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

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_{*}(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_{*}(q) as q \to 0 . Since the eighties, different heuristic perturbation techniques, like formal asymptotic expansions, have conjectured an asymptotic expression of k_{*} (q) which is of the form k_{*}(q) \sim C q^{-1}e\raisebox{-1pt}{${}^{-\frac{\pi}{2n |q|}}$} 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_{*}(q) , providing a rigorous bound for its relative error, which turns out to be k_{*}(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.