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Traveling Waves in the Underdamped Frenkel-Kontorova Model.

1. Introduction

The present work concerns the existence of traveling wave solutions for the following underdamped Frenkel-Kontorova model:

[mathematical expression not reproducible], (1)

with periodic boundary condition

[[psi].sub.j+n] (t) = [[psi].sub.j] (t) + 2m[pi], [for all]j [member of] Z, (2)

where the parameters [GAMMA] > 0, K > 0, F > 0, and m [greater than or equal to] 1.

In the last decades, there has been large growth in the study of the existence and stability of traveling wave solutions for lattice systems including Frenkel-Kontorova model (discrete sine-Gordon equations), which arises from many physical systems, such as circular arrays of Josephson junctions, glassy materials, sliding friction, adsorbate layer on the surface of a crystal, ionic conductors, and mechanical interpretation as a model for a ring of pendula coupled by torsional springs (see [1-4]). When [GAMMA] = 0, system (4) is conservative. Baesens and MacKay [5] proved the existence and also global stability of traveling waves. When [GAMMA] > 0, system (4) is dissipative. Under the condition [GAMMA] > 2[square root of (2K + 1)] and [epsilon] = 1, Baesens and MacKay [6] showed that the traveling wave solution is globally stable if and only if (4) and (2) do not have stationary solutions. Levi in [7] pointed out that the local stability of traveling waves can be obtained by the monotonicity method in [8]. Under the condition [GAMMA] > 2 [square root of 2K] and F [greater than or equal to] [F.sub.0] > 1, Qin et al. [9] investigated the stability of single-wave-form for the underdamped Frenkel-Kontorova model (4) by the monotonicity method.

Recently, by using Schauder fixed point theorem, Mirollo and Rosen [10] and Katriel [11] have obtained a series of results about the existence of traveling waves for (4) with periodic boundary condition (2). Katriel [11] proved the following:

(1) Fixing any [GAMMA] > 0 and K > 0 and given any velocity v > 0, there exists a traveling wave solution of (4) and (2) with velocity v for an appropriate F > 0.

(2) For any F >1 there exists a traveling wave solution of (4) and (2).

(3) Assume that n does not divide m. Fixing any [??] > 0 and [??] > 0, for all K sufficiently large there exists a traveling wave solution of (4) and (2) for any F [greater than or equal to] [??] > 0 and [GAMMA] [greater than or equal to] [??] > 0.

(4) Fixing any [??] > 0 and [??] > 0, for all [GAMMA] > 0 sufficiently small there exists a traveling wave solution of (4) and (2) for any F [greater than or equal to] [??] > 0 and 0 < K [less than or equal to] [??].

In the final of Katriel's paper, he gave several open problems. One of them is the following: Is it true that, fixing [GAMMA] > 0 and K > 0, for sufficiently small F > 0 and small applied force, a traveling wave does exist? If n divides m, what is the situation of the existence of traveling waves for (4) with periodic boundary condition (2)? In fact, if n divides m, there appears the "small divisor." Then, the problem is difficult. Levi et al. [12] showed that, for fixing [GAMMA] > 0, (4) possesses a traveling wave only when F exceeds a positive critical value.

In this paper, we will construct a new Nash-Moser iteration to answer the open problem mentioned above. This method has been used in solving the existence of periodic solutions for nonlinear elliptic equations [13], nonlinear wave equations [14-18], and standing waves [19]. Here, we try to use this method to study the existence of traveling wave solutions for dissipative and conservative lattice systems.

Instead of looking for solutions of (1) in a shrinking neighborhood of zero, it is a convenient device to perform the rescaling

[psi] [right arrow] [epsilon][psi], [epsilon] > 0 (3)

having

[mathematical expression not reproducible]. (4)

To overcome the "small divisor" problem, we need the following nonresonance conditions:

[mathematical expression not reproducible], (5)

where [OMEGA] [subset] R is a bounded region.

It is shown in [20] that if, for some l > 0,

[absolute value of ([v.sup.(l)] ([omega]))] [greater than or equal to] d > 0 on [omega] [member of] [OMEGA], (6)

then the Lebesgue measure

[mathematical expression not reproducible]. (7)

Now, we state our main result.

