685:
some "known" functions available, in terms of which the solutions may be expressed. This notion has no intrinsic meaning, since what is meant by "known" functions very often is defined precisely by the fact that they satisfy certain given equations, and the list of such "known functions" is constantly growing. Although such a characterization of "integrability" has no intrinsic validity, it often implies the sort of regularity that is to be expected in integrable systems.
455:. In classical terminology, this is described as determining a transformation to a canonical set of coordinates consisting of completely ignorable variables; i.e., those in which there is no dependence of the Hamiltonian on a complete set of canonical "position" coordinates, and hence the corresponding canonically conjugate momenta are all conserved quantities. In the case of compact energy level sets, this is the first step towards determining the
511:. This provides, in certain cases, enough invariants, or "integrals of motion" to make the system completely integrable. 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. However, for suitably defined boundary conditions, the spectral transform can, in fact, be interpreted as a transformation to
479:, in which the separation constants provide the complete set of integration constants that are required. Only when these constants can be reinterpreted, within the full phase space setting, as the values of a complete set of Poisson commuting functions restricted to the leaves of a Lagrangian foliation, can the system be regarded as completely integrable in the Liouville sense.
515:, in which the conserved quantities form half of a doubly infinite set of canonical coordinates, and the flow linearizes in these. In some cases, this may even be seen as a transformation to action-angle variables, although typically only a finite number of the "position" variables are actually angle coordinates, and the rest are noncompact.
684:
An imprecise notion of "exact solvability" as meaning: "The solutions can be expressed explicitly in terms of some previously known functions" is also sometimes used, as though this were an intrinsic property of the system itself, rather than the purely calculational feature that we happen to have
434:
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 motion on the invariant tori, expressed in terms of these canonical coordinates, is linear in the angle variables.
356:
Hamiltonian systems (i.e. those for which the
Hamiltonian and Poisson brackets are not explicitly time-dependent) have at least one invariant; namely, the Hamiltonian itself, whose value along the flow is the energy. If the energy level sets are compact, the leaves of the Lagrangian foliation are
400:
and maximal superintegrability. Essentially, these distinctions correspond to the dimensions of the leaves of the foliation. When the number of independent
Poisson commuting invariants is less than maximal (but, in the case of autonomous systems, more than one), we say the system is partially
278:
means that there exists a regular foliation of the phase space by invariant manifolds such that the
Hamiltonian vector fields associated with the invariants of the foliation span the tangent distribution. Another way to state this is that there exists a maximal set of functionally independent
506:
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.
495:(which describes 1-dimensional non-dissipative fluid dynamics in shallow basins), could be understood by viewing these equations as infinite-dimensional integrable Hamiltonian systems. Their study leads to a very fruitful approach for "integrating" such systems, the
665:
In physics, completely integrable systems, especially in the infinite-dimensional setting, are often referred to as exactly solvable models. This obscures the distinction between integrability, in the
Hamiltonian sense, and the more general dynamical systems sense.
99:
sense, the key example being multi-dimensional harmonic oscillators. Another standard example is planetary motion about either one fixed center (e.g., the sun) or two. Other elementary examples include the motion of a rigid body about its center of mass (the
92:. The latter generally have no conserved quantities, and are asymptotically intractable, since an arbitrarily small perturbation in initial conditions may lead to arbitrarily large deviations in their trajectories over a sufficiently large time.
