Bifurcations of traveling wave solutions from KdV equation to Camassa-Holm equation

The dynamics of a 1+1 unidirectional non-linear wave equation which combines the linear dispersion of the Korteweg-de Vries (KdV) equation with the non-linear/non-local dispersion of the Camassa-Holm (CH) equation is explored in this Letter. Phase plane analysis is employed to investigate the bounded traveling-wave solutions. By considering the properties of the equilibrium points and the relative position of the singular line, transition boundaries have been derived to divide the parameter space into regions in which different types of phase trajectories can be observed. The explicit expressions of different types of solutions have been presented, which contain both the KdV solitons and the CH peakons as limiting cases.