Nonlinear Evolution Equations and Dynamical Systems : Needs ’90

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. Front Matter....Pages I-XVI Front Matter....Pages 1-1 Some Remarks on the Hirota Bilinear Identity....Pages 2-5 Hirota Equations of Level > 1....Pages 6-7 Integrable N ≤ O Component Nonlinear Schrödinger Model, Phase Transitions and Supersymmetry....Pages 8-11 A Tri-Hamiltonian Extension of the Boussinesq Equation and Its Modification....Pages 12-15 A Direct Algebraic Method for Solving Nonlinear Integrable Models....Pages 16-21 Multivalued Solitons for a Hybrid Type of the Modified WKI Equation and Related Problems....Pages 22-23 Miura Maps at the Turn of a Handle....Pages 24-28 On the Integration of the Infinite Toda Lattice....Pages 29-32 Front Matter....Pages 33-33 Recent Developments in Multidimensional Inverse Scattering....Pages 34-46 Asymptotic Bifurcation of Multidimensional Solitons....Pages 47-60 Localized Waves in N + 1 Dimensions....Pages 61-67 Recent Development for Integrable Integro-Differential Equations....Pages 68-75 Exactly Solvable Nonlinear Evolution Equations Expressed by Trilinear Form....Pages 76-78 A Derivation of Conserved Quantities and Symmetries for the Multi-Dimensional Soliton Equations....Pages 79-82 Dromion Solutions for Generic NLS- and KdV-Type Equations....Pages 83-85 “Nonstandard” Classes of Integrable Equations in 1+1 and 2+1 Dimensions....Pages 86-88 From Polynomial Solutions to a “General” Solution of the BKP Equation....Pages 89-91 Solutions of the Davey-Stewartson Equation with Non-Zero Boundary Condition....Pages 92-93 Lump Solutions to the BKP Equation....Pages 94-98 Front Matter....Pages 99-99 Geometry of Ermakov Systems....Pages 100-102 Front Matter....Pages 99-99 Multisoliton Adiabatic Perturbation Theory. Algebraic Approach....Pages 103-106 The Algebraic Structure Associated with Systems Possessing Non-Hereditary Recursion Operators....Pages 107-109 Homogeneous Manifolds, Factorisation Problems and Modified KdV Equations....Pages 110-116 Bäcklund Transformations and Spectral Problems: the Korteweg-de Vries Interacting Soliton Equation and the Action-Angle Transformation....Pages 117-120 Integrability of Polynomial-Nonlinear Evolution Equations and Computer Algebra....Pages 121-123 Genetic Codes of Lie Algebras and Nonlinear Evolution Equations....Pages 124-126 On Some Problems Concerning Local Symplectic Operators....Pages 127-129 Infinitesimal Objects of Hypercomplex Systems Generated by Double Adjacent Classes and Nonlinear Differential Equations....Pages 130-132 Local Analysis of Nonlinear Equations....Pages 133-136 Front Matter....Pages 137-137 The q -Deformed Creation and Annihilation Operators as a Realization of the Quantum Superalgebra B q (0∣l)....Pages 138-139 The Heisenberg Quantum Group H (1) q : R -Matrix and Non-Commutative Spaces....Pages 140-142 Phase Transitions in Kuryshkin’s Algebras....Pages 143-149 Semiclassical Quantization of Kowalewski’s Top on 0(4) and 0(3,1) Lie Algebras....Pages 150-153 A Model of Electrodynamics in the Momentum Space of Constant Curvature....Pages 154-160 Front Matter....Pages 161-161 Cellular Automaton to Optical Communication: Diversity of Solitons....Pages 162-167 Integrable Unstable Model for Interaction of Langmuir Waves with Acoustic Waves in Plasmas....Pages 168-174 Self-Localized Excitation in a Polar Medium with Movable Ions....Pages 175-180 Nonlinear Waves Dynamics in Nematics Under the Action of Magnetic Fields....Pages 181-182 The Action of Effects of Dissipation, Dispersion and Nonstationary Kerr Nonlinearity on the Propagation of Solitons in Resonant Media....Pages 183-184 Two-Dimensional Classical Attractors in the Spin Phase Space of the S = 1 Easy-Axis Heisenberg Ferromagnet....Pages 185-193 Front Matter....Pages 161-161 Universal Attractors for Some Dissipative Nonlinear Evolution Equations....Pages 194-196 Evolution of Nonlinear Guided Optical Fields in Planar Layered Structures....Pages 197-201 Numerical Application of the KP Equation to a Particular Oceanographical Problem....Pages 202-203 On a Dimensional Antiferromagnetic Ising Model with Long Range Interaction....Pages 204-206 Coherent State Theory and the Field Lattice Model....Pages 207-211 Self-Consistent System of Equations for Probability Amplitudes and Displacements in the X-Y Model....Pages 212-214 Periodic and Soliton Solutions of the Heat Equation with a Nonlinear Source of Heat....Pages 215-217 Stability Properties of Exact Soliton Solutions of the Parametrically Driven, Damped Nonlinear Schrödinger Equation....Pages 218-222 Equations for the Frustrated Josephson Junction Array....Pages 223-230 Back Matter....Pages 231-244