Amplitude Equation for SPDEs with Quadratic Non-Linearities

In this paper we rigorously derive stochastic amplitude equations for a rather general class of SPDEs with quadratic nonlinearities forced by small additive noise. Near a change of stability we use the natural separation of time-scales to show that the solution of the original SPDE is approximated by the solution of an amplitude equation, which describes the evolution of dominant modes. Our results significantly improve older results. We focus on equations with quadratic nonlinearities and give applications to the one-dimensional Burgers’ equation and a model from surface growth.