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New conservation schemes for the nonlinear Schrödinger equation. (English) Zbl 1094.65095

Summary: 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 \( \frac {dY}{dt}= A(t,Y)Y, A(t,Y)\) 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.

MSC:

65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
35Q55 NLS equations (nonlinear Schrödinger equations)
65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
34A26 Geometric methods in ordinary differential equations
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References:

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