Abstract and Applied Analysis

Volume 2014 (2014), Article ID 436529, 7 pages

http://dx.doi.org/10.1155/2014/436529

Research Article

Homoclinic Solutions of a Class of Nonperiodic Discrete Nonlinear Systems in Infinite Higher Dimensional Lattices

Genghong Lin1,2 and Zhan Zhou1,2

¹School of Mathematics and Information Science, Guangzhou University, Guangzhou, Guangdong 510006, China

²Key Laboratory of Mathematics and Interdisciplinary Sciences of Guangdong Higher Education Institutes, Guangzhou University, Guangzhou, Guangdong 510006, China

Received 26 February 2014; Accepted 12 June 2014; Published 7 July 2014

Academic Editor: Chuangxia Huang

Copyright © 2014 Genghong Lin and Zhan Zhou. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

By using critical point theory, we obtain a new sufficient condition on the existence of homoclinic solutions of a class of nonperiodic discrete nonlinear systems in infinite lattices. The classical Ambrosetti-Rabinowitz superlinear condition is improved by a general superlinear one. Some results in the literature are improved.

1. Introduction

Assume that is a positive integer. Consider the following difference equation in infinite higher dimensional lattices:

where is continuous in , , , is a positive real valued sequence, , and is a Jacobi operator [1] given by

here, and are real valued bounded sequences.

Assume that for ; then is a solution of (1), which is called the trivial solution. As usual, we say that , a solution of (1), is homoclinic (to 0) if

where is the length of multi-index . In addition, if , then is called a nontrivial homoclinic solution. We are interested in the existence of the nontrivial homoclinic solutions for (1). This problem appears when we seek the discrete solitons of nonperiodic discrete nonlinear Schrödinger (DNLS) equation

where and

is the discrete Laplacian in spatial dimension. Typical representatives of power nonlinearities are

Primarily, we are interested in spatially localized, or solitary, standing waves. Such waves are often called breathers or gap solitons. The origin of the last name is that typically such solutions do exist for frequencies in gaps of linear spectrum. Considering (4), we suppose that the nonlinearity is gauge invariant; that is,

and, in addition, for for .

Making use of the standing wave ansatz,

where is a real valued sequence and is the temporal frequency. Then (4) becomes

and (3) holds. This is an equation of the form (1) with and .

When has the form of (6), the homoclinic solutions of (9) were obtained by Karachalios in [2] by assuming that , , for some . We note that implies that . Moreover, (6) satisfies the classical Ambrosetti-Rabinowitz superlinear condition [3], and is nondecreasing with respect to , both of which played important roles in the existence of homoclinic solutions of in [2].

The aim of this paper is to improve both the monotone condition of and the classical Ambrosetti-Rabinowitz superlinear condition by general ones; see Remarks 8 and 9 for details. Moreover, in this paper, we only need . Particularly, our results improved the results in [2]; see Remarks 3 and 7 for details.

In the past years, there has been large growth in the study of DNLS equation, which is a nonlinear lattice system that appears in many areas of physics. Discrete solitons which exist in DNLS systems, that is, solitary waves and localized structures in spatially discrete media, are also of particular interest in their own right. Among these, one can mention photorefractive media [4], biomolecular chains [5], and Bose-Einstein condensates [6]. The experimental observations of discrete solitons in nonlinear lattice systems have been reported [7–11]. To mention that, many authors have studied the existence of discrete solitons of the DNLS equations [12–17]. The fruitful methods include centre manifold reduction [16], variational methods [12, 14], the principle of anticontinuity [13, 17], and the Nehari manifold approach [18]. However, most of the existing literature is devoted to the DNLS equations with constant coefficients or periodic coefficients. Results on such DNLS equations have been summarized in [19–23]. And we also want to mention that, in recent years, the existence of homoclinic solutions for difference equations has been studied by many authors, and we refer to [24–36].

