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\ (« + />) - V(u) - (/, /*)] = 0. 0 such that \\e\\ > r, /?:=
/
\u'(s)\ds=
/
*t+i
u'(a)rfs
for all t s l . Hence, on each interval [t, t + 1] of length one, either v! > 0 or u' < 0. This implies that there exist *i,£2 6 («, t) such that t2 — t\ > 1 and M' E 0 on [^1,^2]. Choose t\ and *2 s o that, in addition, d i s t ^ , { t e l : u'(t) ^ 0}) = 0,
i = 1,2.
Let ^; '
\ S ( t + ( t 2 - * l ) ) , *>*!•
(4.18)
142
Travelling Waves and Periodic Oscillations in FPU Lattices
Since t2 —1\ > 1, we have that U(w) - U(u) = /
X
l/fu(t + 1 + (t2 - «i)) - «(*)) * /•ti
- /
rt2
- I
Jt2-i
U(u(ti)-u(t))dt
U(u(t +
l)-u(ti))dt
= jf * [tf (r^t) + r2{t)) - U(n(t)) - C/(r2(i))] d«, where ri(«)=«(*i)-«(*)>0.
r2(«) = u(t + 1 + (t2 - hj) - u(tt) > 0. By superquadraticity of U, we have that tf(ri+r2)>Cf(n) + tf(r2) for all rx,r 2 > 0, with equality if and only if one of the ri is zero. Now (4.18) implies the existence of io € [t\ — l,fi] such that both r\{to) > 0 and r 2 (i 0 ) > 0. Hence, U(iu) > U(u). Now there exists A G (0,1) such that U(Atu) = U(u) = K and T(Au;) = A2T(iy) < T(w) = T(u) = T(«), that contradicts the fact that w is a minimizer. The proof is complete.
•
Proof of Theorem 4-1. Applying Propositions 4.2, 4.3, Remark 4.2 and Lemma 4.6 to the symmetrized potential, we obtain in this case all the statements of Theorem 4.1, except (p3) and (pi). The last holds in a weaker form: u is a nondecreasing (resp. nonincreasing) function. By Lemma 4.7, the minimizer u we consider is a monotone function and, therefore, either u or — u is a solution of the original problem. Property (pi), i. e. strict monotonicity, follows from Proposition 3.8. If remains to prove (p3), i. e. the fact that the wave is supersonic. Since r~2U(r) increases with \r\, differentiating we obtain \u\r)r>U{r),
r G A.
143
Travelling Waves
Now Eq. (4.16), with h = u, c2 = A"1, and the energy inequality, which holds by Proposition 4.2, imply that
This completes the proof of the theorem. 4.1.5
•
Lennard-Jones type potentials
Here we discuss singular potentials, like the Lennard-Jones potential (see Examples 1.5 and 3.5) U(r) = a[(d + r)~6 - d~6}\
r
>-d,
where a > 0 and d > 0. We start with the following result that concerns the case of solitary waves with small averaged potential energy. Theorem 4.2
Let U{r) = U0(r)
if r> -d,
U{r) = +oo
if r < -d.
with some d > 0, and
Assume that UQ £ C2(— d, oo), Uo > 0, Uo(0) = 0, Uo is superquadratic on (-d,0), U{j(Q) ^ 0 and that (4.2) holds. Then there exists K0>\u"(Q)d2 such that for every K G (0, KQ) the system admits a solitary wave u — UK £ X with the averaged potential energy U(u) = K, with the property -d < AuK{t) < 0,
tGE,
and properties (p2)-(p4) of Theorem 4-1Proof. The proof relies on a cutoff argument which makes Theorem 4.1 applicable. Fix e e (0,d/2) and choose a cutoff function 77 € C°°(R) such that 11
(0 on (-00,-d + e) ~~ \ 1 on (-d + 2e, 00)
144
Travelling Waves and Periodic Oscillations in FPU Lattices
and 77' > 0. Let p be the exponent from assumption (4.2). Choosing C = Ce so large that W£(r) := \ U"(0) r2 + C\r\" > U(r) on {—d + e, 0), we define a new potential on the whole of M. by K)
'
U(r)tf(r) + (l-7/(r))W e (r),
r > -d.
The superquadraticity of U and W, and the fact that r]'(U-W)<0
on (-oo,0)
imply immediately that
which means that Ue is superquadratic on (—00,0). Also it is clear that Ue satisfies (4.2). Hence, Theorem 4.1 applies and for all K > 0 the system with the modified potential Ue possesses a travelling wave u = UK = ue^ such that V£(u) = K. Now we shall show that, for K < i U"(0) d2 and appropriately chosen e,
t e R.
Since U£ = U on (—d + 2e, 00), this completes the proof. By Lemma 3.5 and energy inequality (4.10),
(
9K
\ll2
where Te,K is the minimum of T subject to the modified constraint. Hence, given K
Travelling Waves
145
one can take
( I K \1/21
if, to obtain
•
The following results are obtained in [Friesecke and Matthies (2002)]. Theorem 4.3
In addition to the assumptions of Theorem J^.2, suppose
thatUo GC 3 ((-d,oo)),
U^'(r) < 0 on {-d, 0],
U0(r) < U0(-r) on (0, d)
and U0(r) > a(r + d)" 1
(4.19)
for some a > 0 and all r close to —d. Then the conclusion of Theorem J..2 holds for all K > 0 and u = UK is a solution of problem (FW), with the original (not symmetrized) potential. We skip the proof and mention only that (4.2) follows from UQ"(0) < 0. The assumption U0(r) < Uo(—r) on (0, d) is needed to show that U — UK obtained as a solution of symmetrized problem (FW) is, actually, a solution of unsymmetrized problem (FW). Assumption (4.19) is essential to prove that all K > 0 are allowed. Theorem 4.4 Under the assumptions of Theorem 4-3, let UK S X, K > 0, be any solution obtained by means of problem (FW), and
{
0,
t<0
-dx, 0
uK(t-aK)
146
Travelling Waves and Periodic Oscillations in FPU Lattices
and1 (uK)' -+ {Uooy in L*(R) for allp £ [1, oo), as K —> oo. Furthermore, the corresponding wave speed CK satisfies lim
CK
= oo.
K—>oo
The classical Lennard-Jones potential, as well as its generalized version mentioned in Examples 1.5 and 3.5, satisfies the assumptions of Theorems 4.2 and 4.3. 4.2
Other types of travelling waves
4.2.1
Waves with periodic profile functions
In Section 3.2 we have considered travelling waves whose relative displacement profile is periodic, i. e. r(t)=r(t + 2k), or, equivalently, u'(t)=u'(t + 2k), where k > 1. In the present subsection we impose the boundary condition (4.20)
u(t) = u(t + 2k) which means that the wave profile itself is 2fc-periodic. Let Yk := {u e H{OC(R) : u(t + 2k) =u(t),u(0)
- 0}.
