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

where, represents the complex short wave amplitude, represents the long wave amplitude, and are real parameters. Equations (1) and (2) were considered as a model of the inter- actions between short and intermediate long waves, and were originated in describing the dynamics of Langmuir soliton formation, the interaction in plasma [1] - [4] . Numerical solution of coupled nonlinear Schrödinger equation using different methods can be found in [5] - [8] . Few numerical methods exist in literature for solving the CSBE. Zhang et al. [9] derived a conservative difference scheme to solve the CSBE. Bai et al. [1] [2] proposed the time splitting Fourier spectral method and the quadratic B-spline finite element method for solving the CSBE. Recently, a multi-symplectic scheme for solving the CSBE is developed in [10] .

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 and c are arbitrary constant.

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 [1] [10]

with the initial conditions

and boundary conditions of the forms

where and are given smooth functions and these functions can be extracted from the exact solution, and

By introducing the function such that

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 [10] .

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 [5] - [7] we assume

so, the CSBE can be written as

where

We will consider the numerical solution of the nonlinear system (20)-(23) in a finite interval We assume where and h is the space grid step size, also we assume where k is the time grid step size, we assume We denote the exact and numerical solutions at the grid point by and, respectively.

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

and

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 the exact solution to get

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

It is very interesting to notice that for the local truncation will be reduced to the fourth order

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 and respectively to get

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 [8] [11] .

Lemma 1 For any two discrete functions and

there is the identity

where

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

we multiply (57) by and (58) by and then by adding the resulting equations to obtain

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 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

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