Right and Wrong Ans\"atze for Nonlinear Waves in Stochastic PDEs

TL;DR

This paper critically examines the common method of using stochastic exponents to find explicit solutions of stochastic reaction-diffusion equations, concluding it is generally invalid except for NLS-type equations in the Stratonovich interpretation.

Contribution

It clarifies the limitations of the stochastic exponent Ansatz for nonlinear waves in stochastic PDEs, specifying when it is valid.

Findings

The Ansatz is generally invalid for stochastic reaction-diffusion equations.

It is valid only for NLS-type equations in the Stratonovich interpretation.

Provides guidance on when explicit solutions can be constructed in stochastic PDEs.

Abstract

I investigate the possibility that explicit solutions of stochastic reaction-diffusion equations can be found by multiplying the deterministic travelling waves with a stochastic exponent. This approach has become widespread in the literature in recent years. I will conclude that this approach is, in general, not a valid Ansatz and only works in the case of NLS-type equations in the Stratonovich interpretation.