New Acceleration Principle of Charged Particles for ... - Springer Link

International Journal of Infrared and Millimeter Waves, Vol. 19, No. 1, 1998

NEW ACCELERATION PRINCIPLE OF CHARGED PARTICLES FOR ELECTRONIC APPLICATIONS. THE GENERAL HIERARCHIC DESCRIPTION Victor Kulish,3 Peter B. Kosel,2 and Alexander G. Kailyuk1 1

Sumy State University 2 Rymskii-Korsakov Street Sumy 244007 Ukraine 2 University of Cincinnati Cincinnati, Ohio 45221-0030, USA 3

Temporary address: P.O. Box 84 Worthington Ohio 43085, USA

Received October 13, 1997 ABSTRACT A new principle of acceleration of changed particles and quasineutral plasma beams is proposed and theoretically substantiated. The essence of this new acceleration principle is die utilization of EH-undulated fields, which resemble those of EH freeelectron laser pumping systems. Here the charged particles are moving in a superposition of crossed magnetic and electric vortex undulated fields (EH-accelerator). The advantage of such systems is that both negative (electrons) and positive (ions) charged particles are accelerated simultaneously in the same longitudinal direction. In addition to the concept of the EH-accelerators, a new theoretical approach (the theory of hierarchic oscillations and waves) is given further development here. This approach has been used as a basis for the general nonlinear theory of the EH-accelerators and some other similar isochronous electronic devices with long-time interactions. In addition, several new calculation methods are presented, including the method for nonlinear current density calculations (called the averaged current-density equation hierarchic method) and two versions of hierarchic asymptotic algorithms for the integration of Maxwell's equations. INTRODUCTION

The subject of this paper is somewhat unusual because the results reported here were originally obtained in another scientific field. Namely, the original effort involved the authors' research into new schemes for free electron laser (PEL) pumping systems [1,2]. In the course of that work it was discovered that some specific FEL-pumping systems, such as the EHubitron [2] can be considered as an effective means for accelerating charged particles [3], and, moreover, for the acceleration of quasineutral plasmas [461. Henceforth, our efforts have concentrated on the development of a more 33 0195-927l/98/0100-0033$15.00/0 * 1998 Plenum Publishing Corporation

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Kulish, Kosel, and Kailyuk

general and more detailed theory of a class of accelerators of which we theoretically describe here a variation of the FEL. The main purpose of this paper is to present and further develop a concept of a new class of accelerators for plasma bunches and charged particle beams. The key feature of the new accelerator scheme is the use of a special configuration of crossed transverse periodically-reversed magnetic and vortex electric fields. We refer to such systems as EH-accelerators and to the corresponding field configurations as EH-undulating fields. The major feature, which makes the EH-accelerator attractive, is its potential for widespread applications. This warrants an extensive study of the ideas presented here. In this regard we will consider three aspects. The first aspect relates to the general scientific merit of the proposed new particle acceleration principle. It is well known that the number of efficient means for obtaining accelerated particles is rather limited. Therefore, the appearance of any new principle of acceleration may be considered as a significant event. Since particle accelerators are used in a variety of situations in scientific and technological research, they exert important influences on the developments of new methods for the fundamental studies of physical phenomena in different fields of investigation. The second aspect relates to the potential impact of the material presented here on the fundamental knowledge of some fields of advanced physics. In particular, the theoretical formalism developed here as a general theory of EH-accelerators provides new possibilities for a wide range of scientific branches. For example, for the general description of the nonlinear interactions of single particles and plasma-like beams (charged or quasineutral) with complex electromagnetic field configurations [4-6]. Practical application of this improved approach allows us to discover a number of new interesting physical phenomena. For instance, in the course of this work we have discovered such a new (and unusual) phenomena as the capability of simultaneous acceleration of electrons and ions in the same longitudinal direction, the "cooling" of a beam of charged particles during the acceleration process, the transverse isochronization of interaction in the EH-FELs, and some others. However, it should be noted that of the third (applied) aspects the proposed acceleration principle has the most important significance. But, more detail about this is given below. As an obvious example we propose new ways for solving a number of urgent problems in modern accelerator and relativistic electronic technologies [7]. It is well known that main existing types of accelerators can be categorized into three basic types according to physical acceleration mechanisms that are in use. These are the linear acting accelerators, cyclic

Acceleration Principle of Charged Particles

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accelerators and the linear-resonant systems. Other more exotic systems, such as the plasma-wave accelerator and the oppositely directed storage ring devices are not considered here as comparable accelerator types since they do not play significant roles in discussed here modern technological applications. The basic principles of operation of directed linear-acting particle accelerators could be illustrated by the simple scheme given in Fig.l. The electrostatic and induction accelerators belong to this group. It is well known that electrostatic accelerators are high-voltage systems, which very often turn out to be inconvenient for practice and, like the linear induction accelerators, they are unsuitable for the acceleration of quasineutral plasmas. Besides that, both systems are characterized by relatively low rates of acceleration. There are the magneto-hydrodynamic plasma systems also. Therein, they have some possibilities for acceleration of plasma beams. But, the magneto-hydrodynamic systems do not provide a means for achieving relativistic speeds of the plasma fluxes.

Fig.l Illustration a motion of negative (-e) and positive (+e) charged particles in a longitudinal electric field. Here 1,2 are the trajectories of positive (1) and negative (2) charged particles, respectively, 3 is the line of force of the electric field, E is the vector of the electric field intensity, and u are the particle velocities.

Some similar situation exists in the case of the cyclic systems. The latter include all types of cyclotrons, phasotrons, microtrons, betatrons, etc., which are not fit for accelerating of plasmas, too. The linear-resonant systems (which include all traditional radio frequency driven devices, the surfotrons, etc.) offer some theoretical possibilities for the acceleration of

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Kulish, Kosel, and Kaityuk

quasineutral plasma bunches. But in practice this can not technologicallyeasy be realized [8,9]. The main drawback here is the unrealistically high amplitudes of the accelerated electromagnetic wave fields, which are required here. Thus, it is safe to expect that EH-accelerator systems offer the means for acceleration of plasma beams to relativistic speeds without any competition. Besides that, these systems possess an inherent simplicity of design, at that, even in those cases where they are required to work as accelerators for charged particles or charged plasma beams. This simplicity is inherent in the fact that only a system of electromagnets is required for their operation (see Fig.2). Calculations show (see also the second part of this work) that the proposed EH-accelerator can exhibit relatively high rates of acceleration while maintaining rather moderate dimensions, weight and expense.

