INTRODUCTION
In applied sciences, each physical event may be modeled mathematically. So,
it is very important to have information about analytical solutions of the models
because these solutions provide information about the character of the modeled
event. Therefore, it is very important to find analytical solutions of linear
or nonlinear ordinary and Partial Differential Equations (PDEs) in physics,
chemistry, biology and engineering areas (Ablowitz and Segur,
1981). Among the possible solutions to nonlinear PDEs, certain special form
solutions may depend only on a single combination of variables such as traveling
wave variables. In the literature, there is a wide variety of approaches to
nonlinear problems for constructing traveling wave solutions, such as the inverse
method (Ablowitz and Segur, 1981) Bácklund transformation
(Tam and Hu, 2002) Hirota bilinear method (Hirota,
2004), numerical methods (Borhanifar and Abazari, 2009)
and the Wronskian determinant technique (Freeman and Nimmo,
1983).
It is more difficult to obtain solutions of nonlinear PDEs than those of linear differential equations. Therefore, it may not always be possible to obtain analytical solutions of these equations. In this case, we use semianalytical methods giving series solutions. In these kinds of methods, the solutions are sought in the form of series. Semianalytical methods are based on finding the other terms of the series from given initial conditions for the problem being considered. At this point, we encounter the concept of convergence of the series. So, it is necessary to perform convergence analysis of these methods. As this convergence analysis can be carried out theoretically, one can gain information about the convergence of the series solution by looking at the absolute error between the numerical solution and the analytical solution. In some semianalytic methods, a very good convergence can be achieved with only a few terms of the series, but more terms can be needed in some problems. That is, if the terms of the series increase, this provides better convergence to the analytical solution.
In this case study, similarity transformation has been used to reduce the governing
differential equations into an ordinary nonlinear differential equation. In
most cases, these problems do not admit analytical solution, so these equations
should be solved by using special techniques. In recent years, some researchers
used new methods to solve these kinds of problem (Shabani
and Abazari, 2009; Adomian, 1994; He,
1999; Liao, 2003). Integral transform methods such as
the Laplace and the Fourier transform methods are widely used in engineering
problems. These methods transform differential equations into algebraic equations
which are easier to deal with. However, integral transform methods are more
complex and difficult when applying to nonlinear problems. The differential
transformation method was first applied in the engineering domain by Zhou
(1986). The differential transform method is based on Taylor expansion.
It constructs an analytical solution in the form of a polynomial. It is different
from the traditional high order Taylor series method, which requires symbolic
computation of the necessary derivatives of the data functions. The Taylor series
method is computationally taken long time for large orders. The differential
transform is an iterative procedure for obtaining analytic Taylor series solutions
of differential equations. DTM has been successfully applied to solve many nonlinear
problems arising in engineering, physics, mechanics, biology etc. Ayaz
(2004) applied DTM for solution of system of differential equations. Arikoglu
and Ozkol (2006) employed DTM on differentialdifference equations. Furthermore,
the method may be employed for the solution of partial differential equations.
DTM employed on some PDEs and their coupled version (Borhanifar
and Abazari, 2011, 2010; Abazari
and Borhanifar, 2010; Abazari and Ganji, 2011; Abazari
and Abazari, 2012; Reza and Abazari, 2010) and extended
DTM to solve the first and second kind of the Riccati matrix differential equations
by Abazari (2009).
The propagation of pulses with equal mean frequencies in birefringent nonlinear
fiber is governed by the coupled nonlinear Schrödinger equation (CNLSE)
(Menyuk, 1988):
where, i^{2} = 1 and φ, ψ are the wave amplitudes in two
polarizations and η is the normalized strength of the linear birefringence.
There are various analytical and numerical results on solitary wave solutions
of the general N coupled Schrφdinger equations (Chow,
2001; Chow and Lai, 2003; Cipolatti
and Zumpichiatti, 2000; Biswas and Khalique, 2010;
Cattani, 2005).
