Solitons: Springer Series in Solid-State Sciences, cartea 19

1. Introduction.- 1.1 Why Study Solitons?.- 1.2 Basic Concepts Illustrated by Simple Examples.- 2. The Korteweg-de Vries Equation (KdV-Equation).- 2.1 The Physical Meaning of the KdV Equation.- 2.2 The KdV Equation as a Lagrangian Field Theory: Symmetries.- 2.3 Local Conservation Laws for the KdV System.- 2.4 Simple Solutions of the KdV Equation.- 3. The Inverse Scattering Transformation (IST) as Illustrated with the KdV.- 3.1 The Linear Eigenvalue Problem.- 3.2 Commutation Relations for (KdV)n.- 3.3 Inverse Scattering Theory of Gel'fand-Levitan-Marchenko.- 3.4 Application to the KdV Equation: N Soliton Solution.- 3.5 Squared-Function Systems, or: the Secret of the KdV Equation.- 3.6 Dynamics of the Scattering Data.- 3.7 Birth and Death of Solitons.- 4. Inverse Scattering Theory for Other Evolution Equations.- 4.1 Statement of the Problem.- 4.2 Inverse Scattering Theory for Equation (4.1.1).- 4.3 Orthogonal Systems of Functions, Associated Operators, and Induced Poisson Brackets.- 4.4 Further Nonlinear Evolution Equations.- 4.5 The Simplest Nonpolynomial “Dispersion Relations”.- 4.6 Time Development of the Scattering Data.- 4.7 Transformation Theory: Miura and Bäcklund Transformations.- 4.8 Perturbation Theory and Stability.- 4.9 Summary of Results, Problems, and Simple Extension to Higher Dimensions.- 5. The Classical Sine-Gordon Equation (SGE).- 5.1 Basic Equations.- 5.2 Soliton Solutions of the SGE.- 5.3 Simple Solutions of the PSG.- 5.4 Cauchy Problem for the PSG and Particle Representation.- 5.5 PSG Solitons in the Presence of External Perturbations.- 5.6 Possible Generalizations.- 6. Statistical Mechanics of the Sine-Gordon System.- 6.1 Functional Integrals.- 6.2 Partition Function in the Soliton Picture.- 6.3 Partition Function by a Scale Transformation.- 7.Difference Equations: The Toda Lattice.- 7.1 Basic Considerations.- 7.2 IST for the Toda Lattice.- 7.3 Systems of Squared Functions.- 7.4 Soliton Solutions for the Toda Lattice.- References.