Fig.2

A schematic representation of the fields and trajectories of negative (-e)and positive (+e) charged particles in EH-undulated field system. Here: 1 are electromagnets, 2 is the trajectory of a negative particle, 3 is a trajectory of a positive particle, 4 is the magnetic field B , 5 is the electric field E, and u are the particle velocities.

The potential uses of EH-accelerators are very broad. For instance, low and moderately energy systems can be used in electronic materials processing (for deposition, etching, alloying, surface hardening, and film growth, etc.). In this regard we look to the possibility of new studies which make use of the combined syncretic action of the simultaneous impact of both the negative and positive components of a quasineutral plasma beam. For this application an EH-plasma beam accelerator can provide a very


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narrow spread of thermal particle velocities and, at the same time, it allows to realize good control of locality of an anisotropic action. [6], Moreover, this effort may lead to new discoveries of new reaction mechanisms and the realization of corresponding new technological advances in materials processing. The prospects for the use of electron EH-accelerators merit a separate consideration. For instance, such accelerators could be used both as sources of high-quality electron beam devices such as the isochronous pumping systems in free electron lasers (FELs) [3]. There also exist possible applications of the EH-plasma-accelerators which are significantly more novel and, at first glance, somewhat fantastic. However, preliminary numerical simulation studies and analyses show that evens these applications are quite possible, in principle. For instance, such EH-accelerator system could be configured into a new type of engine for long-distance space travel (for example, from Earth to Mars) or into a new scheme of plasma heating and confinement systems for plasma fusion reactors. The basis to formulating a description and an understanding of EHaccelerator systems has been the hierarchic theory of oscillations and waves [10-13]. However, the complete form of that theory has nowhere yet been given a complete presentation. Therefore, in this paper we undertake the further development of the hierarchic theory. As part of this, a new method for the asymptotic integration of Maxwell's equations is presented. Therein, the EH-accelerator systems are used as examples of its applications. The general theory, which is elaborated here, turn out to be fit for application to other branches of electrodynamics also. In particular, to general nonlinear theory of relativistic electronic devices with long-time interactions. In turn, the theoretical results of these applications can be used for analysis of electron beam systems of other types, too. So, the first stage of our study involves the elaboration of the general hierarchic theory for investigation of electrodynamic nonlinear phenomena, and to show that the this theory gives a powerful means for the quantitative analysis of long-time interacted electronic devices, in particular, the EHaccelerators. In the second part of this article we will use this theory for detail analysis of some concrete models of EH-accelerators and EH-free electron lasers (EH-FELs). The short total variant of this dual article has been delivered in the form of four oral presentations at the 22nd International Conference on Infrared and Millimeter Waves, (20-25 July 1997), Wintergreen Virginia, USA. Besides that, the separate fragments of its have been presented at the 11th IEEE International Pulsed Power Conference, (June 29 - July 2,

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Kulish, Kosel, and Katlyuk

1997), Baltimore, Maryland, USA and at the Symposium on Causality and Locality in Modern Physics and Astronomy: Open Questions and Possible Solutions, (August 25 - 29,1997), York University, North York, Ontario, Canada. PART1 GENERAL HIERARCHIC THEORY OF EHACCELERATORS AND SIMILAR SYSTEMS 1. PRINCIPLES OF OPERATION OF EH-ACCELERATORS As stated above, the essence of the new method of acceleration consists of the use of a special arrangement of transverse crossed undulating magnetic and electric vortex fields (EH-undulated). We, therefore, begin here with a discussion of the peculiarities of EH-undulated fields and the ways for forming of its. 1.1. Ways of forming of EH-undulated fields The simplest example of an EH-undulated field system is shown in Figs. 2 and 3. The Fig. 2 depicts the fields and particle trajectories in threedimension space. This arrangement of fields, in principle, is similar to that used in traditional pumping systems of free-electron lasers (FELs) [1416,48]. However, there are essential differences, too. Namely, instead of the permanent magnets the special electromagnetic coils must be used in our case. Those should be driven with radio frequency (RF) current sources. This feature provides the time varying magnetic fields between electromagnetic poles [2]. As a consequence of the sinusoidal time-variation of the magnetic field a periodically varying vortex electric field is generated by electromagnetic induction. Therein this electric field inherits a spatial periodicity by virtue of the spatial periodicity of the magnetic component of the EH-field. The vector of electric field intensity E here is perpendicular to the magnetic induction vector B. This arrangement is reinforced in the plan view shown in Fig. 3. Here shows the plan view of either the top or bottom faces of the electromagnetic poles that produce the periodically reversed B fields. The latter are perpendicular to the plane of the drawing always. Therewith, the opposite directions of this vector are shown by the dots and crosses enclosed by small circles. The lines of force (strength lines) representing the induced vortex electric fields occur in the plane of the drawing. It should be readily seen that along the axis of symmetry of the

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drawing, the occurrence of the magnetic and electric crossed field reversals is consistent with that required to work form of an undulated field pattern of an EH-accelerator. A charged particle injected from the left will move along a sinewavelike trajectory under the influence of the crossed fields (see Fig.2). To realize a net acceleration effect on the particle the spatial phases of oscillation of the magnetic and electric fields need to be displaced by one quarter of the spatial period relative to each other. It is easy to see that this condition is automatically satisfied by the EH-undulated field arrangement. It should be noted that this field arrangement for the production of the EH-undulated fields is not the only conceivable one. In the general case, both the electric and magnetic fields could be created independently by separate means of excitation. In this case these fields can have different periods of oscillation and spatial phases distinguished of ±DDD. (An example design for separately excited undulated vortex electric fields (E-undulated) has been discussed in [3]; and examples of sources of undulated magnetic (Hundulated) fields can be found in ordinary permanent magnetic wiggler systems [3,14-16,48]). However, the analysis shows that the above-proposed scheme of the EH-undulated field of an accelerator system is the most promising for practical implementation. Nevertheless, for the sake of generality the theory developed below we will assume that, in the general case, the fields of the EH-ubitron are independently created by separate sources.

Fig. 3 Plan view representation of the EH-undulated field pattern. Here: 1 is the flow of a ribbontype charged particle beam, 2 are the lines of force of the vortex electric fields, 3 are the pole faces of the electromagnets, and 4 are the directions of the magnetic field directions.

