,
since the Frechet Jacobian belongs to
M„+l(53[d]).
This proposition motivates the search for solution (in function rings) of the equations « < =5-=, which remain for all time on the manifolds B ——0, where P is an integral. 6u 6P 67> Actually, instead of the equations —6KerB the equations -= = 0 — the extremals of the bu
6a
Lagrangiaa P— and the flows induced on them are usually investigated (cf. Gel'fand and Dikii [3]). Both formulations of the problem are equivalent if KerB = k^-1 and in place of the equation — = c&kN-x it is possible to consider the equation -=(P — c'u) = 0, noting that 8u
P-~c'u is in this case an integral together with merits special investigation.
ou
P .
In the general case the question
Regarding the Hamiltonian property for the induced flows see the work of Bogoyavlenskii and Novikov [1] and Gel'fand and Dikii [3]. 3.14. We shall apply these considerations to the Lax equations Li = \zJci
< Lsi ) , Z.J and
their integrals "S d,v" (r't ''^Q . By Theorem 3.2 the condition 5 - ^ = 0, implies that *• * r / + « ou 52
\"^idl{Lri
>, Z.] = 0. Thus, the search for solutions of the Lax equation lying on the "quasi-
extremals" of their integrals
|5-==0 in place of -==0|, reduces to the joint solution \ ou
of the system of equations Lt=[P,
oa
/
L], [Q, Z.]=0 . The condition
[Q, I]=0 is called the
auxiliary stationary problem (for the nonstationary problem Lt=[P, mainly devoted to the description of such solutions. 3.15.
L\).
Chapter III is
We set N = 2, L = d2-{-u . The equation
Example: The Korteweg-nde Vries Equation.
Lt = u,t-—\ ( L32 ) , L] is called the Korteweg-tle Vries equation, while the equations | 2 {L'i^Ct,
L\
axe. its higher analogues. As s^Q2
Lt =
it suffices to take half integers, since
< Ls ) =LS for integral s (this remark applies to general We observe that Corollary 3.9 assumes the form
N) . *=sv_\(s — 1), so that increasing s
V
by 1 corresponds to "variational integration." 3.16.
The Matrix Case. We shall now briefly describe the changes to be made of the
definitions and results of 3.1-3.12 in order to extend them to the matrix case. We set N
L=£iUkdk
, where t/Ar = diag(c1
ct) is a semisimple matrix of
Mt(k), which is diagonal for
simplicity and UQ,..., f/.v-i are matrices with independent variables as elements: f/i = («i,os); «jv-i,ap = 0 of
for
a, p with c a = Cp .
Having chosen an admissible root of an iV-th power W
UN , we can construct the fractional powers
further investigate equations of the form exhaust a l l Lax equations over the ring We set L^=y^lV_fi(s,W)l-^.
Z£, as in Sec. 2 for
dxL= 2
Ci W
- ( Lw ) •
r ff
*=o
L<*-o
We shall
recall that they now do not
&\d\, $ = Af,($30)> $o = *l a i%l-
A representation analogous to (7) holds as before but only
up to a commutator in the term dxUQ, which dropped out for /v-i
We
S^QN .
1 = 1:
1
AT-1
p-i i
J
L
'
The contribution from this commutator will vanish if we go over to equations for the matrix traces of the Uk in place of the
Uk themselves.
This becomes even more necessary if we wish
to carry over to the matrix case at least a part of the results regarding the Hamiltonian property. UV(m+n+i)
Indeed, the key Lemma 3.3 ceases to hold, since the matrix no
longer need be a total derivative.
W/<m+B+1)—(— l)B,+»+1.
However, this expression differs from a
m n 1
total derivative by the commutator [U, V< + + >] , and therefore after going over to traces we again obtain Tr res [a,'b] 6?33 .
In the considerations of 3.4 and thereafter in place of Ql(3S)
i t is now necessary to take the $[d]-module Af,(21 (S3o)); B^-^M^Q 1 ^))
is defined
wise, and the subsequent constructions are modified in an obvious manner. must represent
coefficient-
In Lemma 3.5 we
5P in the form 2 Qi.*$U-i,*t + du>, where Qi,c£p=g^— (coefficient-wise).
The
proof of the analogue of Lemma 3.8 leads to the result Tr8V_1(s, W) = STT res8Z 0 L OU_1) mod IIIK?.
53
Analogues of Corollary 3.9 and Proposition 3.10 now connect =
TrV^is, W)
with Tr V,(s— 1, W).
ou
l,afl
'
Substitution of expressions analogous to (4) into the equations for Tr
then (under weak additional conditions) L and P are
connected by a polynomial relation with constant coefficients. This reduces the problem of solving the stationary Lax equations to the problem of imbedding one-dimensional commutative rings (i.e., rings of functions on affine algebraic curves) in rings of differential operators. The fruitfulness of such a reduction will be demonstrated in Chap. III. 4.2. We shall actually prove a more general theorem pertaining to rings of operators generated by several differentiations
dn .
Let S3 be a not necessarily commutative
Q -algebra which is free of finite rank as a
module over its center 33a- Let du .. .,dn:33-+33
be differentiations taking 33o into itself
and commuting pairwise. We write i=>(h, • • •. *«). \i 1 = 2 *y d'=dil
• • • dn,
33 [d] = { 2 M ' i *<653}
with the usual rules of multiplication. Let Z,i,..., £r6$[d] be a finite family of operators. We call it independent if the sum of 33 -modules 2 33LPl... if/ is direct. We write LP=LPt...
L"/
and |/>|=2 pj .
4.3. LEMMA. We assume that in the category of free 53-modules the concept of rank is well defined (i.e.,
3?/=3?,S=$T=S)
the module. If Llt...,Lr
, and the rank of a submodule does not exceed the rank of
is an independent family of operators, then r < « . L\,...,Lr,
Proof. There exists a constant /, depending on the degrees of the operators such that for all m > 0 there is the following imbedding of free
^-modules: £i 3hlPa \p]<m
2 33d' .
But the 33 -rank of the module on the left grows asymptotically like c-jn.' and
\l[
that of the module on the r i g h t l i k e c2(lm)n.
Therefore,
r<«.
4 . 4 . We s h a l l c a l l a family of operators Lu ..., Ln£33[d\ maximal if i t i s independent and there e x i s t s a free 33 -module of f i n i t e rank Mcz33[d], such that 33^] = ^ML', and the t
sum on the r i g h t i s d i r e c t . Example;
in the case
n=\
i s maximal if the coefficient
the family consisting of a single operator bNdN-{- ...
bN i s i n v e r t i b l e in 33 .
33bpNdpN+l = 33dpNJrl, i t follows e a s i l y t h a t 33[d\=^ THEOREM. Let
Lu...,Ln
(iu...,i„),
54
q>0,
Since
/
be a maximal family of operators in
be an operator such that [P, Z.i]= . . . = [ / > , £„I = 0 .
33dl .
2 33d> \ D> and the sum on the r i g h t i s d i r e c t .
p_0\i=0
4.5.
Indeed, we s e t M=2
+b0,
Then there e x i s t
33\d] and l e t Pe33[d]
d-constants i q
lying i n the center of 33 and not a l l zero such that 2 aitqL P =0
aiiq, i = .
We f i r s t prove t h e f o l l o w i n g a u x i l i a r y a s s e r t i o n . over 33\d] which commute w i t h one a n o t h e r and w i t h 4 . 4 for t h e family
Lu...,Ln,
LEMMA.
33\d].
We choose a module
Mc33[d],
as i n
and s e t M [h
4.6.
Let X,, . ..,X„ be independent v a r i a b l e s
X„] = { 2 mtV i m&Ai} cz33 [d] [X,
X„J.
There i s t h e d i r e c t sum decomposition n
where the last expression on the right is the left 33[d] [>.l7. ..,/.„] -ideal generated by LL — X;. Proof.
a) We show first that the left side is the sum on the right.
Since
33[d]=^MV, y
i t suffices
to v e r i f y t h a t for any
the r i g h t .
But mV = nOJ -\-m{JJ — W), so t h a t i t remains to check t h a t V — \!
left
i d e a l g e n e r a t e d by L, — X£ .
Jk=£0 .
We s e t j'=(ju
m£M Let
and j
j = Uu..
•••.A-i.O, ...,0) .
t h e element .,/„) and l e t
The f i r s t term on t h e r i g h t i s d i v i s i b l e by
Lk—lk
k be t h e l a r g e s t index for which
L1'>.J* - X; Vt* =
j .
T h e r e f o r e , i n d u c t i o n on t h e number
results.
b) We now show t h a t t h e sum on the r i g h t of (11) i s d i r e c t . v e r i f y t h a t if
^lmlk^^^[d\[).\(Ll—,kl),
then
^=0
t h e n a c c o r d i n g t o t h e f i r s t p a r t of t h e proof a l s o t h e homomorphism of l e f t
l
X i ' . . X " i n t o L\ . . . ! „ " .
l i e s i n t h e k e r n e l of t h i s homomorphism.
4.7.
for a l l
t.
For t h i s i t s u f f i c e s
to
But i f ^,m,lie'^l^ld][k](Li
^imtL'e^^[d\m(Ll
— 'kl).
— h),
We c o n s i d e r
S3[d] -modules $[d][X]->-53[d], which i s t h e i d e n t i t y on
takes the free generator zero i n t e r s e c t i o n w i t h
l i e s in the
, w h i l e t h e second i s analogous to LJ — W,
y" has fewer nonzero components t h a n
of components g i v e s t h e r e q u i r e d
i s c o n t a i n e d i n t h e sum on
Then
Li - V = V'L'k* - l'"Vkk=L' Kk-V'KkJr
b u t t h e index
mV
33[d] and
The submodule 2 ^1^1 IM(^« — h)> c l e a r l y
T h e r e f o r e , i t a s w e l l a s t h e e n t i r e k e r n e l has
$[
Proof of Theorem 4 . 5 .
We c o n s i d e r the
$[X]-module M = 33[d][X] /^~®[d][X](L, — X f ).
According to Lemma 4 . 6 , i t i s f r e e of f i n i t e rank over 33[l], s i n c e i t i s isomorphic to 7W[X] . I t i s t h e r e f o r e f r e e of f i n i t e rank over $?o[X], where S30 i s t h e c e n t e r of t h e r i n g
33.
Since t h e o p e r a t o r P(*33\d] commutes w i t h a l l t h e
£,• , m u l t i p l i c a t i o n by P on t h e r i g h t
induces a
Therefore, the ring
S 0 lX]-endomorphism of t h e module
a c t s n a t u r a l l y on
A!.
M.
On r e p r e s e n t a t i v e s t h i s a c t i o n can be w r i t t e n as f o l l o w s : mobXiP'^bml'PJ,
Since X„,/>]
M
530[Xi> • • •. X„, P\ cz33[d]
b£BQ, m£M.
i s f r e e over 33Q\KU . . . , X„] t h e r e i s a nonzero polynomial Ffa,..
such t h a t
M°F = 0*
.,X„, P)G$?olXi.- ••>
According to Lemma 4 . 6 , t h i s i m p l i e s t h a t 33[d][X\oF(Xu..
.,XniP)ci
55
2 53 [d] [>•] (^ — \.)
.
From the calculations in the f i r s t part of the proof of t h i s lemma i t
i
follows that then 33[d\ [i.]oF (Llt...,
L„, P ) = 0 where the s u b s t i t u t i o n of
be done i n the reduced notation for
F .
Lt in place of
I,
must
Applying F(LU .. .,Ln, P) to l6$]d][X], we find that
F(LU . . . , ! „ , P)G33[d] n 2 ® I d H M ( £ | — h) = {% by the second part of the proof of Lemma 4 . 6 . i
We have thus obtained a polynomial r e l a t i o n between
Lt and P but with coefficients in
330 • We s h a l l now show that for s u i t a b l e F these coefficients are To t h i s end we choose a free b a s i s of the S0['-l-module a
t i a t i o n $b[>-I->-S8b[X] nd i t s l i f t
M and denote by D a differen-
M-*-M with the following p r o p e r t i e s :
a) On 33 the d i f f e r e n t i a t i o n b) DX,=0, i = \,
d-constants.
D coincides with
d.
...,n.
c) The chosen basis of M i s annihilated by
D.
We denote by A the matrix of the endomorphism of m u l t i p l i c a t i o n by P in t h i s basis and by F i t s c h a r a c t e r i s t i c polynomial. In addition to D,
there i s the d i f f e r e n t i a t i o n
the l e f t by d in 53[d][X].
I t also extends
d .
d:M-*-M, induced by m u l t i p l i c a t i o n on
Therefore, D—~d\M-+~M. i s a
330[l] - l i n e a r
operator: (d — D) (bm)=0—D) U—d —D
We set
and identify U
b • m+b (d — D) m = * (d—D) m.
with its matrix in the chosen basis. Since right multi-
plication by P commutes with left multiplication by
d , on calculating the action on this
basis of the composition of d and P in two different ways, we obtain dA-{-ALf=UA
dA=[U,A\
.
This implies that
t r a c e of
i s zero for a l l
dAn = 2 A;(?AAB-'-I = 2 re>0
.
or
n-l
n-l
A
' [ ^ ' A l A a_M = IC/, A"].
Therefore, the
This means that the coefficients of the character-
i s t i c polynomial of A are d -constant by Newton's formula. 4.8.
Integrals of the Stationary Lax Equations. We apply the preceding considerations _
JV-2
N
,
c
i
/
to the case 3S=k [a<>'\.. .,«(/l 2 iy>0] i L=*d +2 «« * /-o factor ring of 33 by the d -closed' ideal generated by [£, P].
We d e n o t e
b
?
_
=
® ®tct)
the
Let d:33-*-33 be the induced
differentiation where L, P(<33[d\ are the images of the operators L and P. It is obvious that L, P is a "general solution" of the stationary Lax equations with operators of the given degrees N,M, if the constants c0,...,cM are taken to be free variables and Si = i/N. Since the leading coefficient of L is equal to 1, Z L,
P
are connected by the relation
is a canonical relation: ^dtjV
l J
^dtJL P =
0, where
generates a maximal family, and dtj
are d -constants in 33. There
are the coefficients of the characteristic polynomial
/ of the endomorphism of m u l t i p l i c a t i o n by P for the 33 [>.]-module 33[d][ky33[d\[l\(L — X). l i f t s dtJ of the elements 7?,, to 33 are i n t e g r a l s of the equation [L, P]=0. 56
The
We note now that by Proposition 3.13 we have another c o l l e c t i o n of i n t e g r a l s of t h i s equation. dTr=0
Namely, for any r£QN,
in
P
r>0 we have
~l f (r)
B
^ j 21 ci ""'J+t * = !£®
so
that
93 .
The r e l a t i o n between the i n t e g r a l s
dtj
and
Ir
i s known for the higher Korteweg—de
Vries equations; in p a r t i c u l a r , the ones are expressed in terms of the o t h e r s . 5.
The Zakharov—Shabat Formalism 5.1.
[13].
In t h i s section we present an i n i t i a l version of the method of Zakharov—Shabat
Our objective here i s to c l a r i f y the algebraic side of t h e i r construction — the
s t r u c t u r e of "dressed" d i f f e r e n t i a l operators — while omitting functional-analytic considerations.
We begin, however, by describing the basic functionals.
5.2.
We consider the space of columns
functions of the v a r i a b l e x and possibly of the additional parameters t K(x,y;
t,z)
and F(x,y;t,z)
be two (NxN)
and
z .
Let
matrix-valued functions. We assign to them the
i n t e g r a l operators K, F: oo
oo
(Sty ( * ) = $ * (x, y) <»j (y) dy, (F| (y) dy. —oo.
X
Similarly, for any operator L we understand by (LK)~ the operator with kernel LK and i n t e g r a t i o n from x to oo, and by (LF)* the analogous operator with i n t e g r a t i o n from — oo to oo .
The dependence on
t, z
i s not indicated here e x p l i c i t l y .
The functions
K, F, and
<> ! are assumed such that a l l c l a s s i c a l formulas used below for d i f f e r e n t i a t i o n of i n t e g r a l s with respect to a parameter are valid (and, of course, the i n t e g r a l s themselves e x i s t ) .
The
following sequence of lemmas on the commutation of various operators i s preparatory to the formulation of the main theorem. 5.3
LEMMA. [dt, F\ = (dtFy,
[d„K\=(dtK)",
and similarly for
dz.
The proof i s obvious. Let FL+
Lx = Zli(x; t, z)dlx . where the U are (NxN)
matrices.
We denote by the symbol
the kernel 2 ( — dy)'(F(x, y)li(x)) and similarly for KL+ . i
5.4. Proof.
LEMMA.
[Lx,F\ =
{LxF-FL+}\
Integrating by p a r t s , we have oo
oo
(l^.^H) (•*) = £, J F(x,y)^(y)dy—oo
oo
J F(x,y)L^{y) = —oo
'oo
= J LXF (x, y) t (y) dy - $ FL+ (x, y) <1< (y) dy= —oo
5.5. LEMMA.
If Lx
{LxF-FLpl
—oo
satisfies the conditions of the preceding lemma, then
57
LfK-(LJO~=oK(f.x), where the d i f f e r e n t i a l operator
i s additive in Lx and for l = l(x,t,
°K(LX)
z) i s defined by
n-i
°K(ld?M= ~ 2 ld'x[{d--^K) (x, *)]. i=0
Proof. of
I t i s clear that the l e f t and r i g h t sides are l i n e a r on matrix-valued functions
x, t, z.
i t therefore suffices to compute
From n to n-\-\
a
ie(dx)-
the computation i s as follows: cK(djfi) d.=«?j+iJ K(x, y)^(y)dy-
(
-**H *
For re=0 the formula i s obvious.
K)(x,y)*Wdy
[ (d^K)(x,
y) *(y) dy=
+ aK(d*ft - J (d»+lK) (x, y) * (sr)rft/= - (
n-1
= - (djjAD (x, x) <•(*) - 2 <^+1 K*r : -'*o <*• •*) * <•*)]• The l a s t expression coincides with the formula for °K(dx+l)ty, indicated in the lemma. 5.6.
LEMMA. We assume t h a t the coefficients of Lx do not depend on x . koLx-(KLpA
Then
= ^(Lx),
where *K i s a d d i t i v e i n Lx and ii—I
** (W3=2 ( - ir* (dj-'-'/o <*. •*) wxi-=0
Proof. As above, for ra=0 the assertion is obvious. We carry out the inductive step from n tora-f-1. Splitting the second integrals into parts, we have: op
lk4dnx+1-K(idnu+1r]*(x)
=$#(*.
u)idnv+1*(y)dy-
X
00
- 5 (-1)"+1 (dl+1K) (x, y) /* (y) dy= OS
00
.=$*-(*, 9)«;+I$(0rfj,+(-l)«+« <<^0(*. x)ty(x) + $(-iy*HdlKH*, X
Here we have used the fact t h a t dy(lty) = ldyV since l = l{t, z) .
The sum of the two i n t e g r a l s
here i s equal to [Koidx - K (ldlr\ dxif = t , (WJ) d/f by the inductive hypothesis.
The e n t i r e expression i s therefore
( - I f " (
58
y)ld^(y)dy.
X
5.7.
We now consider the ring
33 [dx] of matrix-valued operators in dx, the coefficients
of which are polynomials of the r e s t r i c t i o n s to the diagonal of the p a r t i a l derivatives d*K, d$K and the derivatives of these r e s t r i c t i o n s along the diagonal. ( k c o n s i s t s of the constants, i . e . , R or C in the analytic c a s e ) . the same order as Lx, and i t s coefficients depend only on s i b l e to apply
x, t, z .
as i s evident from Lemma 5.5, reduces the order.
LX£MN(k)[dx]
Then {l-\-zK)Lx
has
I t i s therefore permis-
<sK and i t s powers to i t while remaining in 33 [dx] .