Theorem 1. Under the assumption (NR1), fixing any K > 0 and sufficient small F > 0, there exist [[GAMMA].sub.0] > 0, [[epsilon].sub.0] > 0, and 0 < [gamma] < 1 [less than or equal to] [tau], such that, for any [zeta] := v[GAMMA] [member of] [0, [[GAMMA].sub.0]], [epsilon] [member of] [0, [[epsilon].sub.0]], and [omega] [member of] [O.sub.[gamma],[tau]], (4) with periodic boundary condition (2) possesses a unique traveling wave solution u([theta]; [zeta]) + [theta], where [theta] [member of] T := R/2[pi].

When [GAMMA] = 0, (4) is

[mathematical expression not reproducible]. (8)

The corresponding Hamiltonian of (8) is

[mathematical expression not reproducible], (9)

where the nearest-neighbor coupling potential is

[mathematical expression not reproducible]. (10)

We have the following result about the existence of traveling waves for (8).

Theorem 2. Under the assumption (NR2), fixing any K > 0, there exist [F.sub.*] > 0, [[epsilon].sub.0] > 0, and 0 < [gamma] < 1 [less than or equal to] [tau], such that, for any [epsilon] [member of] [0, [[epsilon].sub.0]] and w [member of] [[bar.O].sub.[gamma],[tau]], (4) with periodic boundary condition (2) possesses a unique traveling wave solution u([theta]; [zeta]) + [theta], where [theta] [member of] T := R/2[pi].

This paper is organized as follows. In Section 2, we first establish a Nash-Moser theorem for the case of [GAMMA] > 0. Then, we apply this result to prove our main results. The case of [GAMMA] = 0 is also considered.

2. Proof of the Main Results

2.1. The Case of [GAMMA] > 0. In numerical simulations or experimental works on (4) with periodic boundary condition (2), it is observed that solutions often converge to a traveling wave

[[psi].sub.j] (t) = [phi](t + j[m/n]T), (11)

where the waveform [phi] R [right arrow] R is a function satisfying

[phi](t + T) = [phi](t) + 2[pi], [for all]t [member of] R. (12)

[phi] is a waveform if and only if it satisfies (12) and

[mathematical expression not reproducible]. (13)

Hence, as in [11], we investigate the traveling wave of the type

[phi](t) = u (vt) + vt, (14)

where the wave velocity v = 2[pi]/T = 2[pi][omega] and u satisfies

u([theta] + 2[pi]) = u ([theta]), [for all][theta] [member of] R. (15)

Inserting (14) into (13), we get

[mathematical expression not reproducible]. (16)

Write

sin ([epsilon]d + [epsilon]u ([theta])) = sin ([epsilon]d) + 2[epsilon] cos ([epsilon][theta]) u + g (u). (17)

We consider the following space:

[mathematical expression not reproducible], (18)

where [u.sub.k] denotes the k the Fourier coefficient.

Obviously, for a nested family of Banach spaces {[X.sub.[sigma]] : [sigma] [greater than or equal to] 0}, there holds

[mathematical expression not reproducible]. (19)

For [sigma] [greater than or equal to] 0, the space [X.sub.[sigma]] is Banach algebra with respect to multiplication of functions; that is, if [u.sub.1], [u.sub.2] [member of] [X.sub.[sigma]], then [u.sub.1][u.sub.2] [member of] [X.sub.[sigma]] and there exists a positive constant C, such that

[mathematical expression not reproducible]. (20)

It is obviously that each function in [X.sub.[sigma]] has a bounded analytic extension in the complex multistrip [absolute value of Im [theta]] < [sigma], where [theta] [member of] C. By the definition of the space [X.sub.[sigma]], the following inequality holds:

[mathematical expression not reproducible]. (21)

For uniqueness, we assume that u satisfies

[mathematical expression not reproducible]. (22)

Now we define a function space with zero average by

[mathematical expression not reproducible], (23)

as the closed subspace of [X.sub.[sigma]].

Let [??] > [sigma] > 0. Then, we define

[Y.sub.[sigma]] denotes the set of functions u [member of] [C.sup.[infinity]](T x [0, [[GAMMA].sub.0]) such that, for all [mathematical expression not reproducible].