401:
integrable. When there exist further functionally independent invariants, beyond the maximal number that can be
Poisson commuting, and hence the dimension of the leaves of the invariant foliation is less than n, we say the system is
632:
To explain quantum integrability, it is helpful to consider the free particle setting. Here all dynamics are one-body reducible. A quantum system is said to be integrable if the dynamics are two-body reducible. The
150:
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
413:
When a finite-dimensional
Hamiltonian system is completely integrable in the Liouville sense, and the energy level sets are compact, the flows are complete, and the leaves of the invariant foliation are
239:
An extension of the notion of integrability is also applicable to discrete systems such as lattices. This definition can be adapted to describe evolution equations that either are systems of
527:, which involved replacing the original nonlinear dynamical system with a bilinear system of constant coefficient equations for an auxiliary quantity, which later came to be known as the
669:
There are also exactly solvable models in statistical mechanics, which are more closely related to quantum integrable systems than classical ones. Two closely related methods: the
225:. There is thus a variable notion of the degree of integrability, depending on the dimension of the leaves of the invariant foliation. This concept has a refinement in the case of
637:
is a consequence of this reducibility and leads to trace identities which provide an infinite set of conserved quantities. All of these ideas are incorporated into the
322:
471:
is the dimension of the configuration space), exists in very general cases, but only in the local sense. Therefore, the existence of a complete solution of the
379:
342:
822:
2454:
789:
706:
475:
is by no means a characterization of complete integrability in the
Liouville sense. Most cases that can be "explicitly integrated" involve a complete
681:, provide quantum analogs of the inverse spectral methods. These are equally important in the study of solvable models in statistical mechanics.
503:), which generalize local linear methods like Fourier analysis to nonlocal linearization, through the solution of associated integral equations.
657:. Some other types of quantum integrability are known in explicitly time-dependent quantum problems, such as the driven Tavis-Cummings model.
2406:
2093:
1628:
Sonnad, Kiran G.; Cary, John R. (2004). "Finding a nonlinear lattice with improved integrability using Lie transform perturbation theory".
617:, and the notion of Poisson commuting functions replaced by commuting operators. The notion of conservation laws must be specialized to
190:, is a global property, not a local one, since it requires that the foliation be a regular one, with the leaves embedded submanifolds.
539:
approach, or the
Hamiltonian structure, this nevertheless gave a very direct method from which important classes of solutions such as
76:
the explicit determination of solutions in an explicit functional form (not an intrinsic property, but something often referred to as
175:
2442:
2743:
2371:
725:
558:
approach, in which, typically, the commuting dynamics were viewed simply as determined by a fixed (finite or infinite) abelian
2173:
2059:
1933:
1721:
1618:
1563:
1541:
1488:
1462:
1321:
1299:
1277:
1251:
1228:
1149:
250:
The distinction between integrable and nonintegrable dynamical systems has the qualitative implication of regular motion vs.
523:
Another viewpoint that arose in the modern theory of integrable systems originated in a calculational approach pioneered by
1601:; Markman, E. (1996). "Spectral covers, algebraically completely integrable, Hamiltonian systems, and moduli of bundles".
193:
Integrability does not necessarily imply that generic solutions can be explicitly expressed in terms of some known set of
2311:
869:
551:
254:
and hence is an intrinsic property, not just a matter of whether a system can be explicitly integrated in an exact form.
2226:
1175:
402:
397:
590:, characterizing the PlĂĽcker embedding of the Grassmannian in the projectivization of a suitably defined (infinite)
2548:
1806:
Date, E.; Jimbo, M.; Kashiwara, M.; Miwa, T. (1981). "Operator approach to the Kadomtsev-Petviashvili equation III".
1518:
1411:
1377:
1343:
1206:
779:
678:
638:
528:
2296:
827:
124:
2086:
1773:
405:. If there is a regular foliation with one-dimensional leaves (curves), this is called maximally superintegrable.
887:
794:
622:
115:
systems in physics having an infinite number of degrees of freedom, such as some models of shallow water waves (
2738:
1022:
817:
492:
451:, in which solutions to Hamilton's equations are sought by first finding a complete solution of the associated
116:
2256:
171:
systems, the energy is generally the only one, and on the energy level sets, the flows are typically chaotic.
1682:
1369:
859:
809:
460:
448:
2401:
2355:
2216:
2128:
874:
536:
496:
472:
452:
271:
144:
1834:
487:
A resurgence of interest in classical integrable systems came with the discovery, in the late 1960s, that
197:; it is an intrinsic property of the geometry and topology of the system, and the nature of the dynamics.
