## Abstract

This paper introduces the r-Camassa-Holm (r-CH) equation, which describes a geodesic flow on the manifold of diffeomorphisms acting on the real line induced by the W 1 , r metric. The conserved energy for the problem is given by the full W 1 , r norm. For r = 2, we recover the Camassa-Holm equation. We compute the Lie symmetries for r-CH and study various symmetry reductions. We introduce singular weak solutions of the r-CH equation for r ⩾ 2 and demonstrates their robustness in numerical simulations of their nonlinear interactions in both overtaking and head-on collisions. Several open questions are formulated about the unexplored properties of the r-CH weak singular solutions, including the question of whether they would emerge from smooth initial conditions.

Original language | English |
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Pages (from-to) | 6199-6223 |

Number of pages | 25 |

Journal | Nonlinearity |

Volume | 36 |

Issue number | 11 |

Early online date | 20 Oct 2023 |

DOIs | |

Publication status | Published - 1 Nov 2023 |

### Bibliographical note

Funding Information:We would like to thank our friends and colleagues who have generously offered their attention, thoughts and encouragement in the course of this work during the time of COVID-19. We thank Jonathan Mestel for useful discussions about equation (), which kickstarted this work. C J C is grateful for partial support from EPSRC (EP/W015439/1, EP/W016125/1, EP/R029423/1, EP/R029628/1, EP/L016613/1) and NERC (NE/R008795/1). D H is grateful for partial support from ERC Synergy Grant 856408—STUOD (Stochastic Transport in Upper Ocean Dynamics). T P is grateful for partial support from EPSRC (EP/X017206/1, EP/X030067/1 and EP/W026899/1) and the Leverhulme Trust (RPG-2021-238).

Data availability statement: No new data were created or analysed in this study.

## Keywords

- 35Q53
- 37K05
- 37K06
- 37K58
- Camassa-Holm equation
- peakons
- singular solutions

## ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics
- General Physics and Astronomy
- Applied Mathematics