Singular solutions of the r-Camassa-Holm equation

Colin J. Cotter; Darryl D. Holm; Tristan Pryer

doi:10.1088/1361-6544/ad020a

Singular solutions of the r-Camassa-Holm equation

Colin J. Cotter, Darryl D. Holm, Tristan Pryer

Imperial College London

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1 Citation (SciVal)

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
Pages (from-to)	6199-6223
Number of pages	25
Journal	Nonlinearity
Volume	36
Issue number	11
Early online date	20 Oct 2023
DOIs	https://doi.org/10.1088/1361-6544/ad020a
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.

Funding

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).

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

Access to Document

10.1088/1361-6544/ad020aLicence: CC BY

Cite this

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title = "Singular solutions of the r-Camassa-Holm equation",

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.",

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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.",

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Singular solutions of the r-Camassa-Holm equation

Abstract

Bibliographical note

Funding

Keywords

ASJC Scopus subject areas

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