Since the operator is bounded and self-adjoint in the space (defined in Section 2), we consider (1) as a nonlinear equation in with (3) being satisfied automatically. The spectrum of is closed. Thus, the complement consists of a finite number of open intervals called spectral gaps and two of them are semi-infinite which are denoted by and , respectively. In this paper, we consider the homoclinic solutions of (1) in for the case where and . The case where and is omitted, since, in this case, we can replace by .

The main idea in this paper is as follows. First, we assume that converges to zero at infinity; that is, . After that, we prove a compact inclusion between ordinary sequence spaces and weighted sequence spaces (defined in Section 2), in order to come over lack of compactness for the so-called condition (defined in Section 2). Finally, by making use of the Mountain Pass Lemma [37], we prove the existence of homoclinic solutions of (1) in .

2. Preliminaries

In this section, we first establish the variational setting associated with (1). Let

Then the following embedding between spaces holds:

For , we get the usual Hilbert space of square-summable sequences, with the real scalar product

For a positive real valued bounded sequence , we define the weighted sequence spaces :

It is not hard to see that is a Hilbert space, with the scalar product

For a certain class of weight , we have the following lemmas, which will play a crucial role in our analysis.

Lemma 1. Let be a multiplication operator from to defined by . If , then the operator is compact.

Proof of Lemma 1. Let

We only need to prove that is precompact in . By assumption, for any , there exists such that for any . Define a cutting sequence by

Denote by the anticutting sequence. Then for any

For arbitrary and finite-dimensional and bounded, we know that is precompact. The proof is complete.

Lemma 2. One assumes positive sequence of real numbers with . Then with compact inclusion.

Proof of Lemma 2. Note that for any and is compact. Thus, with compact inclusion by Lemma 1. The proof is complete.

Remark 3. Karachalios [2] proved with compact inclusion assuming that , , for some . Note that implies that . Thus, we find that Lemma 2 improves Lemma 2.1 in [2].

On the Hilbert space , we consider the functional

where

is the primitive function of . Then and

Equation (20) implies that (1) is the corresponding Euler-Lagrange equation for . Therefore, we have reduced the problem of finding a nontrivial homoclinic solution of (1) to that of seeking a nonzero critical point of the functional on .

Let be the distance from to the spectrum ; that is,

Then, we have

We also consider a norm in defined by

Since

norm (23) is an equivalent norm with the usual one of .

In order to obtain the existence of critical points of on , we cite some basic notations and some known results from critical point theory.

Let be a Hilbert space and denote the set of functionals that are Fréchet differentiable and their Fréchet derivatives are continuous on .

Let . A sequence is called a sequence for if for some and as . We say satisfies the condition if any sequence for possesses a convergent subsequence.

Let be the open ball in with radius and center 0, and let denote its boundary. The following lemma is taken from [37].

Lemma 4 (Mountain Pass Lemma). If and satisfies the following conditions: there exist and such that , then there exists a sequence for the mountain pass level which is defined by

where

3. Main Results

In this section, we will establish some sufficient conditions on the existence of nontrivial solutions of (1) in .

Theorem 5. Assume that , , and the following conditions hold.

is continuous in , as uniformly for .There exist , such that

uniformly for and .There exists some such that , for , , and , where . for , and , uniformly for .Positive real valued sequence with .

Then (1) has at least a nontrivial solution in and the solution decays exponentially at infinity. That is, there exist two positive constants and such that

Theorem 5 gives some sufficient conditions on the existence of nontrivial solutions of (1) in . However, (1) may have no nontrivial solutions in . In fact, we have the following proposition.

Proposition 6. Assume that , , and when for all . Then (1) has no nontrivial solutions in .

Proof of Proposition 6. By way of contradiction, we assume that (1) has a nontrivial solution . Then is a nonzero critical point of , and

This is a contradiction as , so the conclusion holds.

