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A Wong--Zakai resonance-based integrator for nonlinear Schrödinger equation with white noise dispersion

Jianbo Cui, Georg Maierhofer

TL;DR

This work tackles the simulation of the nonlinear Schrödinger equation with white-noise dispersion in the low-regularity regime. It develops a Wong–Zakai approximation to regularize the stochastic dispersion and constructs a resonance-based, low-regularity integrator (SDLRI) by twisting with the stochastic dispersion, enabling provable convergence under weaker smoothness assumptions. The authors prove well-posedness and mass conservation for the Wong–Zakai model, derive error bounds relative to the full stochastic system, and establish strong and pathwise convergence results, complemented by numerical experiments that corroborate the theoretical rates. The approach offers a practical and theoretically sound framework for simulating stochastic dispersive PDEs with low regularity, and it lays groundwork for future structure-preserving, resonance-based integrators in this setting.

Abstract

We introduce a novel approach to numerical approximation of nonlinear Schrödinger equation with white noise dispersion in the regime of low-regularity solutions. Approximating such solutions in the stochastic setting is particularly challenging due to randomized frequency interactions and presents a compelling challenge for the construction of tailored schemes. In particular, we design the first resonance-based schemes for this equation, which achieve provable convergence for solutions of much lower regularity than previously required. A crucial ingredient in this construction is the Wong--Zakai approximation of stochastic dispersive system, which introduces piecewise linear phases that capture nonlinear frequency interactions and can subsequently be approximated to construct resonance-based schemes. We prove the well-posedness of the Wong--Zakai approximated equation and establish its proximity to the original full stochastic dispersive system. Based on this approximation, we demonstrate an improved strong convergence rate for our new scheme, which exploits the stochastic nature of the dispersive terms. Finally, we provide numerical experiments underlining the favourable performance of our novel method in practice.

A Wong--Zakai resonance-based integrator for nonlinear Schrödinger equation with white noise dispersion

TL;DR

This work tackles the simulation of the nonlinear Schrödinger equation with white-noise dispersion in the low-regularity regime. It develops a Wong–Zakai approximation to regularize the stochastic dispersion and constructs a resonance-based, low-regularity integrator (SDLRI) by twisting with the stochastic dispersion, enabling provable convergence under weaker smoothness assumptions. The authors prove well-posedness and mass conservation for the Wong–Zakai model, derive error bounds relative to the full stochastic system, and establish strong and pathwise convergence results, complemented by numerical experiments that corroborate the theoretical rates. The approach offers a practical and theoretically sound framework for simulating stochastic dispersive PDEs with low regularity, and it lays groundwork for future structure-preserving, resonance-based integrators in this setting.

Abstract

We introduce a novel approach to numerical approximation of nonlinear Schrödinger equation with white noise dispersion in the regime of low-regularity solutions. Approximating such solutions in the stochastic setting is particularly challenging due to randomized frequency interactions and presents a compelling challenge for the construction of tailored schemes. In particular, we design the first resonance-based schemes for this equation, which achieve provable convergence for solutions of much lower regularity than previously required. A crucial ingredient in this construction is the Wong--Zakai approximation of stochastic dispersive system, which introduces piecewise linear phases that capture nonlinear frequency interactions and can subsequently be approximated to construct resonance-based schemes. We prove the well-posedness of the Wong--Zakai approximated equation and establish its proximity to the original full stochastic dispersive system. Based on this approximation, we demonstrate an improved strong convergence rate for our new scheme, which exploits the stochastic nature of the dispersive terms. Finally, we provide numerical experiments underlining the favourable performance of our novel method in practice.

Paper Structure

This paper contains 24 sections, 15 theorems, 146 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Wong--Zakai approximation
  3. Wong--Zakai approximation of truncated Brownian motion
  4. Wong--Zakai approximation of the NLSE with white noise dispersion
  5. Approximation error of the Wong--Zakai approximation
  6. Low-regularity resonance-based schemes
  7. Pathwise convergence analysis
  8. Stability
  9. Local error
  10. Global error
  11. Improved mean-square error analysis
  12. Numerical experiments
  13. Validation on deterministic regime
  14. Strong convergence to Wong--Zakai approximation
  15. Pathwise convergence to Wong--Zakai approximation
  16. ...and 9 more sections

Key Result

Lemma 2.1

Let $T>0$, $p\ge 1$ and $\delta\in (0,1)$. There exists $R_0=\max(\sqrt{4p|\ln(\delta)|},$$pe^{-1})>0$ such that for any $R\ge R_0$ and any $t\in [0,T],$ it holds that

Figures (5)

  • Figure 1: The approximation error under $H^1$-norm in the linear dispersion regime $\delta=1, R=\infty, N=512$ for a single trajectory with $C^{\infty}$-data.
  • Figure 2: Strong error plots with $N=512$ and $\delta=2^{-12}$ for varying regularity of $u_0$ and $T=1$.
  • Figure 3: Pathwise convergence plots with $N=512$ and $H^2$-data for several choices of Wong--Zakai parameter $\delta$.
  • Figure 4: Pathwise convergence plots with $N=512$ and $\delta=2^{-12}$ for varying regularity of $u_0$.
  • Figure 5: Error in expectation for SDLRI \ref{['eqn:1st_order_lri']} a function of $\delta$ for $N=512$ and $\tau=0.1,0.01,0.001$ for varying regularity of $u_0$.

Theorems & Definitions (34)

  • Lemma 2.1
  • proof
  • Theorem 2.1
  • proof
  • Remark 2.1
  • Lemma 2.2
  • Proposition 2.3
  • proof
  • Lemma 2.4
  • Remark 3.1
  • ...and 24 more