Orbital Stability of Patterns in Semilinear SPDE using a Multiscale Analysis

Abstract

In this thesis we consider orbital stability of certain patterns in stochastic partial differential equations. We study two examples: a rotating wave in a two-dimensional reaction-diffusion equation and a soliton in a parametrically forced nonlinear Schrödinger equation. In both cases, we show that, for small noise, solutions to the stochastic equations remain close to a version of the pattern which is shifted according to some stochastic phase. We give explicit expressions for this phase, and show that it is optimal to first order in the strength of the noise.

To show stability, we construct a multiscale expansion of the solution around an arbitrarily shifted version of the pattern, and show that this expansion is accurate to second order. From this expansion an obvious candidate for the correct phase shift arises. For technical reasons we then construct a sequence of approximations to this phase shift, which is necessary to show the multiscale expansion around the correctly shifted pattern. We then combine this expansion with a deterministic stability result to get stochastic stability.

Finally, we take first steps towards formulating and proving the same results in a more general setting, where the pattern shift is represented by the action of a Lie group. We obtain some estimates necessary for the multiscale expansion, find the correct phase, and formulate necessary assumptions for the stability to hold.