Zakharov Equations, Fourier Spectral Method

One-dimensional nonlinear Zakharov equations is considered in the paper. First, a semi-discrete Fourier spectral scheme for the equations with initial condition and periodic boundary conditions is constructed. And the conservative property of the semi-discrete spectral scheme is proved.We use the conservative property to get the priori estimate of the approximate solutions. And under some conditions, the con-vergence of the semi-discrete Fourier spectral scheme over a finite time interval[O,T] are obtained.Second, a completely discrete Fourier spectral scheme is constructed for the equations with initial condition and periodic boundary conditions. The the conser-vative property of the completely discrete spectral scheme is proved,too. At last we use the conservative property to get the priori estimate and the error estimate of the approximate solutions.