M.P. Coles and D.E. Pelinovsky

Loops of energy bands for Bloch waves in optical lattices

Studies in Applied Mathematics 128 (2012), 300336

We consider stationary Bloch waves in a Bose-Einstein condensate placed in a periodic potential for varying strengths of inter-atomic interactions. Bifurcations of the stationary states are known to occur in this context. These bifurcations generate loops in the energy bands of the Bloch waves near the ends and the center of the Brillouin zone. Using the method of Lyapunov- Schmidt reductions, we show that these bifurcations are of the supercritical pitchfork type. We also characterize the change in stability of the stationary states across the bifurcation point. Analytical results are illustrated by numerical computations.

Gross-Pitaevskii equation, periodic potentials, Bloch waves, energy bands, loop bifurcations, Lyapunov-Shmidt reductions