Hindawi Publishing Corporation e Scientific World Journal Volume 2014, Article ID 593983, 7 pages http://dx.doi.org/10.1155/2014/593983
Research Article Exact Multisoliton Solutions of General Nonlinear Schrödinger Equation with Derivative Qi Li,1 Qiu-yuan Duan,2 and Jian-bing Zhang3 1
Department of Mathematics, East China Institute of Technology, Nanchang 330013, China Department of Mathematics, Fuzhou Vocational College of Technology, Fuzhou 344000, China 3 School of Mathematics and Statitics, Jiangsu Normal University, Xuzhou, Jiangsu 221116, China 2
Correspondence should be addressed to Qi Li; [email protected] Received 31 March 2014; Revised 19 May 2014; Accepted 24 May 2014; Published 12 June 2014 Academic Editor: Fazlollah Soleymani Copyright © 2014 Qi Li 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. Multisoliton solutions are derived for a general nonlinear Schr¨odinger equation with derivative by using Hirota’s approach. The dynamics of one-soliton solution and two-soliton interactions are also illustrated. The considered equation can reduce to nonlinear Schr¨odinger equation with derivative as well as the solutions.
Some nonlinear partial differential equations are integrable models with interesting physical applications. Much work has been focused on those equations such as the celebrating KdV, modified KdV, nonlinear Schr¨odinger equations, and Toda lattice. Inverse scattering transform (IST), Darboux transformation, Hirota’s approach, tanh-function method, and algebraic-geometry method [1–25] have been used to investigate the exact solutions and integrability of those equations. Among these methods, Hirota’s approach is usually used to find 𝑁-soliton solutions for soliton equation. The key is transforming the soliton equations to the bilinear ones by introducing bilinear derivative and appropriate variable transformation [4]. The Hirota approach has been generalized to much more general bilinear equations recently [5]. The invariant subspace method is refined to present more unity and more diversity of exact solutions by taking subspaces of solutions to linear ordinary differential equations as invariant subspaces that evolution equations admit [6]. Associating with Kaup-Newell (KN shortly) spectral problem, there exist three types of derivative nonlinear Schr¨odinger equations [7, 8]. Gauge transformations have been found among them [9, 10]. The first well-known derivative nonlinear Schr¨odinger equation (DNLSE) is 𝑖𝑢𝑡 + 𝑢𝑥𝑥 + 𝑖(𝑢2 𝑢∗ )𝑥 = 0,
(1)
where 𝑢∗ denotes the complex conjugate of 𝑢. This equation models Alfven waves and magnetohydrodynamic waves in
plasmas and also model subpicosecond or femtosecond pulses in single-mode optical fibers in nonlinear optics [11, 12]. The equation is investigated in some literature (see, e.g., [13–15]). Its explicit form of the 𝑁-soliton solutions is also obtained by some algebraic technique [16]. Through the 𝑛fold Darboux transformation the rogue wave solutions are constructed explicitly by seed solutions recently [17]. The standard NLS equation has a tri-Hamiltonian structure [18] and DNLSE equations has some sl(2) generalizations [19] and an so(3) generalization [20]. In the paper, we consider general nonlinear Schr¨odinger equation with derivative (GDNLSE) as follows: 𝑞𝑡 = 𝑞𝑥𝑥 − 𝑖 (𝑞2 𝑟)𝑥 ,
(2a)
𝑟𝑡 = −𝑟𝑥𝑥 − 𝑖 (𝑞𝑟2 )𝑥 .
(2b)
We show that through a variable transformation the bilinear equations for (2a) and (2b) can be derived for constructing its 𝑁-soliton solutions. We also describe that the multisoliton solutions of (1) can be derived by reduction. Firstly, we deduce the Lax pair of GDNLSE (2a) and (2b), which usually assures the complete integrability of a nonlinear equation. From the Kaup-Newell spectral problem 𝜙 𝜙 ( 1) = 𝑀 ( 1) , 𝜙2 𝑥 𝜙2
−𝑖𝜂2 𝜂𝑞 𝑀=( ), 𝜂𝑟 𝑖𝜂2
(3a)
2
The Scientific World Journal (1) = 0, ℎ𝑡(1) + ℎ𝑥𝑥
(11a)
(3) = − (𝐷𝑡 + 𝐷𝑥2 ) ℎ(1) ⋅ 𝑠(2) , ℎ𝑡(3) + ℎ𝑥𝑥
(11b)
time evolution 𝜙 𝜙 ( 1) = 𝑁 ( 1) , 𝜙2 𝑡 𝜙2
𝑁=(
𝐴 𝐵 ), 𝐶 −𝐴
(3b)
(5) = − (𝐷𝑡 + 𝐷𝑥2 ) (ℎ(1) ⋅ 𝑠(4) + ℎ(3) ⋅ 𝑠(2) ) , ℎ𝑡(5) + ℎ𝑥𝑥
and the related zero curvature equation 𝑀𝑡 − 𝑁𝑥 + [𝑀, 𝑁] = 0,
one can derive the GDNLSE (2a) and (2b). Its corresponding Lax pair (3a) and (3b) is governed by 4
2
𝐴 = −2𝜂 − 𝜂 𝑞𝑟,
(5a)
3
2
𝐵 = −2𝑖𝜂 𝑞 + 𝜂 (𝑞𝑥 − 𝑖𝑞 𝑟) , 3
2
𝐶 = −2𝑖𝜂 𝑟 − 𝜂 (𝑟𝑥 + 𝑖𝑞𝑟 ) .
