Effective dynamics of interfaces for nonlinear SPDEs driven by multiplicative white noise

In this work, we investigate the dynamics of the infinite-dimensional stochastic partial differential equation (SPDE) with multiplicative white noise. We derive the effective equation on the approximate slow manifold in detail by utilizing a finite-dimensional stochastic differential equation (SDE) that describes the motion of interfaces. In particular, we verify the equivalence between the full SPDE and the coupled system under small stochastic perturbations. Moreover, we apply our results to the effective dynamics of stochastic models with multiplicative white noise, providing four illustrative examples on the stochastic damped wave equation, the stochastic Allen–Cahn equation, the stochastic nonlinear Schrödinger equation and the stochastic Swift–Hohenberg equation.