Types of Integrable Systems| Types of Integrable Systems |
In General > s.a. integrable quantum systems.
* Types: s-integrable, studied
by spectral methods; c-integrable, solved by changing variables; Superintegrable,
with more than n functionally independent integrals of the motion.
@ 2D: Baleanu & Karasu MPLA(99)gq/00 [Lax, with symmetries];
McLenaghan & Smirnov JMP(00);
Pucacco & Rosquist JMP(05).
@ In arbitrary dimensions: Álvarez et al AIP(99)ht;
Goliath et al JPA(99)si/98.
@ Hamiltonian models: Magri JMP(78);
Calogero & Françoise JMP(96);
Mostafazadeh a1401
[differential integrability condition for 2D Hamiltonian systems].
@ Superintegrable: Kalnins et al JPA(01)mp/01,
JMP(02)mp/01;
Daskaloyannis & Ypsilantis JMP(06)-mp/04 [2D, with integrals quadratic in momenta];
Ballesteros & Herranz JPA(07) [on constant curvature];
Yzaguirre MS-a1209 [geometric structure];
Nucci & Post JPA(12) [and Lie symmetries];
Nikitin JPA(12)-a1205 [new examples];
Gonera & Kaszubska AP(14)-a1311 [on spaces of constant curvature];
> s.a. classical systems; Fock Symmetry.
@ Discrete: Grammaticos et al JPA(01) [integrability];
Kimura et al JPA(02) [and discrete Painlevé];
Quispel et al JPA(05) [duality];
Grammaticos et al JPA(09) [integrability tests].
@ Other types: Sen & Chowdhury JMP(93) [supersymmetric];
Devchand & Ogievetsky ht/94-conf [4D];
Ramani et al JPA(00) [without Painlevé property];
Reshetikhin a1509 [degenerate integrability].
Bi-Hamiltonian and Related Systems
> s.a. Bi-Hamiltonian System.
* Bi-Hamiltonian system:
A bi-Hamiltonian system is integrable if its Nijenhuis tensor vanishes.
@ 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];
Gürses et al JMP(09)-a0903 [all dynamical
systems on \(\mathbb R\)n are (n−1)-Hamiltonian];
Barnich & Troessaert JMP(09)-a0812 [electromagnetism, linearized gravity and Yang-Mills theory];
Mokhov TMP(11) [non-local, of hydrodynamic type];
Bolsinov & Izosimov CMP(14) [singularities];
> s.a. duality in field theory; integrable quantum systems;
types of symplectic structures.
@ Quasi-bi-Hamiltonian systems:
Morosi & Tondo JPA(97).
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 PRA(97)qp;
Gurappa & Panigrahi ht/99,
PRB(00)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];
in Xu a1205 [algebraic approach].
@ 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 mechanics];
Hayashi et al PRS(03) [initial-boundary-value problem];
Bracken PhyA(04) [solutions];
Willink a0710-conf [history of Korteweg-de Vries paper];
Rasin & Schiff JPA(09) [discrete, infinitely-many conservation laws];
Lidsey PRD(12)-a1205 [significance to cosmology];
in Xu a1205-ch [algebraic approach];
Zakharov TMP(13) [Cauchy problem, renormalization method];
Karczewska & Rozmej a1901 [higher-order, solutions];
> s.a. heat kernel; solitons.
@ Other examples:
Vosmischeva 03 [spaces of constant curvature];
Gadella et al JPA(08)-a0711-conf [some 3D systems];
> s.a. Dimer Models;
special potentials [exactly solvable].
> Particle motion in curved spacetimes: see kerr
and generalized kerr spacetimes; spinning particles.
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