Book contents
- Frontmatter
- Contents
- 1 Introduction
- 2 Integrable dynamical systems
- 3 Synopsis of integrable systems
- 4 Algebraic methods
- 5 Analytical methods
- 6 The closed Toda chain
- 7 The Calogero—Moser model
- 8 Isomonodromic deformations
- 9 Grassmannian and integrable hierarchies
- 10 The KP hierarchy
- 11 The KdV hierarchy
- 12 The Toda field theories
- 13 Classical inverse scattering method
- 14 Symplectic geometry
- 15 Riemann surfaces
- 16 Lie algebras
- Index
11 - The KdV hierarchy
Published online by Cambridge University Press: 19 August 2009
- Frontmatter
- Contents
- 1 Introduction
- 2 Integrable dynamical systems
- 3 Synopsis of integrable systems
- 4 Algebraic methods
- 5 Analytical methods
- 6 The closed Toda chain
- 7 The Calogero—Moser model
- 8 Isomonodromic deformations
- 9 Grassmannian and integrable hierarchies
- 10 The KP hierarchy
- 11 The KdV hierarchy
- 12 The Toda field theories
- 13 Classical inverse scattering method
- 14 Symplectic geometry
- 15 Riemann surfaces
- 16 Lie algebras
- Index
Summary
In this chapter we study the Korteweg—de Vries equation, which occupies a central place in the modern theory of integrable systems. All the aspects of integrable systems discussed so far converge in this chapter to draw a particularly rich landscape. In particular, the methods of pseudo-differential operators allow us to easily discuss the formal aspects, the tau-functions yield soliton solutions, and the algebro-geometric methods yield finite-zone solutions. The soliton solutions which we obtained in the Grassmannian setting by using vertex operators are also degenerate cases of these finite-zone solutions. Finally, we use a fermionic fomalism to analyse the structure of the local fields and show that the equations of the hierarchy can be recast in a very compact form. This is used to give a new derivation of the Whitham equations in the KdV case.
The KdV equation
The Korteweg—de Vries (KdV) equation was introduced historically as an approximation of the equations of hydrodynamics, describing unidimensional long waves in shallow water. In their pioneering work, Gardner, Greene, Kruskal and Miura found an unexpected connection with the inverse scattering problem of the Schroedinger equation. More recently, the Hamiltonian aspects of KdV theory connected it to conformal field theory. The KdV equation reads:
The numerical factors in front of each term in eq.
- Type
- Chapter
- Information
- Introduction to Classical Integrable Systems , pp. 382 - 442Publisher: Cambridge University PressPrint publication year: 2003