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Genesis of solitons arising from individual flows of the Camassa-Holm hierarchy


Loubet, E (2006). Genesis of solitons arising from individual flows of the Camassa-Holm hierarchy. Communications on Pure and Applied Mathematics, 59(3):408-465.

Abstract

The present work offers a detailed account of the large-time development of the velocity profile run by a single individual Hamiltonian flow of the Camassa-Holm (CH) hierarchy, the Hamiltonian employed being the reciprocal of any eigenvalue of the underlying spectral problem. In this simpler scenario, I prove some of the conjectures raised by McKean [27]. Notably, I confirm the ultimate shaping into solitons of the cusps that appear, near blowup sites, of any velocity profile emanating from an initial disposition for which breakdown of the wave in finite time is sure to happen. The careful large-time asymptotic analysis is carried from exact expressions describing the velocity in terms of initial data, the integration involving a Lagrangian scale and three theta functions, the rates at which the latter reach their common values at each end of the line characterizing the region where soliton genesis is expected. In fact, the present method also suggests how solitons may arise from initial conditions not leading to breakdown. The full CH flow is nothing but a superposition of such commuting individual actions. Therein lies the hope that the present account will pave the way to elucidate soliton formation for more complex flows, in particular for the CH flow itself. © 2005 Wiley Periodicals, Inc.

The present work offers a detailed account of the large-time development of the velocity profile run by a single individual Hamiltonian flow of the Camassa-Holm (CH) hierarchy, the Hamiltonian employed being the reciprocal of any eigenvalue of the underlying spectral problem. In this simpler scenario, I prove some of the conjectures raised by McKean [27]. Notably, I confirm the ultimate shaping into solitons of the cusps that appear, near blowup sites, of any velocity profile emanating from an initial disposition for which breakdown of the wave in finite time is sure to happen. The careful large-time asymptotic analysis is carried from exact expressions describing the velocity in terms of initial data, the integration involving a Lagrangian scale and three theta functions, the rates at which the latter reach their common values at each end of the line characterizing the region where soliton genesis is expected. In fact, the present method also suggests how solitons may arise from initial conditions not leading to breakdown. The full CH flow is nothing but a superposition of such commuting individual actions. Therein lies the hope that the present account will pave the way to elucidate soliton formation for more complex flows, in particular for the CH flow itself. © 2005 Wiley Periodicals, Inc.

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4 citations in Scopus®
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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Language:English
Date:2006
Deposited On:20 Jan 2010 07:51
Last Modified:05 Apr 2016 13:24
Publisher:Wiley-Blackwell
ISSN:0010-3640
Publisher DOI:https://doi.org/10.1002/cpa.20109

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