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 d*S*/d*t* =
[*S*,*U*], equivalent
to the dynamical equations for the system; As a consequence, the
quantities
*I*_{k}:= *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* = *g*^{ab} *p*_{a} *p*_{b} and *k*^{a} is
a Killing vector field of *g _{ab}*,

**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 S^{2}, 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**

* __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 systems__: Gaeta AP(02) [Nekhoroshev theorem]; Shabat et
al ed-04.

@ __Quasi-integrable systems__: Ferreira & Zakrzewski JHEP(11)-a1011 [example].

@ __Nearly-integrable systems__: Fasso & Sansonetto a1601 [almost-symplectic].

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