Shock-like optical modes trapped by the interface of cubic nonlinear media

We report the existence of new families of TE-polarized stationary waves propagating along the interface between two media with Kerr-like nonlinearity. The interfaces between two self-defocusing media and between self-focusing and self-defocusing media are considered. The eigenmodes found have one or two different nonzero backgrounds and represent (in the transverse direction) shock-like waves that are trapped by the interface. Upon constructing these eigenmodes the singular solutions of the corresponding nonlinear governing equations are utilized and the constraints for the modes existence are identified.