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1.2. Comparison of charged particle motions in longitudinal electric and the transverse EH-undulated fields In this section we make a short qualitative comparison of the processes of motion of charged particles in both a classical longitudinal electric field and the transverse crossed fields of the EH-accelerator. This way is found very convenient for an explanation of peculiarities and possibilities of the proposed acceleration idea. First consider the case of the longitudinal electric field accelerator, which is the simplest form of linear accelerator of direct action. Under the influence of the E field both negative (electron) and positive (ion) particles are forced to move. This situation is illustrated in Fig. 1. It is easy to see in this case that the lines of electric field strength and the trajectories of the charged particles are straight and parallel. In addition, the positive and negative particles are accelerated in opposite directions. Consequently, the possibility of accelerating a plasma bunch (consisting of oppositely charged particle types) is impossible in this case. Now consider the acceleration process in the EH-undulated field. The arrangement of the field vectors and the trajectories of oppositely charged particles are depicted in Fig. 2. Here the magnetic Lorentz force causes the particles to follow sinusoidal trajectories. That is, the electric field lines of force acting on the particles have sinewave-like shapes in the plane coinciding with the plane of the trajectories. This plane lies between the magnetic pole faces. However, at any single local point of the trajectory the action of the local electric field is identical to that in Fig.l. That is, the negatively charged particle moves (and accelerates) against the direction of the electric field line offeree (i.e. the intensity E) while the positive particle moves (and accelerates) in the same direction as the vector E. However, the presence of the local B field causes the paths of both particles to be bent in the same longitudinal direction, which is perpendicular to the E vectors and the B field at the same time. So, in the laboratory coordinate frame of reference the charged particles actually undergo oscillatory motion in mutually opposite directions (depending on the sign of their charge) while simultaneously moving in the same longitudinal direction along the system axis. In the proper (non-inertial) coordinate frame of reference the motion of the particles occur along perpendicular direction that is similar to motion of the particles in a model with longitudinal electric field. Therefore, the major peculiarity of the present acceleration method is the ability to accelerate a plasma bunch as a whole without causing complete separation of the oppositely charged particle species.


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1.3. Free electron lasers based on the EH-accelerator As discussed above, while charged particles are accelerated by the action of the EH-undulated fields their trajectories turn out to be sinwavelike in shape. But this kind of trajectory is essential to the physical mechanism of operation of a free electron laser (PEL). Consequently, the EH-undulated field accelerator can also be used as a means of pumping an EH-FEL [2,3,38]. Such a construction of a FEL not only provides the necessary undulating field characteristics but also provides acceleration of the electron beam to compensate of its energy losses. The latter is associated with the generation of the coherent radiation, which constitutes the electromagnetic signal from the relativistic electron beam. This new EH-FEL scheme, therefore, gives a way of overcoming the physical mechanisms of saturation found in simple FEL designs [3]. In subsequent work we will extend our considerations of the EH-FEL systems. Further in this paper of the joint article we will concentrate on the development of the general theoretical framework, which facilitates the description and analysis of all accelerator systems based on the EHundulated field principle. 2. THE HIERARCHIC METHOD OF ANALYSIS

Mankind has been committed to live in a hierarchic world. Each individual occupies some place in a hierarchic social structure. In places of worship he is reminded that the Supreme Being created our world according to a hierarchic principle, that is, the arrangements of the Universe and man's social order follow similar hierarchic patterns. In everyday life one is constantly reminded of the presence of a hierarchic order be it at the regional, state or national levels, and that all events and situations have a significance depending on the level they occupy in their respective hierarchic system of things. It is rather unexpected that something similar is found in modern physics and technology [1-5]. As in condensed matter physics [3,4], and in systems and coding theories [1,2], there the Fibonacci or Cayley hierarchic trees exist, for instance. The phenomenon of hierarchy has been studied in the theory of oscillations and waves [11, 39, 40] also. But here electrodynamic problems of relativistic electronics did not studied (excepted the [11] and some others), as rule. Besides that, some common mathematical basis for description of hierarchic systems of different physical nature is not demonstrated.

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It is interesting to note that the notion of so modern scientific concept, as it is hierarchy, indeed has an ancient origin. A most vivid abstraction of hierarchy was illustrated by the cosmological representation of Kabbalah [6,7] where the complex idea of a "tree of life" is introduced. According to this concept the Supreme Being created our world (universe and mankind) from the highest hierarchic levels to the lowest ones. Similar ideas are also found in Rosenkeuz's cosmological theory [8] and in some ancient oriental philosophies [9], etc. It is interesting to note that all these ancient references regard our world as a complex (nonlinear) oscillation-wave system that is very unusual for tradition perception of a modern physicist. In this work we at first formulate and extend a new general hierarchic concept to the general theory of dynamic hierarchic systems and to nonlinear problems of electrodynamics, in particular. We begin the wording of our version of theory with the statement that everything in the universe and human society has a hierarchic nature. This admission we regard as the general hierarchic principle. So, all objects in Universe, including our EH-accelerators and other electron devices, should be considered as relevant hierarchic systems. Here it should be mentioned that the first use of the hierarchic approach to problems of nonlinear relativistic electrodynamics was reported in reference [1] for the case of the single-particle theory of free electron lasers, and in reference [11] for the wave-resonant model of an electron beam-system. Later these ideas were extended in references [5,12-18]. In the course of this work a number of new results were derived and recognized as results of the application of the hierarchic theory of oscillations and waves. But, it should be noted that before in all cases, when the hierarchic approach has been used, each system has been analyzed as an independent problem. The main novelties of the presented version of the hierarchic theory are: a) a new specific method for description of hierarchic dynamic systems, e.g. electrodynamic systems of relativistic electronics. The important peculiarity of this method is that a proper specific set of dynamic variables is introduced for each hierarchic level of the system. Besides that, the specific structural and dynamic operators are introduced for characterization of the structural and dynamic relationships between the dynamic variables and the relevant dynamic functions that characterize the different hierarchic system levels; b) a new asymptotic method of calculation. The most important idea of this method is the use of relevant hierarchic scale parameters of a system as expansion parameters of the problem. Further we set forth that more detail.


