Bifurcation studies on travelling wave solutions for nonlinear intensity Klein-Gordon equation

In this paper, we study the travelling waves for the (n + 1)-dimensional Klein-Gordon equation with nonlinear intensity by using the bifurcation theory and methods of planar dynamic systems. With the aid of Maple, we obtain all bifurcations and phase portraits in the parametric space. Moreover we explain the reason for appearance of nonsmooth travelling waves and for persistence of smoothness of smooth travelling waves, and show the existence of smooth and nonsmooth solitary and periodic waves. Under different regions of parametric space, various sufficient conditions to guarantee the existence of the above solutions are given.