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SPACE-TIME EDGE CATASTROPHES AND UNIFORM ASYMPTPTIC DECISIONS OF THE WAVE EQUATIONS DESCRIBING WAVE PROPAGATION IN COLD PLASMA
Andrey S. Kryukovsky, ANO VO Russian New University, Moscow, Russia
Yuliya I. Bova, ANO VO Russian New University, Moscow, Russia
Abstract
The application of the theory of edge wave catastrophes to the problem of describing the propagation of electromagnetic radiation in a cold plasma in the nonstationary case is investigated. The conditions for the formation of edge singularities in the propagation of a frequency-modulated radio pulse in a homogeneous dispersive medium-the ionosphere characterized by a plasma frequency-are studied. It is assumed that the effective dielectric permittivity of the medium (cold plasma) depends on the circular frequency as a unit minus the ratio of the square of the plasma frequency to the operating frequency. The initial phase of the field is proportional to the carrier frequency multiplied by the initial time of the signal output and the nonlinear function characterizing the frequency modulation of the radio signal. This function, from the point of view of catastrophe theory, is responsible for the space-time caustic focusing of the geometric-optical rays, and from the physical point of view characterizes the compression and decompression of the radio signal. The complex frequency response of the receiver filter is taken into account. A semi-infinite radio signal specified by the Heaviside function is considered. Initially, the solution of the problem is presented in the form of a rapidly oscillating integral with respect to the frequency and the initial time of the output of the signal. When space-time focusing is considered, in addition to the contribution of geometric-optical rays, the contribution of space-time edge rays, generated by the initial point of a semi-infinite radio pulse, plays an important role. Ignoring the contribution of edge rays is possible, with some degree of accuracy, only far from the light-shadow boundary of space-time geometro-optical rays, where the contribution of the edge rays is usually much smaller than the contribution of geometro-optical rays. Focusing occurs when the second derivatives of the phase function with respect to frequency and time are zero It is shown that only one-dimensional caspoid focusings of space-time geometric-optical rays of the A-series are possible. Radiation and caustic structures for simple (zero-modal) catastrophes B, C, F and uniform asymptotics for simple space-time edge singularities and unimodal ones are presented and uniform asymptotics are obtained using special functions of wave catastrophes. In the general case, the uniform asymptotics is expressed by a formula containing a special function of the edge wave catastrophe, its derivatives, a special function of restriction the catastrophe to the boundary, and its derivatives.
Keywords: edge catastrophes, field, wave, frequency modulation, dispersion, uniform asymptotics, plasma, propagation, space-time.
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Information about authors:
Andrey S. Kryukovsky, Doctor of Physics and Mathematics, Professor, Dean of the Faculty of Information Systems and Computer Technologies, ANO VO Russian New University, Moscow, Russia
Yuliya I. Bova, Senior Lecturer, Chair of Information Technologies and Natural Science Disciplines, ANO VO Russian New University, Moscow, Russia