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So, in the framework of the developed general hierarchic theory a number of new asymptotic algorithms have been elaborated for solving nonlinear oscillatory and resonant wave problems. The characteristic peculiarity of this method is that, the above discussed general hierarchic principle (and other four fundamental hierarchic principles; see below) here does not just serve to be an interesting method for qualitative analyzing [14], but is an effective computation tool. It allows subdividing a total calculation problem into a few simpler and more particular hierarchic problems, and identifies the solutions of the total problem as aggregate of parts of a realm of these particular problems. As it is mentioned above, the effectiveness of particular application cases of the presented version of the hierarchic theory has been proved by using of its in a number of examples dealing with the nonlinear problems of parametric and superheterodyne electronic devices [5, 10-18]. It should be especially mentioned that an electrodynamics developed there has demonstrated the exceptional versatility and power of the used hierarchic method of analysis. Therein a mathematical architecture of the theory, as a whole, sometimes appears to be very strange for traditional perception. However, high application efficacy of its has been illustrated viability of its. But, the letter it is readily seen from the material that is delivered below. 2.1 Fundamental hierarchic principles and some general features of hierarchic dynamic systems In general case we can define the notion of a hierarchy as some set of preferences with respect of relevant parameters. Such parameters called by us the hierarchic ones. For instance, if af (where K = 1,2,3,..., m+1,) is a specific set of parameters for some dynamic system, and if the latter satisfy the hierarchic series relationship then we say that the system possesses a structural hierarchy. Here m is a total number of hierarchic levels of the system. On the other hand, if for some dynamic system a set of parameters bK easts and the relationship between the parameters is same to that in (2.1) we say that the systems possess the same functional hierarchy. In presented variant of the theory we require the number of terms of both series is to be equal, that is for each term aK there should be exist a corresponding term bK. Besides that, we confine ourselves to discussion of the systems only whose hierarchic series (2.1) is a strong one. The latter means, that aK momenta, and the position vector 7 as functions of the time / can be obtained. The approach taken here is based on Bogolubov's methods [21,22] and hierarchic calculation scheme. Therefore it is applicable only to standard forms of systems of exact differential equations. Here we take as a standard form the system of equations and take into account a hierarchy of fast changing phases (2.14) [10,22]. In this case the hierarchy is obtained by the following procedure. First, we form the vector of the fast phases v|/ from the scalar phases of particle oscillations. Next, we divide all the components of this fast phase vector y/ into groups and assume the rates of change of the scalar components of v|/ which belong to, say, the group number x = 1, to be commensurate with the scale of the velocities of rotation. Suppose that the rates of change of the components from this and other groups are characterized by a set of scale parameters £/, &...,£»... which differ from

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each other. The set of equations, described in this system, can be written in the hierarchic standard form (2.14) as [10-13]:

where \|/K are the partial fast phase vectors composed of the components of the group x, v|/ = {\|/1...,\|/K,...}, £ is the tensor of scale parameters, which is composed of the elements £i,£ 2 >—>£ K ,—, x as above, is the vector of slow variables, and o>K are the corresponding vector-functions, which depend on the vector x components. We now introduce the simplifying assumption that the scale parameters £K, associated with the rates of change of the phase vector \j/ K , can be arranged into a strong hierarchic series (2. IS) A

where m is the last level of the hierarchy. We now show that if condition (4.2) is satisfied, then the solution of our problem is reduced to the procedure considered in the theory of the hierarchic method and in Bogolubov's calculation procedure [10,21,22] (see section 2 and Appendix I). We define the scale parameters £K by where the momentum derivative with respect of time t is designated by the dot; X|/ K/ and xq are relevant components of the vectors \)/and x, respectively. Within the context of the definition of the K -th scale parameter (4.3), the mathematical meaning of the hierarchic series (4.2) is equivalent to the statement that the system possesses a hierarchy of characteristic rotation phase velocities of the fast phases of the given particles, i.e.

where all notations are self-evident. This is interesting in that apart from of the scale parameter £1 (the leading term in the hierarchic series (4.3)), all the other fast phases \j/K (K>1) can be regarded as relatively slow. This point is


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very useful for constructing an asymptotic hierarchic integration algorithm, which can be written in the traditional form as [10,21,22]:

where x1 is the expanded vector of slow variables (here all partial fast phases VJ/K except the first one (K=1) are considered as its components), and so on. We now write the Krylov-Bogolubov substitution (2.11) for the components x1 and xyt according to the traditional scheme as [10,21,22] (see Appendix I for more details):

where the averaged variables x? and ^lt of the first hierarchy can be obtained from a comparison with a reduced (truncated or shortened) system of equations (compare with equations (2.5) and (2.7)) [10,21,22]:

That is, in the present approach, in the solution process of the set (4.1) we look for the "true" particle motions in the electromagnetic fields as a sum of the averaged motion and the "ripples" about this average trajectory. The O(n), u(n), A(n), B(n) are the functions which are computed by well-known procedures (see Appendix I and [10,21,22]). In the next stage of the hierarchic analysis, we separate out the following group of fast scalar phases from "temporary components" of the slow variables of the vector x' .This phases define the dynamics of the partial fast vector phases \j/2 (see (4.4)). Then the first truncated equations (4.7) can be solved by the application of the successive hierarchic calculation scheme described in section 2, i.e., we again reduce obtained equation to the traditional standard form (compare this with equations (2.8)-(2.10)):

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where x"is the part of vector x' where the vector fast phase \|/2 is separated. In view of the above reasoning the meaning of the new functions X", co 2, Y"2

and the others is evident. Then we again treat the system (4.8) in terms of the described asymptotic hierarchic integration algorithm. After that we again separate out the variables from the vector x" (this group of variables describes the dynamics of the vector phase \t/3), and so on. The procedure is repeated in a cyclic manner until no terms remain in the hierarchic series (4.2) and (4.4), i.e. as long as we take into account all the partial vector phases which form the complete fast phase vector \\i in (4.1). As a result we obtain an w-fold averaged (truncated or shortened) equation set which can be solved, as rule, rather easily. The latter circumstance is a major advantage of our version of the hierarchic method, which has made it very attractive in practical applications. The approximate nonaveraged solutions of the initial equations (3.3) or (3.4) are obtained by the successive use of the relevant inverse transformation formulas of the type (4.6). 4.2. Reduction of the equations of particle motion to the hierarchic standard form Thus, the main feature of our hierarchic method is the application of the special successive hierarchic transformations in the three-dimensional coordinate space. This greatly simplifies of the initial formal problem. However, this can be done only with respect to the standard system of equations (4.1) and (4.2) on the basis of the initial system (3.3) or (3.4). Therefore, the first step here consists of finding all the elements of the slow vector x and the fast phase vector \|/, i.e. finding all the variables of the system (3.3) and (3.4) which should be classified into slow and fast elements. This procedure is discussed briefly here. The Hamiltonian H as well as the variables F, J3 (or u = J3) can be considered as variables of the first hierarchic level. Accordingly, for the determination of them in the framework of the slowvarying amplitude method we must construct the specific reduced system of equations [10]. The latter we regard as the relevant equations of the first hierarchic level. Hence, the traditional version of the slow-varying amplitude method can have the hierarchic interpretation. It is readily obvious that here some new possibilities are possible. Namely, according to the procedure of the hierarchic method (see section 2) the reduced equation set in this case can be regarded as a new standard Rabinovich's form of the type (6.19). Consequently, there is the possibility to apply the slow-varying amplitude method repeatedly, and so forth. Such an approach can be applicable, for instance, in the case, when the slow-varying amplitudes turn out to be, in their turn, the periodic functions, i.e. when we deal with slowly modulated amplitudes. Unfortunately, the limited scope of this paper does not allow us to give more attention to this problem. 6.3. Slow-varying amplitude method. Modern version It is should be stated that any attempt to add the traditional version of the slow-varying amplitude method to the non-linear wave-resonant problems of electrodynamic plasma systems proves very soon to be a far less easy task than it seems at first.

