D’Alembert wave and interaction solutions for a (3 + 1)-dimensional generalized Calogero–Bogoyavlenskii–Schiff equation - Feng, Qing-Jiang; Zhang, Guo-Qing (European physical journal plus, 2024)
The research objective of this paper is to improve the exp( − ϕ ( ξ )) expansion method and its application, and some novel D’Alembert wave solutions are derived by applying the Ansӓtze method. A (3 + 1)-dimensional generalized Calogero–Bogoyavlenskii–Schiff equation is used as the research model. With the help of the Ansӓtze method, some new types of D’Alembert wave solutions are derived, such as interaction solutions composed of W -shaped solutions and half periodic wave solutions, envelope solution, and the molecule consisting of kink and antikink. In addition, based on the exp(− ϕ ( ξ )) expansion method, a novel method called the multiple exp(− ϕ ( ξ )) expansion method is proposed to construct multiple solitary wave solutions and mixed periodic wave solutions, as well as complexiton solutions composed of hyperbolic functions, trigonometric functions, exponential functions, and rational functions. Furthermore, these complexiton solutions can explain new interaction phenomenas of nonlinear waves, such as the propagation of bright (dark) solitary waves on periodic wave background, as well as the evolution of bright–dark solitary waves. Therefore, compared to traveling wave solutions and periodic wave solutions derived from exp(− ϕ ( ξ )) expansion method, the complexiton solution obtained in this paper can interpret the interactions between various parts of nonlinear systems more effectively, which helps people better understand the inherent laws in natural science.