Following the discussion of Wadati et al. (1992),
the exact solution of Eq. (1) is:
where, α and v are real parameters and i^{2} = 1.
Recently, Abazari and Ganji (2011) extended RDTM to
study the partial differential equation with proportional delay and Abazari
and Abazari (2012) applied the RDTM on generalized HirotaSatsuma coupled
KdV equation and shown that as a special advantage of RDTM rather than DTM,
the reduced differential transform recursive equations produce exactly all the
Poisson series coefficients of solutions, whereas the differential transform
recursive equations produce exactly all the Taylor series coefficients of solutions.
In this study, we employed the DTM and RDTM on coupled nonlinear Schrödinger
Eq. 1 and compared the obtained results with the exact solution.
As an important result, notwithstanding the simplicity and robustness of the
RDTM, it is depicted that the RDTM results are more accurate in comparison with
those obtained by classic DTM.
BASIC DEFINITIONS
With reference to the studies of Borhanifar and Abazari
(2011) and Reza and Abazari (2010) the basic definitions
of twodimensional differential transformation are introduced as follows:
Twodimensional DTM: Consider a function of two variables w (x, t) and suppose that it can be represented as a product of two singlevariable function, i.e., w (x, t) = f (x) g (t). On the basis of the properties of the onedimensional differential transform, the function w (x, t) can be represented as:
where, w (i, j) = F (i) G (j) is called the spectrum of w (x, t).
The basic definitions and operations for twodimensional differential transform are introduced as follows:
Definition 1: If w (x, t) is analytic and continuously differentiable with respect to time t in the domain of interest, then:
where, the spectrum function w (k, h) is the transformed function, which is also called Tfunction in brief.
In this study, (lower case)w (x, t) represents the original function while (upper case) w (k, h) stands for the transformed function (Tfunction).
The differential inverse transform of w (k, h) is defined as:
Combining Eq. 2 and 3, it can be obtained that:
When (x_{0}, t_{0}) are taken as (0, 0), then Eq.
3 can be expressed as:
In real applications, the function w (x, t) is represented by a finite series of Eq. 4 can be written as:
and Eq. 4 implies that
is negligibly small. Usually, the values of n and m are decided by convergency
of the series coefficients.
From the above definitions, it can be found that the concept of the twodimensional
differential transform is derived from the twodimensional Taylor series expansion.
With Eq. 2 and 3, the fundamental mathematical
operations performed using the twodimensional differential transform be readily
obtained and these are listed in Theorem 1 (Borhanifar and
Abazari, 2011, 2010; Abazari
and Borhanifar, 2010; Abazari and Ganji, 2011; Abazari
and Abazari, 2012; Reza and Abazari, 2010).
Theorem 1: Assume that W (k, h), U (k, h) and V (k, h) are the differential transforms of the functions w (x, t), u (x, t) and v (x, t) respectively, then:
• 
If w (x, t) = u (x, t)±v (x, t) then W (k, h) = U (k,
h)±V (k, h) 
• 
If w (x, t) = cu (x, t) then W (k, h) = cU (k, h) where c∈R 
• 
If w (x, t) = x^{m}t^{n}, then 
• 
If w (x, t) = u (x, t) v (x, t) then 
Proof: From Abazari and Borhanifar (2010), Abazari
and Ganji (2011) and Abazari and Abazari (2012)
and their references.
Twodimensional RDTM: Consider a function of two variables w (x, t) and suppose that it can be represented as a product of two singlevariable function, i.e.,w (x, t) = f (x) g (t). Based on the properties of onedimensional differential transform, the function w (x, t) can be represented as:
where, W(i, j) = F(i)G(j) is called the spectrum of w (x, t).
Remark 1: The poisson function series generates a multivariate Taylor series expansion of the input expression w, with respect to the variables X, to order n, using the variable weights W.
