Validity of the stochastic Landau approximation for super-pattern forming systems with a spatial 1:3 resonance

Abstract: We consider a Kuramoto-Shivashinsky like equation close to the threshold of instability with additive white noise and spatially periodic boundary conditions which simultaneously exhibit Turing bifurcations with a spatial 1:3 resonance of the critical wave numbers. For the description of the bifurcating solutions we derive a system of coupled stochastic Landau equations. It is the goal of this paper to prove error estimates between the associated approximation obtained through this amplitude system and true solutions of the original system. The Kuramoto-Shivashinsky like equation serves as a prototype model for so-called super-pattern forming systems with quadratic nonlinearity and additive white noise.