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The first difficulties arise when one tries to formulate the problem. In particular, when one tries to give an adequate description of a physical model in terms of the slow-varying amplitudes, i.e. when it is necessary to reduce Maxwell's equations to the standard form (6.18) (see Appendix ffi). There, in turn, two main problems arise. They concern the manner of selecting the small parameter e and the problem that the vector U is not strictly a periodic function of r and t, because it consists of the slow-varying Uo and fastoscillating U parts, respectively, i.e. To obtain the relevant equations for U0 and U, it is sufficient to perform a Fourier-series expansion of the left and right hand sides of Maxwell's equations with respect to all periods of the fast phases and to equate the null and non-null harmonics (assuming that in the last case the slow-varying amplitudes are constants). As a result we obtain the system of coupled matrix equations [10,13]:

where V is an operator consisting of the differential three-dimensional nabla-operator V as components, Rot) is a null-harmonic of the function Ku*, Ru is its oscillatory part, and U is defined by equation (6.30). In our variant of the theory we choose the vector U in the form

Then, the vector-function

which in the form of (6.18) can be presented as


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where j is the vector consisting of the three-dimensional current density vector j as components, F is the relevant square matrix; Eou and Ku are the same vectors consisting of the null-harmonics and non-null harmonics of the current density, respectively. Furthermore, we perform a Fourier-series expansion of the function j(r,t) (6.17) (which can be obtained by using the averaged current-density method):

where all definitions are obvious. Consequently, we arrive at the possibility to close the system of equations (6.31) by using the current density equation (6.1) which can be put here into the form:

where f ( j , p ) is a known function of the current density j . The construction of the asymptotic solution for the Ry function is made possible by using the above described (in section (6.1)) calculation procedure (for the current density vector (6.17)), i.e. solutions for the By vector can be represented in the form of the relevant asymptotic converging series:

where

(see definitions (6.34)). In this way we solve the second (of the two) difficulties mentioned above, which concerns the application of the traditional slow-varying amplitude method to nonlinear wave-resonant problems. The latter involves the matter of choosing the small parameter e in the right hand part of the Rabinovich standard form (6.19). As a rule, the greatest amplitude of the waves, correspondingly normalized, is taken traditionally to be e [20, 2632]. But, it should be mentioned that, unfortunately, such a method of separating out a small parameter has no acceptable mathematical foundation and is based purely on considerations of a qualitative nature. While in the low-order nonlinear theories the latter may be tolerable in the high-ordernonlinearity cases the situation is definitely unacceptable. This gives rise to a number of typical disagreeable consequences: vagueness in the problems of convergence, and in the determination of the operating range of the system

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over which the solutions are correct; the loss of terms in the reduced (shortened) equations for the amplitudes, etc. As demonstrated above, using the modernized method of slow-varying amplitudes and the averaged current density equation can solve the given problem. Namely, due to the fact that the method of separating out the small parameter l/£t in Bogolubov's method is strictly correct, the utilization of the expansion (6.37) in the modern standard form (6.31) allows one to solve the problem of finding the value of the small parameter. It is obvious, that here we can substitute for the small parameter e the following value where E1 >>1 is the largest scale parameter of the hierarchic series (4.2). Thus, the essential steps of the modernized procedure of the slowvarying amplitude method are: a) subdividing the Ry) function (in (6.19)) into slow-varying (quasistationary) and the fast-varying (i.e. periodic) parts, respectively (see the (6.30)); b) including among the elements of the vector U the components of the electrodynamic fields (E, D, H, B) and the space charge density p (see the (6.32)); c) employing as the base for finding the explicit form of the relevant expansion of the current-density method the corresponding asymptotic solutions of the type (6.17) (see the (6.37)); and d) utilizing Rabinovich's algorithm for the solution of the periodic part of the problem. As a result, we obtain a rather efficient numerically-analytical calculation approach for solving nonlinear resonant-wave problems of electrodynamics. 6.4. Slow-varying amplitude method. The hierarchic version

Here we restrict attention to some important peculiarities of the traditional and modern versions of the slow-varying amplitude method. As mentioned above, the Lagrange or Euler forms usually describe charged particle beams. The first approach is used for problems involving the motion of an individual particle and the study of its dynamics and kinematics. In the second approach one needs to fix a spatial point within a radius vector 7 (with respect to time /) and to register the velocities of the passing particles. In the first event the coordinates of a space point t always coincide with the electron coordinates and, hence, depend on time t as some function r(t). It is obvious that in the second case t and t are reciprocally independent quantities (because at different moments of time we have different particles).


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As was shown, the averaged current density method is based on the specific transformation of Euler's coordinates (see, for instance, (3.9), (6.1)) into those of Lagrange (see the (5.1)) and then the inverse conversion (see (6.9), (6.15)-(6.17)). That is why we cannot employ the hierarchic technique presented to the asymptotic integration of Rabinovich's standard system (6.19), i.e. to the system of Maxwell's equations (see Appendix HI) immediately because, the latter are written in the Euler formalism. However, the asymptotic integration of the hierarchic scheme can be used here if we employ some indirect approach, which was briefly mentioned above in subsection 6.1. Basically, the method consists of the following. We perform in the standard system (6.19) successive transformations into the mfold averaged coordinate space (see transformation formula (6.11) and Appendix El). Inasmuch as the vector-functions R(n) contain (as components) the current density vector j (see definition (6.34) and taking into account the transformation formulas (6.3)-(6.8)), we can write for the .Ru-function the following transformation equations for the first hierarchy:

where all the quantities have already been defined above. Hence, we can represent the vector [/as where x is the vector of averaged slow values. At the first hierarchic level of our theory we have the following presentation for the standard system:

where V is the differential operator containing the elements of the averaged nabla operator V ; the definition for the latter is presented in Appendix II. At the second hierarchic level we obtain:

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and so forth. Here the definitions of all quantities are quite clear within the context of the above (in subsection 6.1.) discussion. We proceed, until we reach the terminal hierarchic level m. At this hierarchic level we solve the relevant m-fold averaged reduced equation

As mentioned above, an important peculiarity of the reduced equations of the last hierarchic level (of type (6.49)) is the absence of any periodic dependence on both the slow and fast oscillating phases. In this regard, equation (6.49) may be considered to be a quasistationary one, which describes the nonoscillatory dynamics of the system. Here all resonances and oscillations are "hidden" in the transformation formulas of type (6.48). In its mathematics structure equation (6.49) is much simpler than the nonaveraged one (6.36). Consequently, all fields which are excited in this /w-fold averaged system turn out to be quasistationary (i.e. they are nonoscillating). We solve equation (5.49) (for example, by some analytical approximation method) and on this base construct the m-fold averaged standard equation:

where the subscript and superscript m, as before, mean the multiplicity of the hierarchic level, and V(w) is the differential operator consisting of the m-fold averaged nabla-operator VM. Then we solve equation (6.50) and obtain the relevant solutions for the vector-function U(m) (r ( m ) ,t). At the following stage of the calculation scheme we construct the corresponding equations for the functions of the (m-1) order of multiplicity:


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For the (m-2) hierarchic level we get, respectively:

and so on. Thus, the hierarchic procedure of asymptotic integration of the standard system (6.19) consists of: a) extending of the averaged current density method over the calculation ofthe R^-functions; b) transformation of coordinate 7 and the relevant differential operators of the standard form (5.19) into w-fold averaged coordinate space (see the Appendix ED); c) solution of the m-fold averaged equation for the R(n)-functions (see (6.49)) and construction on this basis the m-fold averaged standard equation (6.50); d) solving equation (6.50) and forming the basis for obtaining relevant solutions of the hierarchic (w-l)-level; e) using successive inverse transformations until the solutions of the nonaveraged problem are found. Two important circumstances should be mentioned here. The first is that for solving the standard system of type (6.54), (6.56) we can use the slowvarying amplitude method. The second concerns the level of the difficulties ofthe latter procedure. But more details later. In the case of the normal modernized Rabinovich's method the RU contain all the fast oscillations, which are characterized by Euler's phases v|/1, \|/2, ... and 01, 02 As a result this function turns out to be rather complex, and the corresponding calculation procedure for U becomes too complex [10]. Therefore, the main advantage of the present hierarchic method is following. In the framework of the presented hierarchic calculation

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scheme we subdivide the whole complex problem into a range of much simpler problems because at each hierarchic level the the number of oscillating phases is reduced. Besides that, here we have obtained a way for solving the same boundary and equilibrium problems, which we already discussed above. 7. SELF-CONSISTENT THEORY OF CHARGED PARTICLE BEAM INTERACTIONS WITH ELECTROMAGNETIC FIELDS. THE KINETIC VARIANT

The reason for favoring the kinetic approach to problems of plasma electrodynamics is more efficiency and the broader framework. For the hierarchic variant of the kinetic self-consistent theory all the details have been discussed in [10-13]. Therefore, it remains here to only emphasise those peculiarities which are interesting for our purposes here. In this case we use the kinetic equation, and select as the appropriate choice Boltzman's equation in canonical form (see the (3.51)) instead of the quasihydrodynamic equation (3.9). Here we also use the hierarchic principles as the basis of calculation. 7.1. Averaged kinetic equation method Upon comparing equation (3.5) and the standard hierarchic system (4.1) we see that formally the essence of the new problem (similar to the quasihydrodynamic case) consists of reducing of the equation in the partial derivatives (3.5) to the standard form (4.1), i.e. to the system of equations in momentum derivatives. Here, for the solution of the fluid motion problem we utilise the circumstance that the characteristics of equations (3.5) are the single-particle Hamiltonian equations (3.3), i.e.: where all definitions were given in section 3. The solution process here is similar to the above presented for the case of the quasihydrodynamic theory. Namely, we express the functions dH/cf and - 3H/3F on the left hand side of (3.5) through the Lagrange rotating electron phases p^ (see subsection 4.2) which are used as parameters. Then, using Liouville's theorem and upon passing to the momentum derivatives, we can replace the kinetic equation (3.5) by the new set of equations:


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In its mathematical structure the system (7.2)-(7.5) is similar to the set of equations for the quasihydrodynamic equation (5.1). Therefore, we can employ here the same scheme of calculation as described above. That is, the vectors of slow and fast variables are formed, the largest scale parameter et (see the definitions (4.2), (4.3), (4.11)) is separated out, and the first order solutions (see A. 1.1 in Appendix I) are obtained as:

where for the averaged values are given by the following reduced (truncated) equations (see A. 1.2 in Appendix I):

By using equation (7.9) with (7.10), (7.11 we can obtain the corresponding averaged kinetic equation of the first hierarchy:

where the new collision integral is

At the second hierarchic level we perform relevant transformations by a similar procedure and so on, until the hierarchic series (4.2) has been

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exhausted. The inverse transformations are accomplished by utilization of the scheme described above (see (5.9), (5.10), (6.15)-(6.17)). 7.2 Current and space charge densities According to (3.11), (3.12) the densities of current Jand charge p in the kinetic case are represented by the following:

where we neglect the subscripts a. By virtue of the asymptotic representations (7.6)-(7.11) we can rewrite (7.14), (7.15) in the form of series:

where the definitions of j(n) and p(n) are evident. Thus, in the kinetic case the problem of nonlinear current and space charge determination can be solved also. The latter, in turn, opens up the possibilities for the solution of Maxwell's equations by using the slow-varying amplitude method. 7.3 Method of slowly varying amplitudes. Kinetic case The main peculiarity of the kinetic version of the self-consistent hierarchic theory is that the vector U in (6.31) consists of only the relevant field characteristics (compare with (6.32)):