°K-33[dx]-^33[dx\,
Let
But the operator
Therefore, the following
expression i s meaningful: i ; = 0 +OK)-^KLX=(1 where M>ordLx
..+(-\)Mo*)zKLx63S\dx],
-OK+*K-.
.
The operator
L*x i s called the "dressed" operator
Lx , and the mapping Lx~*-(l +<3k)~hKLx
i s called the "raiment" of Lx . The f i r s t part of the next theorem shows that L*x i s roughly speaking the d i f f e r e n t i a l p a r t of the operator obtained from Lx by means of conjugation with the operator \-\-K : 5.8. K, F
THEOREM, a) L*x{\ +K) — (l+K)Lx
= {LxK — KL+}* . b) We assume that the functions
s a t i s f y the equation oo
K (x, y) + F (x, y) + J K (x, s) F (s, y) ds=0 X
o r , more b r i e f l y ,
K+F-f
K*F=0
.
There i s then the i d e n t i t y
{L'xK-KL+y) +
(LxF-FLl)+{LxK-KL+y)*F+K*(LxF-FL+g)^=0.
c) Under t h i s same condition we have dtK + and s i m i l a r l y for 5.9.
dz
dfF+dtK*F+K*dtF=0
.
Application of Theorem 5.8.
If K+F+K*F=0
some operators R, 5 , we s h a l l say that the pair +
and RK+SF + RK*F+K*SF = 0 for
(R, S) " d i f f e r e n t i a t e s " +
(AT, F) .
to Theorem 5.8, the pair (Z.*— L u + a.d, + $dz, Lx—L y + *dt + $dz) d i f f e r e n t i a t e s
According
(K, F), i f
a, p
are any c o n s t a n t s . We now suppose that the kernel F s a t i s f i e s the system of l i n e a r d i f f e r e n t i a l equations LxF-FL;+zdtF=0, LS-FLl+Wf^O. We find by Theorem 5.8 b) and c) that Vx K - KL*+* dtK + {L'JC - KL\ + *dtK) *F=0, LxK-KLl + ?dzK + {LxK-KL+„ + VdzK)>F=0.
59
We suppose, moreover, that from t h i s i t can be concluded that
LxK-KL*+?dzK=0. Using Theorem 5.8 a) andLemma 5 . 3 , we obtain, on the other hand, (L"x + adt)^+^)~^+f()(Lx+oidt)=lCK-KL+y + oLdtKr=0, {L'x + B
and
l+K
i s i n v e r t i b l e , we find that the d i f f e r e n t i a t i o n operators
I ' - f B ^ are adjoint to Lx+a.d, and Lx+$dz,
We now take the i n i t i a l matrix operators Llx
respectively.
and L2 x to have constant c o e f f i c i e n t s .
We obtain the following r e s u l t . 5.10.
THEOREM.
If [I lx ,Z. 2 ^]=0 and F, K s a t i s f y the conditions of 5.9, then L'x, L*^
s a t i s f y the equations of Zakharov-Shabat [L'lx+adt, L2% + P«?1=0, or
We proceed to the proof of Theorem 5.8. 5.11.
Proof of Theorem 5.8 a ) . According to Lemmas 5.5 and 5.6 L'xoK=(LxKV+cK(Lx), KOLMKLW
+xK{Lx).
Therefore, L
U^+^)-^+k)Lx={LxK--KL+r+Lx-Lx+ok(Lx)-rx(Lx).
But
Lx=(l+c„p(\+-z)Lx. Lx-Lx-oKLx 5.12.
identity
Hence + tKLx H ( l + 8 *)" 1 (1 + V ) - 1 + M 1 + ^ ) _ 1 (1 + t j r ) - v l 4 r « 0 .
Proof of Theorem 5.8 b ) . Applying the operators of Lemmas 5.5 and 5.6 to the K+F+K*F*=0;
we obtain I j r /Sr+I^ 7 + (i x Ar)*F+o J r (^)F=0,
(12)
KL++FL++K*FL+=0. By Lemma 5.6 applied to the columns of
F , we have
K*FL+= -K*(LxF-FL+)+K*LxF=*
60
(13)
-K*(LxF-FL+)+KL+*F
+ xK(Lx)F.
Substituting t h i s expression into (13) and subtracting the r e s u l t from (12), we find LXK-KL+
+ LXF-FL++{LXK-KL+)*F
We s e t LW = LX, LW=a'-i(aK — -zK)Lx for
i>\
+K*(LxF-FL+)
+ (aK-^){Lx)F=0.
(14)
and apply (12) to Z in place of
Lf . We
obtain ^)K+L^F+(HJ)K)*F
+ <,K(LW)F=0.
We take the a l t e r n a t i n g sum of Eqs. (14). with signs
(14) ±
( — 1)' over i = 0,..., M,
M>ordLx.
Noting t h a t M
we obtain L*KK-KL++LxF-FL+
+
(L*xK-KL+)*F+K*(LxF-FL+)=0,
which completes the proof of the theorem. 5.13.
Formal Analogues. Under appropriate analytic assumptions we have
K(x, y)= 2n-<^*H*' *)Q/-*Y and
kr- 2 (d'yK) (*, *) J {1=^ f (y) dy. X
(if
x\l
'.
It is possible to eliminate the factor i
times. Therefore, setting
under the integral by integrating by parts
(d'yK)(x, x) = ut, it is natural to assign to the Volterra
operator R its symbol oo
Here the ut may be considered a r b i t r a r y elements of some d i f f e r e n t i a l ring
3S, or matrices
with independent coefficients as in 3.16. In order to write formulas for the operators <»*•, *#, i t i s useful to introduce further the generator vh corresponding to (d'xK)(x, x). I t i s not hard to see that they are l i n e a r l y expressed in terms of the ttj and t h e i r derivatives (using the formula ((dx-\-dyyK)(x, x) = dx(K(x,x))) :
61
Comparison with formulas (3) and (4) indicates that conversely the u,t are expressed in terms of the
vl .
In these generators we obtain n-l
n-i
'•K-.S3 [d}->33 [d\:ld"- 2 ( - l r 1 " " * * - ! - ^ ' i=0 00
Conjugation by means of K — l + £(—
l)'ttjE~' takes commuting operators in
(53((;_1)), °)
into commuting operators: [I„ i 2 ]=0^" 1 Z:„ /?, ^»Z,/?] = 0
(we here identify
L\, and L2 with their symbols).
Generally speaking, the operators K^LjK
Considering only t h e i r d i f f e r e n t i a l parts ( K~lL[K ) ,
are not purely d i f f e r e n t i a l operators. we find
ord[ ( K^UK > , < K~lL2K > 1 < max ord Lt -1. Thus, t h i s method leads to the construction of c e r t a i n "general" solutions of the equations of Zakharov—Shabat. In conclusion, we s h a l l c l a r i f y which d i f f e r e n t i a l operators are conjugate by way of some K . N
Now l e t that
N
jKufi1, £w£lG$!im
be two symbols
{tlh wt
are any elements).
dUff=0 , «JV i s semisimple in the sense of Sec. 2, and that
We s h a l l assume
d:53+->33+ i s s u r j e c t i v e .
oo
5.14.
THEOREM.
For the existence of a symbol / o o
ff
u
l
2 M +2 i t i s necessary and s u f f i c i e n t that Proof.
\
v
k
/
1+2'°-*^"* \
oo
l
v
with
the
P ro P ert y
»
H
-£~ H + 2 ^~ °2w'6'
tttf = WN and Uw-i—<WN-\
We rewrite the l a s t equation i n the form
ct>(U>l L i=0 JV
We c a l l the k-th
equation (for
'
V
'
/-"
i> ) the equality for the coefficients of
E* on the l e f t
and r i g h t s i d e s . The N-th equation gives Uff = wN ; the (N— 1 ) - s t equation has the form [uN, v_x\ = TOAT-I — tifir-i . This shows the necessity of the condition of the theorem and uniquely determines vzl . The r e s t of the argument i s p a r a l l e l to the proof of Theorem 2.10. We suppose that from the equations with indices N N_k+l the q u a n t i t i e s v_u ..., *CA+1 have been 62
determined.
Then the (N — k)-th
equation uniquely determines the element
UN-lV_k +i — 1 U +i«lAT-l + NUNV'_k+l + [UN, V_k] = W.
Since
JCGS - ,
WN-I—UN-I-{-X,
xv_n+i
.
For d e t e r m i n i n g
t h e f i r s t two terms can be r e w r i t t e n i n t h e form we
"Q—n+i
[ttjv-i, i»^ +1 ] —
obtain the equation
NaN («J:, + 1 )'-(^I* + J )=o' + . which i s s o l v a b l e i f 6.
d:33+-*SS* i s
surjective.
The Benney E q u a t i o n s ; Main R e s u l t s 6.1.
We r e c a l l t h a t t h e system of Benney e q u a t i o n s h a s t h e form y
it + uux-u^ux\y^d^ u.
ht+Kudy)
+ hx=Q,
=0.
The meaning of the notation is explained in Sec. 1 of this chapter.
The formal investiga-
tion of the equations is based on the following lemma of Benney. h
LEMMA. We define the moments An(x, t) = }u(x, y, tfdy, re>Q. From system (15) there o then follows an infinite system of equations for the moments: 6.2.
A„,t + An+Ux+nAn_1A0,x=0, Proof. without
«>0.
M u l t i p l y i n g t h e f i r s t e q u a t i o n of (15) by nnn~i
(16)
and r e g r o u p i n g t e r m s , we o b t a i n
difficulty
(u")t+(W*l)x—\uniuxdvi)
We now i n t e g r a t e t h i s r e l a t i o n on y from
0 to
+nun~1kx=0.
h .
The f o u r t h term becomes
t h e t h i r d [because of t h e second e q u a t i o n of ( 1 5 ) ] becomes W(A, + uhx)\y-h A
f i r s t term of t h e l a s t r e l a t i o n to and
A„+1,x
\(u")tdy,
.
Adding t h e
h
and t h e second to
^(W^) x dy
, we o b t a i n
An,t
, respectively.
I n Benney's work i t i s shown t h a t t h e r e e x i s t two sequences of polynomials [J4 0 , . . . , An]
nAn„xAo,x , and
and F„£Q \A0, ..., Ak+l\
//„6Q
such t h a t (16) i m p l i e s l o c a l c o n s e r v a t i o n laws for
the
system (15) of t h e form Hn.t + Fn.s-O,
«>0.
(17)
In Miura's work [14] it is shown that there exist two further sequences of polynomials Hn, Ft£Q [u, A0,...,
An_i]
such that (16) implies local conservation laws of the form
63
Hn,i + Fn,x + {JRnv)y = 0; n>\,
? = — \uKdn. o
(18)
(In Miura's table the coefficients of v do not coincide with H„ because of misprints and omissions.)
As usual, under the assumptions of rapid decay of solutions^ a t i n f i n i t y , from
t h i s i t i s possible to obtain q u a n t i t i e s conserved in time: these are
\ Hndx,
rt>0.
—00
The next two sections are devoted to a new construction of relations (17) and (18). The description of the generating function of the system of polynomials H„ as solutions of a certain integral equation with parameter plays a central role in the construction.
6.3.
THEOREM.
We set
solution
Then there exists a unique
0
i-0
n(X) of the equation l*<X)+0(p<X)) = X
(ig)
a) in the class of formal s e r i e s of the form K+Q[A0, Ait.. .][[k~1]]', b) in the class of functions analytic i n X of the form h(x,t))
X+o(X""') in a neighborhood of oo (depending on
u{x,y,t),
. 6.4.
THEOREM.
System (15) implies equations for
J*(X) of the form
(1/-(^/2+A)x=0,
(20)
[&*+#i-[*%<*+^l-[$<*+#i\**v= 0 . Theorems 6.3 and 6.4 are proved in Sees. 7-8.
(21)
The Benney conservation laws (17) (up to CO
constant factors) a r e obtained from (20) by s e t t i n g 5jj(—Ty/riX~('+I)
and equating coefficients of
X~(B+I).
ji=X — ^ ( — 1)'//A~ ( ' +I) , \t.2/2+A0=* X2/2 + Miura's conservation laws are deduced
s i m i l a r l y from the r e l a t i o n s (21) i n which (^+«) - 1 i s to he interpreted as
^(—lyu^'^&^lu,
km. 6.5.
The Reduced Equations.
Eqs. (15) by adding the condition depend on height.
As already mentioned, the reduced system i s obtained from «y—0 : the horizontal component of the velocity does not
The reduced system for the functions u(x, t), h(x, t) has the form (the
c l a s s i c a l equations for long waves): u
>=-[!T
+ hlx~'5Z\~drh
This system i s Hamiltonian with Hamiltonian operator '#—(<)
64
(22)
o /'
and
Hami
ltonian
H-
The integrals Hn of the full system (15) are converted into integrals H\ n
0
of the
a
reduced system (22) after substituting An*+hu =A n:
H ndQ.\a, h\ . We explicitly compute these
integrals in Sec. 9, we show there are no others, and we establish their commutativity. [«/21
6.6.
THEOREM,
a) Up to constant multiples we have # J = 2 tnJlun-2k!ikn, A-o i
=
n.k
and any polynomial b
>
H
of
u, A, for which
where
(23^
(n—2k)\k\(k+l)l
HtGdxQ [a, A] i s a l i n e a r combination of t h e
H^.
[ffl,ff°m]6dxQ[u,h].
Theorem 6.6 e n a b l e s us t o i n t r o d u c e t h e " h i g h e r reduced e q u a t i o n s " [*H\
( « , =
u
AT
-ST-
YdHV ff=^cnHl,
c„eR.
(24)
From what has been s a i d , they admit c o n s e r v a t i o n laws of t h e form / / ' , / + P"a,x=0, where the Fn
depend on
c0,
...,cN.
I n S e c . 10 i t i s shown t h a t t h e c o n s t r u c t i o n of commuting H a m i l t o n i a n s {Hn°}
can be
considerably generalized. In S e c . 11 we pass to t h e i n v e s t i g a t i o n of t h e f u l l system (15) o r , more p r e c i s e l y , of t h e system of e q u a t i o n s for t h e moments ( 1 6 ) .
We b e g i n by showing t h a t they a r e Hamiltonian
a s a system of e v o l u t i o n for i n f i n i t e l y many unknown f u n c t i o n s e s t a b l i s h i n g the following We s e t
B=Bid+d°Bif
I , the operator 6.7.
B
—
of two v a r i a b l e s by
facts.
, where
2?i, y=k4, + j_i, i, / ^ 0 .
Then, as was shown i n Sec. 8 of Chap.
i s Hamiltonian.
THEOREM.
and H a m i l t o n i a n
An (x, t)
The Benney e q u a t i o n s f o r t h e moments a r e Hamiltonian w i t h o p e r a t o r
B
y / ^ 2 = _"2"(-^2 + ^o)-
(The H a m i l t o n i a n formalism i s h e r e a p p l i e d to t h e r i n g JL = k\Ai'
J, i, / > 0 , A\!) b e i n g t h e
00
independent v a r i a b l e s .
I t i s a l s o p o s s i b l e to work i n the r i n g
\J C*°(x, A\\i-\-j<&k) *—o
or i n
various intermediate rings.) 6.8.
THEOREM. Let H&A
be any element.
Then a system of equations for u, h of the
form
\/-0
where
^ii)==~rr~<
Ix
0 \,/-0
i m p l i e s t h e system of e q u a t i o n s
Jx >"«
At = B—=-
w i t h Hamiltonian
(25)
H .
65
This theorem is proved in Sec. 11. Theorem 6.7 is a special case of it for
H=—-~H2,
when the system (25) becomes the original system of equations (15) of part 1 and Theorem 6.7 becomes the Benney lemma.
Theorem 6.8 indicates the rather surprising situation: in-
spite of the loss of information on passing from
u, h to A , the equations of evaluation
for A with any Hamiltonian can be lifted to equations of evolution for u, h .
Of course,
the question of lifting solutions requires separate investigation. 6.9.
THEOREM.
Let the elements H&.J, be defined from Benney's conservation laws (17).
Then they commute relative to the Hamiltonian structure with operator B
.
This theorem is proved in Sec. 12. It makes natural the consideration of systems of the form (25) with Hamiltonian H = ^£;//,-, which we call higher Benney equations in analogy with the higher Korteweg-nde Vries equations. By the general formalism the higher equations possess conservation laws of Benney type Hl t-\-Fi x=0,
where the Ft
depend on H . Con-
servation laws of Miura type hold for one of the higher equations. 6.10.
THEOREM.
There exist Miura conservation laws for the system (25) with Hamilton-
ian H=cH3. Unfortunately, we have been unable to determine if this fact holds for the other higher equations.
Theorem 6.10 is proved in Sec. 13.
We began by considering the reduced higher Benney equations (with the additional condition uy=0),
for which Ai = hul . As was mentioned, they are Hamiltonian related to the
operator f J J) in the ring Jt°=Q[uO), 6.11.
h^\j>0\.
The Hamiltonian structures described in the rings Jl and d>° are
THEOREM.
compatible. A precise formulation and proof of this theorem is given in Sec. 14. That section also contains an explanation of the reasons for which the system (25) contains the operators
7.
The Function u(X) In this section Theorem 6.3 is proved, and further information regarding the function
u(X)
and its coefficients needed below is presented. 00
7.1.
|>W as a Formal Series. We set |i(X) = X — 2(~~l)Wi*-('+1) and seek — !«_! from Eq. (19) which we rewrite in the form
The Function
the coefficients Ht oo
»
|~
oo
"|-(i+D
2(_iy// ( X-('+"-2x-('+')(-l)^ i l_2(-iy// / >.-(/+ 2 ) This o b v i o u s l y i m p l i e s t h a t [A0,...,Aa_2,N0,...,Hn_2] 66
f
or
/ / _ i = 0 , H„ = A0, Hi = Ai and f u r t h e r n>2
.
=0.
ffn — An-\-P„,
(26) where
I n d u c t i o n on n immediately shows t h a t t h e
PndZ Hn£An-\-
Z [Aa, ..., An_2]
e x i s t and a r e u n i q u e l y d e t e r m i n e d .
3 4 , Ai. Ai+4A0A2+2A\+2A30 7.2.
The F u n c t i o n
h(x,
t)
t h e moments An
the
\Hn(x,t)\
The f i r s t
Hn a r e : AQ, Au A2 + A20, A3 +
. (A(X) a s an A n a l y t i c F u n c t i o n . become f u n c t i o n s of
For given d i f f e r e n t i a b l e
X, t ; i t would be p o s s i b l e to a t t e m p t to e s t i m a t e Cn(x, t) for a s u i t a b l e f u n c t i o n C(x, t),
and show t h a t they grow no f a s t e r than
and t h i s would e s t a b l i s h t h e a n a l y t i c i t y of
u{X,y,t),
i*(X) i n
X .
However, i t i s i n c o n v e n i e n t t o
o b t a i n such an e s t i m a t e d i r e c t l y , and i n s t e a d of t h i s we apply t h e method of i t e r a t i o n s t o the integral equation (19):
|i(X) + W(X) + a)-1flty = 0 .
of a f u n c t i o n
X w i t h t h e p r o p e r t y (19) and t h e e s t i m a t e
p. a n a l y t i c i n
We s e t
'rhe
K*.) = X-fs(X).
uniqueness
e=0(X _ 1 )
follows h
from the uniqueness of the formal series.
*-N = — jj(« +
To prove existence we set £o = 0.
0
\ + eN-i)~ldy
and show t h a t
e = lime^ e x i s t s ( f o r g i v e n
so l a r g e t h a t t h e f o l l o w i n g i n e q u a l i t i e s a r e
('.*, 0) uniformly i n X, when |X|
satisfied:
|A(jX|-i7)-'(lr-4A(lX|-L0-2r1<(IM-^)/2, \ | X | - f / > 2 V r A ; t/ = s u p { « | 0 < # < A } . In p a r t i c u l a r , if
u, A
is
(27)
a r e bounded t h e r e i s a domain of a n a l y t i c i t y p, n o t depending on
To t h i s end we e s t a b l i s h t h e f o l l o w i n g
x,t.