Endowed with the norm \\u\\k = ||u'|U*(-fc,fc), the space Yk is a Hilbert space and, obviously, this is a closed subspace of the space Xk introduced in Section 3.2. Moreover, u 6 Xk belongs to Yk if 1 It follows from Theorem 4.10 and Remark 4.6 that u'k decay exponentially and, hence, belongs to LP(R) for all p > 1.
147
Travelling Waves
and only if its derivative u' has zero mean value: J-k Therefore, Yy. is 1-codimensional subspace of Xk. The orthogonal complement is exactly the subspace of Xk generated by the function ho(t) = t. We impose the following assumptions already used in Subsection 3.2.3: (»") U(r) = | r2 + V(t), where V € C^R), V(0) = V'(0) and V'(r) = o{r) as r —> 0, (ii1) For some ro £ R we /iave that V(ro) > 0 and there exists 6 > 2 suc/i
that
flV(r) < r V ' ( r ) ,
r£l.
Consider the functional (see Subsection 3.2.1)
Jk(u) = Jk\^u\t?-U(Au{t))^
dt
restricted to the subspace 1/.. We keep the notation Jk for the restricted functional. Since in the proof of Proposition 3.2 we have used only 2kperiodic C°° test functions, we see that any critical point the restriction of Jfe to Yk is, in fact, a solution of Eq. (3.3) that satisfies (4.20). Theorem 4.5 Assume (i") and(ii'). Let c > max(a, 0). Then for every k > 1 there exists a nontrivial travelling wave Wk € Yk- Moreover, there exist constants S > 0 and M > 0 such that 6 < Jk{wk) < M and S < \\wk\\k < M. Proof relies upon the standard mountain pass theorem (Theorem C.I) and goes along the same lines as the proof of Theorem 3.1, with only minor modifications. • Note that since (w'k) = 0, the waves obtained are definitely not monotone, even when a = CQ > 0. In the last case by Theorem 3.1, there exist
148
Travelling Waves and Periodic Oscillations in FPU Lattices
two monotone waves, nondecreasing and nonincreasing, with periodic relative displacement profile and, by Theorem 4.5, an additional nonmonotone wave that has a periodic profile function. 4.2.2
Solitary waves whose profiles vanish at infinity
In contrast to the case of Section 3.3, here we impose the boundary condition u(oo) = 0
(4.21)
that means that the wave profile vanishes at infinity. Let Y be the closure of CQ°(R) with respect to the norm
a
+oo
\ 1/2
u'{t)2dt\ . Obviously, Y is a closed subspace of the space X defined in Subsection 3.3.1. One can think of functions from Y as satisfying boundary condition (4.21) in some generalized sense. Consider the functional (see Subsection 3.3.1) ^ \ju'(t)2-U(Au(t))\dt restricted to the space Y. The restriction of J to Y is still denoted by J. Under assumptions (i") and (ii1) (see Subsection 4.2.1), J is a C 1 functional on Y. Moreover, every critical point of J in Y is a C 2 solution of Eq. (3.3). Indeed, since Cg°(R) C Y, the proof of Proposition 3.5 shows that u is a weak solution of (3.3) and, in fact, a classical solution. Now we have the following existence result. Theorem 4.6 Under assumptions (i") and {ii'), with c > max(a,0), there exists a nontrivial travelling wave w 6 7 . Proof. As we have pointed out in Subsection 3.3.2, the functional J possesses the mountain pass geometry in the space X. But since there exists, obviously, a vector e € Y such that J(e) < 0, the same holds in Y. By the mountain pass theorem without Palais-Smale condition (see Theorem C.4), there exists a sequence wn e Y such that
J K ) -»b
Travelling Waves
149
and J'{wn) —* 0 in Y*
(the dual space),
where b is the mountain pass level, i. e. wn is a Palais-Smale sequence. As usual, assumption (ii1) permits us to conclude that the sequence wn is bounded in Y. Moreover, as in Lemma 3.5, we deduce that ||u>n|| is bounded below by a positive constant. Hence, ||ion|| does not converge to 0. Therefore, we can assume that wn converges to w weakly in Y. Arguing as in Lemma 3.6, we obtain that, for any r > 0 there exists 77 > 0 such that along a subsequence, /
JCn-r
\Awn(t)\ dt > t],
with some £„ € M. Replacing wn(t) by wn(t — £n), we obtain that
I J —T
\Awn{t)\2dt>n
and the new wn still form a Palais-Smale sequence. By the compactness of Sobolev embedding, Awn —* Aw in L^>C(R), i. e. uniformly on compact intervals and we deduce that / \Aw{t)\2dt > T), J—r hence, w ^ 0. Testing J'(wn) with an arbitrary function ip £ C£°(W), we obtain, as in the proof of Proposition 3.6, that J'(w) = 0, i. e. w 7^ 0 is a critical point of J in Y. • Remark 4.4 As it follows from Theorem 4.10 and Remark 4.6, at least in the case when a = CQ > 0 the solution w e Y obtained in Theorem 4.6 decays exponentially at infinity and, hence, satisfies (4.21). Note that for the solution w obtained in Theorem 4.6 we have that 0 < J(w) < b. It is easy to verify that the arguments of the proof of Proposition 3.14 work in the case we consider as well. Therefore, under the additional assumption that V(r) > ao\r\q, with q > 2, we obtain the estimate
\\wf
150
Travelling Waves and Periodic Oscillations in FPU Lattices
provided c > CQ is close to co (here a = cjj > 0). Repeating the proof of Theorem 3.7, we obtain Theorem 4.7 Suppose that V(r) is defined in a neighborhood of 0 and satisfies in that neighborhood assumptions (i") and (ii')t with a = c§ > 0. Assume, in addition, that for \r\ small V(r) > ao\r\^ with a0 > 0 and q > 2. Then there exists c > Co such that for every c € (CQ,~C) there exists a nontrivial travelling wave w £ X with the speed c. Remark 4.5 tion 4.2.1.
Similar result holds for waves considered in Subsec-
Note that Theorem 4.6 applies immediately to FPU a- and /3-models (see Examples 1.2, 1.3 and 3.2, 3.3), while Theorem 4.7 works is the case of Lennard-Jones type potentials (Examples 1.5 and 3.5). Thus we obtain in those cases additional nonmonotone waves. 4.3
Yet another constrained minimization problem
Consider a particular case when the potential is given by 2
V(r) = j
j
+ -\r\p,
p>2,d>0.