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i.e. the vector function which is similar to (6.34) is determined by both the current density vector j and space charge density p, respectively. For the other asymptotic hierarchic integration procedures (for the modified standard system (6.31)) we use the same approach as described above in subsection 6.3. CONCLUSION Thus, a new concept and the general nonlinear theory for EHaccelerators and other related systems have been proposed and elaborated. But the material presented has a broader and more general significance. Namely, a new hierarchic methodology and a corresponding new approach to solve nonlinear problems of self-consistent theory of devices with longterm interactions are presented. They are the current density averaged equation method, a modernized slow-varying amplitude method, a hierarchic asymptotic integration method of Rabinovich's standard system, etc. In a following part of the article we will demonstrate this new approach and show that it opens up very promising prospects for practical uses. REFERENCES 1. Kochmanski S.S., Kulish V.V. On non-linear theory of the free electron laser, Acta Physica Polonica, A68, N5, 725-738 (1985) 2. Kulish V.V. Physics of free electron laser. General principles. Deposited manuscript, Kiev (Ukraine), 1990. Ukr. Inst. Sci. Tech. Inf., 05.09.90, N1226, Uk-90 3. Kulish V.V., Krutko O.B. Amplification properties of free electron lasers with combined transversal EH-ubitron pumping \\ Pisma v Zhurnal Tekhnicheskoy Fiziki, (Letters in Russ. Jaitm. Techn. Phys.), 21, N. 11, pp.47-51 (1995) 4. Kulish V.V., Krutko O.B. Acceleration of charged particles in crossed periodically-reversed electromagnetic fields; \\ Pisma v Zhurnal

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Tekhnicheskoy Fiziki, (Letters in Russ. Journ. Techn. Phys.), 21, N. 9, pp.52-55 (1995) 5. Kulish V.V., Krutko O.B. Acceleration of charged particles in the crossed periodically reversed electric and magnetic fields.\\ Vestnik Swnskogo Universiteta, (Herald of Sumy State University), N 2, pp. 2-9 (1995) 6. Kulish V.V., Kosel P.B., Krutko O.B., Gubanov IV. Effect on charged particle relativistic beams: cooling by their acceleration in crossed EHubitron fields. // Pisma v Zhurnal Tekhnicheskoi Fiziki, (Letters in Russ. Journ, Techn. Phys.), 22, N 17, (1996) 7. Artukh I.G., Kamaldinova G., Sh., Sandalov A.N. Free electron lasers. Acceleration technique for FEL, Reviews on Electron Techniques, 19 (1314), series 1, (1987) 8. Miller M.A. Acceleration of plasma bunches by high-frequency electromagnetic fields. Zhurnal Teoreticheskoi i Experementalnoi Fiziki (Soviet Journal of Theoretical and Experimental Physics), 35, N6, pp. 1909-1917 (1959) 9. Gaponov A.V., Miller M.A. On the application of the moving highfrequency potential hole for the acceleration of charged particles. Zhurnal Teoreticheskoi i Experementalnoi Fiziki (Soviet Journal of Theoretical and Experimental Physics), 34, N3, pp. 751-752 (1958) 10. Kulish V.V. The methods of averaging in nonlinear problems of relativistic electrodynamics. \\ World Federation Publisher Company, Inc. (Tampa-Atlanta), (1997). 11. Kulish V.V. Nonlinear self-consistent theory of free electron lasers. Method of investigation. \\ Ukrainian Physical Journal, 36, N 9, pp. 1318-1325. (1991) 12. Kulish V.V., Lysenko A.V. Method of averaged kinetic equation and its use in the nonlinear problems of plasma electrodynamics. \\ Fizika Plasmy (sov. Plasma Physics), 19, N 2, pp. 216-227 (1993) 13. Kulish V.V., Kuleshov S.A. and Lysenko A.V. Nonlinear self-consistent theory of superheterodyne and free electron lasers. \\The International journal of infrared and millimeter-waves, 14, N 3 (1993). 14. Marshall T.C. Free electron laser, Mac Millan, New York, London (1985) 15. Brau C. Free electron laser, Academic Press. Boston (1990) 16. Luchini P. And Motz U. Undulators and free electron lasers, Clarendonpress, Oxford (1990) 17. Landau L.P., Liftshitz E.M. Theory of field. Nauka, Moscow (1974) 18. Ruhadze A.A., Bogdankevich L.S., Rosinkii S.E. and Ruhlin V.G. Physics of high-current relativistic beams, Atomizdat, Moscow (1980)


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19. Davidson R.C. Theory of nonlinear plasmas. Benjamin, Reading, Mass, (1974) 20. Sitenko A.G. and Malnev V.M. Principles of the plasma theory. Naukova Dumka, Kiev, (1994) 21. Bogolubov N.N. and Mitropolskii. Methods of averaging in the theory of nonlinear oscillations. Pub. House AN USSR, Moscow (1963) 22. Bogolubov N.N. and Zubarev D.N. Asymptotic approximation method for the system with rotating phases and its application to the motion of charged particles in magnetic fields, Ukr. Math. Zhurn. (Ukrainian Mathem. Journal). Iss. 7, pp. 201-221 (1955) 23. Serebriannikov M.G., Pervozvansky A.A. Discovery of Hidden periodicities. Nauka, Moscow (1965) 24. Landtsosh L. Practical methods of applied analysis. Fizmatgis, Moscow (1961) 25. Grebennikov E.A. Introduction to the theory of the reasonable systems, IzdatelstvoMGU, Moscow (1987) 26. Sukhorukov A.P. Nonlinear wave interactions in optics and radiophysics. Nauka, Moscow (1988) 27. Bloembergen N. Nonlinear optics. Benjamin, New York (1965) 28. Weiland J. and Wilhelmsson H. Coherent nonlinear interactions of waves in plasmas. Pergamon Press., Oxford (1977) 29. Vainstein L.A. and Solnzev V.A. Lectures on Microwave electronics, Sov. Radio, Moscow (1973) 30. Gaiduk V.I., Palatov K.I. and Petrov D.M. principles of microwave physical electronics, Sov. Radio, Moscow (1971) 31. Alexandrov A.F., Bogdankevich L.S. and Ruhadze A.A. Principles of plasma electrodynamics, Vyschja Shkola, Moscow (1978) 32. Kondratenko A.N. and Kuklin V.M. Principles of plasma electronics, Energoatomixdat, Moscow (1988) 33. Kulish V.V. On the theory of devices with difference-frequency signal separation in an electron beam. Electronics technology, ser. Microwave devices (Soviet Microwave Electronics), N4, pp. 25-37, (1978) 34. Gaponov A.V., Ostrovsky L.A., Rabinovich M.I. One-dimentianal waves in nonlinear dispersive media. Izv. Vysh. Uchebn., Ser. Radioftzika (Sov. Radiophys.), 13, N2, pp 169-213, 1970 35. Rabinovich M.I. and Talanov V.I. Four lectures on principles of the theory of nonlinear waves and wave interactions. Leningrad: Izd-vo LGU, (1972) 36. Hapaev M.M. Asymptotic methods and equilibrium in the theory of nonlinear oscillations, Vyshchaja shkola, Moscow (1988)

