2019/11/03 by Christian Klein, Klein, C., N. Stoilov +1
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Mathematics #FOS: Physical sciences #Nonlinear Photonic Systems #Numerical Analysis (math.NA) #Numerical methods for differential equations
paper · pdf · doi:10.48550/arxiv.1911.00918
openalex publication_date 2019/11/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Schrödinger (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.