[W.sub.[sigma]] denotes the set [Y.sub.[sigma]] x [C.sup.[infinity]]([0, [[GAMMA].sub.0]).

Denote operator A :='. Then (16) can be written as

[mathematical expression not reproducible]. (24)

We define an operator L : [X.sub.[sigma]] [right arrow] [X.sub.[sigma]] by

[mathematical expression not reproducible]. (25)

Then, (24) can be written as

[mathematical expression not reproducible]. (26)

We have the following properties about operator L.

Lemma 3. Fix the following [GAMMA] > 0 and K > 0. The "diagonal" operator L (on Fourier spaces) satisfies the following:

[mathematical expression not reproducible], (27)

where

[mathematical expression not reproducible]. (28)

(2) Let 0 [less than or equal to] [??] < [sigma] and v [member of] [O.sub.[gamma],[tau]]. The operator L is bounded and invertible, and [L.sup.-1] maps [X.sup.0.sub.a] onto [X.sup.0.sub.[??]],

[mathematical expression not reproducible]. (29)

If n divides m, that is, m/n [member of] Z, then,

[mathematical expression not reproducible], (30)

where

[mathematical expression not reproducible]. (31)

If n does not divide m, then

[mathematical expression not reproducible], (32)

where

[mathematical expression not reproducible]. (33)

Furthermore,

[mathematical expression not reproducible]. (34)

Proof. By the definition of operator L, we can easily get (1) and (29). Now we prove (30), (32), and (34).

If n divides m, that is, m/n [member of] Z and 2K(cos(2mk[pi]/n) - 1) = 0, then we have

[mathematical expression not reproducible]. (35)

Since

[mathematical expression not reproducible], (36)

and [sup.sub.x>0]([x.sup.a][e.sup.-x]) = [(a/e).sup.a], [for all]a [greater than or equal to] 0, we obtain

[mathematical expression not reproducible]. (37)

If n does not divide m, that is, 2K(cos(2mk[pi]/n) - 1) [not equal to] 0, then operator L is invertible and no "small divisor" appears. We have

[mathematical expression not reproducible]. (38)

By [mathematical expression not reproducible], and (36), we obtain

[mathematical expression not reproducible]. (39)

This completes the proof.

Remark 4. By the estimate (39), we have that [GAMMA] [not equal to] 0 in our Nash-Moser algorithm. However, when [omega] = 0, we have v = 0 and [zeta] := v[GAMMA] = 0.

Our method of finding traveling waves comes from the idea of Newton scheme, which is an approximation method. If we choose first step ([u.sub.0], [[zeta].sup.0]) suitable, by finding a "quadratically better approximation," we can move forward a single step to our target. Hence, the critical point is to construct "second step," that is, to get ([u.sub.1], [[zeta].sub.1]); then, the method of making "next step" is the same. Finally, our solution of (26) can be written as

[mathematical expression not reproducible]. (40)

For convenience, we define

[mathematical expression not reproducible], (41)

[mathematical expression not reproducible]. (42)

Now, we construct the "first step approximation" to find ([[mu].sub.1], [[zeta].sub.1]).

Lemma 5. Fix any K > 0, F > 0, and [[GAMMA].sub.0] > 0. Assume that [omega] [member of] [O.sub.[gamma],[tau]]. Then, for any [zeta] [member of] [0, [[GAMMA].sub.0]], one obtains the "first step approximation":

[mathematical expression not reproducible]. (43)

Proof. We define

R := [epsilon] sin ([theta] + [u.sub.0] + [u.sub.1]) - [epsilon] sin ([theta] + [u.sub.0]) + (i[GAMMA] - 1)[u.sub.1]. (44)

Then we have

[mathematical expression not reproducible]. (45)

Based on our approximation method, we need to solve the following equation:

E + L[u.sub.1] + [[zeta].sub.1] = 0. (46)

If n divides m, that is, m/n [member of] Z, operator L is not invertible, the "small divisor" appears. Therefore, the removing of a "small set" (in Lebesgue measure sense) is needed; that is, we require [omega] [member of] [O.sub.[gamma],[tau]]. Then, we construct

[mathematical expression not reproducible]. (47)

If n dose not divide m, operator L is invertible. Then we can also construct ([u.sub.1], [[zeta].sub.1]) as the same form.

