Nonlinear Krönig-Penney model

We study the nonlinear Schrödinger equation with a periodic delta-function potential. This realizes a nonlinear Krönig-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 Krönig-Penney model. We also find 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.

Figure 1

(Color online) Graphical solution of the system of Eqs. (7, 9) for different values of nonlinearity. The allowed values of the chemical potential $\mu$ as a function of momentum $K$ are given by the intersections between the black and the color lines in the filled regions.

Figure 2

Symmetric generalized Bloch states with no nodes (a), and with two nodes (b).

Figure 3

Symmetry broken generalized Bloch states with three nodes (a) and with five nodes (b).

Figure 4

Chemical potential as a function of nonlinearity.