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

• Published in: Nonlinear dynamics 2021-08, Vol.105 (3), p.2525-2538
• Main Authors: Cheng, Chong-Dong, Tian, Bo, Zhang, Chen-Rong, Zhao, Xin
• Format: Article
• Language: English
• Subjects: Amplitudes; Astrophysics; Automotive Engineering; Breathers; Classical Mechanics; Control; Dynamical Systems; Engineering; Fluids; Independent variables; Korteweg-Devries equation; Mechanical Engineering; Oceanography; Original Paper; Solitary waves; Vibration
• ISSN: 0924-090X; 1573-269X
• DOI: 10.1007/s11071-021-06540-x

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 α , β , γ and δ on the amplitude of the second-order breather, 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 or δ 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.