The nonlinear Schroedinger equation for the delta-comb potential: quasi-classical chaos and bifurcations of periodic stationary solutions

摘要

The nonlinear Schroedinger equation is studied for a periodic sequence ofdelta-potentials (a delta-comb) or narrow Gaussian potentials. For thedelta-comb the time-independent nonlinear Schroedinger equation can be solvedanalytically in terms of Jacobi elliptic functions and thus provides usefulinsight into the features of nonlinear stationary states of periodicpotentials. Phenomena well-known from classical chaos are found, such as abifurcation of periodic stationary states and a transition to spatial chaos.The relation of new features of nonlinear Bloch bands, such as looped andperiod doubled bands, are analyzed in detail. An analytic expression for thecritical nonlinearity for the emergence of looped bands is derived. The resultsfor the delta-comb are generalized to a more realistic potential consisting ofa periodic sequence of narrow Gaussian peaks and the dynamical stability ofperiodic solutions in a Gaussian comb is discussed.