Fourier Spectral Method For A Class Of The Fractional Zakharov Systems With A Quantum Correction

The classical Zakharov equation is of great significance in plasma physics.In recent years,some scholars have combined fractional derivatives with Zakharov equation to obtain some generalized forms of fractional Zakharov system.In this paper,Fourier spectral method is applied to solve a class of Zakharov equation with fractional quantum effect.In the first chapter,the research background and physical significance of Zakharov equations are briefly introduced.In the second chapter,a semi-discrete Fourier spectral scheme is established for this kind of space fractional Zakharov equations.It is proved that the space semi-discrete Fourier spectral scheme is conservative.A priori estimate of the approximate solution is given by using the conservative property,and the convergence and stability of the semi-discrete scheme are discussed.In the third chapter,the full discrete Fourier spectral scheme is established based on the semi-discrete scheme in space and Crank-Nicolson discretization in time direction.It is proved that the full discrete scheme is also conservative.Then the stability of the method is analyzed and the unconditional stability of the method is proved.Further,a prior estimation and error analysis of the approximate solution are carried out by using the conservation property.It is proved that the scheme has second-order accuracy in time direction and spectral accuracy in space direction.In the fourth chapter,we use abundant numerical examples to verify the correctness of the theoretical results.Finally,the main work of this paper is summarized.