Integrable system

Many systems studied in physics are completely integrable, in particular, in the Hamiltonian sense, the key example being multi-dimensional harmonic oscillators.In the late 1960s, it was realized that there are completely integrable systems in physics having an infinite number of degrees of freedom, such as some models of shallow water waves (Korteweg–de Vries equation), the Kerr effect in optical fibres, described by the nonlinear Schrödinger equation, and certain integrable many-body systems, such as the Toda lattice.The modern theory of integrable systems was revived with the numerical discovery of solitons by Martin Kruskal and Norman Zabusky in 1965, which led to the inverse scattering transform method in 1967.In the special case of Hamiltonian systems, if there are enough independent Poisson commuting first integrals for the flow parameters to be able to serve as a coordinate system on the invariant level sets (the leaves of the Lagrangian foliation), and if the flows are complete and the energy level set is compact, this implies the Liouville–Arnold theorem; i.e., the existence of action-angle variables.But integrability, in the sense of dynamical systems, is a global property, not a local one, since it requires that the foliation be a regular one, with the leaves embedded submanifolds.Integrability does not necessarily imply that generic solutions can be explicitly expressed in terms of some known set of special functions; it is an intrinsic property of the geometry and topology of the system, and the nature of the dynamics.This concept has a refinement in the case of Hamiltonian systems, known as complete integrability in the sense of Liouville (see below), which is what is most frequently referred to in this context.If the energy level sets are compact, the leaves of the Lagrangian foliation are tori, and the natural linear coordinates on these are called "angle" variables.These thus provide a complete set of invariants of the Hamiltonian flow (constants of motion), and the angle variables are the natural periodic coordinates on the tori.The basic idea of this method is to introduce a linear operator that is determined by the position in phase space and which evolves under the dynamics of the system in question in such a way that its "spectrum" (in a suitably generalized sense) is invariant under the evolution, cf.In the case of systems having an infinite number of degrees of freedom, such as the KdV equation, this is not sufficient to make precise the property of Liouville integrability.Although originally appearing just as a calculational device, without any clear relation to the inverse scattering approach, or the Hamiltonian structure, this nevertheless gave a very direct method from which important classes of solutions such as solitons could be derived.The τ-function was viewed as the determinant of a projection operator from elements of the group orbit to some origin within the Grassmannian, and the Hirota equations as expressing the Plücker relations, characterizing the Plücker embedding of the Grassmannian in the projectivization of a suitably defined (infinite) exterior space, viewed as a fermionic Fock space.
dynamical systemsconserved quantitiesfirst integralsphase spacealgebraic geometrychaotic systemsHamiltonianEuler topLagrange topKorteweg–de Vries equationKerr effectnonlinear Schrödinger equationToda latticesolitonsMartin KruskalNorman Zabuskyinverse scattering transformLagrangian foliationLiouville–Arnold theoremaction-angle variablesFrobenius theoremfoliationspecial functionsfoliationsembedded submanifoldsHamiltonian systemsLiouvilledifferential equations finite difference equationschaotic motionPoisson commutingPoisson bracketssymplectictotally isotropicLagrangiansuperintegrabilitysuperintegrablecanonical coordinatesactioncanonical transformationHamilton–Jacobi methodHamilton–Jacobi equationHamilton–Jacobiseparation of variablesRiemann–Hilbert problemsLax pairτ-functioninverse scatteringMikio SatoKadomtsev–Petviashviligroup actionGrassmann manifolddeterminantprojection operatorgroup orbitPlücker relationsexterior spacefermionic Fock spaceself-adjoint operatorsHilbert spaceeigenstatesYang–Baxter equationquantum inverse scattering methodBethe ansatzLieb–Liniger modelHubbard modelHeisenberg modelYang–Baxter equationsCentral forceexact solutions of classical central-force problemsGeodesic motion on ellipsoidsHarmonic oscillatorLagrange, Euler, and Kovalevskaya topsNeumann oscillatorNewtonian gravitationalVolterra latticeAKNS systemBenjamin–Ono equationBoussinesq equation (water waves)Camassa–Holm equationClassical Heisenberg ferromagnet model (spin chain)Degasperis–Procesi equationDym equationGarnier integrable systemKaup–Kupershmidt equationLandau–Lifshitz equation (continuous spin field)Nonlinear sigma modelsSine–Gordon equationThirring modelThree-wave equationDavey–Stewartson equationIshimori equationKadomtsev–Petviashvili equationNovikov–Veselov equationBelinski–Zakharov transformEinstein field equationsgravitational solitonsSchwarzschild metricKerr metricgravitational wave8-vertex modelGaudin modelIsing modelIce-type modelQuantum Heisenberg modelHitchin systemMathematical physicsSolitonPainleve transcendentsStatistical mechanicsIntegrable algorithmMark AblowitzRodney BaxterPercy DeiftVladimir DrinfeldBoris DubrovinLudvig FaddeevHermann FlaschkaIsrael Gel'fandAlexander ItsMichio JimboIgor M. KricheverPeter LaxRobert MiuraTetsuji MiwaAlan NewellNicolai ReshetikhinEvgeny SklyaninElliott H. LiebGraeme SegalVladimir E. ZakharovArnold, V.I.Audin, M.Cambridge University PressBaxter, R.J.Oxford University PressFaddeev, L.D.Takhtajan, L.A.Fomenko, A.T.Goldstein, H.Harnad, J.Winternitz, P.Sabidussi, G.American Mathematical SocietyJoshi, N.BibcodeKorepin, V. E.Afrajmovich, V.S.Il'yashenko, Yu. S.Sardanashvily, G.Beilinson, A.Drinfeld, V.Donagi, R.Encyclopedia of MathematicsEMS PressKorepin, V.E.Phys. Rev. ACalogero, F.Frobenius integrability Liouville integrabilityCentral force systemsKepler systemTwo body problemIntegrable topsKovalevskayaLagrangeSuperintegrable Hamiltonian systemQuantum harmonic oscillatorHydrogen atomPöschl–Teller potentialKdV equationKdV hierarchySine-Gordon equationGross–Neveu modelBäcklund transformationLax pairsintegrals of motionSoliton solutionsTopological solitonTwistor correspondenceWard conjectureQuantum LiouvilleToda field theoryPrincipal chiral modelFour-dimensional Chern–Simons theoryAffine Gaudin modelsSix-dimensional holomorphic Chern–Simons theorylattice modelsSquare ice modelEight-vertex modelHard hexagon modelChiral Potts modelVladimir ArnoldLeonhard EulerFerdinand Georg FrobeniusNigel HitchinSofia KovalevskayaJoseph-Louis LagrangeJoseph LiouvilleSiméon Denis PoissonClifford S. GardnerJohn M. GreeneMartin David KruskalAlexander ZamolodchikovAlexei ZamolodchikovYang Chen-Ning