In this case one can obtain travelling waves solving another constrained minimization problem. Let
Qk{u) = \ fk [c2{u\t)f - cl(Au{t)f] dt and
*k(u) = ^Jk \Au(t)\pdt. These are well-defined C 1 functionals on the space Xk- Similarly, the functionals
Q(u) = \ [+°° \c2(u'(t))2 - c\{Au{t)f\ dt
Travelling Waves
151
and A /-+oo
*(«) = - /
\Au(t)\pdt
P J —oo
are of class C 1 on X. Obviously, Jfc(u) = Qfc(tx) - * fc (u) and J(«) = Q{u) - #(«), where J^ and J are introduced in Sections 3.2 and 3.3, respectively. Given a > 0, consider the following two minimization problems: h(a) = inf {Qk(v) : veXk, **(«) = a}
(4.22)
I(a) = inf {Q(w) : » e X , f ( » ) = a } .
(4.23)
and
Since \£fc and ^ are positive homogeneous of degree p > 2, and Qk and Q a r e positive homogeneous of degree 2, it is readily verified that problems (4.22) (resp., (4.23)) with different values of a > 0, are equivalent. Moreover, Ik{a) = a 2 / p / fc (l) and I(a) = a2/pl(l). We have the following result Theorem 4.8 Suppose that c > Co > 0. Then there exists a solution vkeXkk>\, (resp., v e X) of problem (4-22) (resp., (4.23)). To prove the result for problem (4.22) one considers an arbitrary minimizing sequence which turns out to be bounded in Xk. The passage to the limit is straightforward and uses the compactness of the Sobolev embedding. In the case of problem (4.23) one has to employ, in addition, a concentration compactness argument based on Lemma B.I. Alternatively, one can obtain a solutions as the limit of appropriately shifted solutions vk of the problem (4.22) as k —» oo.
152
Travelling Waves and Periodic Oscillations in FPU Lattices
Since the functionals Qk and *fc, as well as Q and *, are C 1 , there exist Lagrange multipliers Xk and A such that Q'k{vk) =
\k*'(vk)
and Q'(v) = \*'(v). Actually, Afc —
>0 ap
and
A=
«>0.
ap
These Lagrange multiplies can be scaled out, using the homogeneity properties of the problems. Precisely, letting i
and u — X'p^v, we obtain travelling waves uk £ Xk and u £ X. 4.4
Remark on FPU /3-model
Let us consider the FPU /3-model, with the interaction potential
U(r) = Cjr> + ±r* = ^r* + V(r), in the case when d < 0. In [Friesecke and Wattis (1994)] it is conjectured that nontrivial solitary waves do not exist in this case. The conjecture trivially holds true, as a consequence of following Proposition 4.4
Suppose that
U(r) = Cjr* + V(r)
Travelling Waves
153
where V e C \ V(0) = V'(Q) = 0 and V(r)r < 0. If c > c0 > 0, then nontrivial travelling waves, both periodic and solitary, do not exist. Proof. For the functional Jk introduced in Section 3.2 we have that
{J'k(u),u) = J^ [c2(u'(t))2 - c20(Au(t))2 - V'(Au(t)) Au(t)] dt >jkk[c\u'{t))2-cl{Au{t)f]dt =
c2\\u\\l-c20\\Au\\2LH_ktky
By Lemma 3.1 (J'k(u),u)>(c-co)2\\u\\l Therefore, the only critical point of Jk is the point of minimum u = 0, and there is no nontrivial travelling wave u e Xk. The same reasoning shows that the functional J has the only critical • point u = 0 and nontrivially solitary waves do not exist. However, if 0 < c < CQ, the FPU /3-model with d < 0 possesses periodic travelling waves. More generally, we have Theorem 4.9
Suppose that U, U(r) = ^r2 + V(r),
where V e C^R), F(0) = V'(0) = 0 and V'(r) = o(r) near r = 0, and suppose that there exists 9 > 2 such that O>0V(r)
>rV'(r)
and V(r) < 0 if \r\ > ro for some r$ > 0, i. e. —V satisfies assumptions (ii+) and (ii~) from Section 3.2. Then for every c £ (0, CQ] and every k > 1 there exists a Ik-periodic travelling wave u^ S Xk. The proof goes along the same lines as in the case of Theorem 3.3. We consider the functional
-Jk(u) = -1* ^ (u'{t))2 - | (Au(t))2 - V(A(t))j dt and apply the generalized linking theorem of [Benci and Rabinowitz (1979)] (Theorem C.5).
154
Travelling Waves and Periodic Oscillations in FPU Lattices
Employing the notation used in the proof of Theorem 3.3, we denote by Z the subspace of Xk generated by hf with Xj < 0, while Y stands now for the subspace generated by hf with Xj > 0 and the function ho- So, the quadratic part of —Jk is nonpositive on Y and positive on Z. Clearly, Y±Z and Xk = Y © Z, but now dimY = oo and dimZ < oo. This is the reason to use Theorem C.5 instead of Theorem C.4. The verifications of Palais-Smale condition and of linking geometry are quite similar to the corresponding points of the proof of Theorem 3.3. 4.5
Exponential decay
Here we prove that solitary waves obtained before have exponentially decaying relative displacement profile r(t) = Au(t) = u(t + 1) - u(t). Being rewritten in terms of r, Eq. (3.3) becomes c2r"(t) = -A*AU'(r(t)),
(4.24)
where A*v(t) = v(t) - v(t - 1) is the operator formally adjoint to A, and A*Av(t) = (v(t + 1) + v(t - 1) - 2v(t)). Separating harmonic and unharmonic parts of the potential U, we impose the following assumption:
(h) U(r) = — r 2 + V(r), where V G C 1 in a neighborhood of 0, V(0) = V'(0) = 0 and V'(r) = o(r) as r -* 0. So, we allow locally defined potentials. Now we write Eq. (4.24) in the following form Lr = -A*AV'(r), where Lv(t) = -c2v"(t) + c20{A*A)v(t) = -c2v"{t) + c2 (v(t + 1) + v(t - 1) - 2v{t)).
(4.25)
Travelling Waves
155
The operator L is a pseudodifferential operator with the symbol
while
aA-(t) =
l-e-*
and
= (1 - e-i«)(ei« - 1) = -4sin 2 |
(c/. Subsection 3.5.1). Making use the Fourier transform, we obtain from Eq. (4.25)
VL{Z)nO = -VA-A{O[V'(r)f{Z), i. e. r = TV'(r),
(4.26)
where fn = T
^ ^
^ » A ( Q=
4sin 2 (e/2) c2^-4cgsin 2 (e/2)"
-1
(Formally, T = —L A*A). Actually, this means that T is a convolution operator with the integral kernel K(x) such that K(£) = 0r(£). To study Eq. (4.26) we need the following Lemma 4.8 Let f(t) and g(t) be bounded non-negative functions on R, with limt^ioo g(t) = 0. Suppose that r+oo
/(*)
e-W-Tlg(T)f(T)dT,
with (3 > 0. TTien for every a £ (0,0) there exists a constant C = Ca such that f{t)
156
Travelling Waves and Periodic Oscillations in FPU Lattices
Proof. It suffices to prove the desired estimate for t > 0 because the case t < 0 reduces to the previous one by replacing t by —t. For any integer n > 0 let
Ln= ^
J—oo
e-^-^g(T)f(T)dT,
Fn = sup/(t),
Gn = supg(t).
t>n
t>n
For t > n w e have that rn
/(*)
J—oo
e-Kt-T>g(T)f(T)dT+ />+oo
< Ln + FnGn /
Jn
f+oo Jn
e-^-Tl5(r)/(r)dr r+oo
e-^-^dr
J—oo
e'^dr
= Ln + — FnGn. From there we deduce immediately that Fn
+ ^ FnGn.
