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In this work, we will derive numerical schemes for solving 3-coupled nonlinear Schr ödinger equations using finite difference method and time splitting method combined with finite difference method. The resulting schemes are highly accurate, unconditionally stable. We use the exact single soliton solution and the conserved quantities to check the accuracy and the efficiency of the proposed schemes. Also, we use these methods to study the interaction dynamics of two solitons. It is found that both elastic and inelastic collision can take place under suitable parametric conditions. We have noticed that the inelastic collision of single solitons occurs in two different manners: enhancement or suppression of the amplitude.

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 [

where

with initial conditions

and the homogenous boundary conditions

The exact solution of the 3-coupled nonlinear Schrödinger equation [

where

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 [

To avoid complex computations we assume

where

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

By integrating Equation (18) with respect to x from

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

where h and k are the space and time increments respectively. We denote the exact and numerical solution at the grid point

where

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 [

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 [

where

for arbitrary parameter

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

and

system which can be solved for the unknown vector

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 [

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

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

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

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 [

To apply this method in systematic way, we combine the splitting steps via the standard second order Strang splitting [

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

served the conserved quantities exactly [

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 test, we choose the initial condition as

The following set of parameters are used

The error and the conserved quantities as well as the execution time for all methods are given in Tables 1-6, we have noticed that all method are conserved the conserved quantities exactly and for accuracy the credit goes to the fourth order scheme

Figures 1-3. respectively.

To test the convergent rate in space and time of the proposed schemes. We define the

where

T | |||||
---|---|---|---|---|---|

0.264550 | 0.677249 | 1.058201 | 0.000000 | 0.000000 | 0.0 |

0.264550 | 0.677249 | 1.058201 | 0.007296 | 0.004791 | 2.0 |

0.264550 | 0.677249 | 1.058201 | 0.015365 | 0.009952 | 4.0 |

0.264550 | 0.677249 | 1.058201 | 0.021150 | 0.014046 | 6.0 |

0.264550 | 0.677249 | 1.058201 | 0.028129 | 0.018731 | 8.0 |

0.264550 | 0.677249 | 1.058201 | 0.036204 | 0.023945 | 10.0 |

T | |||||
---|---|---|---|---|---|

0.264550 | 0.677249 | 1.058201 | 0.000000 | 0.000000 | 0.0 |

0.264550 | 0.677249 | 1.058201 | 0.000049 | 0.000035 | 2.0 |

0.264550 | 0.677249 | 1.058201 | 0.000107 | 0.000074 | 4.0 |

0.264550 | 0.677249 | 1.058201 | 0.000154 | 0.000109 | 6.0 |

0.264550 | 0.677249 | 1.058201 | 0.000205 | 0.000145 | 8.0 |

0.264550 | 0.677249 | 1.058201 | 0.000260 | 0.000183 | 10.0 |

T | |||||
---|---|---|---|---|---|

0.264550 | 0.677249 | 1.058201 | 0.000000 | 0.000000 | 0.0 |

0.264550 | 0.677249 | 1.058201 | 0.007296 | 0.004791 | 2.0 |

0.264550 | 0.677249 | 1.058201 | 0.015365 | 0.009951 | 4.0 |

0.264550 | 0.677249 | 1.058201 | 0.021149 | 0.014046 | 6.0 |

0.264550 | 0.677249 | 1.058201 | 0.028128 | 0.018730 | 8.0 |

0.264550 | 0.677249 | 1.058201 | 0.036202 | 0.023944 | 10.0 |

T | |||||
---|---|---|---|---|---|

0.264550 | 0.677249 | 1.058201 | 0.000000 | 0.000000 | 0.0 |

0.264550 | 0.677249 | 1.058201 | 0.000049 | 0.000035 | 2.0 |

0.264550 | 0.677249 | 1.058201 | 0.000107 | 0.000074 | 4.0 |

0.264550 | 0.677249 | 1.058201 | 0.000153 | 0.000109 | 6.0 |

0.264550 | 0.677249 | 1.058201 | 0.000203 | 0.000144 | 8.0 |

0.264550 | 0.677249 | 1.058201 | 0.000259 | 0.000182 | 10.0 |

T | |||||
---|---|---|---|---|---|

0.264550 | 0.677249 | 1.058201 | 0.000000 | 0.000000 | 0.0 |

0.264550 | 0.677249 | 1.058201 | 0.007588 | 0.004977 | 2.0 |

0.264550 | 0.677249 | 1.058201 | 0.015978 | 0.010348 | 4.0 |

0.264550 | 0.677249 | 1.058201 | 0.022000 | 0.014609 | 6.0 |

0.264550 | 0.677249 | 1.058201 | 0.029261 | 0.019486 | 8.0 |

0.264550 | 0.677249 | 1.058201 | 0.037654 | 0.024908 | 10.0 |

T | |||||
---|---|---|---|---|---|

0.264550 | 0.677249 | 1.058201 | 0.000000 | 0.000000 | 0.0 |

0.264550 | 0.677249 | 1.058201 | 0.000251 | 0.000160 | 2.0 |

0.264550 | 0.677249 | 1.058201 | 0.000512 | 0.000326 | 4.0 |

0.264550 | 0.677249 | 1.058201 | 0.000708 | 0.000461 | 6.0 |

0.264550 | 0.677249 | 1.058201 | 0.000936 | 0.000603 | 8.0 |

0.264550 | 0.677249 | 1.058201 | 0.001203 | 0.000773 | 10.0 |

point

To calculate the convergent rate in space, we take the time step k sufficiently small and the error from temporal truncation is relatively small

Rate | Rate | h | ||
---|---|---|---|---|

- | 0.002205 | - | 0.002548 | 0.4 |

4.095 | 0.000192 | 3.936 | 0.000166 | 0.2 |

4.011 | 0.000008 | 4.095 | 0.000010 | 0.1 |

3.000 | 0.000001 | 3.333 | 0.000001 | 0.05 |

we can easily that the rate of convergent is 4 as we claim in this work.

To check the temporal convergent rate, we fix the spatial step h small enough so that the truncation from space is negligible such as h = 0.01. The results are given in

To study the interaction of two solitons with different parameters, we choose the initial condition as a sum of two single solitons of the form

where

For all examples in the case of interaction, we choose the set of parameters

together with different values of

In this test we will consider the two set of parameters (equal and different )

and

For the first set of parameters (36), we have noticed that the amplitudes of soliton 1 and soliton 2 before the interaction are

For the second set of parameters (37), we have noticed that the amplitudes of soliton 1 and soliton 2 before the interaction are

Rate | Rate | h | ||
---|---|---|---|---|

- | 0.006826 | - | 0.009658 | 0.2 |

2.046 | 0.001653 | 1.993 | 0.002426 | 0.1 |

1.993 | 0.000415 | 1.992 | 0.000610 | 0.05 |

1.997 | 0.000104 | 1.986 | 0.000150 | 0.025 |

and in this case we have inelastic collision and we display this scenario in Figures 7-9.

In this test, we choose two different test of parameters

and

For the first set of parameters (38), we have noticed that the amplitudes of soliton 1 and soliton 2 before the interaction are

For the second set of parameters (39), we have noticed that the amplitudes of soliton 1 and soliton 2 before the interaction are

we display the interaction scenario in Figures 13-15.

In this test, we choose the two sets of parameters

and

For the first set of parameters (40), we have noticed that the amplitudes of soliton 1 and soliton 2 before the interaction are

For the second set of parameters, we have noticed that the amplitudes of soliton 1 and soliton 2 before the interaction are

and in this case we have inelastic collision and we display this scenario in Figures 19-21.

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 [

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