The N-soliton solution of a two-component modified nonlinear Schr\"odinger equation

Abstract

The $N$-soliton solution is presented for a two-component modified nonlinear
Schr\"odinger equation which describes the propagation of short pulses in
birefringent optical fibers. The solution is found to be expressed in terms of
determinants. The proof of the solution is carried out by means of an
elementary theory of determinants. The generalization of the 2-component system
to the multi-component system is discussed as well as a (2+1)-dimensional
nonlocal equation arising from its continuum limit.

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arXiv:1107.0776v1 [nlin.SI] 5 Jul 2011
The N-soliton solution of a two-component
modified nonlinear Schr¨ odinger equation
Yoshimasa Matsunoa
Division of Applied Mathematical Science,
Graduate School of Science and Engineering
Yamaguchi University, Ube, Yamaguchi 755-8611, Japan
ABSTRACT
The N-soliton solution is presented for a two-component modified nonlinear Schr¨ odinger
equation which describes the propagation of short pulses in birefringent optical fibers. The
solution is found to be expressed in terms of determinants. The proof of the solution is
carried out by means of an elementary theory of determinants. The generalization of
the 2-component system to the multi-component system is discussed as well as a (2+1)-
dimensional nonlocal equation arising from its continuum limit.
PACS: 05.45.Yv; 42.81.Dp; 02.30.Jr
Keywords: Modified nonlinear Schr¨ odinger equation; N-soliton solution; Two-component
system
aE-mail address: matsuno@yamaguchi-u.ac.jp
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1. Introduction
We consider the following two-component system of nonlinear evolution equations
which is a hybrid of the coupled nonlinear Schr¨ odinger (NLS) equation and coupled deriva-
tive NLS equation
iqj,t+ qj,xx+ µ
?
2
?
k=1
|qk|2
?
qj+ iγ
??
2
?
k=1
|qk|2
?
qj
?
x
= 0,(j = 1,2), (1.1)
where qj= qj(x,t) (j = 1,2) are the slowly varying complex envelopes for polarizations, µ
and γ are real constants and subscripts x and t appended to qjdenote partial differentia-
tions. This system of equations has been derived as a model for describing the propagation
of short pulses in birefringent optical fibers [1]. Its integrability has been established by
constructing the Lax pair and an infinite number of conservation laws [2]. Hence, the
initial value problem can be formulated in principle by means of the inverse scattering
method. A few exact solutions are now available for the system (1.1). Actually, one- and
two-soliton solutions have been presented by employing Hirota’s bilinear transformation
method [3, 4]. The N-soliton solution of the system (1.1) with µ = 0 was also obtained
quite recently by means of the Darboux transformation where N is an arbitrary positive
integer [5].
The purpose of this Letter is to construct the bright N-soliton solution of the system
(1.1) within the framework of the bilinear formalism. We present a compact determinantal
expression of the N-soliton solution which would be suitable for investigating the inter-
action process of solitons. The proof of the solution is performed by using an elementary
theory of determinants. In concluding remarks, the generalization of the system (1.1) to
the n-component system (n ≥ 3) is briefly discussed as well as a (2+1)-dimensional non-
local modified NLS equation arising from its continuum limit. Furthermore, some issues
associated with the system (1.1) will be addressed.
2. Bilinear form and N-soliton solution
2.1. Bilinear form
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We first apply the gauge transformations
qj= ujexp
?
−iγ
2
?x
−∞
2
?
k=1
|uk|2dx
?
,(j = 1,2), (2.1)
