Riassunto
1.1 Why Study Solitons? The last century of physics, which was initiated by Maxwell's completion of the theory of electromagnetism, can, with some justification, be called the era of linear physi cs. ~Jith few excepti ons, the methods of theoreti ca 1 phys ics have been dominated by linear equations (Maxwell, Schrodinger), linear mathematical objects (vector spaces, in particular Hilbert spaces), and linear methods (Fourier transforms, perturbation theory, linear response theory) . Naturally the importance of nonlinearity, beginning with the Navier-Stokes equations and continuing to gravitation theory and the interactions of par ticles in solids, nuclei, and quantized fields, was recognized. However, it was hardly possible to treat the effects of nonlinearity, except as a per turbation to the basis solutions of the linearized theory. During the last decade, it has become more widely recognized in many areas of "field physics" that nonlinearity can result in qualitatively new phenom ena which cannot be constructed via perturbation theory starting from linear ized equations. By "field physics" we mean all those areas of theoretical physics for which the description of physical phenomena leads one to consider field equations, or partial differential equations of the form (1.1.1) ~t or ~tt = F(~, ~x ... ) for one- or many-component "fields" Ht,x,y, ... ) (or their quantum analogs).
Contenuti
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.
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