Ray and Wave Chaos in Ocean Acoustics - Chaos in Waveguides

412 Pages·2009·64.54 MB·English

by  Denis Makarov

RAY AND WAVE CHAOS
IN OCEAN ACOUSTICS
Chaos in Waveguides
Series on Complexity, Nonlinearity and Chaos
Series Editor: Albert C.J. Luo
(Southern Illinois University Edwardsville, USA)
Aims and Scope
The books in this series will focus on the recent developments, findings and progress
on fundamental theories and principles, analytical and symbolic approaches,
computational techniques in nonlinear physical science and nonlinear mathematics.
Topics of interest in Complexity, Nonlinearity and Chaos include but not limited
to:
· New findings and discoveries in nonlinear physics and mathematics,
· Complexity and mathematical structures in nonlinear physics,
· Nonlinear phenomena and observations in nature,
· Computational methods and simulations in complex systems,
· New theories, and principles and mathematical methods,
· Stability, bifurcation, chaos and fractals in nonlinear physical science.
Vol. 1 Ray and Wave Chaos in Ocean Acoustics: Chaos in Waveguides
D. Makarov, S. Prants, A. Virovlyansky & G. Zaslavsky
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Advisory Board
Valentin Afraimovich Michael Shlesinger
San Luis Potosi University, IICO-UASLP Office of Naval Research, Physics and
Av. Karakorum 1470, Lomas 4a Seccion Chemistry Department
San Luis Potosi, SLP 78210, Mexico 800 North Quincy Str, Room 503
Email: [email protected] Arlington, VA 22217-5660, USA
Email: [email protected]
Maurice Courbage
Universite Paris 7-Denis Diderot/L.P.T.M.C. Dietrich Stauffer
Tour 24-14.5eme etage, 4, Place Jussieu Institute for Theoretical Physics
75251, Paris Cedex 05, France University of Cologne, Zülpicher Straße 77
Email: [email protected] D-50937 Köln, Germany
Email: [email protected]
Ben-Jacob Eshel
School of Physics and Astronomy, Jian Qiao Sun
Tel Aviv University, 69978 Tel Aviv, Israel Department of Mechanical Engineering,
Email: [email protected] University of Delaware, Newark
DE 19716, USA
Bernold Fiedler Email: [email protected]
Freie Universität Berlin, Institut für
Mathematik I, Arnimallee 2-6 Dimitry Treschev
14195 Berlin, Germany Department of Mechanics and Mathematics
E-mail: [email protected] Moscow State University, Leninskie Gogy
Moscow 119899, Russia
James A. Glazier E-mail: [email protected]
Biocomplexity Institute
Department of Physics Vladimir V. Uchaikin
Indiana University, Bloomington Ulyanovsk State University
Bloomington, IN 47405-7105, USA L.Tolstoy Str., 42, Ulyanovsk 432700
Email: [email protected] Russia
Email: [email protected]
Nail Ibragimov
Department of Mathematics, IHN, Angelo Vulpiani
Blekinge Institute of Technology, Department of Physics,
S-371 79 Karlskrona, Sweden University La Sapinza, P.le A.Moro 2
Email: [email protected] 00185 Roma, Italy
E-mail: [email protected]
Anatoly Neishtadt
Space Research Institute Pei Yu
Russian Academy of Sciences Department of Applied Mathematics
Profsoyuznaya 84/32 The University of Western Ontario, London
Moscow 117997, Russia Ontario N6A 5B7, Canada
E-mail: [email protected] Email: [email protected]
Leonid Shilnikov
Research Institute for Applied Mathematics &
Cybernetics
10 Ul’yanov Street
Nizhny Novgorod 603005, Russia
Email: [email protected]
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Series on Complexity, Nonlinearity and Chaos – Vol. 1
RAY AND WAVE CHAOS
IN OCEAN ACOUSTICS
Chaos in Waveguides
Denis Makarov, Sergey Prants
Pacific Oceanological Institute, Vladivostok, Russia
Anatoly Virovlyansky
Institute of Applied Physics, Nizhny Novgorod, Russia
George Zaslavsky
New York University, New York, USA
World Scientific
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British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
Series on Complexity, Nonlinearity, and Chaos — Vol. 1
RAY AND WAVE CHAOS IN OCEAN ACOUSTICS
Chaos in Waveguides
Copyright © 2010 by World Scientific Publishing Co. Pte. Ltd.
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ISBN-13 978-981-4273-17-6
ISBN-10 981-4273-17-1
Printed in Singapore.
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To the memory of George Zaslavsky
George Zaslavsky (May 31, 1935–November 25, 2008)
ThisisthelastbookbyGeorgeZaslavsky. Heplannedittowardstheendof
2007whenhealreadyknewabouthismortaldisease. Webeganwritingthis
book in three cities, New York, Nizhny Novgorod, and Vladivostok. His
strongwillandorganizationalskills wasthe forcebehindthe completionof
the book after he died in New York on 25 November, 2008.
GeorgewasborninOdessain1935inafamilyofanartilleryofficerwho
dragged his cannon in World War II, and survived. George received his
education at the University of Odessa and moved in 1957 to Novosibirsk
when a goldenage of Siberianphysics had dawned. In 1965,Georgejoined
the Institute of Nuclear Physics where he became interested in nonlinear
problems of accelerator and plasma physics. Roald Sagdeev and Boris
Chirikov kindled his interest in the theory of dynamical chaos. In 1968,
GeorgewithhisformerwifeN.Filonenkointroducedaseparatrixmapthat
became one of the major tools in theoretical study of Hamiltonian chaos.
Thework“Stochasticalinstabilityofnonlinearoscillations”byG.Zaslavsky
