Exponential integrators for nonlinear Schrödinger equations with white noise dispersion
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.
stochastic partial differential equations
scientific article; zbMATH DE number 6853821
scientific article