Amplitude equations for SPDEs with cubic nonlinearity forced by general multiplicative Gaussian noise

The paper is devoted to a class of SPDEs with cubic nonlinearity driven by multiplicative and degenerate additive noise. Our main goal is to study dynamical behavior of SPDEs close to a change of stability of the corresponding deterministic system. We use the natural separation of timescales to derive amplitude equations describing the essential dynamics of the dominant modes. With the help of the amplitude equations, we show the approximation of solution of SPDEs. A Ginzburg–Landau equation with both kinds of noise will also be presented to illustrate our main result.