Subject Index

Reference is made to the first page of relevant anicles

Anisotropic Kepler Problem 213 Damped Harmonic Oscillator 71 Anisotropy, easy plane III Darboux Theorem 171,283, Anisotropy, one axis 111 289 Anticommuting Variables 111 Darboux Transformation 305 Antikinks 25 Davey-Stewartson Equation 264,312, Antipulses 138 321 Amol'd webs 25 Denaturation of DNA 115 Attractor 218 Diffusion 46, 192 Autosolitons 138 Dirac Operator 298

Dirac System 298

Backlund Transformation 111,289, Direct Methods 255 Discrete Boltzmann Equation 142

312 Discrete Integrable Evolution Equations 227 Bifurcation Theory 25 Discrete KdV Equation 232 Bi-Hamiltonian Systems 171 Discrete Time Garnier System 227 Bilinear Forms 255 Dissipation 71 Bilinear Forms, multidimensional 270 BKP Hierarchy 255

Dissipative 218 Distributivity 58

Boussinesq Equation 246 DNA 115 Boutroux Transformation 99 Drotnions 321 Breathers 25,305 Duality 58 Brownian Motion 183 Dym Hierarchy 289 Burgers Equation 292 Dynamical Entropy 58 Burgers Equation, forced 292 Burgers Equation, two dimensional 246

Entropy 71

Casorati Detertninant 264 First Integral 125 Cauchy Problem 298 Flow, in a cylinder 135 Chaos 25,58 Fourier Integral 298 Chaos, etymology V Fractal Models 25 Chaos, predictability 58,204 Fractal Repeller 46 Chaos, spatiotemporal 129 Free Energy 71 Chaos, transitions to 37 Functional Integral 111 Chaotic Dynamical Systems 213 Chaotic Maps 183,201,

204 Gauge Equivalence 111

Chaotic Maps, partitions for 204 Gauge Transformations 289

Chaotic Pulsations 166 Gaussian Stochastic Processes 183

Chaotic Scattering 46 Generalized Coherent States 108

Classical Attractor 108 Geodesic Motion 213

Complex Eigenvalue 3 Graded Vector Space III

Computer Experiments 108 Gram Detertninant 255

Conservative 218 Coset Space III Hamiltonian Chaos, infinite dimensional 25 Crum Transformation 283 Hamiltonian Systems, autonomous 171 Crystal Defect Movements 37 Hamiltonian Systems, integrable 159,227 oUved Spacetime 71 Hamiltonian Systems, nearly integrable 151

332

Hamiltonian Systems. nonautonomous I71 Localized Structures 138 Harry Dym Equation 289 Long TIlDe Tails 151 Haussdorf Measures 25 Lorenz Model 125 Heinon-Heiles Model 159 Lyapunov Exponent 25.46. Heisenberg Feromagnet 108 192 Heisenberg Model 111 Helmholtz Equation 305 Mapping. area preserving 218 Hirota Bilinear Formalism 264.270. Mapping. measure preserving 218

275.321 Markov. partitions 201 Hirota Satsuma System 255 Markov. processes 201 Homographic Transformations 275 Maxwell-Bloch System. pumped 99 Hopf Bifurcation, supercritical 135 Maya Diagram 255 Hopf -Cole Transformation, generalized 292 Miura Transformation 283 Hydrogen Bonds 115 Mixing Cascades 183 Hyperbolic Billiard 213 Modified Graded NLSE 111 Hyperbolic Upper Half Plane 213 Modified KdV Equation 298

Modularity 58

Information Dynamics 58 M6bius Mapping 213

Information Flow 204 Multifractal Measures 213

Information Operator 58 Multisoliton Solutions 99.275

Instanton Solutions 325 Multisoliton Solutions, KdV-like 270

Integrability 76, 166, 243 N-soliton Test 270

Integrability, concepts V Navier-Stokes Equation, reduction of 135 Integrable Evolution Equation 289 Navier-Stokes Equations 246 Integrable Models 25, III Noise, Iff 151 Integrable Symplectic Maps 232 Nonequilibrium States 46 Integrable System 298 Non-Integrability 3 Intermittency, spatiotemporal 129 Nonlinear Schriidinger, attractive 25 Irreversibility 3,46, Nonlinear Schriidinger, repulsive 25

