Physics 4P06 “Bifurcation of Bloch Waves in the Gross-Pitaevskii Equation”

Stationary Bloch waves are considered in the Gross-Pitaevskii equation with a periodic potential for varying strengths of inter-atomic interactions. Upon a sufficient increase of the inter-atomic interactions one may observe a bifurcation in the number and stability of stationary states. This bifurcation generates 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, the behaviour of stationary states is established close to the linear limit and around the bifurcation value. In particular, the bifurcation for the lowest energy band is shown to be a supercritical pitchfork bifurcation. The change in stability of the stationary states is also examined across the bifurcation point. Analytical results are illustrated by numerical computations for the lowest and excited energy bands.