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On the Camassa–Holm equation and a direct method of solution. II. Soliton solutions

Lookup NU author(s): Dr Allen Parker


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Previous attempts to find explicit analytic multisoliton solutions of the general Camassa–Holm (CH) equation have met with limited success. This study (which falls into two parts, designated II and III) extends the results of the prior work (I) in which a bilinear form of the CH equation was constructed and then solved for the solitary-wave solutions. It is shown that Hirota's bilinear transformation method can be used to derive exact multisoliton solutions of the equation in a systematic way. Here, analytic two-soliton solutions are obtained explicitly and their structure and dynamics are investigated in the different parameter regimes, including the limiting ‘two-peakon’ form. The solutions possess a non-standard representation that is characterized by an additional parameter, and the structure of this key parameter is examined. These results pave the way for constructing the hallmark N-soliton solutions of the CH equation in part III.

Publication metadata

Author(s): Parker A

Publication type: Article

Publication status: Published

Journal: Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

Year: 2005

Volume: 461

Issue: 2063

Pages: 3611-3632

ISSN (print): 1471-2946

ISSN (electronic): 1471-2946

Publisher: The Royal Society Publishing


DOI: 10.1098/rspa.2005.1536


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