AMP Advances in Mathematical Physics 1687-9139 1687-9120 Hindawi Publishing Corporation 10.1155/2014/547692 547692 Research Article N -Soliton Solutions of the Nonisospectral Generalized Sawada-Kotera Equation Zhou Jian Li Xiang-Gui Wang Deng-Shan Punjabi Alkesh School of Applied Science, Beijing Information Science and Technology University, Beijing 100192 China bistu.edu.cn 2014 192014 2014 06 05 2014 10 08 2014 20 08 2014 1 9 2014 2014 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.

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.

1. Introduction

The Hirota method, originating from the work of Hirota in 1971 , 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  and the Darboux transformation . 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.  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 .

Jiang considers the nonisospectral problem  by using the compatibility condition of Lax pairs. In our work, the bilinear form and N -soliton solutions will be considered for a generalized nonisospectral equation.

The nonisospectral generalized Sawada-Kotera (GSK) equation  is written as follows: (1) u t - u x x x x x - 5 u x u x x - 5 u u x x x - u 2 u x - 5 u x x y + 5 x - 1 u y y - 5 u u y - 5 u x x - 1 u y - ω ( 2 u + x u x + 3 y u y ) - a u x - b u y = 0 , where ω , a , and b are real constants. The Lax pair of (1) is (2) L = x 3 + u x + y , M = t + 9 x 5 + ( 15 u + 3 ω y + b ) x 3 + 15 u x x 2 + ( 10 u x x + 5 u 2 - 5 x - 1 u y + ( 3 ω y + b ) u l l l l l l l l u x x + 5 u 2 - 5 x - 1 - ω x - a ) x .

The aim of this paper is to propose a simple method for construction N -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 <inline-formula> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M9"> <mml:mrow> <mml:mi>N</mml:mi></mml:mrow> </mml:math></inline-formula>-Soliton Solutions

Through the dependent variable transformation (3) u = 3 ( 15 - 185 ) ( ln f ) x x . Equation (1) can be written in the bilinear form. Consider (4) D x D t f · f - D x 6 f · f + D x 4 f · f - 5 D x 3 D y f · f - ( a + ω x ) D x 2 f · f + ( 1 - b - 3 ω y ) D x D y f · f + 5 D y 2 f · f - 2 ω f f x = 0 , where the D -operators  is defined by (5) D x m D t n a ( x , t ) · b ( x , t ) = ( x - x ) m ( t - t ) n a ( x , t ) b ( x , t ) | ( t - t ) x = x , t = t .

The perturbation method consists of expanding f with respect to a small parameter ε to obtain (6) f = 1 + ε f ( 1 ) + ε 2 f ( 2 ) + ε 3 f ( 3 ) + , and then finding each coefficient f ( n ) successively for n = 1,2 , 3 , .

