A low regularity exponential-type integrator for the derivative nonlinear Schr\"odinger equation

TL;DR

This paper introduces a first-order exponential integrator for the derivative nonlinear Schrödinger equation that works with low regularity initial data, improving convergence and conservation properties.

Contribution

It presents the first low regularity integrator for the derivative nonlinear Schrödinger equation, including a symmetrized version with enhanced performance.

Findings

Converges with first-order in H^s for s > 1/2

Symmetrized method improves global error and conservation

Numerical experiments confirm theoretical results

Abstract

In this work, we present a first-order unfiltered exponential integrator for the one-dimensional derivative nonlinear Schr\"odinger equation with low regularity. Our analysis shows that for any $s>\frac12$, the method converges with first-order in $H^s(\mathbb{T})$ for initial data $u_0\in H^{s+1}(\mathbb{T})$. Moreover, we constructed a symmetrized version of this method that performs better in terms of both global error and conservation behavior. To the best of our knowledge, these are the first low regularity integrators for the derivative nonlinear Schr\"odinger equation. Numerical experiments illustrate our theoretical findings.