Ames W.F., Roger C. Nonlinear equations in the applied sciences. Volume 185
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Nonlinear Equations in
the Applied Sciences
This is volume
185
in
MATHEMATICS IN SCIENCE
AND
ENGINEERING
Edited
by
William
F.
Ames,
Georgia
Institute
of
Technology
A list
of
recent titles in this series appears
at
the end
of
this volume.
NONLINEAR EQUATIONS
IN
THE APPLIED SCIENCES
Edited
by
W.
F.
Ames
ATLANTA, GEORGIA
C.
Rogers
LOUGHBOROUGH,
LEICESTERSHIRE
ENGLAND
SCHOOL
OF
MATHEMATICS
GEORGIA INSTITUTE
OF
TECHNOLOGY
DEPARTMENT
OF
MATHEMATICAL
SCIENCES
LOUGHBOROUGH
UNIVERSITY
OF
TECHNOLOGY
ACADEMIC
PRESS,
INC.
Harcourt
Brace
Jouanovich,
Publishers
Boston San Diego New York
London Sydney Tokyo Toronto
THIS
BOOK
IS
PRINTED ON ACID-FREE PAPER.
COPYRIGHT
@
1992
BY
ACADEMIC
PRESS,
INC.
ALL RIGHTS RESERVED.
NO PART OF THIS PUBLICATION MAY BE REPRODUCED OR
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ANY
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OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR
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INC.
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92101
United
Kingdom
Edition
published
by
ACADEMIC
PRESS
LIMITED
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7DX
Library
of
Congress Cataloging-in-Publication Data
Nonlinear equations in the applied sciences
/
edited
by
W.E
Ames,
C.
Rogers.
p.
cm.
-
(Mathematics in sciences and engineering;
v.
185)
Includes bibliographical references and index.
ISBN 0-12-056752-0 (alk. paper)
1. Nonlinear theories.
I.
Ames,
William
E
11.
Rogers,
C.
I
I
I.
Series.
QA427.N65 1992
515-dc20
9 1
-
18
3
19
CIP
PRINTED IN THE UNITED STATES OF AMERICA
92939495 987654321
Contents
Contributors
Preface
Karen A. Ames
Improperly Posed Problems for Nonlinear Partial Differential
Equations
W.
F.
Ames
Symmetry in Nonlinear Mechanics
Shui-Nee Chow and Masahiro Yamashita
Geometry
of
the Melnikov Vector
Andrea Donato
Nonlinear Waves
A.
S.
Fokas
Integrable Nonlinear Equations
Benno Fuchssteiner
Hamiltonian Structure and Integrability
Greg King and Ian Stewart
Symmetric Chaos
B.
G. Konopelchenko and C. Rogers
Backlund and Reciprocal Transformations: Gauge
Connections
R.
H.
Martin, Jr. and
M.
Pierre
Nonlinear Reaction-Diffusion Systems
M. C. Nucci
Riccati-type Pseudopotentials and Their Applications
vii
ix
1
31
79
149
175
211
257
317
363
399
V
vi
Contents
R.
W.
Ogden
Nonlinear Elasticity: Incremental Equations and Bifurcation
Phenomena
437
Index
469
List
of
Recent Titles
475
Contributors
Numbers in parentheses indicate the pages
on
which the authors’ contribu-
tions begin.
Karen A. Ames
(l),
Department of Mathematical Sciences, University
of Alabama in Huntsville, Huntsville, Alabama
35899
W.
F. Ames
(31),
School of Mathematics, Georgia Institute of
Technology, Atlanta, Georgia
30332
Shui-Nee Chow
(79),
Center for Dynamical Systems and Nonlinear
Studies, School of Mathematics, Georgia Institute of Technology, Atlan-
ta, Georgia
30332
Andrea Donato
(149),
Department of Mathematics, University
of
Messina,
98010
Sant’Agata, Messina, Italy
A.
S.
Fokas
(175),
Department of Mathematics and Computer Science
and the Institute for Nonlinear Studies, Clarkson University, Potsdam,
New York
13699-5815
Benno Fuchssteiner
(211),
Faculty of Mathematics, University of
Paderborn, D
4790
Paderborn, Germany
Greg King
(257),
Nonlinear Systems Laboratory, Mathematics In-
stitute, University of Warwick, Coventry
CV4
7AL,
England
B.
G.
Konopelchenko
(317),
Institute of Nuclear Physics, Siberian
Division of the Academy of Sciences USSR,
630090
Novosibirsk, USSR
R.
H.
Martin, Jr.
(363),
Department of Mathematics, North Carolina
State University, Raleigh, North Carolina
27695-8205
M.
C.
Nucci
(399),
School
of
Mathematics, Georgia Institute of
Technology, Atlanta, Georgia
30332
vii
viii Contributions
R.
W.
Ogden
(4371, Department
of
Mathematics, University
of
Glasgow, Glasgow G12 8QW, Scotland
M. Pierre
(363), University
of
Nancy, Nancy 54000, France
C.
Rogers
(317), Department
of
Mathematical Sciences, Loughborough
University
of
Technology, Loughborough, Leicestershire LEll 3TU,
England
Ian Stewart
(257), Nonlinear Systems Laboratory, Mathematics In-
stitute, University
of
Warwick, Coventry CV4
7AL,
England
Masahiro Yamashita
(79), Department
of
Mathematics and Statistics,
Wright State University, Dayton, Ohio 45435
Preface
Nature pays no attention to the linearity requirement of much of classical
mathematics. Consequently, the equations describing various phenomena
are usually nonlinear-an ambiguous negative definition. There are a
considerable number of classes of nonlinear equations that have received
attention. The present collection of eleven articles addresses several
physically motivated systems for which much theory is available. These
include reaction-diffusion systems, elasticity and other areas of mechan-
ics, and nonlinear waves. Theories available for wider classes of equa-
tions include discussions of Lie symmetries, improperly posed problems,
integrable nonlinear equations, Backlund and reciprocal transformations,
Hamilton structures and integrability, and Riccati type pseudo-potentials.
A
detailed study of the geometry of the Melnikov vector of dynamical
systems and an article on popular symmetric chaos round out the volume.
While some theory
is
necessary, in general the authors remember that
the main purpose of the book is to address real situations. Thus a plethora
of applications is available for the reader to see how far some of the
studies of the nonlinear world have progressed. We hope that the
developing nonlinear world will be somewhat more understandable as a
result of this volume. The stimulation of additional research is also one of
our objectives.
W.
F.
Ames, Atlanta, Georgia,
USA
C. Rogers, Loughborough, England
ix