It is easy to verify that ([u.sub.1], [[zeta].sub.1]) is the solution of (46) and satisfies condition (22). This completes the proof.

Remark 6. In fact, to obtain sth step approximation ([u.sub.s], [[zeta].sub.s]) (s [greater than or equal to] 1), we need to solve

[E.sub.s] + L[u.sub.s] + [[zeta].sub.s] = 0, (48)

where

[E.sub.s] = F ([J.sup.s] ([u.sub.0], [[zeta].sub.0])). (49)

By the method in Lemma 3, we can construct sth step solution for (48) as

[mathematical expression not reproducible]. (50)

Now, in order to prove the convergence of our algorithm, we need the following KAM estimates.

Lemma 7 (KAM estimates). Assume that ([u.sub.0], [[zeta].sub.0]) [member of] [W.sub.[sigma]]. Then there exist [beta] := [beta]([tau]) > 1 and [C.sub.0] := [C.sub.0]([[GAMMA].sub.0], [tau]) > 1 such that, for any 0 < [alpha] < [sigma] and any [GAMMA] [member of] [0, [[GAMMA].sub.0]], the following estimates hold:

[mathematical expression not reproducible]. (51)

Proof. We first estimate the case that n divides m. It follows from (30) that

[mathematical expression not reproducible]. (52)

By the definition of ([u.sub.1], [[zeta].sub.1]) and (52), we have

[mathematical expression not reproducible], (53)

[mathematical expression not reproducible]. (54)

By (53) and the definition [E.sub.1] in (42), we get

[mathematical expression not reproducible]. (55)

By (52), (53), and (55), there exist [beta] := [beta]([tau]) > 1 and [C.sub.0] := [C.sub.0]([[GAMMA].sub.0], [tau]) > 1 such that

[mathematical expression not reproducible]. (56)

For the case of n not dividing m, we can also get the estimate (51). The method is the same. So we omit it. This completes the proof.

Now, we will give a sufficient condition on the convergence of our algorithm. For s [greater than or equal to] 0 and 0 < [bar.[sigma]] < [sigma] < [??], we set

[mathematical expression not reproducible]. (57)

Then, we have the following result about the convergence of Nash-Moser algorithm.

Lemma 8. Assume that [C.sub.4] [greater than or equal to] [C.sub.0][4.sup.[beta]][([sigma] - [bar.[sigma]]).sup.- [beta]] and [C.sub.4][[parallel]E[parallel].sub.[sigma]] [less than or equal to] i < 1. Then, (u, [zeta]) [member of] [W.sub.[bar.[sigma]]] defined in (40) is a solution of (26); that is, F(u; [zeta]) = 0, [for all][zeta] [member of] [0, [[GAMMA].sub.0]].

Proof. We claim that, for s [greater than or equal to] 1,

[mathematical expression not reproducible], (58)

In fact, if (58) holds, then by the decay of [mathematical expression not reproducible], we obtain that

[mathematical expression not reproducible], (59)

In the following, we will prove (58) by induction. Firstly, we check (58) for the case of s = 1. Let [[alpha].sub.1] := [alpha] and [[sigma].sub.1] := [sigma] - [alpha]. By (51), we have

[mathematical expression not reproducible], (60)

which implies that [mathematical expression not reproducible], so, (58) holds for s = 1.

Let s' [greater than or equal to] 1. Assume that (58) holds true for 1 [less than or equal to] s [less than or equal to] s'. Now we will prove that it also holds for s = s' + 1. Let [alpha] := [[alpha].sub.s+1] and [sigma] := [[sigma].sub.s]. Note that C[[alpha].sup.-[beta].sub.s+1] = (C/[2.sup.[beta]])[2.sup.s[beta]]. By (51), we get

[mathematical expression not reproducible], (61)

which shows that [mathematical expression not reproducible]. Hence, our claim holds. This completes the proof.

Remark 9. In this lemma, we do not care whether n divides m or not. Because the convergence of Nash-Moser algorithm is the same.

Lemma 10 (uniqueness). Assume that ([bar.u], [bar.[zeta]]) and ([mathematical expression not reproducible]) are solutions of (26) in the domain [mathematical expression not reproducible]. Then, [mathematical expression not reproducible]; that is, the solution of (26) is unique.

