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.

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

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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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author = "Cotter, {Colin J.} and Holm, {Darryl D.} and Tristan Pryer",

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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N2 - 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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Singular solutions of the r-Camassa-Holm equation

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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