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37. Hapaev M.M. Averaging in the equilibrium theory of nonlinear oscillations, Nauka, Moscow (1986) 38. Kulish V.V., Kailyuk A.G., Kvak A.A. and Krutko O.B. Ukrainian Patent of Inventory UA 95031123, Priority of 10.03.95. Published 09.01.97 39. Haken H. Advanced Synergetic. Instability Hierarchies of SelfOrganizing Systems and Devices. Springer-Verlag, Berlin-HeidelbergNew York- Tokyo (1983) 40. Nicolis J.S. Dynamics of Hierarchical Systems. An Evolutionary Approach. Springer-Verlag, Berlin-Heidelberg-New York, Tokyo (1986) 41. Olemskoi A.I., Flat A.Ya. Application of the factual concept to the physics of condensed medium. Uspechy Fiz. Nauk, 163, N12 (1963) 42. Rammal R., Toulouse G., Virasoro M.A. Ultrametricity for physicists. Reviews of Modern Physics, 58, N3, July 1986, pp. 765-788 43. Laitman M. Kabbalah. The spiritual secret in Judaism. Printed in Israel, (1984) 44. Fortune D. The mystical Qabalah. Alta Gaia Books, New York. 45. Andre Nataf. Dictionary of the Occult. Wordworth Editions Ltd., Herttordshire (1988) 46. KapraF. Dao of physics. St. Petersburg, ORIS, (1994) 47. Druzshynin V.V., Koutovov D.S. Systemotechnics. Moscow: Radio i sviaz, (1985) APPENDIX I ALGORITHM FOR ASYMPTOTIC INTEGRATION OF STANDARD SYSTEM We look for a solution of the system (4.5) in the form

where: xq, x|/1 are the elements of vectors x and i//, and k, m are the numbers of elements in the latter, respectively. The comparison equation system we take as:


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Thus, the problem of finding the solution of (Al.l) amounts to the determination of unknown function u ( n ) , v(n), A(n) and B(n). This is pursued further. First of all we should mention that the determination of the functions in (Al.l) , (A1.2), generally speaking, are ambiguous. This fact is associated with the arbitrariness available in attributing the different terms of the series. Let us eliminate this arbitrariness by assuming that all M(n) and v(n) are free of null (on vj7,) Fourier harmonics. Thus, we postulate that the whole averaged motion is described by values xq and xj 1 ,. Let us resume the differentiation of (Al.l) and, taking into account (A1.2), substitute the obtained result into (4.5), equating the coefficients of equal powers £-1. As a result of the performed transformations for determining the unknown functions we obtain the infinite sequence of relations:

Hereafter, in the present appendix the averaging sign is omitted for simplicity. Taking into account (4.5), it is easy to see that the functions x? are periodic in \|/,. Using this circumstance, we expand them into the Fourier series of multiplicity m:

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Kulish, Koscl, and Kailyuk

Proceeding from similar considerations as well as (A1.3), we represent the unknown function u(n) in the form:

Here T^ are the periods associated with phases \|/,. Upon substituting (A1.7), (A1.8) into (A1.3) and equating the coefficients with equal exponents, we obtain the expressions for the amplitudes b sk...p (x) and the functions A(0):

where the functions Cqj (x), from the limits on u™, satisfy the normality condition: Accordingly, the definitions for the coefficients A(0) (A 1.10) can be rewritten in the form:

Taking into account the condition of absence of null harmonics on \|/j and u™, for the latter we finally formulate the definition which differs from (A1.8) only in the exclusion of terms in s, k, ..., p=0. Similarly to (A1.10), the second approximation equation is solved, etc. From the structure (A 1.9), in particular, it follows that

i.e. the resonances among the components of the vector of fast phases are absent. (Failure to fulfill (A1.13) means that at the stage of classification of the phases (and their linear combinations) not all-slow phases have been singled out). Following the above-mentioned procedure, we can also easily obtain the expressions for the other unknown functions, in particular:


89

where (...) means the averaging in all fast phases (e.q., refer to (A1.12)); dsk...p (X) are the factors of the expansions in Fourier series of the functions of this form. APPENDIX n TRANSFORMATION PROCEDURE FROM V -OPERATOR TO THE AVERAGED V -OPERATOR

The transformation of the nabla-operator V = d/dr , which is defined in the normal three-dimensional space into the averaged nabla-operator V = d/dr in the three-dimensional averaged (with respect to all fast oscillations) space is:

It is obvious that

Here we use Bogolubov's substitutions of type (Al.l):

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where the calculation procedure for the functions u^ is described above in the Appendix I (see, for example, (A1.14)). Furthermore, we take into account that

i.e. we get a definition of the functions u(n) through the nonaveraged coordinates r by the following series:

Hence, by using (A2.3)-(A2.5) we can write:

i.e. the required transformation function (A2.6) a (r) is given by:

APPENDIX m REDUCTION OF MAXWELL'S EQUATIONS TO THE STANDARD FORM A. 3.1 Kinetic variant We start with Maxwell's equations (3.10) supplemented by the kinetic equation (3.5). Comparing (3.10) to (6.19) we see that Maxwell's equations (3.10) do not satisfy the standard equation requirement (6.19). Therefore, the obvious first step is to standardize the set of equations (3.10). We present the current density J and the charge density p in (3.10) as sums of linear and nonlinear terms with respect to the field amplitudes, i.e.,

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In the most general case, the linear terms J(1) and p(1) may be written as

Explicit expressions for the matrices

can be

easily derived from Maxwell s equations. Fields. We arrange the components of the vectors E, H, D and B as a column vector

For the operator P, we have In terms of the above definitions, Maxwell's equations (3.10) reduce to the form

where


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[0] are 3 x 3 zero matrices; (0) and {0} are the three-dimensional zero row vector and zero column vector, respectively. A. 3.2. Quasihydrodynamic variant

Here the general calculation scheme is preserved. The difference is only that the vector u in this case consists of components of the space charge density p:

Then performing all of the above described mathematical procedures we, as result, obtain the following expressions for the relevant matrix function:


A

where AT = Vj ( 2 ) is a predetermined quantity.

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