This monograph consists of two volumes and provides a unified comprehensive presentation of a new hierarchic paradigm and discussions of various applications of hierarchical methods for nonlinear electrodynamic problems. Volume 1 is the first book, in which a new hierarchical model for dynamic non-linear systems is described and analysed and a set of new hierarchical principles is discussed. The modern hierarchic 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 other. 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, etc.).
Author(s): V. Kulish
Series: Fundamental Theories of Physics
Edition: 1
Publisher: Kluwer Academic Publ
Year: 2002
Language: English
Pages: 385
Contents......Page 6
Acknowledgments......Page 16
Preface......Page 18
1.1 One Illustrative Example of the Multi-Frequency Oscillation–Wave Systems......Page 26
1.2 Essence of the Hierarchical Approach......Page 29
1.3 Classification of the Hierarchical Methods Discussed in the Book......Page 32
2 BASIC CONCEPTS AND DEFINITIONS OF THE OSCILLATION THEORY OF WEAKLY NONLINEAR SYSTEMS......Page 33
2.1 Nonlinear Oscillations and Nonlinear Systems......Page 34
2.2 Hidden and Explicit Oscillation Phases......Page 37
2.3 Resonances......Page 38
2.4 Slowly Varying Amplitudes and Slowly Varying Initial Phases. Complex Amplitudes......Page 45
2.5 Harmonic and Non-Harmonic Oscillations......Page 46
3.1 Definition of the Wave......Page 48
3.2 The Phase and Group Wave Velocities......Page 49
3.4 Transverse and Longitudinal Waves......Page 51
3.6 The Concept of Dispersion......Page 52
3.7 Waves with Negative, Zero and Positive Energy......Page 53
4.2 Linear Momentum and Force......Page 54
4.4 Rotation......Page 55
4.5 Energy. Field of Forces......Page 56
4.6 Motion Integrals......Page 57
4.8 Electromagnetic Field......Page 58
4.9 Lagrange and Euler Variables......Page 59
5.1 What is a System?......Page 60
5.3 General Classification of the Systems......Page 61
5.4 Some General Properties of Complex Systems......Page 62
6.1 Principle of Physicality......Page 63
6.3 Autonomy Postulate......Page 64
6.5 Complementarity Postulate......Page 65
6.7 Uncertainty Postulate......Page 66
6.9 Entropy and Negentropy......Page 67
6.10 Complexity......Page 70
1.1 Why Does Nature Need Hierarchy?......Page 74
1.2 Two Approaches to the Theory of Hierarchical Systems......Page 75
1.3 Self-Modeling Principle......Page 76
1.4 Main Idea of the Hierarchical Method......Page 77
1.5 Again: What is Hierarchy Originally? The Structural Hierarchy......Page 78
1.6 Dynamical Hierarchy......Page 79
2.1 Hierarchical Principles......Page 83
2.2 Dynamical Equation of the Zeroth Hierarchical Level......Page 85
2.3 Structural and Functional (Dynamical) Operators......Page 87
2.4 Classification of the Hierarchical Problems......Page 89
2.5 Hierarchical Tree......Page 91
3. HIERARCHICAL ASYMPTOTIC METHODS. GENERAL IDEAS......Page 96
1.1 Determined Hierarchical Systems......Page 97
1.2 Averaging Operators......Page 101
2.1 Factors of Stochasticity in Dynamical Systems......Page 103
2.2 Example of Hierarchical Model of Stochastic System......Page 104
3 WAVE RESONANT HIERARCHICAL SYSTEMS......Page 106
3.1 Standard Hierarchical Equations in the Case of Hierarchical Wave Problem......Page 107
3.2 Classification of Problems......Page 110
3.3 Krylov–Bogolyubov Substitution......Page 112
3.4 Case b)......Page 113
4.2 Van der Pol's Method's Variables......Page 115
4.3 Truncated Equations and Their Hierarchical Sense......Page 117
5.1 Bogolyubov's Standard System......Page 120
5.2 The Problem of Secular Terms......Page 124
6.1 Two-Level Systems with Slow and Fast Variables. General Case......Page 125
6.2 Two-Level Systems with Fast Rotating Phases......Page 127
1.1 A Few Introductory Words......Page 132
1.2 Formulation of the Hierarchical Single-Particle Electrodynamic Problem......Page 134
1.3 Classification of Oscillatory Phases and Resonances. Hierarchical Tree......Page 137
1.4 Reducing Hierarchical Multi-Level Standard System to the Two-Level Form. The Scheme of Hierarchical Transformations......Page 142
2.1 Formulation of the Problem......Page 144
2.2 Algorithm of Asymptotic Integration......Page 145
2.3 Accuracy of Approximate Solutions......Page 149
2.4 Asymptotic Integration of Initial Equations by Means of Successive Approximations......Page 152
2.5 Peculiarities of Asymptotic Hierarchical Calculational Schemes Based on the Fourier Method......Page 153
3.1 Formulation of Problem......Page 155
3.2 Solutions. Non-Resonant Case......Page 156
3.3 Solutions. Resonant Case......Page 159
4.1 Formulation of the Problem......Page 164
4.2 Algorithm of Asymptotic Integration......Page 165
5.1 Essence of the Problem......Page 168
5.2 Sewing Together of Resonant and Non-Resonant Solutions......Page 169
