Bose-Einstein Condensates in Superlattices

We consider the Gross-Pitaevskii (GP) equation in the presence of periodic and quasiperiodic superlattices to study cigar-shaped Bose-Einstein condensates (BECs) in such potentials. We examine spatially extended wavefunctions in the form of modulated amplitude waves (MAWs). With a coherent structure ansatz, we derive amplitude equations describing the evolution of spatially modulated states of the BEC. We then apply second-order multiple scale perturbation theory to study harmonic resonances with respect to a single lattice wavenumber as well as ultrasubharmonic resonances that result from interactions of both wavenumbers of the superlattice. In each case, we determine the resulting system's equilibria, which represent spatially periodic solutions, and subsequently examine the stability of the corresponding solutions by direct simulations of the GP equation, identifying them as typically stable solutions of the model. We then study subharmonic resonances using Hamiltonian perturbation theory, tracing robust, spatio-temporally periodic patterns.