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Right and Wrong Ansätze for Nonlinear Waves in Stochastic PDEs

C. H. S. Hamster

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.

Right and Wrong Ansätze for Nonlinear Waves 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.

Paper Structure

This paper contains 11 sections, 57 equations.

Table of Contents

  1. Introduction
  2. Origins of flawed Ansätze
  3. A systematic approach to exact solutions
  4. Advective 1D noise
  5. Additive 1D noise
  6. A case study
  7. Several classic PDEs
  8. Burgers' Equation
  9. NLS equation
  10. Stratonovich interpretation
  11. Where to next

Theorems & Definitions (1)

  • Remark 1