Randomized exponential integrators for modulated nonlinear Schr\"odinger equations

TL;DR

This paper introduces a novel randomized exponential integrator for modulated nonlinear Schrödinger equations with fractional Sobolev regularity, achieving improved convergence rates and handling more general modulations than existing methods.

Contribution

A new stratified Monte Carlo-based exponential integrator that enhances convergence and broadens applicability for modulated nonlinear Schrödinger equations.

Findings

Achieves convergence rate of lpha+1/2

Handles a broader class of modulation functions

Numerical results confirm theoretical improvements

Abstract

We consider the nonlinear Schr\"odinger equation with dispersion modulated by a (formal) derivative of a time-dependent function with fractional Sobolev regularity of class $W^{\alpha,2}$ for some $\alpha\in (0,1)$. Due to the loss of smoothness in the problem classical numerical methods face severe order reduction. In this work, we develop and analyze a new randomized exponential integrator based on a stratified Monte Carlo approximation. The new discretization technique averages the high oscillations in the solution allowing for improved convergence rates of order $\alpha+1/2$. In addition, the new approach allows us to treat a far more general class of modulations than the available literature. Numerical results underline our theoretical findings and show the favorable error behavior of our new scheme compared to classical methods.