# Nonequilibrium stationary state of a truncated stochastic NLSE: I. Formulation and mean field approximation

## TL;DR

This paper models a truncated stochastic nonlinear Schrödinger equation to study its nonequilibrium stationary states, revealing a phase transition influenced by wave-breaking effects and nonlinearity strength.

## Contribution

It introduces a novel approach combining wave-breaking with a mean field approximation to analyze nonequilibrium states in a truncated NLSE model.

## Key findings

- System exhibits a transition from low to high field regimes with increasing nonlinearity.
- Wave-breaking reduces large field values compared to the no wave-breaking case.
- The system reaches a unique nonequilibrium stationary state under wave-breaking conditions.

## Abstract

We investigate the stationary state of a model system evolving according to a modified focusing truncated nonlinear Schr\"odinger equation (NLSE) used to describe the envelope of Langmuir waves in a plasma. We restrict the system to have a finite number of normal modes each of which is in contact with a Langevin heat bath at temperature $T$. Arbitrarily large realizations of the field are prevented by restricting each mode to a maximum amplitude. We consider a simple modeling of wave-breaking in which each mode is set equal to zero when it reaches its maximum amplitude. Without wave-breaking the stationary state is given by a Gibbs measure. With wave-breaking the system attains a nonequilibrium stationary state which is the unique invariant measure of the time evolution. A mean field analysis shows that the system exhibits a transition from a regime of low field values at small $|\lambda|$, to a regime of higher field values at large $|\lambda|$, where $\lambda<0$ specifies the strength of the nonlinearity in the focusing case. Field values at large $|\lambda|$ are significantly smaller with wave-breaking than without wave-breaking.