(5b) (5c)
Secondly, we give the bilinear form of GDNLSE and further its 𝑁-soliton solutions. By the variable transformation 𝑔𝑠 , 𝑓2
𝑞=
𝑟=
ℎ𝑓 , 𝑠2
(𝐷𝑡 −
𝐷𝑥2 ) 𝑔
⋅ 𝑓 = 0,
(7a)
(𝐷𝑡 + 𝐷𝑥2 ) ℎ ⋅ 𝑠 = 0,
(7b)
(𝐷𝑡 − 𝐷𝑥2 ) 𝑓 ⋅ 𝑠 = 0,
(7c)
𝑖 𝐷𝑥 𝑓 ⋅ 𝑠 = − 𝑔ℎ, 2
(7d)
where 𝑔, ℎ, 𝑓, and 𝑠 are complex functions and 𝐷 is the wellknown Hirota bilinear operator defined as 𝑚 𝑛 𝐷𝑡𝑚 𝐷𝑥𝑛 𝑎 ⋅ 𝑏 = (𝜕𝑡 − 𝜕𝑡 ) (𝜕𝑥 − 𝜕𝑥 ) 𝑎(𝑡, 𝑥)𝑏(𝑡 , 𝑥 )𝑡 =𝑡,𝑥 =𝑥 . (8) To solve the system (7a), (7b), (7c), and (7d), we expand 𝑓, 𝑔, ℎ, and 𝑠 as ∞
𝑓 = 1 + ∑𝑓(2𝑗) 𝜀2𝑗 , 𝑗=1 ∞
ℎ = ∑ℎ
∞
𝑔 = ∑ 𝑔(2𝑗−1) 𝜀2𝑗−1 , 𝑗=1
(2𝑗−1) 2𝑗−1
𝜀
,
∞
(9) 𝜀 .
(12b)
(6) (6) − (𝑠𝑡(6) + 𝑠𝑥𝑥 ) 𝑓𝑡(6) − 𝑓𝑥𝑥
= (−𝐷𝑡 + 𝐷𝑥2 ) (𝑓(4) ⋅ 𝑠(2) + 𝑓(2) ⋅ 𝑠(4) ) ,
(12c)
.. . 𝑖 𝑓𝑥(2) − 𝑠𝑥(2) = − 𝑔(1) ℎ(1) , 2 𝑓𝑥(4) − 𝑠𝑥(4) = −𝐷𝑥 𝑓(2) ⋅ 𝑠(2) −
(13a)
𝑖 (1) (3) (𝑔 ℎ + 𝑔(3) ℎ(1) ) , (13b) 2
𝑓𝑥(6) − 𝑠𝑥(6) = −𝐷𝑥 (𝑓(2) ⋅ 𝑠(4) + 𝑓(4) ⋅ 𝑠(2) ) −
𝑖 (1) (5) (𝑔 ℎ + 𝑔(3) ℎ(3) + 𝑔(5) ℎ(1) ) , 2
(13c)
.. . In order to get one-soliton of GDNLSE (2a) and (2b), we select 𝑔(1) and ℎ(1) for (10a) and (11a) as follows: 𝑔(1) = 𝑒𝜉1 ,
𝜉1 = 𝜔1 𝑡 − 𝑘1 𝑥 + 𝜉1(0) , 𝜔1 = 𝑘12 ,
(14a)
ℎ(1) = 𝑒𝜂1 ,
𝜂1 = 𝜎1 𝑡 + 𝑙1 𝑥 + 𝜂1(0) , 𝜎1 = −𝑙12 ,
(14b)
where 𝜉1(0) , 𝜂1(0) are all constants. Substituting (14a) and (14b) into (13a), one can obtain 𝑖 𝑓𝑥(2) − 𝑠𝑥(2) = − 𝑒𝜉1 +𝜂1 . 2
(15)
𝑗=1
𝑓(2) =
Substituting (9) into (7a), (7b), (7c), and (7d) yields (1) 𝑔𝑡(1) − 𝑔𝑥𝑥 = 0,
𝑔𝑡(3)
(4) (4) − (𝑠𝑡(4) + 𝑠𝑥𝑥 ) = (−𝐷𝑡 + 𝐷𝑥2 ) 𝑓(2) ⋅ 𝑠(2) , 𝑓𝑡(4) − 𝑓𝑥𝑥
(12a)
Then combining (15) with (12a) yields
(2𝑗) 2𝑗
𝑠 = 1 + ∑𝑠
𝑗=1
(2) (2) ) = 0, − (𝑠𝑡(2) + 𝑠𝑥𝑥 𝑓𝑡(2) − 𝑓𝑥𝑥
(6)
GDNLSE (2a) and (2b) can be transformed to the bilinear form
(11c)
.. .
