Universal features of self-trapping in nonlinear tight-binding lattices

Abstract

We use the discrete nonlinear Schrödinger (DNLS) equation to show that nonlinear tight-binding lattices of different geometries and dimensionalities display a universal self-trapping behavior. First, we consider the problem of a single nonlinear impurity embedded in various tight-binding lattices, and calculate the minimum nonlinearity strength to form a stationary bound state. For all lattices, we find that this critical nonlinearity parameter (scaled by the energy of the bound state), in terms of the nonlinearity exponent, falls inside a narrow band, which converges to e1/2 asymptotically. Then, we examine the self-trapping dynamics of an excitation, initially localized on the impurity, and compute the critical nonlinearity parameter for abrupt dynamical self-trapping. For a given nonlinearity exponent, this critical parameter, properly scaled, is found to be nearly the same for all lattices. Same results are obtained when generalizing to completely nonlinear lattices, suggesting an underlying self-trapping universality behavior for all nonlinear ~even disordered! tight-binding lattices described by DNLS