Geometric singular perturbation analysis to Camassa-Holm Kuramoto-Sivashinsky equation. (English) Zbl 1477.35160
Summary: 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 \(2 k = 0\). If \(2 k > 0\), 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. \(2 k = 0\), 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.
MSC:
| 35Q35 | PDEs in connection with fluid mechanics |
| 76B25 | Solitary waves for incompressible inviscid fluids |
| 35C08 | Soliton solutions |
| 35B44 | Blow-up in context of PDEs |
| 37K10 | Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) |
| 34C37 | Homoclinic and heteroclinic solutions to ordinary differential equations |
Keywords:
geometric singular perturbation theory; Camassa-Holm equation; solitary wave solutions; invariant manifold; homoclinic orbitsReferences:
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