Conservative Finite Difference Schemes And Fourier-spectral Methods For The Solution Of Certain Nonlinear Schr(?)dinger Equation

Many physical phenomena in quantum mechanics , plasma astrophysics , seismology , acoustics and etc are described by the Schro|¨dinger equation. There has been much works on the Schro|¨dinger equation without nonlinear derivative terms, mainly using the finite difference method. These works have focused on the construction of stable schemes, which preserve different energy identities. However, up to our knowledge, no error analysis is available. On the other side, when the nonlinear derivative terms are involved in the Schro|¨dinger equation, it is more difficult to construct stable schemes preserving the energy conservations. The construction and analysis of such schemes for the nonlinear Schro|¨dinger equation is one of our main goal in this research.Our numerical experiences show, as reported by many others, that the classical explicit schemes to the Schro|¨dinger equation are unstable. It is therefore desirable to construct suitable semi-implicit schemes allowing stable large time-stepping. The second goal of this paper consists of considering the Schro|¨dinger equation involving the nonlinear derivative term, subject to the periodic boundary conditions, and proposing a conservative finite difference scheme in time and Fourier spectral method in space. Rigorous numerical analysis is carried out to derive the energy conservation properties of the full discrete problem.Finally, a series of numerical experiments are performed to support our theoretical claims.