The effect of boundaries on the asymptotic wavenumber of spiral wave solutions of the complex Ginzburg–Landau equation

Abstract

In this paper we consider an oscillatory medium whose dynamics are modeled by the complex Ginzburg–Landau equation. In particular, we focus on n-armed spiral wave solutions of the complex Ginzburg–Landau equation in a disk of radius d with homogeneous Neumann boundary conditions. It is well-known that such solutions exist for small enough values of the twist parameter q and large enough values of d. We investigate the effect of boundaries on the rotational frequency of the spirals, which is an unknown of the problem uniquely determined by the parameters d and q. We show that there is a threshold in the parameter space where the effect of the boundary on the rotational frequency switches from being algebraic to exponentially weak. We use the method of matched asymptotic expansions to obtain explicit expressions for the asymptotic wavenumber as a function of the twist parameter and the domain size for small values of q.