Different kinds of exact solutions with two-loop character of the two-component short pulse equations of the first kind

In this paper, by using the integral bifurcation method and the Sakovich’s transformations, we study the two-component short pulse equations of the first kind, different kinds of exact traveling wave solutions with two-loop character, such as two-loop soliton solutions, periodic loop-compacton wave solutions and different kinds of periodic two-loop wave solutions are obtained. Further, we discuss their dynamical behaviors of these exact traveling wave solutions and show their profiles of time evolution by illustrations. This is first time in our research area that we obtain two-soliton solutions of nonlinear partial differential equations under no help of Hirota’s method, inverse scattering method, Darboux transformation and Bächlund transformation.