Electromagnetic oscillations in a coaxial cavity resonator filled with a nonlinear nondispersive medium

Nonlinear resonators are interesting models of physical systems and have been studied extensively in many theoretical and experimental works. A variety of existing electronic devices and materials with nonlinear electromagnetic properties makes it possible to create electrical resonators with different types of nonlinearity. It is well known that fairly complex, e.g., chaotic, oscillations can be excited in nonlinear resonators. Much previous theoretical work on the subject deals with the analysis of nonlinear resonators as lumped systems or merely as a collection of coupled oscillators or modes. Within the framework of such an approach, the problem of oscillations in a nonlinear resonator is reduced to solving a system of ordinary differential equations. Although such a simplified approach is justified in many cases, it is clear that electromagnetic systems should generally be described by the Maxwell equations. In this work, the problem of free oscillations in a nonlinear electrical resonator is considered using a full set of the Maxwell equations. To solve this problem, we apply the method for constructing exact axisymmetric solutions of the Maxwell equations in a nonlinear nondispersive medium, which has been developed in our recent work (E. Yu. Petrov and A. V. Kudrin, Physical Review Letters, 104, 2010, p. 190404), to the oscillations in a bounded coaxial volume.