Solitary Solutions to the Fractional Generalized Hirota–Satsuma-Coupled KdV Equations
In this paper, the complete discrimination system method is used to the fractional generalized Hirota–Satsuma-coupled KdV equations, which depicts the dispersive long wave widely applied in fluid mechanics. First, by the wave transformation, the fractional generalized Hirota–Satsuma-coupled KdV equation can be transformed into ordinary differential equations. Second, a series of new solutions are constructed by the discrimination system method of the fourth order polynomial, these solutions include kink solitary wave solutions, Jacobian elliptic function double periodic solutions, trigonometric function solution, solitary wave solutions, two-period solutions of elliptic function, and implicit solutions formula solutions. Finally, the comparison between our results and others’ works is also given.
In 1981, the Hirota–Satsuma KdV equation was first proposed by Hirota–Satsuma ; this equation is widely used to describe complex phenomena in physics, such as electrodynamics, heat transfer, dynamics of light pulses, solid-state physics, nonlinear optics, etc. So far, many effective methods have been proposed to study it [6–9]. In the study , authors obtained the exact solutions expressed by the Jacobian elliptic functions. In the study [11, 12], the authors constructed new exact wave soliton solutions via the direct algebraic method. In the study , it is proved that (1) has four kinds of soliton solutions that employed the tanh function method. In the study , Fan et al. constructed double periodic wave solutions and proved its degenerate to the soliton solutions at a limit condition. In the study , Ali studied (1) multiple traveling wave solutions. Various initial value systems are considered by the extended homogeneous balance method [16, 17]. Homotopy analysis method  and homotopy perturbation method  are used to study (1). Some articles investigated the fractional partial differential equation by utilizing the bifurcation, stability analysis, traveling wave solutions, and phase portraits (see [16, 20–30]). In 1996, Yang  et al. introduced the complete discriminant system for solving higher-order polynomial equations. Moreover, the method was widely used in the solution of the partial differential equation by Liu . In this paper, in order to obtain a multiform solution and further explore the physical properties of this equation, we intend to investigate equation (1) in accordance with the complete discrimination system method.
The paper is arranged as follows: In Section 2, we use the wave transformation (1) into ordinary differential equations (ODEs) and obtain the solitary solutions. In Section 3, we draw three-dimensional and two-dimensional graphics of some solutions. In Section 4, we present the conclusion.
2. Solitary Solutions of (1)
We use the following wave transformation:in which and .
Aimed at eliminating the coupling relationship of the equations above, the assumption is made as follows:where , and are constants.
When and , we assume that
When and , we assume that
Here, we only consider the case of equation solution when and , and write (6) in following integral form:
The complete discriminant system corresponds to the quartic polynomial as follows:
According to the complete discriminant system, the classification of all solutions of (1) can be obtained, and there are the following nine cases.
Cases 1. Suppose . has a pair of conjugated complex roots, i.e., , where and are real numbers, . The solutions of (4) can be presented as follows:
Cases 2. Suppose . possesses a quadruple real root. Setting , the solutions of (6) can be presented as follows:
Cases 3. Suppose , , there are two differences double real roots, , and are constants, and . Then, , , , and .
When or , the solutions of (6) can be presented as follows:When , the solutions of (6) take the following form:
Cases 4. Suppose , , there are double real roots and two single roots. , where , and are real numbers. When , or when , the solutions of (6) can be presented as follows:When , or when , the solutions of (6) can be presented as follows:When , or when , , the solutions of (6) can be presented as follows:
Cases 5. Suppose that , . So, , or , . There are triple real roots and a single real root, , where or .
When or when , the solutions of (6) take the following form:Meanwhile, we derive extra solutions of (6) as follows:
Cases 6. Suppose , we have , so the solutions of (6) take the following form:where ; this solution is a solitary wave solution.
Cases 7. Suppose , then admits four different roots. We denote , where , are real numbers and . The solutions of (6) take the following form:
Cases 8. Suppose , then we denote , are real numbers and . Taking the transformation , where , , , , , , , the solutions of (6) take the following form:where this solution is a two-period solution of the elliptic function.
Cases 9. Suppose , then we denote , , , , and are real numbers, and . Taking the transformation , where , , , , , , , the solutions of (6) takes the following form:where .
Remark 1. In the previous discussion, we only gave the solutions when and . When and , we can also use the complete discriminant system to get the corresponding solutions.
Remark 2. In this article, only the solutions of equation is obtained, we can easily obtain the solutions of and according to the relationship between and and in (4).
3. Graphical Representation of the Obtained Solutions
In the paper, the complete discrimination system method of the fourth order polynomial is applied to find the solution for equation (1). Compared with the existing literature([1–4]), the Jacobian function solutions and implicit solutions are also obtained. The results show that the complete discrimination system method is a powerful mathematical tool of an efficient way to obtain solutions of many forms using one method to solve the fractional generalized Hirota–Satsuma-coupled KdV equation. As a result, the complete discrimination system method can be thought of as a powerful, robust, and versatile mathematical approach for finding analytical solutions to nonlinear PDEs; it is also a promising method to solve other nonlinear equations.
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
This work was supported by Scientific Research Funds of Chengdu University, under Grant no. 2081920034.
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