Finite Difference Solutions of the Nonlinear Schrödinger Equation and Their Conservation of Physical Quantities

Abstract. The solutions of the nonlinear Schrödinger equation are of great importance for ab initio calculations. It can be shown that such solutions conserve a countable number of quantities, the simplest being the local norm square conservation law. Numerical solutions of high quality, especially for long time intervals, must necessarily obey these conservation laws. In this work we first give the conservation laws that can be calculated by means of Lie theory and then critically compare the quality of different finite difference methods that have been proposed in geometric integration with respect to conservation laws. We find that finite difference schemes derived by writing the Schrödinger equation as an (artificial) Hamiltonian system do not necessarily conserve important physical quantities better than other methods.