Efficient And Accurate Pseudo-spectral Methods For The Nonlinear Schrodinger Equation With Wave Operater

The nonlinear Schr?dinger equation with wave operator(NLSW)has a wide range of applications in many fields of physics,such as Langmuir wave in plasma,nonlinear optics,and modulation plane pulse approximation of Sine-Gordon equation.So far,the research on the NLSW equation has attracted more and more attention,and many numerical methods have been developed to solve it.In this paper,a Fourier pseudo-spectral method and a Sine pseudo-spectral method are used to numerically study the NLSW equation,and the convergence and error estimates of the two methods are analyzed.For the derivation of the numerical methods,this paper adopts the Fourier pseudo-spectral method and the Sine pseudo-spectral method for the spatial discretization,and uses the linearized implicit finite difference method for the time discretization.And two pseudo-spectral schemes with high-precision are consequently obtained.For the analysis of the proposed numerical methods,firstly,a new discrete energy functional is defined by a recursion formula,based on which,it is proved that the two numerical methods preserve the total energy conservation in a discrete sense.Secondly,by using the standard energy method as well as the mathematical induction method and the "lifting" technique,without any restriction on the grid ratio,it is proved that the two proposed numerical methods are of spectral accuracy in space and second-order accuracy in time,respectively.For the implementation of the proposed numerical methods,with the help of the FFT algorithm,they can be explicitly computed in the practical computation.Large number of numerical results are carried out to verify the error estimates and the efficiency of the proposed numerical methods.