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

We investigate the stationary state of a model system evolving according to a modified focusing truncated nonlinear Schrödinger 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 $|λ|$, to a regime of higher field values at large $|λ|$, where $λ<0$ specifies the strength of the nonlinearity in the focusing case. Field values at large $|λ|$ are significantly smaller with wave-breaking than without wave-breaking.

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