Abstract

Through symbolic computation with Maple, fifty-seven sets of rational wave solutions to the generalized Calogero-Bogoyavlenskii-Schiff equation are presented by employing the generalized bilinear operator when the parameter . Via the three-dimensional plots and contour plots with the help of Maple, the dynamics of these solutions are described very well. These solutions have greatly enriched the exact solutions of the generalized Calogero-Bogoyavlenskii-Schiff equation on the existing literature. The result will be widely used to describe many nonlinear scientific phenomena.

1. Introduction

It is well known that nonlinear evolution equations (NLEEs) play an important and significant role in describing nonlinear scientific phenomena, such as fluid dynamics, plasma physics, chemistry, marine engineering, optics, and physics. Rational solutions to NLEEs help us understand the physical phenomena they describe in nature. Searching for rational solutions of NLEEs has become a major concern partly due to the availability of computer symbolic systems like Maple, which allow us to deal with some complicated and tedious algebraic calculation. Through the unremitting efforts of mathematicians, many effective methods for solving NLEEs have been established and developed, including Lie symmetry transformation, inverse scattering transformation, Darboux transformation, and Hirota’s bilinear theory. Particularly, the Hirota bilinear method was proposed by Hirota, a Japanese scholar, in 1971 to solve the problem of N soliton solution of the nonlinear evolution equation [1]. By using a transformation of the potential function of NLEEs and the definition and properties of the D operator, NLEEs are written in bilinear form, and then, the single-double-multiple soliton solutions of NLEEs can be obtained by using the small parameter expansion method. Based on these methods, one can try to find many interesting analytical solutions of NLEEs, such as the rogue wave solutions [2–4], the multiple wave solutions [5, 6], the lump solutions [7–9], the periodic wave solutions [10–13], the Wronskian solutions [14, 15], the rational solutions [16, 17], the high-order soliton solutions [18, 19], the solitary wave solutions [20, 21], and the other solutions [22–27].

The rest of this paper is arranged as follows. We will get the bilinear form of the gCBS equation in Section 2. In Section 3, the rational wave solutions will be gained by using the polynomial method. A few of conclusion and outlook will be given in Section 4.

2. Bilinear Form of the gCBS Equation

We consider a generalized Calogero-Bogoyavlenskii-Schiff (gCBS) equation [28, 29]: where and are two constants, or equivalently, where . This is a generalization of a ()-dimensional CBS equation whose coefficients have a different pattern from the original one. The CBS equation was first constructed by Bogoyavlenskii and Schiff in different ways. Bogoyavlenskii used the modified Lax formalism, whereas Schiff derived the same equation by reducing the self-dual Yang-Mills equation. Recently, the lump solutions of the gCBS equation have been solved [28].

Under dependent variable transformation,

Equation (1) is transformed into the following generalized bilinear form. where being an arbitrarily given natural number [30], often a prime number where are arbitrary nonnegative integers.

If we take , we obtain the Hirota bilinear equation If we take , we obtained the generalized bilinear equation

The transform (4) is also a characteristic transformation for establishing Bell polynomial theory of soliton equation [31]. Its exact relation is as follows:

Hence, if solves the generalized bilinear gCBS equation (7), the gCBS equation (1) will be solved.

3. Rational Wave Solutions for the gCBS Equation

In this section, we want to discuss the rational wave solutions to the gCBS equation by using the polynomial solutions.

Let where the are constants, with the help of the computer algebra system Maple, and substituting (10) into equation (7), we obtain a set of algebraic equations. Solving the set of algebraic equations, we can find 57 solutions, but for lack of space, we will list only the following solutions:

Case 1.

Case 2.

Case 3.

Case 4.

Case 5.

Case 6.

Case 7.

Case 8.

Case 9.

Case 10.

The constant involved are arbitrary as long as the solution is meaningful, considering the transformation of the coefficient .

By observation, we can divide the 57 sets of solutions into two classes, Cases 1–5 belong to the first class, Cases 6–10 belong to the second class.

Here, let us first discuss the solution of Case 1. Through the transformation (4), we get the following solution to the gCBS equation where the functions and are given as follows:

In order to analyze the dynamics properties briefly, we would like to discuss the evolution characteristic. By choosing appropriate values of these parameters in (21), we set

The three-dimensional dynamic graphs and corresponding contour plots of the solution were successfully depicted in Figure 1.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(a)
(b)
(c)
(d)
(e)
(f)
(g)

Figure 1 

3D plots and contour plots of equation (21) by choosing .

Correspondingly, Figure 1 displays the whole process of the creation and disappearance of the rogue wave aroused by solution (21). From the coordinates, that rogue wave is almost uniform over time. When in Figure 1(a), the wave appears gradually, and this obvious wave is moving diagonally toward the center along the - and -axes (Figures 1(a), 1(c), and 1(e)). When in Figure 1(g)), it comes to the center. Then, this wave is moving diagonally away from the center along the - and -axes (Figures 1(f), 1(d), and 1(b)). When in Figure 1(g), this obvious wave disappears gradually. As can be seen, when time approaches zero, the amplitude of rogue wave increases the maximum and then decreases. So the asymptotic behavior of can be obtained, the solution as . We hope our work in this paper contributes to the study of multidimensional and higher-order rogue waves.

Second, let us discuss the solution of Case 6, and the rest of the cases are the same, the rational wave solutions to the gCBS equation read where the functions and are given as follows: