title: Discrete nonlinear Schrödinger equation with defects.
abstract: We investigate the dynamical properties of the one-dimensional discrete nonlinear Schrödinger equation (DNLS) with periodic boundary conditions and with an arbitrary distribution of on-site defects. We study the propagation of a traveling plane wave with momentum k: the dynamics in Fourier space mainly involves two localized states with momenta +/-k (corresponding to a transmitted and a reflected wave). Within a two-mode ansatz in Fourier space, the dynamics of the system maps on a nonrigid pendulum Hamiltonian. The several analytically predicted (and numerically confirmed) regimes include states with a vanishing time average of the rotational states (implying complete reflections and refocusing of the incident wave), oscillations around fixed points (corresponding to quasi-stationary states), and, above a critical value of the nonlinearity, self-trapped states (with the wave traveling almost undisturbed through the impurity). We generalize this treatment to the case of several traveling waves and time-dependent defects. The validity of the two-mode ansatz and the continuum limit of the DNLS are discussed.
authors.affiliation: Theoretical Division and Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA.
authors.forename: A
authors.initials: A
authors.lastname: Trombettoni
authors.forename: A
authors.initials: A
authors.lastname: Smerzi
authors.forename: A R
authors.initials: AR
authors.lastname: Bishop