Table Of ContentIntroduction to Soliton Theory: Applications to Mechanics
Fundamental Theories of Physics
An International Book Series on The Fundamental Theories of Physics:
Their Clarification, Development and Application
Editor:
ALWYN VAN DER MERWE, University of Denver, U.S.A.
Editorial Advisory Board:
GIANCARLO GHIRARDI, University of Trieste, Italy
LAWRENCE P. HORWITZ, Tel-Aviv University, Israel
BRIAN D. JOSEPHSON, University of Cambridge, U.K.
CLIVE KILMISTER, University of London, U.K.
PEKKA J. LAHTI, University of Turku, Finland
ASHER PERES, Israel Institute of Technology, Israel
EDUARD PRUGOVECKI, University of Toronto, Canada
FRANCO SELLERI, Università di Bara, Italy
TONY SUDBURY, University of York, U.K.
HANS-JÜRGEN TREDER, Zentralinstitut für Astrophysik der Akademie der
Wissenschaften, Germany
Volume143
Introduction to Soliton
Theory: Applications
to Mechanics
by
Ligia Munteanu
Institute of Solid Mechanics,
Romanian Academy, Bucharest, Romania
and
Stefania Donescu
Technical University of Civil Engineering,
Department of Mathematics, Bucharest, Romania
KLUWER ACADEMIC PUBLISHERS
NEW YORK,BOSTON, DORDRECHT, LONDON, MOSCOW
eBookISBN: 1-4020-2577-7
Print ISBN: 1-4020-2576-9
©2005 Springer Science + Business Media, Inc.
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Dordrecht
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Contents
Preface ix
Part 1. INTRODUCTION TO SOLITON THEORY
1. MATHEMATICALMETHODS 1
1.1 Scope of the chapter 1
1.2 Scattering theory 1
1.3 Inverse scattering theory 12
1.4 Cnoidalmethod 17
1.5 Hirotamethod 25
1.6 Linear equivalencemethod (LEM) 31
1.7 Bäcklundtransformation 39
1.8 Painlevé analysis 46
2. SOME PROPERTIES OF NONLINEAR EQUATIONS
53
2.1 Scope of the chapter 53
2.2 General properties of the linear waves 53
2.3 Someproperties of nonlinear equations 59
2.4 Symmetrygroups of nonlinear equations 62
2.5 Noether theorem 66
2.6 Inverse Lagrange problem 69
2.7 Recursion operators 73
3. SOLITONS AND NONLINEAR EQUATIONS
78
3.1 Scope of the chapter 78
3.2 Korteweg and de Vries equation (KdV) 78
3.3 Derivation of the KdV equation 86
vi INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS
3.4 Scattering problem for the KdV equation 90
3.5 Inverse scattering problem for the KdV equation 95
3.6 Multi-soliton solutions of the KdV equation 101
3.7 Boussinesq, modified KdV and Burgers equations 107
3.8 The sine-Gordon and Schrödinger equations 112
3.9 Tricomi system and the simple pendulum 115
121
Part 2. APPLICATIONS TO MECHANICS
4. STATICS AND DYNAMICS OF THE THIN ELASTIC ROD 121
4.1 Scope of the chapter 121
4.2 Fundamental equations 122
4.3 The equivalence theorem 132
4.4 Exact solutions of the equilibrium equations 134
4.5 Exact solutions of the motion equations 146
5. VIBRATIONS OF THIN ELASTIC RODS
149
5.1 Scope of the chapter 149
5.2 Linear and nonlinear vibrations 149
5.3 Transverse vibrations of the helical rod 155
5.4 A special class of DRIP media 159
5.5 Interaction of waves 163
5.6 Vibrations of a heterogeneous string 166
6. THE COUPLED PENDULUM
173
6.1 Scope of the chapter 173
6.2 Motion equations. Problem E1 173
6.3 Problem E2 177
6.4 LEM solutions of the system E2 180
6.5 Cnoidal solutions 185
6.6 Modal interaction in periodic structures 191
7. DYNAMICS OF THE LEFT VENTRICLE
197
7.1 Scope of the chapter 197
7.2 The mathematical model 198
7.3 Cnoidal solutions 206
7.4 Numerical results 209
7.5 A nonlinear system with essential energy influx 213
CONTENTS vii
8. THE FLOW OF BLOOD IN ARTERIES
220
8.1 Scope of the chapter 220
8.2 A nonlinear model of blood flow in arteries 221
8.3 Two-soliton solutions 228
8.4 A micropolar model of blood flow in arteries 235
9. INTERMODAL INTERACTION OF WAVES
242
9.1 Scope of the chapter 242
9.2 A plate with Cantor-like structure 243
9.3 The eigenvalue problem 248
9.4 Subharmonic waves generation 249
9.5 Internal solitary waves in a stratified fluid 255
9.6 The motion of a micropolar fluid in inclined open channels 259
9.7 Cnoidal solutions 265
9.8 The effect of surface tension on the solitary waves 269
10. ON THE TZITZEICA SURFACES AND SOME RELATED
