Period doubling, two-color lattices, and the growth of swallowtails in Bose-Einstein condensates

The band structure of a Bose-Einstein condensate is studied for lattice traps of sinusoidal, Jacobi elliptic, and Kronig-Penney form, all in the context of the nonlinear Schr\"odinger equation. It is demonstrated that the physical properties of the system are independent of the type of lattice. The Kronig-Penney potential, which admits a full exact solution in closed analytical form, is then used to understand the swallowtails, or loops, that form in the band structure. The appearance of swallowtails is explained by adiabatically tuning a second lattice with half the period. Such a two-color lattice, which can be easily realized in experiments, has intriguing physical properties. For instance, swallowtails appear even for weak nonlinearity, which is the experimental regime. We determine the stability properties of this system and relate them to current experiments.