Bifurcations and Parametric Representations of Peakon, Solitary Wave and Smooth Periodic Wave Solutions for a Two-Component Degasperis-Procesi Equation

Abstract

By using the bifurcation method of dynamical systems and the method of phase portraits analysis, we consider a two-component Degasperis-Procesi equation: $m_t=-\mathrm{3}m{u}_x-m_{x}u+k\rho {\rho }_x,\mathrm{}{\rho }_t=-{\rho }_{x}u+\mathrm{2}\rho u_x,$ the existence of the peakon, solitary wave and smooth periodic wave is proved, and exact parametric representations of above travelling wave solutions are obtained in different parameter regions.