Advances in Mathematical Physics

Volume 2014 (2014), Article ID 547692, 5 pages

http://dx.doi.org/10.1155/2014/547692

Research Article

-Soliton Solutions of the Nonisospectral Generalized Sawada-Kotera Equation

Jian Zhou, Xiang-Gui Li, and Deng-Shan Wang

School of Applied Science, Beijing Information Science and Technology University, Beijing 100192, China

Received 6 May 2014; Revised 10 August 2014; Accepted 20 August 2014; Published 1 September 2014

Academic Editor: Alkesh Punjabi

Copyright © 2014 Jian Zhou et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 .

Figure 1: The shape and motion of the one-soliton solution for , , , , , .

Figure 2: The shape and motion of the one-soliton solution for , , , , , .

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.

Figure 3: The shape and motion of the two-soliton solutions for , , , , , , , .

Figure 4: The density plots of two-soliton resonance for the nonisospectral GSK equation with parameters , , , , , , , .

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.

References

R. Hirota, “Exact solution of the korteweg-de vries equation for multiple collisions of solitons,” Physical Review Letters, vol. 27, no. 18, pp. 1192–1194, 1971.

L. Pitaevskii and S. Stringari, Bose-Einstein Condensation, Oxford University, New York, NY, USA, 2003.

D.-S. Wang and X.-G. Li, “Localized nonlinear matter waves in a Bose-Einstein condensate with spatially inhomogeneous two- and three-body interactions,” Journal of Physics B: Atomic, Molecular and Optical Physics, vol. 45, no. 10, Article ID 105301, 2012.

M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform, SIAM, Philadelphia, Pa, USA, 1981.

V. B. Matveev and M. A. Salle, Darboux Transformations and Solitons, Springer, Berlin, Germany, 1991.

S. Deng, D. Zhang, and D. Chen, “Exact solutions for the nonisospectral Kadomtshev-Petviashvili equation,” Journal of the Physical Society of Japan, vol. 74, no. 9, pp. 2383–2385, 2005.

Y. Sun, J. Bi, and D. Chen, “-soliton solutions and double Wronskian solution of the non-isospectral AKNS equation,” Chaos, Solitons & Fractals, vol. 26, no. 3, pp. 905–912, 2005.

Y. Sun, “The non-isospectral modified Kadomtsev-Petviashvili equation with self-consistent sources and its coupled system,” Applied Mathematics and Computation, vol. 217, no. 4, pp. 1639–1645, 2010.

X. Li, W. Yan, and C. K. Chan, “Numerical schemes for Hamilton-Jacobi equations on unstructured meshes,” Numerische Mathematik, vol. 94, no. 2, pp. 315–331, 2003.

D. Hua and X. Li, “The finite element method for computing the ground states of the dipolar Bose-Einstein condensates,” Applied Mathematics and Computation, vol. 234, pp. 214–222, 2014.

S. Li, X. Li, and D. Hua, “A time-splitting and sine spectral method for dynamics of dipolar Bose-Einstein condensate,” Advances in Mathematical Physics, vol. 2013, Article ID 517395, 7 pages, 2013.

Z. Jiang, “Nonisospectral problems related to DS and other -dimensional nonlinear evolution equations,” Inverse Problems, vol. 9, no. 3, pp. L1–L8, 1993.

R. Hirota, The Direct Methods in Soliton Theory, Cambridge University Press, Cambridge, UK, 2004.

X. Lü, T. Geng, C. Zhang, H.-W. Zhu, X.-H. Meng, and B. Tian, “Multi-soliton solutions and their interactions for the (2+1)-dimensional sawada-kotera model with truncated painlevé expansion, hirota bilinear method and symbolic computation,” International Journal of Modern Physics B, vol. 23, no. 25, pp. 5003–5015, 2009.

R. Hirota, “Exact envelope-soliton solutions of a nonlinear wave equation,” Journal of Mathematical Physics, vol. 14, no. 7, pp. 805–809, 1973.