(4.27)
Next we estimate L n +i:
Ln+i=e-p
f1 e-«"-^ 5 (r)/(r)dr J — OO
+
Jn
r+1e-^n+^g{T)f{T)dT
= e-0Ln + FnGn.
(4.28)
Since, by assumption, Gn —> 0 as n —> +oo, then there exists an integer M > 0 such that for n > M we have
2Gn < min [|, e " a - e'A . Now (4.27) implies immediately that Fn < Ln + —Fn. Hence, Fn < 2Ln.
(4.29)
157
Travelling Waves
This together with inequalities (4.28) and (4.29) gives us Ln+1
2Gn)
+
The last inequality implies easily that, for n> M, Ln < e~aneaMLM
= Ke~an.
Therefore, Fn < 2Ke-an and this implies the required. Indeed, for t 6 [n, n + 1], with n > M, we have that /(*) < Fn < 2Ke~an = 2Kea^-n^e~at
< (2Kea)e~at
and the proof is complete.
•
Now we obtain an exponential bound for the kernel K{x). Lemma 4.9 Suppose that CQ > 0. Then there exists /?o > 0 such that for every /? € (0,/?0) \K{x)\
with some C > 0. Proof. We start with R
k?;
=
4sin2(£/2) c2e2 - 4 ^ sin2(e/2)
=
J_ sinc2(g/2) c2 d2 - sinc 2 (^/2)'
where d2 = c2/c2 > 1 and sines = z" 1 sinz, and study K(£) for complex values of £. Obviously, K(£) is a meromorphic function of £. Since c?2 > 0, the point f = 0 is a regular point of K(£). The poles of K(£) are exactly the roots of the equation d2-sinc2-=0.
(4.30)
The roots possesses the following symmetries: if £ is a root, then —£, £ and —£ are also. Therefore, it suffice to describe roots of Eq. (4.30) in the first quadrant.
158
Travelling Waves and Periodic Oscillations in FPU Lattices
Let f = a + ib, a > 0, b > 0. The identity £ 2
2
. £ 2
sinC
=
1 . a ,6 1, a ,6 2aSm2COSh2 + 26cOS2Sinh2 ./a a . , a fe a , b\
+ , {- cos- s i n h - - - s i n - cosh-J
shows that in the case a > 0 and 6 > 0 Eq. (4.30) is equivalent to: either a = 0,
sinh(6/2) —±L-L=d,
(4.31)
or tan(a/2) _ tanh(6/2) = a/2 6/2 '
sinh(6/2) _ d = V2 |cos(o/2)|"
(4.32) J
(4
Since d > 1, the second equation in (4.31) has a unique solution /30. The second equation in (4.32) shows that for all other roots b > (3Q. Now we compute the residues of K(£) at the poles ±i/30. They are ^fia0 respectively and since d = sinc(if30/2), we obtain
a0 = i lim vs> (^ - i/30; )K(0 v ; = i lim £-i/30
,f^~^°]
2sinh/3o-d2/3o We write
K(x) = Ko(x) + K1(x), where ^ /c\ -
°
U J
Nx
«-»^o 2d(d - sinc(£/2))
"
ia
°
-L ^Qo
_
^ - i A ) f + i/flb
2a
o/^o
^2 + /3o2
and
#i(O = £(O-#o(OThe inverse Fourier transform gives
The remainder K\ has no singularities in the strip
S = { £ € C : |Ime|
159
Travelling Waves
From the form of K and KQ it is clear that |^i(0| ^ j - ^ j j .
^S.
(4.33)
Now we estimate Ki(x). Fix f3 G (—/3o,/3o). Applying the Fourier inversion formula and shifting the contour of integration by £ = C +»/?, we have, using (4.33), that 1
r+oo
I e0xKx (ar)1I = - = = / tf i (C + */3)c
for any real x. This implies the required.
•
Now we obtain T h e o r e m 4.10 Assume (h) with c2 > CQ > 0. Let u G X be a travelling wave and r = Au its relative displacement profile. Then for every a € (0,/?o), with /?o from Lemma 4-9, there exists a constant C = Ca such that
\r(t)\ < Ce- a | t | . Moreover, u"(t), t —» ± oo.
u'{t) and u{t) — u(±oo) also decay exponentially as
Proof. From Eq. (4.26) we obtain that r+oo
\r(t)\< /
\K(t-r)\g(r)\r(T)\dr,
J — OO
where
Assumption (h) implies that g(t) —> 0 as t —> ±oo. By Lemma 4.9, r+oo
\r(t)\
e-W-^g(r)\r(T)\dT J-oo
for every f3 £ (0,/?o). Applying Lemma 4.8, we obtain the first assertion of the theorem.
160
Travelling Waves and Periodic Oscillations in FPU Lattices
Exponential decay of u" follows immediately from Eq. (3.3), assumption (h) and the first part of the theorem. Since u' G L2(R), we have that /- + OO
ft
u'(t)=
u"(s)ds = -
u"(s)ds. J-oo J-t Together with exponential decay of u", this gives that u'(t) is an exponentially decaying function. Repeating this argument, we obtain the exponential decay of u(t) — u(±oo). The existence of u(£ ±00) follows immediately from the exponential decay of u'. The proof is complete. D Remark 4.6 If co = 0, the situation simplifies considerably and we obtain the same result as in Theorem 4.10, with /?o = +oo, i. e. \r{t)\ < Cae~aW for every a > 0. We expect that the solutions still decay exponentially if we replace c2, in assumption (h) by a < 0 and suppose that c2 > 0. Remark 4.7 Thus, the solitary travelling waves obtained in Theorems 3.4-3.7, 4.1-4.3 decay exponentially. Remark 4.8 Under assumptions of Theorem 4.10, or Remark 4.6, we have the following identity r+00
c2 [u{+oo) - u(-oo)] = /
U'(u(t + 1) - u(t)) dt
J—00
which is a counterpart of identity from Remark 3.1.
