From Stochastic Zakharov System to Multiplicative Stochastic Nonlinear Schr{\"o}dinger Equation

TL;DR

This paper proves the convergence of a stochastic Zakharov system driven by spatially colored white noise to a multiplicative stochastic nonlinear Schrödinger equation as ion-sound speed increases, using diffusion-approximation and perturbed test-function methods.

Contribution

It establishes the first convergence result for a stochastic Zakharov system to a stochastic NLS, including existence and uniqueness of solutions and handling noise-induced singularities.

Findings

Convergence in probability of the Zakharov system to the stochastic NLS

Existence and uniqueness of regular solutions for the stochastic Zakharov system

Limitation of results to one-dimensional case

Abstract

We study the convergence of a Zakharov system driven by a time white noise, colored in space, to a multiplicative stochastic nonlinear Schr{\"o}dinger equation, as the ion-sound speed tends to infinity. In the absence of noise, the conservation of energy gives bounds on the solutions, but this evolution becomes singular in the presence of the noise. To overcome this difficulty, we show that the problem may be recasted in the diffusion-approximation framework, and make use of the perturbed test-function method. We also obtain convergence in probability. The result is limited to dimension one, to avoid too much technicalities. As a prerequisite, we prove the existence and uniqueness of regular solutions of the stochastic Zakharov system.