« i n e q u a l i t i e s by i n d u c t i o n :
• n - i I •*<* u I 5„_i — e „ _ 2 j A
dy
I n d e e d , | ^l | =
\%\—U
< A s u p | a + X|-'
^\%\—U
<'
'
" .
F u r t h e r , for any W > 1 we
0
have for
|SJV+I
— ^ K A s u p K i i + ^ + ^KM + ^ + S A r - O r M ^ - e i v - i l .
i V = l , we f i n d by (27) 2
(» + Jt |
2
2A(|X| — U)~ \ti |<4A(|XI— U)~ \ £i I .
_1
Setting here
N=\
1
< ( | X \ -U)~\
| « + X + > 1 r < 2 ( | ^\-U)~\
This p r o v i d e s t h e b a s i s for i n d u c t i o n .
t h e i n e q u a l i t i e s of (28) a r e s a t i s f i e d for a l l
ft
.
and u s i n g (28) whence
|«2-ti|<
Suppose now t h a t
Then
je/sr+i—ejv^AsupKtf + X + e^Xa + X + SAr-OI-1! e^ — e^r-i |-<8 | &N — e„_i \, by (28) f o r
n=N—\,N.
Hence JV+I B-l
by (27). This completes the proof of Theorem 6.3. Below we shall mainly use
n(k)
as a formal series. The validity of the corresponding
computations in the analytic version can be trivially verified.
We shall establish several
curious algebraic properties of the series for p.(A,) and its coefficients
Hi.
An
the weight of the variable n + 2 , and we
introduce the corresponding graded ring Z[i4B].
In this gradation the polynomials Hn are
7.3.
Homogeneity.
We call the number
67
homogeneous of weight
n+2 . This fact is obtained by an obvious induction on n from the
identity (26). 7.4. The Symmetry At++— //,-, X-*-^.
a
We define a homomorphism
into itself by the conditions a{A,)=—Hh
of the ring Z[-A,]|[X-1j]
o(X) = [i.. it is obviously an automorphism. We 2
shall show that it is an involution: o = id. ' (!*) — 2 "< (° (P))""''4-0 = I 1 •
Applying- a to (19), we obtain
0 n tne
other hand, by the
definition of (J., X — 2 ( — 1)'//(A—f'+1> = fi . But from the first equation, as in 7.2, the element 1=0
8
1
,
(t )Gt*+Z[.AJ]|[ji.- ]] = X-f-Z[.A,][[X-1]l
Hence a{v) = \ . Applying a again
is uniquely determined.
to the .series for ji , we find i»—2(—1)'°(#;)(JL~(i+1, = x; since H + 2 ( — 1)'AtJl~('+1>=* by (19), t^o
<-o
in view of the uniqueness of the expression for h in terms of |t, we obtain o(// i )=— A t , i.e., o(—fft) = At.
7.5. Summation of the Benney—Miura Series. In the work of Benney [27] and Miura [46] the generating series for the conservation laws were constructed in a different way. oo
oo
•O
Benney
oo
V
V
/ o
+
,+1
\T" 'Z'
started from the series T(z)=^Anz't and showed that y>Hnzn = £i\zij^) -(n , ^ ; similar formulas are given in [27, 46] for other series from (20), (21). After the substitution z = — X - 1 and passage to 4>(X) = X_ir( —X - 1 ) all the series of Benney and Miura can be rep00
resented in the form S i l p ^ u ' V ' M I ' M I . where < ? = — 0, ty£Z[At]((X"1)). Our basic integral equation (19) was obtained by applying to these series the following identity which is of independent interest. 7.6. Proposition. Let k be some
Q -algebra, and let ?GA[[*-1]]. ^(Qr1))
.
Then
00
^ l j j r ( g r ) [
'
p& + k[[X-1]]
Proof.
We n o t e f i r s t of a l l t h a t t h e s e r i e s f o r
terms X_2B+const , so t h a t t h e sum converges able
i s t h e unique r o o t of t h e e q u a t i o n
JJ. ==
'
C and d e n o t e by
^
1^-j \yn$i\ b e g i n s a t l e a s t w i t h t h e
*._1 - a d i c a l l y .
t h e r o o t of t h e e q u a t i o n
We i n t r o d u c e t h e a u x i l i a r y v a r i -
pc = c
/«**i+r 0rd q>—l«
Proof.
The mapping V.O-+Z, v(
v(G)=£{0}, ti(A) = {0}; v(»<{') = '»(?)'+«(min(v(9), v(if)) .
I t i s easy to see that v extends
-1
to the quotient f i e l d of O by the formula «(9<|> ) = z> (9) —tf(),and a l l properties described are preserved.
Therefore,
vdefines
a k-valuation
on the f i e l d of functions on
00 be the k -point corresponding to t h i s valuation. for some
I t does not l i e on Spec©
The group of values of ordoo coincides with
of v i s rZ for s u i t a b l e i n t e g r a l
r ; t h i s i s the rank of
This lemma i s usually employed in the following manner.
Z,
9 with
oogC(A)
requirement means that
since z>(
i. Suppose that we are interested L = d2-\-u
In order to ensure the existence in O of a function
orddZ.(j, = 2 , we must have on C a function with i t s only pole at
In the case
Let
while the group of values
in solutions of Lax equations [Q, I ] = 0 with an operator L of low order, e . g . , for the Korteweg-nie Vries equations.
C .
and the closure of
Spec©
00 of second order.
has no s i n g u l a r i t i e s a t i n f i n i t y t h i s
C i s a h y p e r e l l i p t i c curve, possibly degenerate, and
r=l.
We s h a l l now indicate how solutions of Lax equations are constructed on the basis of the bimodule Jf . 2.4. 2.2e)-i).
THEOREM. Suppose that a bimodule M
i s given with s t r u c t u r e 2.2 a)-d) and axioms
Then there e x i s t s an imbedding i:G-+33[d] and an isomorphism of bimodules 33[d]^J(,
which preserves a l l these s t r u c t u r e s . Proof. We denote by S[V] the ring of k -endomorphisms of the space M generated by m u l t i p l i c a t i o n s by elements S3 and the operator V . The canonical mapping S3-*-33[V] i s an N
imbedding by 2.2 g ) . Any element of S [ V ]
can be represented in the form
^ * ( V ' using the
commutation r u l e V* — bS7=db [2.2 h ) ] . This representation i s unique by 2.2 i ) . There i s therefore a unique ring isomorphism 33[V\—33[d], V"-d, which i s the i d e n t i t y on 53. I t follows from 2.2i) that M is a free 53 [V] -module of rank 1, generated by 1, and that the isomorphism 53[Vl—S[d] extends to an isomorphism of the modules J£=53[Vl 1— 53[d]. We point
82
out that in t h i s argument the
O -module s t r u c t u r e on M was nowhere used.
This is important
for Theorem 2.6 below. For each
cpgO we now define an operator
action of O on Jt .
By the foregoing,
Ly&B [d] by the formula
Z.
The mapping f--Lv
is
k -
l i n e a r , has t r i v i a l k e r n e l , and i s m u l t i p l i c a t i v e by 2.2 e) which completes the proof. The next two theorems concern nonstationary equations. 2.5.
THEOREM. Suppose that a bimodule M i s given with the following additional
structures: a differentiation (V, V i ] = 0
di'.S3-^3i which extends to a
and (V, l ) ? = V i ( l f ) for a l l
in 2.4, and we define Proof.
For
P£33[d\ from the equality
We have (£„ — Id-?)1 = 0 a n d
[Vi, Id-?11=0
cpg© .
and [P, Id-9]1=0, while
di -connection ' Vi'-Jt-+Jl,
(V!— P ) 1 = 0 .
|Vi, £
Then
Hence since
[Vi— P, Ly — Id-?]1=0. But [Vi, V] = 0 .
(dilg,— \P, l
Suppose that a
1 is the free
53-module Jt i s given with the structures 2.2 a ) - i )
in the d e s c r i p t i o n of which a l l mention of
O is omitted.
additional s t r u c t u r e s are defined on . Jt :
53 -connections
where V, Vi, V2 commute pairwise. V11=H, V21=P1 .
t i o n by noting that
We define operators
Suppose, moreover, the following Vf.Jt-^-M,
extending dJt j = \, 2,
L, P£53[d] from the conditions
Then (?1P + d2Z. = [P, L].
From the r e l a t i o n
Remark.
Therefore,
over 531G>], by the argument of the preceding s e c t i o n .
THEOREM.
Proof.
with
[Vi— L, V2 — P] 1 = 0 we obtain, as above, the required a s s e r -
[Vi, V 2 ]=0, [Vi, P]=tfiP, [V2. L] = d2L.
Although as i s evident from the formulation of the theorem, the
O -module
s t r u c t u r e on Jt is not e s s e n t i a l , those modules which are constructed by the method of Krichever and Drinfel'd carry t h i s s t r u c t u r e with some O by the very construction. 2.7.
We remark in conclusion t h a t in the ring of symbols
53((?"')) i t i s possible to
obtain additional information on commutative subrings using Theorem 5.14 of Chap. I I .
Let
N
€>,vcS ((;"')) be such a subring and suppose i t contains an element is a
d -constant, i s semisimple, and
tiN-id\iiN, 53]=S~.
L = 2^ufii,
for which ttjv
Theorem 5.14, assuming s u r j e c t i v i t y
OO
of d:5i*-+33*, implies the existence of the symbol 1 + 2 ™-**~* = Q w i t h t h e P r o P e r t y Q~l°L»Q= *=i UN\N . Therefore, the e n t i r e ring Q l°0°Q consists of symbols which commute with UN\N. Now i t i s evident from the r e s u l t s of Sec. 1 of Chap. I I that the space of symbols commuting with uN\N,
c o n s i s t s precisely of symbols of the form ^ vfif, where v£Ker df)33+,
since a l l such symbols commute with terms in the given i n t e r v a l
UN\N , and the subfactors of t h i s space, considering
of o r d e r s , have the required dimension.
This implies several conclusions.
83
a) If the leading term KJV-IGS
-
, and
+
53
UN of the operator L i s a d -constant and is semisimple, if
i s commutative, then any two operators commuting with
L, commute with
one another. Indeed, then the ring (Ker
3S=Mt(%),
au =diag(Ci, .. .,ct), ct=£Cj for
i^=j-
1
In t h i s case the e n t i r e ring of symbols commuting with u^ *, consists of diagonal matrices of symbols with constant c o e f f i c i e n t s .
Therefore,
/ " p a r t i a l " order functions are defined
on i t : the orders of the symbols a t the s i t e ( i i ) ,
1<£
The argument of Lemma 2.3 can
be applied to each of these separately, so we obtain the following r e s u l t . b) If in the commutative subring Ocz3S[d\ with ^ == Af4(.^0X there i s an element L with invertible there are
«Af=diag(c 1 ,.. -,ct),
Ci=f=Cj, UAT-IG®-, then on the smooth projective model of Spec©
i n f i n i t e l y d i s t a n t points the orders of the poles a t which are determined by
the p a r t i a l order functions of the symbols of the corresponding o p e r a t o r s . In the c e n t r a l i z e r
L in 53((?-1)) the p a r t i a l orders do not depend on one another, but
in O they may be r e l a t e d and even completely determined by one of them. the points a t i n f i n i t y there a r e not necessarily 3.
Therefore, among
/ distinct points.
The Standard Realization of a Bimodule Over a Field 3.1.
In t h i s section we assume t h a t the basic d i f f e r e n t i a l ring
53 i s a f i e l d .
This
does not reduce appreciably the generality of the r e s u l t s , since for almost a l l i n i t i a l cond i t i o n s the solutions of the s t a t i o n a r y and nonstationary Lax equations are l o c a l l y a n a l y t i c , and we can seek them i n the f i e l d of germs of meromorphic functions. applications i s R or with the
C ; i t i s assumed to be a l g e b r a i c a l l y closed in
d -constants in
The base field
k in
$3 and to coincide
S3 .
Let 0(Z33[d\ be a commutative subring containing
k .
I t contains no zero d i v i s o r s .
The orders of i t s elements form a semigroup; hence, there e x i s t s an integer «o> such that a l l those orders greater than «o> form an arithmetic progression with difference the proof of Lemma 2.3 i t i s c l e a r that Jf=ffi[d].
The semigroup of orders of
r . From
r coincides with the rank of the (S3, O)-bimodule O is f i n i t e l y generated; choosing i t s generators and
operators of the corresponding orders we find that the ring 6 i s f i n i t e l y generated over k .
I t i s thus the ring of an absolutely irreducible curve.
Let
oo be the point of i t s
smooth model defining the orders of operators of O as in Lemma 2 . 3 .
We s h a l l assume that
the leading coefficient of a t l e a s t one element of O of nonzero order i s a constant.. From formula (1) of Chap. I I i t then follows that a l l leading coefficients of elements of O are constants. In t h i s section we consider i
as a l e f t
53®0=530-module with action
(bf)m = bmf,
k
bgB, m£<M, /GO, and we r e a l i z e M as a submodule of vectors i n the r
ordinate space K
over the quotient field
the standard r e a l i z a t i o n . 84
K of the ring
330 .
r —dimensional co-
This r e a l i z a t i o n i s called
In order to describe i t and carry over to i t s language the basic s t r u c t u r e s r e l a t e d to M , i t i s convenient to use the elements of the theory of algebraic curves. is the field of functions on an algebraic curve over with f i e l d of constants extended to
53 .
The points of the field are l o c a l rings in K con-
53 .
Spec©,
are called constants and the remaining points are v a r i a b l e s .
Those f i e l d points which come from points of the complete smooth model of
constant p o i n t .
The principal part of an element
/ modulo the maximal ideal corresponding to Any k - d i f f e r e n t i a t i o n of
dO={0} .
K
53— the smooth complete model of Spec©
taining
of
The field
The same l e t t e r
fdK
a t a point
For example,
oo i s a
P i s called the class
P.
53, in p a r t i c u l a r
d, extends uniquely to K by the condition
d i s used to denote the coordinate-wise action on
K''.
We can now formulate a theorem on the standard r e a l i z a t i o n . 3.2. r
McK
THEOREM.
The bimodule Jl = 3&\d\ i s canonically isomorphic to the 530 -submodule
with the following p r o p e r t i e s : a) (530)rcJ£; the factor space b) The element
Ml{?BO)r i s finite-dimensional over 53.
\^Jt i s represented by the vector 1=(1,0,.. .,0)'g.^.
c) Let .W(i) denote the subset of elements of M, the j - t h coordinates of which have poles a t
l
oo of order not exceeding
^(i) =(«^i)~ for a l l d)
i>i0,
where
The connection ^:M->M
\£Mr(K)
~'
with leading d .
form a
53-basis of
330 -module M has no t o r s i o n .
(^!b n fn)m = ZJbntnfn of that
[j^bnfnjtn
.
O .
If 2&„/„=£0
is
any
Choosing the g r e a t e s t
i s equal to the leading term
M has no n o n t r i v i a l t o r s i o n .
V =
e emen
l
t
«o with
Indeed, suppose
For each n(
Clearly, the elements {/„} form a k-basis
£0.
I and
r>2)
T i s the semigroup of the orders of elements of fn^P
all
i s induced by a connection on KT of the form
We note f i r s t of a l l that the n
Then M(i)
t0 i s a s u i t a b l e constant.
i s a matrix of the form (for
Proof.
, y'=0, .. . , r — 1 .
of
of
O .
53(5 and m£J(,
b^^O,
They therefore
m^O,
then
we find t h a t the leading term
*„„"*/„„> i . e . , i s different from zero.
This means
This implies t h a t the canonical mapping JI-+K ® M:
mo m^*\®m.
i s an imbedding. K®-M = Kr .
We e s t a b l i s h the isomorphism elements of
Y are d i v i s i b l e by
r,
We s e t M' = 5 3 1 0 + . . . + 53V~11©.
the orders of the elements of /
to ymodr , and therefore the sum of the 53V 1© i s d i r e c t . of M' include a l l i n t e g e r s greater than some space Mn
i s finite-dimensional over
Since a l l
83S7'\0 are a l l congruent
Further, the orders of elements
/to. and therefore
M = M' -\-Jini.
Since the
53 , the 53© -module Ji/JC i s a t o r s i o n module. Indeed, 85
if
tion of the form ^!>&/'£.#',
i.e., Jl'-+Jt
Thus, the imbedding K®M'
^bj1
{ V y l | 0 < / < r — 1} (more precisely, {l^V^l}) • We identify
with the space Kr
and K®M
mmodJC.
is annihilated by
induces an isomorphism K®M'' -*K®M . In view of the above,
has the canonical basis
K®JC'
of columns of height r with coordinates in K by
This defines the canonical imbedding Jt->Kr.
means of this basis. Jt.
m£Jl, then there is always a nontrivial rela-
/G
We denote its image by
r
The inclusion (33Q)
already been proved: (330) space Jl^M'^M^.
is the image of Jt',
r
while Jll(330)
is isomorphic to the factor
The element IQJC is represented by the vector (1,0,...,0)' .
The filtration with respect to order is easily described on (*o/o> • • •»^r-i/r-i)'' (bfi33, ffcO) if and only if
not exceed — —
bj=£Q.
for
we may remove the condition Thus Jfwcz(Jfi)" tary to (630y .
j — rordM/;
to the vector
by Lemma 2.3.
(b0/0,...,
Therefore,
, i.e., the order of the pole of f}
at oo does
Taking the order of the pole of the zero element to be — oo, bj^O.
. In M we now choose a finite-dimensional subspace over 93 complemen-
The orders of the poles at oo of all the coordinates of elements of this They are therefore contained in Jt . for the sufficient-
subspace are uniformly bounded. ly large i.
(330)r:
there corresponds the operator 2 */V'l//» the order of which
is equal to max(y'+ordZ./.l ^/^O) =max(y— rotdoJj\bj=^Q) br-.1fr-iyG(Jti)~,
over 33 have actually
This implies that J£(/) = (^i)* for sufficiently large i.
We have now verified asser-
tions a)-c) of Theorem 3.2. In order to establish d) we note first of all that the connection V extends uniquely to a connection \7:K®Jt-+K®Jt
with V (f®m) = df®m-\-f®S7m
for all f&K,
m£Jt (this is
a standard fact regarding the extension of connections on a localization). With the identification of V — d
difference Since
/
is a
K®Jl
with K'
we obtain a connection T
/C-linear mapping K'-+K .
S7'-Kr-*Kr.
The
Let A be the matrix of this mapping.
+1
V(V 1) = V' 1 , we have on setting e, = (0.. .0 1 0.. .())< a representative of
VJl
in Kr,
i (tj+i
for j
— 2,
2 V A W ) f°r J = r-\.
V-o
Therefore, A has the form indicated in the theorem. 3.3. Remark.
The standard realization of Jt
This completes the proof.
shows that the matrix A, or its last
row (A0, ..., Ar_j) is essentially the unique invariant of Jt
(for given S3 and O ) : Jt
is
OO
recovered from it as the connection tions that M
V-
^33(d+Ay(\,Q,.
..,0)', together with the filtration, the action of 33 and
However, (A0, ..., A,_,) cannot be chosen arbitrarily, because the condi-
be invariant with respect to multiplication by
and that the filtration of Jt
O , have finite type over O ,
be described in terms of the behavior at oo, as in Theorem
3.2, impose strong and nontrivial restrictions on A . We shall occupy ourselves with them
86
in the next s e c t i o n s .
Here we note only t h a t part of these r e s t r i c t i o n s have the character
of d i f f e r e n t i a l equations: of
A i s to be considered as a c o l l e c t i o n of functions on the model
Spec©, depending on parameters x, (x, t), (x,u,t)
and satisfying c e r t a i n d i f f e r e n t i a l
equations in these parameters. 4.
Bimodules of Rank 1 4.1.