4.6 Travelling waves in chains of oscillators The approach developed in Chapter 3 applies, with minor modifications, in the case of travelling waves in a homogeneous chain of nonlinear oscillators governed by Eq. (1.38). An equivalent equation of motion is (1.39). In this section we present results by S. Bak and the author (unpublished). We consider a homogeneous chain of nonlinear oscillators with the potential
U(r) = -^r2
+ V(r)
161
Travelling Waves
and linear coupling between nearest neighbors (a stands for the coupling constant). The equation of motion becomes q(n) = aAdq(n) + c0 q(n) - V (q(n)),
(4.34)
where q{n) = q(t, n) and (Adq){n) = q{n + 1) + q(n - 1) - 2q{n) is the one-dimensional discrete Laplacian. Making use the travelling wave Ansatz q(t,n) — u{n — ct), we obtain the equation c2u"{t)=a(u(t + l)+u(t-l)-2u{t))+Cou(t)-V'(u(t))
(4.35)
governing travelling waves. This equation has, actually, a variational structure. Let
Hi = {u E HU«) • u{t + 2k) = «(*)} endowed with the norm IMU = \\\u\\h(-k,k) + IKIIz,2(-fc,fc)) . i. e. the Sobolev space of 2/c-periodic functions, and let H1(W) be the standard Sobolev space on K. On these spaces we consider the functional
Mu) = fk [y u'{tf - \ (Au(t))2 + I u(t)2 - V(u(t))] dt (4.36) and
j(u)=
ri
[Tu'(i)2 • \ ( A u ( ° ) 2 + 1 u { t ) 2 ~ v(u( * }) ] dt> (4-37)
respectively, where t+i
/
We assume that
u{s) ds.
162
Travelling Waves and Periodic Oscillations in FPU Lattices
(hi) the function V is C1, V(0) = V'(0) = 0 and V'(r) = o(r) as r -> 0, (h2) there exist r0 £ R and 6 > 2 such that V(r0) > 0 and 9V(r)
r € R.
The functionals Jfc and J are well-defined C 1 functionals on Hi and H (M), respectively. One can verify that their critical points are weak solutions of Eq. (4.35). In fact, they are C 2 solutions (cf. Propositions 3.4 and 3.5). Moreover, by Theorem A.I in the case of functional J any its critical point u € H1(R) satisfies l
u(oo) := lim u(t) = 0. t—»oo
Under the additional assumption that c2 > max(o, 0) and CQ > 0, the quadratic parts of Jfc and J are positive defined, and the functionals themselves possess the mountain pass geometry. Moreover, the functional Jfc satisfies also the Palais-Smale condition (here the compactness of Sobolev embedding plays a crucial role). Thus, applying the mountain pass theorem (Theorem C.I), we obtain the existence of periodic travelling waves in the system considered: Theorem 4.11 Assume (h\) and (/i2)- Suppose that CQ > 0. Then for every k > 1 and c2 > max(a, 0) there exists a nontrivial Ik-periodic solution Wfc 6 Hi of Eq. (4- 35), i. e. a 2k-periodic travelling wave with the speed c. Notice that, in contrast to Section 3.2, the waves considered in this theorem have periodic profile functions. Arguing as in Section 3.3, one can pass to the limit as k —» oo and obtain a nontrivial solitary wave. We have Theorem 4.12 Assume (hi) and(h^) and suppose thatco > 0. Then for every c2 > max(a, 0) there exists a nontrivial solution u G X of Eq. (4.35), i. e. a solitary travelling wave with the speed c. Moreover, the solution u decays exponentially: \u(t)\
Travelling Waves
163
which is easy to derive by means of Fourier transform. Remark 4.9 Using linking instead of mountain pass, one can extend the result of Theorem 4.11 to the case when c2 > 0, and a and c are arbitrary real number. However, it is very likely that nontrivial solitary waves do not exist in this case. 4.7
Comments and open problems
Travelling solitary waves in FPU lattices were observed numerically long time ago (see, e. g. [Eilbeck (1991); Eilbeck and Flesch (1990); Flytzanis et. al (1989); Hochstrasser et. al (1989); Peyrard et. al (1986)]). Another area of work in this field deals with the so-called continuum approximations. Here the equation of motion reduces to a patrial differential equation and the equation governing travelling waves becomes an ordinary differential equation. The results are of perturbative nature and concern the waves with near sonic speed (see [Flytzanis et. al (1989); Friesecke and Pego (1999); Peyrard et. al (1986); Rosenau (1989); Wattis (1993a)]). The first rigorous result on the existence of solitary waves on general FPU lattices was obtained in [Friesecke and Wattis (1994)] (we present this approach in Section 4.1). Note that this approach is based on appropriate constrained minimization problem and relies essentially on the concentration compactness method introduced in [Lions (1984)]. Similar kind of constrained minimization was used before in [Valkering (1978)] to establish the existence of periodic travelling waves for some class of Lennard-Jones type potentials. Certainly, in the last case the problem simplifies considerably and no concentration compactness is needed. The one dimensional Sobolev embedding theorem is enough for that purpose. Later, in [Smets and Willem (1997)], the existence of solitary waves with prescribed speed was considered and the techniques based on the mountain pass theorem without (PS) condition was introduces. Another approach to the same problem was proposed in [Pankov and Pniiger (2000b)]. This approach relies on a combination of periodic approximations and the standard mountain pass theorem and is technically simpler than that of [Smets and Willem (1997)]. In addition, it gives a local convergence of periodic waves to solitary ones. Moreover, employing the Nehari manifold method originated in [Nehari (1960); Nehari (1961)] one obtains an improved con-
164
Travelling Waves and Periodic Oscillations in FPU Lattices
vergence of periodic ground waves to ground solitary waves. The notion of ground wave was also introduced in [Pankov and Pfliiger (2000b)]. Note that the existence of ground solitary waves is not obvious at all. This approach is discussed in Sections 3.1-3.4 in full details. Remark that Theorem 3.3 is new. We conjecture that under assumptions of Theorem 3.3, nontrivial subsonic solitary waves do not exist. Problem 4.1
Prove, or disprove, this conjecture.
In Proposition 3.8 we have shown that under natural assumptions monotone solitary waves are strictly monotone. However, the following problem remains open. Problem 4.2 In the setting of Section 3.2, are monotone periodic waves strictly monotone? The results of Section 3.5 are obtained by the author and nowhere published before. These results deal with the near sonic limit c —» CQ. On the other hand, the following problem is completely open. Problem 4.3 Study the behavior of ground travelling waves, periodic and solitary, in the high speed limit c —> oo. Another result on near sonic waves is obtained in [Priesecke and Pego (1999)]. Those authors employ a technique that relies on perturbation from an appropriate continuous limit, in fact, KdV-equation. Actually, under certain assumptions they show that for c > Co close to CQ there exists a unique, up to translations and adding constants, solitary wave u(t) such that u'(t) > 0 (u'(t) < 0) and u' is an even function, i. e. the wave itself is monotone and symmetric. In this connection we offer the following Problem 4.4 Under assumptions of Section 3.2, is an increasing (resp., decreasing) Ik-periodic travelling wave unique up to translations and adding constants? Same question about monotone solitary waves. What's about uniqueness of ground waves in the setting of Section 3.4? Are all such waves symmetric? We mention also the paper [Iooss (2000)], where bifurcation analysis of travelling waves on FPU type lattices is carried out. As we already mentioned, in Section 4.1 we present the results of [Priesecke and Wattis (1994)], while Theorems 4.3 and 4.4 are borrowed from [Priesecke and Matthies (2002)]. Note also that a result similar to
Travelling Waves
165
Theorem 4.4 is obtained in [Treschev (2004)], where the method of integral equations is employed. Theorem 4.4 deals with high energy limit of solitary waves when the averaged potential energy K —> oo in the case of Lennard-Jones type potentials. Problem 4.5 What can one say about high energy limit for systems like FPU a- and (3-models? Another question in this direction is the following. Let UK be a solitary wave with the average potential energy K and CK its wave speed. Suppose that Ko = 0 in Theorem 4.1. Problem 4.6 Is it true that en —> CQ as K —•> 0, where CQ = y/V"(Q) is the speed of sound?
