Nonequilibrium stationary state of a truncated stochastic nonlinear Schrödinger equation: Formulation and mean-field approximation

We investigate the stationary state of a model system evolving according to a modified focusing truncated nonlinear Schrödinger equation 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.