In recent years, the concept of soliton has been receiving considerable attention in optical communications, since soliton is capable of propagating over long distances without change of shape and velocity. It has been found that the soliton propagating through optical fiber arrays is governed by a set of equations related to the coupled nonlinear Schrödinger equation [1] [2] .

where, is the envelope or the amplitude of the jth wave packets. Equation (1) reduces to the standard nonlinear Schrödinger equation for, to Manakov integrable system for, and recently for this case the exact two soliton solution obtained and novel shape changing in elastic collision property has been brought out. The system for is of physical interest, in optical communication, and in bio- physics this system can be used to study the lunching and propagation of solitons along the three spines of an alpha-helix shape changing in protein [1] [2] [3] [4] . In this work, we are going to derive some numerical methods for solving the three coupled nonlinear Schrödinger equations (3-CNLS)

with initial conditions

and the homogenous boundary conditions

The exact solution of the 3-coupled nonlinear Schrödinger equation [1] [2] is given by

where are four arbitrary complex parameters. Further gives the amplitude of the jth mode and the soliton velocity.

Many numerical methods for solving the coupled nonlinear Schrödinger equation are derived in the last two decades. Finite difference and finite element methods are used to solve this system by Ismail [5] - [10] . A conservative compact finite difference schemes are given in [11] [12] . In [3] [4] , Xing Lü studied the bright soliton collisions with shape change by intensity for the coupled Sasa-Satsuma system in the optical fiber communications. A higher order exponential time differencing scheme for system of coupled nonlinear Schrödinger equation is given in [13] . A semi-explicit multi-sypm- lectic splitting scheme for 3-coupled nonlinear Schrödinger equation is given in [14] . Many researchers are used time splitting method for solving the coupled non-linear schrödinger equation, the basic idea in this method is to split the original system into a linear subsystem and nonlinear subsystem, The splitting simplify the problem, since the linear problem is uncoupling, relatively easy to solve and the nonlinear problem can be solved exactly due to their point-wise conservation law, for more details see [15] - [23] .

To avoid complex computations we assume

where are real functions, the systems (2)-(4) can be written as

where

The resulting systems (8)-(10) can be written in a matrix-vector form as

where

Proposition 1 The three coupled nonlinear Schöridinger equations have the con- served quantities

Proof : we consider the first conserved quantity (13), from (8) we have

by multiplying (16) by and (17) by and by adding the resulting equ- ations to obtain

By integrating Equation (18) with respect to x from to and using the vanishing boundary conditions to obtain

and this is the proof of the first conserved quantity (13). The other two conserved quantities (14) and (15) can be proved in the same way.

The exact values of the conserved quantities using the exact soliton solution (6) are given by the following formula

The paper is organized as follows. In Section 2, we derived a second order Crank- Nicolson scheme for solving the proposed system. The fourth order compact difference scheme is derived in Section 3. In Section 4, we present two fixed point schemes to solve the block nonlinear tridiagonal systems obtained in Sections 2 and 3. To avoid the nonlinearity obtained in the previous sections, we present time splitting method to solve the 3-CNLS in Section 5. The numerical comparison of the derived methods are reported in Section 6. Finally, we draw some conclusions in Section 7.

In order to develop a numerical method for solving the system given in (12), the region will be covered with a rectangular mesh of points with coor- dinates,

where h and k are the space and time increments respectively. We denote the exact and numerical solution at the grid point by and, respectively. We approximate the space derivative by the second order central difference formula

where is the second order central difference operator. The Crank-Nicolson scheme for the 3-CNLS equation is given by

where

The scheme in (20) is of second order accuracy in both directions space and time, and it is unconditionally stable using von-Neumann stability analysis, see Ismail [8] . A nonlinear block tridiagonal system must be solved at each time step. Fixed point method is used to do this job and this will be discussed later.

