The Scientific World Journal

Volume 2014 (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

¹Department of Mathematics, East China Institute of Technology, Nanchang 330013, China

²Department of Mathematics, Fuzhou Vocational College of Technology, Fuzhou 344000, China

³School of Mathematics and Statitics, Jiangsu Normal University, Xuzhou, Jiangsu 221116, China

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.

Abstract.

Multisoliton solutions are derived for a general nonlinear Schrödinger 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 Schrödinger 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 Schrödinger 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 -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 Schrödinger equations [7, 8]. Gauge transformations have been found among them [9, 10]. The first well-known derivative nonlinear Schrödinger equation (DNLSE) is

where denotes the complex conjugate of . 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 -soliton solutions is also obtained by some algebraic technique [16]. Through the -fold 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 Schrödinger equation with derivative (GDNLSE) as follows:

We show that through a variable transformation the bilinear equations for (2a) and (2b) can be derived for constructing its -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

time evolution

and the related zero curvature equation

one can derive the GDNLSE (2a) and (2b). Its corresponding Lax pair (3a) and (3b) is governed by

Secondly, we give the bilinear form of GDNLSE and further its -soliton solutions. By the variable transformation

GDNLSE (2a) and (2b) can be transformed to the bilinear form

where , and are complex functions and is the well-known Hirota bilinear operator defined as

To solve the system (7a), (7b), (7c), and (7d), we expand , and as

Substituting (9) into (7a), (7b), (7c), and (7d) yields

In order to get one-soliton of GDNLSE (2a) and (2b), we select and for (10a) and (11a) as follows:

where are all constants. Substituting (14a) and (14b) into (13a), one can obtain

Then combining (15) with (12a) yields

Assuming that , one can find that (10a), (10b) and (10c)–(13a), (13b), and (13c) are still hold. Thus, let , substituting (9), (14a) and (14b) and (16a) and (16b) into (6), one can arrive at one soliton solution for GDNLSE (2a) and (2b):

where are defined by (14a) and (14b), are all arbitrary constants, and . We depict and in Figure 1. For convenience, we replace by and by .

(a)

(b)

Figure 1: The shape of the one-soliton solution of GDNLSE given by (17). and with , and . (a) , (b) .

To get two-soliton of GDNLSE (2a) and (2b), we select and for (10a) and (11a) as follows:

where are all constants. Substituting (18a) and (18b) into (13a), one can obtain

Then combining (19) with (12a) yields

Substituting (18b) and (20a) into (10b), by some computations, we obtain

where

Then substituting (18b) and (20b) into (11b), by some computations, we obtain

where .  Substituting (21) and (23) into (12b) and (13b), we obtain

Assuming that , one can find that (10a), (10b) and (10c)–(13a), (13b), and (13c) are still hold. Thus, let , we have two-soliton solution for GDNLSE (2a) and (2b):

Figure 2 gives the interaction of two-soliton solution.

(a)

(b)

Figure 2: The interaction of two-soliton solution of GDNLSE given by (6). and with , and . (a) , (b) .

So, by the standard Hirota’s approach, one can derive -soliton () in terms of , and :

where

are all arbitrary constants; take over all possible combinations of and satisfy the following condition:

respectively.

We replace and by and ( and are arbitrary real constants), respectively. Then the two-soliton solution (25) under the limit of leads to the limit solution

where

and are arbitrary constants. This is the so-called one-double-pole solution. This kind of limit procedure can be found in [21, 22], which builds a bridge between Hirota’s approach and the inverse scattering transform on the level of double-pole solution. Zhou and the coauthors find that the limit solutions for classical -solitons are nothing but the -double-pole solutions [23].

Now, we consider the derivative nonlinear Schrödinger equation (1). We shall give its bilinear equation and -soliton solutions by reduction. Setting and replacing by and by in (2a) and (2b), one can find that (2a) and (2b) reduce to DNLSE (1). Taking and replacing by and by , (7a), (7b), (7c), and (7d) reduce to the bilinear forms of DNLSE (1):

which can be also directly obtained from (1) through the transformation . If we take in (26a), (26b), (26c), and (26d) and (27a), (27b), (27c), (27d), (27e), (27f), and (27g), then . Thus we can also have , and obtain -soliton solutions of DNLSE (1) by reduction:

where

are all arbitrary constants; take over all possible combinations of and satisfy the condition (28). If replacing for , for , and for , (32a) and (32b) are in accord with the -soliton solutions in [14], where the solutions of DNLSE (1) are reduced by a multicomponent modified nonlinear Schrödinger equation. Dynamics for one- and two-soliton solutions for DNLSE (1) are described in Figures 3 and 4. Figure 4 depicts 2D plot of two-soliton of DNLSE.

(a)

(b)

(c)

Figure 3: The shape of the one-soliton and the interaction of two-soliton of DNLSE (1). (a) One-soliton with and , (b) one-soliton with and , and (c) two-soliton with , and .

(a)

(b)

(c)

(d)

Figure 4: The 2D plot of two-soliton of DNLSE. with , and at , and .

In summary, we present multisoliton solutions for a general nonlinear Schrödinger equation with derivative by Hirota’s approach. By reductions, we also directly obtain the multisoliton solutions for nonlinear derivative Schrödinger equation. We demonstrate that the solitons of general nonlinear Schrödinger equation with derivative and nonlinear derivative Schrödinger equations result in elastic scattering.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This project is supported by the Research Foundation of Education Bureau of Jiangxi Province of China (no. GJJ13459) and the National Natural Science Foundation of China (no. 11101350).

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