Mixed boundary-value problems for the two-dimensional Schrِdinger-Poisson system

The time-dependent two-dimensional nonlinear Schrِdinger-Poisson system is studied on a rectangle with various boundary conditions including those of Floquet type. Using the Galerkin approximation method with bases chosen according to the boundary conditions, it is shown that the problems are well-posed and that the solutions can be approximated by the Galerkin sequence.