Continuous approximation of breathers in one and two dimensional DNLS lattices

TL;DR

This paper develops a method to approximate breather solutions in one- and two-dimensional discrete nonlinear Schrödinger (DNLS) lattices by starting from the continuous limit and using finite element interpolation.

Contribution

It introduces a novel approach to construct breather solutions in DNLS lattices via perturbations of the continuous NLS ground state, including new hybrid modes in 2D.

Findings

Recovered known ST and P modes in 1D and 2D

Constructed new hybrid modes in 2D

Validated the approach using finite element interpolation

Abstract

In this paper we construct and approximate breathers in the DNLS model starting from the continuous limit: such periodic solutions are obtained as perturbations of the ground state of the NLS model in $H^1(\RR^n)$, with $n=1,2$. In both the dimensions we recover the Sievers-Takeno (ST) and the Page (P) modes; furthermore, in $\RR^2$ also the two hybrid (H) modes are constructed. The proof is based on the interpolation of the lattice using the Finite Element Method (FEM).