Nonlinear impurity modes in a lattice

Discreteness effects on nonlinear impurity modes are discussed in the framework of an approximation given by the integrable lattice nonlinear Schro{}dinger equation (the so-called Ablowitz-Ladik model). It is shown that the nonlinear impurity mode may be treated as an intrinsic localized mode captured by the impurity. Considering a local impurity in the lattice model as a perturbation, analytical results describing interaction of a nonlinear localized mode (as a soliton in the discrete model) with the impurity are presented. It is shown that discreteness may change the character of the soliton-impurity interaction, e.g., the soliton may be trapped by the impurity which is repulsive in the continuum limit.