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In this paper we are going to derive two numerical methods for solving the coupled nonlinear Schrodinger-Boussinesq equation. The first method is a nonlinear implicit scheme of second order accuracy in both directions time and space; the scheme is unconditionally stable. The second scheme is a nonlinear implicit scheme of second order accuracy in time and fourth order accuracy in space direction. A generalized method is also derived where the previous schemes can be obtained by some special values of l . The proposed methods will produced a coupled nonlinear tridiagonal system which can be solved by fixed point method. The exact solutions and the conserved quantities for two different tests are used to display the robustness of the proposed schemes.

In this work we are going to derive a highly a accurate schemes for the coupled nonlinear Schrödinger- Boussinesq equations (CSBE)

where

To derive the exact solution of the given system (1)-(2), we assume the solution of the CSBE of the form

where

By substituting (3) and (4) into (1) and (2), and after lengthy calculations, we found that the solution exists if we have the following relations

where

The system (1)-(2) also has a plane wave solution

where A, k, and d are constants.

In order to study the properties of the coupled nonlinear Schrödinger-Boussinesq equation, we consider the initial boundary value problem [

with the initial conditions

and boundary conditions of the forms

where

By introducing the function

and the boundary conditions

the CSBE coupled system can be written as

For the initial-boundary value problem (13)-(14), there are at least three conservation laws [

1) The Langmuir Plasmon number

2) The total perturbed number density

3) The total energy

Physically, these conserved laws play major roles in all physical theories, and can be useful tools for qualitative analysis. Trapezoidal rule and the numerical solution are used to calculate the conserved quantities. The conservation of the conserved quantities for the proposed system using the numerical methods presented in this work is a good indication for the efficiency and robustness these methods.

In order to avoid the complex computation [

so, the CSBE can be written as

where

We will consider the numerical solution of the nonlinear system (20)-(23) in a finite interval

The Crank Nicolson like scheme for the system (20)-(23) can be displayed as follows

where

The scheme in (25)-(28) is a nonlinear implicit scheme with block nonlinear tridiagonal structure. The fixed point method is used to solve this system. The scheme is of second order accuracy in both direction space and time. The scheme is unconditionally stable using the von Neumann stability analysis.

In order to improve the accuracy in space direction, we approximate the second space derivative using the compact approximation

by using this , we will can derive the highly accurate compact finite difference scheme

The scheme in (29)-(32) is a nonlinear implicit scheme of fourth order accuracy in space and second order in time. To obtain the numerical solution, we need to solve a block nonlinear tridiagonal system at each time step. We have done this by using fixed point method. Using von Neumann stability analysis the scheme is also unconditionally stable.

In this subsection we present the generalized finite difference scheme of the form

where

values of

subsection we present a fixed point iterative scheme to solve the nonlinear system obtained.

In order to get the numerical solution for the nonlinear system (33)-(36), we propose the following fixed point iterative scheme of the following form

where

We apply the iterative scheme (37)-(40) until the following condition

is satisfied. Tol is a very small prescribed value.

To study the accuracy of the proposed scheme, we will consider only Equation (33), the other equations can be analyzed in the similar fashion. By replacing the numerical solution

By using Taylor’s series expansions of all terms in Equation (44), we will end with the local truncation error (LTE)

and this indicates that, the scheme is of second order accuracy in space and time directions for arbitrary

accuracy in space and second order accuracy in time.

To prove that the decomposed system

satisfies the conserved quantities

and

we multiply Equation (46) and Equation (47) by

by adding (51) and (52), this will lead us after some manipulation to the following equation

By integrating (53) with respect to x, we get

By imposing the vanishing boundary conditions, Equation (54) will be reduced to

which is Equation (49).

To prove the second conserved quantity , by integrating Equation (48) with respect to x and this will lead us to the following

and by imposing the vanishing boundary conditions, this will lead us to the second conserved quantity (50).

