0

Optical Solitons in a Nonlinear Fiber

Medium with Higher-Order Effects
Deng-Shan Wang
School of Science, Beijing Information Science and Technology University, Beijing, 100192
China
1. Introduction
Nowadays we can see many interesting applications of solitons in different areas of physical
sciences such as plasma physics (1), nonlinear optics (2; 3), Bose-Einstein condensate (4; 5),
uid mechanics (6), and so on. Solitons are so robust particles that they are unlikely to
breakdown under small perturbations. The most interesting factor about the soliton, however,
is that their interactions with the medium through which it propagates is elastic. Recent
researches on nonlinear optics have shown that dispersion-managed pulse can be more useful
if the pulse is in the form of a power series of a stable localized pulse which is called soliton.
Optical solitons have been the objects of extensive theoretical and experimental studies
during the last four decades, because of their potential applications in long distance
communication. In 1973, the pioneering results of Hasegawa and Tappert (7) proved that
the major constraint in the optical ber, namely, the group velocity dispersion (GVD) could be
exactly counterbalanced by the self-phase modulation (SPM). SPM is the dominant nonlinear
effect in silica bers due to the Kerr effect. The theoretical results of Hasegawa and Tappert
were greatly supported by the experimental demonstration of optical solitons by Mollenauer
et al. (8) in 1980. Since then many theoretical and experimental works have been done to
achieve a communication system based on optical solitons.
The solitons, localized-in-time optical pulses, evolve from a nonlinear change in the refractive
index of the material, known as Kerr effect, induced by the light intensity distribution.
When the combined effects of the intensity-dependent refractive index nonlinearity and
the frequency-dependent pulse dispersion exactly compensate for one another, the pulse
propagates without any change in its shape, being self-trappedby the waveguide nonlinearity.
The propagation of optical solitons in a nonlinear dispersive optical ber is governed by the
well-known completely integrable nonlinear Schr odinger (NLS) equation
i
q
z
+

2
q

2
+|q|
2
q = 0, = 1, (1)
where q is the complex amplitude of the pulse envelope, and z represent the spatial and
temporal coordinates, and the + and sign of before the dispersive term denote the
anomalous and normal dispersive regimes, respectively. In the anomalous dispersive regime,
this equation possesses a bright soliton solution, and in the normal dispersive regime it
possesses dark solitons. The bright soliton and dark soliton solutions can be derived by
3
www.intechopen.com
2 Will-be-set-by-IN-TECH
the inverse-scattering transform method with vanishing (9; 11) and nonvanishing boundary
conditions (10).
However, if optical pulses are shorter, the standard NLS equation becomes inadequate.
Therefore, some higher-order effects such as third-order dispersion, self-steepening, and
stimulated Raman scattering, will play important roles in the propagation of optical pulses.
In such a case, the governing equation is the one known widely as the higher-order NLS
equation, rst derived by Kodama and Hasegawa (12). The effect of these effects in uncoupled
and coupled systems for bright solitons is well explained (13; 14). Inelastic Raman scattering
is due to the delayed response of the medium, which forces the pulse to undergo a frequency
shift which is known as a self-frequency shift. The effect of self-steepening is due to the
intensity-dependent group velocity of the optical pulse, which gives the pulse a very narrow
width in the course of propagation. Because of this, the peak of the pulse will travel more
slowly than the wings.
In practice, the refractive index or the core diameter of the optical ber are fucntions of the
axial coordinate, which means that the ber is actually axially inhomogeneous. In this case,
the parameters which characterize the dispersive and nonlinear properties of the ber exhibit
variations and the corresponding nonlinear wave equations are NLS equations with variable
coefcients. Moreover, the problem of ultrashort pulse propagation in nonlinear and axially
inhomogeneous optical bers near the zero dispersion point is more complicated because
the high order effects have to be taken into account as well. In order to understand such
phenomena, we consider the higher-order NLS (HNLS) equation with variable coefcients
u
z
= i(d
1

2
u

2
+ d
2
|u|
2
u) + d
3

3
u

3
+ d
4
(u|u|
2
)

+ d
5
u
|u|
2

+ d
6
u, (2)
where u is the slowly varying envelope of the pulse, d
1
, d
2
, d
3
, d
4
, d
5
and d
6
are the z-dependent
real parameters related to GVD, SPM, third-order dispersion (TOD), self-steepening, and
stimulated Raman scattering (SRS), and the heat-insulating amplication or loss, respectively.
Though Eq. (2) was rst derived in the year 1980s, only for the past few years, it has
attracted much attention among the researchers fromboth theoretical and experimental points
of view. For example, Porsezian and Nakkeeran (13) derive all parametric conditions for
soliton-type pulse propagation in HNLS equation using the Painlev e analysis, and generalize
the Ablowitz-Kaup-Newell-Segur method to the 3 3 eigenvalue problem to construct the
Lax pair for the integrable case. Papaioannou et al. (15) give an analytical treatment of the
effect of axial inhomogeneity on femtosecond solitary waves near the zero dispersion point
which governed by the variable-coefcient HNLS equation. The exact bright and dark soliton
wave solutions of this variable-coefcient equation are derived and their behaviors in the
presence of the inhomogeneity are analyzed. Mahalingam and Porsezian (16) analyze the
propagation of dark solitons with higher-order effects in optical bers by Painlev e analysis
and Hirota bilinear method. Xu et al. (17) investigate the modulation instability and solitons
on a cw background in an optical ber with higher-order effects. In addition, there have
recently been several papers giving W-shaped solitary wave solution in the HNLS equation.
However, in recent years the studies of Eq. (2) have not been widespread. In this
chapter, we consider equation (2) again and derive some exact soliton solutions in explicit
form for specied soliton management conditions. We rst change the variable-coefcient
HNLS equation into the well-known constant-coefcient HNLS equation through similarity
transformation. Then the Lax pairs for two integrable cases of the constant-coefcient
HNLS equation are constructed explicitly by prolongation technique, and the novel exact
50 Recent Progress in Optical Fiber Research
www.intechopen.com
Optical Solitons in a Nonlinear Fiber
Medium with Higher-Order Effects 3
bright N-soliton solutions for the bright soliton version of HNLS equation are obtained by
Riemann-Hilbert formulation. Finally, we examine the dynamics and present the features
of the optical solitons. It is seen that the bright two-soliton solution of the HNLS equation
behaves in an elastic manner characteristic of all soliton solutions. These results are useful in
the design of transmission lines with spatial parameter variations and soliton management to
future research.
2. Similarity transformation
A direct and efcient method for investigating the variable-coefcient nonlinear wave
equation is to transform them into their constant-coefcient counterparts by similarity
transformation. To do so, we rstly take the similarity transformation (18; 19)
u = q (T, X) e
i(a
1
+a
2
)
, (3)
to reduce Eq. (2) to the constant-coefcient HNLS equation
q
T
= i(
1

