Continuation of spatially localized periodic solutions in discrete NLS lattices via normal forms

TL;DR

This paper investigates how localized and periodic solutions in 1D discrete nonlinear Schrödinger lattices can be continued as the interaction strength varies, using normal form methods to analyze stability and bifurcations.

Contribution

It applies a novel normal form algorithm to study the continuation and stability of localized solutions and invariant tori in discrete NLS models, providing both existing and new insights.

Findings

Recovered known results on localized solutions.

Provided new stability insights for invariant tori.

Analyzed bifurcations in discrete NLS lattices.

Abstract

We consider the problem of the continuation with respect to a small parameter $\epsilon$ of spatially localised and time periodic solutions in 1-dimensional dNLS lattices, where $\epsilon$ represents the strength of the interaction among the sites on the lattice. Specifically, we consider different dNLS models and apply a recently developed normal form algorithm in order to investigate the continuation and the linear stability of degenerate localised periodic orbits on lower and full dimensional invariant resonant tori. We recover results already existing in the literature and provide new insightful ones, both for discrete solitons and for invariant subtori.