1-Soliton Solution for Zakharov Equations

Transcription

1 Adv. Studies Theor. Phys., Vol. 6, 2012, no. 10, Soliton Solution for Zakharov Equations Bouthina S. Ahmed Department of Mathematics, Collage of Girls Ain Shams University, Cairo, Egypt Abstract The Zakharov equations are one of the important mathematical models in plasma physics. The solitary wave ansatz method is applied to obtain the exact solution. We use a finite difference scheme which is fourth order in space and second order in time to get a numerical solution. We show that the scheme is unconditionally stable. Numerical tests for single and interaction of two solitary waves are given to show the validity of the methods. Keywords: Zakharove equations, ansatz method, soliton solutions, finite difference methods 1 Introduction The evolution equation describing the interaction between Langmuir and ion acoustic waves in plasma [15] are where is a complex unknown function represent the slowly varying envelope of the highly oscillatory electric field, and ( ) a real scalar function denotes the fluctuation of the ion density about its equilibrium value. We supplement (1) by prescribing the initial-boundary value conditions:

2 458 B. S. Ahmed The theoretical properties of the Zakharov equations, such as the existence and uniqueness of smooth solution, and collision of solution have been studied by several authors [2, 5, 7, 8]. There have been many numerical methods used to solve the (generalized) Zakharov equations. Numerical studies include Fourier spectral method [12] time-splitting spectral methods [1, 13], conservative finite difference schemes [3, 4, 6], etc. In this paper we use the ansatz method to get the exact solutions of eq. (1) in section 2. In section 3 we derive the numerical solution by the finite difference scheme. In section 4 we discuss the accuracy of the scheme and we prove that the scheme is of fourth order in space and second order in time. In section 5 we show that the scheme is unconditionally stable by using a linear stability analysis. Numerical tests for single and two interaction solitary waves are given to show the validity of the methods in section 6. 2 Mathematical analysis In order to get the exact solution of (1) we use the ansatz method [9, 11, 14] we suppose that where and the amplitudes of the and solution respectively, where B is the inverse width of the solution and q is the soliton velocity. In (3) represents the phase of the solution that is defined as

3 1-Soliton solution for Zakharov equations 459 Substituting from (7) (11) into (1) we get By setting the imaginary part to zero in (12) we get (14) By balancing the power of and in (12) we have By balancing the power of and in (13) we get By setting the coefficients of to zero in (12) we get Also, from (12) we have By setting the coefficients of and to zero in (13) we get Substituting in (3) and (4) we have, 3 Numerical method We compose the function in its real parts and imaginary parts by writing

4 460 B. S. Ahmed Also we set Substituting in (1) we get where, 3.1 Finite difference method The and coordinates are discretized by a grid spacing a time step. This gives the the grid points with and, used to denote an approximation solution used to the exact solution. The finite difference scheme of (22). where, 4 Accuracy of the scheme To study the accuracy of the scheme we replace the numerical solution exact solution in (23). Doing this the proposed scheme will be of the form by the (24)

5 1-Soliton solution for Zakharov equations 461 where, Taylor expansion of all terms in (24) can be displayed as follows Substituting from (25)-(27) in (24) we get By using the differential eq. (22) all terms inside the brackets equal zero, then we have We deduce that the proposed scheme is a second order in time and a fourth order in space, it is consistent since the local truncation error in (28) tends to zero as and tends to zero.

6 462 B. S. Ahmed 5 Stability To study the stability of the proposed scheme, the von Neumann stability analysis will be used. This method can only be applied for linear scheme. By freezing all terms which make the scheme nonlinear [10], then (22) has the form where The finite difference scheme of (29) is + where. Assume that be the test function β R R and R be the amplification matrix. The necessary condition for stability of the scheme is Substituting in (31) we have where The eigenvalues of the matrix are

7 1-Soliton solution for Zakharov equations 463, It is clear that all the modulus of the eigenvalues equal 1, then the scheme is unconditionally stable, the scheme is also consistent, according to Lax theorem the scheme is convergent. 6 Numerical results To investigate the performance of the proposed scheme, we study two tests, single solitary and interaction of two solitary wave. The accuracy of the scheme is measured by calculating, error of and which are defined as 6.1 Single solitary wave We begin our computation by simulating the evolution of single soliton. In this test we take the initial condition as The single solitary wave is located at and moving to the right with spatial speed and. We set the spatial domain to be considered and the interval Figure 1. Table 1 show that the error is small as we expected, the numerical method is higher order accuracy.

8 464 B. S. Ahmed T E E E E E E E E E E E E E E E E E E E E E E E E-05 Table 1 : Accuracy of the method h=0.1, k=0.01, B=0.5, q=0.5 Figure 1 : Single solton solution left right 6.2 Two solitary waves In this test we take the intersection of two solitary waves, where the intial condition are assumed of the form Case 1 : Interaction of two solitary solution with equal amplitude and opposite velocities, we set and located at. The computation is performed with and time. Two solitary waves of equal amplitude are placed along the x-axis, one on the left is moving in the right direction, while the other is moving in the left direction. Figure 2 display the numerical solutions for the interaction process. We can say that the two solitary waves keep their own shapes and velocities unchanged after interacation.

9 1-Soliton solution for Zakharov equations 465 Case 2 : Interaction of two solitary waves with different speed and opposite direction. We set and located at. The computation is performed with and time. Figure 3 display the numerical solution of interacation. Figure 2: Interaction of two solitary waves case 1, left right Figure 3: Interaction of two solitary waves case 2, left right 7 Conclusions In this paper we used the ansatz method to obtain the exact solution of Zakharov equations. We derive the numerical solution by the finite difference scheme which is of fourth order in space and second order in time. The accuracy of the scheme is discussed; also we show that the scheme is unconditionally stable. Numerical tests for single and two interaction solitary waves are given to show the validity of the methods. References [1] W. Bao, F.F. Sun, G.W.Wei, Numerical methods for the generalized Zakharov system,

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