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Strong Approximation of Stochastic Semiclassical Schroedinger Equation with Multiplicative Noise

Lihai Ji, Zhihui Liu

TL;DR

This work tackles the stochastic semiclassical nonlinear Schrödinger equation with multiplicative noise in the small-$\varepsilon$ regime, where high-frequency oscillations Challenge numerical methods. It develops a numerical framework combining spectral Galerkin spatial discretization with a midpoint time integrator to efficiently approximate the solution and observables, and it establishes strong convergence rates that depend explicitly on $\varepsilon$. The main contributions include well-posedness and moment estimates for the stochastic problem, sharp strong error bounds for both spatial and spatio-temporal discretizations, and practical meshing guidelines linking $N$ and $\tau$ to a desired accuracy. These results provide rigorous guidance for accurate simulation of semiclassical stochastic wave dynamics and observation of physically meaningful quantities.

Abstract

We consider the stochastic nonlinear Schroedinger equation driven by a multiplicative noise in a semiclassical regime, where the Plank constant is small. In this regime, the solution of the equation exhibits high-frequency oscillations. We design an efficient numerical method combining the spectral Galerkin approximation and the midpoint scheme. This accurately approximates the solution, or at least of the associated physical observables. Furthermore, the strong convergence rate for the proposed scheme is derived, which explicitly depends on the Planck constant. This conclusion implies the semiclassical regime's admissible meshing strategies for obtaining "correct" physical observables.

Strong Approximation of Stochastic Semiclassical Schroedinger Equation with Multiplicative Noise

TL;DR

This work tackles the stochastic semiclassical nonlinear Schrödinger equation with multiplicative noise in the small- regime, where high-frequency oscillations Challenge numerical methods. It develops a numerical framework combining spectral Galerkin spatial discretization with a midpoint time integrator to efficiently approximate the solution and observables, and it establishes strong convergence rates that depend explicitly on . The main contributions include well-posedness and moment estimates for the stochastic problem, sharp strong error bounds for both spatial and spatio-temporal discretizations, and practical meshing guidelines linking and to a desired accuracy. These results provide rigorous guidance for accurate simulation of semiclassical stochastic wave dynamics and observation of physically meaningful quantities.

Abstract

We consider the stochastic nonlinear Schroedinger equation driven by a multiplicative noise in a semiclassical regime, where the Plank constant is small. In this regime, the solution of the equation exhibits high-frequency oscillations. We design an efficient numerical method combining the spectral Galerkin approximation and the midpoint scheme. This accurately approximates the solution, or at least of the associated physical observables. Furthermore, the strong convergence rate for the proposed scheme is derived, which explicitly depends on the Planck constant. This conclusion implies the semiclassical regime's admissible meshing strategies for obtaining "correct" physical observables.
Paper Structure (5 sections, 9 theorems, 78 equations)

This paper contains 5 sections, 9 theorems, 78 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Well-posedness and Moment's Estimates
  4. Spatial Semi-discretization
  5. Spatio-temporal Full Discretization

Key Result

Theorem 3.1

Let $p \ge 2$, $\mu>0$, $u_0^\epsilon \in L^p(\Omega; \dot H^\mu)$, and Assumption ap-fg hold. Then Eq. semi-NLS possesses a unique solution $\{u(t):t \ge 0\}$ and there exists a positive constant $C$ such that and for any $t,s \in [0, T]$, In particular if $F(u)={ \bf i} \alpha u$ with $\alpha>0$ and $G={\rm Id}$ such that con-add holds, then there exists a positive constant $C$ such that for a

Theorems & Definitions (19)

  • Remark 2.1
  • Remark 2.2
  • Theorem 3.1
  • proof
  • Theorem 4.1
  • proof
  • Lemma 5.1
  • Theorem 5.1
  • proof
  • Theorem 5.2
  • ...and 9 more