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In this paper we analyze the existence, stability, dynamical formation andmobility properties of localized solutions in a one-dimensional systemdescribed by the discrete nonlinear Schr\"{o}dinger equation with a linearpoint defect. We consider both attractive and repulsive defects in a focusinglattice. Among our main findings are: a) the destabilization of the on--sitemode centered at the defect in the repulsive case; b) the disappearance oflocalized modes in the vicinity of the defect due to saddle-node bifurcationsfor sufficiently strong defects of either type; c) the decrease of theamplitude formation threshold for attractive and its increase for repulsivedefects; and d) the detailed elucidation as a function of initial speed anddefect strength of the different regimes (trapping, trapping and reflection,pure reflection and pure transmission) of interaction of a moving localizedmode with the defect.

Solitons in one-dimensional nonlinear Schr\"{o}dinger lattices with a local inhomogeneity