Convergence of a $$\theta $$ θ -scheme to solve the stochastic nonlinear Schrödinger equation with Stratonovich noise

Abstract

We propose a θ -scheme to discretize the d-dimensional stochastic cubic Schrödinger equation in Stratonovich sense. A uniform bound for the Hamiltonian of the discrete problem is obtained, which is a crucial property to verify the convergence in probability towards a mild solution. Furthermore, based on the uniform bounds of iterates in H 2 (O) for O ⊂ R 1 , the convergence order 1 2 in strong local sense is obtained.

Strong convergence rate of finite difference approximations for stochastic cubic Schrödinger equations

“…These difficulties are common features to obtain strong convergence rates for numerical approximations appearing in many situations, see e.g. [CHP16,dBD06] for stochastic nonlinear Schrödinger equations and [BJ13,KLM11] for other SPDEs with non-monotone coefficients. Our main idea is applying Gronwall-Bellman inequality to (8) before taking expectation.…”

“…(1); see e.g. [CHP16,dBD04,dBD06] and references therein. To deal with the nonlinearity, one usually apply the truncation technique.…”

Fluctuations and temperature effects in bose-einstein condensation

“…The argument is again to use a cut-off in the highest modes of the operator A = −∆+|x| 2 which would correspond to the use of a spectral space discretization. Note that a different argument was used in [9], for the standard, cubic nonlinear Schrödinger equation with multiplicative noise, consisting in slightly more impliciteness in the scheme, and leading to better stability properties. However, the argument does not apply in the present situation, were the lack of stability is due to a lack of "good" commutator properties between the noise and the linear operator.…”

New scheme for nonlinear Schrödinger equations with variable coefficients

“…The Strang-type splitting scheme is proposed to NSE with multiplicative noise [13]. A θ-scheme is analyzed for NSE with Stratonovich noise [3].…”