O. Babelon, D. Bernard, M. Talon.
INTRODUCTION TO CLASSICAL INTEGRABLE SYSTEMS
Contents
1 Introduction
2 Integrable dynamical systems
2.1 Introduction.
2.2 The Liouville theorem.
2.3 Action-angle variables
2.4 Lax pairs
2.5 Existence of an r-matrix.
2.6 Commuting flows.
2.7 The Kepler problem.
2.8 The Euler top.
2.9 The Lagrange top.
2.10 The Kowalevski top.
2.11 The Neumann model.
2.12 Geodesics on an ellipsoid.
2.13 Separation of variables in the Neumann model.
3 Synopsis of integrable systems
3.1 Examples of Lax pairs with spectral parameter.
3.2 The Zakharov-Shabat construction.
3.3 Coadjoint orbits and Hamiltonian formalism.
3.4 Elementary flows and wave function.
3.5 Factorization problem.
3.6 Tau functions.
3.7 Integrable field theories and monodromy matrix.
3.8 Abelianization.
3.9 Poisson brackets of the monodromy matrix.
3.10 The group of dressing transformations.
3.11 Soliton solutions.
4 Algebraic methods
4.1 The classical and modified Yang-Baxter equations.
4.2 Algebraic meaning of the classical Yang-Baxter equations.
4.3 Adler-Kostant-Symes scheme.
4.4 Construction of integrable systems.
4.5 Solving by factorization.
4.6 The open Toda Chain.
4.7 The r-matrix of the Toda models.
4.8 Solution of the open Toda chain.
4.9 Toda system and Hamiltonian reduction.
4.10 The Lax pair of the Kowalevski top.
5 Analytical methods
5.1 The spectral curve.
5.2 The eigenvector bundle.
5.3 The adjoint linear system.
5.4 Time evolution.
5.5 Theta functions formulae.
5.6 Baker-Akhiezer functions.
5.7 Linearization and the factorization problem.
5.8 Tau-functions.
5.9 Symplectic form.
5.10 Separation of variables and the spectral curve.
5.11 Action-angle variables.
5.12 Riemann surfaces and integrability.
5.13 The Kowalevski top.
5.14 Infinite dimensional systems.
6 The closed Toda chain
6.1 The model.
6.2 The spectral curve.
6.3 The eigenvectors.
6.4 Reconstruction formula.
6.5 Symplectic structure.
6.6 The Sklyanin approach.
6.7 The Poisson brackets.
6.8 Reality conditions.
7 The Calogero-Moser model
7.1 The spin Calogero-Moser model.
7.2 Lax pair.
7.3 The r-matrix
7.4 The scalar Calogero-Moser model.
7.5 The spectral curve.
7.6 The eigenvector bundle.
7.7 Time evolution.
7.8 Reconstruction formulae.
7.9 Symplectic structure.
7.10 Poles systems and double-Bloch condition.
7.11 Hitchin systems.
7.12 Examples of Hitchin systems.
7.13 The trigonometric Calogero-Moser model.
8 Isomonodromic deformations
8.1 Introduction
8.2 Monodromy data.
8.3 Isomonodromy and Riemann-Hilbert problem.
8.4 Isomonodromic deformations.
8.5 Schlesinger transformations
8.6 Tau-functions.
8.7 Ricatti equation.
8.8 Sato formula.
8.9 Hirota equations.
8.10 Tau-functions and Theta-functions.
8.11 Painlevé equations
9 Grassmannian and integrable hierarchies
9.1 Introduction.
9.2 Fermions and GL(infinity)
9.3 Boson-fermion correspondence.
9.4 Tau-functions and Hirota bilinear identities.
9.5 The KP hierarchy and its soliton solutions.
9.6 Fermions and Grassmannian.
9.7 Schur polynomials.
9.8 From fermions to pseudo-differential operators.
9.9 The Segal-Wilson approach.
10 The KP hierarchy
10.1 The algebra of pseudo-differential operators.
10.2 The KP hierarchy.
10.3 The Baker-Akhiezer function of KP.
10.4 Algebro-geometric solutions of KP.
10.5 The tau function of KP.
10.6 The generalized KdV equations.
10.7 KdV Hamiltonian structures.
10.8 Bihamiltonian structure.
10.9 The Drinfeld Sokolov reduction.
10.10 Whitham equations.
10.11 Solution of the Whitham equations.
11 The KdV hierarchy
11.1 The KdV equation.
11.2 The KdV hierarchy.
11.3 Hamiltonian structures and Virasoro algebra.
11.4 Soliton solutions.
11.5 Algebro-geometric solutions.
11.6 Finite zone solutions.
11.7 Action-angle variables.
11.8 Analytical description of solitons.
11.9 Local fields.
11.10 Whitham's equations.
12 The Toda field theories
12.1 The Liouville equation.
12.2 The Toda systems and their zero-curvature representations.
12.3 Solution of the Toda field equations.
12.4 Hamiltonian formalism.
12.5 Conformal structure.
12.6 Dressing transformations.
12.7 The affine sinh-Gordon model.
12.8 Dressing transformations and soliton solutions.
12.9 N-soliton dynamics.
12.10 Finite zone solutions.
13 Classical inverse scattering method
13.1 The sine-Gordon equation.
13.2 The Jost solutions.
13.3 Inverse scattering as a Riemann-Hilbert problem.
13.4 Time evolution of the scattering data.
13.5 The Gelfand-Levitan-Marchenko equation.
13.6 Soliton Solutions.
13.7 Poisson brackets of the scattering data.
13.8 Action-angle variables.
14 Symplectic geometry
14.1 Poisson manifolds and symplectic manifolds.
14.2 Coadjoint orbits.
14.3 Symmetries and Hamiltonian reduction.
14.4 The case M = T* G.
14.5 Poisson-Lie groups.
14.6 Action of a Poisson-Lie group on a symplectic manifold.
14.7 The groups G and G*.
14.8 The group of dressing transformations.
15 Riemann surfaces
15.1 Smooth algebraic curves.
15.2 Hyperelliptic curves.
15.3 The Riemann-Hurwitz formula.
15.4 The field of meromorphic functions of a Riemann surface.
15.5 Line bundles on a Riemann surface.
15.6 Divisors.
15.7 Chern class.
15.8 Serre duality.
15.9 The Riemann-Roch theorem.
15.10 Abelian differentials.
15.11 Riemann Bilinear Identities.
15.12 Jacobi Variety.
15.13 Theta functions.
15.14 The genus one case.
15.15 The Riemann-Hilbert factorization problem.
16 Lie algebras
16.1 Lie groups and Lie algebras.
16.2 Semi-simple Lie algebras.
16.3 Linear representations.
16.4 Real Lie algebras.
16.5 Affine Kac-Moody algebras.
16.6 Vertex operator representations.
Index