2
q
X
2
+
2
|q|
2
q) + (
3

3
q
X
3
+
4
(q|q|
2
)
X
+
5
q
|q|
2
X
), (4)
where q = q(T, X) is the complex amplitude of the pulse envelope, the parameter (0 < <
1) denotes the relative width of the spectrum that arises due to the quasi-monochromocity,

1
,
2
,
3
,
4
and
5
are the real constant parameters. In Eq. (3), , T, a
1
and a
2
are functions of
z, and X is a function of and z.
Substituting Eq. (3) into Eq. (2) and asking q (T, X) to satisfy the constant-coefcient HNLS
equation (4), we have a set of partial differential equations (PDEs)
d
1
X

+ 3 d
3
X

a
1
= 0, d
3
X

3
=
3
T
z
,
z
= d
6
,
2 d
1
X

a
1
+ X
z
+ 3 d
3
X

a
1
2
= d
3
X

,
2
d
4
a
1
+
2
d
2
=
2
T
z
,
2
2
X

d
4
+
2
X

d
5
= 2
4
T
z
+
5
T
z
,
2
X

d
4
+
2
X

d
5
=
4
T
z
+
5
T
z
,
d
3
a
1
3
+ a
1z
+ a
2z
+ d
1
a
1
2
= 0, 3 d
3
X

2
a
1
+ d
1
X

2
=
1
T
z
, X
,
= 0,
where the subscript denotes the derivative with respect to z and . Solving this set of PDEs,
we have X = k + f and
a
1
= c, d
1
=
T
z
(k
1
3
3
c)
k
3
, d
2
=
T
z
(
2
k
4
c)

2
k
, d
3
=

3
T
z
k
3
,
d
4
=

4
T
z

2
k
, d
5
=

5
T
z

2
k
, f =
c (3
3
c 2 k
1
) T
k
2
, a
2
=
(2
3
c k
1
) c
2
T
k
3
,
where =
0
e
_
d
6
z
, k,
0
and c are constants, and T and d
6
are arbitrary functions of z. So the
similarity transformation (3) becomes
u =
0
e
_
d
6
dz
q
_
T,
k
3
2 ck
1
T + 3 c
2

3
T
k
2
_
e
ic(k
3
+2 c
2

3
Tck
1
T)/k
3
. (5)
Therefore, if we can get the exact soliton solutions of the constant-coefcient HNLS equation
(4) we can obtain the exact soliton solutions for HNLS equation (2) through Eq. (5). In the next
section, we will investigate the integrable condition of equation (4) by prolongation technique.
51 Optical Solitons in a Nonlinear Fiber Medium with Higher-Order Effects
www.intechopen.com
4 Will-be-set-by-IN-TECH
3. Prolongation structures of the constant-coefcient HNLS equation
In this section, we investigate the prolongation structures of the constant-coefcient HNLS
equation (4) by means of the prolongation technique (2022). Firstly, the complex conjugate
of the dependent variable q in Eq. (4) is denoted as q

= u. Then, Eq. (4) and its conjugate

become
i
1
q
XX
+ i
2
q
2
u + [
3
q
XXX
+ (
4
+
5
)q
2
u
X
+ (2
4
+
5
)quq
X
] q
T
= 0, (6a)
i
1
u
XX
i
2
u
2
q + [
3
u
XXX
+ (
4
+
5
)u
2
q
X
+ (2
4
+
5
)quu
X
] u
T
= 0. (6b)
Next we introduce four new variables p, r, v and w by
q
X
= p, p
X
= r, u
X
= v, v
X
= w, (7)
and dene a set of differential 2-form I = {
1
,
2
,
3
,
4
,
5
,
6
} on solution manifold M =
{T, X, u, v, w, p, q, r} , where

1
= dq dT + pdT dX,
2
= dp dT + rdT dX,

3
= du dT + vdT dX,
4
= dv dT + wdT dX,

5
= dq dX +
3
dr dT +
1
dX dT,
6
= du dX +
3
dw dT +
2
dX dT,
with

1
= i
1
r + i
2
q
2
u + [(
4
+
5
)q
2
v + (2
4
+
5
)qup],

2
= i
1
wi
2
u
2
q + [(
4
+
5
)u
2
p + (2
4
+
5
)quv].
When these differential 2-forms restricted on the solution manifold M become zero, we
recover the original constant-coefcient HNLS equation (4). It is easy to verify that I is a
differential closed idea, i.e. dI I.
We further introduce n differential 1-forms

i
= d
i

F
i
dX

G
i
dT, (8)
where i = 1, 2, , n,

F
i
and

G
i
are functions of u, v, w, p, q, r,
i
and are assumed to be both
linearly dependent on
i
, namely