Proof. Let u := [bar.u] - [??] and [zeta] := [bar.[zeta]] - [??]. Then

[mathematical expression not reproducible], (62)

where

[mathematical expression not reproducible]. (63)

Note that F([bar.u], [bar.[zeta]]) = 0. Therefore, by (62), we have

Lu + [zeta] + [bar.R] = 0. (64)

It follows from condition (22) that

[zeta] = (i[GAMMA] - 1)<[L.sup.-1] [bar.R]>, (65)

u := -[L.sup.-1][bar.R] + <[L.sup.-1][bar.R]>. (66)

Note that [[parallel][bar.R][parallel].sub.[sigma]] [less than or equal to] [C.sub.6][[parallel]u[parallel].sub.[sigma]], so we have that

[absolute value of [zeta]] [less than or equal to] [C.sub.7][[alpha].sup.-[beta]] [[parallel]u[parallel].sub.[sigma]]. (67)

Next we will estimate (66). By (63) and the similar estimates in (53), for 0 < [alpha] < [sigma] [less than or equal to] [bar.[sigma]], we have

[mathematical expression not reproducible]. (68)

If we take as [[sigma].sub.s] := [bar.[sigma]]/[2.sup.s] and [[alpha].sub.s] := [bar.[sigma]]/[2.sup.s+1] (s [greater than or equal to] 0). Then, it follows from (68) that

[mathematical expression not reproducible], (69)

where [mathematical expression not reproducible] and b := [2.sup.[beta][tau]].

By (69), we have

[mathematical expression not reproducible], (70)

which shows that

[mathematical expression not reproducible]. (71)

There exists [chi] := [chi](s) [member of] [C.sup.1](1, [infinity]) such that

[mathematical expression not reproducible]. (72)

Then, it follows from (71) that

[mathematical expression not reproducible]. (73)

Note our assumption [mathematical expression not reproducible]. Therefore, by (73), we obtain

[[parallel]u[parallel].sub.0] = 0, (74)

which implies that

u = 0, that is., [bar.u] = [??]. (75)

This together with (65) means the uniqueness of solutions for (26). This completes the proof.

The following result can be seen as a Nash-Moser theorem for dissipative lattice systems.

Theorem 11. Let 0 < [gamma] < 1 [less than or equal to] [tau], 0 < [bar.[sigma]] < [sigma] [less than or equal to] 1, [omega] [member of] [O.sub.[gamma],[tau]], and [zeta] [member of] [0, [[GAMMA].sub.0]] for some [[GAMMA].sub.0] > 0. Assume that "initial approximate solution" ([u.sub.0], [[zeta].sub.0]) [member of] [C.sup.[infinity]](T x [0, [[GAMMA].sub.0]]) x [C.sup.[infinity]][0, [[GAMMA].sub.0]] and [mathematical expression not reproducible], and [??] > 1. Then, (26) possesses solutions (u, [zeta]) [member of] [C.sup.[infinity]](T x [0, [[GAMMA].sub.0]]) x [C.sup.[infinity]]([0, [[GAMMA].sub.0]]) and u [member of] [X.sup.0.sub.[bar.[sigma]]]. Moreover, if ([bar.u], [bar.[zeta]]) [member of] [C.sup.[infinity]](T x [0, [[GAMMA].sub.0]]) x [C.sup.[infinity]]([0, [[GAMMA].sub.0]]) is also the solution of (26) and satisfies [mathematical expression not reproducible in ascii], then, (u, [zeta]) = ([bar.u], [bar.[zeta]]); that is, the solution of (26) is unique.

Proof. This result is the conclusion of Lemmas 5-14. Let [beta]([tau]) and [C.sub.0] be defined in Lemma 7. We choose [??] such that [mathematical expression not reproducible]. Then, by our assumption, Lemmas 8 and 10, we can get the existence and uniqueness of solutions of (26). This completes the proof.