2728:
2246:
2079:
1867:
1677:
500:
324:
and the maximal number of independent Poisson commuting invariants (including the Hamiltonian itself) is
361:, and the natural linear coordinates on these are called "angle" variables. The cycles of the canonical
2733:
1672:
1587:
1480:
1438:
1395:
1190:
1167:
654:
2327:
1949:
Sinitsyn, N.A.; Li, F. (2016). "Solvable multistate model of Landau-Zener transitions in cavity QED".
2590:
2448:
2195:
804:
784:
774:
2553:
2521:
942:
837:
674:
634:
444:
160:
535:. Although originally appearing just as a calculational device, without any clear relation to the
298:(i.e., the center of the Poisson algebra consists only of constants), it must have even dimension
2667:
2116:
1533:
1243:
891:
476:
263:
39:, that its motion is confined to a submanifold of much smaller dimensionality than that of its
2615:
104:) and the motion of an axially symmetric rigid body about a point in its axis of symmetry (the
2426:
2411:
2376:
2221:
976:
832:
646:
610:
456:
427:
382:
349:
345:
240:
164:
2605:
2301:
1551:
981:
895:
736:
618:
419:
226:
491:, which are strongly stable, localized solutions of partial differential equations like the
2672:
2641:
2291:
2020:
1970:
1891:
1879:
1746:
1637:
1498:
1442:
1119:
986:
966:
899:
554:
hierarchy, but then for much more general classes of integrable hierarchies, as a sort of
218:
156:
1692:, a conference devoted to the study of integrable difference equations and related topics.
587:
381:-form are called the action variables, and the resulting canonical coordinates are called
8:
2631:
2348:
2151:
1580:
1530:
Statistical Field Theory. An Introduction to Exactly Solved Models of Statistical Physics
1088:
847:
717:
571:
301:
295:
167:. General dynamical systems have no such conserved quantities; in the case of autonomous
2024:
1974:
1883:
1750:
1641:
1446:
645:
can be used to obtain explicit solutions. Examples of quantum integrable models are the
550:
and his students, at first for the case of integrable hierarchies of PDEs, such as the
2600:
2498:
2493:
2488:
2338:
2004:
1986:
1960:
1903:
1417:
1138:
1037:
907:
364:
327:
222:
168:
159:), and if the flows are complete and the energy level set is compact, this implies the
96:
66:
32:
131:. The modern theory of integrable systems was revived with the numerical discovery of
2421:
2055:
1990:
1929:
1907:
1895:
1758:
1717:
1653:
1614:
1584:
1559:
1537:
1514:
1484:
1458:
1421:
1407:
1373:
1339:
1331:
1317:
1295:
1273:
1247:
1224:
1202:
1171:
1145:
1017:
864:
348:
with respect to the symplectic form and such a maximal isotropic foliation is called
244:
206:
194:
187:
1774:"Soliton equations as dynamical systems on infinite dimensional Grassmann manifolds"
84:
Integrable systems may be seen as very different in qualitative character from more
2610:
2183:
2166:
2028:
1978:
1921:
1887:
1846:
1815:
1788:
1754:
1645:
1606:
1506:
1472:
1450:
1399:
1357:
1194:
1032:
757:
591:
431:
267:
24:
1159:
2698:
2580:
2381:
2049:
1711:
1502:
1309:
1287:
1133:
1106:
1096:
284:
280:
95:
Many systems studied in physics are completely integrable, in particular, in the
2703:
2693:
2636:
2585:
2483:
2469:
2416:
2306:
2200:
2161:
1982:
1851:
1649:
1361:
1265:
1261:
1083:
1057:
1052:
1027:
954:
936:
920:
842:
251:
140:
136:
36:
27:. While there are several distinct formal definitions, informally speaking, an
2033:
2008:
625:
has an infinite set of conserved quantities given by projectors to its energy
2722:
2688:
2595:
2284:
2251:
2156:
1899:
1454:
1216:
1198:
1042:
1004:
999:
702:
650:
614:
2651:
2543:
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2279:
2188:
1951:
1657:
1430:
1111:
1078:
1073:
1047:
925:
799:
752:
730:
670:
642:
563:
559:
236:(see below), which is what is most frequently referred to in this context.
