Traveling wave solutions of a highly nonlinear shallow water equation. (English) Zbl 1397.35209
Summary: Motivated by the question whether higher-order nonlinear model equations, which go beyond the Camassa-Holm regime of moderate amplitude waves, could point us to new types of waves profiles, we study the traveling wave solutions of a quasilinear evolution equation which models the propagation of shallow water waves of large amplitude. The aim of this paper is a complete classification of its traveling wave solutions. Apart from symmetric smooth, peaked and cusped solitary and periodic traveling waves, whose existence is well-known for moderate amplitude equations like Camassa-Holm, we obtain entirely new types of singular traveling waves: periodic waves which exhibit singularities on both crests and troughs simultaneously, waves with asymmetric peaks, as well as multi-crested smooth and multi-peaked waves with decay. Our approach uses qualitative tools for dynamical systems and methods for integrable planar systems.
MSC:
| 35Q35 | PDEs in connection with fluid mechanics |
| 34C37 | Homoclinic and heteroclinic solutions to ordinary differential equations |
| 35C07 | Traveling wave solutions |
| 76B15 | Water waves, gravity waves; dispersion and scattering, nonlinear interaction |
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