Bifurcations of traveling wave solutions for the mixed Korteweg-de Vries equation

In this paper, the bifurcation theory of planar dynamical systems is employed to investigate the mixed Korteweg-de Vries (KdV) equation. Under different parameter conditions, the bifurcation curves and phase portraits of corresponding Hamiltonian system are given. Furthermore, many different types of exact traveling waves are obtained, which include hyperbolic function solution, triangular function solution, rational solution and doubly periodic solutions in terms of the Jacobian elliptic functions. Furthermore, as all parameters in the representations of exact solutions are free variables, the solutions obtained show more complex dynamical behaviors, and could be applicable to explain diversity in qualitative features of wave phenomena.