Full Description
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 Schrödinger, 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.
Contents
Global solutions to non linear Dirac equations in Minkowski space.- Statistical mechanics of the NLS models and their avatars.- Asymptotic behavior of solutions and universal attractors for a system of nonlinear hyperbolic equations.- Solitons in two dimensions.- Nonlinear wave propagation through a random medium and soliton tunneling.- Trace formulae and singular spectra for the Schrödinger operator.- Hunting of the quarton.- Nekhoroshev's theorem and particle channeling in crystals.- Linearized maps for the Davey-Stewartson I equation.- On the role of nonlinearities in classical electrodynamics.- Upper bounds on the Lyapunov exponents for dissipative perturbations of infinite dimensional Hamiltonian systems.- The Cauchy problem for the generalized Korteweg-de-Vries equation.- Effective stability in Hamiltonian systems in the light of Nekhoroshev's theorem.- Analysis of the linearization around a critical point of an infinite dimensional Hamiltonian system.- On numerical chaos in the nonlinear Schrödinger equation.- Singular solutions of the cubic Schrödinger equation.- Hamiltonian deformations and optical fibers.- The Josephson fluxon mechanics.- Real lattices modelled by the nonlinear Schrödinger equation and its generalizations.- Modulation of trapped waves giving approximate two-dimensional solitions.- Solvable nonlinear equations as concrete realizations of the same abstract algebra.- Properties of solutions of dispersive equations.- Multichannel nonlinear scattering theory for nonintegrable equations.- On the linear stability of solitary waves in Hamiltonian systems with symmetry.- Nonlinear Schrödinger equations with magnetic field effect: Existence and stability of solitary waves.
