Bifurcation analysis and solutions of a three-dimensional Kudryashov-Sinelshchikov equation in the bubbly liquid

Three-dimensional Kudryashov-Sinelshchikov (KS) equation, which describes the propagation of nonlinear waves in a bubbly liquid, is investigated in this paper. Bifurcations and phase portraits for such an equation are discussed. The parameter χ, which depends on the viscosity of such a liquid, affects the types and stability of the equilibrium points in phase portraits. With the different values of χ, five cases of the phase portraits can be obtained, and periodic, homoclinic and heteroclinic orbits can be found. Inspired by the characteristics of those orbits, we derive the corresponding solutions including the periodic, solitary wave solutions for χ=0, and kink wave solutions for χ≠0. Effects of χ on the dilation factor, steepness and velocity of the kink wave solutions are discussed. With the increasing magnitude of χ, the dilation factor and steepness of the kink wave solutions increase, while the velocity of the kink wave solutions first decreases and then increases.