Remark 2: The relationship introduce in (7) is the poisson series form of the input expression w (x, t) with respect to the variables x and t, to order N, using the variable weights W_{k} (x).
Similar on previous section, the basic definitions of twodifferential reduced differential transformation are introduced as follows:
Definition 2: If w (x, t) is analytical function in the domain of interest, then the spectrum function:
is the reduced transformed function of w (x, t).
Similarly on previous sections, the lowercase w (x, t) respect the original function while the uppercase W_{k} (x) stand for the reduced transformed function. The differential inverse transform of W_{k} (x) is defined as:
Combining Eq. 7 and 8, it can be obtained
that
From the above proposition, it can be found that the concept of the reduced
twodimensional differential transform is derived from the twodimensional differential
transform method. With Eq. 7 and 8, the
fundamental mathematical operations performed by reduced twodimensional differential
transform can readily be obtained and listed in Theorem 2.
Theorem 2: Assume that W_{k} (x),U_{k} (x) and V_{k} (x), are the differential transforms of the functions w (x, t), u (x, t) and v (x, t), respectively, then:
• 
If w (x, t) = u (x, t)±v (x, t), then W_{k}
(x) = U_{k} (x)±V_{k} (x) 
• 
If w (x, t) = cu (x, t) then W_{k} (x) = cU_{k}
(x) where c∈R 
• 
If w (x, t) = x^{m}t^{n} then 
• 
If w (x, t) = u (x, t) v (x, t) then 
Proof: From Abazari and Borhanifar (2010), Abazari
and Ganji (2011) and Abazari and Abazari (2012)
and their references.
Description of the methods: Consider the nonlinear coupled Schrödinger
equation (Menyuk, 1988; Chow, 2001;
Chow and Lai, 2003).
subject to initial conditions
where, φ (x) and φ (x), are complex functions. For our numerical
work, we decompose the complex functions φ and Ψ into their real and
imaginary parts by writing (Borhanifar and Abazari, 2010):
where, u_{j} and v_{j}, (j = 1, 2) are real functions. Therefore the coupled equation given in (9), can be written in a following form:
Where:
where, we apply the both DTM and RDTM on the coupled nonlinear Schrödinger
eq. 1. We will shown that the DTM convert the eq.
(1) to a two parameters recursive equation where is the RDTM convert to
a oneparameter recursive equation.
Twodimensional DTM: According to twodimensional differential transform
operators listed in Theorem 1, the differential transform version of system
given in Eq. (12), will be:
where, U_{j} (k, h), V_{j} (k, h) and Z_{j} (k, h) (j = 12) are the differential transform version of u_{j} (x, t), v_{j} (x, t) and z_{j} (x, t) respectively.
In order to obtain the unknowns of U_{j} (k, h),, and V_{j} (k, h), k, h = 1,1,2,..., N we must construct and solve the above equations by using initial conditions (10) and substitute in Eq. 5. The following corresponding algorithm can be introduced to the case below:
DTM Algorithm:
Step 1: 
Choose N∈N as the degree of approximate solution 
Step 2: 
Determine the initial value U_{1} (k,0), V_{1} (k,0),
U_{2} (k,0) and V_{2} (k,0) for k = 0, 1, 2, ..., N 
Step 3: 
Set
and 
Step 4: 
Set
and 
Step 5: 
For k = 0,1,2,..., N do for h = 0,1,2,..., N do: 
Towdimensional RDTM: Now, according to twodimensional reduced differential
transform operators listed in Theorem 2, the coupled equation given in Eq.
