Nonlinear Evolution Equations and Dynamical Systems

Nonlinear Evolution Equations and Dynamical Systems (NEEDS) provides a presentation of the state of the art. Except for a few review papers, the 40 contributions are intentially brief to give only the gist of the methods, proofs, etc. including references to the relevant litera- ture. This gives a handy overview of current research activities. Hence, the book should be equally useful to the senior resercher as well as the colleague just entering the field.

Texte du rabat

Nonlinear Evolution Equations and Dynamical Systems (NEEDS) provides a presentation of the state of the art. Except for a few review papers, the 40 contributions are intentially brief to give only the gist of the methods, proofs, etc. including references to the relevant litera- ture. This gives a handy overview of current research activities. Hence, the book should be equally useful to the senior resercher as well as the colleague just entering the field. Keypoints treated are: i) integrable systems in multidimensions and associated phenomenology ("dromions"); ii) criteria and tests of integrability (e.g., Painlevé test); iii) new developments related to the scattering transform; iv) algebraic approaches to integrable systems and Hamiltonian theory (e.g., connections with Young-Baxter equations and Kac-Moody algebras); v) new developments in mappings and cellular automata, vi) applications to general relativity, condensed matter physics, and oceanography.

Contenu
I Integrable Systems in (2+1)-Dimensions.- Solitons and Dromions, Coherent Structures in a Nonlinear World.- Boundary Value Problems in 1+1 and in 2+1, the Dressing Method, and Cellular Automata.- Exponentially Localized Solitons in 2+1 Dimensions.- On the Boundary Conditions of the Davey-Stewartson Equation.- Rational Solutions to the Two-Component K-P Hierarchies.- Construction of Inverse Data in Multidimensions.- II Criteria and Tests of Integrability: Painlevé Property, Hirota Method, Lie-Bäcklund Symmetries.- Examples of Nonclassical Similarity Reductions.- Equations That Pass Hirota's Three-Soliton Condition and Other Tests of Integrability.- Selection of Solvable Nonlinear Evolution Equations by Systematic Searches for Lie Bäcklund Symmetries.- III Spectral Methods and Related Topics, C-Integrable Systems.- Inverse Problems of Spectral Analysis and the Integration of Nonlinear Equations.- The Inverse Scattering Transform for the Elliptic Sinh-Gordon Equation.- Reflection Coefficients and Poles.- A N × N Zakharov-Shabat System with a Quadratic Spectral Parameter.- On Integration of the Korteweg-de Vries Equation with a Self-consistent Source.- On the Initial Value Problem of the Third Painlevé Equation.- Nonlinear Equations for Soliton Eigenfunctions Are the IST Integrable Equations.- The Geometry and Completeness of the Two-Phase Solutions of the Nonlinear Schrödinger Equations.- N Double Pole Solution and Its Initial Value Problem for the Modified Korteweg-de Vries Equation.- C-Integrable Generalization of a System of Nonlinear PDE's Describing Nonresonant N-Wave Interactions.- The Burgers Equation: Initial/Boundary Value Problems on the Semiline.- IV Algebraic Approach to Integrability and Hamiltonian Theory.- The Tangent Bundle for Multisolitons:Ideal Structure for Completely Integrable Systems.- Action-Angle Variables and Asymptotic Data.- The Action-Angle Transformation for the Korteweg-de Vries Equation.- Algorithms to Detect Complete Integrability in 1+1 Dimension.- GN Manifolds, Yang-Baxter Equations and ILW Hierarchies.- Integral and Discrete Evolution Equations: A Unified Approach.- An Abstract Tri-Hamiltonian Lax Hierarchy.- On Symplectic and Hamiltonian Differential Operators.- On a Non-Standard Hamiltonian Description of NLEE.- Energy Dependent Spectral Problems: Their Hamiltonian Structures, Miura Maps and Master Symmetries.- Super Hamiltonian Operators and Lie Superalgebras.- Higher (Non-isospectral) Symmetries of the Kadomtsev-Petviashvili Equations and the Virasoro Action on Riemann Surfaces.- A Combinatorial Rule to Hirota's Bilinear Equations.- Liouville-Arnold Integrability for Scattering Under Cone Potentials.- V Mappings, Cellular Automata and Solitons.- Lattice Equations and Integrable Mappings.- Recent Developments in Soliton Cellular Automata.- Cubic Equation, Newton's Method and Analytic Functions.- Singularity of Differential Mappings and Stability of Solitons.- VI Physical Applications.- Action-Angle Variables in the Quantum Wess-Zumino-Witten Model.- On the Derivation of Propagator and Bound State Equations and S-Matrix Elements for Composite States.- Resonant Flow over Topography.- Taxonomy of Ocean Stability Conditions.- Kinetic Equations and Soliton Diffusion in Low-Dimensional Magnets.- On Einstein's Equations with Two Commuting Killing Vectors.- List of Participants.- Index of Contributors.