N-Soliton Solutions of the Nonisospectral Generalized Sawada-Kotera Equation

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 N-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.


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][10][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.
let us choose where  1 = − 1 () +  3 1 () − (/) 1 () + (/3) 3 1 () + (9/5) 5  1 () +  0 1 .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  () = 0,  = 2, 3, . ... 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 timedependent top trace In fact, the top trace of the solution (14) is a line with timedependent 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  = 1 and  = 2.
We begin here by finding a two-soliton solution.It is a solution describing the interaction of two solitons.
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.
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.

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.