Periodic Solitary Wave Solutions of the (2 + 1)-Dimensional Variable-Coefficient Caudrey-Dodd-Gibbon-Kotera-Sawada Equation

In this paper, through symbolic computations, we obtain two exact solitary wave solitons of the (2 + l)-dimensional variable-coefficient Caudrey-Dodd-Gibbon-Kotera-Sawada equation. We study basic properties of l-periodic solitary wave solution and interactional properties of 2-periodic solitary wave solution by using asymptotic analysis.


Introduction
Nonlinear evolution equations appear in many fields of physics, such as fluids, quantum mechanics, condensed matter, superconductivity and nonlinear optics.
Due to the fact that most systems in nature are complicated, many nonlinear evolution equations may possess variable coefficients. Recently, the investigation on exact solutions of the variable-coefficient nonlinear evolution equations has become the focus in the study of complex nonlinear phenomena in physics and engineering [1] [2] [3] [4].
In this paper, we study the (2 + 1)-dimensional variable-coefficient Caudrey-Dodd-Gibbon-Kotera-Sawada (vc-CDGKS) Equation (see [5]  For the vc-CDGKS Equation (1.1), the bilinear form, bilinear Bäcklund transformation, Lax pair and the infinite conservation laws have been studied by Bell polynomials in [5] and N-soliton solutions have been constructed with the help of the Hirota bilinear method. In [6], non-traveling lump and mixed lump-kink solutions were investigated by Hirota bilinear form and symbolic computational software of Maple.
In this paper, we consider periodic solitary wave solutions of the (2 + 1)-dimensional vc-CDGKS Equation (1.1). The periodic solitary wave solution in this paper comes from Zaitsev [11] and this kind of solution is periodic in the direction of propagation and decays exponentially along the transverse direction.
In [12] [13], some generalizations were given and the interactions between two y-periodic solitons were studied for the (2 + 1)-dimensional Kadomt- [16], respectively. In this paper, we present some generalizations and interactional properties between two periodic solitons for the (2 + 1)-dimensional vc-CDGKS Equation (1.1). The interactional properties will be analyzed based on the ideas in [17] [18], where the analysis was performed for constant-coefficient equations.
In the following section, we deduce the 1-periodic solitary wave solution which is periodic in the direction of one curve and decays exponentially along the proper transverse direction of the corresponding curve. We analyze the propagating curve and the center of the periodic solutions. We also deduce the

t t a t t a t t a t a t a t a t c a t a t c a t c a t c a t a t a t a t c c
where 0 c and 1 c are nonzero real constants. To obtain periodic solitary wave solutions, by using of the transformation in [5] ( ) ( ) and j k , j p and 0 j η are arbitrary constants.
In the following discussion, we let

1-Periodic Solitary Wave Solution
In order to get 1-periodic solitary wave solution of the The above solution describes a sequence of lumps, which is periodic in the direction of 1 1 ln 0 The centers of the lumps are located at

Conclusion
In this paper, we considered periodic solutions of the (2 + 1)-dimensional variable

Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper.