Discrete embedded solitary waves and breathers in one-dimensional nonlinear lattices

TL;DR

This paper explores the existence and stability of embedded solitary waves and breathers in one-dimensional nonlinear lattices, extending linear potential concepts into nonlinear regimes and proposing experimental observation methods.

Contribution

It introduces the extension of embedded eigenmodes into nonlinear discrete systems, specifically the nonlinear Schrödinger and Klein-Gordon models, and analyzes their stability and dynamical behavior.

Findings

Embedded modes become unstable as nonlinearity increases.

Nonlinear regimes lead to dynamical delocalization of solitary waves.

Proposes an experiment using a bi-inductive electrical lattice.

Abstract

For a one-dimensional linear lattice, earlier work has shown how to systematically construct a slowly-decaying linear potential bearing a localized eigenmode embedded in the continuous spectrum. Here, we extend this idea in two directions: The first one is in the realm of the discrete nonlinear Schrodinger equation, where the linear operator of the Schrodinger type is considered in the presence of a Kerr focusing or defocusing nonlinearity and the embedded linear mode is continued into the nonlinear regime as a discrete solitary wave. The second case is the Klein-Gordon setting, where the presence of a cubic nonlinearity leads to the emergence of embedded-in-the-continuum discrete breathers. In both settings, it is seen that the stability of the modes near the linear limit turns into instability as nonlinearity is increased past a critical value, leading to a dynamical delocalization of the solitary wave (or breathing) state. Finally, we suggest a concrete experiment to observe these embedded modes using a bi-inductive electrical lattice.