Analytical Solutions to the Caudrey-Dodd-Gibbon-Sawada-Kotera Equation via Symbol Calculation Approach

In this paper, we derive analytical solutions of the Caudrey-Dodd-Gibbon-Sawada-Kotera (CDGSK) equation via symbol calculation approach. Applying the expð−φðzÞÞ-expansion method, we achieve the trigonometric, exponential, hyperbolic, and rational function solutions for the CDGSK equation. By choosing the appropriate parameters, we give some computer simulation to the analytical solutions of the CDGSK equation.

Sawada and Kotera [33] proposed one of basic models in soliton theory as follows: which is also introduced by Caudrey et al. independently [34]; so in literature, Equation (1) is called the Caudrey-Dodd-Gibbon-Sawada-Kotera equation. Many years have passed by, lots of research results for the CDGSK equation have been developed. As to this equation, the finite dimensional reduction was investigated by Enolski et al. [35], and N-soliton solutions were discovered by Parker [36] via the dressing method. Darboux transformation [37] and Bäcklund transformation in bilinear forms [38] were applied to study the CDGSK equation. Riemann theta function solutions of the CDGSK equation were also established [39]. In this paper, we employ the expð−φðzÞÞ-expansion method [30][31][32] to obtain the exact solutions of the CDGSK equation.

The expð−φðzÞÞ-Expansion Method
In this section, we give the main steps of the mentioned method.
Step 1. Inserting traveling wave transform into a nonlinear PDE P u, u x , u y , u t , u xx , u yy , u tt ,⋯ in which F is the polynomial of u along with its derivatives.
Step 3. Insert Equation (5) into Equation (4), and collect the function expð−φðzÞÞ to yield the polynomial to expð−φðzÞÞ. Let all coefficients with the same power of expð−φðzÞÞ be zero to obtain a system of algebraic equations. In solving these equations, we achieve the values of B τ ≠ 0, γ, ϑ and substitute them into Equation (5) as well as Equations (7)- (12) to accomplish the determination for analytical solutions of the original PDE.

Exact Solutions of the CDGSK Equation
Inserting into Equation (1) and then integrating it, we obtain where δ is the integration constant.

Conclusions
In this paper, abundant analytical solutions of the CDGSK equation are obtained via symbol calculation approach. The properties of the solutions are shown by some graphs. It shows that the expð−φðzÞÞ-expansion method is an effective method to seek analytical solutions for nonlinear differential equations.

Data Availability
The data used to support the findings of this study are included within the article.

Conflicts of Interest
The authors declare that there is no conflict of interest regarding the publication of this paper.   Journal of Function Spaces