Dynamics and Stability of Bose-Einstein Condensates: The Nonlinear Schrödinger Equation with Periodic Potential

The cubic nonlinear Schrödinger equation with a lattice potential is used to model a periodic dilute-gas Bose-Einstein condensate. Both two- and three-dimensional condensates are considered, for atomic species with either repulsive or attractive interactions. A family of exact solutions and corresponding potential is presented in terms of elliptic functions. There are two main topics I'm going to touch in the talk: The first is the linear stability analysis of the solutions and the second one is the idea "ground state" for nonlinear equations.