(4)
−
(3) 𝑔𝑥𝑥
= (−𝐷𝑡 +
𝐷𝑥2 ) 𝑔(1)
(10a) (2)
⋅𝑓 ,
2
𝑒𝜉1 +𝜂1 ,
(16a)
2
𝑒𝜉1 +𝜂1 .
(16b)
2(𝑙1 − 𝑘1 ) 𝑖𝑙1
(10b)
2(𝑙1 − 𝑘1 )
(10c)
Assuming that 𝑔(𝑖) = ℎ(𝑖) = 𝑓(𝑗) = 𝑠(𝑗) = 0, (𝑖 = 3, 5, 7, . . . , 𝑗 = 4, 6, 8, . . .), one can find that (10a), (10b) and (10c)–(13a), (13b), and (13c) are still hold. Thus, let 𝜖 = 1, substituting (9), (14a) and (14b) and (16a) and (16b) into (6),
(5) = (−𝐷𝑡 + 𝐷𝑥2 ) (𝑔(1) ⋅ 𝑓(4) + 𝑔(3) ⋅ 𝑓(2) ) , 𝑔𝑡(5) − 𝑔𝑥𝑥
.. .
𝑠(2) =
𝑖𝑘1
The Scientific World Journal
3
|q1 |
10
5
|r1 |
10
10
10
5 5
5 0 −10
0 −10
0
0 −5
−5 −5
0 5
−5
0 5
−10
10
10
(a)
−10
(b)
Figure 1: The shape of the one-soliton solution of GDNLSE given by (17). 𝑞1 and 𝑟1 with 𝑘1 = 1 + 0.3𝑖, 𝑙1 = −1 + 0.2𝑖, and 𝜉1(0) = 𝜂1(0) = 0. (a) |𝑞1 |, (b) |𝑟1 |.
one can arrive at one soliton solution for GDNLSE (2a) and (2b): 𝑞1 =
𝑟1 =
𝑒𝜉1 (1 + (𝑙1 /2) 𝑒𝜉1 +𝜂1 +(𝜋/2)𝑖+𝜃13 ) 2
(1 + (𝑘1 /2) 𝑒𝜉1 +𝜂1 +(𝜋/2)𝑖+𝜃13 )
𝑙 + 2 (𝑒𝜉1 +𝜂2 +(𝜋/2)𝑖+𝜃14 + 𝑒𝜉2 +𝜂2 +(𝜋/2)𝑖+𝜃24 ) . 2
,
𝑒𝜂1 (1 + (𝑘1 /2) 𝑒𝜉1 +𝜂1 +(𝜋/2)𝑖+𝜃13 ) 2
(1 + (𝑙1 /2) 𝑒𝜉1 +𝜂1 +(𝜋/2)𝑖+𝜃13 )
(17)
𝜉𝑗 = 𝜔𝑗 𝑡 − 𝑘𝑗 𝑥 + 𝜉𝑗(0) ,
𝜔𝑗 = 𝑘𝑗2 , (𝑗 = 1, 2) ,
ℎ(1) = 𝑒𝜂1 + 𝑒𝜂2 ,
𝜂𝑗 = 𝜎𝑗 𝑡 + 𝑙𝑗 𝑥 + 𝜂𝑗(0) ,
𝜎𝑗 = −𝑙𝑗2 ,
𝑔(3) =
𝑘1 𝜉1 +𝜂1 +(𝜋/2)𝑖+𝜃13 + 𝑒𝜉1 +𝜂2 +(𝜋/2)𝑖+𝜃14 ) (𝑒 2 𝑘 + 2 (𝑒𝜉2 +𝜂1 +(𝜋/2)𝑖+𝜃23 + 𝑒𝜉2 +𝜂2 +(𝜋/2)𝑖+𝜃24 ) , 2
(21)
where 𝑒𝜃13 = 𝑒𝜃14 =
(18a)
1 2
,
𝑒𝜃23 =
2
,
𝑒𝜃24 =
(𝑙1 − 𝑘1 ) 1 (𝑙2 − 𝑘1 )
1 2
,
2
,
(𝑙1 − 𝑘2 ) 1 (𝑙2 − 𝑘2 )
(22)
2
𝑒𝜃12 = (𝑘1 − 𝑘2 ) . (18b)
Then substituting (18b) and (20b) into (11b), by some computations, we obtain ℎ(3) =