71 Nonlinear SchrOdinger Equation 111,243 Isospectra1 Equations 289 255 Iterated Functions Systems 213 Nonlinear Schr!ldinger Equation,

generalized 246

Kac-Moody-Virasoro Algebra 99 Nonlinear Schriidinger Equation,

KAMTheory 25 perturbed 85 KAMTori 218 Nonlinear SchrOdinger Equation,

KdV Equation 159 multidimensional 305 KdV Hierarchy 270,289 Normal Forms 76 Kinetic Theory 192 Kinks 25 ODE, classification 76 Kolmogorov-Sinai Entropy 46 ODE, deconpling 76 KPEquation 246,255, Optical Fibers 85,243

264,321 Order 25 KP Hierarchy 255 Order Parameter 111

Landau-Lifschitz Hamiltonian 108 Painlev~ Analysis 166 Landau-Lifshitz Model 25 Painlev~ Equations 246 Lattice Deformation 37 Painlev~ Property 243 Lattice Theory 58 Painlev~ Test 99 Lax Equation 289 Painlev~ Transcendents 99 Lax Form 264 Partial Difference Equations 232 Lax Pairs 159,227 Participatory Observer 58 Lie Method, classical 246 Percolation 129 Lie Method, nonclassical 246 Period Doubling 218 Lie Point Symmetry Groups 99 Periodic Flow, transition to 135 Lieb-Mattis Theorem 111 Perturbed Toda Lattice 115 Liouville Dynamics 46 Pfaffians 255,283 Liouville-von Neumann Equation 3 Phase Transition 111

333

Phonon Fluctuations 151 Soliton Transcent Motion 37 Phonons 25 Soliton Transmission 243 Pliicker Relation 255,264 Solitons 25,85, Poincart Nonintegrability Theorem 3 275,312, Poincart Sections 25 325 Prediction 58 Solitons, annihilation 298 Propositional Logic 58 Solitons, coupled in rotation 37 Pseudo Differential Operator 289 Solitons, creation 298

Spectral Transform 25,312

Quantum Constraints 25 Squeezed States 71

Quantum Integrable Systems 25 Standard Map 192

Quantum Statistics, fractional 25 Stationary KdV Flows 227

Quartz Oscillators 151 Statistical Mechanics 3,25

Quasi-Monomial Transformations 76 Stimulated Raman Scattering 99 Strange Attractors 25

r-Matrix 289 Subdynamics 192

Raman Pumping 85 Superalgebra III

Rayleigh-Benard Convection, ID 129 Supergroup 111

Resonance 3,46, Supersymmetry III

76, 85 Symbolic Code 213

Reversible Mapping 218 Symmetry 218

Riccati Equation 125 Symmetry Reductions, nonclassical 246

Riemannian Matrix 275 Symplectic Maps, Integrable 227

I7-Models 325 Target Patterns 138

Scattering 312 Thermalisation ll5

Schwarzian Derivative 125 Three Mode Coupling 166

Second Harmonic Generation 166 Tlffie Reversal 218

Segre Characteristic 171 Toda Equation, relativistic 264

Self -Consistent Source 298 Toda Lattice 227

Similarity Reductions 246 Transition to Chaos 183

Similarity, methods 99 Transpon Properties 46

Similarity, shock waves 142 Trial Function Method 108

Similarity, solutions 99 Trilinear Form 255,264

Sine-Gordon Equation 298 Turbulence, transition to 129

Sine-Gordon, classical 25 Singularity Analysis 243 Unstable Particles 71 SK Equation 255,283 SK Hierarchy 270 Variable Stars Skyrmion Scattering 325

