A Fourier integrator for the cubic nonlinear Schr\"{o}dinger equation with rough initial data

TL;DR

This paper introduces a Fourier-based integrator for the cubic nonlinear Schrödinger equation that maintains second-order convergence even with low-regularity initial data, outperforming standard methods like Strang splitting.

Contribution

A novel Fourier integrator based on the variation-of-constants formula and resonance approximations, effective for low-regularity initial data in nonlinear Schrödinger equations.

Findings

Achieves second-order convergence with fewer regularity requirements.

Demonstrates superior performance over Strang splitting for rough initial data.

Efficient implementation via fast Fourier methods.

Abstract

Standard numerical integrators suffer from an order reduction when applied to nonlinear Schr\"{o}dinger equations with low-regularity initial data. For example, standard Strang splitting requires the boundedness of the solution in $H^{r+4}$ in order to be second-order convergent in $H^r$, i.e., it requires the boundedness of four additional derivatives of the solution. We present a new type of integrator that is based on the variation-of-constants formula and makes use of certain resonance based approximations in Fourier space. The latter can be efficiently evaluated by fast Fourier methods. For second-order convergence, the new integrator requires two additional derivatives of the solution in one space dimension, and three derivatives in higher space dimensions. Numerical examples illustrating our convergence results are included. These examples demonstrate the clear advantage of the Fourier integrator over standard Strang splitting for initial data with low regularity.