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. Keypoints treated are: i) integrable systems in multidimensions and associated phenomenology ("dromions"); ii) criteria and tests of integrability (e.g., Painleve 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.

Table of Contents:
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.