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Tables of Contents for Peyresq Lectures on Nonlinear Phenomena
Chapter/Section Title
Page #
Page Count
Preface
v
 
Addresses of Contributors
vii
 
Elasticity and Geometry
1
36
B. Audoly
Y. Pomeau
Introduction
1
1
Differential geometry of 2D manifolds
2
9
Developable surfaces
5
2
Geometry of the Poincare half-plane
7
4
Thin plate elasticity
11
17
Euler-Lagrange functional
13
2
Scaling of the FvK equations
15
2
Geometry and the FvK equations
17
6
Thin shells: an example
23
5
Buckling in thin film delamination
28
9
The straight sided blister
29
4
Telephone cord delamination
33
4
Quantum Chaos
37
40
D. Delande
What is Quantum Chaos?
37
9
Classical chaos
37
1
Quantum dynamics
38
2
Semiclassical dynamics
40
2
Physical situations of interest
42
2
A simple example: the hydrogen atom in a magnetic field
44
2
Time scales---Energy scales
46
2
Statistical properties of energy levels---random matrix theory
48
12
Level dynamics
48
3
Statistical analysis of the spectral fluctuations
51
2
Regular regime
53
3
Chaotic regime---random matrix theory
56
2
Random matrix theory---continued
58
2
Semiclassical approximation
60
12
Regular systems---EBK/WKB quantization
60
3
Semiclassical propagator
63
2
Green's function
65
1
Trace formula
66
3
Convergence properties of the trace formula
69
2
An example: the Helium atom
71
1
Conclusion
72
5
The Water-wave Problem as a Spatial Dynamical System
77
18
G. Iooss
Introduction
77
1
Formulation as a reversible dynamical system
77
4
Case of one layer with surface tension at the free surface
77
2
Case of two layers without surface tension
79
2
The linearized problem
81
2
Basic codimension one reversible normal forms
83
6
Case (i)
84
1
Case (ii)
85
1
Case (iii)
86
2
Case (iv)
88
1
Typical results for finite depth problems
89
1
Infinite depth case
90
5
Spectrum of the linearized problem
90
1
Normal forms in infinite dimensions
91
1
Typical results
91
4
Cold Atoms and Multiple Scattering
95
32
R. Kaiser
Classical model of Doppler cooling
95
15
Internal motion: elastically bound electron
96
3
Radiation forces acting on the atom: ``classical approach''
99
4
Resonant radiation pressure
103
2
Dipole force
105
1
Doppler cooling
106
4
Interferences in multiple scattering
110
14
Scattering cross section of single atoms
110
2
Multiple scattering samples in atomic physics
112
1
Dwell time
113
1
Coherent backscattering of light
114
5
Strong localization of light in atom?
119
5
Conclusion
124
3
An Introduction to Zakharov Theory of Weak Turbulence
127
34
M. L. Bellac
Introduction
127
4
Hamiltonian formalism for water waves
131
6
Fundamental equations
131
2
Hamilton's equations of motion
133
2
The pertubative expansion
135
2
The normal form of the Hamiltonian
137
9
Ho=sum of harmonic oscillators
137
1
Nonlinear terms: three wave interactions
138
4
Nonlinear terms: four wave interactions
142
2
Dimensional analysis and scaling laws
144
1
Miscellaneous remarks
145
1
Kinetic equations
146
6
Derivation of the kinetic equations
147
3
Conservation laws
150
2
Stationary spectra of weak turbulence
152
9
Dimensional estimates
152
3
Zakharov transformation
155
2
Examples and final remarks
157
4
Phenomena Beyond All Orders and Bifurcations of Reversible Homoclinic Connections near Higher Resonances
161
40
E. Lombardi
Introduction
161
10
Phenomena beyond all orders in dynamical systems
161
3
A little toy model: from phenomena beyond any algebraic order to oscillatory integrals
164
7
Exponential tools for evaluating mono frequency oscillatory integrals
171
11
Rough exponential upper bounds
171
2
Sharp exponential upper bounds
173
2
Exponential equivalent: general theory
175
4
Exponential equivalent: strategy for nonlinear differential equation
179
3
Resonances of reversible vector fields
182
6
Definitions
182
1
Linear classification of reversible fixed points
183
1
Nomenclature
183
5
The O2+iω resonance
188
5
The (iω0)2iω1 resonance
193
8
Exponential asymptotics of bi-oscillatory integrals
196
2
Strategy for non linear differential equations
198
3
Mathematical Modelling in the Life Sciences: Applications in Pattern Formation and Wound Healing
201
38
P. K. Maini
Introduction
201
1
Models for pattern formation and morphogenesis
202
18
Chemical pre-pattern models
202
5
Cell movement models
207
2
Cell rearrangement models
209
1
Applications
210
5
Coupling pattern generators
215
2
Domain growth
217
2
Discussion
219
1
Models for wound healing
220
10
Corneal wound healing
220
3
Dermal healing
223
7
Discussion
230
1
Conclusions
230
9
Relative Equilibria and Conserved Quantities in Symmetric Hamiltonian System
239
1
J. Montaldi
Introduction
239
1
Hamilton's equations
239
2
Examples
241
2
Symmetry
243
1
Central force problem
243
3
Lie group actions
246
1
Noether's theorem and the momentum map
247
2
Noether's theorem
249
2
Equivariance of the momentum map
251
4
Reduction
255
1
Singular reduction
256
1
Sympletic slice and the reduced space
257
1
Relative equilibria
257
4
Bifurcations of (relative) equilibria
261
1
One degree of freedom
262
1
Higher degrees of freedom
263
1
Geometric bifurcations
264
2
Examples
266
1
Point vortices on the sphere
266
6
Point vortices in the plane
272
3
molecules
275