to the system (1.1) and transform it to the system of equations for uj
iuj,t+ uj,xx+ µ
?
2
?
k=1
|uk|2
?
uj+ iγ
?
2
?
k=1
u∗
kuk,x
?
uj= 0,(j = 1,2), (2.2)
where the asterisk denotes complex conjugate. Note that we have imposed the boundary
conditions qj→ 0,uj→ 0 as |x| → ∞ to give the bright soliton solutions. By means of
the dependent variable transformations
uj=gj
f,
(j = 1,2), (2.3)
the system (2.2) can be transformed to the following system of bilinear equations for f
and gj[4]
(iDt+ D2
x)gj· f = 0,(j = 1,2), (2.4)
Dxf · f∗=iγ
2
2
?
k=1
|gk|2, (2.5)
D2
xf · f∗= µ
2
?
k=1
|gk|2+iγ
2
2
?
k=1
Dxgk· g∗
k. (2.6)
Here, the bilinear operators Dxand Dtare defined by
Dm
xDn
tf · g =
?∂
∂x−
∂
∂x′
?m?∂
∂t−
∂
∂t′
?n
f(x,t)g(x′,t′)
???
x′=x,t′=t, (2.7)
where m and n are nonnegative integers. It follows from (2.3) and (2.5) that
−iγ
2
2
?
k=1
|uk|2=
∂
∂xlnf∗
f,
(2.8)
which, substituted into (2.1), yields the solution of the system (1.1) in the form qj =
gjf∗/f2(j = 1,2).
2.2. N-soliton solution
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We now state the main result in this Letter:
The N-soliton solution of the system of bilinear equations (2.4), (2.5) and (2.6) is
given by the determinants f and gj(j = 1,2) where
f =
????
A
−I
I
B
????,gj=
??????
A
−I
0
I
B
zT
0T
0−a∗
j
??????
,(j = 1,2). (2.9)
Here, A,B and I are N × N matrices and z,aj and 0 are N-component row vectors
defined below and the symbol T denotes the transpose:
A = (ajk)1≤j,k≤N,ajk=1
2
zjz∗
pj+ p∗
k
k
,zj= exp(pjx + ip2
jt), (2.10a)
B = (bjk)1≤j,k≤N,bjk=(µ + iγpk)cjk
p∗
j+ pk
,cjk=
2
?
s=1
αsjα∗
sk, (2.10b)
I = (δjk)1≤j,k≤N,: N × N unit matrix, (2.10c)
z = (z1,z2,...,zN),
aj= (αj1,αj2,...,αjN),
0 = (0,0,...,0). (2.10d)
The above N-soliton solution involves 3N complex parameters pj and α1j,α2j (j =
1,2,...,N). The former parameters determine the amplitude and velocity of the solitons
whereas the latter ones determine the polarizations and the envelope phases of the solitons.
In the special case of N = 1,2, (2.9) and (2.10) reproduce the 1- and 2-soliton solutions
presented in [4]. Note, however that the compact determinantal expression of the solution
has been obtained here for the first time. It is also remarked that in the single-component
case (i.e., the system (1.1) with q2= 0), our solutions reduce to the soliton solutions given
in [6].
2.3. Remark
If we introduce the transformations
f =˜f,gj= exp
?
i
?
µ
γ˜ x +
?µ
γ
?2
˜t
??
˜ gj,(j = 1,2), (2.11a)
x = ˜ x +2µ
γ
˜t,t =˜t,(2.11b)
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then the bilinear equations (2.4), (2.5) and (2.6) recast to
(iD˜t+ D2
˜ x)˜ gj·˜f = 0,(j = 1,2), (2.12)
D˜ x˜f ·˜f∗=iγ
2
2
?
k=1
|˜ gk|2, (2.13)
D2
˜ x˜f ·˜f∗=iγ
2
2
?
k=1
D˜ x˜ gk· ˜ g∗
k, (2.14)
respectively. Thus, the form of Eqs. (2.4) and (2.5) is invariant whereas (2.6) becomes a
simplified equation with µ = 0. Consequently, the proof of the N-soliton solution may be
performed for the corresponding solution with µ = 0. This reduces the total amount of
calculations considerably, in particular for Eq. (2.6). Hence, in the analysis developed in
Sec. 3, we put µ = 0 without loss of generality.
3. Notation and formulas for determinants
To perform the proof of the N-soliton solution concisely, we introduce some notations
and write the basic formulas for determinants.
3.1. Notation
We define the following matrices associated with the N-soliton solution (2.9):
D =
?A






I
B−I
?
, (3.1)
D(a;b) =
A
−I
0
I
B bT
a
0T
0

,

(3.2)
D(a;z) =
A
−I
0
I
B 0T
a
zT
0
,

(3.3)
D(z∗;z) =
A
−I
z∗
I
B 0T
0
zT
0
. (3.4)
Note the position of the vectors a, b,z and z∗in the above expressions where a and b
represent a∗
jand aj(j = 1,2), respectively. The matrices which include more than two
vectors will be introduced as well.
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