F
i
= F
i

i
,

G
i
= G
i

i
. For the sake of simplication, we drop
the indices by rewriting
i
as , F
i
as F and G
i
as G. When restricting on solution manifold, the
differential 1-forms
i
are null, i.e.
i
= 0 which is just the linear spectral problem
X
= F
and
T
= G.
Following the well-known prolongation technique, the extended set of differential form

I =
I

i
_
must be a closed ideal under exterior differentiation, i.e. d

I

I. Because dI I

I,
we only need to let d
_

i
_


I, which denotes that
d
i
=
6

j=1
f
i
j

j
+
i

i
, i = 1, 2, , n, (9)
where f
i
j
(j = 1, 2, 3, 4, 5, 6) are functions of (T, X), and
i
= g
i
(T, X)dX + h
i
(T, X)dT are
differential 1-forms.
52 Recent Progress in Optical Fiber Research
www.intechopen.com
Optical Solitons in a Nonlinear Fiber
Medium with Higher-Order Effects 5
When Eq. (9) is written out in detail, after dropping the indices we have the following PDEs
about F and G as
G
r
=
3
F
q
, G
w
=
3
F
u
,
G
q
p + G
p
r + G
u
v + G
v
w F
q
_
i
1
r +i
2
q
2
u + (
4
+
5
) q
2
v + (2
4
+
5
) qup
_
(10)
F
u
_
i
1
w i
2
u
2
q + (
4
+
5
) u
2
p + (2
4
+
5
) quv
_
[F, G] = 0,
with [F, G] = FGGF.
Solving Eq. (10), we have the expressions of F and G as
F = x
0
+ x
1
q + x
2
u, (11)
G =
3
x
1
r +
3
x
2
w + vq
3
x
5
+ v
3
x
4
pu
3
x
5
+ p
3
x
3
iqu
1
x
5
+i px
1

1
+ qx
2
u
2

4
+
2
3
qx
2
u
2

5
+
1
2

3
u
2
x
13
+ q
2
x
1
u
4
+
2
3
q
2
x
1
u
5
+qu
3
x
8
+ qu
3
x
10
+
3
x
7
u +
1
2
q
2

3
x
9
+ q
3
x
6
+ iq
1
x
3
i
1
x
4
u ivx
2

1
+ x
15
,
where L = {x
0
, x
1
, x
2
, , x
15
} is an incomplete Lie algebra which is called prolongation
algebra and it satises the following commutation relations
[x
2
, x
5
] = x
14
, x
2

5
= 3
3
x
14
, 2 x
8
+ x
10
= x
12
, x
1

5
+3
3
x
11
= 0,
[x
0
, x
1
] = x
3
, [x
0
, x
2
] = x
4
, [x
0
, x
3
] = x
6
, [x
0
, x
4
] = x
7
, [x
0
, x
5
] = x
8
,
[x
1
, x
2
] = x
5
, [x
1
, x
3
] = x
9
, [x
1
, x
4
] = x
10
, [x
1
, x
5
] = x
11
, [x
2
, x
3
] = x
12
,
[x
2
, x
4
] = x
13
,
3
[x
0
, x
9
] + 2 i
1
x
9
+ 2
3
[x
1
, x
6
] = 0, [x
1
, x
15
] + i
1
x
6
+
3
[x
0
, x
6
],

3
[x
1
, x
9
] = 0,
3
[x
2
, x
13
] = 0, [x
0
, x
15
] = 0, 2
3
[x
2
, x
7
] +
3
[x
0
, x
13
] = 2 i
1
x
13
,
[x
2
, x
15
] +
3
[x
0
, x
7
] = i
1
x
7
, 6
3
([x
0
, x
10
] + [x
1
, x
7
] + [x
2
, x
6
] + [x
0
, x
8
]) + 6 i
1
x
8
= 0,
(6 [x
1
, x
8
] +6 [x
1
, x
10
] + 3 [x
2
, x
9
])
3
2
+ (4
5
x
3
+ 6
4
x
3
+ 6 ix
1

2
)
3
+ 2 i
1
x
1

5
= 0,
(6 [x
2
, x
10
] +3 [x
1
, x
13
] + 6 [x
2
, x
8
])
3
2
+ (4
5
x
4
+ 6
4
x
4
6 ix
2

2
)
3
2 i
1
x
2

5
= 0.
It is known that nontrivial matrix representations of prolongation algebra L correspond to
nontrivial prolongation structures. To nd the matrix representation of L, following the
procedure of Fordy (23), we try to embed it into Lie algebra sl(n, C). Starting from the
case of n = 2, we found that sl(2, C) is the whole algebra for some special coefcients

j
(j = 1, 2, 3, 4, 5). For the case of n = 3, we can also nd that sl(3, C) will be the whole
algebra for some other special coefcients
j
(j = 1, 2, 3, 4, 5). In this paper, we only examine
the case of sl(2, C) algebra.
From the above commutation relations, we have the special relations among elements x
1
, x
2
and x
5
as
[x
2
, x
5
] =

5
3
3
x
2
, [x
1
, x
5
] =

5
3
3
x
1
, [x
1
, x
2
] = x
5
, (12)
from which we know that x
1
and x
2
are nilpotent elements and x
5
is a neutral element. So we
have
5
= 6
2

3
and
x
1
=
_
0
0 0
_
, x
2
=
_
0 0
0
_
, x
5
=
_

2
0
0
2
_
, (13)
53 Optical Solitons in a Nonlinear Fiber Medium with Higher-Order Effects
www.intechopen.com
6 Will-be-set-by-IN-TECH
with a nonzero constant. Substituting (13) into the commutation relations of prolongation
algebra L, we nally get the 2 2 matrix representations of F and G. Therefore, we obtain two
integrable HNLS equations with 2 2 spectral problems.
When
2
= 2
2

1
,
4
= 6
2

3
and
5
= 6
2

3
, Eq. (4) becomes the bright soliton version of
Hirota equation
q
T
= i
1
q
XX
+ 2 i
1

