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Tables of Contents for Proceedings of the Euroconference on Nonlinear Klein-Gordon and Schrodinger Systems
Chapter/Section Title
Page #
Page Count
Preface
v
Lectures on Nonlinear Klein-Gordon Equation
1
38
M. Balabane
1 The Hamiltonian Framework for the Wave Equation
1
4
1.1 Hamiltonian framework in finite dimension
1
2
1.2 Wave equations as Hamiltonian systems
3
1
1.3 Conservation of energy
4
1
2 Estimates for the Linear Klein-Gordon Equation
5
7
2.1 Energy estimates
5
1
2.2 Uniform estimates and decay
6
1
2.3 The Strichartz Lp - Lq estimates for (LW)
7
1
2.4 Marshall-Strauss-Wainger Lp - Lq estimate for (LKG)
8
3
2.5 A space-time Lr estimate
11
1
3 Finite Speed of Propagation and Localization
12
2
4 On Local Existence
14
2
4.1 A basic example
14
2
5 Conservation Laws and Multipliers
16
5
5.1 Introduction: conservation of energy for (LKG)
16
1
5.2 Conservation of energy
17
1
5.3 Conservation of momentum
17
1
5.4 Lorentz transformations
17
1
5.5 Dilatations
18
1
5.6 Inversions
18
1
5.7 Klainerman estimates
19
2
6 Blow-Up Results
21
8
6.1 The basic example
21
1
6.2 Levine's result
22
1
6.3 Glassey's result
23
2
6.4 Kato's results for the wave equation
25
2
6.5 Optimal results for weak nonlinearities
27
2
7 Global Existence in the Dispersive Case
29
3
7.1 Existence of solutions
29
1
7.2 Uniqueness
30
1
7.3 The critical power for the wave equation
31
1
8 Global Existence for Small Cauchy Data
32
7
8.1 How smallness of Cauchy Data comes into the game
32
1
8.2 The basic example: The cubic nonlinearity
33
1
8.3 The normal forms technique according to Shatah
34
2
8.4 Klainerman's technique
36
1
8.5 For d = 2 and d = 3
36
3
Mathematical Aspects of the Nonlinear Schrodinger Equation
39
29
G. Velo
1 Introduction
39
2
2 Properties of the Schrodinger Group
41
4
3 The Cauchy Problem in L2 (Rn)
45
4
4 The Critical Values of the Power of the Nonlinearity
49
3
5 The Cauchy Problem in H1
52
5
6 Non Existence of Global Solutions
57
5
7 Scattering Theory
62
6
Stochastic Partial Differential Equations, Infinite Dimensional Stochastic Processes and Random Fields: A Short Introduction
68
20
S. Albeverio
F. Russo
1 Introduction
68
1
2 Stochastic Parabolic Partial Differential Equations: Solutions by the Method of Dirichlet Forms
69
7
2.1 How Dirichlet forms arise
69
1
2.2 The general setting
70
2
2.3 Examples
72
4
3 Remark on Other Methods for Stochastic Evolution Problems
76
1
4 Methods for Stochastic Wave Equations
77
2
4.1 The case d = 1
78
1
4.2 The case d > 2
78
1
5 The Colombeau Setting for the Product of Distributions
79
9
Stochastic Differential Equations. A Pedagogical Random Walk
88
22
L. Streit
1 From Brownian Motion to Stochastic Analysis
88
8
1.1 White noise and Brownian motion
88
1
1.2 Stochastic integration
89
4
1.3 Solving SDEs
93
3
2 From Paths to Fields: Some Remarks on SPDEs
96
4
2.1 Stochastic partial differential equations
96
1
2.2 Gradient coupled noise
97
1
2.3 The case of space time dependent noise
98
2
3 From White Noise to Oksendal's Method
100
5
3.1 Nonlinear functions of white noise
100
4
3.2 B. Oksendal's regularization of SPDEs
104
1
4 Bibliography
105
5
