INTRODUCTION
In recent studies, it has been shown that WavelengthDivision Multiplexing (WDM) can be a successful method to increase the transmission capacity of optical fiber systems (Evangelides and Gardon, 1996; Etrich et al., 2001). However, there are some limitations on the performance of these systems. One of the major ones is due to the nonlinear interchannel interaction of data signals from different channels (Agrawal, 2002). The increasing demand for the development of optical communication systems for high data rate transmission and high quality information leads to the increase in the number of channels used and a decrease in the width of the pulses launched. As a result, the importance of the interchannel collisions between optical pulses is expected to increase and an accurate description of the effects of these collisions is required.
In WDM method, solitons at different wavelengths are injected at the input of the fiber. Since, the spectral separation between these solitons (channels) is λ<=0.5 nm and they have different velocities in different channels, collisions among them inevitably take place (Agrawal, 2002). In here, full attention is given to the effect of thirdorder dispersion on the interchannel collisions between solitons, which is expected to be dominant (in comparison with other inelastic effects) near the zerodispersion wavelength.
In an ideal fiber, interchannel collisions between solitons can be modeled
by using the nonlinear Schrödinger equation (NLSE) (Hasegawa and Matsumoto,
2002). In that case since the collisions are elastic no radiation is emitted.
Moreover, by using NLSE the amplitude, frequency and shape of the solitons do
not change. Using this model for an ideal collision only leaves us a phase shift
proportional to
and a position shift proportional to
where, β is the frequency difference between the solitons. In real optical
fibers, however, this ideal elastic nature of soliton collisions breaks down
owing to the presence of highorder corrections (perturbations), such as thirdorder
dispersion, to the ideal NLSE. In this case, collisions between solitons from
different frequency channels might lead to the emission of radiation, change
in the soliton’s amplitude and frequency, corruption of the soliton’s shape,
stronger shift in the soliton’s position and other undesirable effects.
Accurate analysis of the effects of perturbations on interchannel collisions is a very complicated and longstanding problem. The main technical issue in this case is how to develop a perturbation theory around the multisoliton solutions of the ideal NLSE. In spite of the existence of exact expressions for the multisoliton solutions of the ideal NLSE, direct perturbative analysis around the complex multisoliton solutions has not yet been successfully implemented.
Our aim in this work is to study the effects of the second collision the perturbative
soliton from the origin channel (It means a soliton that has shifted its phase
and position because of a collision with another soliton from the other channel
and partially its energy is lost at continuous radiation) with stationary soliton
from the other channel. Moreover, we calculate the dynamics and the total intensity
of the continuous radiation emitted as a result of two collisions and also the
change, induced by the collision, in the soliton parameters. That before is
shown (Plege et al., 2003) interaction between stationary twosoliton
in presence of third order dispersion lead to O(1/β) phase shift, O(1/β^{2})
position shift. In addition, the amplitude of the emitted radiation is proportional
to O(d_{3}/β^{2}). It should also be mentioned that the
propagation of a single pulse in the presence of thirdorder dispersion was
studied in detail by Kodama, (1985) and Horikis and Elgin (2001). It was found
that even if the pulse launched into the fiber is not exactly of the stationary
form it evolves into the stationary form after a transient (Elgin et al.,
1995).
Notice that the major technical tool used in the analytical calculations is singular perturbation theory that is an appropriate extension of the technique developed (Kaup, 1990).
MATERIALS AND METHODS
Propagation of an electrical field wave packet ψ (t, z) through an optical fiber under the influence of third order dispersion is described by the following modification of the nonlinear Schrφdinger (Agrawal, 2001)):
Where, z is the dimensionless position along the fiber
x is the actual position along the fiber, P_{0} is the peak soliton
power and κ is the kerr nonlinearity coefficient. The dimensionless retarded
time is t = τ/τ_{0}, where, τ is the retarded time associated
with the reference channel and τ_{0} is the soliton width. Figure
1 describe these parameters. The term
(where, d_{3 }is a small constant) on the righthand side of Eq.
1 accounts for the effect of third order dispersion.
For d_{3}≠0, Eq. 1 is not integrable. However, in
many particle cases d_{3}<<1, allowing a perturbative calculation
about the integrable d_{3} = 0 limit. Fiber losses in Eq.
1 are neglected. In practice, this can be achieved by compensating for losses
in a fiber span by means of distributed optical amplification, e.g., Raman amplification.
Let us assume that d_{3}<<1 and derive perturbatively a zindependent
(stationary) singlesoliton solution of Eq. 1. When d_{3}
= 0, the singlesoliton solution of Eq. 1 in a β frequency
channel is described by

Fig. 1: 
Propagation two solitons in channels 0 and β at a fiber 
Where, α_{β}, η_{β} and y_{β}
are the soliton phase, amplitude and position, respectively. Assuming that d_{3}<<1,
we will be looking for a stationary perturbative singlesoliton solution of
Eq. 1 in the form
Where
and
The term
is ideal singlesoliton soluton and is the first order (in d_{3}) correction.
