Several Difference Schemes For The Non-linear Schr(?)dinger Equation

In this paper, the conservative finite difference methods for the nonlinear Schr?dinger equation, which exists two conservative laws and solitary solutions are investigated. First we generalized some results about the nonlinear Schr?dinger equation which people have got. Then we proposes three finite difference scheme through adding a dissipation term to the nonlinear Schr?dinger equation, and all of them have error of O (τ~2 + h~2) (whereτis time step andηis space step). It is proved that the new schemes preserves two conservative quantities that the continuous equation owns. And the convergence and stability of these schemes are proved by means of Mr. Zhou Yu-Lin's methods of discrete function analysis. Numerical experiments results show that the precision of the new schemes with suitable parameter are better than the known schemes. At the same time the scheme is computed easily and has higher efficiency. In a word, our schemes are practicable and efficient. Then we get the conclusion that it is a significant job to add a dissipation term.