2
|q|
2
q +
3
q
XXX
+ 6
2

3
|q|
2
q
X
, (14)
with linear spectral problem

X
= F,
T
= G, (15)
and
F =
_
i q
q

i
_
, (16)
G = 4 i
3

3
_
1 0
0 1
_
2
2
_
i
1
2
3
q
2
3
q

i
1
_
+ 2
_
i
3

2
|q|
2

1
q i
3
q
X

1
q

i
3
q

X
i
3

2
|q|
2
_
(17)
+
_

3

2
q
X
q

2
q

X
q + i
1

2
|q|
2

3
q
XX
+ i
1
q
X
+ 2
3

3
|q|
2
q
i
1
q

X

3
q

XX
2
3

3
|q|
2
q

2
q

X
q
3

2
q
X
q

i
1

2
|q|
2
_
,
where is a spectral parameter and (T, X, ) is a vector or matrix function.
When
2
= 2
2

1
,
4
= 6
2

3
and
5
= 6
2

3
, Eq. (4) becomes the dark soliton version
of Hirota equation
q
T
= i
1
q
XX
2 i
2

1
|q|
2
q +
3
q
XXX
6
2

3
|q|
2
q
X
, (18)
with linear spectral problem Eq. (15) and
F =
_
i q
q

i
_
, (19)
G = 4 i
3

3
_
1 0
0 1
_
2
2
_
i
1
2
3
q
2
3
q

i
1
_
+ 2
_
i
3

2
|q|
2
i
3
q
X
+
1
q

1
q

+ i
3
q

X
i
3

2
|q|
2
_
(20)
+
_

3

2
q

X
q
3

2
q
X
q

i
1

2
|q|
2

3
q
XX
+ i
1
q
X
2
3

3
|q|
2
q

3
q

XX
i
1
q

X
2
3

3
|q|
2
q

2
q
X
q

2
q

X
q + i
1

2
|q|
2
_
.
54 Recent Progress in Optical Fiber Research
www.intechopen.com
Optical Solitons in a Nonlinear Fiber
Medium with Higher-Order Effects 7
4. The bright soliton solutions for Eq. (14)
In this section, we propose the N-bright soliton solutions of Eq. (14) using the
Riemann-Hilbert formulation (2428). Let us consider Eq. (14) for localized solutions, i.e.
assuming that potential function q decay to zero sufciently fast as X, T . In the
Riemann-Hilbert formulation, we treat as a fundamental matrix of the two linear equations
in (15). From (15) we note that when X, T , one has = e
iX+(4i
3

3
2i
2

1
)T
with
= diag(1, 1). This motivates us to introduce the variable transformation
= Je
iX+(4i
3

3
2i
2

1
)T
, (21)
where J is (X, T)-independent at innity. Inserting (21) into (15) with (16)-(17), we get
J
X
= i[, J] + QJ, (22a)
J
T
= (2i
1

2
4 i
3

3
)[, J] + VJ, (22b)
with
Q =
_
0 q
q

0
_
, V = (2
1
4
2

3
)Q + 2
_
i
3

2
|q|
2
i
3
q
X
i
3
q

X
i
3

2
|q|
2
_
+
_

3

2
q
X
q

2
q

X
q + i
1

2
|q|
2

3
q
XX
+i
1
q
X
+ 2
3

3
|q|
2
q
i
1
q

X

3
q

XX
2
3

3
|q|
2
q

2
q

X
q
3

2
q
X
q

i
1

2
|q|
2
_
.
Here [, J] = J J is the commutator, tr(Q) = tr(V) = 0 and
Q

= Q, V

= V, (23)
where represents the Hermitian of a matrix.
In what folows, we consider the scattering problem of the Eq. (22a). By doing so, the variable
T is xed and is a dummy variable. We rst introduce the matrix Jost solutions J

(X, ) of
(22a) with the asymptotic condition
J

I, when X , (24)
where I is a 2 2 unit matrix. Here the subscripts in J

refer to which end of the X-axis the

boundary conditions are set. Then due to tr(Q) = 0 and Abels formula we have det(J

) = 1
for all X. Next we denote E = e
iX
. Since J
+
E and J

E are both solutions of the

rst equation in (15), they must be linearly related, i.e.
J

E = J
+
ES(), R (25)
where
S() =
_
s
11
s
12
s
21
s
22
_
, R
is the scattering matrix, and R is the set of real numbers. Notice that det(S()) = 1 since
det(J

) = 1. If we denote (, ) as a collection of columns,

= [
1
,
2
], = [
1
,
2
], (26)
55 Optical Solitons in a Nonlinear Fiber Medium with Higher-Order Effects
www.intechopen.com
8 Will-be-set-by-IN-TECH
by using the same formulation as (24; 25; 27), we have the Jost solution
P
+
= [
1
,
2
]e
iX
= J

H
1
+ J
+
H
2
, (27)
is analytic in C
+
, and Jost solution
P

= e
iX
_

2
_
= H
1
J
1

+ H
2
J
1
+
, (28)
is analytic in C

, with

1
=
_

2
_
,
1
=
_

2
_
,
and
H
1
= diag(1, 0), H
2
= diag(0, 1).
In addition, it is easy to see that
P
+
(X, ) I, as C
+
, (29)
and
P

(X, ) I, as C

. (30)
In addition, if we express S
1
as
S
1
=
_
s
11
s
12
s
21
s
22
_
, R,
from det(S()) = 1 we have
s
11
= s
22
, s
22
= s
11
, s
12
= s
12
, s
21
= s
21
. (31)
Hence we have constructed two matrix functions P
+
and P

which are analytic in C

+
and
C

, respectively. On the real line, using Eqs. (25), (27) and (28), it is easily to see that
P

(X, )P
+
(X, ) = G(X, ), R, (32)
with
G = E(H
1
+ H
2
S)(H
1
+ S
1
H
2
)E
1
= E
_
1 s
12
s
21
1
_
E
1
.
This determines a matrix Riemann-Hilbert problem with asymptotics
P

