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

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.