273
PROBLEMS
10.1 Scope of the chapter 273
10.2 Tzitzeica surfaces 273
10.3 Symmetry group theory applied to Tzitzeica equations 276
10.4 The relation between the forced oscillator and a Tzitzeica 283
curve
10.5 Sound propagation in a nonlinear medium 285
10.6 The pseudospherical reduction of a nonlinear problem 291
References 298
Index 305
Preface
This monograph is planned to provide the application of the soliton theory to solve
certain practical problems selected from the fields of solid mechanics, fluid mechanics
and biomechanics. The work is based mainly on the authors’ research carried out at
their home institutes, and on some specified, significant results existing in the published
literature. The methodology to study a given evolution equation is to seek the waves of
permanent form, to test whether it possesses any symmetry properties, and whether it is
stable and solitonic in nature.
Students of physics, applied mathematics, and engineering are usually exposed to
various branches of nonlinear mechanics, especially to the soliton theory. The soliton is
regarded as an entity, a quasi-particle, which conserves its character and interacts with
the surroundings and other solitons as a particle. It is related to a strange phenomenon,
which consists in the propagation of certain waves without attenuation in dissipative
media. This phenomenon has been known for about 200 years (it was described, for
example, by the Joule Verne's novel Les histoires de Jean Marie Cabidoulin, Éd.
Hetzel), but its detailed quantitative description became possible only in the last 30
years due to the exceptional development of computers.
The discovery of the physical soliton is attributed to John Scott Russell. In 1834,
Russell was observing a boat being drawn along a narrow channel by a pair of horses.
He followed it on horseback and observed an amazing phenomenon: when the boat
suddenly stopped, a bow wave detached from the boat and rolled forward with great
velocity, having the shape of a large solitary elevation, with a rounded well-defined
heap of water. The solitary wave continued its motion along the channel without change
of form or velocity. The scientist followed it on horseback as it propagated at about
eight or nine miles an hour, but after one or two miles he lost it. Russell was convinced
that he had observed an important phenomenon, and he built an experimental tank in his
garden to continue the studies of what he named the wave of translation.
The wave of translation was regarded as a curiosity until the 1960s, when scientists
began to use computers to study nonlinear wave propagation. The discovery of
mathematical solutions started with the analysis of nonlinear partial differential
equations, such as the work of Boussinesq and Rayleigh, independently, in the 1870s.
Boussinesq and Rayleigh explained theoretically the Russell observation and later
reproduction in a laboratory experiment. Korteweg and de Vries derived in 1895 the
equation for water waves in shallow channels, and confirmed the existence of solitons.
x INTRODUCTION TO SOLITON THEORY : APPLICATIONS TO MECHANICS
An explosion of works occurred when it was discovered that many phenomena in
physics, electronics, mechanics and biology might be described by using the theory of
solitons. Nonlinear mechanics is often faced with the unexpected appearance of chaos
or order. Within this framework the soliton plays the role of order. The discovery of
orderly stable pulses as an effect of nonlinearity is surprising. The results obtained in
the linear theory of waves, by ignoring the nonlinear parts, are most frequently too far
from reality to be useful. The linearisation misses an important phenomenon, solitons,
which are waves, which maintain their identity indefinitely just when we most expect
that dispersion effects will lead to their disappearance. The soliton as the solution of the
completely integrable partial differential equations are stable in collision process even if
interaction between the solitons takes place in a nonlinear way.
