Existence and multiplicity of wave trains in 2D lattices

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:257

Article

Zhang, Ling¹  Guo, Shangjiang¹ 

[1] Hunan Univ, Coll Math & Econometr, Changsha 410082, Hunan, Peoples R China

关键词: Wave train;    Bifurcation;    Resonance;    Singularity theory;   

DOI  :  10.1016/j.jde.2014.04.016

来源: Elsevier

【 摘 要 】

We study the existence and branching patterns of wave trains in a two-dimensional lattice with linear and nonlinear coupling between nearest particles and a nonlinear substrate potential. The wave train equation of the corresponding discrete nonlinear equation is formulated as an advanced-delay differential equation which is reduced by a Lyapunov-Schmidt reduction to a finite-dimensional bifurcation equation with certain symmetries and an inherited Hamiltonian structure. By means of invariant theory and singularity theory, we obtain the small amplitude solutions in the Hamiltonian system near equilibria in non-resonance and p : q resonance, respectively. We show the impact of the direction theta of propagation and obtain the existence and branching patterns of wave trains in a one-dimensional lattice by investigating the existence of traveling waves of the original two-dimensional lattice in the direction theta of propagation satisfying tan theta is rational. (C) 2014 Elsevier Inc. All rights reserved.

【 授权许可】

Free