Nonlinear scattering and its transfer matrix formulation in one dimension

.We present a systematic formulation of the scattering theory for nonlinear interactions in one dimension and develop a nonlinear generalization of the transfer matrix that has a composition property similar to that of its linear analog. We offer alternative characterizations of spectral singularities, unidirectional reflectionlessness and invisibility, and nonreciprocal transmission for nonlinear scattering systems, and examine the application of our general results in addressing the scattering problem for nonlinear single- and double--function potentials.