Multidomain spectral method for Schr\"odinger equations

TL;DR

This paper introduces a multidomain spectral method combined with advanced time integrators for high-precision numerical solutions of Schrödinger equations on the entire real line, including non-vanishing solutions like rogue wave models.

Contribution

The paper presents a novel spectral method with compactified exterior domains and stable time integrators, enabling accurate simulations of Schrödinger equations with non-zero boundary conditions.

Findings

Achieves high-precision propagation of Peregrine breather

Effectively handles asymptotically non-vanishing solutions

Provides a comparison with boundary condition methods

Abstract

A multidomain spectral method with compactified exterior domains combined with stable second and fourth order time integrators is presented for Schr\"odinger equations. The numerical approach allows high precision numerical studies of solutions on the whole real line. At examples for the linear and cubic nonlinear Schr\"odinger equation, this code is compared to transparent boundary conditions and perfectly matched layers approaches. The code can deal with asymptotically non vanishing solutions as the Peregrine breather being discussed as a model for rogue waves. It is shown that the Peregrine breather can be numerically propagated with essentially machine precision, and that localized perturbations of this solution can be studied.