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