Solitons as a signature of the modulation instability in the discrete nonlinear Schrodinger equation

TL;DR

This paper investigates how modulation instability influences solitary wave propagation in the discrete nonlinear Schrödinger equation, developing a quasicontinuum approximation to analyze soliton stability and behavior.

Contribution

It introduces a self-contained quasicontinuum approximation for small-amplitude solitons and analytically studies their stability under modulation instability, especially in dark and high-velocity regions.

Findings

Solitons are less prone to instabilities at high velocities.

An analytical upper boundary for self-defocusing instability is established.

The method provides insights where standard modulation analysis fails.

Abstract

The effect of the modulation instability on the propagation of solitary waves along one-dimensional discrete nonlinear Schr\"odinger equation with cubic nonlinearity is revisited. A self-contained quasicontinuum approximation is developed to derive closed-form expressions for small-amplitude solitary waves. The notion that the existence of nonlinear solitary waves is a signature of the modulation instability is used to analytically study instability effects on solitons during propagation. In particular, we concern with instability effects in the dark region, where other analytical methods as the standard modulation analysis of planewaves do not provide any information on solitons. The region of high-velocity solitons is studied anew showing that solitons are less prone to intabilities in this region. An analytical upper boundary for the self-defocusing instability is defined.