- m at . o th er ] 2 4 Ju n 20 05 Stationary solutions of the nonlinear Schrödinger equation for the delta - comb

The nonlinear Scrödinger equation is studied for a periodic sequence of deltapotentials (a delta-comb resp. Kronig-Penney potential) or narrow Gaussian potentials. For the delta-comb the time-independent nonlinear Schrödinger equation can be solved analytically in terms of Jacobi elliptic functions and thus provides useful insight into the features of nonlinear eigenstates of periodic potentials. Phenomena well-known from classical chaos are found, such as a bifurcation of periodic eigenstates and a transition to spatial chaos. The nonlinear Bloch bands exhibit fundamentally new features, such as looped and period doubled bands. The relation of these features to the periodic solutions is analyzed in detail, leading to an analytic expression for the critical nonlinearity for the emergence of looped bands. Finally the results for the delta-comb are generalized to a more realistic potential consisting of a periodic sequence of narrow Gaussian peaks. Furthermore the dynamical stability of periodic solutions in a Gaussian comb is discussed.