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Two-component Camassa-Holm system and its reductions

Presented by: 
Yoshimasa Matsuno Yamaguchi University
Wednesday 9th August 2017 - 11:30 to 12:30
INI Seminar Room 1
My talk is mainly concerned with an integrable two-component Camassa-Holm (CH2) system which describes the propagation of nonlinear shallow water waves.  After a brief review of strongly nonlinear models for shallow water waves including the Green-Naghdi and related systems, I develop a systematic procedure for constructing soliton solutions of the CH2 system.  Specifically, using a direct method combined with a reciprocal transformation, I obtain the parametric representation of the multisoliton solutions, and investigate their properties. Subsequently, I show that the CH2  system reduces to the CH equation and the two-component Hunter-Saxton (HS2) system by means of appropriate limiting procedures. The corresponding expressions of the multisoliton solutions are presented in parametric forms, reproducing the existing results for the reduced equations. Also, I discuss the reduction from the HS2 system to the HS equation. Last, I comment on an interesting issue associated with peaked wave (or peakon) solutions of the CH, Degasperis-Procesi, Novikov and modified CH equations.  
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University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons