Structure and stability of two-dimensional Bose-Einstein condensates   under both harmonic and lattice confinement
  In this work, we study pancake-shaped Bose-Einstein condensates confined by both a cylindrically symmetric harmonic potential and an optical lattice with equal periodicity in two orthogonal directions. We first identify the spectrum of the underlying two-dimensional linear problem through multiple-scale techniques. Then, we use the results obtained in the linear limit as a starting point for a nonlinear existence and stability analysis of the lowest energy states, emanating from the linear ones, in the nonlinear problem. Two-parameter continuations of these states are performed for increasing nonlinearity and optical lattice strengths, and their instabilities and temporal evolution are investigated. It is found that the ground state as well as one of the excited states are either stable or weakly unstable for both attractive and repulsive interatomic interactions. 
K.J.H. Law, P.G. Kevrekidis, B.P. Anderson, R. Carretero-Gonzalez, and   D.J. Frantzeskakis