Existence and continuous approximation of small amplitude breathers in 1D and 2D Klein--Gordon lattices

TL;DR

This paper demonstrates the existence of small amplitude breathers in 1D and 2D Klein-Gordon lattices and shows they can be approximated by solutions of the nonlinear Schrödinger equation, using analytical techniques.

Contribution

It establishes the existence and approximation of small amplitude breathers in Klein-Gordon lattices via a novel application of Lyapunov-Schmidt decomposition and continuum approximation methods.

Findings

Breathers exist in 1D and 2D Klein-Gordon lattices.

Breathers are well approximated by the ground state of the nonlinear Schrödinger equation.

The approach links Klein-Gordon and nonlinear Schrödinger lattices through analytical techniques.

Abstract

We construct small amplitude breathers in 1D and 2D Klein--Gordon infinite lattices. We also show that the breathers are well approximated by the ground state of the nonlinear Schroedinger equation. The result is obtained by exploiting the relation between the Klein Gordon lattice and the discrete Non Linear Schroedinger lattice. The proof is based on a Lyapunov-Schmidt decomposition and continuum approximation techniques introduced in [7], actually using its main result as an important lemma.