Remark 7. It is easy to see that the function defined by

where and , , for some , satisfies all conditions in Theorem 5. This case was studied by [2], and we find that Theorem 5 improves Theorem 2.3 in [2].

Remark 8. We will introduce another condition : is nondecreasing with respect to . We want to point out that condition is equivalent to when and gives “better monotony” when , since implies (see [38]). Moreover, we can find that satisfies but not for some .

Remark 9. As we know, the condition

is often called Ambrosetti-Rabinowitz superlinear condition [3]. Clearly, implies . Actually, is more general than . Let be a positive sequence, and . Then satisfies . However, does not satisfy .

The proof of Theorem 5 is based on a direct application of the following lemmas. The key points read as follows.

Lemma 10. Assume that the conditions of Theorem 5 hold; then one has the following.

There exist two constants and such that .There exists an such that as .

Proof of Lemma 10. Let and . By and , there exists , such that

for all and , and (31) implies that

By (32) and the Hölder inequality, we have

Since , we have

where .

Let be the eigenvector of corresponding to the eigenvalue ; that is to say, . There exists , such that

Let

By , for any , there exists such that

Taking large enough, such that for all , then, combining (35), (36), and (37), we have

where . Letting be large enough, such that , we obtain that as . The proof is complete.

Lemma 11. Assume that the conditions of Theorem 5 hold; then the functional satisfies the condition for any given .

Proof of Lemma 11. Let be a sequence of ; that is,

First, we prove that is bounded in . By way of contradiction, assume that as . Set . Up to a sequence, we have

Case 1   (). By , where as , we have

Let . Obviously, is nonempty. Then, for some , it follows from (41) that

Combining () and , we have

However,

as . This contradicts (42).

Case 2  (). Let

For any given , let be large enough such that and . Combining (32), (41), and , it is easy to see that

Thus, for large enough, we have

By (47), (48), and , we have

Noting that and , as , then when is big enough. Thus, . In view of (), it follows that

By (50), we have

This contradicts (49), so is bounded in .

Second, we show that there exists a convergent subsequence of . In fact, there exists a subsequence, still denoted by the same notation, such that

By Lemma 2, we have

By direct calculation, we obtain

Therefore, combining (11), (24), (52), (53), (54), and the boundedness of , it is clear that

and this means satisfies condition. The proof is complete.

Now, we are ready to prove Theorem 5.

Proof of Theorem 5. Let , , and be obtained in Lemma 10.

Since as , there exists a real number such that

Immediately, we obtain

Now that we have verified all assumptions of Lemma 4, we know possesses a sequence for the mountain pass level with

where

By Lemma 11, has a convergent subsequence such that as for some bounded . Since , we have

as . By the uniqueness of limit and the fact that is bounded, we obtain that is a nontrivial critical point of as the corresponding critical value . Hence, (1) has at least one nontrivial solution in .

Finally, we show that satisfies (28). In fact, similar to [39], for , let

then

where

Clearly, . Thus, the multiplication by is a compact operator in , which implies that

where stands for the essential spectrum. Equation (62) means that is an eigenfunction that corresponds to the eigenvalue of finite multiplicity of the operator . Equation (28) follows from the standard theorem on exponential decay for such eigenfunctions [1]. Now the proof of Theorem 5 is complete.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is supported by the Program for Changjiang Scholars and Innovative Research Team in University (no. IRT1226), the National Natural Science Foundation of China (no. 11171078), the Specialized Fund for the Doctoral Program of Higher Education of China (no. 20114410110002), and the Project for High Level Talents of Guangdong Higher Education Institutes.

References

G. Teschl, Jacobi Operators and Completely Integrable Nonlinear Lattices, vol. 72, The American Mathematical Society, Providence, RI, USA, 2000.

N. I. Karachalios, “A remark on the existence of breather solutions for the discrete nonlinear Schrodinger equation in infinite lattices: the case of site-dependent anharmonic parameters,” Proceedings of the Edinburgh Mathematical Society II, vol. 49, no. 1, pp. 115–129, 2006.

M. Willem, Minimax Theorems, vol. 24 of Progress in Nonlinear Differential Equations and Their Applications, Birkhäuser, Boston, Mass, USA, 1996.

W. Królikowski, B. Luther-Davies, and C. Denz, “Photorefractive solitons,” IEEE Journal of Quantum Electronics, vol. 39, no. 1, pp. 3–12, 2003.

G. Kopidakis, S. Aubry, and G. P. Tsironis, “Targeted energy transfer through discrete breathers in nonlinear systems,” Physical Review Letters, vol. 87, no. 16, Article ID 165501, pp. 1–4, 2001.

R. Livi, R. Franzosi, and G. Oppo, “Self-localization of Bose-Einstein condensates in optical lattices via boundary dissipation,” Physical Review Letters, vol. 97, no. 6, Article ID 060401, 2006.

M. H. Jakubowski, K. Steiglitz, and R. Squier, “Information transfer between solitary waves in the saturable Schrödinger equation,” Physical Review E, vol. 56, no. 6, pp. 7267–7272, 1997.

A. Khare, K. Rasmussen, M. Samuelsen, and A. Saxena, “Exact solutions of the saturable discrete nonlinear Schrodinger equation,” Journal of Physics A, vol. 38, no. 4, pp. 807–814, 2005.

W. Królikowski and S. A. Holmstrom, “Fusion and birth of spatial solitons upon collision,” Optics Letters, vol. 22, no. 6, pp. 369–371, 1997.

J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of discrete solitons in optically induced real time waveguide arrays,” Physical Review Letters, vol. 90, no. 2, p. 023902/4, 2003.

J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature, vol. 422, no. 6928, pp. 147–150, 2003.

G. Arioli and F. Gazzola, “A remark on the existence of breather solutions for the discrete nonlinear Schrodinger equation in infinite lattices: the case of site-dependent anharmonic parameters,” Proceedings of the Edinburgh Mathematical Society II, vol. 26, no. 6, pp. 1103–1114, 1996.

S. Aubry, “Breathers in nonlinear lattices: existence, linear stability and quantization,” Physica D: Nonlinear Phenomena, vol. 103, no. 1–4, pp. 201–250, 1997.

S. Aubry, G. Kopidakis, and V. Kadelburg, “Variational proof for hard discrete breathers in some classes of Hamiltonian dynamical systems,” Discrete and Continuous Dynamical Systems B, vol. 1, no. 3, pp. 271–298, 2001.

J. Cuevas, P. G. Kevrekidis, and D. J. Frantzeskakis, “Discrete solitons in nonlinear Schrodinger lattices with a power-law nonlinearity,” Physica D: Nonlinear Phenomena, vol. 238, no. 1, pp. 67–76, 2009.

G. James, “Centre manifold reduction for quasilinear discrete systems,” Journal of Nonlinear Science, vol. 13, no. 1, pp. 27–63, 2003.

R. S. MacKay and S. Aubry, “Proof of existence of breathers for time-reversible or Hamiltonian networks of weakly coupled oscillators,” Nonlinearity, vol. 7, no. 6, pp. 1623–1643, 1994.

A. Pankov and V. Rothos, “Periodic and decaying solutions in discrete nonlinear Schrödinger with saturable nonlinearity,” Proceedings of the Royal Society A: Mathematical, Physical and Engineering, vol. 464, no. 2100, pp. 3219–3236, 2008.

S. Aubry, “Discrete breathers: localization and transfer of energy in discrete Hamiltonian nonlinear systems,” Physica D. Nonlinear Phenomena, vol. 216, no. 1, pp. 1–30, 2006.

S. Flach and A. V. Gorbach, “Discrete breathers—advances in theory and applications,” Physics Reports, vol. 467, no. 1–3, pp. 1–116, 2008.