𝑘1 𝜉1 +𝜂1 +𝜂2 +(𝜋/2)𝑖+𝜃13 +𝜃14 +𝜃34 𝑒 2 𝑘 + 2 𝑒𝜉2 +𝜂1 +𝜂2 +(𝜋/2)𝑖+𝜃23 +𝜃24 +𝜃34 , 2
(19)
(23)
where 𝑒𝜃34 = (𝑙1 − 𝑙2 )2 . Substituting (21) and (23) into (12b) and (13b), we obtain
Then combining (19) with (12a) yields 𝑓(2) =
𝑙1 𝜉1 +𝜉2 +𝜂1 +(𝜋/2)𝑖+𝜃13 +𝜃23 +𝜃12 𝑒 2 𝑙 + 2 𝑒𝜉1 +𝜉2 +𝜂2 +(𝜋/2)𝑖+𝜃14 +𝜃24 +𝜃12 , 2
where 𝜉𝑗(0) , 𝜂𝑗(0) are all constants. Substituting (18a) and (18b) into (13a), one can obtain 𝑖 𝑓𝑥(2) − 𝑠𝑥(2) = − (𝑒𝜉1 +𝜂1 + 𝑒𝜉1 +𝜂2 + 𝑒𝜉2 +𝜂1 + 𝑒𝜉2 +𝜂2 ) . 2
(20b)
Substituting (18b) and (20a) into (10b), by some computations, we obtain
,
where 𝜉1 , 𝜂1 are defined by (14a) and (14b), 𝑘1 , 𝑙1 are all arbitrary constants, and 𝑒𝜃13 = (1/(𝑙1 − 𝑘1 )2 ). We depict |𝑞1 | and |𝑟1 | in Figure 1. For convenience, we replace 𝑡 by −𝑖𝑡 and 𝑥 by −𝑥. To get two-soliton of GDNLSE (2a) and (2b), we select 𝑔(1) and ℎ(1) for (10a) and (11a) as follows: 𝑔(1) = 𝑒𝜉1 + 𝑒𝜉2 ,
𝑙1 𝜉1 +𝜂1 +(𝜋/2)𝑖+𝜃13 + 𝑒𝜉2 +𝜂1 +(𝜋/2)𝑖+𝜃23 ) (𝑒 2
𝑠(2) =
(20a)
𝑓(4) =
𝑘1 𝑘2 𝜉1 +𝜉2 +𝜂1 +𝜂2 +𝜋𝑖+𝜃12 +𝜃13 +𝜃14 +𝜃23 +𝜃24 +𝜃34 , 𝑒 4
(24a)
𝑠(4) =
𝑙1 𝑙2 𝜉1 +𝜉2 +𝜂1 +𝜂2 +𝜋𝑖+𝜃12 +𝜃13 +𝜃14 +𝜃23 +𝜃24 +𝜃34 . 𝑒 4
(24b)
4
The Scientific World Journal
|q2 |
10
5 5 0 −10
|r 2 |
10
10
10
5 5 0 −10
0
0 −5
−5 −5
0 5 10
−5
0 5
−10
10
(a)
−10
(b)
Figure 2: The interaction of two-soliton solution of GDNLSE given by (6). 𝑞2 and 𝑟2 with 𝑘1 = 1 + 0.3𝑖, 𝑘2 = 1 + 0.9𝑖, 𝑙1 = −1 + 0.2𝑖, 𝑙2 = −1 + 0.8𝑖, and 𝜉1(0) = 𝜉2(0) = 𝜂1(0) = 𝜂2(0) = 0. (a) |𝑞2 |, (b) |𝑟2 |.
