  We give explicit criteria for the reflectionlessness, transparency, and
invisibility of a finite-range potential in the presence of an incoherent
(intensity-dependent) nonlinearity that is confined to the range of the
potential. This allows us to conduct a systematic study of the effects of such
a nonlinearity on a locally periodic class of finite-range potentials that
display perturbative unidirectional invisibility. We use our general results to
examine the effects of a weak Kerr nonlinearity on the behavior of these
potentials and show that the presence of nonlinearity destroys the
unidirectional invisibility of these potentials. If the strength of the Kerr
nonlinearity is so weak that the first-order perturbation theory is reliable,
the presence of nonlinearity does not affect the unidirectional
reflectionlessness and transmission reciprocity of the potential. We show that
the expected violation of the latter is a second order perturbative effect.
