Book contents
- Frontmatter
- Contents
- Acknowledgments
- Introduction
- 1 The KdV Hierarchy
- 2 The Combined sine-Gordon and Modified KdV Hierarchy
- 3 The AKNS Hierarchy
- 4 The Classical Massive Thirring System
- 5 The Camassa—Holm Hierarchy
- A Algebraic Curves and Their Theta Functions in a Nutshell
- B Hyperelliptic Curves of the KdV-Type
- C Hyperelliptic Curves of the AKNS-Type
- D Asymptotic Spectral Parameter Expansions and Nonlinear Recursion Relations
- E Lagrange Interpolation
- F Symmetric Functions, Trace Formulas, and Dubrovin-Type Equations
- G KdV and AKNS Darboux-Type Transformations
- H Elliptic Functions
- I Herglotz Functions
- J Spectral Measures and Weyl—Titchmarsh m-Functions for Schrödinger Operators
- List of Symbols
- Bibliography
- Index
Introduction
Published online by Cambridge University Press: 23 December 2009
- Frontmatter
- Contents
- Acknowledgments
- Introduction
- 1 The KdV Hierarchy
- 2 The Combined sine-Gordon and Modified KdV Hierarchy
- 3 The AKNS Hierarchy
- 4 The Classical Massive Thirring System
- 5 The Camassa—Holm Hierarchy
- A Algebraic Curves and Their Theta Functions in a Nutshell
- B Hyperelliptic Curves of the KdV-Type
- C Hyperelliptic Curves of the AKNS-Type
- D Asymptotic Spectral Parameter Expansions and Nonlinear Recursion Relations
- E Lagrange Interpolation
- F Symmetric Functions, Trace Formulas, and Dubrovin-Type Equations
- G KdV and AKNS Darboux-Type Transformations
- H Elliptic Functions
- I Herglotz Functions
- J Spectral Measures and Weyl—Titchmarsh m-Functions for Schrödinger Operators
- List of Symbols
- Bibliography
- Index
Summary
It often happens that the understanding of the mathematical nature of an equation is impossible without a detailed understanding of its solutions.
Freeman J. DysonBackground: The discovery of solitary waves of translation goes back to Scott Russell in 1834, and during the remaining part of the 19th century the true nature of these waves remained controversial. It was only with the derivation by Korteweg and de Vries in 1895 of what is now called the Korteweg—de Vries (KdV) equation, that the one-soliton solution and hence the concept of solitary waves was put on a firm basis. An extraordinary series of events took place around 1965 when Kruskal and Zabusky, while analyzing the numerical results of Fermi, Pasta, and Ulam on heat conductivity in solids, discovered that pulselike solitary wave solutions of the KdV equation, for which the name “solitons” was coined, interact elastically. This was followed by the 1967 discovery of Gardner, Greene, Kruskal, and Miura that the inverse scattering method allows one to solve initial value problems for the KdV equation with sufficiently fast-decaying initial data. Soon thereafter, in 1968, Lax found a new explanation of the isospectral nature of KdV solutions using the concept of Lax pairs and introduced a whole hierarchy of KdV equations. Subsequently, in the early 1970s, Zakharov and Shabat (ZS), and Ablowitz, Kaup, Newell, and Segur (AKNS) extended the inverse scattering method to a wide class of nonlinear partial differential equations of relevance in various scientific contexts ranging from nonlinear optics to condensed matter physics and elementary particle physics.
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- Publisher: Cambridge University PressPrint publication year: 2003