Nonlinear Evolution Equations and Dynamical Systems

Nonlinear Evolution Equations and Dynamical Systems (NEEDS) provides a presentation of the state of the art. But for some exceptions, the contributions are intentionally brief to give only the gist of the methods, proofs, etc. including references to the relevant literature. This gives a handy overview of current research activities. Hence, the book should be equally useful to the senior researcher as well as the colleague just entering the field. Topics treated: One- and multidimensional (integrable) models, geometric and algebraic methods, quantum field theory, applications to nonlinear optics, condensed matter physics, oceanography, and many others. Further keywords: Hirota bilinearity, Hamiltonians, Toda lattice, multi-dimensional inverse scattering, bifurcations, dromions, polynomial solutions, Ermakov systems, computer algebra, symplectic operators, (quantum) superalgebras, groups, Ising model.

I One-Dimensional Integrable Models.- Some Remarks on the Hirota Bilinear Identity.- Hirota Equations of Level > 1.- Integrable N ? 0 Component Nonlinear Schrödinger Model, Phase Transitions and Supersymmetry.- A Tri-Hamiltonian Extension of the Boussinesq Equation and Its Modification.- A Direct Algebraic Method for Solving Nonlinear Integrable Models.- Multivalued Solitons for a Hybrid Type of the Modified WKI Equation and Related Problems.- Miura Maps at the Turn of a Handle.- On the Integration of the Infinite Toda Lattice.- II Multi-Dimensional (Nonlinear) Models.- Recent Developments in Multidimensional Inverse Scattering.- Asymptotic Bifurcation of Multidimensional Solitons.- Localized Waves in N+1 Dimensions.- Recent Development for Integrable Integro-Differential Equations.- Exactly Solvable Nonlinear Evolution Equations Expressed by Trilinear Form.- A Derivation of Conserved Quantities and Symmetries for the Multi-Dimensional Soliton Equations.- Dromion Solutions for Generic NLS- and KdV-Type Equations.- “Nonstandard” Classes of Integrable Equations in 1+1 and 2+1 Dimensions.- From Polynomial Solutions to a “General” Solution of the BKP Equation.- Solutions of the Davey-Stewartson Equation with Non-Zero Boundary Condition.- Lump Solutions to the BKP Equation.- III Geometric and Algebraic Methods.- Geometry of Ermakov Systems.- Multisoliton Adiabatic Perturbation Theory. Algebraic Approach.- The Algebraic Structure Associated with Systems Possessing Non-Hereditary Recursion Operators.- Homogeneous Manifolds, Factorisation Problems and Modified KdV Equations.- Bäcklund Transformations and Spectral Problems: the Korteweg-de Vries Interacting Soliton Equation and the Action-Angle Transformation.- Integrability of Polynomial-Nonlinear Evolution Equations and Computer Algebra.- Genetic Codes of Lie Algebras and Nonlinear Evolution Equations.- On Some Problems Concerning Local Symplectic Operators.- Infinitesimal Objects of Hypercomplex Systems Generated by Double Adjacent Classes and Nonlinear Differential Equations.- Local Analysis of Nonlinear Equations.- IV Quantum Field Theory.- The q-Deformed Creation and Annihilation Operators as a Realization of the Quantum Superalgebra Bq(0\l).- The Heisenberg Quantum Group H(l)q: R-Matrix and Non-Commutative Spaces.- Phase Transitions in Kuryshkin’s Algebras.- Semiclassical Quantization of Kowalewski’s Top on 0(4) and 0(3,1) Lie Algebras.- A Model of Electrodynamics in the Momentum Space of Constant Curvature.- V Applications: Nonlinear Optics, Condensed Matter.- Cellular Automaton to Optical Communication: Diversity of Solitons.- Integrable Unstable Model for Interaction of Langmuir Waves with Acoustic Waves in Plasmas.- Self-Localized Excitation in a Polar Medium with Movable Ions.- Nonlinear Waves Dynamics in Nematics Under the Action of Magnetic Fields.- The Action of Effects of Dissipation, Dispersion and Nonstationary Kerr Nonlinearity on the Propagation of Solitons in Resonant Media.- Two-Dimensional Classical Attractors in the Spin Phase Space of the S = 1 Easy-Axis Heisenberg Ferromagnet.- Universal Attractors for Some Dissipative Nonlinear Evolution Equations.- Evolution of Nonlinear Guided Optical Fields in Planar Layered Structures.- Numerical Application of the KP Equation to a Particular Oceanographical Problem.- On a Dimensional Antiferromagnetic Ising Model with Long Range Interaction.- Coherent State Theory and the Field Lattice Model.- Self-Consistent System of Equations for Probability Amplitudes and Displacements in the X-Y Model.- Periodic and Soliton Solutions of the Heat Equation with a Nonlinear Source of Heat.- Stability Properties of Exact Soliton Solutions of the Parametrically Driven, Damped Nonlinear Schrödinger Equation.- Equations for the Frustrated Josephson Junction Array.- List of Participants.- Index of Contributors.