New conservation schemes for the nonlinear Schrödinger equation

New explicit square-conservation schemes of any order for the nonlinear Schrödinger equation are presented. The basic idea is to discrete the space variable of the nonlinear Schrödinger equation approximately so that the resulting semi-discrete equation can be cast into an ordinary differential equation dYdt = A(t, Y)Y, A(t, Y) is a skew symmetry matrix. Then the Lie group methods, which can preserve the modulus square-conservation property of the ordinary differential equation, are applied to the ordinary differential equation. Numerical results show the effective of the Lie group method preserving the modulus square-conservation of the discrete nonlinear Schrödinger equation.