Semiclassical radiation theory and the inverse method

A relationship between semiclassical radiation theory and the inverse method of solution for nonlinear dispersive waves is developed through two physical examples. The Josephson transmission line is modeled by Maxwell's equations coupled to a phenomenological quantum mechanics. It is shown that this quantum mechanics contains the same linear problem used in the inverse method to solve the sine-Gordon equation, the equation which governs the evolution of the electromagnetic wave. This (nonlinear) wave equation and the linear quantum equations are of equal importance in the physical description of this system. This same relationship exists among the self-induced transparency (SIT) equations of nonlinear optics. This second example, due to Lamb, is discussed in a manner which again displays the precise relationship of the linear problem of the inverse method to the quantum physics. In addition, analogies between SIT and the Josephson transmission line are discussed.