Nonlinear Dynamical Systems of Mathematical Physics: Spectral and Symplectic Integrability Analysis

563 Pages·2011·2.292 MB·English

by  Denis Blackmore

7960tp.indd 1 8/24/10 9:27 AM
TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk
This page intentionally left blank
Denis Blackmore
New Jersey Institute of Technology, USA
Anatoliy K Prykarpatsky
AGH University of Science and Technology, Poland
The Ivan Franko State Pedagogical University, Ukraine
Valeriy Hr Samoylenko
Kyiv National Taras Shevchenko University, Ukraine
World Scientific
NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI
7960tp.indd 2 8/24/10 9:27 AM
Published by
World Scientific Publishing Co. Pte. Ltd.
5 Toh Tuck Link, Singapore 596224
USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601
UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
Library of Congress Cataloging-in-Publication Data
Nonlinear dynamical systems of mathematical physics : spectral and symplectic integrability
analysis / by Denis Blackmore ... [et al.].
p. cm.
Includes bibliographical references and index.
ISBN-13: 978-981-4327-15-2 (hardcover : alk. paper)
ISBN-10: 981-4327-15-8 (hardcover : alk. paper)
1. Differentiable dynamical systems. 2. Nonlinear theories. 3. Symplectic geometry.
4. Spectral analysis--Mathematics. I. Blackmore, Denis L.
QA614.8.N656 2011
530.15'539--dc22
2010028336
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
Copyright © 2011 by World Scientific Publishing Co. Pte. Ltd.
All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means,
electronic or mechanical, including photocopying, recording or any information storage and retrieval
system now known or to be invented, without written permission from the Publisher.
For photocopying of material in this volume, please pay a copying fee through the Copyright
Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to
photocopy is not required from the publisher.
Printed in Singapore.
EH - Nonlinear Dynamical Systems.pmd 1 1/6/2011, 4:09 PM
December22,2010 18:4 WorldScientificBook-9inx6in BPSbk
We dedicate this book to our families, whose love, devotion
and patience sustained us; our teachers and mentors, who
enlightened and guided us; our colleagues, students and
friends, who aided and encouraged us; and the pioneers in the
field, whose magnificent contributions inspired us.
v
December22,2010 18:4 WorldScientificBook-9inx6in BPSbk
Preface
This book grew out of our lecture notes for advanced undergraduate and
graduate students of mathematics and physics over the last several years
at the New Jersey Institute of Technology (Newark, NJ, USA), the AGH-
University of Science and Technology (Krakow, Poland), the Ivan Franko
StatePedagogicalUniversity(Drohobych,Ukraine)andT.ShevchenkoNa-
tionalUniversity(Kyiv,Ukraine),aswellasourrecentresearch. Wefirstin-
troducereaderstothestandardmodernbackgroundsofdynamicalsystems
theory on finite-dimensional symplectic manifolds; in particular, the main
differential-geometric and Lie-algebraic aspects of the abelian Hamilton–
Jacobi and Liouville–Arnold integrability theories along with a modern
grounding in the non-abelian Mishchenko–Fomenko integrability aspects
of nonlinear Hamiltonian systems on Poisson manifolds with symmetries.
We have also devoted a chapter to a rather extensive introduction to the
differential-geometric properties of reduced canonically-symplectic mani-
folds with symmetry, and their relationship with structures on principal
fiberbundles. Asnaturalapplications,weanalyzetheHamiltonianproper-
ties of classical electromagnetic Maxwell and Yang–Mills equations. Keep-
inginmindthenumerousapplicationsoftheresultsforstudyingquantum-
mechanical problems, we also describe some of Lie-algebraic aspects of in-
tegrable dynamical systems related to Hopf and quantum algebras.
Next, we provide a detailed introduction to the theory of infinite-
dimensional Hamiltonian dynamical systems on functional manifolds and
describe the formulation of their Lax type integrability analysis in the
framework of a new and very effective gradient-holonomic algorithm, and
present some typical examples of testing the integrability of nonlinear
infinite-dimensional dynamical systems. Of particular note is the integra-
bilityanalysisofaWhithamtypenonlocaldynamicalsystemdescribingthe
vii
December22,2010 18:4 WorldScientificBook-9inx6in BPSbk
viii NONLINEAR DYNAMICAL SYSTEMS OF MATHEMATICAL PHYSICS
