Traveling wave solutions of a highly nonlinear shallow water equation

Anna Geyer, Ronald Quirchmayr

Research output: Contribution to journalArticleScientificpeer-review

6 Citations (Scopus)
15 Downloads (Pure)


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.
Original languageEnglish
Pages (from-to)1567-1604
Number of pages38
JournalDiscrete and Continuous Dynamical Systems A
Issue number3
Publication statusPublished - 2018


  • Shallow water equation
  • traveling waves
  • phase plane analysis


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