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This monograph consists of two volumes and provides a unified comprehensive presentation of a new hierarchical paradigm and discussions of various applications of hierarchical methods for nonlinear electrodynamic problems. Volume one offers an alternative hierarchical model for dynamic non-linear systems, which is described and analysed and a set of hierarchical principles is discussed. The modern hierarchical asymptotic methods are set forth systematically, taking into account specific features of electrodynamic problems, and the phenomenon of hierarchy in electrodynamics, in itself, is thoroughly discussed from a new point of view. A set of hierarchical asymptotic calculative methods of two types is discussed in detail. The methods of the first type are destined for asymptotic integration of non-linear differential equations with total derivatives and with multifrequency (including multi-scale) non-linear right hand parts. These are the Van der Pol method, Krylov-Bogolyubov method, Bogolyubov-Zubarev method and their hierarchical versions. The methods of the second type include the method of slowly varying amplitudes, the method of averaged characteristics, the methods of averaged kinetic and quasihydrodynamic equations, and some others. These methods are intended for asymptotic integration of non-linear differential equations with partial derivatives and multifrequency (including multi-scale) right hand parts. Detailed calculative technologies for practical application of all mentioned methods are illustrated by examples of real electrodynamic systems (free electron lasers, undulative induction accelerators, systems for transformation of laser signals, and so forth). Volume two describes and analyses a hierarchical treatment of Undulative Electrodynamic Systems. The hierarchical one-particle and self-consistent theory of Undulative Eletrodynamic Systems of different types is set forth. Those include, the single-particle theory of EH-accelerators, EH-formers, EH-coolers, traditional (parametrical) Free Electron Lasers (FELs) wi H-ubitron, Dopplertron and EH-ubitron pumping which are exposed from the hierarchical point of view. The self-consistent nonlinear hierarchical theory of parametrical and two-stream superheterodyne FELs is developed. Special attention is paid to phenomena such as explosive instability, phase and polarization discrimination, nonlinear generation of electric and magnetic fields, superheterodyne amplification of electromagnetic waves. The nonlinear theory of multi-harmonic two-stream superheterodyne FELs is given as a demonstration of high practical efficacy of the presented hierarchical methods.