(X, ) I, as , (33)
which provide the canonical normalization condition for this Riemann-Hilbert problem. If
this problemcan be solved, one can readily reconstruct the potential q(X, T) as follows. Notice
that P
+
is the solution of the spectral problem (22a). Thus if we expand P
+
at large as
P
+
(X, ) = I +
1

P
+
1
(X) +O(
2
), , (34)
and inserting this expansion into (22a), then comparing O(1) terms in (34), we nd that
Q = i[, P
+
1
] =
_
0 2iP
12
2iP
21
0
_
. (35)
56 Recent Progress in Optical Fiber Research
www.intechopen.com
Optical Solitons in a Nonlinear Fiber
Medium with Higher-Order Effects 9
Thus, recalling the denition of Q the potentials q is reconstructed immediately as
q = 2iP
12
/, (36)
where P
+
1
= (P
ij
). In addition, from the denitions of P
+
, P

and Eq. (25) we have

detP
+
= s
22
= s
11
, detP

= s
22
= s
11
. (37)
The symmetry properties of the potential Q and V in (23) give rise to symmetry properties in
the scattering matrix as well as in the Jost functions. In fact, after some computation we have
J

satises the involution property

J

(X,

) = J
1

(X, ), (38)
analytic solutions P

satisfy the involution property

(P
+
)

) = P

(), (39)
and S satises the involution property
S

) = S
1
(). (40)
Let
k
and

k
are zero points of detP
+
and detP

, respectively. We see from (37) that (

k
,

k
)
are zeros of the scattering coefcients s
22
() and s
22
(). Due to the above involution property,
we have the symmetry relation

k
=

k
. (41)
For simplicity, we assume that all zeros
_
(
k
,

k
), k = 1, 2, , N
_
are simple zeros of s
22
() and
s
22
(), then each kernal of P
+
(
k
) and P

k
) contains only a single column vector v
k
and
row vector v
k
,
P
+
(
k
)v
k
= 0, v
k
P

k
) = 0.
Taking the Hermitian of the above equations and using the involution properties, we have
v
k
= v

k
. (42)
To obtain the soliton solutions, we set G = I in (32). In this case, the solutions to this special
Riemann-Hilbert problem have been derived in (25; 26) as
P
+
1
(T, X, ) =
N

j,k=1
v
j
_
M
1
_
jk
v
k
, (43)
where
M
jk
=
v
j
v
k

k
. (44)
The zeros
k
and

k
are T-independent. To nd the spatial and temporal evolutions for vectors
v
k
(T, X), we take the X-derivative to equation P
+
v
k
= 0. By using (22a), one gets
P
+
(X,
k
)(
v
k
X
+ i
k
v
k
) = 0, (45)
57 Optical Solitons in a Nonlinear Fiber Medium with Higher-Order Effects
www.intechopen.com
10 Will-be-set-by-IN-TECH
thus we have
dv
k
dX
+i
k
v
k
= 0. (46)
Similarly, taking T-derivative to equation P
+
v
k
= 0 and using (22b), one has
P
+
(T, X,
k
)(
v
k
T
+ (2i
1

k
2
4 i
3

k
3
)v
k
) = 0, (47)
thus we have
v
k
T
+ (2i
1

k
2
4 i
3

k
3
)v
k
= 0. (48)
Solving (46) and (48) we get
v
k
(T, X) =e
i
k
X+(4i
3

k
3
2i
k
2

1
)T
v
k0
, (49a)
v
k
(T, X) = v
k0
e
i
k
X+(4i
3

k
3
+2i
k
2

1
)
, (49b)
where (v
k0
, v
k0
) are constant vectors.
In summary, the N-bright soliton solutions to Eq. (14) are obtained from the analytical
functions P
+
1
in (43) together with the potential reconstruction formula (36) as
q(T, X) = 2iP
12
/ = 2i
_
_
N

j,k=1
v
j
_
M
1
_
jk
v
k
_
_
12
/, (50)
where the vectors v
j
are given by (49). Without loss of generality, we take v
k0
= [b
k
, 1]

with
b
k
constants. And if we denote

k
= i
k
X + (4 i
3

k
3
2i
k
2

1
)T, (51)
the general N-soliton solution to Eq. (14) can be written out explicitly as
q(T, X) =
2i

j,k=1
b
j
e

k
(M
1
)
jk
, (52)
with
M
jk
=
1

j

k
_
b

j
c
k
e

k
+

j
e

j
_
. (53)
In what follows, we investigate the dynamics of the one-soliton and two-soliton solutions in
Eqs. (14) in detail.
4.1 Examples of single and two bright solitons in Eq. (14)
To get the single bright soliton solution for Eq. (14), we set N = 1 in (52) to have
q(T, X) =
2i(

1

1
)

b
1
e

1
e

1
+|b
1
|
2
e

1
+

1
. (54)
If setting
1
=
1
+ i
1
, b
1
= e
2
1
X
0
+i
0
, the single soliton solution (54) can be rewritten as
q(T, X) =
2
1

sech[2
1
(X +
_
4
3

1
2
+ 4
1

1
12
3

1
2
_
T X
0
)] exp
i
, (55)
58 Recent Progress in Optical Fiber Research
www.intechopen.com
Optical Solitons in a Nonlinear Fiber
Medium with Higher-Order Effects 11
Fig. 1. (color online). Evolution of single soliton |q(T, X)| in (55) with parameters (56). It is
similar to single soliton in standard NLS equation.
Fig. 2. (color online). The shapes of two-soliton solutions |q(T, X)| in (52) with (53). (a)
soliton collision with parameters (57); (b) bound state with parameters (58).
with = 2
1
X +
_
4
1