Substituting the expansion formula of f into the bilinear equation (4) and arranging it at each order of ε , we have (7) ε : f t x ( 1 ) - f x x x x x x ( 1 ) + f x x x x ( 1 ) - 5 f x x x y ( 1 ) - ( a + ω x ) f x x ( 1 ) + ( 1 - b - 3 ω y ) f x y ( 1 ) + 5 f y y ( 1 ) - ω f x ( 1 ) = 0 , (8) ε 2 :  2 [ f t x ( 2 ) - f x x x x x x ( 2 ) + f x x x x ( 2 ) - 5 f x x x y ( 2 ) - ( a + ω x ) f x x ( 2 ) kkkk + ( 1 - b - 3 ω y ) f x y ( 2 ) + 5 f y y ( 2 ) - ω f x ( 2 ) ] + D x D t f ( 1 ) · f ( 1 ) - D x 6 f ( 1 ) · f ( 1 ) + D x 4 f ( 1 ) · f ( 1 ) - 5 D x 3 D y f ( 1 ) · f ( 1 ) - ( a + ω x ) D x 2 f ( 1 ) · f ( 1 ) + ( 1 - b - 3 ω y ) D x D y f ( 1 ) · f ( 1 ) + 5 D y 2 f ( 1 ) · f ( 1 ) - 2 ω f ( 1 ) · f x ( 1 ) = 0 , (9) ε 3 : f t x ( 3 ) - f x x x x x x ( 3 ) + f x x x x ( 3 ) - 5 f x x x y ( 3 ) - ( a + ω x ) f x x ( 3 ) + ( 1 - b - 3 ω y ) f x y ( 3 ) + 5 f y y ( 3 ) - ω f x ( 3 ) + D x D t f ( 1 ) · f ( 2 ) - D x 6 f ( 1 ) · f ( 2 ) + D x 4 f ( 1 ) · f ( 2 ) - 5 D x 3 D y f ( 1 ) · f ( 2 ) - ( a + ω x ) D x 2 f ( 1 ) · f ( 2 ) + ( 1 - b - 3 ω y ) D x D y f ( 1 ) · f ( 2 ) + 5 D y 2 f ( 1 ) · f ( 2 ) - ω ( f ( 1 ) · f x ( 2 ) + f x ( 1 ) · f ( 2 ) ) = 0 , llllllllllllllllllllllllllllllllllllllll let us choose (10) f ( 1 ) = e ξ 1 , where ξ 1 = - k 1 ( t ) x + k 1 3 ( t ) y - ( a / ω ) k 1 ( t ) + ( b / 3 ω ) k 1 3 ( t ) + ( 9 / 5 ω ) k 1 5 ( t ) + ξ 1 0 .

Since substituting this into the left-hand side of (7) gives (11) k 1 , t ( t ) = ω k 1 ( t ) , then (11) is an ordinary differential system and it can be solved exactly. The solutions of (11) are written as (12) k 1 ( t ) = c 1 e ω t .

Therefore, we are able to choose f ( j ) = 0 , j = 2,3 , . This shows that the expansion of f may be truncated as the finite sum (13) f = 1 + f ( 1 ) = 1 + e ξ 1 . Substituting (13) into (3), the one-soliton solution of the nonisospectral GSK equation (1) can be obtained (14) u = 3 ( 15 - 185 ) k 1 2 ( t ) 4 sech 2 ( ξ 1 2 ) . Here u 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 (15) ξ 1 = 0 .

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: (16) A = 3 ( 15 - 185 ) k 1 2 ( t ) 4 .

Figures 1 and 2 describe the different amplitudes of the one-soliton solution at t = 1 and t = 2 .

The shape and motion of the one-soliton solution for ω = - 0.8 , a = 1 , b = 1 , c 1 = 1 , ξ 1 0 = 1 , t = 1 .

The shape and motion of the one-soliton solution for ω = - 0.8 , a = 1 , b = 1 , c 1 = 1 , ξ 1 0 = 1 , t = 2 .

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 (17) f ( 1 ) = e ξ 1 + e ξ 2 , where ξ j = - k j ( t ) x + k j 3 ( t ) y - ( a / ω ) k j ( t ) + ( b / 3 ω ) k j 3 ( t ) + ( 9 / 5 ω ) k j 5 ( t ) + ξ j 0 for j = 1,2 .

Substituting (17) into the left-hand side of (7), we have (18) k j , t ( t ) = ω k j ( t ) , ( j = 1,2 ) . The solutions of (18) are written as (19) k j ( t ) = c j e ω t , ( j = 1,2 ) .

We here set that (20) f ( 2 ) = e ξ 1 + ξ 2 + θ 12 .

From (20), we might assume that the relations e θ 12 = F ( t ) . Equation (20) may also be written as (21) f ( 2 ) = F ( t ) e ξ 1 + ξ 2 . Substituting (17), (21) into the left-hand side of (8) and using (18), we have (22) F ( t ) t - 3 k 1 ( t ) k 2 ( t ) [ k 1 ( t ) + k 2 ( t ) ] F ( t ) + 3 k 1 ( t ) k 2 ( t ) [ k 1 ( t ) - k 2 ( t ) ] 2 [ k 1 ( t ) + k 2 ( t ) ] = 0 . Substitution of (19) into (22) gives (23) F ( t ) = [ k 1 ( t ) - k 2 ( t ) ] 2 [ k 1 ( t ) + k 2 ( t ) ] 2 = e θ 12 .

