We study the nonlinear Schr\"odinger equation with a periodic delta-function potential. This realizes a nonlinear Kr\"onig-Penney model, with physical applications in the context of trapped Bose-Einstein condensate alkaly gases and in the transmission of signals in optical fibers. We find analytical solutions of zero-current Bloch states. Such wave-functions have the same periodicity of the potential, and, in the linear limit, reduce to the Bloch functions of the Kr\"onig-Penney model. We also find new classes of solutions having a periodicity different from that of the external potential. We calculate the chemical potential of such states and compare it with the linear excitation spectrum.
Categories: cond-mat.other