The transmission through a linear nondispersive–nonlinear dispersive interface is investigated. Propagation-mode and nonpropagation-mode solutions are obtained, and the stabilities of these solutions are checked. A complete figure of the dependence of the transmitted wave (output) on the incident (input) wave shows hysteresis and bistability. This analysis may be used to study similar nonlinear dispersive systems and to search for some nonlinear-optical elements serving as switching, logic, and memory devices.