Bifurcation and the exact smooth,cusp solitary and periodic wave solutions of the generalized Kudryashov–Sinelshchikov equation

The main aim of this paper is to study the exact traveling wave solutions of the generalized Kudryashov–Sinelshchikov equation by using the auxiliary equation method based on the conclusion of qualitative analysis. The advantage of this method is to choose the effective and proper auxiliary equation on the base of the behaviors and traits of solutions revealed by analysis of phase portraits to study the solution of differential equations. By applying the proposed approach to the generalized Kudryashov–Sinelshchikov equation, the number, behavior and existence of smooth and non-smooth traveling wave solutions are gained, at the same time, the new exact smooth solitary, periodic wave solutions and cusp solitary, periodic wave solutions are obtained. From the dynamic point of view, the behavior of traveling wave solutions is analyzed. The profile,type and the form of exact expression of traveling wave solutions are influenced by the order of nonlinear term and nonlinear terms.