166

Skyrmions 325 Variable White Dwarfs 166

Soliton Dynamics 37 Soliton Lattices 232 Zakharov-Shabat Problem 111 Soliton Statistical Mechanics 25 z:z Ceti Stars 166

Index of Contributors

Ablowitz, M., 7.3 Aizawa, Y., 4.1 Atmanspacher, H., 1.5

Beck, C., 5.1 Bessis, D., 5.5 Boiti, M., 8.2 Bonetti, M., 3.2 Borckmans, P., 3.4 Brenig, L., 1.7 Brons, M., 3.3 Bullough, R., 1.2

Capel, H., 5.6, 5.8 Celeghini, E., 1.6 Chen, Yu. 1.2 Christensen. E., 3.3 Christiansen, P., 2.5, 3.3 Clarkson, P.A., 6.2 Conte, R., 3.1 Comille. H.. 3.5

Daviaud, F .• 3.2 de Lillo. S., 7.3 Dewel. G., 3.4 Dubois, M., 3.2

Fordy. A., 4.2

Gaspard. P .• 1.4 Goriely, A., 1.7 Grammaticos, B., 6.1

Hasegawa. H .• 1.1. 5.2 Hereman, W., 4.3

Hietarinta, J., 8.3 Hirota, R., 6.3

Jawarski, M., 6.6

Kajiwara, K., 6.4 Kaup. D.J., 2.1

Lambert, F., 6.5 Lomdahl. P.S .• 2.5

Mac Kernan. D., 5.3 Makhankov. V.G .• 2.3 Mantica. G .• 5.5 Martina. L.. 8.2 Matsukidaira, J.. 6.4 Mel'nikov, V.K .• 7.4 Musette, M., 3.1 Muto, V., 2.5

Nicolis. G., 5.3 Nijboff, F., 5.8 Nimmo, J.J.C., 7.1

Oevel, W., 7.2 Olver, P., 4.4

Papageorgiou, V., 5.8 Pashaev, 0., 2.4, 8.2 Pempinelli, F .• 8.2 Perrone, D., 8.2 Petrovsky, T., 1.1 Piette, B .• 8.4 Pompe, B .• 5.4 Post, T., 5.6

336

Pouget, 1., 1.3 Prigogine, I., 1.1

Quispel, G., 5.6, 5.8

Ragnisco, 0., 5.7 Ramani, A., 6.1 Rasetti, M., 1.6 Roberts, J., 5.6

Sabatier, P.C., 8.1 Saphir, W., 5.2 Satsuma, J., 6.4 Sayadi, M., 1.3

Scott, A.c., 2.5 Sorensen, J., 3.3

Tanaka, K., 4.1 Tasaki, S., 1.1 Timonen, J.,1.2

Verheest, F., 4.3 Vitiello, G., 1.6

Willox, R., 6.5 Winternitz, P., 2.2 Wyser, K., 6.6

Zagrodzinski, J., 6.6 Zakrzewski, W.J., 8.4

N. G. Chetaev

Theoretical Mechanics 1989.407 pp. 190 figs. Hardcover ISBN 3·540·51379-5

This university-level textbook reflects the extensive teaching experience ofN. G. Chataev, one of the most influential teachers of theoreti­cal mechanics in the Soviet Union. The mathe­matically rigorous presentation largely follows the traditional approach, supplemented by material not covered in most other books on the subject. To stimulate active learning numerous carefully selected exercises are provided. Attention is drawn to historical pitfalls and errors that have led to physical misconceptions. Extensive appendices contain material from additional lectures on optics and mechnics analogies, Poincare's equation and the special theory of elasticity.

