 Integrable Systems

In General > s.a. integrable quantum systems; types of integrable 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 Concepts and Techniques
@ General references: Haak JMP(94) [symmetries, generalized Bäcklund transformations]; Fokas & Gelfand ed-96 [algebraic]; Mironov ht/96-conf [group theoretic]; Rasin & Schiff JPA(13) [Gardner method to generate symmetries and conservation laws]; > s.a. random matrices.
@ Symmetries: Chavchanidze JPA(04)mp/03 [non-Noether]; Rastelli a1001 [on S2, with Platonic symmetries]; > s.a. symmetries.
@ Approximating a non-integrable one: Kaasalainen & Binney PRL(94).
@ Relationships between wave equations, AKS theorem: Symes PhyD(80).
@ 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]; Maciejewski & Przybylska PLA(11) [integrable deformations].
> Related topics: see harmonic maps; KAM Theorem and weak chaos [perturbations]; Painlevé Analysis; symplectic structures [KdV].

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]; Zuparic PhD(09)-a0908 [and quantum]; Gómez-Ullate et al ed-JPA(10)#43; Tudoran JGP(12)-a1106 [unified formulation]; Doikou IJMPA(12)-a1110 [rev]; Torrielli JPA-a1606-ln.
@ 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]; Cariglia et al PRD(13)-a1210 [on a curved manifold, geometric formulation].
@ Geometrical aspects: Pyatov & Solodukhin ed-96; Strachan JGP(97)ht/96 [deformed differential calculus]; Prykarpatsky & Mykytiuk 98 [and Lie-algebraic approach]; Lesfari JGP(99) [algebraic geometry]; Grant JGP(01) [Grassmannn structures]; Clementi & Pettini ap/01; Motter & Letelier PRD(02) [coordinate invariance of integrability]; Cieslinski et al ed-JPA(09) [non-linearity and geometry]; Ibort & Marmo TMP(12)-a1203.
@ Space of integrable theories: Mironov & Morozov PLB(02)ht/01 [canonical transformations and flows].

Generalizations
* Quasi-integrable systems: Classical quasi-integrable systems are ones that can be considered ads integrable systems with an added non-integrable small perturbation; They have Lyapunov times orders of magnitude shorter than their ergodic time, the most clear example being the Solar System.
* Nekhoroshev theorem: An n-degree-of-freedom system, with k constants of the motion in involution, kn, 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 systems: Gaeta AP(02) [Nekhoroshev theorem]; Shabat et al ed-04.
@ Quasi-integrable systems: Ferreira & Zakrzewski JHEP(11)-a1011 [example]; Goldfriend & Kurchan a1909 [quantum].
@ Nearly-integrable systems: Fasso & Sansonetto a1601 [almost-symplectic].