Abstract

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.