Homoclinic solutions in periodic nonlinear difference equations with superlinear nonlinearity. (English) Zbl 1284.39006

The existence of homoclinic solutions in periodic nonlinear difference equations with superlinear nonlinearity is considered. The classical Ambrosetti-Rabinowitz superlinear condition is improved by a general superlinear one. The main idea in this paper is an application of a linking theorem combined with an approximation technique.


39A10 Additive difference equations
39A70 Difference operators
39A12 Discrete version of topics in analysis
39A23 Periodic solutions of difference equations
37C29 Homoclinic and heteroclinic orbits for dynamical systems
Full Text: DOI


[1] Arioli, G., Gazzola, F.: Periodic motions of an infinite lattice of particles with nearest neighbor interaction. Nonlinear Anal., 26, 1103–1114 (1996) · Zbl 0867.70004
[2] Aubry, S.: Breathers in nonlinear lattices: existence, linear stability and quantization. Phys. D, 103, 201–250 (1997) · Zbl 1194.34059
[3] Aubry, S., Kopidakis, G., Kadelburg, V.: Variational proof for hard discrete breathers in some classes of Hamiltonian dynamical systems. Discrete Contin. Dyn. Syst. Ser. B, 1, 271–298 (2001) · Zbl 1092.37523
[4] Aubry, S.: Discrete breathers: localization and transfer of energy in discrete Hamiltonian nonlinear systems. Phys. D, 216, 1–30 (2006) · Zbl 1159.82312
[5] Bruno, G., Pankov, A., Tverdokhleb, Yu.: On almost-periodic operators in the spaces of sequences. Acta Appl. Math., 65, 153–167 (2001) · Zbl 0993.39013
[6] Christodoulides, D. N., Lederer, F., Silberberg, Y.: Discretizing light behaviour in linear and nonlinear waveguide lattices. Nature, 424, 817–823 (2003)
[7] Cuevas, J., Kevrekidis, P. G., Frantzeskakis, D. J., et al.: Discrete solitons in nonlinear Schrödinger lattices with a power-law nonlinearity. Phys. D, 238, 67–76 (2009) · Zbl 1153.82321
[8] Efremidis, N. K., Sears, S., Christodoulides, D. N., et al.: Discrete solitons in photorefractive optically induced photonic lattices. Phys. Rev. E, 66, 046602 (2002)
[9] Flach, S., Gorbach, A. V.: Discrete breathers – Advance in theory and applications. Phys. Rep., 467, 1–116 (2008) · Zbl 1218.37107
[10] Flach, S., Willis, C. R.: Discrete breathers. Phys. Rep., 295, 181–264 (1998)
[11] Fleischer, J. W., Carmon, T., Segev, M., et al.: Observation of discrete solitons in optically induced real time waveguide arrays. Phys. Rev. Lett., 90, 023902 (2003)
[12] Fleischer, J. W., Segev, M., Efremidis, N. K., et al.: Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices. Nature, 422, 147–150 (2003)
[13] Gorbach, A. V., Johansson, M.: Gap and out-gap breathers in a binary modulated discrete nonlinear Schrödinger model. Eur. Phys. J. D, 29, 77–93 (2004)
[14] James, G.: Centre manifold reduction for quasilinear discrete systems. J. Nonlinear Sci., 13, 27–63 (2003) · Zbl 1185.37158
[15] Livi, R., Franzosi, R., Oppo, G.-L.: Self-localization of Bose-Einstein condensates in optical lattices via boundary dissipation. Phys. Rev. Lett., 97, 060401 (2006)
[16] MacKay, R. S., Aubry, S.: Proof of existence of breathers for time-reversible or Hamiltonian networks of weakly coupled oscillators. Nonlinearity, 7, 1623–1643 (1994) · Zbl 0811.70017
[17] Mawhin, J., Willem, M.: Critical Point Theory and Hamiltonian Systems, Springer-Verlag, New York, 1989 · Zbl 0676.58017
[18] Pankov, A.: Gap solitons in periodic discrete nonlinear Schrödinger equations II: A generalized Nehari manifold approach. Discrete Contin. Dyn. Syst., 19, 419–430 (2007) · Zbl 1220.35164
[19] Pankov, A.: Gap solitons in periodic discrete nonlinear Schrödinger equations. Nonlinearity, 19, 27–40 (2006) · Zbl 1220.35163
[20] Pankov, A.: Gap solitons in periodic discrete nonlinear Schrödinger equations with saturable nonlinearities. J. Math. Anal. Appl., 371, 254–265 (2010) · Zbl 1197.35273
[21] Rabinowitz, P. H.: Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Regional Conference Series in Mathematics, 65, American Mathematical Society, Providence, RI, 1986 · Zbl 0609.58002
[22] Sukhorukov, A. A., Kivshar, Y. S.: Generation and stability of discrete gap solitons. Opt. Lett., 28, 2345–2347 (2003)
[23] Teschl, G.: Jacobi Operators and Completely Integrable Nonlinear Lattices, Mathematical Surveys and Monographs, 72, American Mathematical Society, Providence, RI, 2000 · Zbl 1056.39029
[24] Wang, Y., Shen, Y.: Existence of sign-changing solutions for the p-Laplacian equation from linking type theorem. Acta Math. Sin., Engl. Series, 26, 1355–1368 (2010) · Zbl 1200.35165
[25] Willem, M.: Minimax Theorems, Birkhäuser, Boston, 1996
[26] Yu, J., Guo, Z.: On boundary value problems for a discrete generalized Emden-Fowler equation. J. Differential Equations, 231, 18–31 (2006) · Zbl 1112.39011
[27] Zhou, Z., Yu, J.: On the existence of homoclinic solutions of a class of discrete nonlinear periodic systems. J. Differential Equations, 249, 1199–1212 (2010) · Zbl 1200.39001
[28] Zhou, Z., Yu, J., Chen, Y.: On the existence of gap solitons in a periodic discrete nonlinear Schrödinger equation with saturable nonlinearity. Nonlinearity, 23, 1727–1740 (2010) · Zbl 1193.35176
[29] Zhou, Z., Yu, J., Chen, Y.: Periodic solutions of a 2n-th-order nonlinear difference equation. Sci. China Math., 53, 41–50 (2010) · Zbl 1232.39019
[30] Zhou, Z., Yu, J., Guo, Z.: Periodic solutions of higher-dimensional discrete systems. Proc. Roy. Soc. Edinburgh Sect. A, 134, 1013–1022 (2004) · Zbl 1073.39010
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.