The coefficient e θ 12 obtained in (23), which are similar to the SK equation (see ), can also be KdV type.

Therefore, we are able to choose f ( j ) = 0 , j = 3,4 , . The two-soliton solutions are obtained by (3) in which f is defined as (24) f = 1 + f ( 1 ) + f ( 2 ) = 1 + e ξ 1 + e ξ 2 + e ξ 2 + ξ 2 + θ 12 .

Its shape and motion are shown in Figures 3 and 4.

The shape and motion of the two-soliton solutions for ω = - 1 , a = 1 , b = 1 , c 1 = 1 , c 2 = 2 , ξ 1 0 = 1 , ξ 2 0 = 1 , t = 2 .

The density plots of two-soliton resonance for the nonisospectral GSK equation with parameters ω = - 1 , a = 1 , b = 1 , c 1 = 1 , c 2 = 2 , ξ 1 0 = 1 , ξ 2 0 = 1 , t = 2 .

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 (25) f ( 1 ) = e ξ 1 + e ξ 2 + e ξ 3 , where ξ j = - k j ( t ) x + k j 3 ( t ) y - ( a / ω ) k j ( t ) + ( b / 3 ω ) k j 3 ( t ) + ( 9 / 5 ω ) k j 5 ( t ) + ξ j 0 for j = 1,2 , 3 .

Substituting (25) into the left-hand side of (7), we have (26) k j , t ( t ) = ω k j ( t ) , ( j = 1,2 , 3 ) . The solutions of (26) are written as (27) k j ( t ) = c j e ω t , ( j = 1,2 , 3 ) .

We here set that (28) f ( 2 ) = e ξ 1 + ξ 2 + θ 12 + e ξ 1 + ξ 3 + θ 13 + e ξ 2 + ξ 3 + θ 23 . Substitution of (25), (28) into (8) gives (29) e θ 12 = [ k 1 ( t ) - k 2 ( t ) ] 2 [ k 1 ( t ) + k 2 ( t ) ] 2 , e θ 13 = [ k 1 ( t ) - k 3 ( t ) ] 2 [ k 1 ( t ) + k 3 ( t ) ] 2 , e θ 23 = [ k 2 ( t ) - k 3 ( t ) ] 2 [ k 2 ( t ) + k 3 ( t ) ] 2 .

Let (30) f ( 3 ) = e ξ 1 + ξ 2 + ξ 3 + θ 123 . Substituting (25), (28), and (30) into (9), one obtains (31) e θ 123 = [ k 1 ( t ) - k 2 ( t ) ] 2 [ k 1 ( t ) - k 3 ( t ) ] 2 [ k 2 ( t ) - k 3 ( t ) ] 2 [ k 1 ( t ) + k 2 ( t ) ] 2 [ k 1 ( t ) + k 3 ( t ) ] 2 [ k 2 ( t ) + k 3 ( t ) ] 2 .

Therefore, we are able to choose f ( j ) = 0 , j = 4,5 , . The three-soliton solutions are obtained by (3) in which f is defined as (32) f = 1 + f ( 1 ) + f ( 2 ) + f ( 3 ) = 1 + e ξ 1 + e ξ 2 + e ξ 3 + e ξ 1 + ξ 2 + θ 12 + e ξ 1 + ξ 3 + θ 13 + e ξ 2 + ξ 3 + θ 23 + e ξ 1 + ξ 2 + ξ 3 + θ 123 .

The nonisospectral GSK equation  has been shown to be integrable. It can be represented as the compatibility condition in the Lax form [ L , M ] = 0 . Therefore, it would be reasonable to continue to find the N -soliton solutions ( N > 3 ) with the help of symbolic computation (see ).

