A Finite Difference Method For The One-dimensional Time-dependent Nonlinear Schr(?)dinger Equation

It is well known that the nonlinear Schr(o|¨)dinger equation has been widely used in various application areas,e.g.,high-energy physics,quantum mechanics,laser bean scan in the nonlinear medium and concentrate problems.However,it is difficult to solve numerically this kind of problems.In this paper,we concern a finite difference method for the nonlinear Schr(o|¨)dinger equation with soliton solution.For the model problem,we propose a two-layer linear finite-difference scheme basing on reduction order -relaxation method,which leads to a truncation error order of O(h~2+Υ~2),whereΥand h are time and spatial steplength respectively.This new scheme has been proved to be uniquely solvable and unconditionally stable.Also,the convergence is partly proved.The numerical example shows that the scheme should be convergent and has optimal convergent orders.