In this section we give a classification of bimodules of rank 1 which satisfy two
additional conditions. We can formulate the first one immediately: it is that the ring 530 consist precisely of all functions of the field K, having a pole only at the point oo . This condition is equivalent to the condition that complete model by excising the point at
oo.
Spec© be obtained from its smooth
The second condition will be formulated later.
Xv
We fix Jl and i t s standard r e a l i z a t i o n
Jt(*K.
Let Afc/X" be defined as i n Theorem 3 . 2 . We c a l l a pole of
M any point
f(*Jl, having a pole of order order of the pole of 4.2. field
THEOREM,
P of the field
> 1 at
f(*Jl a t
P.
K, for which there e x i s t s an element
We c a l l the order of the pole P
P (oo i f there i s no g r e a t e s t o r d e r ) .
a) There e x i s t s a f i n i t e number of nonconstant points Px,...,Pt
K and positive integers ax,..., as such that
having poles of order < a ;
at
P;(l
b) The degree of the divisor
and a pole of any order at
D = £alPt
a l l the points Px, ..., Ps, oo and such t h a t
i s equal to the genus
Ph oo, r e s p e c t i v e l y .
respective forms
Proof.
Pit oo.
z—2(co)
oo.
g
of the field
K-
O , which i s f i n i t e a t
z— z(oo) are local parameters a t the
Then the principal part of
-[J \p' y +£;•
classes of the points
z — z(Pt),
of the
M consists of a l l functions of the field
c) Let z be an element of the field of fractions of the ring points
the greatest
A a t the points Pi,<x>, have the
+ c °°' where cu c«= l i n e in the fields of the residue
The divisor of the poles of A i s precisely
Let P be any point of the f i e l d
K,
and l e t
Op
oo +
2 ^
be i t s local r i n g .
It is
oo
known that d takes Op poles of M
into itself (cf. [20, Lemma 2]).
Since •M'='2i
must be contained among the poles of A, so that M
We shall first show that at oc M
has a finite number of poles.
has a pole of exactly first order. If A had no pole
at oo then neither would Jl, and this would contradict the inclusion S30cz.M. of
A
at oo were of order
precisely
ja
a > 2 , then the order of the pole of
(an easy induction on j using the fact that
of a pole at a constant point).
But
(d-fA)-'l
d
must coincide with
would be
does not increase the order
corresponds to the operator di^[d\=JC,
by Theorem 3.2 c) the order of the pole at oo of the function large j
(d-\-L)i\
If the pole
/ . Therefore, the case a > 2
(d^-A^l
and
for sufficiently
is impossible.
87
Suppose now t h a t A and hence of shows t h a t
P=£<x> i s a constant p o i n t .
Jl .
We s h a l l show t h a t i t cannot be a pole of
Indeed, otherwise, the same argument as in the preceding paragraph
JC contains functions with a pole a t
i s f i n i t e l y generated over
P
of a r b i t r a r i l y higher o r d e r .
530, while elements of SBO have no poles a t
P
But M
at a l l ;
this
i s thus impossible. We now consider a nonconstant pole Pt of the module otherwise, as above,
M could not be f i n i t e l y generated over
3BO have no poles a t
Pi .
the f i e l d of f r a c t i o n s a t feM
.
Since
Suppose t h a t at>l
I t s order i s f i n i t e ,
for
SBO , since the elements of
i s the order of t h i s pole.
We choose z
in
O, as in part b) of the formulation of Theorem 4.2 and an element
with a pole of order
((z — z(Pi))~"i+1)
,M .
at
at
Pi .
I t may be assumed t h a t
(d + k)ff<.M must a l s o have a pole a t
Pt
—z(Pt))~aur
f=(z
0
of order no greater than
o-i, and aidz(Pi
0 + A)/-A/+
we find t h a t the expansion of A a t
\l+l
0{{z-z(Pi))-a^,
+
Pi must begin with —
gifg\pr
s
We set D^^a^i
and denote by X(D-\-joo)
in the linear space over SB of all func-
tions of K, the divisor of the poles of which does not exceed
D4-Joo.
According to
Theorem 3.2 a) and c), for sufficiently large /' we have OC(joo)czJijC%(D+Joo). .A(j=Z(D + joo)
for / > 0 , since in Jt}
Therefore,
there are elements with principal parts at P, of
order exactly at, and taking linear combinations of them with suitable coefficients in 530, we can make these principal parts whatever we wish. Now the dimension of Mj
over SB is equal to y' + l, while the dimension of
for y > 0 by the Riemann-Roch theorem is equal to degZ) — g + l+jdeg a point of first degree and AegD=g.
oc.
Therefore, oo is
This completes the proof.
4.3. We now formulate the second condition imposed on oar bimodule divisor Pi+...+P,
£(D-\-Joo)
•$>. It is that the
defined in the preceding section be nonspecial, i.e., there exists no
differential of first kind that vanishes at Pi+'...+Ps» nonconstant function
But then from the existence of a
A with a divisor of poles Pi +...+Ps+co
g, ft,=l; the function
it follows that
deg(SPj) =
A is uniquely defined up to an additive constant.
The divisor Pi + ...+Ps
x=x0,
is clearly nonspecial if this is so at the initial point
and the set of such initial conditions is dense in the g -fold symmetric product of spec 0. We shall now show that the conditions on A of the Jacobian coordinates of Px-\-..
.^P,
become a condition on the linear variation
with respect to X in the functional case. We
shall assume that SB is the ring of germs of meromorphic functions of X Pt
as germs of holomorphic paths on the Riemanri surface of Spec©, parametrized by
may be assumed that they are distinct, i.e.,
88
and represent the
s=g.
x.
It
We choose a basis of d i f f e r e n t i a l s of f i r s t kind O ,' a function 2 as in Theorem 4.2 c ) , and set points Pi, ...,Pg.
We set
/j = ./* (JC) = 2
\ mt.
">,, ...,10^ in the f i e l d of fractions of
«!>i = Uidz.
The functions
We define the element
ut are f i n i t e a t the
b&i from the condition
7—1 oo
A=
r- Z (oc) + Q < 1 ) 4.4.
near
THEOREM,
the divisor
°°-
a) dxJi = bU:(oc) for a l l
i = l,...,g.
P\-]r • •••\-Pe move in a constant d i r e c t i o n
Hence the Jacobian coordinates of (Ui(oo)) with speed
b(x) a t the point
x . b) Conversely, suppose D = P i + . . . + P e is a nonspecial divisor of the field K, Jacobian coordinates of which vary with x as in part a) with some function
the
b£33 . Then
there exists a unique (up to a constant of 33 ) function A with principal part
z_z,00\
oo
at
oo and divisor of poles
D+oo,
and £ 33 (d-\-&) dl i s a
(53, ©)-bimodule.
7=0
Proof,
a) For any d i f f e r e n t i a l
be poles of «;A
only a t the points
u> of the field Pu...,Pg,
K we have ^res e (joA)=0 . •
oo, so that
*
There can
Q
,
]£,• res p (i»jA) = —res^^A) . 7=1
°°
'
Therefore, £
£
dxJ'i^^iMP'j)oxz(P;)=2
S
res p .
y-i
7-1
Ui
f_f{P)J
'
7
7=1
b) Conversely, suppose that Pi + . ..-\-Pg dji
dz = - ^ resp (^A) = resoo(oJiA) = tei(oo). varies as in part a).
read in the reverse order shows that at the points Pj ^ ^
with
Then the calculation of
the principal part of A
, since the matrix («;(P/)) is nondegenerate because Pj+.. -+Pg
begins
is nonspecial.
Arguments analogous to those given in the UX(PX 7>o
proof of Theorem 4.2 show that 2 33(o +... +P e -\-J<x>), and all the axioms of a bimodule for this space are verified without
difficulty. 4.5. field.
Theorem 4.4 gives almost a complete c l a s s i f i c a t i o n of bimodules of rank 1 over a
In order to apply i t to find solutions of nonstationary equations, i t is useful to
have in mind the following s i t u a t i o n s . a) Let us assume that 33 i s a ring of germs of functions of ary Lax equations we must extend
dt to M
of V, must have the form dt + kf where kfijt
.
by the condition
&t€K .
The commutation condition has the form
Since
x, t .
[V*, V/]=0.
To solve nonstationThis extension
(dt + bt)\eJl , we necessarily have
dxkt= — dt&.
uniquely determined by i t s principal part a t oo (because
Since each element of Jl
Pi-\-... +Pg
is
is nonspecial), A,
X
can only have the form — )\ dtkdx + (an element with a
^-constant principal part at
On the other hand, since the connection V< must take A
oo ) .
into itself, the behavior of A,
near P;- is determined by the same sort of conditions as in Theorem 4.2 c). The argument used in the proof of Theorem 4.4 shows that the motion of Pi+.-.+P* with respect to t also becomes rectilinear in Jacobian coordinates; its direction and speed are determined by the principal part of A, at oo.
The same is true for the motion in y for solution of the
Zakharov—Shabat equations.
89
jArfx
b) We s e t
x
$=e '
.
Then for any m^M we have \(dx + K)m]-b = dx(rti<§ ,
can also be r e a l i z e d as the space
Therefore, M
Jtif, to which a l l s t r u c t u r e s carry over i n an obvious
way, while the connection Vx goes over in to ordinary d i f f e r e n t i a t i o n with respect to x . The function
f i s called the (stationary) Akhiezer function.
Prescribing t h i s function i s
equivalent to prescribing A and hence the bimodule Jt i n the standard r e a l i z a t i o n .
In the
papers of Krichever [16] and Matveev [44] t h i s function i s taken as the i n i t i a l object.
If
A does not depend on t nor A, on x , then i n the nonstationary case the function ty(x, t) = expi \ Arfjc + 5 bjAt J .
On M§ the connections 'Vx and V* become dx and dt,
respectively.
c) Since A, A* a r e determined by t h e i r divisors and principal p a r t s a t oo, they can be written out e x p l i c i t l y i n terms of the c l a s s i c a l Riemann theta functions.
The coefficients
of a l l operators which enter i n the s o l u t i o n of Lax equations a r e expressed i n terms of A, At and t h e i r d e r i v a t i v e s .
This leads to e x p l i c i t formulas for the s o l u t i o n s .
For further
d e t a i l s we refer the reader to the papers [16], [ 4 4 ] , and Sec. 6 of Chap. IV. .5.
Bimodules of Higher Rank dver a Rational Curve with Double Points In t h i s section a c l a s s of bimodules i s constructed which may have a r b i t r a r y rank r ^ l .
For the motivation see 1.4 and 2 . 2 . The r e a l i z a t i o n s given here a r e close to the standard r e a l i z a t i o n s but do not coincide with them. 5.1.
The I n i t i a l Objects.
We s e t & = R or C, and we l e t 3$o be a r i n g of germs of
k - a n a l y t i c or i n f i n i t e l y d i f f e r e n t i a b l e functions of the variable x (for s t a t i o n a r y Lax equations), of (x, t) (for nonstationary Lax equations), or of (x, t, y) (for the equations of Zakharov-Shabat); $ = Af,(5?n) (the matrix algebra of order / over 530); d = dx = dldx,
d2=dy=didy. To construct O we choose 2N d i s t i n c t numbers ah f^eC, i — \,...,
a,=JFj
i f k=U
N with the condition
and we s e t 0={/,(X)6A[M|v; = l, ...,N,
(1
/(«>)=/«
>
I t i s obvious t h a t O i s a r i n g of functions on the affine l i n e with N pairs of identifiedpoints realized as a subring of the functions on the l i n e i t s e l f , i . e . , &[>•]• Similarly, we construct M as a submodule of the t r i v i a l
(53, *M)-bimodule
JV=3$\\\r
of rank r with i d e n t i f i c a t i o n conditions a t the points (a*, P,) . We s h a l l write elements of 53r and jr = §imr as columns of height in
53, 53 [X], respectively ( X commutes with 53 ) .
by the formula
90
r with coordinates
We introduce on Jf the l e f t action of 53
where t
S = Af/(,<S0)-
is the transpose in
We define the right action of
ordinate-wise multiplication on the right by GL(r, 53) .
k\k\
on JV as co-
£[M^r » where Er i s the identity matrix in
Finally, we introduce on JV the natural left action of the group GL(r, 53).
Obviously, the actions of 53 and k\k\ on JV are effective and commute with one another and with the action of GL(r, 53). The identification rules defining M
are described by a collection of N
g;gQL(r, C®53), on which additional conditions will later be imposed.
matrices
Having chosen this col-
lection, we set M = {m(\)£
k=R
N, m(ai)=gim(^)}-
pV-=a<, this implies the condition
and
(2)
gi=gf ! .
The remainder of this section i s devoted to describing those conditions on (*/• p\)>
(gt) and
which enable us to introduce on JC the bimodule structures with the axioms of 2.2.
5.2. on JV.
The Actions of
5? and
O.
We have described above the actions of k\K\ and 53
I t i s obvious from definitions (1) and (2) that
of 53 and
O on M commute.
The F i l t r a t i o n .
and the actions
That these actions are effective will become clear below
following construction of the element 5.3.
SSJtttzJl, JtGaJt
l£M.
We f i r s t describe an auxiliary f i l t r a t i o n on JV by setting
^f_1 = {0}, and for any a>0, 0 < £ < r ; l bi
( \ JG53[X]', b ...,b u k
* /,/ "ar^ = )j "
are polynomials in
X of degree
; (3)
^*+i, • • •, br
are polynomials in \ of degree < a— 1}. 00
(A polynomial
of degree
It is obvious that Ji"c:JVui
< — 1 is zero.)
and Ji"=\JJl", . 1=0
/,
r
Further, ^ ,Ti = " (©53eJ+i, where el+1 has and zeros elsewhere.
Therefore,
JVul/jVl
a
X Er at the s i t e
k + l for l = ar + k, a>0, 0<&
is free of rank 1.
The f i l t r a t i o n we need on M will be induced by this f i l t r a t i o n on JV and the followoo
ing s h i f t .
We temporarily set
!
Mw=J\ 'k(\dl
.
I t is obvious that J(=[J^ik)
We impose the following conditions on the collections 5.4. i
The Nondegeneracy Condition.
and 3HM{k)aJ(w.
(gi), (a,-, Pj) .
The block matrix in MrN(33): G = {a.[Er — ^[gi), 1<
0
i
Lemma.
If the condition of nondegeneracy 5.4 is satisfied, then Jtw = {0} for
and the natural mapping <MlJr\\Mi-+J\'*lJ:XiJVl is an isomorphism of
l>rN + l
53 -modules for
.
The f i l t r a t i o n g where any
J^=J£(*+rAr+i)
for k> — 1 therefore satisfies conditions 2.2e, f, and
53-generator of the module Jf(r,v+i) may be taken as
1 .
91
Proof.
According to ( 3 ) , Ji"rN = { 2
=
m k
i ' Imfi®>T\'
Therefore, by (2)
m lJ
2
lXJV-1
J I V/ = 1, . . ., iV, 2 W*r~ P/Sl) % = 0 • y=0
J
The condition i n braces may be w r i t t e n
Since G i s nondegenerate by hypothesis, i t follows that Jt^N) = {^} (and hence jt(t)—0
for
*
is a
33-isomorphism for
l>rN,
£,+i of the form et+i =s £/ mod Jft,
it
suffices
where e^i
is
described i n 5 . 3 . To find
ei+i = 2 O T A ;
we
™"st write down the system of l i n e a r equations for the nij, cor-
responding to the conditions ( 2 ) .
After displaying them over the commutative ring
$?o we
find i n the system a nondegenerate minor of maximum possible rank flN corresponding to the matrix
G.
The system i s therefore solvable.
5.6.
The Explicit Form of
1.
From the proof of the preceding lemma it is clear that
we may set _y-i
AN
l=\.m;\J+l° where the
ttij form a solution of the system
5.7.
The Connection
V = VX .
(i = \,
J, m&P,
W
...,N):
The following r e s u l t s with obvious modifications w i l l
also be applied to the construction of the connections Vi = V«, V 2 = V y . We f i r s t continue dx:3?,->- 63 coordinate-wise to t
Inasmuch as dx(blb 2) = dx(bi)b'2+b1(dxb2y Any other connection extending r
EndS[A.]
.
( T r i v i a l i t y on
for
63r and then to 8tf\l\
bu b2(<33 , t h i s action i s a
dx and t r i v i a l on
X, has the form
X gives the condition Vx(m'j)={S7xm)y
so that
dx\=0.
dx -connection on J" .
dx + dx, where dxGM,(53\)<]) = for
meJ",
We use the freedom in the choice of d.{ to ensure the condition Vx~MaJ(. 5.8.
LEMMA. Let
x
7x = dx + dx(l).
Then S?xM
V/ = 1,...,A', d^-g^AM-dA'dSi-
92
(6)
Proof. According to (2), m(\)£Jf,
if and only if m(ni)=-g,mQi)
N ,
for all i = l
whence (djn) (a,) = dxgim (&) + gidxm (,3,.)Tlierefore, \(dx+dx)
m\ (a,) = dxgirn (fc) + gtdxm (fc)+rfx (a;) g,/» (p,).
On t h e o t h e r hand,
Si Wx + dx) m] (M = g t d,m (f>,) + g
S o l u t i o n s of E q s . ( 6 ) .
t i v e l y with
[dx{at,x),
commute r e s p e c dx{a.i,y)\
*=[dx$„x),
(Jf)=exp ( - Jrf^a,, E)rf6j gl (0) exp ( j <*,(&, ?)rf?J.
I n t h e examples we s h a l l t a k e
5.10.
(x) = e x p ( - JM*, (a,)) gi (0) exp (xrf,(p,)).
Tlie Connection of Then SJMiCM^
Vx with the F i l t r a t i o n i n
for a l l
if the following conditions are
/>—1
where d0x£Mr(5?)
Ji.
(8)
LEMMA.
Let
and V* i n d u c e s an isomorphism
S/xJKczJ(,
Vx=
JillMl_\'^.MlJ!xlMl,
satisfied:
for r = l : dx = d0x+dlx\, for
(7)
dv(X)6Mr(M,(fe[X])), and t h e n gi
.
and dx($t,x)
Then an e x p l i c i t s o l u t i o n of (6) can be w r i t t e n i n t h e form
gi
^x rdx
dx(a.it x)
e x p K r f ^ a , , S)rfM m d e x p K d x ( M ) d S K t h i B I s BO I f
dx($i,y)]=0).
J
We assume t h a t
rflre$*=GL(l,33);
r > 2 : ^==rf 0t . + rfi.v>.,
h a s z e r o s on the main d i a g o n a l and above and i n v e r t i b l e elements ( i n 53*)
a l o n g t h e d i a g o n a l s below the main d i a g o n a l , w h i l e
dlx^Mr(3S)
h a s an i n v e r t i b l e element i n
t h e r i g h t upper c o r n e r and z e r o s e l s e w h e r e . Proof.
Since
Jfl = JCf\^"i+rN+i
and Jll+llJll^J''i+rN-\-2l'd/'i+rN+i
v e r i f y t h a t under the hypotheses of t h e lemma V x ^ ' i C ^ + i and JV^I^^^t+ol^i+i since
.
for
L>rN,
i t suffices
to
V x induces an isomorphism
But t h e a c t i o n of VA- on t h e l a s t f a c t o r c o i n c i d e s w i t h t h e a c t i o n
dx,
d ^ ' j C - ^ j , and t h e r e q u i r e d r e s u l t i s checked by d i r e c t c o m p u t a t i o n . We have now completed t h e c o n s t r u c t i o n of t h e c l a s s of bimodules
t h e s e c t i o n i s devoted to some f u r t h e r 5.11.
The Rank of
M.
n , then
Jt-ficMi+w,
The r e s u l t of
remarks.
