Periodic Wave Solutions and Solitary Wave Solutions of the (2+1)-Dimensional Korteweg-de-Vries Equation

In this paper, we investigate the periodic wave solutions and solitary wave solutions of a (2+1)-dimensional Korteweg-de Vries (KDV) equation by ap-plying Jacobi elliptic function expansion method. Abundant types of Jacobi elliptic function solutions are obtained by choosing different coefficients p, q and r in the elliptic equation. Then these solutions are coupled into an auxiliary equation and substituted into the (2+1)-dimensional KDV equation. As a result, a large number of complex Jacobi elliptic function solutions are obtained, and many of them have not been found in other documents. As 1 m → , some complex solitary solutions are also obtained correspondingly. These solutions that we obtained in this paper will be helpful to understand the physics of the (2+1)-dimensional KDV equation.


Introduction
Solitary wave phenomena were first discovered by British scientist Russell in 1834 [1]. In 1965, it was found that the particle velocity and waveform can remain unchanged after the interaction of solitary waves when solving KdV equation, which is called soliton [2]. After then, the concepts of soliton and solitary waves are widely used in various fields of physics. From hydrodynamics, plasma, optics, condensed matter physics to basic particle physics and even to astrophysics, everywhere [3]- [8], it is all found that there are experimental facts or physical mechanisms for the existence of solitons. Most physical laws can establish mathematical models under certain approximations, and many studies on nonlinear identification can be reduced to nonlinear evolution equations (NLEEs) finally. Therefore, seeking their exact solutions such as the breather, solitary wave and periodic wave solutions is very significant for the exploration of related nonlinear problems, which also has always been an important focus on the study of mathematics and physics. Significant progress has been made in recent centuries and many strong and effective methods have been proposed in the documents to obtain the exact solutions of NLEEs. For example, algebraic method [9], homogeneous balance method [10], tanh/sech method and the extended tanh/coth method [11] [12], the sine-cosine method [13], F-expansion method [14] [15], Jacobi elliptic function expansion method [16] [17], Expfunction method [18], the modified extended mapping method [19] [20] [21], auxiliary equation method [22] [23] [24], and so on.
In this paper, we consider revealing the new periodic wave and solitary solutions for the (2+1)-dimensional KDV equation [25] [26] [27] which was first derived by Boiti et al. by using the idea of the weak Lax pair [26].
If v = u and y = x, Equation (1) degenerates into the (1 + 1)-dimensional KdV equation [2]. The (2+1) dimensional case involves more complex nonlinear phenomena, and this scan describes certain physics phenomena in plasmas and fluids, where ( ) , , u x y t and ( ) , , v x y t are real differential wave functions depending on the 2-dimensional space variable x and y and 1-dimensional time variable t. Equation (1) was investigated in different methods. In Refs. [28] and [27], the variable separation solution of Equation (1) was obtained and some special types of solitary wave solutions were given in Ref. [29]. In Ref. [30], the exact periodic cross-kink wave solutions are obtained by using Hirota's bilinear form and a generalized three-wave approach. In Ref. [30] Higherorder KDV equation is considered, many travelling solitary wave solutions are found.

Method
We assume Equation (1) and Equation where u' means du/dξ and v' means dv/dξ. Integrating the above two equation once and setting the integration constant in Equation (4) to zero yields ( ) It is assumed that Equation (4) has the following formal solution where a i , b i , c i and d i are constants to be determined later. The positive integer n can be determined by the homogeneous balance method in Equation (6). f(ξ) expresses the solutions of the following elliptic equation where p, q, r are parameters to be selected for Jacobi elliptic function. By selecting different p, q, r, the different Jacobi elliptic function solutions of Equation (9) are shown in Table 1. Furthermore, these solutions include hyperbolic function solutions when 1 m → and trigonometric function solutions when 0 m → .  Table 1, the following new type of periodic wave solutions of Equation (4) can be obtained.
Case 5 American Journal of Computational Mathematics Table 1. Jacobi elliptic function solutions for Equation (9).   Equations (17), (18) and (19). It can be seen in Figure 1 that the solution expressed by Equation (15) changes periodically with the spatial position and the amplitude will change accordingly, while the solution expressed by Equation (16) has no change in period, but the amplitude changes greatly compared with Equation (15). The period of the solution expressed by Equation (17) is longer than that expressed in Equation (15), and the amplitude increases slightly. The period of the solution expressed by Equation (18) is similar to that expressed American Journal of Computational Mathematics  (17), (18) and (19).
by Equation (17) while the amplitude increases greatly. The period and amplitude of the solution expressed in Equation (19) are similar to that expressed by Equation (18). These five types of solitary wave solutions of (2+1)-dimensional KDV equation are shown in Figure 2, where C = 0, t = 0, "−" is chosen in Equation (23), "+" is chosen in Equation (22) and "+" is chosen for the previous "±" and "−" is chosen for the next. It can be seen that these five solitary wave solutions are all    by Equation (28), (e) solution expressed by Equation (29), where m = 0.2, t = 0, C = 0 and "−" are chosen in Equations (27), (28) and (29).   There are still a large number of periodic wave solutions of (2+1)-dimensional KDV equation, according to Equations (9), (10), (11), (12), (13), (14) and Table   1. These solutions may also have solitary solutions under the conditions of 1 m → . The corresponding solutions of v(ξ) can be obtained from Equation (7).
Limited to this scope, we will not give examples one by one.

Conclusion
In this paper, we apply the extended Jacobi elliptic function expansion method to explore the exact solution of the (2+1)-dimensional KDV equation. With the cooperation of the auxiliary Equations (8), (9) and its Jacobi elliptic function solution set Table 1