Bifurcations of travelling wave solutions for a general Sine–Gordon equation

Using the method of planar dynamical systems to study a general Sine–Gordon equation, the existence of solitary wave, periodic wave, kink and anti-kink wave solutions is proved in different regions of the parametric space. Sufficient conditions to guarantee the existence of the above solutions are given. Moreover, all possible exact explicit parametric representations of solitary wave, periodic travelling wave, kink and anti-kink wave solutions are obtained.