I t i s e q u a l to
r
i n the sense described i n the introduction
and a l s o i n t h e sense of Lemma 2 . 3 and i t s c o r o l l a r y . degree
M .
and m u l t i p l i c a t i o n by
Indeed, i f
= induces t h e isomorphism
Jt,/Jt,_i~ 93
•MUrnl'Mi+Tn-\
•
I t suffices to verify t h i s on the corresponding f i l t r a t i o n submodules of
J where a l l i s obvious from the d e f i n i t i o n s . 5.12.
The Connections
Vt, V y .
We s e t S7t = d(+dt,
V , = d , , - f rfr For solving nonsta-
tionary equations these connections must s a t i s f y conditions analogous to (6) but not Lemma 5.10.
In place of i t we need the
(cf. Theorem 2 . 5 , 2 . 6 ) .
commutation
conditions
IVX, Vt] =[V JC . Vy] = [V<, Vy]=0
They can be w r i t t e n i n the form dxdt-dtdx+[dx,dt]=0
and s i m i l a r l y for the remaining p a i r s . [^JC>^I=0
etc.
(9)
In the case dx, dtf>Mr (Mt{k[k])) they become simply
If these conditions are s a t i s f i e d ,
then the matrices
gi(x, t) for the Lax
equations have the form gi(x,t)=exp(-xdx(at)-tdt(lxi))xgi{0,0)exp(xdjctfi)
+ tdt$i)).
(10)
Conditions (9) with constant dx, dt, dy are most simply satisfied by taking dt, dy to be polynomials in 5.13.
dx . The Order
of the Operators. According to 2.3 and 5.11, the order of Lv, is
equal to rdeg-f . It is not hard to see that the order of P, found from the condition V/1=/>1, does not exceed
rdegdt
(degree in X ) and similarly for
Vy . These considera-
tions determine the choice of O, V/ in the next section. The case where O contains polynomials of degree 2 and r=l. leads to multisoliton solutions of the Korteweg-nde Vries equation and its higher analogues and has been treated in the literature in various ways (by the method of inverse scattering, the method of Zakharov-Shabat, Hirota's method).
The next most difficult case is the case r = 2.
6. Example: Solitons of Rank 2 6.1.
Parameters of an Ar-Soliton Solution. We shall construct an Af-soliton solu-
tion of a nonstationary Lax equation of the form Lt<=\P, L] , where the order of L is equal to four and the order of P is two.
The parameters of the N -soliton solution are con-
stants which go into the construction of the appropriate bimodule M. We take £ = R, SS — Sio be the germs of meromorphic functions of x, t of solitons N
R ; the number
is the number of pairs (*> PA) of dual points.
In order that O
contain a polynomial of degree 2, it is necessary and sufficient
that <*ft+P* not depend on for all k;
over
a
k . Replacing
X by
/.-[-const , we may assume that
a
ft+P* = 0
2
then X g<9 . Real a* lead to nonlocalized solutions. We therefore take a*
to be pure imaginary.
For convenience of subsequent computations we set o.k=2ia2kb, $k—-
— 2ia\b, ak, b&R, where b is chosen as follows. According to Lemma 5.10 and (8), the connection V ^ has the form u Q), a, *6R* • Replacing X by may assume that 94
dx-\-dx,
where tfx =
ar'X and modifying the value of ak, correspondingly, we
a—1; b remains free.
We s u b j e c t the c h o i c e of have o r d e r 2 .
V , to t h e c o n d i t i o n t h a t t h e o p e r a t o r
According t o 5 . 1 2 , i t s u f f i c e s t h a t for Vf = dt-\-dt t h e m a t r i x
X l i n e a r l y ; moreover, f o r c o n s t a n t c o e f f i c i e n t s of V<
commute l e a d s to t h e c o n d i t i o n
term
P, f o r which V , 1 = P 1 ,
dt=
dx
and
dt
dt depend on
the condition that
where (»GR, CGR*
( t h e a priori
V* and
admissible
i t i s n e c e s s a r y t o f u r t h e r choose i n i t i a l c o n d i t i o n s gk(0, 0)GGL(2, C) . 1
must have g*(0, 0)=g f t (0, 0)" . We f u r t h e r s e t
We r e p r e s e n t
$ = b(x + cf),
T=
gk(0, 0) i n t h e form
M,
According t o 5 . 1 , we
gk(0, 0)=TT1-T*. T*eQL(2, C) .
Then formula (10) becomes
gk(x,t)=Gt(x,trGk(x,t),
(11)
Gk(x,t)=e<\exp(°l2'f'iy-
<">
where
Thus,
— c has the interpretation of a certain "group velocity" of the
N-soliton
solution; co is the common frequency of oscillation, and b is a scale factor;
The Element
and V>
k -th soliton.
determine the shape of the 6.2.
ah
1 . The coefficients m3-
of the element
1, defined according to
formula (4) are found from the system of equations (5) by transforming with the use of relations (11) and (12)
(/ — I,.:., AT):
2l(2^2/*)*Gy-(-2w^)*G;)]m/i =[{2ia)brG1-(-2ia)by'G]{
6.3.
The O p e r a t o r s
we s o l v e i n
M
L and
P.
The c o e f f i c i e n t s of and
/KAr_2 = ^ 3 J
= -
4d
+ wx+wVx
«, v, w, z
+ z)\ = ll2b2.
(1*) coefficients
by t h e formulas v = — 4dx\>.2, w = — 6d2xV.2—Abd&i + ^ d . ^ ;
xf12 - 6 A ^ 2 - 4MX|*, + 8 (dxV.2f + 4&M*ft! + 6^2d2x^+4fy2
To o b t a i n them i t s u f f i c e s to e q u a t e t h e c o e f f i c i e n t s of r i g h t s i d e s of
To find
L a r e u n i q u e l y e x p r e s s e d i n terms of t h e components of t h e
u=0; z
L = d*x + ud3 +vd2 +wdx + z.
(13)
the equation (Vx+uS7x
mff-i = r1\
We s e t
? J.
(15 )
Xw+2, Xw+', Xw on t h e l e f t and
(14).
S i m i l a r l y , for t h e e q u a t i o n
V / 1 = / > 1 , we f i n d P=mb~ldl + cdx—2
(16)
95
The condition
Lt — [P, L\ has the form of a system of equations for
v, w, z:
—bu>~lvt=2vxx—2wx—bcsTxvx\ —b«rlwt = —wxx+2vxxx + vvx - bc^~lwx—2zx;
(17 )
l
- b«r zt = ^ vxxxx + 2 vvxx - zxx + j vxw - bc
ba~l and the "speed" of the solution
equations; the remaining constants of 6 . 4 . A Single Soliton. acquires the form
c determine the form of the
N - s o l i t o n solutions can vary.
We solve Eq. (13) for the case N = l, ai = a, yi=iE2 .
The system
(iGQ + iG0) (*) = (2a2G0 - 2a2G0) (£), whence
(jj)=2d»(ReO^r'ImO0(J),
(18)
G0=e«"^xp§2ioa2)l.
(19)
where
/02ia2\2 (2ia2 0 \ i(l+i)a 0 \2 Since ^i Q ) —y 0 2ia2j \ 0 (1 -\-i)a) ' exp we obtain without d i f f i c u l t y
bt y
se
P 3 r a t l n g even and odd powers in the s e r i e s for
n _ . j i . - « ( c h ( 1 + < ) f l S . 0 + i)«sh(l + i)al \ «o-« '\(1 + i)-ia-i Sh(1 + i)al, ch (1 + i)&)•
(20) u u ;
To simplify the notation we set ch = cha£, sh = sha;, c = cpsac, s = sina;. cos = cos2a 2 t, sin=sin2a 2 t. Computing (20) e x p l i c i t l y , we obtain 0
/ch c cos — sh s sin, a (sh e cos—ch s cos — sh c sin — ch s sin)\ ~\ 2^(shccos+chscos — chssin+shcsin), chccos — shssinj'
,„.,..
ImG _/ch£sin-f-shscos, a(shcsin+shccos+shscos—chssin) \ I ^ ( s h c s i n + c h s sin+chscos —shccos), chcsin-fshscos] # After rather lengthy but straightforward transformations, we find from this
det Re G0 = j (ch 2aS + cos 2a\+2 cos 4a2t) and then, using (18), (21), and (22), _ n 2 ch 2a£-^cos 2ag +. 2 sin 4a2r (x, — zffla c h 2(j g + CQS 2 a g + 2 ^ os 4ffl!i ,, ji 2 ==— 2a
96
sh 2ag—sin 2ag eh 2a£ + cos 2ag + 2 cos 4a 2 T'
(22)
The qualitative behavior of the corresponding (v, w, z)
was described in the introduction
where, for simplicity, we set b = \ . 7.
Solutions of the Reduced Benney Equations and Their Analogues
fc—
•••—
—,.i
7.1.
i i —
i
...
-
•••—
- — — —
' ••——
The reduced Benney equations are Eqs. (22) of Chap. I I . 2
solutions of them for which ft = (p(u), we find that h=(u+c) /4, compatibility condition. 7.2.
cGR i s a constant from the
We have, moreover, the following r e s u l t .
THEOREM. Let c6R, fc=(«+c)74» and l e t u be a s o l u t i o n of the equation
2
(3u /4+cu/2)x Proof.
In attempting to find
.
Then the pair
ut = —
(«, h) i s a s o l u t i o n of system (22) of Chap. I I .
Under the hypotheses of the theorem we have ht = ("+c) «,/2 = - (a+c) (3a + c) ux/4 = - (u (u + cfi4)x = - («A),, ut= -(3u2l4+uc/2)x
= -(u2l2+{u+c)2/4)x=
We now consider the equation ut + (3u2/4+cu/2)x. 7.3.
Proposition.
~(a2/2 + h)x.
The following method i s c l a s s i c a l .
Let if> be d i f f e r e n t i a b l e , l e t
0).
u=u(x, t) be a smooth s o l u t i o n of the functional equation u = ^{x—
{x, t) .
Then in t h i s range .[tt* +
COROLLARY. Under the hypotheses of the proposition u(x, t)
i s a solution of the equa-
t i o n ut +
Differentiating the r e l a t i o n
u=ty(x—
obtain ux = ty'(x~
t , we
We multiply the f i r s t
equation by q/("), replace q>'(u)ux by f(w)x, twice, add the r e l a t i o n s obtained, and take a l l terms to the l e f t .
We obtain a s s e r t i o n 7 . 3 .
If the Cauchy problem i s posed for the equation «(+
We s h a l l investigate for what i n i t i a l conditions 1>|
the reduced system has a unique solution for a l l *eR and t>Q.
We must put
(obviously, any constant can be taken i n place of 3 / 4 ) . 7.4. and
Proposition.
The equation u=$(x—(3«/2 + c2)> has a unique solution for a l l JCGR
t>0, if and only if the smooth function Proof.
For given X and
t>0
i|> • i s nondecreasing everywhere.
the graph of
(•*— 3«/2—ctl2) as a function of u i s
obtained from the graph of
<> !
horizontally 3/2 times, and t r a n s l a t i n g to the l e f t by x — ct/2.
In order
that a f t e r any compression with positive coefficient and any t r a n s l a t i o n the graph of <> j have a unique i n t e r s e c t i o n with the diagonal, i t i s necessary and s u f f i c i e n t that nonincreasing.
Indeed, a small neighborhood of any l o c a l maximum or minimum of <^ under
compression and t r a n s l a t i o n can be made to i n t e r s e c t the diagonal twice. monotone.
<»| be
Hence
If i t i s nondecreasing everywhere, then i t cannot be a nonhorizontal l i n e , since
a s u i t a b l e compression and t r a n s l a t i o n w i l l take t h i s l i n e into the diagonal. a horizontal l i n e , then there i s a point
u0 with
If i t i s not
" f'(oo)="">0. f*("o)¥=0 »
both sides of the tangent a t the point u0 . sect <}> twice. Remarks, solution
A s u i t a b l e compression and t r a n s l a t i o n takes t h i s secant into the diagonal. a) The i n i t i a l condition
u=
A small t r a n s l a t i o n of t h i s tangent w i l l i n t e r -
2 + ct
y ^ = (2 +cii"
wllic
${x) = x on some segment leads to the c l a s s i c a l
h i s called the "ruptured dike" s o l u t i o n .
b) If the velocity p r o f i l e if a t the i n i t i a l time has an i n f l e c t i o n point with a horizont a l tangent, then by an a r b i t r a r i l y small deformation i t i s possible to obtain from i t a p r o f i l e with a maximum and minimum which leads to nbnuniqueness of the solution i n f i n i t e time. 7.5.
We now present a method for finding invariant manifolds of the form h= 9(a) for
the Hamiltonian system described i n Sec. 10.of Chap. I I .
Our c a l c u l a t i o n s w i l l show that
the manifold A=—^— i s invariant also for the higher reduced Behney equations.
I t would
be of i n t e r e s t to find an analogue of i t for the unreduced equations. In the notation of Sec. '10 of Chap. I I we seek solutions of the equations tlt — t\llix, ht = filuj[ subject to a r e l a t i o n of the form S(ti, A)=cGR, where tion.
S i s a s u i t a b l e dif ferentiable func-
We must ensure the consistency of the system
0=Sx=Sttux+Shkx.
(24)
We multiply (24) by — T J " ^ and add to (23); into the r e s u l t we S u b s t i t u t e the r e l a t i o n s C y l M ^
and ^iUl=aCk1)V2i^ and divide by a(Xj)Vfy? .
We obtain the equation
S A V>,+S tt V 2 A,=0.
(25)
In order that the system of equations (24) and (25) have nontriVial solutions necessary t h a t 0
5 J ^ , - ^ V 2 = = 6 , i . e . , SalV~Vx(«)±ShlVV2(A)=0.
(ux, hx) i t i s
This i s s a t i s f i e d if 5 i s a
h
function of K \rVxdii ± \ VV2dh possible to s e t simply
.
Since we are interested in the r e l a t i o n
Sts^W^du
± [VVjh
=const.
For the higher Benney equations the conditions (26) reduce to A = (a+c) 2 /4, as above. We express A i n terms of u from Eq. (26): u in terms of A and argue analogously).
S=conSt , i t i s
(26) u±2Vh
= const, i . e . ,
A = A(tt) (when t h i s i s impossible we express
Under condition (26) the equation ut = rnhx becomes an equation u-t = (ylih\h=inu))x °f t n e type considered in 7 . 3 . The equation ht = i\lax i s automatically s a t i s f i e d . Indeed, from Eq. (23) and then (25) we find (always with A = A(w)): 98
k
< = ~~Sft
U =
'
_
Sj, ^ *
=
_
Sft ft'**** + VlhuUx) =
From l i n e a r i t y c o n s i d e r a t i o n s i t i s c l e a r t h a t t h e same formalism remains i n f o r c e i f formal s e r i e s ian) .
"Hi i s r e p l a c e d by any l i n e a r combination of i t s c o e f f i c i e n t s
This shows t h a t the manifold
\YV^du
the
(as t h e Hamilton-
± \VV?dh = const i s i n v a r i a n t w i t h r e s p e c t t o
e q u a t i o n s w i t h any H a m i l t o n i a n from t h e c o r r e s p o n d i n g s p a c e . CHAPTER IV INDIVIDUAL RESULTS 1«
The H i r o t a Formalism 1.1.
I n t h i s s e c t i o n we d e s c r i b e t h e s e m i h e u r i s t i c method of H i r o t a for s o l v i n g non-
linear equations.
The method c o n s i s t s of two s t e p s : r e d u c t i o n of t h e e q u a t i o n t o s o - c a l l e d
b i l i n e a r form by f i n d i n g a s u i t a b l e s u b s t i t u t i o n and t h e n s o l v i n g t h e b i l i n e a r e q u a t i o n by means of some v e r s i o n of p e r t u r b a t i o n t h e o r y o r a lucky g u e s s . g r e a t d e a l i n common with t h e Zakharov-Habat
The H i r o t a formalism has a
formalism d e s c r i b e d i n Sec. 5 of Chap. I I , but
t h e i r e x a c t r e l a t i o n s h i p h a s n o t been c l a r i f i e d .
Our p r e s e n t a t i o n i s based on t h e papers
[ 3 8 , 3 9 ] ; s e e a l s o t h e l i t e r a t u r e c i t e d i n t h e s e works. 1.2.
The H i r o t a O p e r a t o r s .
L e t f(x, t), g(x, t) be two f u n c t i o n s of two v a r i a b l e s which
a r e d i f f e r e n t i a b l e an a p p r o p r i a t e number of t i m e s .
For a p a i r of i n t e g e r s
m, nwe
define
the expression
A system of e q u a t i o n s i n b i l i n e a r form i s o b t a i n e d by e q u a t i n g t o zero some system of l i n e a r combinations of such e x p r e s s i o n s f o r t h e unknown f u n c t i o n s
f
and
g ( t h e c o e f f i c i e n t s of
t h e l i n e a r c o m b i n a t i o n s i n a l l examples a r e c o n s t a n t s ) . 1.3.
The H i r o t a S u b s t i t u t i o n s .
We s h a l l p r e s e n t a sequence of s u b s t i t u t i o n s which
r e d u c e well-known n o n l i n e a r e q u a t i o n s t o b i l i n e a r a) The Korteweg-de V r i e s e q u a t i o n
form.
ut+6uux + uxxx — 0. S u b s t i t u t i o n :
u=2(\ogf)xx
.
Bilinear
form: Dx(Dt + b) The e q u a t i o n
D»x)f.f=0.
ut + 45a2tix + \5(uxuxx + uxxxu) + uxXXXX=0.
S u b s t i t u t i o n : - « = 2(logf) x x .
Bilinear
form: Dx{Dt+Dx)/./=0. This e q u a t i o n i s s i m i l a r to t h e second of t h e h i g h e r Korteweg-de V r i e s e q u a t i o n s (corresponding
99
to the operator
< {dx+uf2
> i n the notation of Chap. 2) which has the form
10(2uxuxx+uuxxx)+tixxxxx=0.
ut+3Qu2ux +
However, as Hirota observes, the l a t t e r does not reduce to 1.3 b)
by l i n e a r transformations. c) The equation for waves on shallow water: 00
u
3
3
"/ — xxt — ""< + "JC S atd^ + Ux = 0. X
Substitution: «=2(log/),.t.
Bilinear form:
Dx{Dt-DxDt+Dx)f.f=0. d) The Boussinesq equation: utt — Uxx—3(tt2)^—Uxxxx—0.
Substitution: « = 2 ( I o g / ) ^ .
Bilinear form
(Z>5-Di-Dl)/./-0. e) The two-dimensional Kbrteweg—de Vries equation: utx±uyy t i o n : u=2(logf)xx . Bilinear form:
+6(uux)x+uxxxx=0.
Substitu-
(D,£>,±D2+£M)/./=0. vt+6v2vx-{-vxxx=0.
f) The modified Korteweg-de Vries equation:
\
f—ig
Substitution:
)x
Bilinear form: (Dt+Dx)(f+ig).{f-ig)=0, Dx(f+ig).(f-ig)=0. g) The sine-Gordon equation: wt/ = sin« .
Substitution: v= —2i log(f+ig)l(f—ig)
.
Bilinear form: DxDtg-f=gf,
DxDt(f.f-g.g)=0.
h) The two-dimensional sine-Gordon equation: vxx + vyy —
Substitution:
(Dx+D*y-D*)(f.f-g.g)=0.
i ) The equation ity, + 3ta |<|»|2<|>x +?<[>„ + n
(DX-\)F>F=-2\G\*. Substitution ty = G/F , F a r e a l function. Bilinear
form (f£>( + D 2 — X)G./ 7 =0, (Dx — \)FF=— 100
Substitution:
2|G| 2 .
The c o n s t a n t
X i s determined by t h e b e h a v i o r of
<> ] as
|JC|-»-OO.
k) The e q u a t i o n s of two waves
Substitution: (fl=Gl/F . Bilinear form: (D < +© i D J )O r F=0, i = l,2. 1) The e q u a t i o n s of t h r e e waves: fit+«!?i* -=^i?yT*. cpi = G i // ? .