1
2
+ 4
1

1
2
+ 8
3

1
3
24
3

1

1
2
_
T +
0
, and X
0
,
0
are
constants. This solution is similar to the solitary wave solution in the standard NLS equation
(1). Its amplitude function has the shape of a hyperbolic secant with peak amplitude 2
1
/,
and its velocity depends on several parameters, which is 12
3

1
2
4
3

1
2
4
1

1
. The
phase of this solution depends linearly both on space X and time T. We show this single
soliton solution in Fig. 1 with parameters

1
= 0.5,
1
= 0.1, X
0
= 1.5,
0
= 2, = 1,
1
= 0.5,
3
= 1, = 1. (56)
The two-soliton solution in Eq. (14) corresponds to N = 2 in the general N-soliton solution
(52) with (53). This solution can also be written out explicitly, however, we prefer to showing
59 Optical Solitons in a Nonlinear Fiber Medium with Higher-Order Effects
www.intechopen.com
12 Will-be-set-by-IN-TECH
Fig. 3. (color online). Evolution of single soliton solutions |u(z, )| in HNLS equation (2) with
controlable coefcients (59) and (62), respectively. (a) Soliton solution (61) with parameter
(56) and
0
= 0.5, c = 1, k = 2. (b) Soliton solution (64) with parameter (59) and

0
= 0.5, c = 1, k = 2.
their behaviors by gures, see Fig. 2(a)-(b). Below we take
1
=
1
+ i
1
and
2
=
2
+ i
2
and examine this solution with various velocity parameters: one is 12
3

1
2
4
3

1
2

4
1

1
= 12
3

2
2
4
3

2
2
4
1

2
, i.e. the collision between two solitons, and the other
is 12
3

1
2
4
3

1
2
4
1

1
= 12
3

2
2
4
3

2
2
4
1

2
, i.e. bound state. In Fig. 2(a),
the two soliton parameters in Eq. (52) with (53) are

1
= 0.5,
3
= 0.8, = 1, = 1,
1
= 0.2 + 0.7i,
2
= 0.1 + 0.5i, b
1
= 1, b
2
= 1. (57)
Under these parameters, the velocity of the two solitons are different. It is observed that
interactions between two soliton dont change the shape and velocity of the solitons, and there
is no energy radiation emitted to the far eld. Thus the interaction of these solitons is elastic,
which is a remarkable property which signals that the HNLS equation (14) is integrable.
Fig. 2(b) displays a bound state in Eq. (14), and the soliton parameters here are

1
= 0.5,
3
= 0.8, = 1, = 1,
1
= 0.3i,
2
= 0.1 +0.4272i, b
1
= 1, b
2
= 1. (58)
Under these parameters, the two constituent solitons have equal velocities, thus they will stay
together to form a bound state which moves at the common speed. It can be seen that the
width of this solution changes periodically with time, thus this solution is called breather
soliton.
5. Dynamics of solitons in HNLS equation (2)
In what follows, we investigate the dynamic behavior of solitons in the variable-coefcients
HNLS equation (2) with special soliton management parameters d
j
(j = 1, 2, 3, 4, 5, 6).
5.1 Single soliton solutions
We choose two cases of soliton management parameters d
j
(j = 1, 2, 3, 4, 5, 6) to study
the dynamics of the single solitons in HNLS equation (2). Firstly, if we take the soliton
management parameters to satisfy
d
1
= 1.6 (k
1
3
3
c) z/k
3
, d
2
= 1.6 (
2
k
4
c) z/
0
2
k,
60 Recent Progress in Optical Fiber Research
www.intechopen.com
Optical Solitons in a Nonlinear Fiber
Medium with Higher-Order Effects 13
Fig. 4. (color online). The two-soliton solutions |u(z, )| in HNLS equation (2) with
coefcients (59). (a) soliton collision with parameter (57) and
0
= 0.5, c = 1, k = 2; (b) bound
state with parameter (58) and
0
= 0.5, c = 1, k = 2.
d
3
= 1.6
3
z/k
3
, d
4
= 1.6
4
z/
0
2
k, d
5
= 1.6
5
z/
0
2
k, d
6
= 0, (59)
the variables , T and X in similarity transformation (3) are
=
0
, T = 0.8 z
2
, X = k + (2.4 c
2

3
1.6 ck
1
)z
2
/k
2
. (60)
So the single soliton solution in HNLS equation (2) with coefcients (59) is
u(z, ) =
0
q (T, X) e
ic(k
3
+1.6 c
2

3
z
2
0.8 ck
1
z
2
)/k
3
, (61)
where q (T, X) satises Eq. (55) and T, X satisfy Eq. (60).
Secondly, if we take the soliton management parameters to satisfy
d
1
= 0.8 cos (0.8 z) (k
1
3
3
c) /k
3
, d
2
= 0.8 cos (0.8 z) (
2
k
4
c) /
0
2
k, d
6
= 0,
d
3
= 0.8
3
cos (0.8 z) /k
3
, d
4
= 0.8
4
cos (0.8 z) /
0
2
k, d
5
= 0.8
5
cos (0.8 z) /
0
2
k, (62)
the variables , T and X in similarity transformation (3) are
=
0
, T = sin(0.8 z) , X = k + (3 c
2

3
2 ck
1
) sin(0.8 z) /k
2
. (63)
In this case the single soliton solution in HNLS equation (2) with coefcients (62) is
u(z, ) =
0
q (T, X) e
ic(k
3
+2 c
2

3
sin(0.8 z)ck
1
sin(0.8 z))/k
3
, (64)
where q (T, X) satises Eq. (55) and T, X satisfy Eq. (63).
In Fig. 3, we show the single soliton solutions (61) and (64) in HNLS equation (2) with
coefcients (59) and (62), respectively. Here the solution parameters are given in (56) and