Assuming that 𝑔(𝑖) = ℎ(𝑖) = 𝑓(𝑗) = 𝑠(𝑗) = 0, (𝑖 = 5, 7, . . . , 𝑗 = 6, 8, . . .), one can find that (10a), (10b) and (10c)–(13a), (13b), and (13c) are still hold. Thus, let 𝜖 = 1, we have two-soliton solution for GDNLSE (2a) and (2b):
𝑞=
𝑟=
(𝑔(1) + 𝑔(3) ) (1 + 𝑠(2) + 𝑠(4) ) 2
(1 + 𝑓(2) + 𝑓(4) )
2
𝑓𝑁 (𝑡, 𝑥) =
2𝑁
∑ 𝐴 1 (𝜇) exp [ ∑ 𝜇𝑗 𝜉𝑗 𝜇=0,1 𝑗=1 [
2𝑁
2𝑁
2𝑁
+ ∑ 𝜇𝑗 𝜇𝜌 𝜃𝑗𝜌 ] , 1≤𝑗
Research Article Exact Multisoliton Solutions of General Nonlinear Schrödinger Equation with Derivative Qi Li,1 Qiu-yuan Duan,2 and Jian-bing Zhang3 1
Department of Mathematics, East China Institute of Technology, Nanchang 330013, China Department of Mathematics, Fuzhou Vocational College of Technology, Fuzhou 344000, China 3 School of Mathematics and Statitics, Jiangsu Normal University, Xuzhou, Jiangsu 221116, China 2
Correspondence should be addressed to Qi Li; [email protected] Received 31 March 2014; Revised 19 May 2014; Accepted 24 May 2014; Published 12 June 2014 Academic Editor: Fazlollah Soleymani Copyright © 2014 Qi Li 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. Multisoliton solutions are derived for a general nonlinear Schr¨odinger equation with derivative by using Hirota’s approach. The dynamics of one-soliton solution and two-soliton interactions are also illustrated. The considered equation can reduce to nonlinear Schr¨odinger equation with derivative as well as the solutions.
Some nonlinear partial differential equations are integrable models with interesting physical applications. Much work has been focused on those equations such as the celebrating KdV, modified KdV, nonlinear Schr¨odinger equations, and Toda lattice. Inverse scattering transform (IST), Darboux transformation, Hirota’s approach, tanh-function method, and algebraic-geometry method [1–25] have been used to investigate the exact solutions and integrability of those equations. Among these methods, Hirota’s approach is usually used to find 𝑁-soliton solutions for soliton equation. The key is transforming the soliton equations to the bilinear ones by introducing bilinear derivative and appropriate variable transformation [4]. The Hirota approach has been generalized to much more general bilinear equations recently [5]. The invariant subspace method is refined to present more unity and more diversity of exact solutions by taking subspaces of solutions to linear ordinary differential equations as invariant subspaces that evolution equations admit [6]. Associating with Kaup-Newell (KN shortly) spectral problem, there exist three types of derivative nonlinear Schr¨odinger equations [7, 8]. Gauge transformations have been found among them [9, 10]. The first well-known derivative nonlinear Schr¨odinger equation (DNLSE) is 𝑖𝑢𝑡 + 𝑢𝑥𝑥 + 𝑖(𝑢2 𝑢∗ )𝑥 = 0,
(1)
where 𝑢∗ denotes the complex conjugate of 𝑢. This equation models Alfven waves and magnetohydrodynamic waves in
plasmas and also model subpicosecond or femtosecond pulses in single-mode optical fibers in nonlinear optics [11, 12]. The equation is investigated in some literature (see, e.g., [13–15]). Its explicit form of the 𝑁-soliton solutions is also obtained by some algebraic technique [16]. Through the 𝑛fold Darboux transformation the rogue wave solutions are constructed explicitly by seed solutions recently [17]. The standard NLS equation has a tri-Hamiltonian structure [18] and DNLSE equations has some sl(2) generalizations [19] and an so(3) generalization [20]. In the paper, we consider general nonlinear Schr¨odinger equation with derivative (GDNLSE) as follows: 𝑞𝑡 = 𝑞𝑥𝑥 − 𝑖 (𝑞2 𝑟)𝑥 ,
(2a)
𝑟𝑡 = −𝑟𝑥𝑥 − 𝑖 (𝑞𝑟2 )𝑥 .
(2b)
We show that through a variable transformation the bilinear equations for (2a) and (2b) can be derived for constructing its 𝑁-soliton solutions. We also describe that the multisoliton solutions of (1) can be derived by reduction. Firstly, we deduce the Lax pair of GDNLSE (2a) and (2b), which usually assures the complete integrability of a nonlinear equation. From the Kaup-Newell spectral problem 𝜙 𝜙 ( 1) = 𝑀 ( 1) , 𝜙2 𝑥 𝜙2
−𝑖𝜂2 𝜂𝑞 𝑀=( ), 𝜂𝑟 𝑖𝜂2
(3a)
2
The Scientific World Journal (1) = 0, ℎ𝑡(1) + ℎ𝑥𝑥
(11a)
(3) = − (𝐷𝑡 + 𝐷𝑥2 ) ℎ(1) ⋅ 𝑠(2) , ℎ𝑡(3) + ℎ𝑥𝑥
(11b)
time evolution 𝜙 𝜙 ( 1) = 𝑁 ( 1) , 𝜙2 𝑡 𝜙2
𝑁=(
𝐴 𝐵 ), 𝐶 −𝐴
(3b)
(5) = − (𝐷𝑡 + 𝐷𝑥2 ) (ℎ(1) ⋅ 𝑠(4) + ℎ(3) ⋅ 𝑠(2) ) , ℎ𝑡(5) + ℎ𝑥𝑥
and the related zero curvature equation 𝑀𝑡 − 𝑁𝑥 + [𝑀, 𝑁] = 0,
one can derive the GDNLSE (2a) and (2b). Its corresponding Lax pair (3a) and (3b) is governed by 4
2
𝐴 = −2𝜂 − 𝜂 𝑞𝑟,
(5a)
3
2
𝐵 = −2𝑖𝜂 𝑞 + 𝜂 (𝑞𝑥 − 𝑖𝑞 𝑟) , 3
2
𝐶 = −2𝑖𝜂 𝑟 − 𝜂 (𝑟𝑥 + 𝑖𝑞𝑟 ) .