Substitution:
Bilinear
{'./.*} = 0 . 2 , 3 } ;
form:
{Dt + vfi^OrF^qft/lt. 1.4.
Sample S o l u t i o n s .
fc=±l.
i = l,2,3.
We p r e s e n t two examples of s o l u t i o n s of t h e b i l i n e a r
equations
and r e f e r t h e r e a d e r t o t h e p a p e r s c i t e d of H i r o t a f o r o t h e r s o l u t i o n s . a) The i n t e r a c t i o n of t h r e e waves ( E q s . 1 . 3 1 ) . a r y s m a l l parameter
We expand
F, G,{ i n powers of a n a u x i l i -
e:
F = l+2* 2fl /2«,
Gi=gio+2 s2 "£i»'
G y =2^ +1 g/,2«+i,
y==i,2.
We s u b s t i t u t e t h e s e expansions i n t o t h e b i l i n e a r e q u a t i o n s 1.31 and e q u a t e c o e f f i c i e n t s of l i k e powers of
e.
W r i t i n g o u t t h e c o n d i t i o n s for t h e v a n i s h i n g of t h e l e a d i n g
coefficients,
we find two t y p e s of s o l u t i o n s : The s o l u t i o n w i t h
gio=0 .
Here *l ?2
__ =
™
I + «„ exp 2tj2 + 60, exp 2t|a » a,, exp t)8 l+«,2exp2Tia +ft0,exp2T|j' a.i exp t|, 1 + «0j exp 2r|s + 6o2 exp 2T|, '
where ri2=*Pt(x — Vtt),
v\3=P3(x — V3t),
fl
22~<7l 2 ( 0 , - 0 ft'
The l e t t e r s
a2]
and «3i
denote a r b i t r a r y c o n s t a n t s , while
Pi and p3 obey t h e c o n d i t i o n s
(«i—v2)P2=(vi—v3)p3-
101
This s o l u t i o n was obtained by V. E. Zakharov and S. V. Manakov by methods of i n v e r s e s c a t t e r ing t h e o r y . The s o l u t i o n with
gio^O .
Here li—gio T2—
«
1— exp2t|, i + exp2n, q,expt|, l+exp2t|, ' *iexpii
* * - I + ex P 2t ll ' where
(Sj - z>2A) (2, - fsA)=ftftffw a\=—giqy4: (2 t — vtfx) (fy — z ^ ) , *i = — q\q% • 4 (2,—«>,/>,) ( 2 , — ^ A ) c) The two-dimensional sine-Gordon equation (Eq. 1.3 h ) . s o l u t i o n s of t h r e e - s o l i t o n t y p e .
For t h i s equation Hirota found
I t i s i n t e r e s t i n g t h a t there are apparently no known
s o l u t i o n s w i t h a d i f f e r e n t number of s o l i t o n s .
/ + t e = H—0,1 2exP
2
The Hirota s o l u t i o n has the form
A
13>*>;'»1
«l*il,7+2(1lJi + ',c/2)fr|.
where ili=PtX+9iy
— &it,
and the c o n s t a n t s obey the f o l l o w i n g r e s t r i c t i o n s :
mA P
1.5.
li
(Pi-Pi)* + (gi—?/)'—(Pi—Q/)' (Pi + Pi)2 + (It + »y)»-(Oi + Qy)1 dij=PiPj+<]i
~~
The Hirota I d e n t i t i e s .
dU— 1 ~du+V
In seeking s u i t a b l e s u b s t i t u t i o n s r e l a t i n g the equations
i n the usual form to b i l i n e a r e q u a t i o n s , Hirota makes use of i d e n t i t i e s samples of which are given below.
The reader w i l l have no d i f f i c u l t y i n extending t h i s l i s t by i n d u c t i o n . Dxta-b = D^ab
=
(—iy*Dxtb-a; D«-1(ax-b—abx);
D™ exp fax) • exp {p2x)={p1
-p2)m
Dxab -c=^bc+a(Dxb D2xab.c=d^bc+2pxDxb.c+a(Dlb.c);
102
exp fa+p2) x\ • c);
exp (Wt) [exp (2zDx) a-b]-cd= = exp (eDJ [exp (eD, + W,) a • d] • [exp (tDx - Wt) c • b]; exp (id/dx) alb = [exp (zDx) a • 6]/cos h (sDx) b • b; da
Dxa-l\
dx b ~~ ~1F~' d' t a\ D^a-b a Dxb-b dx2\bj * ) • b2 3 d* a _D xa-b dx3 b b*
b b* ' p^.b^b-b b* b' '
2 cos h (ed/dx) log / = l o g cos h „ d* , Dxf.f
{iDx)f-f\
2^1og/=-ir-=tt;
D*J.fip~ihx+U-; 2.
P o l e s of t h e S o l u t i o n s 2.1.
I n [40] Kruskal s u g g e s t e d c o n s i d e r i n g t h e p r o c e s s of s o l i t o n i n t e r a c t i o n by t r a c -
ing t h e p o l e s of t h e a n a l y t i c c o n t i n u a t i o n of a m u l t i s o l i t o n s o l u t i o n i n t h e complex domain. For t h e e q u a t i o n
Vt + vxJr'Vxxx'
which i s c l o s e l y r e l a t e d to t h e Korteweg-rle V r i e s e q u a t i o n , ,—
t h e s i n g l e - s o l i t o n s o l u t i o n has t h e form
v(x, t) = 3y c t h - — ^2
i n a sum of p r i n c i p a l p a r t s near t h e p o l e s h a s t h e form
v(x,t) = 6^
l/^T-( x
(X-ct-tiKi/Vc
thz = ^
ct\
'-. I
The expansion of 2-
-?)
thz
* whence
)-'.
T h e r e f o r e , a c c o r d i n g t o K r u s k a l , t h e e v o l u t i o n of t h e s o l i t o n may b e c o n s i d e r e d a s a "parade of p o l e s " moving i n s t r i c t f i l e w i t h speed
c.
The t w o - s o l i t o n s o l u t i o n can a l s o be r e p r e s e n t e d i n t h e form of a sum of p r i n c i p a l p a r t s :
v(x,t)=62i(x-xW-^H)~^6^i(x-x^-^tr\ ««5l<2)
B=sl(2)
where xW, /=1,2 are constants, and the \W behave like c^t-\-nxilVc^i\
where
c
W)
are functions of time which for large
is the speed of the j-th soliton.
\t\
The speed of a
soliton is proportional to the square of the density of its poles. When a fast soliton overtakes one which is twice as slow, pairs of poles of the first soliton momentarily coalesce with the poles of the second.
If the speeds are close part of the poles of the fast soliton
spring into the gaps between poles of the slow soliton. 2.2.
In the joint workof Airault, McKean, and Moser [25] Kruskal's remark was developed
and led to the discovery that the poles of multisoliton solutions evolve in correspondence with known Hamiltonian equations of the type introduced earlier by Calogero and investigated by Moser. One of the results of their work which is most simply formulated is the following. We suppose that the function
103
*(*.*)-22(*-*,<<))-»
(i)
i-l
i s a solution of the equation vt+3vvx—f'0xxx=^.
Then the functions
Xj(t) s a t i s f y the
system •*/= 6 2 (x/—•**)"*' 7 = 1
re,
*#/ with the additional condition n
2(*;-**)~3=°. J=1
»•
«)
ft-1
The s e t (2) i s empty if i t i s r e s t r i c t e d to real values.
However, for
a=—^—.
the s e t of i t s complex points i s isomorphic to a Zariski-open, dense part of complex space (after symmetrization in 2.3.
d>\,
d -dimensional
(xu ..., xn)).
In [30], which was completed almost simultaneously with the work of A i r a u l t ,
KcKean, and Moser, D. V.Choodnovsky and G. V. Choodnovsky obtained analogous r e s u l t s for other equations.
Especially i n t e r e s t i n g i s t h e i r remark that the evolution of the poles of
the Burgers-Hopf equation ut=2uux+uxx i s free from r e s t r i c t i o n s of type ( 2 ) . 2 . 4 . Proposition. Function u(x, t) = £i(x — Xj(t))~1 i s a solution of the equation . ; ut=2uux-\-uxx, i f and only if the Xj{t) s a t i s f y the system
Xj(t)= - 2 2 (*,(<) -XtWr1n+i (For an i n f i n i t e set of indices j the following computations are formal; convergence r e quires a separate i n v e s t i g a t i o n . ) Proof.
If
u(x, *) = 2 ( J C — -MO)-1 » then i
2uax+axx=-2
2
(x-Xj(t)r(x-xk(t))-\
1, A; l+k
«,=2-M')(*--M0)-2. Comparing the poles of these expressions (assuming t h a t the Xj(t) do not pairwise coincide), we immediately obtained the required a s s e r t i o n . 2.5. For the modified Korteweg—de Vries equations algebraic r e s t r i c t i o n s on the motion of the poles again a r i s e : the function u(x, Q = 2g<(JC~JC/(*))~1» cf= ± 1 s a t i s f i e s the equation ut=6u?ux — uxxx, if and only if Xj^^iXj
— X,,)-2,
2M*y--**)",=0. k+i
104
2.6.
The results briefly considered here are close in spirit to classical expansions
in the theory of elliptic functions and also to the approach for obtaining solutions of Lax equations of algebraic type described in Chap. III. We recall that for the bimodule of rank 1 introduced there the invariant A also evolves in such a way that the poles of its derivative cancel in a particular manner with the poles of the function A that the poles considered there refer
itself.
It is true
to the auxiliary "spectral parameter" — the variable
point on the Riemann surface of the curve Spec O — rather than to * as a function of t. However, it appears likely that recalculation of the evolution of A in terms of the poles of solutions will lead to a generalization of the results of 2.2 and 2.6. 3. Pseudopotentials and Generalized Conservation Laws 3.1.
This section gives an introduction to the interesting formalism proposed by
Estabrook and Wahlquist [54, 32] and further investigated, in particular, in the work of Corones [31], Corones and Tests [26], and Morris [49]. The central feature of these papers is a certain generalization of the concept of a conservation law for evolution equations of the form Ut — K(u,tl',...)
where U = (ult.. .,«„) .
As we have repeatedly mentioned, an ordinary ("algebraic" in the terminology of Chap. I) conservation law for such an equation is a relation of the form Ft — Gx = 0, which is a formal consequence of the equation U( — K.
More precisely, let ^ = ^[a{;)] or
(J C°°(«F' i-o
!|y|<0< ^G<^> and l e t dt:J,-*Jl be an evolution d i f f e r e n t i a t i o n corresponding to F,G§.JL with Ft=Gx c o n s t i t u t e a conservation law. us
If
then
i s a solution of the system, then in terms of i t and a conservation law i t i s v : a solution of the system vx — F(as)
possible to construct a p o t e n t i a l
Suppose F, G are vector-valued functions of variables
ut=K;
v = (t»i,..., vN) .
u\!) of height
N and the auxiliary
We assume that the following system of equations i s consistent
(the r i g h t sides do not contain derivatives of vt ) : ut=K(u,~u',...); vx=F(u, u', ...;©); vt = G(«,«', ...;©). Then for any solution
(us, :vs) the r e l a t i o n ^-tF(us, .. .,'vs)=jiG(us,
...,VH)
• 3.2.
F and
satisfied.
I t is
Vs i s called the corresponding pseudopotential.
called a generalized conservation law, and In a l l examples considered
.. . ^ t s
G l i e in A®&N, where & = R, C and
3^ = Cx(vu
An algebraic model of t h i s s i t u a t i o n can be formed as follows. We consider the ring
dx:uy)®\*~u\J+u®l
and
A®$& .
dt:u®\*~K®l;
On the subring A®\
[dx, dt] = 0 .
there act the d i f f e r e n t i a t i o n s
The r i g h t sides of the two l a s t equations
of (3) can be considered as the extension of these d i f f e r e n t i a t i o n s to a l l of
d~t:\®v-+G.
105
The formal consistency condition of the system
vx=F
means simply t h a t the d i f f e r e n t i a t i o n s
dt commute.
dx and
and vt = G under the condition
ut=K
We may therefore pose the
following problem. 3.3. For some tion
Problem.
Given a
A-algebra
ft-algebra
A and two commuting
ft-differentiations
du d2: A->A-
33 describe the s e t of extensions du d~2-A<&33->-A®!® with the condi-
[di, d 2 ]=0. 3.4.
Without p r a c t i c a l loss of g e n e r a l i t y , we may assume that du d2£A®D(53) + D(A)®33.
In order to solve Problem 3 . 3 , under c e r t a i n r e s t r i c t i o n s on A we construct a Lie algebra ft over
X=%(j&, dl,'d2),
such that the s e t of a l l extensions
&>$ i s in b i j e c t i v e correspondence
with some subset of Lie algebra morphisms of Hom(X, D (33)) , where D(33) i s the algebra of k d i f f e r e n t a t i o n s of 3.5.
33 into i t s e l f .
We consider a basis
The r e s t r i c t i o n s on A
are as follows.
(£/,), j&J of the algebra A
as a l i n e a r space over ft, which
determines the s e t of s t r u c t u r e constants by
'&
'&
(4)
UjUk=2dmUt, where the r i g h t sides are f i n i t e l i n e a r combinations with coefficients in
ft.
We suppose, moreover, t h a t these coefficients possess thefollowing property: for each 1(<J there e x i s t s only a f i n i t e number of pairs (/, ft)g/XJ (respectively, elements /(*/), such t h a t tf^T^O (respectively, cft^O, cf>=£0). 3.6.
For the algebra
(4); moreover, djkl^0 we define the weight
(&[#<'>], dx,dt) the basis of monomials in
only if w(Uj)
Uj, Uk, divide
a^
obviously s a t i s f i e s
Ut, and these are of f i n i t e number.
by a d d i t i v i t y , s e t t i n g
w(uV)) — j -
Further,
Then dx i s homogeneous and
increases the weight by one, and the space of polynomials of a given weight i s f i n i t e dimensional; therefore,
Ut can enter i n only a f i n i t e number of the derivatives
Thus, only the condition on the c'jf (for d2 = dt) can be n o n t r i v i a l . to Korteweg-rie Vries (ut = uu' +u'") i t can be s a t i s f i e d as for < i)
weight wl(u ~ ) = i-{-2.
==
Then the field ^ 2^
and increases t h i s weight by 3, so t h a t
Ut
ttU
a
( )
'"*" '") ' r77r
is
d\Uj.
For evolution according
dx, by introducing the new homogeneous with respect to o>i
can enter the expansion of d2Uj only for
w1(Uj) = wi(Ul) — 3 ; there are a f i n i t e number of such j
.
Similar considerations are pro-
bably applicable to other Lax equations. 3.7. We now fix the algebra (A, di, d2) and basis (£//), j&J, satisfying the conditions of 3 . 5 . We consider the free Lie algebra £, generated by Xk, Yp j , k£j, with r e l a t i o n s
2 c ii ) J 'y-2 c ^y=S r f 7«l J , y. *J> &•
106
(5)
The condition of 3.5 ensures that a l l these l i n e a r combinations are f i n i t e . 3.8.
Proposition.
The s e t of commuting extensions
(di, d2) of the pair
(du d2) to
Jl®D(3is)-\-D(.A)®3$ corresponds b i j e c t i v e l y to the s e t of homomorphisms of the Lie algebra X to the Lie algebra D(53), which vanish on a l l but a f i n i t e number of the Xk, Y} . Proof.
Any extension has the form
d, =
where almost all the Zj, T)/ are equal to zero. The condition [du d2] — 0 means that
J
(6)
l.k
i
where the commutators are evaluated in the Lie algebra D(33).
Substituting here the r i g h t
sides of the i d e n t i t i e s ( 4 ) , we find
2 «$>®C/j®ty-2 c(fiUi®ll= 21 dmUi® hy **!• Equating coefficients of the £/,, we obtain the i d e n t i t i e s (5) with Xk, Yj in place of
»*. f\j
This completes the proof. 3.9.
In 3.7 an almost invariant d e f i n i t i o n of the Lie algebra
to the pair (-A, dlt d2) was given.
k
The dependence on the choice of basis
over £ with respect (Uj) with conditions
3.5 i s probably not e s s e n t i a l i n the sense that for a c e r t a i n class of bases a l l Lie algebras X obtained are canonically isomorphic. We c a l l
X the EW algebra for (
The search for pseudopotentials for the evolution equation represented by (..-#, du d2), naturally breaks into two s t e p s : a) description of the
EW Lie algebra X ; b) description
of i t s representations i n the Lie algebras of vector fields D(C'°(Vi, .. ,,vN)). Both problems are far from being completely solved even for the Korteweg-^de Vries equation.
Apparently, even a finite-dimensional
EW algebra for i t i s unknown.
In the papers
cited a t the beginning of the section a sequence of factors of EW algebras i s constructed for some i n t e r e s t i n g t r i p l e s (X ^2). which are generated by a f i n i t e number of generators and r e l a t i o n s , and some p a r t i c u l a r representations of them by vector fields are given. Factors of a f i n i t e - t y p e EW algebra if
Uj, Uh contain derivatives of
ly high" usually means
ut
St are most easily obtained by s e t t i n g Xj, Yh = 0,
of s u f f i c i e n t l y high order.
In examples " s u f f i c i e n t -
>3, and the additional wonder i s t h a t then by v i r t u e of the r e l a -
tions (5) a l l but a f i n i t e number of the Xh Yh automatically vanish. 107
If the factor obtained has a n o n t r i v i a l , finite-dimensional factor Lie algebra
S?o ,
then i t i s not hard to construct i t s representation by invariant vector f i e l d s on the corresponding Lie group.
We observe t h a t representations with an Abelian image, as i s evident
from ( 6 ) , lead to the usual conservation laws, so that non-Abelian representations and the corresponding pseudopotentials a r e of special i n t e r e s t . Another c l a s s of representations which can be constructed i s obtained i n the Lie algebra of fields over a one-dimensional base. p o t e n t i a l s simple.
Corones suggested c a l l i n g the corresponding pseudo-
This c l a s s i s nice i n that over a one-dimensional base
[£, t)] =0 implies
t h a t £=cn, c a constant, which makes i t possible to avoid the many r e l a t i o n s (5) and their consequences. With these rudiments of a systematic theory, we now present several samples of computations from t h e works cited above. 3.10.
The Korteweg-Tie Vries Equation:
Ui + uXXK+l2uux = 0.
i n the work of Wahlquist and
Estabrook [54] (see also Corones and Testa [26]) a factor EW-algebra £? i s constructed which i s given by the following generators and r e l a t i o n s : [Xu X3\ = [X2, Xa]~[Xlt X4\ = [X2, X6\=0, [XuX2\ + X,=0; [XuX7]-X5=0; [Xh X7]-X6=0; [Xu XS\ + [X2, X 4 1=0; [*3, X^ + lXu X6]+X7=0. The generalized'conservation law Ft — Gx
(7)
has the form
F=2Xl+3uX2 + Su2X3; G=-2(uxx +6a2) X2+3 (u2x - 8u3 - 2uuxx) X3 + +8Xi+8uX5+4u2X6+4uKX7. Any representation of (7) i n the algebra of vector fields on an N-dimensional fold gives
mani-
r
N concrete generalized conservation laws(A i , F, G are expanded in terms of
components). In order to obtain a n o n t r i v i a l factor of the algebra ( 7 ) , Wahlquist and Estabrook s e t . 8
Xg = [AT4, X3\ and impose the new r e l a t i o n [^1.^5] = ^ cmXm. They then verify that t h i s leads m=l
to the zero algebra in all cases except corresponding factor by
Ci-=... = e 6 = 0, c 7 = — c 8 = X, X^O. We denote the
Xt .
According to computations of S. I. Gel'f and, Xx has as a direct component the Lie algebra sl(2), engendered by the generators 2Xs + lXs, X6, X7 — Xs , and the representation of Wahlquist and Estabrook coincides (up to a formal diffeomorphism) with a representation of the leftinvariant fields on SL(2).
