Variational approximations for travelling solitons in a discrete nonlinear Schr\"{o}dinger equation

TL;DR

This paper develops a variational approximation for travelling solitons in a discrete nonlinear Schrödinger equation with saturable nonlinearity, providing analytical predictions validated by numerical simulations, including stability and bound state properties.

Contribution

The paper introduces an analytical variational approximation for travelling solitons and embedded solitons in the DNLSE, enhancing understanding of their stability and interactions.

Findings

VA accurately predicts soliton parameters

Embedded solitons are shown to be stable

Good agreement between analytical and numerical results

Abstract

Travelling solitary waves in the one-dimensional discrete nonlinear Schr\"{o}dinger equation (DNLSE) with saturable onsite nonlinearity are studied. A variational approximation (VA) for the solitary waves is derived in an analytical form. The stability is also studied by means of the VA, demonstrating that the solitons are stable, which is consistent with previously published results. Then, the VA is applied to predict parameters of travelling solitons with non-oscillatory tails (\textit{embedded solitons}, ESs). Two-soliton bound states are considered too. The separation distance between the solitons forming the bound state is derived by means of the VA. A numerical scheme based on the discretization of the equation in the moving coordinate frame is derived and implemented using the Newton--Raphson method. In general, a good agreement between the analytical and numerical results is obtained. In particular, we demonstrate the relevance of the analytical prediction of characteristics of the embedded solitons.