12, can be written in a matrixvector form as:
Where:
Then the differential transform of system given in Eq. 15,
will be:
where,
is the reduced differential transform of
and for
therefore, for k = 0,1,2,..., N Equation 16, can be rewritten
as follows iteration method
with initial condition
where,
and
In order to obtain the unknowns of
for
we must construct iteration eq. 17 and by substituting in
the following series, we can obtain the Nth term approximation series form
of exact solutions:
Therefore, the corresponding algorithm can be introduced to the case below:
RDTM Algorithm:
Step 1: 
Choose N∈N as the degree of approximate solution 
Step 2: 
Determine the initial value
from initial conditions (10) 
Step 3: 
Set 
Step 4: 
For
do 
Step 5: 

Step 6: 

Numerical Examples: To compare the numerical results of DTM and RDTM,
we consider the coupled nonlinear Schrödinger Eq. 1 with
exact wave solution (2), when
and
(Borhanifar and Abazari, 2010)
subject to the initial conditions
From the initial conditions (20) and according to Eq. 11,
we get:
In these kind of initial conditions, we first obtain the DTM recurrence relation
and secondly, the RDTM version of equation and solve both recurrence relations
by programming in MATLAB environment. The results of the test example show that
the RDTM results are more powerful than DTM results. These numerical results
are given in Table 1 4.
Numerical results of DTM: For finite values of
the differential transform version of initial conditions (21) obtain from following
recursing relationship
from the recursive Eq. 22, the following initial values can be obtained easily
Table 1: 
The error of real and imaginary parts of Φ (x, t)respect
to first 3terms approximation solutions obtained by DTM and RDTM at some
points in the intervals 0≤x≤5 and 0≤t≤1 

Table 2: 
The error of real and imaginary parts of Φ (x, t) respect
to first 6terms approximation solutions obtained by DTM and RDTM at some
points in the intervals 0≤x≤5 and 0≤t≤1 

Table 3: 
The error of real and imaginary parts of Ψ(x, t) respect
to first 3terms approximation solutions obtained by DTM and RDTM at some
points in the intervals 0≤x≤5 and 0≤t≤1 

Table 4: 
The error of real and imaginary parts of Ψ(x, t) respect
to first 6terms approximation solutions obtained by DTM and RDTM at some
points in the intervals 0≤x≤5 and 0≤t≤1 

then by utilize the differential transform of initial condition (23) in recursive
method (14), the three term DTM approximation solution,
and
in a series form obtain as follow:
which are the same as the partial sum of the Taylor series form of real and
imaginary parts of traveling wave solution (1.2), when
Numerical results of RDTM: Similarity, we consider the Eq.
(1) subject to vector form of initial conditions:
Then by utilize the initial value (25) in recursive method (17) for
the first three terms of
obtain as follow:
Where:
and
Where:
and
Where:
In the same manner, the rest of components can be obtained using the recurrence relation (26). Substituted the obtained quantities (27), (28) and (29) in the approximation series form solution (18), the following 3th approximation solution in the Poisson series form can be obtained
where,
for
and
are listed in (27), (29) and (31). The closed form solution of (32) can be obtain
as follow:
and substitute in (11), the traveling wave solution of coupled nonlinear Schrödinger
eq. 1, obtained as follow:
which are the same as the closed form solutions of exact solutions.
CONCLUSIONS
In this study, we presented the definition and operation of both twodimensional Differential Transformation Method (DTM) and their reduced form, reducedDTM (RDTM). These methods used for finding soliton solutions of nonlinear coupled Schrodinger equation. These methods have been applied directly without using bilinear forms, Wronskien, or inverse scattering method. It is worth pointing out that the both DTM and RDTM have convergence for the solutions, actually, the accuracy of the series solution increases when the number of terms in the series solution is increased. From computational precess of DTM and RDTM, we find that the RDTM is more easier to apply. In the other word, it is seem that DTM have very complicated computational precess rather than RDTM. The RDTM reduces the computational difficulties of the DTM and all the calculations can be made with simple manipulations MATLAB. Actually, as a special advantage of RDTM rather than DTM, the reduced differential transform recursive equations produce exactly all the Poisson series coefficients of solutions, whereas the differential transform recursive equations produce exactly all the Taylor series coefficients of solutions.