and B. Chirikov, published in Physics Uspekhi in 1971, was the first re-
view paper that “opened the eyes” of many physicists to the power of the
theory of dynamical systems and modern ergodic theory. It was realized
v
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vi Ray and Wave Chaos inOcean Acoustics
that very complicated behavior is possible in dynamical systems with only
a few degrees of freedom. This complexity cannot be adequately described
in terms of individual trajectories, and requires statistical methods. Typ-
ical Hamiltonian systems are not integrable but chaotic, and such chaos
is not homogeneous. For the same values of the control parameters, there
coexistregionsinphasespacewithregularandchaoticmotion. Theresults
obtainedinthe60thweresummarizedinthebookStatistical Irreversibility
in Nonlinear Systems (Nauka, Moscow, 1970).
The late 1960s was a hard time for George. He was forced to leave
the Institute of Nuclear Physics in Novosibirsk for signing letters in de-
fense of some Soviet dissidents. George got a position at the Institute of
Physics in Krasnoyarsk,not far away fromNovosibirsk. There, he founded
a laboratory engaged in the theory of nonlinear processes which exists up
to now. In Krasnoyarsk, George became interested in the theory of quan-
tum chaos. The first rigorous theory of quantum resonance was developed
in 1977 in collaboration with Gennady Berman. They introduced the im-
portant notion of quantum break time (the Ehrenfest time) after which
quantum evolution begins to deviate from a semiclassical one. The results
obtainedinKrasnoyarskweresummarizedinthebookChaos in Dynamical
Systems (Nauka, Moscow and Harwood, Amsterdam, 1985).
In 1981, George and Sadrilla Abdullaev published the first paper on
chaotic instability of sound rays in idealized underwater waveguides. The
first results of their studies on this topic were summarized in a review
paper published in Physics Uspekhi in 1991. It was one of his main focus
of researchup to the last days.
In1984,R.SagdeevinvitedGeorgetotheInstituteofSpaceResearchin
Moscow. There he worked on the theory of degenerate and almost degen-
erate Hamiltonian systems, anomalous chaotic transport, plasma physics,
and theory of chaos in waveguides. The book Nonlinear Physics: from the
Pendulum to Turbulence and Chaos (Nauka, Moscow and Harwood, New
York, 1988), written with R. Sagdeev, is now a classical textbook for
everybody who studies chaos theory. When studying interaction of a
charged particle with a wave packet, George with colleagues from the
institutediscoveredthatstochasticlayersofdifferentseparatricesindegen-
eratedHamiltoniansystemsmaymergeproducingastochasticweb. Unlike
the famous Arnolddiffusioninnon-degeneratedHamiltoniansystems,that
appears only if the number of degrees of freedom exceeds 2, diffusion in
the Zaslavsky webs is possible at one and half degrees of freedom. This
diffusion is a rather universal phenomenon and its speed is much greater
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To the Memory of George Zaslavsky vii
than that of Arnold diffusion. Beautiful symmetries of the Zaslavsky webs
and their properties in different branches of physics have been described
in the book Weak Chaos and Quasi-Regular Structures (Nauka, Moscow,
1991 and Cambridge University Press, Cambridge, 1991) coauthored with
R. Sagdeev, D. Usikov and A. Chernikov. A funny review on this publica-
tionwaspublishedinaCambridgestudentnewspaper. Thebookappeared
duringthetimeofdisintegrationoftheSovietUnionandthereviewerwrote:
“Once again those Russians with their chaos even if it is a weak one!”.
In 1991, George emigrated to the USA and became a Professor of
Physics and Mathematics at the Department of the New York University
andattheCourantInstituteofMathematicalSciences. Thelast17yearsof
his life weredevotedto principalproblemsofHamiltonianchaosconnected
withanomalouskinetics andfractionaldynamics,foundations ofstatistical
mechanics, chaotic advection, quantum chaos, and long-range propagation
of acoustic waves in the ocean. During his New York period, George pub-
lishedtwoimportantbooksontheHamiltonianchaos: Physics of Chaos in
Hamiltonian Systems(ImperialCollegePress,London,1998)andHamilto-
nian Chaos and Fractional Dynamics(OxfordUniversityPress,NY,2005).
George was a very creative scientist and a very good teacher whose
former students and collaborators are now working in America, Europe,
and Asia. He authored and coauthored 9 books and more than 300 papers
in journals. Many of his works are widely cited. George worked hard all
hislife. Helovedmusic,theater,literatureandwasanexpertongoodwine
andfood. Onlyafewpeople knewthathe lovedtopaint. Inthe lastyears,
he spent every summer in Provence (France) working, writing books and
papers and painting in water-colors; the album with his water-colors was
issued in 2009 in Moscow.
George Zaslavsky was one of the key persons in the theory of dynam-
ical chaos and made many important contributions to a variety of other
topics. His books and papers have influenced very much the advancement
of modern nonlinear science.
We wish to dedicate this book to the memory of George Zaslavsky.
Denis Makarov
Sergey Prants
Anatoly Virovlyansky
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