Meromorphic exact solutions of the (2 + 1)-dimensional generalized Calogero-Bogoyavlenskii-Schiff equation

Abstract In this article, meromorphic exact solutions for the (2 + 1)-dimensional generalized Calogero-Bogoyavlenskii-Schiff (gCBS) equation are obtained by using the complex method. With the applications of our results, traveling wave exact solutions of the breaking soliton equation are achieved. The dynamic behaviors of exact solutions of the (2 + 1)-dimensional gCBS equation are shown by some graphs. In particular, the graphs of elliptic function solutions are comparatively rare in other literature. The idea of this study can be applied to the complex nonlinear systems of some areas of engineering.

The (2 + 1)-dimensional generalized Calogero-Bogoyavlenskii-Schiff (gCBS) equation [39] is the potential form of Eq. (1.1) that is given as: where a b c , , are constants. In the past decade, several methods have been used to obtain the exact solutions of the gCBS equation, for example, Chen and Ma [40] by considering the Hirota bilinear form of the gCBS equation explicitly generated a class of lump solutions; Li and Chen [41] by using the generalized Riccati equation expansion method found some exact analytical solutions; Al-Amr [42] applied the modified simple equation method to construct the exact solutions of the gCBS equation; Wang and Yang [43] implemented the Hirota bilinear method for the construction of the quasi-periodic wave solutions in terms of theta functions; Najafi et al. [44] established the ( ′/ ) G G -expansion method to find travelling wave solutions for the gCBS equation; and Zhang et al. [45] studied the integrability of this equation by the Painlevé test and derived its symmetry reductions.
Recently, Yuan et al. [46] have proposed an effective method named complex method to seek the exact solutions of nonlinear differential equations. The complex method is based on complex differential equations and complex analysis. More details of the complex method can be seen in [47][48][49][50]. In this article, the complex method is utilized to obtain meromorphic exact solutions of the (2 + 1)-dimensional gCBS equation.
Substituting traveling wave transform into Eq. (1.2), and then integrating it we get where d is the integration constant. If a meromorphic function ξ is a rational function of z, or an elliptic function, or a rational function of ∈ e μ , μz , then ξ is said to belong to the class W.  , g 3 is arbitrary and ∈ z 0 .
Breaking soliton equation is given by .
(iii) The elliptic function solutions and g 3 is arbitrary.

Preliminary theory
The constants a b c , , may differ in different places.
The differential polynomial is given by: Considering the following ordinary differential equation: where ≠ a d 0, are constants, and ∈ n . Set ∈ p q , , and assume that meromorphic solutions v of Eq. (2.2) have at least one pole. Substituting the Laurent series 2 3 is the Weierstrass elliptic function with double periods satisfying: Weierstrass zeta function ( ) ζ z is a meromorphic function which satisfies These two Weierstrass functions admit the addition formulas as follows:   . Therefore, we can determine that 1 , Thus, the rational solutions of Eq.

Conclusions
In this article, the complex method is utilized to construct meromorphic solutions to the complex (2 + 1)dimensional gCBS equation, then exact solutions to the (2 + 1)-dimensional gCBS equation are obtained. By the applications of our results, traveling wave exact solutions to the breaking soliton equation are achieved. To our knowledge, the solutions of this study have not been reported in former literature. The dynamic behaviors of these solutions are shown by some graphs. Figures 1-4 obviously present soliton phenomena and show that soliton interaction will contain oscillations. Figures 1-3 also display that by changing the time, the solutions of the (2 + 1)-dimensional gCBS equation continue to move forward.
The complex method is an efficient method to get the solutions of a BBEq through its undetermined forms. This method is applied to obtain the exact solutions of the (2 + 1)-dimensional gCBS equation and breaking soliton equation which enrich the studies of the mentioned equations. However, for the differential equation which is not a BBEq, how to solve it by the complex method? It will be considered in future studies.