128:
105:
89:
46:
Three features are often referred to as characterizing integrable systems:
2054:. London Mathematical Society. Vol. 255. Cambridge University Press.
1866:
Calabrese, Pasquale; Essler, Fabian H L; Mussardo, Giuseppe (2016-06-27).
1605:. Lecture Notes in Mathematics. Vol. 1620. Springer. pp. 1–119.
1497:
1403:
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1819:
1387:
1353:
1009:
930:
903:
769:
712:
626:
575:
567:
423:
291:
182:(i.e., is generated by an integrable distribution) if, locally, it has a
120:
40:
1868:"Introduction to 'Quantum Integrability in Out of Equilibrium Systems'"
1792:
1610:
1598:
1101:
595:
547:
1737:
Hirota, R. (1986). "Reduction of soliton equations in bilinear form".
2646:
2332:
2178:
2121:
2071:
1062:
609:
In the quantum setting, functions on phase space must be replaced by
393:
232:
214:
183:
101:
1588:"Quantization of Hitchin's integrable system and Hecke eigensheaves"
1965:
1689:
540:
508:
132:
1314:
Integrable Hamiltonian Systems: Geometry, Topology, Classification
287:
with the Hamiltonian of the system, and with each other, vanish).
257:
186:
by maximal integral manifolds. But integrability, in the sense of
2343:
971:
488:
283:
invariants (i.e., independent functions on the phase space whose
221:
of the smallest possible dimension that are invariant under the
629:. However, this does not imply any special dynamical structure.
1713:
Integrable Systems: Twistors, Loop Groups, and Riemann Surfaces
430:, such that the invariant tori are the joint level sets of the
499:
and more general inverse spectral methods (often reducible to
1690:"SIDE - Symmetries and Integrability of Difference Equations"
688:
459:. In the general theory of partial differential equations of
415:
358:
174:
A key ingredient in characterizing integrable systems is the
518:
54:
set of conserved quantities (the usual defining property of
1926:
Quantum Inverse Scattering Method and Correlation Functions
1477:
Quantum Inverse Scattering Method and Correlation Functions
1166:. Cambridge Studies in Advanced Mathematics. Vol. 51.
396:
sense, and partial integrability, as well as a notion of
1352:
1920:
1471:
482:
418:. There then exist, as mentioned above, special sets of
127:, and certain integrable many-body systems, such as the
1872:
Journal of Statistical Mechanics: Theory and Experiment
1865:
1805:
1428:
606:
There is also a notion of quantum integrable systems.
992:
367:
330:
304:
2051:
Symmetries and Integrability of Difference Equations
463:
type, a complete solution (i.e. one that depends on
2266:
Integrable PDEs/Classical integrable field theories
1184:
790:
Classical Heisenberg ferromagnet model (spin chain)
707:
exact solutions of classical central-force problems
1137:
373:
336:
316:
111:In the late 1960s, it was realized that there are
2720:
1835:"Solitons and infinite-dimensional Lie algebras"
1710:Hitchin, N.J.; Segal, G.B.; Ward, R.S. (2013) .
1709:
1394:. Cambridge Monographs on Mathematical Physics.
1260:
823:Landau–Lifshitz equation (continuous spin field)
722:Integrable Clebsch and Steklov systems in fluids
2455:Six-dimensional holomorphic Chern–Simons theory
2047:
1308:
438:
258:Hamiltonian systems and Liouville integrability
213:refers to the existence of invariant, regular
2435:
2048:Clarkson, Peter A.; Nijhoff, Frank W. (1999).
1942:
1221:Exactly solved models in statistical mechanics
2087:
1597:
1579:
1550:
1366:Integrable Systems: From Classical to Quantum
1292:Symplectic Geometry. Methods and Applications
1270:Hamiltonian Methods in the Theory of Solitons
1164:Spinning Tops: A Course on Integrable Systems
601:
266:, we have the notion of integrability in the
31:is a dynamical system with sufficiently many
1386:
1187:Introduction to classical integrable systems
1185:Babelon, O.; Bernard, D.; Talon, M. (2003).