To calculate this term, we adopt the perturbation method (Kaup, 1990). In Kaup^{’}s
theory, the differential operator L^{^}_{η} is used to
describe a linear perturbation around the ideal soliton solution. We expand
in terms of the eigenfunctions of L^{^}_{η} and calculate
the coefficients of this expansion. It is shown in (Plege et al., 2003)
that
is stable and localized. Stationary of the solution Eq. 3 means
that each of the solitons propagates without any change in their parameters
and without shedding any radiation; thus effects of radiation emission and parameter
change are due only to soliton collisions. We develop a double perturbation
theory with the two small parameters d_{3} and 1/β. The perturbation
theory presented here is valid for any value of β, provided d_{3}<<1
and d_{3}<<1+d_{3}β. For simplicity andwithout any
loss of generality, one of the two channels is chosen as a reference one with
β = 0. We also assume that for the second channel β>>1.The
stationary twosoliton solution Eq. 1 with d_{3}...
is calculated (Plege et al., 2003). In this study we investigate the
effects of the perturbative soliton from the origin channel in order to its
collision again with stationary soliton from a typical β channel up to
the third order of the theory, in which appear in terms of Φ΄ (Fig.
2).

Fig. 2: 
Non simultaneous, Threesoliton collision:
η_{2} = η_{1} = η_{0} = 1, β_{2} = 10, β_{1} = 10, β_{0} = 0, y_{2} = 0, y_{1} = 0, y_{0} = 10 
Symbolically we assume the perturbative soliton as an ideal soliton Ψ_{0}
in which there are some changes in the former one due to the first collision
that can be used as initial conditions for the second collision. We are looking
for a twosoliton solution of Eq. 1 in the form, Ψ_{two}
= Ψ_{0}+Ψ_{β}+Φ’ where, ψ_{0}
and ψ_{β} are described by Eq. 3 with β
= 0 and β, respectively and Φ΄ is a small correction due to collision.
It is straightforward to check that the exact twosoliton solution of Eq.
1 at d_{3} = 0 acquires the form
Where, Φ_{0} and Φ_{β} are corrections of the
leading order 1/β in the channels 0 and β, respectively. The terms
Φ_{β} and Φ_{2β} correspond to O (1/β^{2})
corrections in channelsβ and 2β, respectively; the two latter corrections
are exponentially small outside the collision region. By analogy with the ideal
d_{3} = 0 case, one substitutes a solution of the form Eq.
4 into Eq. 1 and calculates Φ΄_{0}. Since
Φ΄_{0} oscillates together with ψ_{0} and d_{3}<<1,
one neglects the exponentially small contributions from the terms rapidly oscillating
with t and z. Then the equation describing Φ΄_{0} is
Where:
and
Vicinity (in z) of the collision event is given by
Where1
and is naturally separated from the region before and after the collision. In
the collision region,
acquires a fast (with respect to z) change. Since for this region Δz~1/β,
theand
terms
give leading contributions to Eq. 5, whereas the
term can be neglected together with all the order terms. In the successive orders
of the perturbation theory, one should carefully consider contributions coming
from terms such as
and
. In the region before and after the collision, the interaction between the
solitons is exponentially small, so that all the interaction terms there can
be neglected. Formally, separation into three welldefined regions means that
one can replace all the in Eq. 5, except for
with
where, δ (z) is the Dirac delta function and the constant C is simply the
integral of all these terms over z. This separation results in the three wellformulated
Cauchy problems for Φ΄_{0} in the three regions. The precollision
region is
we use this region as
Even though the rigorous calculation of the effects of the collision in successive orders of the perturbation theory is quite complicated, the main result can be derived in a straightforward manner by use of just a few equations. The initial condition for Φ΄^{(0)}_{0} before the second collision is equal to solution of after the first collision in the first order perturbation theory
Where:
In Eq. 7 and the following equations the superscript in stands
for initial values of phase, position, etc., while the superscript out represents
final values of the same parameters.
In the collision region Eq. 5 reduces to
Integrating Eq. 8 over the collision region and using the
initial condition 6 at
one arrives at
And
Comparing Eq. 9 and 6, we see that the only
effect of the collision in the first order of the perturbation theory is a change
of the soliton phase
Notice that Eq. 11 is also equal to the result obtained
from the first collision.
Calculation of higher order terms requires knowledge of the complete z dependence
of.
To achieve this aim, integrating Eq. 8 from 8 to some general
z, one obtains
For second order perturbation theory similar to the first order perturbation
theory, the initial condition for the 0(1/β^{2}) term
in the precollision region is:
Where:
In the collision region the 0(1/β^{2}) part of Eq.
5, one can show that the only change in the solitons’s parameters comes
from the term:
Where:
Integrating over the collision region and using the initial condition (13)
at
, one derives
Where:
and
Because of, we
neglected result obtained from last two terms on the rhs of Eq.
15.
Comparing Eq. 17 and 13, we see that the
only effect of the collision in 1/β^{2} order is a position shift
(time retardation) give by
Also, we shown that after each collision the rate of position shift is the
same.