nontrivial wave processes in a medium with spatial memory: its Riemann
hydrodynamical generalized regularization and related infinite hierarchies
of conservation laws are constructed, and the Lax integrability is proved.
In an effort to generalize the class of integrable dynamical systems, we
also develop a new parametric-spectral version of the gradient-holonomic
integrability algorithm for nonuniform and non-autonomous nonlinear dy-
namical systems on functional manifolds.
Also presented are some interesting and useful results on the versal de-
formationstructureofaone-dimensionalDiractypelinearspectralproblem
onanaxis. Awholesectionisdevotedtoananalysisoftheclassicalintegra-
bilityinquadraturesofRiccatiandRiccati–Abelordinarydifferentialequa-
tions. By making novel use of B¨acklund transformations in the context of
a Lie-algebraic integrability scheme, we study integrable three-dimensional
couplednonlineardynamicalsystemsrelatedtocentrallyextendedoperator
Lie algebras.
The Lie-algebraic theory of Lax integrable nonlinear dynamical sys-
tems is used to thoroughly analyze the integrability aspects of differential-
difference nonlinear dynamical systems. We show how this approach can
be employed to obtain new results on tensor Poisson structures related to
factorized operator dynamical systems, and in the process give a detailed
Hamiltonian analysis of the systems and provide a complete description of
their integrability properties.
A chapter is devoted to modern aspects of a generalized de Rham–
Hodge theory related to the classical problem of describing Delsarte–Lions
transmutation operators in multi-dimensions within a spectral reduction
approach. This theory naturally gives rise to characteristic Chern classes,
whichweshowcanbeusedtodevelopanewdifferential-geometricintegra-
bility analysis of multi-dimensional differential systems of Gromov type on
Riemannian manifolds.
Developments over past century have clearly demonstrated that the
methods of quantum physics provide powerful tools for studying many
mathematical problems, so we also included some introductory material
thatillustratesthisbywayofrecentresultsobtainedbymeansofquantum
mathematicalapproaches. Inparticular,wepresentaregularizedapproach
based on the classical Fock space embedding technique that allows a com-
pletelinearizationofawideclassofnonlineardynamicalsystemsinHilbert
spaces. A brief description of how the geometric approach can be used to
obtain some relatively new results on quantum holonomic computing is in-
cluded as well. We also revisit in detail modern relativistic electrodynamic
December22,2010 18:4 WorldScientificBook-9inx6in BPSbk
Preface ix
andhadronicstringmodelsusingLagrangianandHamiltonianformalisms.
In order to make the overall treatment as self contained as possible,
we have included at the end a short Supplement covering the essential
differential-geometric preliminaries used throughout the text.
Many parts of the manuscript were discussed with our students and
colleagues, to whom we express our heartfelt thanks. Especially, we ex-
press our cordial appreciation to Maciej B(cid:195)laszak, Nikolai Bogolubov (Jr.),
Vasyl Gafiychyk, Roy Glauber, Vladislav Goldberg, Roy Goodman, Lech
G´orniewicz, Ilona Gucwa, Petro Holod, Yuriy Kozicky, Vasyl Kuybida,
Chjan Lim, Miros(cid:195)law Lu´styk, Anatoliy Logunov, Robert Miura, Ryszard
Mruga(cid:195)la, Ihor Mykytyuk, Jolanta Napora-Golenia, Maxim Pavlov, Zbig-
niewPeradzyn´ski,ZiemowitPopowicz,YaremaPrykarpatsky,MykolaPry-
tula, Anthony Rosato, Anatoliy Samoilenko, Roman Samulyak, Andrey
Shoom, Jan S(cid:195)lawianowski, Ufuk Taneri, John Tavantzis, Lu Ting, Xavier
TricocheandKevinUrban. SpecialthanksareduetoNataliaPrykarpatsky
for her constant support and help in editing the bulk of the manuscript.
Andlast,butnotleast,weexpressthankstoWorldScientificPublishingfor
the opportunityof publish this book as part of their outstanding collection
of titles in mathematics and mathematical physics.
ThecontributiontothisbookbyDenisBlackmorewassupportedinpart
bytheNationalScienceFoundation(GrantCMMI-1029809)andCenterfor
Applied Mathematics and Statistics at NJIT.
The research by Anatoliy K. Prykarpatsky was supported in part by
theEuropeanScientificFoundationthroughanESF-grant-06attheSISSA
and the ICTP-scholarship-07 in Trieste, Italy.
Newark-Cracow:Drohobych-Kyiv
Denis Blackmore
Anatoliy K. Prykarpatsky
Valeriy H. Samoylenko