In their notation this is a representation in D(C°°(;/2, y3, ye)),
which is described as follows:
108
The generalized conservation law corresponding to -j—, has the form {2a + yl-l)t+4[(u If
+ X)(2u + yl-}.)+2uxx-2axysl=0.
(8)
a i s a solution of the Korteweg—de Vries equation and ya i s a solution of ( 8 ) , then
a = —a—y\-\-\ speed
>• .
i s a new solution which i s obtained by "adding" to
u
a single s o l i t o n with
The reasons for the appearance of the "Backlund transformation" i n such a con-
text remain somewhat puzzling. 3.11.
The Hirota Equation
function and a
ut = — 3auuux — $uxxx + i~(tixx + izu?u. Here
a
i s a complex
i s i t s conjugate, they are considered algebraically independent, and the
Hirota equation corresponds to a pair of equations for complex conjugation from the f i r s t .
u, u: the second i s obtained by formal
In [31] Corones constructs the following factor EW
algebra for i t : -<»*, + [»[*„ [ ^ . A - J I - O ; aX2 + HX2,[XuX2\\=0; [X3,[XuX2]]=0; / e ^ + iTl^i, [X2, AMJ-p [Xx, [X2, [X3, X,]]] - i P \X„ [Xlt [X3, X^-a -hX2-ii\X2,
[X2, Xft-WXi,
[X3, *,]=();
[X2, [X3, Xl]\] — j P[-^i. [X* [Xz, X2]]]-*[X3,
X2]=0,
—yP[Jf„[A' 1 ,[A'a,Jf ,111=0; -^[X2AX2,[X3,X2]]}=0; - * T [ * i . [Xu * 3 H - P l * 3 . [Xu [X3, ^,111-Pl^i, [Xz, [X3, X41H=0; -h[X2,
[X2, X3]]-HX2,
[X3, [X3, X2\\]-
-jnX3,[X2,[X3,X2\\]=0; -WXU [X2, X3]] + i-({X2, [X3, Xl]] + ^\XZ, \X,i X\]\- p [Xu [X3, [X3, JfJH-P [X2, [X3, [X3, X,]]]- p [X3, [X2, \X3, JMU + lXs, X<]=0; -il[X3, \XU X3]\-HX3, [X3, [X3, XJU + IX,, XA}=0; il[X3, [X2, X3]}-$ [Xz, [Xa, [X3, X2]\] + [X2, X 4 ]=0;
The corresponding conservation law does not depend on UU), u(i) for
j>3-
Corones observes that i t has non-Abelian representations by one-dimensional f i e l d s only i n the case
Be—av = 0.
I t i s i n t e r e s t i n g t h a t t h i s i s precisely the condition obtained by
Hirota for m u l t i s o l i t o n solutions of h i s equation. 3.12.
The Equation
the s u b s t i t u t i o n
3utt + uxxxx+6(uux)x = 0. This equation was investigated by Morris [49];
v = ut reduces i t to the usual system of evolution.
The EW factor algebra constructed by Morris i s defined by the r e l a t i o n s
109
[A-!, Xs\=A%;
[Xi, X10\ = j A^ -f -y pX3\
[Xu Xn] = ±*4-3^2; [ A 2,
A5] = — - j
[*2, * J - -AT U ;
A 8;
[X2, X 8 ] = — -g- X3\
[X2, A"ig] = — -£ X2;
[X2, Xn] = — -^ A"8;
[Ao, A 8 J = — -g- X3;
[X2, A n ]
=
— 4" X55
J A 3, A"i 1] = A^;
[ A J , A^IOJ = — 4- X 2 ;
1X3, A 4] = Tj A"10;
IX4, X 5 ] = — 3[iX8;
[X 4 , A'uJ = ~2 Xt—3X2; [X~5, Xs] = j X2;
[A 3 , A10] = — - j A"3;
[X 4 , A%] = — -j- [1X3—j Xi; |X 4 , Xn] = —311X5;
[X5, Xn] = — -jg XI — -pr X3;
[Xit Xl0] = — -j
[ A 8 , Xn] = — 4- A 10;
A8;
[A*j0l Xn] = — j X n . Morris c o n s t r u c t s i t s r e p r e s e n t a t i o n i n t h e a l g e b r a
£)(C°°(A:I,
JC2, J£3))
by s e t t i n g
(
^1^3;
A 3=
X4=x%dx+p (x^+x 2 d 3 ); X&=j
(Xidl +
X&=-^(Xid2 XlQ=-£
x3d3—2x2d2); — .*2
(Xidi — JC3O3);
Xn = -j- (JC3(?2 — JC2O1).
From t h e e q u a t i o n for t h e g e n e r a l i z e d c o n s e r v a t i o n law Morris deduces t h e r e l a t i o n s
x['" + -2ux[ + iju' + w)x'l = vxu Xi,t+3x'l + uxl=0, 3
where
TW i s a t r i v i a l p o t e n t i a l :
wx= —jUt,
1
(9J
3
wt= 4-«'"-+-j(uu')'.
Equations (9) make i t p o s -
s i b l e to apply t h e t e c h n i q u e of s c a t t e r i n g t h e o r y t o t h e o r i g i n a l problem. 3.13. conjugate.
The Nonlinear S c h r o d i n g e r E q u a t i o n ; The f o l l o w i n g f a c t o r
iut-\-uxx—2"Su«2=0,
5[A- 2l Z 2 ] + [K 1 .Z 1 ]-eZ 1 =.0;
[ X ^ R 2 1 ^ , ^ 1 = 0;
[A'1,K1l + [A'2,K2l = 2[Z 1 ,Z 2 ]-2[Z 1 ,i;i 110
i s
the formal complex
EW a l g e b r a for t h i s e q u a t i o n was found i n t h e work of
Estabrook and Wahlquist [ 3 2 ] : [Jf 1 ,Jfjl = [X 1 ,J' 2 ] = [X ! | i r i ]»=[A'„Z l ] = [ 2 1 , Z a i - 0 ; [X„Z,] = Z 2 , 2
u
\ZX,Z,\=\Y,\ (10)
(To these relations are to be added their formal complex conjugates.) Denoting complex variables by
tfu U-z and by *?i = 37-, dt = -=— te dyi
the standard
differentia-
tions, we obtain the following representation ( ft is any complex constant): X1 = 2(kyldl+kyldl); Vi=t[yld1—'yl'd1 —
Td2+-yd2); ;
___
(ID
Zx = — 2" {tfidi—E
fields.
Backlund Transformations 4.1.
E(u)=0
A Ba'cklund transformation
B, relating two systems of differential equations
and F(v) = Q, i s a system of differential equations B(u, v) — 0 such that E and B
formally imply F, and F and B formally imply
E .
In the language of
algebra, suppose that E corresponds to an ideal / E C ^ , F to an ideal ideal
IstzA®^'
Then IB^IE®^
+ 1®//=-.
differential 7pc53 and 53 to an
Thus, a Backlund transformation is an analogue of
a "correspondence" in algebraic geometry.
(In our case Jl, 33 are differential rings over
a common base differential ring K, over which a l l tensor products are taken; the ideals are assumed to be differentially closed and to be radical for analytic applications.) There is no systematic theory of Backlund transformations, and we limit ourselves to describing a portion of the experimental material at hand taken from [26]. 4.2.
The Backlund Transformation for the sine-Gordon Equation.
While investigating
surfaces of constant negative curvature, Backlund in 1880 found the following transformation relating the sine-Gordon equation uxv = sinu to i t s e l f :
Ba: where a is any constant.
2
(12>
a -a
This transformation makes it possible to obtain a sequence of
solutions of the equation in quadratures starting from the solution
U0—0.
Bianchi observed
that the pair of transformations B0l, Ba, commutes in the following sense: starting from the initial solution «0 , the compositions
Bai°Ba,
and Ba,°Bai
generate a certain common
solution tt3, for which
111
where «i
and
4.3.
« 2 a r e t h e s o l u t i o n s g e n e r a t e d by
A Remark of Rund.
suppose t h a t t h e e q u a t i o n
I n h i s work i n
Bai
and
Ba,
from
[26] Rund put f o r t h t h e f o l l o w i n g i d e a .
E i s t h e Euler-Lagrange e q u a t i o n w i t h Lagrangian
Chap. I ) , and we a r e i n t e r e s t e d i n Backlund t r a n s f o r m a t i o n s of t h e n seek i d e a l s Lagrangians
IB
ca(u) and
simultaneously.
a0 .
E
We
into i t s e l f .
such t h a t (o(«)— ©(u)Glmxd+ (/B) ; o n s o l u t i o n s of t h e system
One can B
a>(v) d i f f e r by a d i v e r g e n c e and t h e r e f o r e have zero f i r s t
Rund shows t h a t t h e t r a n s f o r m a t i o n (12) b e l o n g s t o t h i s c l a s s .
the variation A certain
m o d i f i c a t i o n of h i s i d e a i s a p p l i c a b l e t o t h e Korteweg-nde V r i e s e q u a t i o n . 4.4.
Backlund T r a n s f o r m a t i o n s f o r t h e Korteweg—de V r i e s E q u a t i o n .
We c o n s i d e r t h e
s p e c t r a l problem r e l a t e d to t h i s e q u a t i o n d e f i n e d by t h e Lax p a i r
\(dt+4d3x + 6udx + 6dxou)
w = <J»Jt/<|»,
we
see t h a t i t i s possible to eliminate
modified Korteweg-de V r i e s e q u a t i o n for
(13)
u from (13) .
The r e s u l t i s t h e
v.
vt-6v2vx+6\vx+vxxx=0, d i s c o v e r e d by Miura.
(14)
E q u a t i o n (14) h a s t h e t r i v i a l t r a n s f o r m a t i o n
i>>-»-— v; then a p p l y i n g t h e
t r a n s f o r m a t i o n i n v e r s e to (13) we o b t a i n a Backlund t r a n s f o r m a t i o n of t h e Korteweg-nde V r i e s e q u a t i o n i n t o i t s e l f which we d e n o t e by
B% .
Estabrook and Wahlquist found t h a t i f
ui=B),i(u0),
i = l,2,
then
B^B^tt^)
and £;!,,£>., (a0)
contain the s o l u t i o n
an analogoue of B i a n c h i ' s formula for t h e sine-Gordon e q u a t i o n . 4.5.
F l a s c h k a and MacLaughliri i n t h e i r paper c o n t a i n e d i n [26] s t u d i e d how t h e t r a n s -
f o r m a t i o n (12) a c t s on t h e spectrum of t h e Schrodinger o p e r a t o r .
Changing t h e n o r m a l i z a t i o n
s l i g h t l y , we w r i t e t h e e q u a t i o n i n t h e form ut—6uux + uxxx = 0 and t h e Schrodinger o p e r a t o r i n t h e form U=2WX
dx2 + u=l. , and
W
Setting
u=2wx,
we o b t a i n t h e new Schrodinger o p e r a t o r
dx2+U = L, where
i s found from t h e e q u a t i o n s
(Wx = - wx+(W - w)2+k \Wt==—wt+4[\Wx Here W—w=—i|>*/i|>, where If
wxx(W.—w)].
|x|-»-oo, t h e s c a t t e r i n g d a t a for / = —d x 2 +u a r e determined
For any complex number k w i t h lm&>0 t h e r e a r e s o l u t i o n s of t h e problem
d i s t i n g u i s h e d by t h e a s y m p t o t i c b e h a v i o r a t
112
+
/i(>=A,i|>. .
u i s r a p i d l y decreasing as
as follows.
+ wx + wx(W-w)2
infinity:
Lf=k2f,
elkx, x-+-\-o&; /i(x, k)^[b(&je-ikx+a^^ Mx,
x_+_00.
e 1X + a e k)^[zf^ " ^ ~"'X' e Utx, x ->- — oo.
x^ + oo;
On the imaginary axis there may be a finite number of points of the discrete spectrum of l:k2= —rf., J=l, .. .,N, which determine constants
Cj, j = \,. ..,N,
by conditions on their
oo
eigenfunctions: <}'/~]/c;exp( — rtjx),
if \y*(x)dx=l.
The collection (a(k), b(k), •»),, Cj) is called
—00
the scattering data for b. If
%(x, X0) i s any solution of /ij»0 = X0<|»0, and ty(x, X) i s a solution of lty = \ty , then
*P(x, \) = Y(x, X) —tjj(jc, X) ° *' x* i s a solution of the equation not vanish i t suffices that
JL*' = X'F .
In order that <M*> X0)
>-0 l i e to the l e f t of the spectrum of /.
We choose X0= — rf and denote by (A(k), B(k), 11r Cy)the scattering data for L . According to the analysis of Flaschka and MacLaughlin, there are the following facts. a) If %=fi(x, ty , then
A(k)=a(k), a<*)«3=£*<*).
Cj=^cP
and the discrete spectra of I and L coincide. b) If %=f2(x,
in) , then
A(k) = a(k), B{k)=^b(k),
C y =5±^c /r
and the discrete spectra of I and L coincide. c) If % = Dlfl(x,
if)-\-D2f2(x,ir),
Dx, D2 nonzero constants, then
A(k) = ^a(k), N
B(k)=-b(k),
Cj=^cj,
points of the discrete spectra for / and L coincide and, moreover, L has an additional
point — T]2, with normalization constant depending on Dx and D2 . This implies several curious conclusions. First of all, there are Backlund transformations which add no solitons (these correspond to points of the discrete spectrum). They only shift the phase of solitons present in the old solution (cases a) and b): Cj'&Cj) . Secondly, iteration of the transformations defined by h(x, ir\) and then D\Fi(x, ii\) + D2F2(x, tr\) : u-+U does not change
b(k)/a(k)
and Cj, but adds the eigenvalues — r,2.
For it
we have
U(x)=a(x)-2D/l(x,iti)ll
+ D J/»(2, hfldzj ,
113
where D is a certain constant. Finally, if tt(x, t) is a solution of the Korteweg-^de Vries equation, then, as is known t)=8ik3b(k,
jfb(k,
t), jfCj(t)=8^c/0
/i(x, if\\ t), changing
. This implies that Backlund transformation by means of
b(k, t) into *j~'.. b(k, f), takes the solution
a into the new solution U,
i . e . , i t commutes with theKorteweg-nde Vries flow. With this we conclude our brief discussion, referring the reader to the literature for further information. 5.
Lax's Method for Generating the Algebra of Korteweg-^de Vries Integrals 5.1.
We consider the algebra
.A=&[ap] and some operator
B which is Hamiltonian in
the sense of Sec. 7 of Chap. I .
I t generates a Lie algebra structure on -A/KerS —= . Ex-
plicit description of operators
B, for which in this Lie algebra there exists large, e . g . ,
infinite-dimensional, Abelian subalgebras is a very interesting question.
Equations
tlt=B—= , where F l i e s in such a subalgebra have infinitely many conservation laws. In ou Chap. I I we described such subalgebras for the Gel'fand—Dikii operators corresponding to Lax equations and for both the reduced and unreduced Benney operator. In this section we present the recurrence method of Lax for the operator B = dx, which leads to the Korteweg-nde Vries algebra.
The exposition i s based on notes of I . Ya. Dorfman
who has kindly permitted the author to use them. 5.2. by /
Let A = k\u.W \j>OJ,
the image of / i n
A.
d:u^>-*a^+1K
We set A = A/dA
We shall write
structure on A: {/, i"} = (d-g£---g|-] •
L e t
f~g , if
and for any f£A we denote
f=g-
There is a Lie algebra
H:A-*A be a linear operator which is formally anti-
symmetric in the sense that ///g-f g7//~0 for a l l / , g£A. We consider a sequence of elements /_i, /o, • •., fn£A, f_x^=cuy c=£Q. The following result is motivated by the remark in 3.15 of Chap. I I . 5.3.
Proposition.
If # - g j / i ^ - g j / i + i .
—l
a) {/;-//}=0 f<>r
Proof. H
~L~f>-v^f
—Ki,j
From the antisymmetry of H and d i t follows that for l < y < i<«: d4p--4p = ^Er,d^ET~d^W"^i^'
•
Iteratin
S
t h i s
argument, we obtain
<*Sfs 6/, 1° 6« * 6a '
Both expressions on the right l i e in Imd (the second by the antisymmetry of H) which proves assertion a ) . Further, H Ma-
~6u~
114
c v
- i - | ^ . . / / - ^ i = — c~J^--d -^f-~0 . 6a
6u
6tt " 6«
This completes the proof.
5.4.
Proposition 5.3 makes it possible to continue the commuting collection of elements
/_i, • ••,/ n 6^ to the commuting collection /_i, • • •, /n, fn+u if d~lH-^~/„glm-r— : as /n+1 we take any element of f-r— j d '//-£—/„. In order to guarantee the solvability of the equation for /n+i. we impose additional conditions on H. d'.
We recall that the Frechet operator D(g)
We define a mapping [D, //]:
A->-Z,(J4,
for g£A has the form D(g) =
.A), where L(A, A) is the space of linear
operators on A by the formula
\D,H\f=D(Hf)-HoD(f) (cf. Sec. 7 of Chap. I). If H=^htd'
» then
[D,H]f=
2 <*7-D(*i)-
We c a l l the antisymmetric operator / / Laxitive if for any pair / , d i t i o n Hf=dg 5.5
the operator
Proposition.
If
g(«A with the con-
\D, H\g»d — [D, H\f°H i s formally symmetric.
H i s a Laxitive operator and /_i,..., / B s a t i s f y the condition of
Proposition 5 . 3 , then the equation # §^/n = dg^/ n+ i i s solvable. Proof.
We s e t gt = -T- ft, i = — 1 , . . . , rt, and l e t
gnJrX be a solution of Hgn = dgn+i, which
e x i s t s by 5.3 b ) . According to the r e s u l t s of Sec. 7 of Chap. I , i t suffices to verify that the operator D(gn+1) i s symmetric: t h i s implies that g"n+i6g- A. to the symmetry of the operator
This, in turn, i s equivalent
d°D(gn^)°d.
We begin with the case « = — 1 . L a x i t i v i t y of H (for the pair f—0,
The operator
[D, H\cod
i s symmetric because of the
g = c i n the d e f i n i t i o n of t h i s property).
This means
that the following operator i s symmetric: doD (g0)od = D (dg0)od = D (Hc)°d = [D,H] cod. Now l e t « > — 1 . . Inasmuch as Hgn = dgn+U we have, D{Hgn) = d°D(gn+x}, i . e . , [D,H\gn + H°D(gn) = d°D(gn^). Similarly, from Hgn_^dgn we find [D, H\gn_x +HoD(#„_,) =d<>D(gn) . Multiplying the f i r s t equation on the r i g h t by d, the second by —H and adding, we obtain doD(gn+1)°d = [D, H\gn°d-[D,H\g^oH+HoD(gn)*d From the L a x i t i v i t y of H operators D(gn) Finally, 5.6.
+
d°D(ga)off-H°D(g^H.
i t follows that the sum of the f i r s t two terms i s symmetric. The
and D(g„_i) are symmetric, since gn and gn_i are v a r i a t i o n a l d e r i v a t i v e s .
d and H are antisymmetric.
This completes the proof.
Thus, for any Laxitive operator
H and
c(
115
d e f i n e s t h e g e n e r a t o r s of an A b e l i a n L i e s u b a l g e b r a i n 5.7.
Proposition.
Let
c0, c&k.
A.
Then t h e o p e r a t o r
H = d o 3 + 2 (cQu + Cj) da + c0u' is Laxitive. The proof i s o b t a i n e d by d i r e c t v e r i f i c a t i o n .
D i r e c t computations show t h a t t h i s i s t h e
g e n e r a l form of L a x i t i v e o p e r a t o r s of t h i r d o r d e r l y i n g i n Setting
c — -^, c0=— 2,
cl=0,
k [u, u'\ [
we o b t a i n by t h e formulas of 5.6 a sequence of
of t h e Korteweg-de V r i e s e q u a t i o n ( i n t h e form
u.f — §aa' — u'").
integrals
The f i r s t members have t h e
form a*
a
o
an"
•§-; —2-; a3—j-; 6.
