Traveling wave solutions of a highly nonlinear shallow water equation

Traveling wave solutions of a highly nonlinear shallow water equation

Geyer, A. (TU Delft Mathematical Physics)
Quirchmayr, Ronald (KTH Royal Institute of Technology)

2018

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.

Shallow water equation
traveling waves
phase plane analysis

http://resolver.tudelft.nl/uuid:a8970e80-04d5-4ddb-808b-ebc2286ff705

https://doi.org/10.3934/dcds.2018065

1078-0947

Discrete and Continuous Dynamical Systems A, 38 (3), 1567-1604

Institutional Repository

journal article

© 2018 A. Geyer, Ronald Quirchmayr