On the Spectral Problem $${\mathcal{L} u=\lambda u'}$$ and Applications

Abstract

We develop a general instability index theory for an eigenvalue problem of the type \({\mathcal{L} u=\lambda u'}\), for a class of self-adjoint operators \({\mathcal{L}}\) on the line R1. More precisely, we construct an Evans-like function to show (a real eigenvalue) instability in terms of a Vakhitov–Kolokolov type condition on the wave. If this condition fails, we show by means of Lyapunov–Schmidt reduction arguments and the Kapitula–Kevrekidis–Sandstede index theory that spectral stability holds. Thus, we have a complete spectral picture, under fairly general assumptions on \({\mathcal{L}}\). We apply the theory to a wide variety of examples. For the generalized Bullough–Dodd–Tzitzeica type models, we give instability results for travelling waves. For the generalized short pulse/Ostrovsky/Vakhnenko model, we construct (almost) explicit peakon solutions, which are found to be unstable, for all values of the parameters.