Periodic Travelling Wave Solutions of Discrete Nonlinear Schrödinger Equations

Dirk Hennig

doi:10.1155/2017/3694103

Abstract

The existence of nonzero periodic travelling wave solutions for a general discrete nonlinear Schrödinger equation (DNLS) on one-dimensional lattices is proved. The DNLS features a general nonlinear term and variable range of interactions going beyond the usual nearest-neighbour interaction. The problem of the existence of travelling wave solutions is converted into a fixed point problem for an operator on some appropriate function space which is solved by means of Schauder’s Fixed Point Theorem.

Highlights

The problem of the existence of travelling wave solutions is converted into a fixed point problem for an operator on some appropriate function space which is solved by means of Schauder’s Fixed Point Theorem
Coherent structures arising in the form of travelling waves, solitons, and breathers in systems of coupled oscillators have attracted considerable interest not least due to the important role they play for applications in physics, biology, and chemistry
For the existence statement we introduce some appropriate function space on which the original problem is converted into a fixed point problem for a corresponding operator

Summary

Introduction

Coherent structures arising in the form of travelling waves, solitons, and breathers in systems of coupled oscillators have attracted considerable interest not least due to the important role they play for applications in physics, biology, and chemistry (for reviews see [1,2,3]). In this context a variety of nonlinear lattice systems has been studied including Fermi-Pasta-Ulam systems, discrete nonlinear Klein-Gordon systems, phase oscillator lattices, Josephson junction systems, reaction-diffusion systems, and the discrete nonlinear Schrodinger equation. The presented method can readily be applied to prove the existence of periodic TWs in general DNLS systems of higher dimensions and in various other nonlinear lattice systems

General DNLS Systems

Statement of the Existence Problem

Proof of the Existence Theorem

Summary