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

Abstract

In this work, $n$-armed Archimedean spiral wave solutions of the complex Ginzburg–Landau equation are considered. These solutions are shown to depend on two characteristic parameters, the so-called twist parameter $q$ and the asymptotic wavenumber $k$. The existence and uniqueness of the value of $k=k_{\ast }(q)$ for which $n$-armed Archimedean spiral wave solutions exist is a classical result, obtained back in the eighties by Kopell and Howard. In this work, we deal with a different problem, that is, the asymptotic expression of $k_{\ast }(q)$ as $q\to 0$. Since the eighties, different heuristic perturbation techniques, like formal asymptotic expansions, have conjectured an asymptotic expression of $k_{\ast }(q)$ which is of the form $k_{\ast }(q)\sim C{q}^{-1}e{}^{-\frac{\pi }{2n|q|}}$ with a known constant $C$. 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 for $k_{\ast }(q)$, providing a rigorous bound for its relative error, which turns out to be $k_{\ast }(q)=C{q}^{-1}{e}^{-\frac{\pi }{2nq}}(1+\mathcal{O}(|\log |q|{|}^{-1}))$. Moreover, such approach can be used in more general equations such as the celebrated $\lambda -\omega$ systems.