Traveling Waves in the Underdamped Frenkel – Kontorova Model

This paper studies a damped Frenkel–Kontorova model with periodic boundary condition. By using Nash–Moser iteration scheme, we prove that such model has a family of smooth traveling wave solutions.


Introduction
The present work concerns the existence of traveling wave solutions for the following underdamped Frenkel-Kontorova model: with periodic boundary condition where the parameters Γ > 0,  > 0,  > 0, and  ≥ 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][2][3][4]).When Γ = 0, system (4) is conservative.Baesens and MacKay [5] proved the existence and also global stability of traveling waves.When Γ > 0, system (4) is dissipative.Under the condition Γ > 2 √ 2 + 1 and  = 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 Γ > 2 √ 2 and  ≥  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 Γ > 0 and  > 0 and given any velocity V > 0, there exists a traveling wave solution of ( 4) and (2) with velocity V for an appropriate  > 0.
In the final of Katriel's paper, he gave several open problems.One of them is the following: Is it true that, fixing Γ > 0 and  > 0, for sufficiently small  > 0 and small applied force, a traveling wave does exist?If  divides , what is the situation of the existence of traveling waves for (4) with periodic boundary condition (2)?In fact, if  divides , there appears the "small divisor."Then, the problem is difficult.
Levi et al. [12] showed that, for fixing Γ > 0, (4) possesses a traveling wave only when  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][15][16][17][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 having To overcome the "small divisor" problem, we need the following nonresonance conditions: where Ω ⊂ R is a bounded region.
It is shown in [20] that if, for some  > 0, then the Lebesgue measure Now, we state our main result.
When Γ = 0, (4) is The corresponding Hamiltonian of ( 8) is where the nearest-neighbor coupling potential is We have the following result about the existence of traveling waves for (8).

Proof of the Main Results
2.1.The Case of Γ > 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 where the waveform  : R → R is a function satisfying is a waveform if and only if it satisfies (12) and Hence, as in [11], we investigate the traveling wave of the type where the wave velocity V = 2/ = 2 and  satisfies Inserting ( 14) into (13), we get Write sin ( +  ()) = sin () + 2 cos ()  +  () .(17) We consider the following space: where   denotes the the Fourier coefficient.
Obviously, for a nested family of Banach spaces {  :  ≥ 0}, there holds For  ≥ 0, the space   is Banach algebra with respect to multiplication of functions; that is, if  1 , For uniqueness, we assume that  satisfies Now we define a function space with zero average by as the closed subspace of   .Let σ >  > 0.Then, we define Denote operator A fl  .Then ( 16) can be written as We define an operator L :   →   by Then, (24) can be written as We have the following properties about operator L.
Our method of finding traveling waves comes from the idea of Newton scheme, which is an approximation method.If we choose first step ( 0 ,  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 ( 1 ,  1 ); then, the method of making "next step" is the same.Finally, our solution of (26) can be written as For convenience, we define Now, we construct the "first step approximation" to find ( 1 ,  1 ).Lemma 5. Fix any  > 0,  > 0, and Γ 0 > 0. Assume that  ∈ O , .Then, for any  ∈ [0, Γ 0 ], one obtains the "first step approximation": Proof.We define Then we have Based on our approximation method, we need to solve the following equation: If  divides , that is, / ∈ 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  ∈ O , .Then, we construct If  dose not divide , operator L is invertible.Then we can also construct ( 1 ,  1 ) as the same form.
It is easy to verify that ( 1 ,  1 ) is the solution of (46) and satisfies condition (22).This completes the proof.Remark 6.In fact, to obtain th step approximation (  ,   ) ( ≥ 1), we need to solve where By the method in Lemma 3, we can construct th step solution for (48) as Now, in order to prove the convergence of our algorithm, we need the following KAM estimates.
Lemma 7 (KAM estimates).Assume that ( 0 ,  0 ) ∈ W  .Then there exist  fl () > 1 and  0 fl  0 (Γ 0 , ) > 1 such that, for any 0 <  <  and any Γ ∈ [0, Γ 0 ], the following estimates hold: By the definition of ( 1 ,  1 ) and ( 52), we have For the case of  not dividing , 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  ≥ 0 and 0 <  <  < σ, we set Then, we have the following result about the convergence of Nash-Moser algorithm.
The following result can be seen as a Nash-Moser theorem for dissipative lattice systems.
Remark 12.In fact, in this abstract result, we do not need any assumption on  > 0 in the case of  = 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  > 0, Γ > 0 and sufficient small  > 0, there is a unique traveling wave solution for (4).However, it is difficult to find the initial approximation solution ( 0 ,  0 ) which must make the error function  satisfying ‖‖  ≤ 1 ( > 1).Now, we will use Theorem 11 to prove our main result.
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  = 0.It suffices to take  0 ≤ 1/ and  ∈ [0,  0 ].

2.2.
The Case of Γ = 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  satisfies the following equation: Hence, as in [11], we investigate the traveling wave of the type where the wave velocity V = 2/ = 2 and  satisfies Inserting ( 79) into (78), we get We will also use the idea of Newton scheme to obtain the solution of (83).Firstly, we need to give some notations: where (  ,   ) denotes the th step approximation solution.
Next, the spectrum analysis of operator M is essential.
Then we have For getting ( 1 ,  1 ), we need to solve the following equation: By condition (22), we can construct "the first approximation solution": 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.
It follows from Theorem 19 that our result holds.This completes the proof.