Numerical study of the transverse stability of the Peregrine solution

TL;DR

This paper develops an advanced numerical method combining spectral and explicit schemes to analyze the transverse stability of the Peregrine solution in 2D NLS, revealing its instability and potential for blow-up under perturbations.

Contribution

It introduces a fully explicit 4th order time integration scheme combined with Fourier spectral methods for transverse variables in NLS equations, applied to stability analysis of the Peregrine solution.

Findings

Peregrine solution is unstable to standard perturbations.

Some perturbations cause blow-up in elliptic NLS.

The method effectively studies transverse stability in 2D NLS.

Abstract

We generalise a previously published approach based on a multi-domain spectral method on the whole real line in two ways: firstly, a fully explicit 4th order method for the time integration, based on a splitting scheme and an implicit Runge--Kutta method for the linear part, is presented. Secondly, the 1D code is combined with a Fourier spectral method in the transverse variable both for elliptic and hyperbolic NLS equations. As an example we study the transverse stability of the Peregrine solution, an exact solution to the one dimensional nonlinear Schr\"odinger (NLS) equation and thus a $y$-independent solution to the 2D NLS. It is shown that the Peregrine solution is unstable against all standard perturbations, and that some perturbations can even lead to a blow-up for the elliptic NLS equation.