In this talk we present some results on stochastic symplectic (multi-symplectic) methods for solving stochastic nonlinear Schroedinger (NLS) equation. The equation considered is charge conservative and has the multi-s...
In this talk we present some results on stochastic symplectic (multi-symplectic) methods for solving stochastic nonlinear Schroedinger (NLS) equation. The equation considered is charge conservative and has the multi-symplectic conservation law. Based a stochastic version of variational principle, we show that the phase flow of the equation,considered as an evolution equation, preserves the symplectic structure on the phase space. We give some symplectic integrators and multi-symplectic methods for the equation. By constructing control system and invariant control set, it is proved that a symplectic method, based on the central difference scheme, possesses a unique invariant measure on the unit sphere. Furthermore, the midpoint scheme is applied further to get a full discretization, which possesses the discrete charge conservation law and the discrete multi-symplectic conservation law. Utilizing the Poisson equation corresponding to the finite dimensional approximation, the convergence error between the temporal average of the full discretization and the ergodic limit of the symplectic method is derived (In collaboration with Dr. Chuchu Chen, Dr. Xu Wang and Dr. Liying Zhang).
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