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

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. (C) 2021 Elsevier Inc. All rights reserved.

Automated summary

In this study, a singularly perturbed Camassa-Holm equation with Kuramoto-Sivashinsky perturbation is analyzed using methods of geometric singular perturbation theory. The persistence of smooth and peaked solitons is studied, and it is found that the presence or absence of linear dispersion determines whether these solitons will persist under singular perturbation. The behavior of the system differs when there is linear dispersion compared to when there is none.