  We analyze the existence and stability of nonlinear localized waves described
by the Kronig-Penney model with a nonlinear impurity. We study the properties
of such waves in a homogeneous medium, and then analyze new effects introduced
by periodicity of the medium parameters. In particular, we demonstrate the
existence of a novel type of stable nonlinear band-gap localized states, and
also reveal an important physical mechanism of the oscillatory wave
instabilities associated with the band-gap wave resonances.