The unexpected results obtained in 1955 by Fermi, Pasta and Ulam in the study of a
nonlinear anharmonic oscillator, generate much of the work on solitons. Their attempt
to demonstrate that the nonlinear interactions between the normal modes of vibrations
lead to the energy of the system being evenly distributed throughout all the modes, as a
result of the equipartition of energy, failed. The energy does not spread throughout all
the modes but recollect after a time in the initial mode where it was when the
experiment was started.
In 1965, Zabusky and Kruskal approached the Fermi, Pasta and Ulam problem from
the continuum point of view. They rederived the Korteweg and de Vries equation and
found its stable wave solutions by numerical computation. They showed that these
solutions preserve their shape and velocities after two of them collide, interact and then
spread apart again. They named such waves solitons.
Gardner, Green, Kruskal and Miura introduced in 1974 the Inverse Scattering
Transform to integrate nonlinear evolution equations. The conserved features of solitons
become intimately related to the notion of symmetry and to the construction of
pseudospherical surfaces. The Gauss–Weingarten system for the pseudospherical
surfaces yields sine-Gordon equation, providing a bridge to soliton theory.
A privileged surface related to the certain nonlinear equations that admit solitonic
solutions, is the Tzitzeica surface (1910). Developments in the geometry of such surface
gave a gradual clarification of predictable properties in natural phenomena.
A remarkable number of evolution equations (sine-Gordon, Korteweg de Vries,
Boussinesq, Schrödinger and others) considered by the end of the 19th century,
radically changed the thinking of scientists about the nature of nonlinearity. These
equations admit solitonic behavior characterized by an infinite number of conservation
laws and an infinite number of exact solutions.
In 1973, Wahlquist and Estabrook showed that these equations admit invariance
under a Bäcklund transformation, and possess multi-soliton solutions expressed as
simple superposition formulae relating explicit solutions among themselves.
The theory of soliton stores the information on some famous equations: the
Korteweg de Vries equation, the nonlinear Schrödinger equation, the sine-Gordon
equation, the Boussinesq equation, and others. This theory provides a fascinating
glimpse into studying the nonlinear processes in which the combination of dispersion
and nonlinearity together lead to the appearance of solitons.
This book addresses practical and concrete resolution methods of certain nonlinear
equations of evolution, such as the motion of the thin elastic rod, vibrations of the initial
deformed thin elastic rod, the coupled pendulum oscillations, dynamics of the left
PREFACE xi
ventricle, transient flow of blood in arteries, the subharmonic waves generation in a
piezoelectric plate with Cantor-like structure, and some problems of deformation in
inhomogeneous media strongly related to Tzitzeica surfaces. George Tzitzeica is a great
Romanian geometer (1873–1939), and the relation of his surfaces to the soliton theory
and to certain nonlinear mechanical problems has a long history, owing its origin to
geometric investigations carried out in the 19th century.
The present monograph is not a simple translation of its predecessor which appeared
at the Publishing House of the Romanian Academy in 2002. Major improvements
outline the way in which the soliton theory is applied to solve some engineering
problems. In each chapter a different problem illustrates the common origin of the
physical phenomenon: the existence of solitons in a solitonic medium.
The book requires as preliminaries only the mathematical knowledge acquired by a
student in a technical university. It is addressed to both beginner and advanced
practitioners interested in using the soliton theory in various topics of the physical,
mechanical, earth and life sciences. We also hope it will induce students and engineers
to read more difficult papers in this field, many of them given in the references.
Authors
