Geometric singular perturbation analysis to Camassa-Holm Kuramoto-Sivashinsky equation

Abstract

We analyze a singularly Kuramoto-Sivashinsky perturbed Camassa-Holm equation with methods of the geometric singular perturbation theory. Especially, we study the persistence of smooth and peaked solitons. Whether a solitary wave of the original Camassa-Holm equation is smooth or peaked depends on whether there is linear dispersion, i.e. whether 2k=0. If 2k>0, then a unique smooth solitary wave persists with selected wave speed under singular Kuramoto-Sivashinsky perturbation just as what happens in the KS-KdV equation. On the other hand, we show that if there is no linear dispersion, i.e. 2k=0, then any observable peaked soliton fails to persist. This case is non-typical since the related slow manifold blows up and the classical geometric singular perturbation theory is not available.