Exponential integrators for nonlinear Schrödinger equations with white noise dispersion
type: Item
instance of: scholarly article
title: Exponential integrators for nonlinear Schrödinger equations with white noise dispersion
zbMATH Open document ID: 1386.65036
author: Peng Zhang
published in: Stochastic and Partial Differential Equations. Analysis and Computations
publication date: 2018-03-28T00:00:00Z
full work available at URL: http://urn.kb.se/resolve
review text: The authors consider the following nonlinear Schrödinger equation with white noise dispersion in the Stratonovich sense \[ {i} du + c \Delta u \circ d\beta + |u|^{2 \sigma} u \, dt =0, \] \[ u(0)=u_0, \] where the unknown \(u=u(x,t)\) with \(t \geq0\) and \(x \in \mathbb R^{d}\) is a complex valued random process, \[ \Delta u = \sum_{j=1}^d \frac{ \partial^2 u }{ \partial x_j^2 } \] denotes the Laplacian in \(\mathbb R^d\), \(c\) is a real number, \(\sigma\) is a positive number, and \(\beta=\beta(t)\) is a real valued standard Brownian motion. A new exponential integrator is introduced that integrates the noisy part of the equation exactly. This new scheme has mean-square order one. In contrast to methods such as Lie-Trotter splitting and the Crank-Nicolson method, it does not preserve the \(L^2\)-norm exactly. A second exponential integrator is proposed, which is implicit and symmetric, and in contrast to the first one preserves the \(L^2\)-norm of the solution.
reviewed by: Rózsa Horváth-Bokor
Mathematics Subject Classification ID: 65C30
Mathematics Subject Classification ID: 35Q55
Mathematics Subject Classification ID: 60H15
zbMATH DE Number: 6853821
zbMATH Keywords: stochastic partial differential equations
zbMATH Keywords: nonlinear Schrödinger equation
zbMATH Keywords: white noise dispersion
zbMATH Keywords: numerical methods
zbMATH Keywords: geometric numerical integration
zbMATH Keywords: exponential integrators
zbMATH Keywords: mean-square convergence
MaRDI profile type: Publication
OpenAlex ID: W2599234101
cites work: Exponential Integrators for Stochastic Schrödinger Equations Driven by Itô Noise
cites work: Full discretization of semilinear stochastic wave equations driven by multiplicative noise
cites work: Numerical analysis of the nonlinear Schrödinger equation with white noise dispersion
cites work: Conservation of phase space properties using exponential integrators on the cubic Schrödinger equation
cites work: High Order Exponential Integrators for Nonlinear Schrödinger Equations with Application to Rotating Bose--Einstein Condensates
cites work: Exponential time integration of solitary waves of cubic Schrödinger equation
cites work: Projected explicit lawson methods for the integration of Schrödinger equation
cites work: Symmetric exponential integrators with an application to the cubic Schrödinger equation
cites work: One-stage exponential integrators for nonlinear Schrödinger equations over long times
cites work: A Trigonometric Method for the Linear Stochastic Wave Equation
cites work: A fully discrete approximation of the one-dimensional stochastic wave equation
cites work: Stochastic symplectic and multi-symplectic methods for nonlinear Schrödinger equation with white noise dispersion
cites work: The nonlinear Schrödinger equation with white noise dispersion
cites work: 1D quintic nonlinear Schrödinger equation with white noise dispersion
cites work: Exponential Runge-Kutta methods for the Schrödinger equation
cites work: Probability and Pathwise Order of Convergence of a Semidiscrete Scheme for the Stochastic Manakov Equation
cites work: Geometric Numerical Integration
cites work: Exponential integrators for quantum-classical molecular dynamics
cites work: Exponential integrators
cites work: The numerical approximation of stochastic partial differential equations
cites work: A numerical scheme for stochastic PDEs with Gevrey regularity
cites work: Stochastic exponential integrators for the finite element discretization of SPDEs for multiplicative and additive noise
cites work: On a splitting scheme for the nonlinear Schrödinger equation in a random medium
cites work: An exponential integrator scheme for time discretization of nonlinear stochastic wave equation
Wikidata QID: Q59603552
DOI: 10.1007/S40072-017-0098-1
description: scientific article; zbMATH DE number 6853821