Integrable Systems  

In General > s.a. quantum systems.
* Idea: (Liouville) A system with n degrees of freedom is integrable if it has n conserved quantities in involution (commuting); Then in principle the Hamilton equations can be reduced to quadratures.
* Lax pair/equation: An integrable Hamiltonian system has an associated Lax pair of matrices S and U, satisfying the equation dS/dt = [S,U], equivalent to the dynamical equations for the system; As a consequence, the quantities Ik:= k–1 tr S k are a sequence of invariants of the system.
* Relationships: The non-linear equations of motion / field equations are the integrability conditions for systems of linear differential equations with a spectral parameter.
* Conserved quantities: If the system is in the form H = gab pa pb and ka is a Killing vector field of gab, I1 = ka pa; If kab is a Killing tensor, ...

Special Types, Techniques > s.a. harmonic maps; KAM Theorem [perturbations]; Painlevé Analysis; symplectic structures [KdV].
* Types: s-integrable, studied by spectral methods; c-integrable, solved by changing variables; Superintegrable, with more than n functionally independent integrals of the motion.
* Bi-Hamiltonian system: It is integrable if its Nijenhuis tensor vanishes.
@ 2D: Baleanu & Karasu MPLA(99)gq/00 [Lax, with symmetries]; McLenaghan & Smirnov JMP(00); Pucacco & Rosquist JMP(05).
@ In arbitrary dimensions: Alvarez et al ht/99-in; Goliath et al JPA(99)si/98.
@ Relationships between wave equations, AKS theorem: Symes PhyD(80).
@ Approximating a non-integrable one: Kaasalainen & Binney PRL(94).
@ Symmetries: Chavchanidze JPA(04)mp/03 [non-Noether]; > s.a. symmetries.
@ Hamiltonian models: Magri JMP(78); Calogero & Françoise JMP(96).
@ Bi-Hamiltonian systems: Smirnov LMP(97) [constructive]; Sergyeyev AAM(04)nl/03 [construction]; Nutku & Pavlov JMP(02) [multiple Lagrangians]; Praught & Smirnov Sigma(05)n.SI [history, Lenard recursion formula]; Bogoyavlenskij DG&A(07) [identity for Schouten tensor]; > s.a. quantum systems, symplectic structures.
@ Quasi-bi-Hamiltonian systems: Morosi & Tondo JPA(97).
@ Superintegrable: Kalnins et al JPA(01)mp/01, JMP(02)mp/01; Daskaloyannis & Ypsilantis mp/04 [2D, with integrals quadratic in momenta]; Ballesteros & Herranz JPA(07) [on constant curvature].
@ Discrete: Grammaticos et al JPA(01) [integrability]; Kimura et al JPA(02) [and discrete Painlevé]; Quispel et al JPA(05) [duality].
@ Other types: Sen & Chowdhury JMP(93) [supersymmetric]; Devchand & Ogievetsky ht/94-in [4D]; Ramani et al JPA(00) [without Painlevé property].
@ Techniques: Haak JMP(94) [symmetries, generalized Bäcklund transformations]; Fokas & Gelfand 96 [algebraic]; Mironov ht/96-in [group theoretic]; > s.a. random matrices.

Specific Examples > s.a. types of field theories [integrable]; non-commutative systems; self-dual fields; toda lattice.
* Examples: The Toda lattice and Korteweg-de Vries (KdV) equation; 3-body ones include the Kaluza-Klein two-center problem [@ Cornish & Gibbons CQG(97)gq/96].
@ Calogero-Moser: Calogero in(91), JMP(93); Gonera JMP(98); Bordner et al PTP(98)ht, PTP(99)ht/98, Bordner & Sasaki PTP(99)ht/98; Bordner et al PTP(99) [generalized]; Bordner et al PTP(00).
@ Calogero & Sutherland models: Rühl & Turbiner MPLA(95); Efthimiou & Spector qp/97; Gurappa & Panigrahi ht/99, ht/99; Forger & Winterhalder ht/99; Jonke & Meljanac PLB(01) [symmetry algebra]; Guhr & Kohler PRE(05)mp/04 [supersymmetric extension]; Sasaki & Takasaki JMP(06) [explicit solutions, any root system]; Polychronakos JPA(06)ht [rev].
@ Celestial mechanics: Vosmischeva 03 [spaces of constant curvature].
@ KdV: Nakamura JMP(81) [Bäcklund transformation]; Dimakis & Müller-Hoissen PLA(00)ht [non-commutative]; Kersten & Krasil'shchik n.SI/00 [KdV-mKdV]; Khare & Sukhatme PRL(02)mp/01 [superposition of solutions]; Gieseker JDG(03) [deformation]; Carroll qp/03 [KP/KdV and quantum mechanic]; Hayashi et al PRS(03) [initial-boundary-value problem]; Bracken PhyA(04) [solutions]; Willink a0710-in [history of Korteweg-de Vries paper]; > s.a. heat kernel.
@ Other examples: Gadella et al a0711-in [some 3D systems].

References
@ General: Das 89, & Okubo AP(89), & Huang JMP(90); Perelomov 90; Zakharov ed-91; Dorfman 93; Frønsdal FP(93); Fiorani IJGMP(08)-a0802 [rev, including recent developments].
@ Lax equations/pairs: Cariñena & Martínez IJMPA(94); Rosquist gq/94, Rosquist & Goliath GRG(98)gq/97 [tensorial interpretation]; Baleanu & Baskal MPLA(00)gq/01; Przybylska JGP(01) [generalization]; Sakovich nlin.SI/01 [true and fake].
@ Geometrical aspects: Pyatov & Solodukhin ed-96; Strachan JGP(97)ht/96 [deformed differential calculus]; Lesfari JGP(99) [algebraic geometry]; Grant JGP(01) [Grassmannn structures]; Clementi & Pettini ap/01; Motter & Letelier PRD(02) [coordinate invariance of integrability].
@ Space of integrable theories: Mironov & Morozov PLB(02)ht/01 [canonical transformations and flows].
@ Related topics: Dullin & Wittek JPA(94) [actions, numerical calculation]; Bogoyavlenskij CMP(96) [tensor invariants]; Labrunie & Conte JMP(96) [finding integrals]; Alvarez et al NPB(98)ht/97 [new approach]; Fock et al JHEP(00)ht/99 [duality]; Leach et al JNMP(00)n.SI [from Yang-Baxter]; Nutku & Pavlov JMP(02)ht/01 [multi-Lagrangians].

Generalizations
* Nekhoroshev theorem: An n-degree-of-freedom system, with k constants of the motion in involution, k n, has persistent k-dimensional invariant tori, and local partial action-angle coordinates, under suitable non-degeneracy conditions; Thus, it interpolates between the Poincaré-Lyapunov theorem (k = 1) and the Liouville-Arnold theorem (k = n); The crucial tool for the proof is a generalization of the Poincaré map.
@ Partially solvable: Gaeta AP(02) [Nekhoroshev theorem]; Shabat et al ed-04.

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