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 that the interaction between two perpendicular first-order breathers on the x-y and x-z planes and the periodic line wave interacts with the first-order breather on the y-z plane, where x, y and z are the independent variables in the equation. Furthermore, we discuss the effects of α, β, γ and δ on the amplitudes of the second-order breathers, where α, β, γ and δ are the constant coefficients in the equation: Amplitude of the second-order breather decreases as α increases; Amplitude of the second-order breather increases as β increases; Amplitude of the second-order breather keeps invariant as γ and δ increase. 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. Furthermore, we find that the periodic-wave solutions approach to the one-soliton solutions under certain limiting condition.