Periodic travelling wave solutions of discrete nonlinear Schr\"odinger equations

TL;DR

This paper proves the existence of nonzero periodic travelling wave solutions in a general discrete nonlinear Schrödinger equation with variable interactions, using fixed point theory.

Contribution

It extends the existence results of travelling waves to DNLS with general nonlinearities and interaction ranges beyond nearest neighbors.

Findings

Existence of periodic travelling wave solutions is established.

The approach uses Schauder's Fixed Point Theorem.

Results apply to a broad class of DNLS models.

Abstract

The existence of nonzero periodic travelling wave solutions for a general discrete nonlinear Schr\"odinger equation (DNLS) on finite 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.