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.

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

A great monograph! I’ve got its hardcopy, fortunately.