arXiv subjects
Publications and source records attributed to C. H. S. Hamster.
I present an overview of different phases used in the literature to describe the phase of travelling waves in reaction-diffusion equations. The discussion is tailored to understand the effectiveness of these different phase descriptions in a stochastic setting in 1D. The explicit computation of the autonomous isochronal phase for the Nagumo equation is a new result.
I investigate the possibility that explicit solutions of stochastic reaction-diffusion equations can be found by multiplying the deterministic travelling waves with a stochastic exponent. This approach has become widespread in the literature in recent years. I will conclude that this approach is, in general, not a valid Ansatz and only works in the case of NLS-type equations in the Stratonovich interpretation.
We consider reaction-diffusion equations that are stochastically forced by a small multiplicative noise term. We show that spectrally stable travelling wave solutions to the deterministic system retain their orbital stability if the amplitude of the noise is sufficiently small. By applying a stochastic phase-shift together with a time-transform, we obtain a semilinear sPDE that describes the fluctuations from the primary wave. We subsequently develop a semigroup approach to handle the nonlinear stability question in a fashion that is closely related to modern deterministic methods.