Numerical simulation [Friesecke and Matthies (2002)] supports that this should be true. The results of Section 4.2 are, actually, modifications of those presented in Sections 3.2 and 3.3. They shows that different types of travelling waves may occur in FPU lattices. If the potential of interaction has more complicated shape, the situation is completely unclear. For instance, suppose that V(r) has at least two local (or global) maxima at r = r\ and r = r^, and a local minimum point r$ in between. Applying Theorem 3.7, we see that, under some mild assumptions on the potential near r = ro, for every c > Co — V'(0) and c close to CQ there exists a solitary wave that looks like rot + c± near ±oo, with some constants c±. However, the following problem is completely open. Problem 4.7 Does there exist a travelling wave that connects the equilibria r = r\ and ri, i- e. looks like r±t + c_ (resp., r^t + C-) at negative infinity and r^t + c+ (resp., r\t + c+) at positive infinity? Such waves can be considered as travelling transition layers. Problem 4.7 is inspired by the results of [Rabinowitz and Stredulinsky (2003)]. The result on exponential decay of solitary waves (Section 4.5) is a simplified and less precise version of that obtained in [Friesecke and Pego (1999)] in which one can find more information. Systems of nonlinear oscillators considered in Section 4.6 seem to be similar to and even simpler than FPU lattices. Nevertheless, not so many
166
Travelling Waves and Periodic Oscillations in FPU Lattices
rigorous results about travelling waves in such systems are known. We mention here the paper [looss and Kirschgassner (2000)] in which bifurcation of such waves is considered. The following problem seems to be interesting. Problem 4.8 Do there exist travelling waves with different behavior at ±oo in the case when the anharmonic potential has many (local) extreme points? A prototypical example is the Frenkel-Kontorova potential. In Chapters 3 and 4 we consider only monoatomic lattices. However, Anatomic FPU lattices also may support travelling waves. Some results on periodic waves in diatomic lattices can be found in [Georgieva et. al (1999); Georgieva et. al (2000)], but the problem is still not studied in details, especially in the case of solitary waves. Also we believe that the variational approach presented here can be extended to the case of twoand iV-dimensional lattices, as well as to the case of lattices with second nearest neighbor interaction. Finally, we would like to point out one of the most challenging problems that concern lattice travelling waves — stability. In the series of papers [Friesecke and Pego (1999); Friesecke and Pego (2002); Friesecke and Pego (2004a); Friesecke and Pego (2004b)] G. Friesecke and R. Pego develop an interesting approach to this problem and obtain some results about stability of FPU solitary waves, in particular, near sonic waves. Nevertheless, the problem is still far to be understood well.
Appendix A
Functional Spaces
We recall here definitions of basic functional spaces just to fix the notation. We consider spaces of real valued functions. However, each such a space has its natural "complex valued" counterpart for which we use the same notation. A.I
Spaces of sequences
Two-sided sequences of real numbers are considered as real valued functions on the set of integers. We write
« = W«)}B6z = M")} for such a sequence. Let p > 1. We denote by V the vector space of all sequences u = {u(n)} such that the norm
||«||(, =
r oc
£
|u(n)|p
-IVP
is finite. By Z°° we denote the vector space of all bounded sequences endowed with the norm ||u||j~ = sup|u(n)|. It is known that lp, 1 < p < oo, is a Banach space. The space V is reflexive if 1 < p < oo, and its dual space, (lp)', is identified with V , P
P' ~ ' 167
168
Travelling Waves and Periodic Oscillations in FPU Lattices
via the bilinear form oo
(u,v) = ^
u(n)v(n).
(A.I)
n=—oo
This means that every v e lp> generates a bounded linear functional fv(u) = (u,v) and the map v t—> fv is a linear isometric isomorphism of lp onto (lp)' provided 1 < p < oo. In fact, one has the following Holder inequality |(U,W)|
(A.2)
for all p € [l,oo] (we set 1' = 00 and oo' = 1). The space I2 is a Hilbert space with respect to the inner product defined by Eq. (A.I). By l0 we denote the vector space of all finite sequences, i. e. sequences u = {u(n)} such that supp u = {n 6 Z : u(n) ^ 0} is a finite set. Obviously, IQ is a dense subspace of lp, 1 < p < 00. Recall that lpClq,
1
00,
and the embedding is continuous. Moreover, the embedding is dense if q < 00.
A.2
Spaces of functions on real line
We denote by C([a,b]) the Banach space of all continuous functions endowed with the standard supremum norm | | / | | c = sup |/(x)|, x€[a,b]
while C((a,b)) stands for the vector space of all continuous functions on the open interval (a, b). The last space is a Prechet space with the topology of uniform convergence on compact subsets of (a, b). For any natural number n we denote by Cn([a, b]) the space of all n times continuously differentiable functions on [a,b]. This is a Banach space
169
Functional Spaces
with the norm
ll/llc- = t\\f{n)\\ck=0
Denote by C n ((a, 6)) the Prechet space of all n times continuously differentiable functions on (a,b). We set
C°°([a,b])=f)C»([a,b)) n>l
and
C°°((a,b))=f}Cn((a,b)). n>l
These are the spaces of all infinitely differentiable functions on [a, b] and (a,b), respectively. We also denote by C£°((a, 6)) the vector space of all compactly supported infinitely differentiable functions on (a, 6). We denote by Cb(]R) the Banach space of all bounded continuous functions on M equipped with the norm
ll/l|cb=sup|/(x)|. Let us also denote by Co(M) a closed subspace of Cb(M) that consist of all functions vanishing at infinity, i. e. lim f(x) = 0.
I—»OO
Let Lv{a, b), 1 < p < oo, be the Banach space of all Lebesgue measurable functions on (a, b) with finite norm
/ r»
V/p
11/11^= ( / \f(x)\Pdx\
.
