Cover -- Half-title -- Series-title -- Title -- Copyright -- 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 -- References -- 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 -- References -- 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 -- References -- 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 -- References -- 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 -- References -- 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 -- References -- 8 Isomonodromic deformations -- 8.1 Introduction -- 8.2 Monodromy data -- 8.3 Isomonodromy and the Riemann-Hilbert problem -- 8.4 Isomonodromic deformations -- 8.5 Schlesinger transformations -- 8.6 Tau-functions -- 8.7 Ricatti equation -- 8.8 Sato's formula -- 8.9 The Hirota equations -- 8.10 Tau-functions and theta-functions -- 8.11 The Painlevé equations -- References -- 9 Grassmannian and integrable hierarchies -- 9.1 Introduction -- 9.2 Fermions… -- 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 Grassmannians -- 9.7 Schur polynomials -- 9.8 From fermions to pseudo-differential operators -- 9.9 The Segal-Wilson approach -- References -- 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 -- References -- 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 -- References -- 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 -- References -- 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 -- References -- 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 -- References -- 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 1 case -- 15.15 The Riemann-Hilbert factorization problem -- References -- 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 -- References -- Index.