Nonlinear Evolution Equations and Dynamical Systems

✍ Scribed by P. M. Santini, A. S. Fokas (auth.), Sandra Carillo, Orlando Ragnisco (eds.)

• Publisher: Springer-Verlag Berlin Heidelberg
• Year: 1990
• Tongue: English
• Leaves: 246
• Series: Research Reports in Physics
• Edition: 1
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

Front Matter....Pages I-XIV
Front Matter....Pages 1-1
Solitons and Dromions, Coherent Structures in a Nonlinear World....Pages 2-13
Boundary Value Problems in 1 + 1 and in 2 + 1, the Dressing Method, and Cellular Automata....Pages 14-25
Exponentially Localized Solitons in 2 + 1 Dimensions....Pages 26-28
On the Boundary Conditions of the Davey-Stewartson Equation....Pages 29-31
Rational Solutions to the Two-Component K-P Hierarchies....Pages 32-35
Construction of Inverse Data in Multidimensions....Pages 36-40
Front Matter....Pages 41-41
Examples of Nonclassical Similarity Reductions....Pages 42-45
Equations That Pass Hirota’s Three-Soliton Condition and Other Tests of Integrability....Pages 46-50
Selection of Solvable Nonlinear Evolution Equations by Systematic Searches for Lie Bäcklund Symmetries....Pages 51-54
Front Matter....Pages 55-55
Inverse Problems of Spectral Analysis and the Integration of Nonlinear Equations....Pages 56-63
The Inverse Scattering Transform for the Elliptic Sinh-Gordon Equation....Pages 64-67
Reflection Coefficients and Poles....Pages 68-72
A N × N Zakharov-Shabat System with a Quadratic Spectral Parameter....Pages 73-76
On Integration of the Korteweg-de Vries Equation with a Self-consistent Source....Pages 77-81
On the Initial Value Problem of the Third Painlevé Equation....Pages 82-86
Nonlinear Equations for Soliton Eigenfunctions Are the IST Integrable Equations....Pages 87-93
The Geometry and Completeness of the Two-Phase Solutions of the Nonlinear Schrödinger Equations....Pages 94-97
N Double Pole Solution and Its Initial Value Problem for the Modified Korteweg-de Vries Equation....Pages 98-101
C-Integrable Generalization of a System of Nonlinear PDE’s Describing Nonresonant N-Wave Interactions....Pages 102-104
The Burgers Equation: Initial/Boundary Value Problems on the Semiline....Pages 105-111
Front Matter....Pages 113-113
The Tangent Bundle for Multisolitons: Ideal Structure for Completely Integrable Systems....Pages 114-122
Action-Angle Variables and Asymptotic Data....Pages 123-126
The Action-Angle Transformation for the Korteweg-de Vries Equation....Pages 127-130
Algorithms to Detect Complete Integrability in 1 + 1 Dimension....Pages 131-135
GN Manifolds, Yang-Baxter Equations and ILW Hierarchies....Pages 136-139
Integral and Discrete Evolution Equations: A Unified Approach....Pages 140-143
An Abstract Tri-Hamiltonian Lax Hierarchy....Pages 144-147
On Symplectic and Hamiltonian Differential Operators....Pages 148-151
On a Non-Standard Hamiltonian Description of NLEE....Pages 152-156
Energy Dependent Spectral Problems: Their Hamiltonian Structures, Miura Maps and Master Symmetries....Pages 157-160
Super Hamiltonian Operators and Lie Superalgebras....Pages 161-164
Higher (Non-isospectral) Symmetries of the Kadomtsev-Petviashvili Equations and the Virasoro Action on Riemann Surfaces....Pages 165-169
A Combinatorial Rule to Hirota’s Bilinear Equations....Pages 170-172
Liouville-Arnold Integrability for Scattering Under Cone Potentials....Pages 173-180
Front Matter....Pages 181-181
Lattice Equations and Integrable Mappings....Pages 182-185
Recent Developments in Soliton Cellular Automata....Pages 186-189
Cubic Equation, Newton’s Method and Analytic Functions....Pages 190-194
Singularity of Differential Mappings and Stability of Solitons....Pages 195-199
Front Matter....Pages 201-201
Action-Angle Variables in the Quantum Wess-Zumino-Witten Model....Pages 202-204
On the Derivation of Propagator and Bound State Equations and S-Matrix Elements for Composite States....Pages 205-208
Front Matter....Pages 201-201
Resonant Flow over Topography....Pages 209-211
Taxonomy of Ocean Stability Conditions....Pages 212-215
Kinetic Equations and Soliton Diffusion in Low-Dimensional Magnets....Pages 216-218
On Einstein’s Equations with Two Commuting Killing Vectors....Pages 219-223
Back Matter....Pages 225-238

✦ Subjects

Statistical Physics, Dynamical Systems and Complexity;Quantum Information Technology, Spintronics;Quantum Physics