Moreover, we see that the only effect of the collision in O(d_{3}/β)
is a O(1/β) change of phase, on top of the O(d_{3}) stationary
solution.
This change also is equal to those from the first collision.
Emission of radiation comes from the O(d_{3}/β^{2}) term. To analyze the O(d_{3}/β^{2}) correction, we first write it in the form
Where,
is the leading, O(d_{3}/β^{2}), contribution to the radiation
emitted due to the collision and
corresponds to the nonradiative part. The initial collision for is
the changes that is caused by the first collision
The initial condition for is taken to be
Where:
i.e., the initial condition contains radiation.
The equation of source term for the emitted radiation Φ^{~´(1)R}_{03} is
Substituting Eq. 12 into Eq. 24 and integrating
over the collision region, one obtains
Where, the coefficient B is defined by
and
The functions
are eigenfunctions of operator L^{^}_{η}; The expansion
coefficients
appearing in Eq. 27 are given by
Where, s = k/η_{0}. After second collision, we find
Dynamics of the coefficients a_{k}(x) are given by
Equations 29 and 30 describe the dynamics
of the term ,
which is the leading contribution responsible for radiation.
The emitted radiation after each collision is of order d_{3}/β^{2}.
The absolute value of this emission for four different values of z = z_{1},
z = z_{1}+1, z = z_{1}+3 and z = z_{1}+7 are shown in
Fig. 3.
Since ,
the leading contribution to the radiation intensity emitted due to the collision
is of order d^{2}_{3}/β^{4}. This contribution
give by

Fig. 3: 
Absolute value of the radiation profile
function normalized to B after the second collision, i.e., , is shown as a function of t, for four values of z: z = z _{1}(—),
z = z _{1}+1 (), z = z _{1}+3(−−−), z
= z _{1}+7(......) 

Fig. 4: 
Absolute value of the radiation profile
function normalized to B, i.e., , is shown as a function of t, for four values of z: z = z _{1}+10
(—), z = z _{1}+13 (−−−), z = z _{1}+15
( ), z = z _{1}+17 (......) 
So, according to the Fig. 4, for z››z_{1}+1,
all z dependent contributions to E^{R}_{O} decay algebraically
with zz_{1}. Thus, far away from the collision region the only nonvanishing
contribution to E^{R}_{O} is
Using Eq. 26, we can calculate the radiation energy emitted
E^{R}_{O} by the reference channel soliton after each collision
in far away from the collision region:
RESULTS AND DISCUSSION
We finds that the only changes in the pulse parameters up to the third order of the theory are the O(1/β) phase shift, Δα_{0}~ 4η_{β}(1+3d_{3}β)^{1/2 }[(1+3d_{3}β/2)β]^{1} and the O (1/β^{2}) position shift, Δy_{0} =  4η_{β }(1+3d_{3}β)^{1/2 }[(1+3d_{3}β/2)^{2}ββ]^{1}.
Thus, the radiations propagate away from the soliton (in t) with velocity, is of the first order. The soliton retains its shape (such that, at each instant, the soliton is close to a stationary solution, width phase anaphase velocity) while evolving slowly. Note that the rate of emitted energy after each collision is always the same. Also, the rate of changes soliton parameters after each collision is always the same. Moreover, neglecting the decrease in the soliton amplitude, the total energy emitted by the reference channel soliton
as a result of many collisions with solitons from the β channel grows linearly
with the number of collisions. Taking η_{0} = 1 and requiring that
the widths of the colliding solitons are equal, we obtain η_{β}
= (1+3d_{3}β)^{1/2} with the assumption that (typical setup
for a short pulse optical fiber experiment) τ_{0} = 0.5 ps, β_{2}
= 1 ps^{2} km^{1}, β_{3} = 0.1 ps^{3}
km^{1}, d_{3} = 0.07, β = 10, p_{0} = 0.4 W, κ
= 10 W^{1} km^{1}, Δv = 2.03x10^{12} Hz. Thus
for the parameters introduced, we calculated that the mean distance passed by
the soliton until it experiences 20000 collisions and loses about 10% of its
energy is approximately 2500 km (Plege et al., 2003). The soliton amplitude
and phase velocity do not acquire any change up to third order of dispersion
in WDM systems. The result for the soliton amplitude is consistent with the
conservation law for the total energy, which requires η = 1+O (d^{2}_{3}/β^{4})
for both solitons. Results obtained from the second collision the perturbative
soliton from origin channel with stationary soliton from other channel at third
order perturbation collision is similar with ones after the first collision
the ideal soliton from origin channel with stationary soliton from other channel.
Moreover, after the second collision at this order of theory, we obtained also,
the O(1/β) phase shift on top of the O(d_{3}) stationary solution.
An interesting feature of the collision is that the leading contributions to
the observed effects come from terms in the equations that involve.
Thus, the leading, O(1/β), contribution to the phase shift, which is due
to the term
is simply given by.
Then, the leading, O(1/β^{2}), contribution to the position shift
is due to the
term. Finally, the leading, O(d_{3}/β^{2}), contribution
to the radiation emission is due to
the term that does not exist in the ideal twosoliton collision problem.