The Dual Systems And Peakon Solutions Of Multi-component Bi-Hamiltonian Integrable Systems

Soliton theory is a significant research area in nonlinear science.Integrable systems in soliton theory have become the common concerned focus of the mathematical physics.In the previous study,it has been found that some classical integrable bi-Hamiltonian systems can be constructed by recombining the Hamiltonian operators.For instance,the Camassa-Holm equation can be reconstructed by the Hamiltonian operators of the Korteweg-de Vries equation.The resulting so-called dual integrable system often has a nonlinear dispersion structure and allows for non-smooth soliton solutions.In this paper,we consider the two classical two-component bi-Hamiltonian integrable systems.The method of Hamiltonian operators recombination is applied to build its dual integrable system.Further more,the soliton solutions of the dual integrable system are considered respectively.Using weak formulation form of the system and multiple integration theory,we lead to the single peakon solution of the dual integrable system.This method can be applied to N-peakon solution.The main results are as follows:1.The dual integrable system of the complex Korteweg-de Vries system is constructed,which is complex Camassa-Holm system.Peakon solution,N-peakon solution of the complex CH integrable system are obtained.2.The dual integrable system of the symmetric coupled Korteweg-de Vries system is constructed,and the peakon solution,N-peakon solution of the resulting dual integrable system are obtained.