0
= 0.5, c = 1, k = 2. It is observered that when the soliton management parameters
d
j
(j = 1, 2, 3, 4, 5) are linearly dependent on variable z and d
6
= 0 (see Eq. (59)), the trajectory
of the optical soliton is a localized parabolic curve, as shown in Fig. 3(a). When the soliton
management parameters d
j
(j = 1, 2, 3, 4, 5) are periodically dependent on variable z and
d
6
= 0 (see Eq. (62)), the trajectory of the optical soliton is a periodical localized nonlinear
wave, as shown in Fig. 3(b).
61 Optical Solitons in a Nonlinear Fiber Medium with Higher-Order Effects
www.intechopen.com
14 Will-be-set-by-IN-TECH
Fig. 5. (color online). The two-soliton solutions |u(z, )| in HNLS equation (2) with
coefcients (62). (a) soliton collision with parameter (57) and
0
= 0.5, c = 1, k = 2; (b) bound
state with parameter (58) and
0
= 0.5, c = 1, k = 2.
5.2 Collisions of the two-solitons
We now demonstrate various collision scenarios in HNLS equation (2) with coefcients (59)
and (62), respectively. As in Section 4.1, we consider the two-soliton collisions and bound
states in equation (2).
When the coefcients of equation (2) satises (59), its two-soliton solution is
u(z, ) =
0
q (T, X) e
ic(k
3
+1.6 c
2

3
z
2
0.8 ck
1
z
2
)/k
3
, (65)
where T, X satisfy Eq. (60), and q (T, X) satises Eq. (52) with (53) and N = 2.
When the coefcients of equation (2) satises (62), its two-soliton solution is
u(z, ) =
0
q (T, X) e
ic(k
3
+2 c
2

