Reciprocal conditions in one-dimensional nonlinear wave systems

We study wave reciprocity in one-dimensional asymmetric systems constructed by multiple nonlinear delta-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 delta scatter, or a symmetric nonlinear barrier configuration. As such, the reciprocity appears periodically by changing widths of linear constant potentials between neighboring nonlinear delta scatters. Moreover, the wave reciprocity will not be violated if one replaces the linear constant potential between two delta-nonlinear scatters with any other kind of transparent scatterers.