By L°°(a,b) we denote the Banach space of all essentially bounded measurable functions, endowed with the norm ||/||Lo= =esssup|/(a;)|. x€(a,b)
p
If 1 < p < oo, the space L (a, b) is reflexive and its dual space is identified with the space Lp'(a,b), p"1 + (p'^1 = 1. The duality between these two
170
Travelling Waves and Periodic Oscillations in FPU Lattices
spaces is given by the following bilinear form
(f,9)= I f(x)g(x)dx.
(A.3)
Ja
Recall that
|(/,$)| < H/IUHIfflliP'
(A.4)
(the Holder inequality) for all p G [l,oo]. The space L2(a,b) is a Hilbert space with the inner product (A.3). If (a, b) is a finite interval, then Lp(a,b) cLq(a,b),
1 < q < p < oo,
and the embedding is continuous and dense. Moreover, the embedding C([a,b]) C Lp(a,b) is continuous. It is also dense provided p < oo. Local Lebesgue spaces are denned as follows. The space Lpoc(a, b), 1 < p < oo, consists of all measurable functions / such that the restriction of / to every relatively compact subinterval (a, /3) C (a, b) belongs to Lp(a,f3). Endowed with the topology of //-convergence on relatively compact subintervals, Lpoc(a, b) is a Frechet space. Now let us recall basic facts about Sobolev spaces. Denote by H1(a,b) the space of all functions u 6 L2(a, b) such that the weak derivative, u', of u belongs to L2(a, b). Recall that a function v is called the weak derivative of u if /
u ip'dx = — I vipdx Ja
1
for all ip € CQ°((a, b)). The space H (a, b) is a Hilbert space with the norm
IMlH> = ( l l * + ||t>'||LO 1/2 The corresponding inner product in H1^^) (UI,U2)HI
(A-5)
is
= {u\,u2) + {u'l,u'2).
If (a, b) is a finite interval, H1(a,b) coincides with the closure of C ([a, b}) with respect to the norm defined by (A.5). The closure of Co°((a,6)) with respect to (A.5) is denoted by /^o(a,6). This is a closed linear subspace of Hl{a,b). Recall that Hl(W) = H&(M). 1
171
Functional Spaces
The local space #/oc(M) consists of all functions u £ Lfoc(R) such that for any ip £ CQ°(W) the function
t-.±oo
V
'
1
for every u £ iJ (R). The assertion concerning Cb(lR) is probably less known. Therefore, we sketch its proof. Let u £ H1(R). Since u £ L2(M), there exists a sequence tn —* +oo such that u(tn) —> 0. For x > tn the identity
1 \u(x)\2 - 1 \u(tn)\2 = \ ^ (u\t))'dt =
J\(t)u'(t)dt
yields
/ rx \1/2 / rx \x/2 | u ( i ) | < | t i W | + 2 f / u2(t)dt) I I (u'(t))2dt) and we obtain that lim u(x) = 0. X—>-|-OO
Similarly, lim ulx) = 0. X—>-OO
Remark A.I
Certainly, for any finite interval the embedding
Hl(a,b) C Lp(a,b), is compact.
l
172
Travelling Waves and Periodic Oscillations in FPU Lattices
The definitions of the functional spaces we consider extend naturally to the case of vector valued functions. In particular, for any Banach space E, there are well-defined spaces lp{E), C([a,b],E), Cn([a,b],E), Cb(R,E), Lp(a, b; E) of E-valued functions. One considers also the space Hl(a, b; H) of H-valued functions, where H is a Hilbert space. The statements of Theorem A.I are still valid in this case, with only one exception: the embedding H\a,b;H)cC(\a,b},H) is compact if and only if the space H is finite dimensional. References: [Adams and Fournier (2003); Dunford and Schwartz (1988a); Evans and Gariepy (1992); Lions and Magenes (1972)].
Appendix B
Concentration Compactness
Concentration compactness is a powerful techniques for studying variational problems without compactness introduced in [Lions (1984)]. We sketch here few technical lemmas. Lemma B.I Suppose that {pk} is a sequence of nonnegative functions in Ll(W) such that \\pk\\L1 = A > 0. Then, after passing to a subsequence, one of the three following statements holds true: (i) (concentration) there is a sequence {yk} C K such that for every e > 0 there exists R > 0 with the property that j
rVk+R
Jyk-R
pk{x) dx > X - e;
(u) (vanishing) i-Vk+R
lim sup / k ->°°ymJyk-R
pk{x)dx = 0
for all R > 0; (in) (dichotomy) there exist a € (0, A) and sequences of compactly supported nonnegative functions {p}} },{pk } C L1(R) such that
dist Isupp (pj^^supp (/9j.2)) —> oo as k —> oo,
173
174
Travelling Waves and Periodic Oscillations in FPU Lattices
and
^limj^ll^ =X-a. Proof. See [Lions (1984); Chabrowski (1997)]. L e m m a B.2
Let r > 0. If {wfc} is a bounded sequence in H1(R) and if ry+r
sup /
2
|ufc(i)| dx —> 0, fc —> oo,
y£Rjy-r
then uk -> 0 in LP(R) for
allp>2.
Proof. See [Lions (1984)]. Now we present discrete versions of Lemmas B.I and B.2 (see [Pankov and Zakharchenko (2001)]). L e m m a B . 3 Let {vk} be a sequence of nonnegative elements of I1 such that ||i>A:||ii —> A > 0. Then there exists a subsequence, still denoted by v/., such that one of the three following possibilities holds: (i) (concentration) there is a sequence {m^} of integers such that for every e > 0 there exists an integer r > 0 with the property that mfc+r
J2
v
k{n) > A - e;
n=mk—r
(ii)
(vanishing) lim |KI|z°° = 0 ;
(in) (dichotomy)
{ujj. } C ^o
there exist a £ (0, A) and two sequences {v^ }, suc
h that
dist supp (t^ )iSupp (y^ ) —> oo as k —• oo,
feii^ii.--
Concentration Compactness
and
Lemma B.4
Let {u*.} be a bounded sequence in I2 such that k—»oo p
Then vk -> 0 m / , 2 < p < oo.
175
Appendix C
Critical Point Theory
C.I
Differentiable functionals
Let
(C.I)
Any Frechet derivative is a Gateaux derivative. Therefore, for Frechet derivatives we use the same notation as for Gateaux derivatives. If X is a Hilbert space with the inner product (•, •) and ip has a Gateaux derivative at u 6 X, the gradient, V
178
Travelling Waves and Periodic Oscillations in FPU Lattices
L e L(X, X') atueX
if, for every h, v £ X,
lim - (
lim J L (^( U + A) _ „'(„) _ Lft) = 0. Certainly, any second Frechet derivative is a second Gateaux derivative and we apply the same notation
Mountain pass theorem
Let ip be a C 1 functional on a Banach space X. We say that
inf ¥>(«) >
||u||=r
(C.2)
and ip satisfies the Palais-Smale condition. Let
(c.3)
179
Critical Point Theory
where
r := {7 G C([0,1], X) : 7(0) = 0, 7 (l) = e}.