Distribution rights for the socialist countries India and Iran: ' V 10 "Mezhdunarodnaya Kniga", Moscow

D.Park

Classical Dynamics and Its Quantum Analogues 2nd enl. and updated ed. 1990. IX, 333 pp. 101 figs. Hardcover ISBN 3-540-51398-1

The primary purpose of this textbook is to introduce students to the principles of classical dynamics of particles, rigid bodies, and contin­uous systems while showing their relevance to subjects of contemporary interest. Two of these subjects are quantum mechanics and general relativity. The book shows in many examples the relations between quantum and classical mechanics and uses classical methods to derive most of the observational tests of general rela­tivity. A third area of current interest is in nonlinear systems, and there are discussions of instability and of the geometrical methods used to study chaotic behaviour. In the belief that it is most important at this stage of a student's education to develop clear conceptual under­standing, the mathematics is for the most part kept rather simple and traditional. This book devotes some space to important transitions in dynamics: the development of analytical methods in the 18th century and the invention of quantum mechanics.

A.Hasegawa

Optical Solitons in Fibers 2nd enl. ed. 1990. XII, 79 pp. 25 figs. Softcover ISBN 3-540-51747-2

Already after six months high demand made a new edition of this textbook necessary. The most recent developments associated with two topical and very important theoretical and prac­tical subjects are combined: Solitons as analyti­cal solutions of nonlinear partial differential equations and as lossless signals in dielectric fibers. The practical implications point towards technological advances allowing for an economic and undistorted propagation of signals revolutionizing telecommunications. Starting from an elementary level readily accessible to undergraduates, this pioneer in the field provides a clear and up-to-date exposi­tion of the prominent aspects of the theoretical background and most recent experimental results in this new and rapidly evolving branch of science. This well-written book makes not just easy reading for the researcher but also for the interested physicist, mathematician, and engineer. It is well suited for undergraduate or graduate lecture courses.

A. G. Sitenko

Scattering Theory 1991. XI, 294 pp. 32 figs. (Springer Series in Nuclear and Particle Physics) Hardcover ISBN 3-540-51953-X

This book is an introduction to nonrelativ­istic scattering theory. The presentation is mathematically rigorous, but is accessible to upper level undergraduates in physics. The relationship between the scattering matrix and physical observables, i. e. transition probabilities, is discussed in detail. Among the emphasized topics are the stationary fonnulation of the scattering problem, the inverse scattering problem, dispersion rela­tions, three-particle bound states and their scattering, collisions of particles with spin and polarization phenomena. The analytical properties of the scattering matrix are discussed. Problems round off this volume.

B.N.Zakhariev,A.A.Suzko

Direct and Inverse Problems Potentisls In Qusntum Scattering 1990. XIII, 223 pp. 42 figs. Softcover ISBN 3-540-52484-3

This textbook can almost be viewed as a "how-to" manual for solving quantum inverse problems, that is, for deriving the potential from spectra or scattering data and also, as somewhat of a quantum "picture book" which should enhance the reader's quantum intuition. The fonnal exposition of inverse methods is paralleled by a discussion of the direct problem. Differential and finite­difference equations are presented side by side. The common features and (dis)advan­tages of a variety of solution methods are analyzed. To foster a better understanding, the physical meaning of the mathematical quantities are discussed explicitly. Wave confinement in continuum bound states, resonance and collective tunneling, energy shifts and the spectral and phase equivalence of various interactions are some of the physi­cal problems covered.'

P. C. Sabatier (Ed.)

Inverse Methods inAction Proceedings of the Multicentennials Meeting on Inverse Problems, Montpellier, November 27th - December 1, 1989 1990. XIV, 636 pp. 125 figs. (Inverse Problems and Theoretical Imaging) Hardcover ISBN 3-540-51994-7

The basic idea of inverse methods is to extract from the evaluation of measured signals the details of the object emitting them. The applications range from physics and engineering to geology and medicine (tomography). Although most contributions are rather theoretical in nature, this volume is of practical value to experimentalists and engineers and as well of interest to mathe­maticians. The review lectures and contrib­uted papers are grouped into eight chapters dedicated to tomography, distributed param­eter inverse problems, spectral and scattering inverse problems (exact theory), wave propa­gation and scattering (approximations); miscellaneous inverse problems and applica­tions and inverse methods in nonlinear mathematics.

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