(5b) (5c)
Secondly, we give the bilinear form of GDNLSE and further its 𝑁-soliton solutions. By the variable transformation 𝑔𝑠 , 𝑓2
𝑞=
𝑟=
ℎ𝑓 , 𝑠2
(𝐷𝑡 −
𝐷𝑥2 ) 𝑔
⋅ 𝑓 = 0,
(7a)
(𝐷𝑡 + 𝐷𝑥2 ) ℎ ⋅ 𝑠 = 0,
(7b)
(𝐷𝑡 − 𝐷𝑥2 ) 𝑓 ⋅ 𝑠 = 0,
(7c)
𝑖 𝐷𝑥 𝑓 ⋅ 𝑠 = − 𝑔ℎ, 2
(7d)
where 𝑔, ℎ, 𝑓, and 𝑠 are complex functions and 𝐷 is the wellknown Hirota bilinear operator defined as 𝑚 𝑛 𝐷𝑡𝑚 𝐷𝑥𝑛 𝑎 ⋅ 𝑏 = (𝜕𝑡 − 𝜕𝑡 ) (𝜕𝑥 − 𝜕𝑥 ) 𝑎(𝑡, 𝑥)𝑏(𝑡 , 𝑥 )𝑡 =𝑡,𝑥 =𝑥 . (8) To solve the system (7a), (7b), (7c), and (7d), we expand 𝑓, 𝑔, ℎ, and 𝑠 as ∞
𝑓 = 1 + ∑𝑓(2𝑗) 𝜀2𝑗 , 𝑗=1 ∞
ℎ = ∑ℎ
∞
𝑔 = ∑ 𝑔(2𝑗−1) 𝜀2𝑗−1 , 𝑗=1
(2𝑗−1) 2𝑗−1
𝜀
,
∞
(9) 𝜀 .
(12b)
(6) (6) − (𝑠𝑡(6) + 𝑠𝑥𝑥 ) 𝑓𝑡(6) − 𝑓𝑥𝑥
= (−𝐷𝑡 + 𝐷𝑥2 ) (𝑓(4) ⋅ 𝑠(2) + 𝑓(2) ⋅ 𝑠(4) ) ,
(12c)
.. . 𝑖 𝑓𝑥(2) − 𝑠𝑥(2) = − 𝑔(1) ℎ(1) , 2 𝑓𝑥(4) − 𝑠𝑥(4) = −𝐷𝑥 𝑓(2) ⋅ 𝑠(2) −
(13a)
𝑖 (1) (3) (𝑔 ℎ + 𝑔(3) ℎ(1) ) , (13b) 2
𝑓𝑥(6) − 𝑠𝑥(6) = −𝐷𝑥 (𝑓(2) ⋅ 𝑠(4) + 𝑓(4) ⋅ 𝑠(2) ) −
𝑖 (1) (5) (𝑔 ℎ + 𝑔(3) ℎ(3) + 𝑔(5) ℎ(1) ) , 2
(13c)
.. . In order to get one-soliton of GDNLSE (2a) and (2b), we select 𝑔(1) and ℎ(1) for (10a) and (11a) as follows: 𝑔(1) = 𝑒𝜉1 ,
𝜉1 = 𝜔1 𝑡 − 𝑘1 𝑥 + 𝜉1(0) , 𝜔1 = 𝑘12 ,
(14a)
ℎ(1) = 𝑒𝜂1 ,
𝜂1 = 𝜎1 𝑡 + 𝑙1 𝑥 + 𝜂1(0) , 𝜎1 = −𝑙12 ,
(14b)
where 𝜉1(0) , 𝜂1(0) are all constants. Substituting (14a) and (14b) into (13a), one can obtain 𝑖 𝑓𝑥(2) − 𝑠𝑥(2) = − 𝑒𝜉1 +𝜂1 . 2
(15)
𝑗=1
𝑓(2) =
Substituting (9) into (7a), (7b), (7c), and (7d) yields (1) 𝑔𝑡(1) − 𝑔𝑥𝑥 = 0,
𝑔𝑡(3)
(4) (4) − (𝑠𝑡(4) + 𝑠𝑥𝑥 ) = (−𝐷𝑡 + 𝐷𝑥2 ) 𝑓(2) ⋅ 𝑠(2) , 𝑓𝑡(4) − 𝑓𝑥𝑥
(12a)
Then combining (15) with (12a) yields
(2𝑗) 2𝑗
𝑠 = 1 + ∑𝑠
𝑗=1
(2) (2) ) = 0, − (𝑠𝑡(2) + 𝑠𝑥𝑥 𝑓𝑡(2) − 𝑓𝑥𝑥
(6)
GDNLSE (2a) and (2b) can be transformed to the bilinear form
(11c)
.. .
