Analysis of a Splitting Scheme for Damped Stochastic Nonlinear Schrödinger Equation with Multiplicative Noise

Abstract

In this paper, we investigate the damped stochastic nonlinear Schrödinger(NLS) equation with multiplicative noise and its splitting-based approximation. When the damped effect is large enough, we prove that the solutions of both the damped stochastic NLS equation and the splitting scheme are exponentially stable and possess some exponential integrability. These properties show that the strong order of the scheme is 1 2 and independent of time. Additionally, we analyze the regularity of the Kolmogorov equation with respect to the stochastic nonlinear Schrödinger equation. As a consequence, the weak order of the scheme is shown to be 1 and independent of time. 1This manuscript is for review purposes only.2. Some properties for damped stochastic NLS equation. We first introduce some frequently used notation and assumptions. The norm and inner product of H := L 2 (R; C) are denoted by · and u, v := ℜ R u(x)v(x)dx , respectively. This manuscript is for review purposes only.

Strong convergence rates of semidiscrete splitting approximations for the stochastic Allen–Cahn equation

“…The following lemma states standard a priori estimates for the processes X, X N and Y δt defined by Eq. (1), (7) and (9) respectively. For convenience, throughout this paper, we omit the mollification procedure to get the evolution of · L q .…”

“…We would like to mention that these splitting-up based methods have many applications on approximating SPDEs with the Lipschitz nonlinearity, and are also used for approximating SPDEs with non-Lipschitz or non-monotone nonlinearities (see e.g. [7,9,11,15]).…”

“…To the best of our knowledge,this is the first result with strong convergence order 1 about the temporal numerical schemes approximating the stochastic Allen-Cahn equation. For similar approaches to derive the strong convergence rates of numerical schemes, we refer to [7,16,18] and the references therein. We first study stability and exponential integrability properties of the exact solution, in dimensions d ≤ 3, and obtain results of their own interest beyond analysis of numerical schemes.…”

On global existence and blow-up for damped stochastic nonlinear Schrödinger equation

“…Moreover, we can get the following exponential integrability of u, which is useful for studying the continuous dependence on initial data and noise, exponential tail estimate of the solution, strong and weak convergence rates of numerical approximations (see e.g. [7,8,9]). Again by the Gagliardo-Nirenberg inequality (6) and the Sobolev embedding theorem, we obtain…”

“…with α depending on u 0 , a and Q, we obtain the strong continuous dependence on the initial data in one dimensional case, which is not a trivial property for stochastic partial differential equation with non-global coefficients, see [6,8] and references therein. We would like to mention that this exponential integrability is useful for studying the continuous dependence on noises, exponential tail estimate of the solution, strong and weak convergence rates of numerical approximations, see [6,7,8,9,15] and references therein. Next we consider the influence of damped term and noise on the blow-up.…”