Abstract
The soliton interaction is investigated based on solving the nonisospectral generalized Sawada-Kotera (GSK) equation. By using Hirota method, the analytic one-, two-, three-, and -soliton solutions of this model are obtained. According to those solutions, the relevant properties and features of line-soliton and bright-soliton are illustrated. The results of this paper will be useful to the study of soliton resonance in the inhomogeneous media.
1. Introduction
The Hirota method, originating from the work of Hirota in 1971 [1], is a powerful method for constructing solutions for integrable systems. The soliton theory is presented in several monographs and review papers (see [2, 3]). In the literature, various approaches have been proposed to find a soliton solution for a given equation, for instance, the inverse scatting transform [4] and the Darboux transformation [5]. It is remarked that the Hirota method is very efficient for construction of soliton solutions.
The nonisospectral equations describe solitary waves in inhomogeneous media. Recently, much attention has been paid on the analytic solutions of the nonisospectral equations. Deng et al. [6] and Sun et al. [7, 8] develop a systematic procedure to find soliton solutions of the nonisospectral equations. Based on exact solutions, numerical methods can be presented well for the nonisospectral nonlinear problem [9–11].
Jiang considers the nonisospectral problem [12] by using the compatibility condition of Lax pairs. In our work, the bilinear form and -soliton solutions will be considered for a generalized nonisospectral equation.
The nonisospectral generalized Sawada-Kotera (GSK) equation [12] is written as follows: where , and are real constants. The Lax pair of (1) is
The aim of this paper is to propose a simple method for construction -soliton solutions. The main tool is the Hirota method. Then we apply the idea to the nonisospectral GSK equation.
This paper is organized as following: In Section 2, with the aid of symbolic computation, the bilinear form of (1) is obtained by use of Hirota method. Some special solutions are explicitly presented based on their bilinear form (4) and the soliton resonance is illustrated. The final section contains some discussion.
2. Bilinear Form and -Soliton Solutions
Through the dependent variable transformation Equation (1) can be written in the bilinear form. Consider where the -operators [13] is defined by
The perturbation method consists of expanding with respect to a small parameter to obtain and then finding each coefficient successively for .
Substituting the expansion formula of into the bilinear equation (4) and arranging it at each order of , we have let us choose where .
Since substituting this into the left-hand side of (7) gives then (11) is an ordinary differential system and it can be solved exactly. The solutions of (11) are written as
Therefore, we are able to choose . This shows that the expansion of may be truncated as the finite sum Substituting (13) into (3), the one-soliton solution of the nonisospectral GSK equation (1) can be obtained Here is the one-soliton solution. By the form of the solution (14), one can see that the one-soliton travels with a time-dependent top trace
In fact, the top trace of the solution (14) is a line with time-dependent slope. Equation (14) provides a line-soliton with the following time-dependent amplitude:
Figures 1 and 2 describe the different amplitudes of the one-soliton solution at and .
We begin here by finding a two-soliton solution. It is a solution describing the interaction of two solitons.
To this end, we choose the solution to the linear differential equation (7) to be where for .
Substituting (17) into the left-hand side of (7), we have The solutions of (18) are written as
We here set that
From (20), we might assume that the relations . Equation (20) may also be written as Substituting (17), (21) into the left-hand side of (8) and using (18), we have Substitution of (19) into (22) gives
The coefficient obtained in (23), which are similar to the SK equation (see [14]), can also be KdV type.
Therefore, we are able to choose , . The two-soliton solutions are obtained by (3) in which is defined as
Its shape and motion are shown in Figures 3 and 4.
In Figures 3 and 4, the line-soliton characters are shown in two-soliton solutions, where the black areas denote zero value and the white lines denote bright-soliton. In this case, the amplitudes and slopes of the two-soliton will vary with time and this time-dependent property comes from the effects of inhomogeneous media.
Let us choose where for .
Substituting (25) into the left-hand side of (7), we have The solutions of (26) are written as
We here set that Substitution of (25), (28) into (8) gives
Let Substituting (25), (28), and (30) into (9), one obtains
Therefore, we are able to choose . The three-soliton solutions are obtained by (3) in which is defined as
The nonisospectral GSK equation [12] has been shown to be integrable. It can be represented as the compatibility condition in the Lax form . Therefore, it would be reasonable to continue to find the -soliton solutions with the help of symbolic computation (see [15]).
This process can be extended to the four-soliton solutions, and so on. Generally, the -soliton solutions are expressed as where the coefficients and are defined by respectively.
In formula (33), the first means a summation over all possible combinations of , , , and means a summation over all possible pairs chosen from the set , with the condition that .
Substituting (33) into (3), we obtain the -soliton solutions for the nonisospectral GSK equation.
3. Conclusion
In this paper, we have obtained the -soliton solutions of the nonisospectral GSK equation by the Hirota method. Under transformation (3), (1) has been transformed into bilinear form (4) directly. Based on formula (33), -soliton solutions have been constructed. A KdV-type solution has also been obtained. Soliton resonance and interaction for (1) can be regarded as the combination of the effects of various variable coefficients, as shown in Figures 1–3. Effects of the line-soliton, bright-soliton, and soliton resonance have been summarized. Finally, according to Figure 4, the possible applications of soliton resonance in the inhomogeneous media have been discussed.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work is supported by National Natural Science Foundation of China under Grant nos. 11171032, 11271362, and 11375030 and Beijing special project from Beijing education committee. The third author is supported by Beijing Natural Science Foundation under Grant no. 1132016 and Beijing Nova program no. Z131109000413029.