Spectral Methods for Solving Nonlinear Klein-Gordon Equation
110
25
B. -Y. Guo
Li Xun
1 Introduction
110
1
2 Fourier Spectral Method for Periodic Problem
111
9
3 Fourier Pseudospectral Method
120
6
4 Legendre Pseudospectral Method
126
9
Some Existence and Stability Problems of Solitary Waves in Nonlinear Schrodinger and Klein-Gordon Systems
135
26
K. H. Spatschek
1 Introduction
135
6
1.1 Scalar NLS equation
135
2
1.2 Zakharov equations
137
1
1.3 Optical solitons
138
2
1.4 Coupled waveguides
140
1
2 Existence of Localized Solutions in Discrete Systems
141
3
2.1 The concept of generating functions
141
2
2.2 Results for discrete Schrodinger equations
143
1
3 Some Stability Results in One Space Dimension
144
5
3.1 Variational principles for instability
144
2
3.2 Results for discrete Schrodinger equations
146
2
3.3 Results for discrete Klein-Gordon equations
148
1
4 Remarks on Robustness of Solitons
149
3
4.1 Structural perturbations
149
1
4.2 Low-dimensional attractors
150
2
5 Multidimensional Effects
152
6
5.1 Modulational instability
152
1
5.2 Wave collapse
153
2
5.3 Liapunov methods
155
1
5.4 Example of a stable three-dimensional Schrodinger soliton
156
2
6 Summary and Outlook
158
3
Mode Interactions in Nonlinear Discrete Klein-Gordon Models
161
30
V. V. Konotop
1 Introduction
161
1
2 Statement of the Problem
162
2
3 Properties of Linear Lattices
164
6
3.1 Dispersion relation
164
1
3.2 Group velocity and its dispersion: Monoatomic lattice
164
2
3.3 Peculiarities of diatomic lattices
166
2
3.4 Group velocity and its dispersion: Diatomic lattice
168
2
4 Multiscale Expansion
170
4
4.1 General remarks
170
1
4.2 Single-mode dynamics
171
3
5 Envelope Solitons in Lattices with Long-Range Interactions
174
5
5.1 Coexistence of bright and dark solitons
174
1
5.2 Two-mode solitons
175
4
6 Coupled Solitons in a Diatomic Lattice with Quartic Nonlinearity
179
7
6.1 Soliton in a lattice with small gap: Equation for the envelope
180
2
6.2 Soliton in a lattice with small gap: Role of the nonlinearity and examples
182
3
6.3 Coupled solitons in the diatomic lattice with finite gap
185
1
7 Second-Harmonic Generation in a Diatomic Lattice with Cubic Nonlinearity
186
3
8 Conclusion
189
2
Nonlinear Equations and Structural Phase Transitions
191
24
A. R. Bishop
1 Introduction
191
1
2 Nonlinear Equations and Continuous Structural Phase Transitions
192
12
3 Hierarchical Structures in Martensitic Materials
204
9
4 Discussion
213
2
On Some NLS Systems and Their Applications
215
19
P. L. Christiansen
K. O. Rasmussen
M. Johansson
Yu. B. Gaididei
O. Bang
1 The Nonlinear Schrodinger Equation and Temperature
215
7
2 A Model of the Scheibe Aggregate
222
8
3 A Nonlocal Nonlinear Schrodinger Equation
230
4
Modulational Instability, a Source for Solitons and Localized Modes
234
23
M. Remoissenet
1 Introduction
234
2
2 Modulational Instability of a Single Wave
236
8
2.1 Modulational instability as a resonant-wave interaction
236
1
2.2 Modulational instability in the framework of the nonlinear Schrodinger equation
237
1
2.3 Modulational instability conditions
238
2
2.4 Experimental observations of modulational instability
240
3
2.5 Generalizations of the nonlinear Schrodinger model
243
1
3 Modulational Instability of Two Coupled Waves
244
4
3.1 Coupled nonlinear Schrodinger equations
244
1
3.2 Modulational instability conditions
245
1
3.3 Experiments
246
2
4 Modulational Instability in One-dimensional Discrete Systems
248
3
4.1 The discrete nonlinear Schrodinger equation
248
1
4.2 The generalized discrete nonlinear Schrodinger equation
249
1
4.3 Experiments and localized modes
250
1
5 Modulational Instability in a Two-dimensional Lattice
251
2
5.1 Model equations
251
1
5.2 Modulational instability conditions and numerical simulations
252
1
6 Summary and Concluding Remarks
253
4
Contributions
257
88
On a Nonlocal sine-Gordon Equation
257
5
G. L. Alfimov
Stochastic Effects on NLS Solitons
262
4
G. Biondini
S. de Lillo
Artificial Boundary Conditions for Nonlinear Schrodinger Equations
266
5
C. -H. Bruneau
L. Di Menza
Wave Dynamics in a Stratified Medium
271
5
S. A. Bulgakov
Instability of Running Waves in Anharmonic Lattices
276
8
S. A. Darmanyan
V. M. Burlakov
Design of a "Piston" for the Generation of Large Amplitude Solitary Water Waves
284
9
W. A. B. Evans
Numerical Study of Various sine-Gordon "Breathers"
293
10
W. A. B. Evans
M. D. Cunha
V. V. Konotop
L. Vazquez
Generalized Singular Manifold Method for a Dispersive Long Wave Equation in 2+1 Dimensions
303
5
P. R. Gordoa
P. G. Estevez
Stability Properties of Stationary Integrable and Nonintegrable Discrete Nonlinear Schrodinger Equations
308
5
D. Hennig
Polaron Solutions in the Two-Dimensional Holstein Model
313
4
G. Kalosakas
G. P. Tsironis
S. Aubry
Breathers in Nonlinear Lattices: Numerical Methods Based on the Anti-integrability Concept
317
7
J. L. Marin
S. Aubry
A Wave Equation with a Dirac Distribution
324
5
Y. Martel
Existence of Stationary Solutions for Certain Nonlinear Wave Equations
329
5
J. Matos
The Schrodinger-Hirota Equation: A Model of Ultrashort Optical Pulse Propagation
334
5
A. S. Rodrigues
M. Santagiustina
E. M. Wright
On Some Conjectures about Surface Smoothing
339
6
A. Sanchez
A. R. Bishop
D. Cai
N. Gronbech-Jensen
Posters
345
14
Ellipse Rotation in a Stochastic Fiber
345
1
C. Balslev Clausen
P. L. Christiansen
J. H. Povlsen
K. Rottwitt
Spatial Solitons in Cascadable Nonlinear Media
346
1
A. D. Boardman
K. Xie
Discreteness Effects in Hydrogen-Bonded Chains
347
1
T. Cretegny
M. Peyrard
Transport Properties of Nonlinear Superlattices
348
1
E. Diez
A. Sanchez
F. Dominguez-Adame
Modulational Instability in Real Discrete Systems
349
1
P. Marquie
J. M. Bilbault
B. Michaux
S. Dos Santos
Parallel Implementation of Multigrid Methods on the CRAY T3D
350
3
S. Molina
I. Martin
J. C. Fabero
Propagation of Trigger Waves in a Disordered Medium
353
1
J. M. Noriega
M. A. Rodriguez
L. Pesquera
Raman Effect Induced by Modulational Instability for Normal Dispersion in a High Birefringent Fiber
354
1
E. Seve
G. Millot
Soliton Solutions of the Quintic Complex Ginzburg-Landau Equation
355
1
J. M. Soto-Crespo
N. N. Akhmediev
V. V. Afanasjev
Discrete sine-Gordon Breathers: A Variational Approach and a Peierls-Nabarro Energy Analysis
356
1
J. A. D. Wattis
Symplectic Methods for the Nonlinear Schrodinger Equation
357
2
Y. -F. Tang
V. M. Perez-Garcia
L. Vazquez
List of Participants
359