Bose-Einstein condensates in the presence of a magnetic trap and optical lattice

In this paper we consider solutions of a nonlinear Schrodinger equation with a parabolic and a periodic potential motivated from the dynamics of Bose-Einstein condensates. Our starting point is the corresponding linear problem which we analyze through regular perturbation and homogenization techniques. We then use Lyapunov-Schmidt theory to establish the persistence and bifurcation of the linear states in the presence of attractive and repulsive nonlinear inter-particle interactions. Stability of such solutions is also examined and a count is given of the potential real, complex and imaginary eigenvalues with negative Krein signature that such solutions may possess. The results are corroborated with numerical computations.