Loop solutions, breaking kink (or anti-kink) wave solutions, solitary wave solutions and periodic wave solutions for the two-component Degasperis–Procesi equation

In this paper, the method of bifurcation of planar dynamic systems is used to investigate bounded traveling wave solutions for the two-component Degasperis–Procesi shallow water equation. By using the phase portrait bifurcation of the traveling wave system, solitary wave solutions, periodic wave solutions and breaking kink (or anti-kink) wave solutions are considered under different parameter conditions. And the loop soliton solution is also discussed. By the use of numerical simulation, the graphs of some solutions are obtained under specific parameter conditions.