Existence of exponentially spatially localised breather solutions for lattices of nonlinearly coupled particles: Schauder's fixed point theorem approach

TL;DR

This paper introduces a fixed point theory method using Schauder's Fixed Point Theorem to prove the existence and spatial localization of breather solutions in nonlinear lattices of coupled particles.

Contribution

It provides a general, concise approach to establish the existence and localization rate of breather solutions in nonlinear lattices, expanding mathematical tools in the field.

Findings

Existence of exponentially localized breather solutions proven.

Solutions exhibit even or odd parity symmetry.

Energy bounds for solutions established.

Abstract

The problem of showing the existence of localised modes in nonlinear lattices has attracted considerable efforts from the physical but also from the mathematical viewpoint where a rich variety of methods has been employed. In this paper we prove that a fixed point theory approach based on the celebrated Schauder's Fixed Point Theorem may provide a general method to establish concisely not only the existence of localised structures but also a required rate of spatial localisation. As a case study we consider lattices of coupled particles with nonlinear nearest neighbour interaction and prove the existence of exponentially spatially localised breathers exhibiting either even-parity or odd-parity symmetry under necessary non-resonant conditions accompanied with the proof of energy bounds of the solutions.