Bilinear form, soliton, breather, hybrid and periodic-wave solutions for a (3+1)-dimensional Korteweg–de Vries equation in a fluid

Abstract

Fluids are studied in such disciplines as atmospheric science, oceanography and astrophysics. In this paper, we investigate a (3+1)-dimensional Korteweg–de Vries equation in a fluid. Bilinear form and N-soliton solutions are obtained, where N is a positive integer. Via the N-soliton solutions, we derive the higher-order breather solutions. We observe the interaction between the two perpendicular first-order breathers on the $$x-y$$ and $$x-z$$ planes and the interaction between the periodic line wave and the first-order breather on the $$y-z$$ plane, where x, y and z are the independent variables in the equation. We discuss the effects of $$\alpha $$ , $$\beta $$ , $$\gamma $$ and $$\delta $$ on the amplitude of the second-order breather, where $$\alpha $$ , $$\beta $$ , $$\gamma $$ and $$\delta $$ are the constant coefficients in the equation: Amplitude of the second-order breather decreases as $$\alpha $$ increases; amplitude of the second-order breather increases as $$\beta $$ increases; amplitude of the second-order breather keeps invariant as $$\gamma $$ or $$\delta $$ increases. Via the N-soliton solutions, hybrid solutions comprising the breathers and solitons are derived. Based on the Riemann theta function, we obtain the periodic-wave solutions, and find that the periodic-wave solutions approach to the one-soliton solutions under a limiting condition.