Solitons and Chaos

"Solitons and Chaos" is a response to the growing interest in systems exhibiting these two complementary manifestations of nonlinearity. The papers cover a wide range of topics but share common mathematical notions and investigation techniques. An introductory note on eight concepts of integrability has been added as a guide for the uninitiated reader. Both specialists and graduate students will find this update on the state ofthe art useful. Key points: chaos vs. integrability; solitons: theory and applications; dissipative systems; Hamiltonian systems; maps and cascades; direct vs. inverse methods; higher dimensions; Lie groups, Painleve analysis, numerical algorithms; pertubation methods.

Contenu

I General Questions on Chaos and Integrability.- Integration of Non-Integrable Systems.- Order and Chaos in the Statistical Mechanics of the Integrable Models in 1+1 Dimensions.- Soliton Dynamics and Chaos Transition in a Microstructured Lattice Model.- What is the Role of Dynamical Chaos in Irreversible Processes?.- A Propositional Lattice for the Logic of Temporal Predictions.- Damping, Quantum Field Theory and Thermodynamics.- Quasi-Monomial Transformations and Decoupling of Systems of ODE's.- II Physical Systems with Soliton Ingredients.- Solitons in Optical Fibers: First- and Second-Order Perturbations.- Similarity Solutions of Equations of Nonlinear Optics.- Heisenberg Ferromagnet, Generalized Coherent States and Nonlinear Behaviour.- Integrable Supersymmetric Models and Phase Transitions in One Dimension.- Denaturation of DNA in a Toda Lattice Model.- III Dissipative Systems.- A Simple Method to Obtain First Integrals of Dynamical Systems.- Transition to Turbulence in 1-D Rayleigh-Bénard Convection.- Modelling of Low-Dimensional, Incompressible, Viscous, Rotating Fluid Flow.- Spatial Coherent Structures in Dissipative Systems.- Hierarchies of (1+1)-Dimensional Multispeed Discrete Boltzmann Model Equations.- IV Hamiltonian Systems.- Universality of the Long Time Tail in Hamiltonian Dynamics.- Why some Hénon-Heiles Potentials are Integrable.- Chaotic Pulsations in Variable Stars with Harmonic Mode Coupling.- Canonical Forms for Compatible BiHamiltonian Systems.- V Maps and Cascades.- Transitions from Chaotic to Brownian Motion Behaviour.- Kinetic Theory for the Standard Map.- Probabilistic Description of Deterministic Chaos: A Local Equilibrium Approach.- State Prediction for Chaotic 1-D-Maps.- Exact and Approximate Reconstruction of Multifractal Coding Measures.- Conservative Versus Reversible Dynamical Systems.- A Simple Method to Generate Integrable Symplectic Maps.- Integrable Mappings and Soliton Lattices.- VI Direct Methods Applicable to Soliton Systems.- Integrable Higher Nonlinear Schrödinger Equations.- Nonclassical Symmetry Reductions of a Generalized Nonlinear Schrödinger Equation.- Direct Methods in Soliton Theories.- Trilinear Form - an Extension of Hirota's Bilinear Form.- On the Use of Bilinear Forms for the Search of Families of Integrable Nonlinear Evolution Equations.- From Periodic Processes to Solitons and Vice-Versa.- VII Inverse Methods Related to a Linearization Scheme.- The Crum Transformation for a Third Order Scattering Problem.- Darboux Theorems Connected to Dym Type Equations.- Forced Initial Boundary Value Problems for Burgers Equation.- Creation and Annihilation of Solitons in Nonlinear Integrable Systems.- VIII Nonlinear Excitations in more than one Space Dimension.- Multidimensional Nonlinear Schrödinger Equations Showing Localized Solutions.- New Soliton Solutions for the Davey-Stewartson Equation.- 2+1 Dimensional Dromions and Hirota's Bilinear Method.- Skyrmions Scattering in (2+1) Dimensions.- Index of Contributors.