3
sin(0.8 z)ck
1
sin(0.8 z))/k
3
, (66)
where T, X satisfy Eq. (63), and q (T, X) satises Eq. (52) with (53) and N = 2.
In Fig. 4, we display the evolutions of the two-soliton solutions (65) in HNLS equation (2) with
coefcients (59). Fig. 4(a) shows the soliton collision with parameter (57) and
0
= 0.5, c =
1, k = 2, and Fig. 4(b) shows the bound state with parameter (58) and
0
= 0.5, c = 1, k = 2. In
Fig. 5, we display the evolutions of the two-soliton solutions (66) in HNLS equation (2) with
coefcients (62). Fig. 5(a) shows the soliton collision with parameter (57) and
0
= 0.5, c =
1, k = 2, and Fig. 5(b) shows the bound state with parameter (58) and
0
= 0.5, c = 1, k = 2.
6. Conclusions
In summary, we have studied the variable-coefcient higher order nonlinear Schr odinger
equation which describes the wave propagation in a nonlinear ber medium with
higher-order effects such as third order dispersion, self-steepening and stimulated Raman
scattering. By means of similarity transformation, we rst change this variable-coefcient
equation into the constant-coefcient HNLS equation. Then we investigate the integrability
of the constant-coefcient HNLS equation by prolongation technique and nd two Lax
integrable HNLS equations. The exact bright N-soliton solutions for the bright soliton version
of HNLS equation are obtained using Riemann-Hilbert formulation. Finally, the dynamics
of the optical solitons in both constant-coefcient and variable-coefcient HNLS equations is
examined and the effects of higher-order effects on the velocity and shape of the optical soliton
62 Recent Progress in Optical Fiber Research
www.intechopen.com
Optical Solitons in a Nonlinear Fiber
Medium with Higher-Order Effects 15
are observed. In addition, it is seen that the bright two-soliton solution of the HNLS equation
behaves in an elastic manner characteristic of all soliton solutions.
7. Acknowledgments
This work was supported by NSFC under grant No. 11001263 and China Postdoctoral Science
Foundation.
8. References
[1] K. E. Lonngren, Soliton experiments in plasmas, Plas. Phys. 25, 943 (1983).
[2] G.P. Agrawal, Nonlinear Fiber Optics, Academic Press, San Diego (1989).
[3] A. Hasegawa and Y. Kodama, Solitons in Optical Communications, Clarendon, Oxford,
1995.
[4] J. Denschlag, J. E. Simsarian, D. L. Feder, Charles W. Clark, L. A. Collins, J. Cubizolles1,
L. Deng, E. W. Hagley, K. Helmerson, W. P. Reinhardt1, S. L. Rolston, B. I. Schneider and
W. D. Phillips, Generating Solitons by Phase Engineering of a Bose-Einstein Condensate,
Science, 287, 97-101 (2000).
[5] B. P. Anderson, P. C. Haljan, C. A. Regal, D. L. Feder, L. A. Collins, C. W. Clark, and E. A.
Cornell, Watching Dark Solitons Decay into Vortex Rings in a Bose-Einstein Condensate,
Phys. Rev. Lett. 86, 2926-2929 (2001).
[6] M. A. Helal, Soliton solution of some nonlinear partial differential equations and its
applications in uid mechanics, Chaos, Solitons and Fractals, 13, 1917-1929 (2002).
[7] A. Hasegawa and F. Tappert, Transmission of stationary nonlinear optical pulses in
dispersive dielectric bers. I. Anomalous dispersion, Appl. Phys. Lett. 23, 142-144 (1973);
Transmission of stationary nonlinear optical pulses in dispersive dielectric bers. II.
Normal dispersion, Appl. Phys. Lett. 23, 171-172 (1973).
[8] L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, Experimental observation of picosecond
pulse narrowing and solitons in optical bers, Phys. Rev. Lett. 45, 1095-1098 (1980).
[9] V. E. Zakharov and A. B. Shabat, Exact theory of two-dimensional self-focusing and
one-dimensional self-modulation of manes in nonlinear media, Zh. Eksp. Teor. Fiz. 61,
118 (1971) (Sov. Phys. JETP 34, 62, 1972).
[10] V. E. Zakharov and A. B. Shabat, On interactions between solitons in a stable medium,
Zh. Eksp. Teor. Fiz. 64, 1627 (1973) (Sov. Phys. JETP 37, 823, 1974).
[11] J. Yang, Nonlinear waves in integrable and non-integrable systems, Society for Industrial
and Applied Mathematics, U.S. (2010).
[12] Y. Kodama and A. Hasegawa, Nonlinear pulse propagation in a monomode dielectric
guide, IEEE J. Quantum Electron. 23, 510 (1987).
[13] K. Porsezian and K. Nakkeeran, Optical Solitons in Presence of Kerr Dispersion and
Self-Frequency Shift, Phys. Rev. Lett. 76, 3955 (1996).
[14] K. Nakkeeran, K. Porsezian, P. Shanmugha Sundaram, and A. Mahalingam, Optical
solitons in N-coupled higher order nonlinear Schrodinger equations, Phys. Rev. Lett.
80, 1425 (1998).
[15] E. Papaioannou, D. J. Frantzeskakis, K. Hizanidis, An analytical treatment of the effect
of axial inhomogeneity on femtosecond solitary waves near the zero dispersion point,
32, 145-154 (1996).
[16] A. Mahalingam and K. Porsezian, Propagation of dark solitons with higher-order effects
in optical bers, Phys. Rev. E 64, 046608 (2001).
63 Optical Solitons in a Nonlinear Fiber Medium with Higher-Order Effects
www.intechopen.com
16 Will-be-set-by-IN-TECH
[17] Z. Y. Xu, L. Li, Z. H. Li, and G. S. Zhou, Modulation instability and solitons on a cw
background in an optical ber with higher-order effects, Phys. Rev. E 67, 026603 (2003).
[18] D. S. Wang, X. H. Hu, J. P. Hu, and W. M. Liu, Quantized quasi-two-dimensional
Bose-Einstein condensates with spatially modulated nonlinearity, Phys. Rev. A 81,
025604 (2010).
[19] D. S. Wang, X. H. Hu, and W. M. Liu, Localized nonlinear matter waves
in two-component Bose-Einstein condensates with time- and space-modulated
nonlinearities, Phys. Rev. A 82, 023612 (2010).
[20] H.D. Wahlquist and F.B. Estabrook, Prolongation structures of nonlinear evolution
equations. J. Math. Phys. 16, 1-7 (1975).
[21] F.B. Estabrook and H.D. Wahlquist, Prolongation structures of nonlinear evolution
equations II, J.Math.Phys.17, 1293-1297 (1976 ).
[22] D. S. Wang, Integrability of the coupled KdV equations derived from two-layer uids:
Prolongation structures and Miura transformations, Nonlinear Analysis, 73, 270-281
(2010). Deng-Shan Wang
[23] R. Dodd and A. P. Fordy, The prolongation structure of quasi-polynomial ows Proc. R.
Soc. London A 385, 389-429 (1983).
[24] V.S. Gerdjikov, Basic aspects of soliton theory, Sixth International Conference on
Geometry, Integrability and Quantization June 3-10, 2004, Varna, Bulgaria Ivailo M.
Mladenov and Allen C. Hirshfeld, Editors, Soa 1-48 (2005).
[25] V. E. Zakharov, S. V. Manakov, S. P. Novikov, and L. P. Pitaevskii, The Theory of Solitons:
The Inverse Scattering Method, Consultants Bureau, New York, 1984.
[26] V.S. Shchesnovich and J. Yang, General soliton matrices in the Riemann-Hilbert problem
for integrable nonlinear equations. J. Math. Phys. 44, 4604 (2003).
[27] E.V. Doktorov, J. Wang, and J. Yang, Perturbation theory for bright spinor Bose-Einstein
condensate solitons, Phys. Rev. A. 77, 043617 (2008).
[28] D. S. Wang, D. J. Zhang and J. Yang, Integrable properties of the general coupled
nonlinear Schrodinger equations, J. Math. Phys. 51, 023510 (2010).
64 Recent Progress in Optical Fiber Research
www.intechopen.com
Recent Progress in Optical Fiber Research
Edited by Dr Moh. Yasin
ISBN 978-953-307-823-6
Hard cover, 450 pages
Publisher InTech
Published online 25, January, 2012
Published in print edition January, 2012
InTech Europe
University Campus STeP Ri
Slavka Krautzeka 83/A
51000 Rijeka, Croatia
Phone: +385 (51) 770 447
Fax: +385 (51) 686 166
www.intechopen.com
InTech China
Unit 405, Office Block, Hotel Equatorial Shanghai
No.65, Yan An Road (West), Shanghai, 200040, China
Phone: +86-21-62489820
Fax: +86-21-62489821
This book presents a comprehensive account of the recent progress in optical fiber research. It consists of four
sections with 20 chapters covering the topics of nonlinear and polarisation effects in optical fibers, photonic
crystal fibers and new applications for optical fibers. Section 1 reviews nonlinear effects in optical fibers in
terms of theoretical analysis, experiments and applications. Section 2 presents polarization mode dispersion,
chromatic dispersion and polarization dependent losses in optical fibers, fiber birefringence effects and spun
fibers. Section 3 and 4 cover the topics of photonic crystal fibers and a new trend of optical fiber applications.
Edited by three scientists with wide knowledge and experience in the field of fiber optics and photonics, the
book brings together leading academics and practitioners in a comprehensive and incisive treatment of the
subject. This is an essential point of reference for researchers working and teaching in optical fiber
technologies, and for industrial users who need to be aware of current developments in optical fiber research
areas.
How to reference
In order to correctly reference this scholarly work, feel free to copy and paste the following:
Deng-Shan Wang (2012). Optical Solitons in a Nonlinear Fiber Medium with Higher-Order Effects, Recent
Progress in Optical Fiber Research, Dr Moh. Yasin (Ed.), ISBN: 978-953-307-823-6, InTech, Available from:
http://www.intechopen.com/books/recent-progress-in-optical-fiber-research/optical-solitons-in-a-nonlinear-
fiber-medium-with-higher-order-effects