To prove that the proposed schemes preserve the discrete analog of the invariant (49) and (50), we borrow the following lemma [

Lemma 1 For any two discrete functions

where

We will only prove the Crank Nicolson (25)-(28).

we multiply (57) by

or

which is the discrete analog of the conserved quantity (49), this indicates that no blow up in the numerical solution, and it it is a good indication the scheme is unconditionally stable. The second discrete conserved quantity can be easily obtained.

In this section , we will test the proposed schemes for two different problems. The infinity error norm is used to calculate the error, and this can be defined by

Trapezoidal rule is used to approximate the conserved quantities. We will present some numerical results for the solitary wave solutions and the plane wave solutions for the coupled Schrodinger-Boussinesq equations.

In this test, we choose the initial conditions

where

In

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

0 | 0.00000 | 0.000000 | 0.000000 | 0.500000 | −2.000000 | 0.200175 |

2 | 0.000028 | 0.000033 | 0.000080 | 0.500000 | −2.000000 | 0.200155 |

4 | 0.000037 | 0.000005 | 0.000122 | 0.500000 | −2.000000 | 0.200138 |

6 | 0.000068 | 0.000066 | 0.000164 | 0.500000 | −2.000000 | 0.200121 |

8 | 0.000111 | 0.000078 | 0.000211 | 0.500000 | −2.000000 | 0.200121 |

10 | 0.000122 | 0.000118 | 0.000263 | 0.500000 | −2.000000 | 0.200116 |

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

0 | 0.00000 | 0.000000 | 0.000000 | 0.500000 | −2.000000 | 0.200175 |

2 | 0.000003 | 0.000002 | 0.000001 | 0.500000 | −2.000000 | 0.200175 |

4 | 0.000004 | 0.000005 | 0.000001 | 0.500000 | −2.000000 | 0.200175 |

6 | 0.000005 | 0.000011 | 0.000001 | 0.500000 | −2.000000 | 0.200176 |

8 | 0.000004 | 0.000020 | 0.000001 | 0.500000 | −2.000000 | 0.200176 |

10 | 0.000011 | 0.000031 | 0.000001 | 0.500000 | −2.000000 | 0.200175 |

The initial conditions in this case are chosen as

In this test we select the set of parameters

together with the boundary conditions

The conserved quantities and the infinity error norm are given in

In this work, we transform the coupled Schrödinger-Boussinesq equations into a first order differential system in time. We derived two different numerical schemes. Using these methods, a coupled nonlinear block tridiagonal system is obtained. A fixed point iterative method is used to solve this system. The numerical tests and the conserved quantities show the efficiency and robustness of the schemes. To sum up, the proposed schemes are

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

0.0 | 0.000000 | 3.141593 | 6.283185 | 1.571566 |

2.0 | 0.000000 | 3.141593 | 6.283185 | 1.571566 |

4.0 | 0.000000 | 3.141592 | 6.283185 | 1.571566 |

6.0 | 0.000000 | 3.141592 | 6.283185 | 1.571566 |

8.0 | 0.000000 | 3.141592 | 6.283185 | 1.571566 |

10.0 | 0.000000 | 3.141592 | 6.283185 | 1.571566 |

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

0.0 | 0.000000 | 3.141593 | 6.283185 | 1.571566 |

2.0 | 0.000000 | 3.141593 | 6.283185 | 1.571566 |

4.0 | 0.000000 | 3.141593 | 6.283185 | 1.571566 |

6.0 | 0.000000 | 3.141593 | 6.283185 | 1.571567 |

8.0 | 0.000000 | 3.141593 | 6.283185 | 1.571567 |

10.0 | 0.000000 | 3.141593 | 6.283185 | 1.571567 |

reliable and capable to solve like systems.

M. S. Ismail,H. A. Ashi, (2016) A Compact Finite Difference Schemes for Solving the Coupled Nonlinear Schrodinger-Boussinesq Equations. Applied Mathematics,07,605-615. doi: 10.4236/am.2016.77056