5
. . 5
,, , 10 , „
1
—jtt^+ytta^-f j - a V - ^ s i t " " .
Solutions of Algebraic Type and Theta Functions 6.1.
This section is to be considered as an appendix to Chap. II: the formulas pre-
sented here can be obtained from the facts proved there regarding the structure of bimodules over a field if they are augmented to include the classical results from the analytic theory of Riemann surfaces and Jacobian varieties. Our exposition is based on the survey of Matveev [44]; for the proofs the reader should see this survey, the literature cited there, and also the paper of Krichever [16]. 6.2.
The Topology of Riemann Surfaces. To each field
and one-dimensional over
K , which is finitely generated
C, there corresponds a one-dimensional compact, complex variety
T , the Riemann surface of K . The field K is isomorphic to the field of meromorphic functions on
T . The genus of K and T is
a) half the first Betti number of
T;
b) the dimension of the space of holomorphic differential 1-forms on T : Abelian differentials of the first kind. A Riemann surface of genus zero is the Riemann sphere or the set of points of P 1 (C)of the projective line over C
. The genus of the curve
T:y2=]J_
ferentials of first kind on T have the form f(z)y-ldz curves are called hyperelliptic (elliptic for £ = 1).
(z—E.)
is
equal to g ; the dif-
, where /€C[z], deg/(2)
elliptic curve is given by either of the following two conditions (provided the genus > 1 ) . a) On T there is a function with a single pole of second order. b) The ratios of differentials of first kind generate a field of kind zero. On a compact Riemann surface T of genus g>\
it is possible to choose a basis of the
homology group #i(I\ Z) of the form {a„ bj}, i, j = 1 , .. .,g,
116
where (ah aj) = (bh bj)=0,
(ai,bJ) = Zlj.
2g+t
A typical choice of
{at, bj) on T: y2=i[
(z—£;); Et r e a l , i s as follows.
copies of the Riemann sphere with cuts along the intervals
(£\, E2), (E3, £4), . . .,(E2g+i, co) and
we glue them together crosswise along the edges of the cuts. the cut (E2j-i, E2j) is
Differentials and Periods.
Abelian.
A clockwise contour around
a;- ; a contour which on each of the sheet joins an interior point of
(£2;-i, E2J) with an interior point of (E2g+i, °°) is 6.3.
bj.
Any meromorphic 1-differential
to on T is called
The numbers A}(u>)= ^w, Bj(c«) = ^u> are called i t s periods (along a-
1 C
Cj = -^-y<>
We take two
a/t bj).
The numbers
b.
(over a circle around a logarithmic singularity) are its residues.
If all the
residues of *> are trivial <» is called a differential of second kind; all differentials of first kind are also differentials of second kind. A basis of differentials of first kind j¥=k,
(k)=^ik (0 for
and 1 for ; =fe) . Any Abelian differential is uniquely determined by its A -periods and principal parts
at singular points. For differentials of second kind it is possible to uniquely define by the conditions
Aj(a>)=0
©
and any value of the principal parts. For general differentials
the" same is true if the sum of the residues of the prescribed principal parts is equal to zero. 6.4.
The Riemann Theta Function. We consider g-dimensional complex space Qe and the
lattice Zg^Cg.
Let B be some complex
(gXg)
matrix for which there exists a constant
g
c>0,
such that
(\mBk, k)>cj?ik2
there i s a theta function
for a l l
k£Zs, k = {ku .. .,kg).
Corresponding to this matrix
6:0->C : 6
(P)= 2
exp{n(Bk, k)+2r.i(p, k)}, p£&.
If Bjk =B):{<.%) it is called the theta function of the Riemann surface F, referred to the {di,bj}
basis
of the group Hi(V,Z)
and the normalized basis of differentials of first kind
(ij);.) on T . The matrix B is called the matrix of periods of
T ; it is symmetric.
The following properties of the theta function are easily verified: a) 6,(_p)=:9(p). b) Q(p+v)=HP) c)
f
oT all vQZz.
%(pJirBi)=e''t'B'r2,lip'^{,p)
, where
We now assign to each point PQ
is a fixed point of
I\
Pff
B1
is the j-th column of B.
its Jacobian coordinates: P^
Here
The Jacobian coordinates of P are defined up to the lattice
of periods of V spanned by the columns of the matrix B , since different paths of integration from P0 to P differ by some cycle representing a homology class in H1(T, Z). The multivalued function r~>-C:P>-*8(u)(P) — e ) , where e(*C£, is called the Riemann theta function.
It is meromorphic and is either identically zero or has exactly g zeros on T.
117
The Jacobian coordinates of these zeros are
£—*, where the vector x is defined by the
relations p
i+m. aj
P,
(the Riemann c o n s t a n t s ) . If the Riemann theta function i s nonzero, then the divisor of i t s zeros i s nonspecial, and any nonspecial divisor of degree g i s a divisor of the zeros of a s u i t a b l e theta function. From properties b) and c) of the theta function i t follows that dlog6(w(/5)—e) i s a meromorphic d i f f e r e n t i a l on T, having as i t s divisor of poles the zeros of
9-
All poles
have f i r s t o r d e r . 6.5.
Analytic Description of the Akhiezer Function.
Riemann surface
V of genus
l o c a l parameter
ft-1
g , a point Po6l\
i n a neighborhood of
We fix the following data: a
a nonspecial divisor D = ^Pi
on T , a
P 0 , and two polynomials P, Q€C [ft] of degrees
n
and m , r e s p e c t i v e l y . 6.6.
THEOREM.
There e x i s t s a unique meromorphic function
<[>:C3X(r\P0)->-Cwith the
following p r o p e r t i e s : a) The d i v i s o r of the poles of b) At the point
P<> the function
f in
P
coincides with D for any X, y, t£C3.
<>| has an e s s e n t i a l s i n g u l a r i t y of the form
4 (x, y,t; P) = exp [ft (P) + P (ft (P)) + Q (ft (P))] • (1 + 0 (ft (P)"1). c)
This function has the form
4(*, jf, *, P) - e(o,o))e(-«(x,y.o)exp \ .H 01 i x + U, ^+ ( V)J> where u>i,
and principal parts d(ft-f clogft), dR(k),
dQ(k),
respectively, and e
e{x, y . /)=s/(B(»,)j: +fi(»2)ff+flW ) + 2 « ( / > y ) + * . 5(co i ) = A ( o A ,
6.7.
x = (xffl)
is the Riemann vector.
The connection of this construction with the results of Sec. 3 of Chap. Ill is
as follows. Let 33 be a field of meromorphic functions in x,y,t, of meromorphic functions on T with pole at
and let O denote the ring
P 0 . Then on the space of functions
it is possible to introduce the structure of a (53,0)-bimodule with <J» as
118
33\dx\$
1 , in which the
filtration is induced by the filtration of
V*, V y , Vt
33 [
coincide with dx,dy,dt
,
respectively; multiplication by O is the natural multiplication. Further calculations along the lines of Chap. Ill but using the analytic information now at hand lead to the following explicit formulas. 6.8. The Korteweg-rde Vries Equation ut = 6uux — uxxx.
As was explained in Chap. Ill, in
order that under the imbedding . G-+3?l[dx] the image of O contain a Schrodinger operator of second order
— dx-\-u,
it is necessary that Y be hyperelliptic.
surface of the curve y1 — II(z— E}), Ej=£Ek.
Let T be the Riemann
We choose a normalized basis of differentials of
first kind
where the
cjk
as in 6.2.
are found from the system of equations
Then for
\o).=8,.
and the ak
P0= oo and a nonspecial divisor D=P\-\- . u(x,t)=-2dxlogB{xg
+
are defined
-\-Pg we have
tv+l)+c,
where g, V, /gO, c£C and gj=2icn;
i>, = 8/1 ^cnj
2«+l
^Em)
+ cJ2);
I
c=^Ej-2
2
$;•
f-1
(The nonuniqueness in the choice of path of i n t e g r a t i o n in the formulas for affect u(x,t)).
lj does not
The Akhiezer function 4>lx t P)-
BXaW + ^+.fa + Oe.Cfa+i)
i s a solution of the Schrodinger equation
{ — dx+u(x,t))y
normalized d i f f e r e n t i a l of second kind with i = dVz
/. f = y(P)ty.
3/2
+0(z~ dz)
as
) Here P e r
and
v is a
z->oo, and having no
other poles. If a l l or part of the pairs
(Em, Em+l) coalesce the l i m i t solutions w i l l be
g -solitons
or s o l i t o n s on an almost-periodic background. 6.9.
The Kadomtsev—Petviashvili Equations.
These correspond to the Akhiezer function
R(k,) = k2, Q(k) = k3.
In the ring
3S[dx] with the help of the bimodule S [ ^ H w e seek
operators 119
L = dx + u(x, y, t), P=d3x + vl(x, y, t)dx + v,(x,y, satisfying the Zakharov—Shabat equation u, Vu V2
t),
dyP—dtL =[L, P]. In terms of the coefficients
this system can be rewritten in the form ( v2y—ut + V2xx—uxx—zi,ax=0; «ux+2v2x—3ttxx+'Oiy=0; [2wijr='3ttx.
Solutions constructed on the basis of the bimodule actually even satisfy the condition 2t»i=3a .
Eliminating
V\ and
w2 from this system, we obtain the Kadomtsev—Petviashvili
equation 4 u*y + axt+T
¥xxx + 6" «A = 0,
which i s also called the two-dimensional Korteweg—de Vries equation. written in the form
I t s solution can be
i
2dx log8(xg+tv+yh+l). LITERATURE CITED
1.
2.
3. 4. 5. 6.
7. 8.
9. 10. 11. 12. 13. 14. 15. 16.
120
0. I . Bogoyavlenskii and S. P. Novikov, "On the connection of the Hamiltonian formalisms of stationary and nonstationary problems," Funkts. Analiz Prilozhen., 10, No. 1, 9-13 (1976). I . M. Gel'fand and L. A. Dikii, "Asymptotic resolvents of Sturm—Liouville equations and the algebra of the Korteweg-de Vries equation," Usp.Mat. Nauk, ^0, No. 5, 67-100 (1975). I . M. Gel'fand and L. A. Dikii, "The structure of Lie algebras in the formal calculus of v a r i a t i o n s , " Funkts. Analiz Prilozhen., 10_, No. 1, 28-36 (1976). I . M. Gel'fand and L. A. Dikii, "Fractional powers of operators and Hamiltonian systems," Funkts. Analiz Prilozhen., .10, No. 4, 13-29 (1976). I . M. Gel'fand and L. A. Dikii, "Resolvents and Hamiltonian systems," Funkts. Analiz Prilozhen., 11, No. 2, 11-27 (1977). I . M. Gel'fand, Yu. I . Manin, and M. A. Shubin, "Poisson brackets and the kernel of the variational derivative inthe formal calculus of variations," Funkts. Analiz Prilozhen., 10_, No. 4, 30-34 (1976). V. G. Drinfel'd, "On commutative subrings of some noncommutative rings," Funkts. Analiz Prilozhen., _U, No. 1, 11-14 (1977). B. A. Dubrovin, V. B. Matveev, and S. P. Novikov, "Nonlinear equations of Korteweg—de Vries type, finite-zone linear operators, and Abelian manifolds," Usp.Mat. Nauk, 31, No. 1, 55-136 (1976). V. E. Zakharov, "A kinetic equation for s o l i t o n s , " Zh. Eksp. Teor. Fiz., jj0_, No. 3, 993-1000 (1971). V. E. Zakharov and S. V. Manakov, "On complete integrability of the KdV equation and the nonlinear Schrodinger equation," Preprint IYaF, No. 68-73, Novosibirsk (1973). V. E. Zakharov, L. A. Takhtadzhyan, and L. D. Faddeev, "A complete description of sineGordan solutions," Dokl. Akad. Nauk SSSR, 219, No. 6, 1334-1337 (1974). V. E. Zakharov and L. D. Faddeev, "The Korteweg-de Vries equation — a completely integrable Hamiltonian system," Funkts. Analiz Prilozhen., _5, No. 4, 18-27 (1971). V. E. Zakharov and A. B. Shabat, "A scheme for integrating nonlinear equations of mathematical physics by the method of inverse scattering," Funkts. Analiz Prilozhen., 8., No. 3, 43-53 (1974). A. P. I t s and V. B. Matveev, "On a class of solutions of the Korteweg-^de Vries equat i o n s , " in: Probl. Mat. F i z . , No. 8, Leningr. Univ., Leningrad (1976), pp. 70-92. I . M. Krichever, "Potentials with zero reflection coefficients on a background of finite-zone potentials," Funkts. Analiz Prilozhen., 9_, No. 2, 77-78 (1975). I . M. Krichever,""Integration of nonlinear equations by methods of algebraic geometry," Funkts. Analiz Prilozhen., 11_, No. 1, 15-31 (1977).
17. 18. 19.
20. 21. 22. 23. 24. 25. 26.
27. 28. 29.
30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41.
42. 43. 44. 45.
E. A. Kuznetsov and A. V. Mikhailov, "On the complete integrability of the classical two-dimensional Thirring model," Teor. Mat. Fiz., _3_9_» No - 3, 303-314 (1977). B. A. Kupershmidt, "The Lagrangian formalism in the calculus of variation," Funkts. Analiz Prilozhen., 10, No. 2, 77-78 (1976). B. A. Kupershmidt and Yu. I. Manin, "Equations of long waves with a free surface. I. Conservation laws and solutions. II. The Hamiltonian structure and higher equations," Funkts. Analiz Prilozhen., 1^, No. 3, 31-42 (1977). Yu. I. Manin, "Rational points of algebraic curves over function fields," Izv. Akad. Nauk SSSR, Ser. Mat., 2]_, No. 6, 1397-1442 (1963). S. P. Novikov, "The periodic problem for the Korteweg-de Vries equation. I," Funkts. Analiz Prilozhen., 8_, No. 3, 54-66 (1974). L. A. Takhtadzhyan and L. D. Faddeev, "An essentially nonlinear one-dimensional model of classical field theory," Teor. Mat. Fiz., 21_, No. 2, 160-174 (1974). L. D. Faddeev, "The inverse problem of quantum scattering theory. II," in: Sovr. Probl. Mat. (Itogi Nauki i Tekhn. VINITI AN SSSR), Moscow (1974), pp. 93-180. L. D. Faddeev, "In search of multidimensional solitons," in: Nonlocal, Nonlinear, and Nonrenormalizable Field Theories," JINR (1976), pp. 207-223. H. Airault, H. P. McKean, and J. J. Moser, "Rational and elliptic solutions of the Korteweg—de Vries equation and a related many-body problem," Preprint (1976). R. M.Miura (editor), Backlund Transformations, the Inverse Scattering Method, Solitons, and Their Applications, NSF Research Workshop on Contact Transformations, Lecture Notes in Mathematics, No. 515, Springer-Verlag, Berlin (1976). F. Calogero, "Exactly solvable one-dimensional many-body problems," 1st. di Fisica G. Marconi, Univ. di Roma, Preprint (1975). F. Calogero and A. Degasperis, "Nonlinear equations solvable by the inverse spectral transform method. II," 1st. di Fisica G. Marconi, Univ. di Romn, Preprint No. 20 (1976). F. Calogero and A. Degasperis, "Backlund transformations, nonlinear superposition principles, multisoliton solutions, and conserved quantities for the boomeron nonlinear evolution equation," Lettere Nuovo Cimento, J^, No. 14, 434-438 (1976). D. V. Choodnovsky and G. V. Choodnovsky, "Pole expansion of nonlinear partial differential equations," Preprint (1977). J. Corones, "Solitons and simple pseudopotentials," J. Math. Phys., 17, No. 5, 18671872 (1976). F. B. Estabrook and H. D. Wahlquist, "Prolongation structures of nonlinear evolution equations," J. Math. Phys., 17, No. 7, 1293-1297 (1976). Exact Treatment of Nonlinear Lattice Waves, Progr. Theor. Phys., 59_, (1976). L. D. Faddeev, "Quantization of solitons," Preprint, Inst, for Adv. Study, Princeton (1976). L. D. Faddeev, "Some comments on many-dimensional solitons," Preprint, CERN, TH 2188 (1976). C. S. Gardner, "The Korteweg-xle Vries equation and generalizations. IV. The Kortewegde Vries equation as a Hamiltonian system," J. Math. Phys., 12_, No. 8, 1548-1551 (1971). P. Hasenfratz and D. A. Ross, "Are quarks short-range solitons?" Phys. Lett., 64B, No. 1, 78-80 (1976). R. Hirota, "A direct method of finding exact solutions of nonlinear evolution equations, Lecture Notes in Mathematics, No. 515, Springer-Verlag, Berlin (1976), pp. 40-68. R. Hirota and J. Satsuma, "A variety of nonlinear equations generated from the Backlund transformation for the Toda lattice," Progr. Theor. Phys., No. 59, 64-100 (1976). M. D. Kruskal, "The Korteweg—de Vries equation and related evolution equations," Lect. Appl. Math., 15, 61-83 (1974). M. D. Kruskal, R. M. Miura, and C. S. Gardner, "Korteweg-de Vries equation and generalizations. V. Uniqueness and nonexistence of polynomial conservation laws," J. Math. Phys., 11, 952-960 (1970). P. D. Lax, "Integrals of nonlinear equations of evolution and solitary waves," Comm. Pure Appl. Math., _21, No. 5, 467-490 (1968). P. D. Lax, "Periodic solutions of the KdV equations," Comm. Pure Appl. Math., ^8_, No. 1, 141-188 (1975). V. B. Matveev, "Abelian functions and solitons," Inst. Theor. Phys., Univ. Wroclaw, Preprint No. 373 (1976). H. P. McKean and E. Trubowitz, "Hill's operator and hyperelliptic function theory in the presence of infinitely many branch points," Comm. Pure Appl. Math., 29, No. 2, 143-226 (1976). 121
46.
R. M. Miura, "Conservation laws for the f u l l y nonlinear long wave e q u a t i o n s , " Stud. Appl. Math., 5 3 , No. 1 , 45-56 ( 1 9 7 4 ) . 4 7 . R. Miura, "The Korteweg-de V r i e s equation: a survey of r e s u l t s , " SLAM Rev., 1 8 , No. 3 , 412-459 ( 1 9 7 6 ) . 4 8 . R. M. Miura, C. S. Gardner, and M. D. Kruskal, "Korteweg-de Vries equations and g e n e r a l i z a t i o n s . I I . E x i s t e n c e of conservation laws and constants of the motion," J. Math. P h y s . , 9 , No. 8 , 1204-1209 ( 1 9 6 8 ) . 4 9 . H. C. Morris, "Prolongation s t r u c t u r e s and a generalized i n v e r s e s c a t t e r i n g problem," J . Math. P h y s . , 1 7 , No. 1 0 , 1867-1869 ( 1 9 7 6 ) . 50. Nonlinear Waves, L e c t . Appl. Math. 1974, AMS, Providence ( 1 9 7 4 ) . 5 1 . A. P a t a n i , M. Schlinwein, and Q. S h a f i , "Topological charges i n the f i e l d theory," J . Phys. A: Math. and Gen., 9, 1513-1520 ( 1 9 7 6 ) . 52. K. Pohlmeyer, "Integrable Hamiltonian systems and interaction through quadratic constraints," Comm. Math. Phys., 46, 207-221 (1976). 53. A. C. Scott, F. Y. F. Chu, and D. W. MacLaughlin, "The soliton: a new applied s c i e n c e , " Proc. IEEE, 61, 1443-1483 ( 1 9 7 3 ) . 54.
122
H. D. Wahlquist and F. B. Estabrook, Prolongation s t r u c t u r e s of nonlinear e v o l u t i o n e q u a t i o n s . I , " J. Math. P h y s . , 16_, No. 1 , 1-7 ( 1 9 7 5 ) .