A rigorous derivation of the asymptotic wavenumber of spiral wave solutions of the complex Ginzburg-Landau equation

Abstract

In this work n-armed Archimedian spiral wave solutions of the complex Ginzburg-Landau equation are considered. These solutions are showed to depend on two characteristic parameters, the so called twist parameter and the asymptotic wavenumber. The existence and uniqueness of the value of the asymptotic wavenumber, depending on the twist parameter, for which n-armed Archimedian spiral wave solutions exist is a classical result, obtained back in the 80s by Kopell and Howard. In this work we deal with a different problem, that is, the asymptotic expression of the asymtptotic wavenumer for small values of the twist parameter. Since the eighties, different heuristic perturbation techniques, like formal asymptotic expansions, have conjectured an asymptotic expression of which is exponentially small with respect to the twist parameter. However, the validity of this expression has remained opened until now, despite of the fact that it has been widely used for more than 40 years. In this work, using a functional analysis approach, we finally prove the validity of the asymptotic formula, providing a rigorous bound for its relative error