Finite Difference Method For A Class Of The Fractional Zakharov Systems With A Quantum Correction

Since the standard Zakharov system which describes the long wave Langmuir turbulence was proposed by the famous physical mathematician V. E. Zakharov in 1972, it has become a heated topic among scholars. This system is one of the best models in plasma physics community, in which the high-frequency waves are nonlinearly coupled with the low-frequency waves. Scholars have continu-ously improved the Zakharov equations, and proposed various kinds of Zakharov equations according with the experimental results and the physical phenomena,which include the simplified two-fluid model, the magnetic field Zakharov equa-tions and the generalized Zakharov equations etc, and they are the important dynamic models about the interaction between particle and wave.For decades, because of the interest in the study of differential equations of fractional order, some researchers combine the Zakharov system with fractional order derivatives, and propose some classes of Zakharov equations in the nonlocal sense. They studied their well-posedness. On the basis we will consider the numerical method of them, and mainly study the finite difference method of a class of the fractional Zakharov systems with a quantum correction.Firstly, this paper briefly introduced the research background and physical significance of the various types of Zakharov systems and the fractional order dif-ferential equations. In the section 2, a conservative linearly difference scheme for the fractional modified Zakharov system with a quantum correction is proposed.The existence and uniqueness of the numerical solution and the fact that this scheme conserves the mass and energy in the discrete level are proved. In the section 3, the stability and convergence of the scheme are discussed, and the fact that the discrete scheme is unconditionally stable is obtained. On the basis of some priori estimates and inequalities about norms, the solution of the difference scheme converges to the exact solution with order O(?2+h2) in the maximum norm. Lastly, we present the numerical examples. And numerical results are given to demonstrate the accuracy of the difference scheme and the theoretical results. In addition, we show and analyze some of the peculiar physical phenom-ena of Zakharov system in the numerical examples. Particularly, the scheme in this paper has efficient computational capabilities. In fact, the system discussed in this paper is the nonlinear coupled equations, and we usually solve the cou-pling nonlinear algebraic system by iteration, so the computational efficiency is low. But the method in this paper decouples the equations, and transforms the nonlinear term into linear term. During calculations only linear system is solved,and this greatly improves the computational efficiency.