The bifurcation and exact peakons, solitary and periodic wave solutions for the Kudryashov–Sinelshchikov equation

In this paper, the Kudryashov–Sinelshchikov equation is studied by using the bifurcation method of dynamical systems and the method of phase portraits analysis. From dynamic point of view, the existence of peakon, solitary wave, smooth and non-smooth periodic waves is proved and the sufficient conditions to guarantee the existence of the above solutions in different regions of the parametric space are given. Also, some new exact travelling wave solutions are presented through some special phase orbits.

► We investigate the Kudryashov-Sinelshchikov equation by using the bifurcation method of dynamical systems and the method.
► The bifurcation of all travelling wave solutions is given.
► The existence of peakon, solitary wave, smooth and non-smooth periodic waves is proved.
► We obtain some exact explicit and implicit parametric representations of the above waves.