The features of propagation of intense waves are of great interest for theory and experiment in electrodynamics and acoustics. The behavior of nonlinear waves in a bounded volume is of especial importance and, at the same time, is an extremely complicated problem. It seems almost impossible to find a rigorous solution to such a problem even for any model of nonlinearity. We obtain the first exact solution of this type. We present a new method for deriving exact solutions of the Maxwell equations in a nonlinear medium without dispersion and give examples of the obtained solutions that describe propagation of cylindrical electromagnetic waves in a nonlinear nondispersive medium and free electromagnetic oscillations in a cylindrical cavity resonator filled with such a medium. Wave propagation in nonlinear media is a fundamental and wide-ranging problem in physics [1, 2]. The possible self-steepening and formation of shock discontinuities in large-amplitude pressure waves is well known in fluid mechanics, being a typical nonlinear phenomenon. A similar phenomenon (formation of surfaces of discontinuity for the electric and magnetic fields) can also be observed during propagation of intense electromagnetic waves in certain media, and there is an elegant physical analogy between the fluid mechanics and electrodynamics in this case. The discovery of materials with well-pronounced nonlinearity of electromagnetic properties (for example, ferrites and ferroelectrics) has attracted considerable attention to essentially nonlinear electromagnetic phenomena, and some important advances have been made [3,4]. Further theoretical progress, however, has met serious difficulties because such phenomena cannot be described satisfactorily by perturbation theory, and rigorous solutions of field equations are required in order to get theoretical predictions. In view of the above, finding new, physically important exact solutions of nonlinear partial differential equations (PDEs) that describe the behavior of waves in nonlinear media is very topical [5,6]. In most papers on the subject, plane nonlinear waves are considered. At the same time, features of propagation of nonlinear cylindrical and spherical waves, as well as the properties of the related nonlinear PDEs in the corresponding curvilinear coordinates, remain poorly studied.In what follows, we present a new method for constructing exact axisymmetric solutions of the Maxwell equations in a nonlinear nondispersive medium. It is assumed that the medium considered lacks a center of inversion and the dependence of the electric displacement on the electric field can be approximated by an exponential function.Consider electromagnetic fields in a loss-free nonmagnetic medium. With a view to analyzing uniaxial crystals, we assume that the medium possesses an axis of symmetry, hereafter taken as the z axis of a cylindrical coordinate system (r, φ, z). If the fields are independent of φ and z, the Maxwell equations admit solutions in which only the E z and H φ components are nonzero (E wav...

Abstract-A study is made of the characteristics of a perfectly conducting cylindrical antenna insulated from the surrounding cold collisionless magnetoplasma by an isotropic coaxial cylindrical sheath for the case where the antenna is aligned with an external magnetic field and is excited by means of a delta-function voltage generator. A rigorous representation is obtained for the current distribution on an infinitely long antenna. It is shown that in the whistler frequency range, the current distribution of a sufficiently thin antenna is determined mainly by the eigenmode whose guided propagation is found to be supported along the antenna. Based on the results obtained for an infinitely long antenna, a generalized transmission-line theory is developed for determining the current distribution and the input impedance of an insulated antenna of finite length located in a resonant magnetoplasma. The influence of the sheath parameters on the antenna characteristics is analyzed.

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The behavior of electromagnetic fields in nonlinear media has been a topical problem since the discovery of materials with a nonlinearity of electromagnetic properties. The problem of finding exact solutions for the source-excited nonlinear waves in curvilinear coordinates has been regarded as unsolvable for a long time. In this work, we present the first solution of this type for a cylindrically symmetric field excited by a pulsed current filament in a nondispersive medium that is simultaneously inhomogeneous and nonlinear. Assuming that the medium has a power-law permittivity profile in the linear regime and lacks a center of inversion, we derive an exact solution for the electromagnetic field excited by a current filament in such a medium and discuss the properties of this solution.

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