Numerical study of transverse (in-)stability of solitary waves in the cubic-quintic nonlinear Schr\"odinger equation

TL;DR

This paper numerically investigates the transverse stability of line solitary waves in a cubic-quintic nonlinear Schrödinger equation on a waveguide, identifying a critical domain size where instability occurs.

Contribution

It provides the first numerical analysis of transverse stability in this model, revealing the critical domain size for instability onset.

Findings

Existence of a critical torus length $L_y$ for instability

Numerical confirmation of stability below the critical length

Observation of instability emergence above the critical length

Abstract

We study the nonlinear Schr\"odinger equation with a competing cubic-quintic power law nonlinearity on the waveguide domain $\mathbb R_x \times \mathbb T_{L_y}$. This model is globally well-posed and admits line solitary wave solutions, whose transverse (in-)stability is numerically investigated. We consider both spatially localized perturbations and periodic deformations of the line solitary wave and numerically confirm that there exists a critical torus length $L_y>0$ above which instability appears.