S. Flach and C. R. Willis, “Discrete breathers,” Physics Reports, vol. 295, no. 5, pp. 181–264, 1998.

A. A. Sukhorukov and Y. S. Kivshar, “Generation and stability of discrete gap solitons,” Optics Letters, vol. 28, no. 23, pp. 2345–2347, 2003.

A. V. Gorbach and M. Johansson, “Gap and out-gap breathers in a binary modulated discrete nonlinear Schrödinger model,” European Physical Journal D, vol. 29, no. 1, pp. 77–93, 2004.

A. Mai and Z. Zhou, “Discrete solitons for periodic discrete nonlinear Schrodinger equations,” Applied Mathematics and Computation, vol. 222, pp. 34–41, 2013.

G. Sun, “On standing wave solutions for discrete nonlinear Schrödinger equations,” Abstract and Applied Analysis, vol. 2013, Article ID 436919, 6 pages, 2013.

D. Ma and Z. Zhou, “Existence and multiplicity results of homoclinic solutions for the DNLS equations with unbounded potentials,” Abstract and Applied Analysis, vol. 2012, Article ID 703596, 15 pages, 2012.

X. Tang and J. Chen, “Infinitely many homoclinic orbits for a class of discrete Hamiltonian systems,” Advances in Difference Equations, vol. 2013, article 242, 2013.

X. H. Tang and X. Lin, “Infinitely many homoclinic orbits for discrete Hamiltonian systems with subquadratic potential,” Journal of Difference Equations and Applications, vol. 19, no. 5, pp. 796–813, 2013.

X. H. Tang and X. Y. Lin, “Homoclinic solutions for a class of second order discrete Hamiltonian systems,” Acta Mathematica Sinica, vol. 28, no. 3, pp. 609–622, 2012.

Z. Zhou and J. Yu, “On the existence of homoclinic solutions of a class of discrete nonlinear periodic systems,” Journal of Differential Equations, vol. 249, no. 5, pp. 1199–1212, 2010.

J. Kuang and Z. Guo, “Homoclinic solutions of a class of periodic difference equations with asymptotically linear nonlinearities,” Nonlinear Analysis, vol. 89, pp. 208–218, 2013.

G. Chen and S. Ma, “Discrete nonlinear Schrödinger equations with superlinear nonlinearities,” Applied Mathematics and Computation, vol. 218, no. 9, pp. 5496–5507, 2012.

G. Chen and S. Ma, “Ground state and geometrically distinct solitons of discrete nonlinear Schrödinger equations with saturable nonlinearities,” Studies in Applied Mathematics, vol. 131, no. 4, pp. 389–413, 2013.

A. Mai and Z. Zhou, “Ground state solutions for the periodic discrete nonlinear Schrödinger equations with superlinear nonlinearities,” Abstract and Applied Analysis, vol. 2013, Article ID 317139, 11 pages, 2013.

S. Ma and Z. Wang, “Multibump solutions for discrete periodic nonlinear Schrödinger equations,” Journal of Applied Mathematics and Physics, vol. 64, no. 5, pp. 1413–1442, 2013.

Z. Zhou and J. S. Yu, “Homoclinic solutions in periodic nonlinear difference equations with superlinear nonlinearity,” Acta Mathematica Sinica, vol. 29, no. 9, pp. 1809–1822, 2013.

C. A. Stuart, “Locating Cerami sequences in a mountain pass geometry,” Communications in Applied Analysis, vol. 15, no. 2–4, pp. 569–588, 2011.

S. B. Liu and S. J. Li, “Infinitely many solutions for a superlinear elliptic equation,” Acta Mathematica Sinica, vol. 46, no. 4, pp. 625–630, 2003.

A. Pankov, “Gap solitons in periodic discrete nonlinear Schrödinger equations,” Nonlinearity, vol. 19, no. 1, pp. 27–40, 2006.