By V. Kulish
This monograph comprises volumes and offers a unified complete presentation of a brand new hierarchic paradigm and discussions of varied functions of hierarchical tools for nonlinear electrodynamic difficulties. quantity 1 is the 1st e-book, during which a brand new hierarchical version for dynamic non-linear platforms is defined and analysed and a suite of new hierarchical rules is mentioned. the trendy hierarchic asymptotic equipment are set forth systematically, considering particular good points of electrodynamic difficulties, and the phenomenon of hierarchy in electrodynamics, in itself, is completely mentioned from a brand new standpoint. a collection of hierarchical asymptotic calculative equipment of 2 forms is mentioned in aspect. The tools of the 1st sort are destined for asymptotic integration of non-linear differential equations with overall derivatives and with multifrequency (including multi-scale) non-linear correct hand components. those are the Van der Pol approach, Krylov-Bogolyubov technique, Bogolyubov-Zubarev process and their hierarchical types. The tools of the second one kind comprise the tactic of slowly various amplitudes, the strategy of averaged features, the equipment of averaged kinetic and quasihydrodynamic equations, and a few different. those equipment are meant for asymptotic integration of non-linear differential equations with partial derivatives and multifrequency (including multi-scale) correct hand components. designated calculative applied sciences for functional software of all pointed out tools are illustrated through examples of actual electrodynamic platforms (free electron lasers, undulative induction accelerators, structures for transformation of laser signs, etc.).
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Extra info for Hierarchical Methods: Hierarchy and Hierarchical Asymptotic Methods in Electrodynamics
Example text
Both these cases can occur in the above discussed FEL model. For instance, the phase and group velocities of the signal wave in traditional FEL H-ubitron arrangements, as a rule, are coincided [2–4]. However, some other rather exotic physical situations could be realized in some Dopplertron type of FELs. Their characteristic design feature is the use intense electromagnetic waves (including microwaves) as FEL pumping [1]. Therefore the situations with opposite directions of the phase and group velocities can occur really [16, 24, 25].
5) represents the wide class of mathematical objects which are called nonlinear equations. Generalizing, we can say that all equations which contain some nonlinear functions in their right part (in the case above we have the trigonometric sine), could be classified as nonlinear equations. Besides the differential equations the integral and mixed integral–differential nonlinear equations are known also. However, we will take an interest in the differential nonlinear equations only. All physical systems whose dynamics are described by nonlinear equations of any type are called nonlinear dynamical systems.
This characteristic process can be illustrated by nonlinear pendulum. 5). 9). In the second, third, and so on orders of approximation, oscillation harmonics appear in the solution (see further Chapter 5 for more details). The nonlinear pendulum represents the evident example of so called multi-harmonic periodic oscillations. Similarly, higher harmonics are generated really in FEL by pumping and signal oscillation electron phases, simultaneously (see Volume II, for instance). 20)) could be exited, too.