673:approach, in its modern sense, based on the
467:independent constants of integration, where
200:
88:dynamical systems, which are more typically
1140:Mathematical Methods of Classical Mechanics
915:Exactly solvable statistical lattice models
2682:Classical and quantum statistical lattices
2094:
2080:
1948:
1924:; Bogoliubov, N.M.; Izergin, A.G. (1997).
1627:
1556:Handbook of Integrable Hamiltonian Systems
1475:; Bogoliubov, N.M.; Izergin, A.G. (1997).
689:List of some well-known integrable systems
660:
519:Hirota bilinear equations and Ď„-functions
2032:
1964:
1850:
1832:
1330:
408:
2003:
1808:Journal of the Physical Society of Japan
1527:
1237:
1286:
231:complete integrability in the sense of
2721:
2101:
1736:
1215:
1132:
764:Integrable systems in 1 + 1 dimensions
726:Lagrange, Euler, and Kovalevskaya tops
546:Subsequently, this was interpreted by
2075:
1603:Integrable systems and quantum groups
1158:
566:. The Ď„-function was viewed as the
483:Solitons and inverse spectral methods
16:Property of certain dynamical systems
2508:Exactly solvable quantum spin chains
2443:Four-dimensional Chern–Simons theory
1826:
1799:
1771:
1392:Tau functions and Their Applications
388:There is also a distinction between
123:in optical fibres, described by the
2372:Anti-self-dual Yang–Mills equations
1765:
1294:(2nd ed.). Gordon and Breach.
881:Integrable PDEs in 3 + 1 dimensions
854:Integrable PDEs in 2 + 1 dimensions
531:. These are now referred to as the
13:
2227:Superintegrable Hamiltonian system
1573:
1435:Discrete systems and integrability
993:Some key contributors (since 1965)
439:The Hamilton–Jacobi approach
344:. The leaves of the foliation are
14:
2755:
2549:Quantum inverse scattering method
2395:Integrable Quantum Field theories
1781:Kokyuroku, RIMS, Kyoto University
1665:
1240:Solitons, Instantons and Twistors
780:Boussinesq equation (water waves)
679:quantum inverse scattering method
639:quantum inverse scattering method
582:within the Grassmannian, and the
205:In the context of differentiable
2574:Classical mechanics and geometry
1892:10.1088/1742-5468/2016/06/064001
1338:(2nd ed.). Addison-Wesley.
960:
513:completely ignorable coordinates
178:, which states that a system is
2312:Kadomtsev–Petviashvili equation
2041:
894:; general solutions are termed
870:Kadomtsev–Petviashvili equation
699:Calogero–Moser–Sutherland model
217:; i.e., ones whose leaves are
69:(a property known sometimes as
2744:Partial differential equations
2297:Nonlinear Schrödinger equation
1997:
1928:. Cambridge University Press.
1914:
1859:
1739:Physica D: Nonlinear Phenomena
1730:
1703:
828:Nonlinear Schrödinger equation
653:and several variations on the
473:Hamilton–Jacobi equation
453:Hamilton–Jacobi equation
125:nonlinear Schrödinger equation
65:invariants, having a basis in
1:
2468:Exactly solvable statistical
1878:(6). IOP Publishing: 064001.
1370:American Mathematical Society
1126:
890:generates a Lax pair for the
713:Geodesic motion on ellipsoids
290:In finite dimensions, if the
2356:Inverse scattering transform
1833:Jimbo, M.; Miwa, T. (1983).
1759:10.1016/0167-2789(86)90173-9
694:Classical mechanical systems
497:inverse scattering transform
449:Hamilton–Jacobi method
145:inverse scattering transform
7:
2247:Quantum harmonic oscillator
1716:. Oxford University Press.
