Integrable Systems and Applications: Proceedings of a Workshop Held at Oléron, France June 20–24, 1988

Alain Bachelot (auth.), M. Balabane, P. Lochak, C. Sulem (eds.)3540516158, 9783540516156, 0387516158

In this volume nonlinear systems related to integrable systems are studied. Lectures cover such topics as the application of integrable systems to the description of natural phenomena, the elaboration of perturbation theories, and the statistical mechanics of ensembles of objects obeying integrable equations. The more physical lectures center largely around the three paradigmatic equations: Korteweg de Vries, Sine-Gordon and Nonlinear Schr?dinger, especially the latter. These have long been of great mathematical interest, and also exhibit a “universality” which places them among the most frequently encountered integrable equations in the description of physical systems. Tidal waves, optical fibers and laser beams are among the topics discussed. Lectures are also devoted to multidimensional solitons, integrability of Hamiltonian systems of ODEs and dissipative systems of PDEs.

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Table of contents :
Global solutions to non linear Dirac equations in Minkowski space….Pages 1-11
Statistical mechanics of the NLS models and their avatars….Pages 12-26
Asymptotic behavior of solutions and universal attractors for a system of nonlinear hyperbolic equations….Pages 27-30
Solitons in two dimensions….Pages 31-48
Nonlinear wave propagation through a random medium and soliton tunneling….Pages 49-64
Trace formulae and singular spectra for the Schrödinger operator….Pages 65-81
Hunting of the quarton….Pages 82-86
Nekhoroshev’s theorem and particle channeling in crystals….Pages 87-94
Linearized maps for the Davey-Stewartson I equation….Pages 95-104
On the role of nonlinearities in classical electrodynamics….Pages 105-112
Upper bounds on the Lyapunov exponents for dissipative perturbations of infinite dimensional Hamiltonian systems….Pages 113-126
The Cauchy problem for the generalized Korteweg-de-Vries equation….Pages 127-141
Effective stability in Hamiltonian systems in the light of Nekhoroshev’s theorem….Pages 142-153
Analysis of the linearization around a critical point of an infinite dimensional Hamiltonian system….Pages 154-191
On numerical chaos in the nonlinear Schrödinger equation….Pages 192-206
Singular solutions of the cubic Schrödinger equation….Pages 207-217
Hamiltonian deformations and optical fibers….Pages 218-231
The Josephson fluxon mechanics….Pages 232-260
Real lattices modelled by the nonlinear Schrödinger equation and its generalizations….Pages 261-270
Modulation of trapped waves giving approximate two-dimensional solitions….Pages 271-281
Solvable nonlinear equations as concrete realizations of the same abstract algebra….Pages 282-294
Properties of solutions of dispersive equations….Pages 295-311
Multichannel nonlinear scattering theory for nonintegrable equations….Pages 312-327
On the linear stability of solitary waves in Hamiltonian systems with symmetry….Pages 328-335
Nonlinear Schrödinger equations with magnetic field effect: Existence and stability of solitary waves….Pages 336-342