In this post, we will see the book Nonlinear Phenomena in Plasma Physics and Hydrodynamics edited by R.Z.Sagdeev. This book is part of the Advances
in Science and Technology in the USSR Physics Series

About the book (From the Preface)

The modern theory of nonlinear phenomena is going through a period of explosive growth. Each “invention” in the field is disseminated rapidly, generating general interest and unexpected applications. This was for instance the case with solitons, strange attractors, and stochasticity. That scientists from each of the different branches of the theory of nonlinear phenomena should cooperate needs no proof, and the value of this cooperation could be seen at a conference held in Kiev on nonlinear and turbulent processes in physics.* The same approach has been adopted for this collection, which includes articles on regular nonlinear phenomena (vortices, solitons, auto-waves) and some on stochasticity and turbulence. In addition, the collection includes two mathematical articles that develop the Kolmogorov-Amold-Moser theory, the importance of which for the theory of nonlinear dynamic systems is undoubted.

The articles are to some degree grouped around the nonlinear problems of plasma physics and hydrodynamics, in their broadest sense.

Hydrodynamics and plasma physics are traditional sources of exciting problems and ideas for the physics of nonlinear phenomena. We need only recall the classical examples of the discovery of solitons in shallow water, the exact integrability of the Korteweg-de Vries equation, and the discoveries of a strange attractor in the Lorcntz system and stochasticity in Hamiltonian systems with small numbers of degrees of freedom.

These ideas and results, all stimulated by problems in hydrodynamics and plasma physics, quickly gained more general significance for physics as a whole. In turn, many new and efficient techniques are tested on such traditionally difficult subjects as turbulence. For instance, there is the recent application to turbulence of the renormalization group methods, which were successfully employed first in field theory and the physics of phase transitions.

Finally, all this culminates in general fund of knowledge in the
physics of nonlinear phenomena. I hope that the publication of
this collection will advance the progress of physics.

The book was translated from the Russian by Valerii Ilyushchenko, and was first published by Mir in 1986.

Many thanks to Akbar Azimi for providing the raw 2-in-1 scans. We cleaned and OCRed the scans.

The Internet Archive Link

This book has an essay by A. Zhabotinsky of the Belousov–Zhabotinsky reaction

Contents

Preface 7

Vortices In Plasma and Hydrodynamics by A.B. Mikhailovskii

Introduction 8
Nonlinear Equations for Rossby Waves 12
The Simplest Nonlinear Equations for Drift Waves 14
Vector Vortical Structures 16
Scalar Vortical Structures 18
Further Development of Concepts Concerning Electrostatic Vortices in Plasma 19
Electromagnetic Vortices 22
Conclusion 24
References 28

Oscillations and Bifurcations in Reversible Systems by V. I. Arnol’d and M. B. Sevryuk 31

Introduction 31
Reversible Mappings 32
Reversible Flows 33
Integrable Reversible Mappings and Vector Fields 34
Kolmogorov’s Tori 35
Weak Reversibility 37
The Local Theory 37
Weak Reversibility in a Local Situation 42
Periodic Solutions 42
Kolmogorov’s Tori for Additional “Even” Coordinates 44
The Local Theory for Additional “Even” Coordinates 45
Application to Reversible Equations 48
Kon-Autonomour Reversible Systems 49
The Lyapunov-Devaney Theorem 51
The Resonance 1:1 52
Further Resonances l:N (N > 2) 54
References 83

Regular and Chaotic Dynamics of Particles In a Magnetic Field by R. Z. Sagdeev and C. U. Zaslavskii 85

Introduction 65
Equations of Motion 67
The Resonances of Longitudinal Motion 69
The Overlapping of the Resonances of Longitudinal Motion 72
A Kinetic Description 75
Equations of Transverse Motion 78
The Resonances of High-Energy Particles 83
Resonances in a Weak Magnetic Field 84
Generalization to a Wave Packet 85
A Kinetic Description of Transverse Motion 86
Quasi-Resonance Particles 88
References 92

The Renormalization Group Method and Kolmogorov-Arnold-Moser Theory by K. M. Khanin and Ya. G. Sinai 93

Introduction 93
Rectification of the Nonlinear Rotation of a Circle 97
Construction of Invariant KAM Curves by the Renormalization
Group Method 110
References 118

Nonlinear Problems of Turbulent Dynamo by Ya B, Zel’dovich and A. A. Rukmalkin 119

Introduction 119
Nonlinear Mean Field Dynamo 121
MHD Turbulence 130
References 135

Problems of the Theory of Strong Turbulence and Topological Solitons by R. Z. Sagdeev, S. S. Molseev, A. V. Tur, and V. V. Yanovskii 137

Introduction 137
The Scaling Group and Functional Method 140
“Null-Modes” and the Self-Similar Spectrum 152
Invariant Properties of Hydrodynamic Models and Topological Solitons 163
References 180

Self-Oscillations and Auto-Waves in Chemical Systems by A. M. Zhabotinskii 183

Introduction 183
Experimental Studies 184
Theoretical Studies 195
Conclusion 207
References 208

Auto-Waves In Biologically Active Media by V. I. Krinskii 210

Introduction 210
Mathematical Description 211
Local Sources of Auto-Waves 213
Cardiac Disorders 215
Mathematical Simulation of Auto-Wave Sources 216
A Chemically Active Medium 216
New Auto-Wave Modes 217
Wave Sources in Three-Dimensional Active Media 217
The Effect of Medium Parameters on Auto-Wave Sources 218
An Anomalous Reverberator 219
Theoretical Studies of Reverberators 220