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 $\sigma \in [0,2)$ and $\sigma \in [0,\mathrm{\infty })$ and prove the polynomial mixing which implies the uniqueness of the invariant measure by using a coupling method.