Solitary waves in the Ablowitz-Ladik equation with power-law nonlinearity

TL;DR

This paper introduces a generalized Ablowitz-Ladik model with power-law nonlinearity, analyzing discrete soliton states, their stability, and dynamics, bridging the gap between continuum and discrete nonlinear Schrödinger equations.

Contribution

It extends the Ablowitz-Ladik model to include power-law nonlinearity, providing new insights into soliton behavior and stability in discrete systems with variable nonlinearity.

Findings

Identification of stationary discrete-soliton states for various nonlinearity powers

Analysis of stability changes with frequency variations

Prediction of bistability and instability development through variational methods

Abstract

We introduce a generalized version of the Ablowitz-Ladik model with a power-law nonlinearity, as a discretization of the continuum nonlinear Schr\"{o}dinger equation with the same type of the nonlinearity. The model opens a way to study the interplay of discreteness and nonlinearity features. We identify stationary discrete-soliton states for different values of nonlinearity power $\sigma $, and address changes of their stability as frequency $\omega $ of the standing wave varies for given $\sigma $. Along with numerical methods, a variational approximation is used to predict the form of the discrete solitons, their stability changes, and bistability features by means of the Vakhitov-Kolokolov criterion (developed from the first principles). Development of instabilities and the resulting asymptotic dynamics are explored by means of direct simulations.