5.3 Example for the Solution 'Sewing': The 'Stimulated' Duffing Equation......Page 171
5. HIERARCHICAL SYSTEMS WITH FAST ROTATING PHASES. EXAMPLES OF PRACTICAL APPLICATIONS......Page 182
1.1 Systems for Transformation of Optical Signals into Microwave Signals as a Convenient Illustrative Examples......Page 183
1.2 Formulation of the Problem of Electron Motion in the Field of Two Oppositely Propagating Electromagnetic Waves......Page 189
2.1 Reducing Initial Motion Equations to the Standard Forms with Two Rotation Phases......Page 192
2.2 Zeroth Hierarchical Level. Parametrical Resonance......Page 195
2.3 Passage to First Hierarchical Level. Nonlinear Pendulum......Page 196
2.4 Nonlinear Pendulum. The Miller–Gaponov Potential......Page 201
2.6 Nonlinear Pendulum. Exact Solutions and Analysis......Page 203
2.7 Full Solutions of the Initial System......Page 208
3.1 Transition to the Second Hierarchical Level......Page 210
3.2 Duffing Oscillator......Page 213
4.1 Stimulated Oscillation of a Charged Particle......Page 217
4.2 Stimulated Oscillations of an Electron Ensemble......Page 221
6. HIERARCHICAL SYSTEMS WITH PARTIAL DERIVATIVES. METHOD OF AVERAGED CHARACTERISTICS......Page 232
1.1 Motion Equations......Page 233
1.2 Field Equations......Page 235
1.3 Some General Information about Equations with Partial Derivatives......Page 236
2.1 Concept of the Standard Form......Page 238
2.2 General Scheme of the Method......Page 239
3.1 Method of Characteristics. The Scalar Case......Page 247
3.2 Method of Characteristics. The Vector Case......Page 252
4.1 Initial Equations......Page 256
4.3 Passage to the First Hierarchical Level......Page 257
4.4 Back Transformations......Page 259
5 HIERARCHICAL METHOD OF AVERAGED QUASI-HYDRODYNAMIC EQUATION......Page 261
5.1 Averaged Quasi-Hydrodynamic Equation......Page 262
6 THE METHOD OF AVERAGED CURRENT–DENSITY EQUATION......Page 265
6.1 Averaged Current–Density Equation......Page 266
7.1 Averaged Kinetic Equation......Page 269
7. EXAMPLE: APPLICATION OF THE METHOD OF AVERAGED CHARACTERISTICS IN NONLINEAR THEORY OF THE TWO-STREAM INSTABILITY......Page 274
1 PROBLEM OF MOTION OF A TWO-VELOCITY ELECTRON BEAM IN GIVEN ELECTROMAGNETIC FIELDS......Page 277
1.1 Statement of the Motion Problem......Page 278
1.2 Averaged Characteristics for the Motion Problem......Page 279
1.3 Back Transformations......Page 285
1.4 Integration of the Averaged Quasi-Linear Equation for the Beam Velocity......Page 286
2 FIELD PROBLEM. HIERARCHICAL ASYMPTOTIC INTEGRATION OF THE CONTINUITY EQUATION......Page 287
2.1 Continuity Equation of the Two-Velocity Electron Beam......Page 288
2.2 Averaged Characteristics and the Averaged Quasi-Linear Equation......Page 289
2.3 Back Transformation......Page 293
2.4 Characteristics of the Averaged Continuity Equation......Page 295
3.1 Averaged Maxwell's Equations......Page 296
3.2 Back Transformations......Page 300
3.3 Solving the Averaged Quasilinear Equation for the Electric Field......Page 301
3.4 Truncated Equations for the Harmonic Amplitudes of a Space Charge Wave......Page 302
3.5 Some Commentaries for the Obtained Results......Page 304
8. HIERARCHICAL SYSTEMS WITH PARTIAL DERIVATIVES. SOME OTHER ASYMPTOTIC METHODS......Page 308
1 MAIN IDEAS OF THE METHOD OF SLOWLY VARYING AMPLITUDES......Page 309
1.1 General Calculational Scheme of the Slowly Varying Amplitudes Method......Page 310
1.2 Simplified Version of the Slowly Varying Amplitude Method. Example: Effect of Parametric Amplification of a Wave......Page 313
2 TRADITIONAL VARIANT OF THE SLOWLY VARYING AMPLITUDES METHOD. RIGOROUS VERSION......Page 318
2.1 Case of Spatially One-Dimensional Model......Page 319
2.3 Model with Moderate Inhomogeneity......Page 331
2.4 Method of Parabolic Equation......Page 333
3.1 Field Problem......Page 334
3.2 Current Density Problem......Page 336
3.3 Current Density Problem in Framework of the Kinetic Approach......Page 338
4.1 Main Idea of the Hierarchical Transformations......Page 339
4.2 Hierarchical Equations......Page 342
4.3 Averaged Operator [equation omitted]......Page 345
5.1 Reduction of a Partial Differential Equation to the Standard Form with Fast Rotating Phases......Page 347
5.2 Basic Solutions......Page 348
5.3 Truncated Equations......Page 349
6.1 Kinetic Version......Page 351
6.2 Quasi-Hydrodynamic Case......Page 353
7.1 Statement of the Problem......Page 354
7.2 Motion Problem. The Averaged Quasi-hydrodynamic Equation......Page 355
7.3 Motion Problem. The Back Transformations......Page 365
7.4 Field Problem. The Method of Slowly Varying Amplitudes......Page 366
Appendix A Results of calculations in the second approximation for Chapter 8, Subsection 7.2......Page 374
D......Page 376
I......Page 377
P......Page 378
S......Page 379
Z......Page 380
