Influence of additive white noise forcing on solutions of mixed NLS equations

Abstract

In this paper, the influence of an additive white noise forcing term on the numerical solution for a class of deterministic nonlinear one-dimensional Schrödinger equations with mixed concave convex was studied, sub-super nonlinearities, that is, the stationary states and the blowing-up solutions. Such a perturbation occurs when the size of the noise, described by the real-value parameter  is positive. The size of the noise is controlled by the parameter  We also proved that as  approaches zero, the solution of the perturbed problem converges to the unique trajectory of the deterministic equation, which is the solitary wave. The stochastic model appears to be more realistic, and one can observe, for small values of , a similar evolution phenomena about the solution as that given by the deterministic case. However, an explosion of the solution and a blow-up phenomena can be noted as  becomes bigger.

Keywords: Nonlinear Schrödinger equation, Mixed nonlinearity, Blow-up phenomena, Finite difference scheme, White noise, Solvability.