Loop structure of the lowest Bloch band for a Bose-Einstein condensate

We investigate analytically and numerically Bloch waves for a Bose-Einstein condensate in a sinusoidal external potential. At low densities the dependence of the energy on the quasimomentum is similar to that for a single particle, but at densities greater than a critical one the lowest band becomes triple valued near the boundary of the first Brillouin zone and develops the structure characteristic of the swallowtail catastrophe. We comment on the experimental consequences of this behavior.