(C.4)
Then b is a critical value of tp and b > /3.
The following theorem of mountain pass type can be found in [Berestycki et. al (1995)]. Theorem C.2 Under the assumptions of Theorem C.I let P : X —> X be a continuous mapping such that tp(Pu) <
u£X,
P(0) = 0 and P(e) = e. Then there exists a critical point u S PX (the closure of PX) of ip with the critical value b. Remark C.I Typically, in applications the functional (p has a local minimum at 0, a trivial critical point. In this case Theorem C.I gives the existence of a nontrivial critical point. There is also a version of the mountain pass theorem without condition (PS). Theorem C.3 Suppose that all the assumptions of Theorem C.I except condition (PS) are satisfied. Let b be defined by (C.3), (C.4). Then there exists a sequence {un} C X such that lim
n—>oo
and lim
n—>oo
i. e. {un} is a Palais-Smale sequence at the level b. We say that the functional
Linking theorems
Let H be a Hilbert space decomposed into the direct orthogonal sum H = Y®Z. Let g > r > 0 and let z e Z be such that ||z|| = r. Define M:={u
= y + \z:y<=Y, \\u\\ < g, A > 0}
180
Travelling Waves and Periodic Oscillations in FPU Lattices
and Mo := {u = y + Xz :yeY,
\\u\\ =g and A > 0, or ||u|| < g and A = 0},
i. e. Mo is the boundary dM of M. Let i V : = {ueZ
: \\u\\
=r}.
Consider a functional tp on H and suppose that /? := inf
(C.5)
In this situation we say that the functional 9? possesses the linking geometry. Theorem C.4 Suppose that the functional ip of class C1 satisfies the Palais-Smale condition and possesses the linking geometry. Let
6:={inf sup
(C.6)
where r := {7
G
C(M; H) : 7 = id on M 0 }. 2
(C.7)
If dimy < 00, then b is a critical value of
Remark C.2 The additional assumption dimy < 00 in Theorem C.4 implies that the suprema in (C.5) and (C.6) are attained and, hence, can be replaced by maxima. Under further restrictions on the functional tp one can drop the assumption dim Y < 00 in the previous theorem. Theorem C.5 Suppose that the functional
As usual, id denotes the identity map.
Critical Point Theory
181
(Hi) if {un} € H is a sequence such that (p(un) is bounded above and
Appendix D
Difference Calculus
Let I be the vector space of all real, or complex, two-sided sequences, i. e. functions on Z. Difference operations are linear maps from I into itself. The simplest are left and right shifts defined by (5-«)(n):=u(n-l)
(D.I)
(S+u)(n):=u(n + l),
(D.2)
and
respectively. Operations of left and right differences are defined by (d~u) (n) := u(n) - u(n - 1)
(D.3)
(d+u)(n) :=u(n + l)-u(n),
(D.4)
and
respectively. In other words, d+ = I - S~ and d~ =S+
-1.
Here I stands for the identity operation. The following identity is useful
d~ (ad+u) = d+ [(S-o)fl-«], 183
(D.5)
184
Travelling Waves and Periodic Oscillations in FPU Lattices
where a £ I. We mention also the product rules d+(fg)(n) = f(n) (d+g)(n)+g(n + 1) (d+f)(n), d-(fg)(n) = f(n) (d-g)(n)+g(n - 1) {d~f){n). We do not consider general difference operations here. Instead, we restrict ourself to the case of divergence form difference operations of second order. These are operations of the form Ru = d~(ad+u)+bu
(D.6)
where a £ I and 6 G I are given sequences (coefficients). Alternatively, Ru = d+ \(S-a)d-u\ + bu,
(D.7)
as it follows from Eq. (D.5). Abel's summation by parts formula reads
E M) (d+f) d) = 9(n) f(n + 1) ~ 9(m - 1) f{m)
-it(d-g)(j)f(j).
(D.8)
3—m
It implies that the operations d+ and — d~ are formally adjoint, i. e.
E 9U) (d+f) V) = E ( - d~9) (i) M j
i
for all / G Z and g € lo (see Appendix A.I for the definition of l0). As consequence, if the coefficient sequences in Eq. (D.6) are real, then the operation R is formally self-adjoint, i. e.
^2g(j)(Rf)U) = J2(R9)U)fU) j
j
for all / £ I and g G ZoThe operation A:=d+d~ ='0-0+ is called (one dimensional) discrete Laplacian. References: [Atkinson (1964); Gel'fond (1971); Samarskii (2001); Teschl (2000)].
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Index
Abel's summation by parts formula, 186 average kinetic energy, 123 average potential energy, 123
expansion wave, 80 Floquet condition, 16 Floquet transform, 16 FPU o-model, 7 FPU /3-model, 7 Frechet derivative, 179 Frenkel-Kontorova model, 22 functional of class C 1 , 179 functional of class C 2 , 180
band function, 17 Bloch condition, 16 Bloch eigenfunction, 17 Bloch wave, 17 chain of coupled nonlinear oscillators, 18 compression wave, 80 critical point, 180 critical value, 180
gap breather, 76 gap solution, 76 Gateaux derivative, 179 Gelfand transform, 16 geometrically distinct solutions, 63 gradient, 179
diatomic lattice, 8 difference operation, 185 difference operation in divergence form, 186 discrete iy34-equation, 23 discrete breather, 29 discrete Laplacian, 7, 186 discrete nonlinear Klein-Gordon equation, 22 discrete nonlinear wave equation, 22 discrete wave equation, 7 dispersion relation, 17 DNKG equation, 22 DNW equation, 22
harmonic lattice, 12 homoclinic, 74 lattice with impurities, 8 left difference, 185 left shift, 185 Lennard-Jones potential, 8 linking geometry, 182 linking theorems, 181 monoatomic lattice, 8 mountain pass geometry, 181 mountain pass theorem, 180 mountain pass value, 181
energy inequality, 133 193
194
Travelling Waves and Periodic Oscillations in FPU Lattices
multiatomic lattice, 8 multibump solution, 62 Nehari functional, 108 Nehari manifold, 108 Palais-Smale condition, 180 periodic ground wave, 106 periodic travelling wave, 78 plane wave, 18 profile, 77 profile function, 77 profile function for relative displacements, 78 quasimomentum, 16 relative displacement, 5 right difference, 185 right shift, 185 second Frechet derivative, 180 second Gateaux derivative, 179 Sobolev embedding theorem, 173 Sobolev spaces, 172 solitary ground wave, 111 solitary travelling wave, 78 spectral band, 17 spectral gap, 17 speed of sound, 80 speed of wave, 77 subadditivity inequality, 132 subsonic periodic travelling wave, 88 Toda lattice, 8 travelling transition layer, 167 travelling wave, 77 weak derivative, 172
