Polynomial Mixing for a Weakly Damped Stochastic Nonlinear Schrödinger Equation

This paper is devoted to proving the polynomial mixing for a weakly damped stochastic nonlinear Schrödinger equation with additive noise on a 1D bounded domain. The noise is white in time and smooth in space. We consider both focusing and defocusing nonlinearities, respectively, with exponents of the nonlinearity $σ\in[0,2)$ and $σ\in[0,\infty)$ and prove the polynomial mixing which implies the uniqueness of the invariant measure by using a coupling method.