Bifurcations of travelling wave solutions for the nonlinear dispersion Drinfel’d–Sokolov (D(m,n)D(m,n)) system

By using the bifurcation theory of planar dynamical systems to the nonlinear dispersion Drinfel’d–Sokolov (D(m,n)D(m,n)) system, the existence of solitary wave solutions, kink and anti-kink wave solutions, compacton solutions and uncountably infinite many smooth and non-smooth periodic wave solutions is proved. Under different regions of parametric spaces, various sufficient conditions to guarantee the existence of above solutions are given. Some exact explicit parametric representations of the above waves are determined.