Remark 12. In fact, in this abstract result, we do not need any assumption on [epsilon] > 0 in the case of [epsilon] = 1. Then, the problem of finding traveling wave solutions for (4) with periodic boundary condition (2) is another open problem in [11]. By Theorem 11, we can see that, for fixing K > 0, [GAMMA] > 0 and sufficient small F >0, there is a unique traveling wave solution for (4). However, it is difficult to find the initial approximation solution ([u.sub.0], [[zeta].sub.0]) which must make the error function E satisfying [??][[parallel]E[parallel].sub.[sigma]] [less than or equal to] 1 ([??] > 1).

Now, we will use Theorem 11 to prove our main result.

Proof of Theorem 1. Let 0 < [bar.[sigma]] < [sigma], [for all][bar.[sigma]], [sigma] [member of] [R.sup.+]. We choose the initial approximation solution

([u.sub.0], [[zeta].sub.0]) = (0, 0). (76)

Let [[epsilon].sub.0] [less than or equal to] 1/[??] - F and [epsilon] [member of] [0, [[epsilon].sub.0]] ([??] > 1). Then, the error function E defined in (42) is given by

[mathematical expression not reproducible]. (77)

Here, we require that F > 0 be sufficiently small so that 1/[??] - F > 0.

It follows from Theorem 11 that our result holds. This completes the proof.

Remark 13. By the proof of Theorem 1, we can see that our result also holds for the case of F = 0. It suffices to take [[epsilon].sub.0] [less than or equal to] 1/[??] and [epsilon] [member of] [0, [[epsilon].sub.0]].

2.2. The Case of [GAMMA] = 0. We now focus on the proof of Theorem 2 by the same method.

By strong monotonicity arguments, Baesens and MacKay have obtained the existence and stability of traveling waves for (8) with periodic boundary condition (2). Here, we will use Nash-Moser iteration to study the existence and uniqueness of traveling wave solutions for (8) with periodic boundary condition (2).

Note that a waveform [phi] satisfies the following equation:

[mathematical expression not reproducible]. (78)

Hence, as in [11], we investigate the traveling wave of the type

[phi](t) = u (vt) + vt, (79)

where the wave velocity v = 2[pi]/T = 2[pi][omega] and u satisfies

u([theta] + 2[pi]) = u ([theta]), [for all][theta] [member of] R. (80)

Inserting (79) into (78), we get

[mathematical expression not reproducible]. (81)

Define the operator M : [X.sub.[sigma]] [right arrow] [X.sub.[sigma]] as

[mathematical expression not reproducible]. (82)

Then, (81) can be written as

G (u, F) = Mu + [epsilon] sin ([theta] + u([theta])) - u([theta]) - F = 0. (83)

We will also use the idea of Newton scheme to obtain the solution of (83). Firstly, we need to give some notations:

[mathematical expression not reproducible], (84)

where ([u.sub.s], [F.sub.s]) denotes the sth step approximation solution.

Next, the spectrum analysis of operator M is essential.

Lemma 14. Fix K > 0. The "diagonal" operator M (on Fourier spaces) satisfies the following:

(1) [for all]u [member of] [X.sup.0.sub.[sigma]],

[mathematical expression not reproducible], (85)

where

[mathematical expression not reproducible]. (86)

(2) Let 0 [less than or equal to] [??] < [sigma] and v [member of] [[bar.O].sub.[gamma],[tau]]. The operator M is bounded and invertible, and [M.sup.-1] maps [X.sup.0.sub.[sigma]] onto [X.sup.0.sub.[??]],

[mathematical expression not reproducible]. (87)

If n divides m, that is, m/n [member of] Z, then

[mathematical expression not reproducible], (88)

where

[mathematical expression not reproducible]. (89)

If n does not divide m, then

[mathematical expression not reproducible], (90)

where

[mathematical expression not reproducible]. (91)

Furthermore,

[mathematical expression not reproducible]. (92)

Proof. The idea of this proof is similar to the proof of Lemma 3. Here, we only need to verify (88), (90), and (92).

Note that [sup.sub.x>0] ([x.sup.a][e.sup.-x]) = [(a/e).sup.a], [for all]a [greater than or equal to] 0, and

[mathematical expression not reproducible]. (93)

Hence, in the case that n divides m,

[mathematical expression not reproducible]; (94)

in the case that n does not divide m,

[mathematical expression not reproducible], (95)

where

[mathematical expression not reproducible]. (96)

This completes the proof.

