Numerical study of transverse (in-)stability of solitary waves in the cubic-quintic nonlinear Schrödinger equation

We study the nonlinear Schrödinger 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.