Abstract
This article deals with the numerical integration in time of the nonlinear Schrödinger equation with power law nonlinearity and random dispersion. We introduce a new explicit exponential integrator for this purpose that integrates the noisy part of the equation exactly. We prove that this scheme is of mean-square order 1 and we draw consequences of this fact. We compare our exponential integrator with several other numerical methods from the literature. We finally propose a second exponential integrator, which is implicit and symmetric and, in contrast to the first one, preserves the L^2-norm of the solution.
Highlights
We consider the time discretisation of the following nonlinear Schrödinger equation with white noise dispersion idu + c u ◦ dβ + |u|2σ u dt = 0 u(0) = u0, (1.1)where the unknown u = u(x, t), with t ≥ 0 and x ∈ Rd, is a complex valued random process, u = d j =1 ∂2u ∂ x2 j denotes the Laplacian inRd, c is a real number, σ positive real number, and β = β(t) is a real valued standard Brownian motion
We review the literature on the numerical analysis of the nonlinear Schrödinger equation with white noise dispersion (1.1)
This subsection presents convergence plots for the above mentioned numerical methods applied to the nonlinear Schrödinger equation with white noise dispersion (1.1); space-time evolution plots; experiments illustrating the influence of the power nonlinearity σ supporting a conjecture proposed in [3]; and illustrations of the preservation of the L2-norm along numerical solutions
Summary
The authors of [3] studied an implicit Crank–Nicolson scheme for the time integration of (1.1) They show that this scheme preserves the L2-norm and has order one of convergence in probability. Exponential integrators for the time integration of deterministic semi-linear problems of the form y = L y + N (y), are nowadays widely used and studied, as witnessed by the recent review [22] Applications of such numerical schemes to the deterministic (nonlinear) Schrödinger equation can be found in, for example, [4,5,6,7,8,9,10,17,21] and references therein. 2. After that, we present our explicit exponential integrator for the numerical approximation of the above stochastic Schrödinger equation and analyse its convergence in Sect. We discuss the preservation of the mass, or L2-norm, by symmetric exponential integrators
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