(4)
−
(3) 𝑔𝑥𝑥
= (−𝐷𝑡 +
𝐷𝑥2 ) 𝑔(1)
(10a) (2)
⋅𝑓 ,
2
𝑒𝜉1 +𝜂1 ,
(16a)
2
𝑒𝜉1 +𝜂1 .
(16b)
2(𝑙1 − 𝑘1 ) 𝑖𝑙1
(10b)
2(𝑙1 − 𝑘1 )
(10c)
Assuming that 𝑔(𝑖) = ℎ(𝑖) = 𝑓(𝑗) = 𝑠(𝑗) = 0, (𝑖 = 3, 5, 7, . . . , 𝑗 = 4, 6, 8, . . .), one can find that (10a), (10b) and (10c)–(13a), (13b), and (13c) are still hold. Thus, let 𝜖 = 1, substituting (9), (14a) and (14b) and (16a) and (16b) into (6),
(5) = (−𝐷𝑡 + 𝐷𝑥2 ) (𝑔(1) ⋅ 𝑓(4) + 𝑔(3) ⋅ 𝑓(2) ) , 𝑔𝑡(5) − 𝑔𝑥𝑥
.. .
𝑠(2) =
𝑖𝑘1
The Scientific World Journal
3
|q1 |
10
5
|r1 |
10
10
10
5 5
5 0 −10
0 −10
0
0 −5
−5 −5
0 5
−5
0 5
−10
10
10
(a)
−10
(b)
Figure 1: The shape of the one-soliton solution of GDNLSE given by (17). 𝑞1 and 𝑟1 with 𝑘1 = 1 + 0.3𝑖, 𝑙1 = −1 + 0.2𝑖, and 𝜉1(0) = 𝜂1(0) = 0. (a) |𝑞1 |, (b) |𝑟1 |.
one can arrive at one soliton solution for GDNLSE (2a) and (2b): 𝑞1 =
𝑟1 =
𝑒𝜉1 (1 + (𝑙1 /2) 𝑒𝜉1 +𝜂1 +(𝜋/2)𝑖+𝜃13 ) 2
(1 + (𝑘1 /2) 𝑒𝜉1 +𝜂1 +(𝜋/2)𝑖+𝜃13 )
𝑙 + 2 (𝑒𝜉1 +𝜂2 +(𝜋/2)𝑖+𝜃14 + 𝑒𝜉2 +𝜂2 +(𝜋/2)𝑖+𝜃24 ) . 2
,
𝑒𝜂1 (1 + (𝑘1 /2) 𝑒𝜉1 +𝜂1 +(𝜋/2)𝑖+𝜃13 ) 2
(1 + (𝑙1 /2) 𝑒𝜉1 +𝜂1 +(𝜋/2)𝑖+𝜃13 )
(17)
𝜉𝑗 = 𝜔𝑗 𝑡 − 𝑘𝑗 𝑥 + 𝜉𝑗(0) ,
𝜔𝑗 = 𝑘𝑗2 , (𝑗 = 1, 2) ,
ℎ(1) = 𝑒𝜂1 + 𝑒𝜂2 ,
𝜂𝑗 = 𝜎𝑗 𝑡 + 𝑙𝑗 𝑥 + 𝜂𝑗(0) ,
𝜎𝑗 = −𝑙𝑗2 ,
𝑔(3) =
𝑘1 𝜉1 +𝜂1 +(𝜋/2)𝑖+𝜃13 + 𝑒𝜉1 +𝜂2 +(𝜋/2)𝑖+𝜃14 ) (𝑒 2 𝑘 + 2 (𝑒𝜉2 +𝜂1 +(𝜋/2)𝑖+𝜃23 + 𝑒𝜉2 +𝜂2 +(𝜋/2)𝑖+𝜃24 ) , 2
(21)
where 𝑒𝜃13 = 𝑒𝜃14 =
(18a)
1 2
,
𝑒𝜃23 =
2
,
𝑒𝜃24 =
(𝑙1 − 𝑘1 ) 1 (𝑙2 − 𝑘1 )
1 2
,
2
,
(𝑙1 − 𝑘2 ) 1 (𝑙2 − 𝑘2 )
(22)
2
𝑒𝜃12 = (𝑘1 − 𝑘2 ) . (18b)
Then substituting (18b) and (20b) into (11b), by some computations, we obtain ℎ(3) =
𝑘1 𝜉1 +𝜂1 +𝜂2 +(𝜋/2)𝑖+𝜃13 +𝜃14 +𝜃34 𝑒 2 𝑘 + 2 𝑒𝜉2 +𝜂1 +𝜂2 +(𝜋/2)𝑖+𝜃23 +𝜃24 +𝜃34 , 2
(19)
(23)
where 𝑒𝜃34 = (𝑙1 − 𝑙2 )2 . Substituting (21) and (23) into (12b) and (13b), we obtain
