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A low regularity exponential-type integrator for the derivative nonlinear Schrödinger equation

Lun Ji, Hang Li, Alexander Ostermann

TL;DR

This work develops a low-regularity, first-order exponential-type integrator for the 1D derivative nonlinear Schrödinger equation on the torus, built via a gauge transformation that relocates the derivative in the nonlinearity. It proves convergence in $H^s(\mathbb{T})$ for $s>1/2$ with data in $H^{s+1}(\mathbb{T})$, and introduces a time-symmetric second-order variant that improves conservation properties while retaining first-order accuracy at low regularity. The analysis combines Bourgain-space techniques, oscillatory integral approximations, and careful error decompositions to achieve unconditional stability without filtering or CFL constraints. Numerical experiments corroborate the theoretical results, show favorable long-time behavior, and demonstrate robust mass and energy conservation, particularly for the symmetric scheme. The results advance low-regularity numerical treatment of derivative nonlinearities and provide practical tools for simulating dNLS dynamics with rough data.

Abstract

In this work, we present a first-order unfiltered exponential integrator for the one-dimensional derivative nonlinear Schrödinger 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 Schrödinger equation. Numerical experiments illustrate our theoretical findings.

A low regularity exponential-type integrator for the derivative nonlinear Schrödinger equation

TL;DR

This work develops a low-regularity, first-order exponential-type integrator for the 1D derivative nonlinear Schrödinger equation on the torus, built via a gauge transformation that relocates the derivative in the nonlinearity. It proves convergence in for with data in , and introduces a time-symmetric second-order variant that improves conservation properties while retaining first-order accuracy at low regularity. The analysis combines Bourgain-space techniques, oscillatory integral approximations, and careful error decompositions to achieve unconditional stability without filtering or CFL constraints. Numerical experiments corroborate the theoretical results, show favorable long-time behavior, and demonstrate robust mass and energy conservation, particularly for the symmetric scheme. The results advance low-regularity numerical treatment of derivative nonlinearities and provide practical tools for simulating dNLS dynamics with rough data.

Abstract

In this work, we present a first-order unfiltered exponential integrator for the one-dimensional derivative nonlinear Schrödinger equation with low regularity. Our analysis shows that for any , the method converges with first-order in for initial data . 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 Schrödinger equation. Numerical experiments illustrate our theoretical findings.
Paper Structure (13 sections, 8 theorems, 112 equations, 3 figures)

This paper contains 13 sections, 8 theorems, 112 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Gauge Transformation and Cauchy Problem for \ref{['dnls']}
  3. The gauge transformation
  4. Cauchy problem for (\ref{['dnls']})
  5. Construction of the Numerical Method
  6. Construction of the basic method
  7. Construction of a symmetric method
  8. Main theorem
  9. Local error analysis
  10. Some technical tools
  11. Local error analysis
  12. Global error analysis
  13. Numerical Experiments

Key Result

Lemma 2.2

For any $\gamma\ge0$, the transformation is a gauge transformation, i.e., the map $\mathcal{G}$ is a homeomorphism that satisfies where $C$ depends on $\|u\|_{L^\infty([0,T],H^\gamma(\mathbb{T}))}$ and $\|v\|_{L^\infty([0,T], H^\gamma(\mathbb{T}))}$. Furthermore, the inverse map is given by and $\mathcal{G}^{-1}$ satisfies a similar estimate as gaugeprop on $[0,T]$.

Figures (3)

  • Figure 1: $H^s$ error of our low regularity exponential integrators for rough initial data $u_0\in H^{s+1}$. Left: $s=0.5$; right: $s=1$.
  • Figure 2: Mass conservation behavior of low regularity exponential integrators for rough initial data $u_0\in H^2$. Left: relative mass error of the basic method \ref{['meth1']}; right: relative mass error of the symmetric method \ref{['meth2']}.
  • Figure 3: Energy conservation behavior of low regularity exponential integrators for rough initial data $u_0\in H^2$. Left: relative energy error of the basic method \ref{['meth1']}; right: relative energy error of the symmetric method \ref{['meth2']}.

Theorems & Definitions (14)

  • Definition 2.1
  • Lemma 2.2: herr06
  • Corollary 2.3
  • Proposition 2.4: herr06
  • Theorem 3.1
  • Lemma 4.1
  • proof
  • Lemma 4.2: Integration by parts
  • proof
  • Lemma 4.3
  • ...and 4 more