Abstract
In this paper, we derive analytical solutions of the Caudrey-Dodd-Gibbon-Sawada-Kotera (CDGSK) equation via symbol calculation approach. Applying the exp-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.
1. Introduction
Nonlinear fractional and integer order differential equations are widely utilized in fluid dynamics, solid state physics, plasma physics, biology, nonlinear optics, chemistry, etc. It has aroused the strong interest of researchers to these differential equations [1–18]. The study to exact solutions of various NLDEs is extremely important in modern mathematics with ramifications to some areas of physics, mathematics, and other sciences. There are many systematic methods to seek exact solutions of nonlinear differential equations, for example, the Hirota bilinear method [19], Tanh method [20], Lie symmetry [21], modified Kudryashov method [22], Exp-function method [23], sine-Gordon expansion method [24], complex method [25–29], and exp-expansion method [30–32].
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 exp-expansion method [30–32] to obtain the exact solutions of the CDGSK equation.
2. The exp-Expansion Method
In this section, we give the main steps of the mentioned method.
Step 1. Inserting traveling wave transform into a nonlinear PDE yields in which is the polynomial of along with its derivatives.
Step 2. Assume that Equation (4) has analytical solutions as follows:
where are constants which will be ascertained subsequently, such that and satisfy
The solutions of Equation (6) are given in the following:
When ,
When ,
When ,
When ,
When ,
In Equations (7)–(12), are constants. Taking the homogeneous balance between the nonlinear terms and highest order derivatives of Equation (4) yields the positive integer .
Step 3. Insert Equation (5) into Equation (4), and collect the function exp to yield the polynomial to exp. Let all coefficients with the same power of exp be zero to obtain a system of algebraic equations. In solving these equations, we achieve the values of and substitute them into Equation (5) as well as Equations (7)–(12) to accomplish the determination for analytical solutions of the original PDE.
3. Exact Solutions of the CDGSK Equation
Inserting into Equation (1) and then integrating it, we obtain where is the integration constant.
Taking the homogeneous balance between and of Equation (14) yields where and and are constants.
Substituting into Equation (14) and letting the coefficients about be zero, we obtain
We solve the above algebraic equations and then obtain two different cases:
Case 1. where and are arbitrary constants.
Substituting Equation (17) into Equation (15) yields
Applying Equations (7)–(12) into Equation (18), respectively, yields analytical solutions of the CDGSK equation as follows:
Case 1.1. When ,
Case 1.2. When ,
Case 1.3. When ,
Case 1.4. When ,
Case 1.5. When ,
Case 2. where and are arbitrary constants.
Substituting Equation (24) into (15) yields
Applying Equations (7)–(12) into Equation (25), respectively, yields analytical solutions of the CDGSK equation as follows:
Case 2.1. When ,
Case 2.2. When ,
Case 2.3. When ,
Case 2.4. When ,
Case 2.5. When ,
Figures 1–5 show the properties of the solutions.
4. 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 exp-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.
Acknowledgments
This work is supported by the NSF of China (11901111) and the Visiting Scholar Program of Chern Institute of Mathematics.