Ray Methods and Special Functions of Wave Catastrophes

New asymptotic methods for solving problems of diffraction, focusing and propagation of electromagnetic waves in inhomogeneous anisotropic media are considered: the bicharacteristic method and the wave catastrophe theory. Hamilton-Lukin's bicharacteristic method was used to perform mathematical modeling of ray and caustic structures arising from the propagation of radio waves in the Earth's ionosphere. The results of mathematical modeling of caustic sections of the main, edge and corner wave catastrophes are given. Uniform asymptotic solutions of wave problems leading to special functions of wave catastrophes are considered, and amplitude-phase structures of wave catastrophe special functions are presented.