Then combining (19) with (12a) yields 𝑓(2) =
𝑙1 𝜉1 +𝜉2 +𝜂1 +(𝜋/2)𝑖+𝜃13 +𝜃23 +𝜃12 𝑒 2 𝑙 + 2 𝑒𝜉1 +𝜉2 +𝜂2 +(𝜋/2)𝑖+𝜃14 +𝜃24 +𝜃12 , 2
where 𝜉𝑗(0) , 𝜂𝑗(0) are all constants. Substituting (18a) and (18b) into (13a), one can obtain 𝑖 𝑓𝑥(2) − 𝑠𝑥(2) = − (𝑒𝜉1 +𝜂1 + 𝑒𝜉1 +𝜂2 + 𝑒𝜉2 +𝜂1 + 𝑒𝜉2 +𝜂2 ) . 2
(20b)
Substituting (18b) and (20a) into (10b), by some computations, we obtain
,
where 𝜉1 , 𝜂1 are defined by (14a) and (14b), 𝑘1 , 𝑙1 are all arbitrary constants, and 𝑒𝜃13 = (1/(𝑙1 − 𝑘1 )2 ). We depict |𝑞1 | and |𝑟1 | in Figure 1. For convenience, we replace 𝑡 by −𝑖𝑡 and 𝑥 by −𝑥. To get two-soliton of GDNLSE (2a) and (2b), we select 𝑔(1) and ℎ(1) for (10a) and (11a) as follows: 𝑔(1) = 𝑒𝜉1 + 𝑒𝜉2 ,
𝑙1 𝜉1 +𝜂1 +(𝜋/2)𝑖+𝜃13 + 𝑒𝜉2 +𝜂1 +(𝜋/2)𝑖+𝜃23 ) (𝑒 2
𝑠(2) =
(20a)
𝑓(4) =
𝑘1 𝑘2 𝜉1 +𝜉2 +𝜂1 +𝜂2 +𝜋𝑖+𝜃12 +𝜃13 +𝜃14 +𝜃23 +𝜃24 +𝜃34 , 𝑒 4
(24a)
𝑠(4) =
𝑙1 𝑙2 𝜉1 +𝜉2 +𝜂1 +𝜂2 +𝜋𝑖+𝜃12 +𝜃13 +𝜃14 +𝜃23 +𝜃24 +𝜃34 . 𝑒 4
(24b)
4
The Scientific World Journal
|q2 |
10
5 5 0 −10
|r 2 |
10
10
10
5 5 0 −10
0
0 −5
−5 −5
0 5 10
−5
0 5
−10
10
(a)
−10
(b)
Figure 2: The interaction of two-soliton solution of GDNLSE given by (6). 𝑞2 and 𝑟2 with 𝑘1 = 1 + 0.3𝑖, 𝑘2 = 1 + 0.9𝑖, 𝑙1 = −1 + 0.2𝑖, 𝑙2 = −1 + 0.8𝑖, and 𝜉1(0) = 𝜉2(0) = 𝜂1(0) = 𝜂2(0) = 0. (a) |𝑞2 |, (b) |𝑟2 |.
Assuming that 𝑔(𝑖) = ℎ(𝑖) = 𝑓(𝑗) = 𝑠(𝑗) = 0, (𝑖 = 5, 7, . . . , 𝑗 = 6, 8, . . .), one can find that (10a), (10b) and (10c)–(13a), (13b), and (13c) are still hold. Thus, let 𝜖 = 1, we have two-soliton solution for GDNLSE (2a) and (2b):
𝑞=
𝑟=
(𝑔(1) + 𝑔(3) ) (1 + 𝑠(2) + 𝑠(4) ) 2
(1 + 𝑓(2) + 𝑓(4) )
2
𝑓𝑁 (𝑡, 𝑥) =
2𝑁
∑ 𝐴 1 (𝜇) exp [ ∑ 𝜇𝑗 𝜉𝑗 𝜇=0,1 𝑗=1 [
2𝑁
2𝑁
2𝑁
+ ∑ 𝜇𝑗 𝜇𝜌 𝜃𝑗𝜌 ] , 1≤𝑗
