Soliton behavior for a generalized mixed nonlinear Schr
€
odinger model
with N-fold Darboux transformation
Xing L
€
u
a)
Department of Mathematics, Beijing Jiao Tong University, Beijing 100044, China and State Key Laboratory
of Rail Traffic Control and Safety, Beijing Jiao Tong University, Beijing 100044, China
(Received 19 May 2013; accepted 29 August 2013; published online 18 September 2013)
A spectral problem, the x-derivative part of which is a simple generalization of the standard
Ablowitz-Kaup-Newell-Segur and Kaup-Newell spectral problems, is presented with its associated
generalized mixed nonlinear Schr
€
odinger (GMNLS) model. The N-fold Darboux transformation with
multi-parameters for the spectral problem is constructed with the help of gauge transformation.
According to the Darboux transformation, the solution of the GMNLS model is reduced to solving a
linear algebraic system and two first-order ordinary differential equations. As an example of
application, we list the modulus formulae of the envelope one- and two-soliton solutions. Note that
our model is a generalized one with the inclusion of four coefficients (a, b, c,andd), which involves
abundant NLS-type models such as the standard cubic NLS equation, the Gerdjikov-Ivanov equation,
the Chen-Lee-Liu equation, the Kaup-Newell equation, and the mixed NLS of Wadati and/or Kundu,
among others.
V
C
2013 AIP Publishing LLC.[http://dx.doi.org/10.1063/1.4821132]
The generalized mixed nonlinear Schr
€
odinger equation
(GMNLSE) in the form of i
›u
›t
¼
›
2
u
›x
2
þ i a juj
2
›u
›x
þ i bu
2
›u
*
›x
1c juj
4
u1d juj
2
u; arises in several physical applica-
tions such as quantum field theory, weakly nonlinear dis-
persive water waves, and nonlinear optics. With different
choices of the multiple parameters a, b, c and d, a series
of celebrated nonlinear evolution equations in mathemat-
ical physics can be included by the GMNLSE. Based on
the spectral problem associated with the GMNLS E, the
x-derivative part of which is a simple generalization of
the standard Ablowitz-Kaup-Newell-Segur and Kaup-
Newell spectral problems, soliton behavior will be investi-
gated with the symbolic construction of the N-fold
Darboux transformation.
I. INTRODUCTION
For describing various complex nonlinear phenomena in
our realistic world, the nonlinear Schr
€
odinger equation
(NLSE) and NLS-type equations appear very attractive in
many fields of physical and engineering sciences.
1–25
In the
absence of optical losses, the wave dynamics of nonlinear
pulse propagation in a monomode fiber can be described by
the standard cubic NLSE,
2,3
and the optical soliton technique
has been widely used for ultra-high bit rate, long haul and
transoceanic optical communication systems.
2–6
In the plas-
mas physics, it is important to study the dynamics of Alfv
en
waves, which are the most “robust” plasma oscillations in a
magnetized system and widespread in space and laboratory
plasmas.
7,8
In the one-dimensional approximation, the deriva-
tive NLSE is employed as a general model to govern the dy-
namical behavior of weakly nonlinear and weakly dispersive
Alfv
en waves.
9
With the experimental observation and
theoretical studies of Bose-Einstein condensates, intense inter-
est has been attracted in the nonlinear excitations of the
atomic matter waves, such as dark and bright solitons.
10
Studying the behaviour of these solitons requires solving the
known Gross-Pitaevskii equation, which governs the evolu-
tion of the macroscopic wave function of Bose-Einstein
condensates.
10–12
In the physically important case of the
cigar-shaped Bose-Einstein condensates, the Gross-Pitaevskii
equation reduces into the one-dimensional inhomogeneous
NLSE (e.g., see Eq. (1) in Ref. 11 or Eq. (2) in Ref. 12).
Additionally, many literatures have devoted to other NLS-
type equations such as the Gerdjikov-Ivanov equation, the
Chen-Lee-Liu equation, the Kaup-Newell equation, and the
mixed NLSE of Wadati and/or Kundu, among others.
13–24
In this paper, we will consider the following generalized
mixed nonlinear Schr
€
odinger equation (GMNLSE) in the
form of
13,14,20
i
@u
@t
¼
@
2
u
@x
2
þ i a juj
2
@u
@x
þ i bu
2
@u
@x
þ c juj
4
u þ d juj
2
u;
(1)
where * means the complex conjugate, u ¼ u(x,t) is the com-
plex function of (x,t), and a, b, c, and d are all real constants.
Equation (1) arises in several physical applications including
quantum field theory, weakly nonlinear dispersive water
waves, and nonlinear optics.
13
It is shown to enjoy the
Painlev
e property only if c ¼
1
4
b ð2 b aÞ holds, regardless
of the value of d.
14
With different choices of the multiple
parameters a, b, c, and d, a series of celebrated nonlinear
evolution equations in mathematical physics are included by
Eq. (1), as follows:
•
When d ¼ 0, Eq. (1) reduces to the generalized derivative
NLSE, i.e., Eq. (1.1) in Ref. 14.
•
When a ¼ b ¼ c ¼ 0 and d ¼ 6
1
2
, Eq. (1) reduces to the
celebrated standard cubic NLSE.
2,3
a)
Author to whom correspondence should be addressed. Electronic mail:
xlv@bjtu.edu.cn
1054-1500/2013/23(3)/033137/8/$30.00
V
C
2013 AIP Publishing LLC23, 033137-1
CHAOS 23, 033137 (2013)