This dissertation deals with a class of nonlinear wave equations of the type discovered by R. Camassa and D. D. Holm which includes the Camassa-Holm, the Degasperis-Procesi, and the two component Camassa-Holm equations. All these equations admit certain non-smooth soliton-like solutions, called peakons as well as other non-smooth solutions like cuspons. We apply the techniques of the theory of distributions of L. Schwartz to study these solutions. In particular, every non-smooth traveling wave which is a distributional solution of the two component Camassa-Holm equation is a distributional solution of the Camassa-Holm equation if the set of points where the height of the wave equals its speed, is of measure zero. This includes peakon or cuspon traveling wave solutions.<p>We also develop a suitable modification of the classical Lax pair formalism to deal with singular solutions. We show that the Lax pair formalism can be extended to a distributional weak Lax pair which is appropriate for dealing with the peakon solutions of the Camassa-Holm equation.
Identifer | oai:union.ndltd.org:LACETR/oai:collectionscanada.gc.ca:SSU.etd-09162008-132121 |
Date | 18 September 2008 |
Creators | Mohajer, Keivan |
Contributors | Sowa, A., Smolyakov, Andrei I., Patrick, George W., Misiolek, G., Szmigielski, Jacek |
Publisher | University of Saskatchewan |
Source Sets | Library and Archives Canada ETDs Repository / Centre d'archives des thèses électroniques de Bibliothèque et Archives Canada |
Language | English |
Detected Language | English |
Type | text |
Format | application/pdf |
Source | http://library.usask.ca/theses/available/etd-09162008-132121/ |
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