Lemma 15. Fix any K > 0 and [F.sub.0] > 0. Assume that [omega] [member of] [[bar.O].sub.[gamma],[tau]]. Then, for any F [member of] [0, [F.sub.0]], one obtains the "first step approximation":

[mathematical expression not reproducible]. (97)

Proof. Define

[??] := [epsilon] sin ([theta] + [u.sub.0] + [u.sub.1]) - [epsilon] sin ([theta] + [u.sub.0]) + [u.sub.1] ([theta]). (98)

Then we have

[mathematical expression not reproducible]. (99)

For getting ([u.sub.1], [F.sub.1]), we need to solve the following equation:

[bar.E] + M[u.sub.1] + [F.sub.1] = 0. (100)

By condition (22), we can construct "the first approximation solution":

[mathematical expression not reproducible]. (101)

This completes the proof.

Remark 16. In fact, we can construct the sth step approximation solution as

[mathematical expression not reproducible], (102)

by solving the following equation:

[[bar.E].sub.s] + M[u.sub.s] + [F.sub.s] = 0. (103)

Lemma 17 (KAM estimates). Assume that ([u.sub.0], [F.sub.0]) [member of] [W.sub.[sigma]]. Then, there exist [bar.[beta]] := [bar.[beta]] > 1 and [C.sub.9] := [C.sub.9]([F.sub.*], [tau], [gamma]) > 1 such that, for any 0 < [alpha] < [sigma] and any F [member of] [0, [F.sub.*]], the following estimates hold:

[mathematical expression not reproducible]. (104)

Proof. The proof is the same as Lemma 7, so we omitted it.

Lemma 18. Assume that [mathematical expression not reproducible]. Then, [mathematical expression not reproducible] is a solution of (83); that is, F(u; F) = 0, [for all]F [member of] [0, [F.sub.*]]. Furthermore, in the domain

[mathematical expression not reproducible], (105)

(83) admits a unique solution (u, F).

Proof. This proof is also similar to Lemmas 8 and 10, so we omitted it.

Based on Lemma 18, we show the following Nash-Moser theorem for the conservative lattice systems.

Theorem 19. Let 0 < [gamma] < 1 < [tau], 0 < [bar.[sigma]] < [sigma] < 1, [omega] [member of] [O.sub.[gamma],[tau]], and F [member of] [0, [F.sub.*]] for some [F.sub.*] > 0. Assume that "initial approximate solution" [mathematical expression not reproducible] and [mathematical expression not reproducible], and [bar.[??]] > 1. Then, (83) possesses solutions (u, F) [member of] [C.sup.[infinity]](T x [0, [F.sub.*]])x[C.sup.[infinity]]([0, [F.sub.*]]) and u [member of] [X.sup.0.sub.[bar.[sigma]]]. Moreover, the solution of (83) is unique in the domain [mathematical expression not reproducible].

Proof of Theorem 2. Let 0 < [bar.[sigma]] < [sigma], [for all][bar.[sigma]], [sigma] [member of] [R.sup.+]. We choose the initial approximation solution

([u.sub.0], [F.sub.0]) = (0, 0). (106)

Let [[epsilon].sub.0] [less than or equal to] 1/[bar.[??]] and [epsilon] [member of] [0, [[epsilon].sub.0]] ([bar.[??] > 1). Then, the error function [bar.E] defined in (84) is given by

[mathematical expression not reproducible]. (107)

It follows from Theorem 19 that our result holds. This completes the proof.

https://doi.org/10.1155/2018/7081804

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

The authors express their thanks to Dr. Weiping Yan for his suggestion and giving this problem to them.

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Hengyan Li (iD) (1) and Shaowei Liu (iD) (2)

(1) School of Mathematics and Information Sciences, North China University of Water Resources and Electric Power, Zhengzhou 450011, China

(2) School of Energy Science and Engineering, Henan Polytechnic University, Jiaozuo 454000, China

Correspondence should be addressed to Shaowei Liu; lsw770320@hpu.edu.cn

Received 11 January 2018; Accepted 13 March 2018; Published 30 April 2018

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Title Annotation:Research Article
Author:Li, Hengyan; Liu, Shaowei
Publication:Discrete Dynamics in Nature and Society
Date:Jan 1, 2018
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