This process can be extended to the four-soliton solutions, and so on. Generally, the N -soliton solutions are expressed as (33) f = exp [ j = 1 N μ j ξ j + j < l ( N ) θ j l μ j μ l ] , where the coefficients θ j l and ξ j are defined by (34) θ j l = [ k j ( t ) - k l ( t ) ] 2 [ k j ( t ) + k l ( t ) ] 2 , (35) ξ j = - k j ( t ) x + k j 3 ( t ) y - a ω k j ( t ) + b 3 ω k j 3 ( t ) + 9 5 ω k j 5 ( t ) + ξ j 0 , j = 1,2 , , N , respectively.

In formula (33), the first means a summation over all possible combinations of μ 1 = 0,1 , μ 2 = 0,1 , , μ N = 0,1 , and j < l ( N ) means a summation over all possible pairs ( j , l ) chosen from the set { 1,2 , , N } , with the condition that j < l .

Substituting (33) into (3), we obtain the N -soliton solutions for the nonisospectral GSK equation.

3. Conclusion

In this paper, we have obtained the N -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), N -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 13. 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.

Hirota R. Exact solution of the korteweg-de vries equation for multiple collisions of solitons Physical Review Letters 1971 27 18 1192 1194 10.1103/PhysRevLett.27.1192 2-s2.0-35949040541 Pitaevskii L. Stringari S. Bose-Einstein Condensation 2003 New York, NY, USA Oxford University MR2012737 Wang D.-S. Li X.-G. 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 2012 45 10 105301 10.1088/0953-4075/45/10/105301 2-s2.0-84860820896 Ablowitz M. J. Segur H. Solitons and the Inverse Scattering Transform 1981 Philadelphia, Pa, USA SIAM MR642018 Matveev V. B. Salle M. A. Darboux Transformations and Solitons 1991 Berlin, Germany Springer 10.1007/978-3-662-00922-2 MR1146435 Deng S. Zhang D. Chen D. Exact solutions for the nonisospectral Kadomtshev-Petviashvili equation Journal of the Physical Society of Japan 2005 74 9 2383 2385 10.1143/JPSJ.74.2383 MR2186616 2-s2.0-25144525589 Sun Y. Bi J. Chen D. N -soliton solutions and double Wronskian solution of the non-isospectral AKNS equation Chaos, Solitons & Fractals 2005 26 3 905 912 10.1016/j.chaos.2005.01.032 MR2145426 2-s2.0-18844458902 Sun Y. The non-isospectral modified Kadomtsev-Petviashvili equation with self-consistent sources and its coupled system Applied Mathematics and Computation 2010 217 4 1639 1645 10.1016/j.amc.2009.08.053 MR2727406 ZBL1202.35261 2-s2.0-77957268399 Li X. Yan W. Chan C. K. Numerical schemes for Hamilton-Jacobi equations on unstructured meshes Numerische Mathematik 2003 94 2 315 331 10.1007/s00211-002-0418-9 MR1974558 ZBL1029.65111 2-s2.0-0142118613 Hua D. Li X. The finite element method for computing the ground states of the dipolar Bose-Einstein condensates Applied Mathematics and Computation 2014 234 214 222 10.1016/j.amc.2014.01.085 MR3190533 Li S. Li X. Hua D. A time-splitting and sine spectral method for dynamics of dipolar Bose-Einstein condensate Advances in Mathematical Physics 2013 2013 7 517395 MR3084967 10.1155/2013/517395 Jiang Z. Nonisospectral problems related to DS and other ( 2 + 1 ) -dimensional nonlinear evolution equations Inverse Problems 1993 9 3 L1 L8 10.1088/0266-5611/9/3/001 MR1222588 2-s2.0-33947101065 Hirota R. The Direct Methods in Soliton Theory 2004 Cambridge, UK Cambridge University Press 10.1017/CBO9780511543043 MR2085332 X. Geng T. Zhang C. Zhu H.-W. Meng X.-H. Tian B. 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 2009 23 25 5003 5015 10.1142/S0217979209053382 2-s2.0-70449120088 Hirota R. Exact envelope-soliton solutions of a nonlinear wave equation Journal of Mathematical Physics 1973 14 7 805 809 10.1063/1.1666399 MR0338587 2-s2.0-0000492769