A highly accurate finite difference scheme can be obtained by using the fourth order Padè compact difference approximation for the spatial discretization

together with the Crank-Nicolson scheme, this will lead us to the compact finite diffe- rence scheme

where

The scheme (21) can be written in a block tridiagonal form as

where

The method is of second order accuracy in time and fourth order in space, it is implicit unconditionally stable, see Ismail [9] . The resulting system is a block nonlinear tridiagonal system and can be solved by fixed point method and this will be discussed next. From the previous methods, we can derive the generalized finite difference scheme

where

for arbitrary parameter. The scheme is second order in time and space for.

It is very easy to see that the previous methods can be recovered by selecting

and, respectively. The resulting system is again a block nonlinear tridiagonal

system which can be solved for the unknown vector by any iterative solver, the fixed point method is adopted in this work.

Since the generalized compact finite difference scheme (23) is nonlinear and implicit, an iterative method is needed to solve it. Two iterative algorithms are implemented to perform this job [11] .

Algorithm 1

where

where the superscript s denotes the sth iterate for solving the nonlinear system of equations for each iteration. The system in (24) can be solved by Crout’s method, where we need only one LU factorization for the block tridiagonal matrix at the beginning, and the solutions of lower and upper bi-diagonal block systems at each iteration are required only .

Algorithm 2

where

where the superscript s denotes the sth iterate for solving the nonlinear system of equations for each time. The block tridiagonal system (25) can be solved by Crout's method. Note that in this method we need to do factorization at each iteration. The initial iterate can be chosen as We apply the iterative schemes till the following condition

is satisfied. The convergence of the iterative schemes, Algorithm 1 and Algorithm 2 is given in [11] .

In this work we are going to present the time splitting method for solving the 3-coupled nonlinear Schrödinger Equation (1). The basic idea in the time splitting method is to decompose the original problem into subproblems, which are simpler than the original problem and then to compose the approximate solution of the original problem by using the exact or approximate solutions of the subproblems in a given sequential order. To display this method for our system

with initial conditions

and the homogenous boundary conditions

The system in (26) can be written as

where

We solve the system (27) from to in two splitting steps. We solve first the linear system equation

with the homogenous Dirichlet boundary conditions

using the finite difference method for the time step k, followed by solving the nonlinear system

for the same time step. Equation (30) can be integrated exactly in time [15] , the exact solution is

To apply this method in systematic way, we combine the splitting steps via the standard second order Strang splitting [21] . The flowchart of this method can be des- cribed by the following steps.

Step 1: Solution of the nonlinear problem

Step 2: Solution of the linear problem

The solution of the linear can be obtained using the generalized difference scheme

The solution of this system can be obtained by solving linear block tridiagonal system with constant coefficients using Crouts method, and this can be executed in very efficient way.

Step 3: Solution of the nonlinear problem

The numerical scheme is of second order accuracy in time, second and fourth order in space for and, respectively. It is unconditionally stable and con-

served the conserved quantities exactly [15] [20] [23] . We denote this method by the time splitting finite difference method by (TSFDM).

In this section, we conduct some typical numerical examples to verify the accuracy, conservation laws, computational efficiency and some physical interaction phenomena described by 3-coupled nonlinear Schrödinger equations.

In this work, we have derived different methods for solving the 3-coupled nonlinear Schrödinger equation using finite difference method and time splitting method with finite difference methods. All schemes are unconditionally stable and highly accurate and conserve the conserved quantities exactly. The interaction of two solitons is discussed in details for different parameters. We have noticed that to have elastic interaction the following constraint

must be satisfied, and for other values the interaction is inelastic, and different behaviors occur (enhancement,suppression) in the amplitude of each soliton. This behavior is in agreement with [1] [2] [3] [4] . The derived methods can be used to solve similar nonlinear problems.

Ismail, M.S. and Alaseri, S.H. (2016) Computational Methods for Three Coupled Nonlinear Schrödinger Equations. Applied Mathematics, 7, 2110- 2131. http://dx.doi.org/10.4236/am.2016.717168