On the inviscid limit of the singular stochastic complex Ginzburg-Landau equation at statistical equilibrium

TL;DR

This paper investigates how, under small viscosity and noise, the stochastic complex Ginzburg-Landau equation converges to the deterministic nonlinear Schrödinger equation at equilibrium, using advanced analytical techniques.

Contribution

It demonstrates the convergence of the singular stochastic complex Ginzburg-Landau equation to the nonlinear Schrödinger equation in a specific regime, combining heat and Schrödinger analysis methods.

Findings

Convergence of the stochastic Ginzburg-Landau to NLS at equilibrium.

Identification of the limiting dynamics in small viscosity/noise regimes.

Application of Fourier restriction norm method for analysis.

Abstract

We study the limiting behavior of the two-dimensional singular stochastic stochastic cubic nonlinear complex Ginzburg-Landau with Gibbs measure initial data. We show that in the appropriate small viscosity and small noise regimes, the limiting dynamics is given by the deterministic cubic nonlinear Schr\"odinger equation at Gibbs equilibrium. In order to obtain this convergence, our approach combines both heat and Schr\"odinger analysis, within the framework of the Fourier restriction norm method of Bourgain (1993).