Abstract:
A general algorithm for constructing solutions of the type of Gaussian wave beams and wave packets for linear systems of differential and pseudodifferential equations is discussed. Our main example is a linearized system of equations in a cold plasma in a toroidal region (TOKAMAK). Such solutions are constructed using Maslov's complex sprout theory, developed to construct short-wave or quasi-classical asymptotics with complex phases. The term “quasi-classical” asymptotics is understood in a broad sense: asymptotic solutions of a wide class of evolutionary and stationary scalar and vector partial differential equations describing wave processes (in quantum mechanics, optics, continuum mechanics, etc.) are expressed through solutions of the corresponding equations of classical mechanics - the Hamilton system and its linearization. This, in particular, allows us to use useful and illustrative geometric considerations that are absent in other approaches to constructing Gaussian beams based, for example, on the "parabolic" Fock-Leontovich equation. In addition, our approach allows us to consider problems in which Gaussian beams have focal points. In our opinion, the proposed approach is very practical and leads to effective formulas based on solving systems of ordinary differential equations, and serves as the basis for a fast analytical-numerical algorithm for modeling Gaussian packets and bundles using programs such as Wolfram Mathematica, Maple, MatLab, etc. We will also discuss the possibility of reconstructing the characteristics of a cold plasma in a toroidal region from the parameters of a Gaussian beam passed through it.