Reciprocal conditions in one-dimensional nonlinear wave systems.

We study wave reciprocity in one-dimensional asymmetric systems constructed by multiple nonlinear δ-function scatters embedded within linear scatters. A general reciprocal condition is proposed, in terms of the rotation symmetry between forward and backward transfer matrices. We then derive various resonance conditions, under which all scatterers behave as merging into either a single nonlinear δ scatter, or a symmetric nonlinear barrier configuration. As such, the reciprocity appears periodically by changing widths of linear constant potentials between neighboring nonlinear δ scatters. Moreover, the wave reciprocity will not be violated if one replaces the linear constant potential between two δ-nonlinear scatters with any other kind of transparent scatterers.