Interaction of a soliton with point impurities in an inhomogeneous, discrete nonlinear Schrödinger system

We develop a comprehensive perturbation theory for the inhomogeneous, discrete one-dimensional nonlinear Schrödinger equation based on the inverse scattering transform. We also discuss single-soliton dynamics within the adiabatic approximation and derive higher order corrections to this approximation. Using this perturbation theory, we study in detail the motion of a soliton interacting with a point impurity, either nondissipative or dissipative, in the presence of a spatially linear potential. We predict that there are two kinds of dynamical localization of a soliton in the presence of the nondissipative impurity, depending on the impurity strength. One is the usual dynamical localization, which is qualitatively the same as the one in the absence of the impurity, and the other is the pinning of a soliton by an impurity of sufficient strength. The predictions of these phenomena and their various dynamical properties are confirmed by numerical simulations of the full system.