1678:Encyclopedia of Mathematics
1528:Mussardo, Giuseppe (2010).
948:
888:Belinski–Zakharov transform
795:Degasperis–Procesi equation
262:In the special setting of
245:finite difference equations
10:
2760:
2480:in one- and two-dimensions
1983:10.1103/PhysRevA.93.063859
1839:Publ. Res. Inst. Math. Sci
1650:10.1103/PhysRevE.69.056501
1481:Cambridge University Press
1439:Cambridge University Press
1396:Cambridge University Press
1191:Cambridge University Press
1168:Cambridge University Press
1144:(2nd ed.). Springer.
818:Korteweg–de Vries equation
814:Krichever–Novikov equation
602:Quantum integrable systems
562:on a (finite or infinite)
493:Korteweg–de Vries equation
163:; i.e., the existence of
143:in 1965, which led to the
117:Korteweg–de Vries equation
2681:
2660:
2624:
2591:Ferdinand Georg Frobenius
2573:
2566:
2535:
2514:
2507:
2467:
2394:
2364:
2320:
2272:
2265:
2239:
2209:
2196:Garnier integrable system
2144:
2137:
2109:
2034:10.4249/scholarpedia.7216
1312:; Bolsinov, A.V. (2003).
860:Davey–Stewartson equation
810:Kaup–Kupershmidt equation
805:Garnier integrable system
744:Integrable lattice models
621:conservation laws. Every
201:General dynamical systems
23:is a property of certain
2522:Quantum Heisenberg model
2365:ASDYM as a master theory
2217:Liouville–Arnold theorem
1852:10.2977/prims/1195182017
1696:
1455:10.1017/CBO9781107337411
1199:10.1017/CBO9780511535024
943:Quantum Heisenberg model
892:Einstein field equations
875:Novikov–Veselov equation
501:Riemann–Hilbert problems
445:canonical transformation
272:Liouville–Arnold theorem
161:Liouville-Arnold theorem
2668:Alexander Zamolodchikov
2328:Bäcklund transformation
2257:Pöschl–Teller potential
2129:Liouville integrability
2117:Frobenius integrability
2110:Geometric integrability
2009:"Calogero-Moser system"
1534:Oxford University Press
1244:Oxford University Press
910:solutions are examples.
737:Newtonian gravitational
661:Exactly solvable models
477:separation of variables
276:Liouville integrability
71:algebraic integrability
2427:Principal chiral model
2377:Twistor correspondence
2222:Action-angle variables
2138:In classical mechanics
1433:; Nijhoff, F. (2016).
1316:. Taylor and Francis.
977:Painleve transcendents
933:in 1- and 2-dimensions
896:gravitational solitons
833:Nonlinear sigma models
749:Ablowitz–Ladik lattice
611:self-adjoint operators
552:Kadomtsev–Petviashvili
457:action-angle variables
428:action-angle variables
409:Action-angle variables
390:complete integrability
383:action-angle variables
375:
338:
318:
241:differential equations
165:action-angle variables
56:complete integrability
2739:Hamiltonian mechanics
2606:Joseph-Louis Lagrange
2157:Central force systems
1404:10.1017/9781108610902
1390:; Balogh, F. (2021).
1238:Dunajski, M. (2009).
982:Statistical mechanics
785:Camassa–Holm equation
775:Benjamin–Ono equation
675:Yang–Baxter equations
574:from elements of the
556:universal phase space
461:Hamilton–Jacobi
447:theory, there is the
420:canonical coordinates
376:
339:
319:
219:embedded submanifolds
113:completely integrable
2673:Alexei Zamolodchikov
2642:Martin David Kruskal
2616:Siméon Denis Poisson
2554:Yang–Baxter equation
2449:Affine Gaudin models
2292:Sine-Gordon equation
2240:In quantum mechanics
1820:10.1143/JPSJ.50.3806
1120:Vladimir E. Zakharov
987:Integrable algorithm
967:Mathematical physics
900:Schwarzschild metric
838:Sine–Gordon equation
641:where the algebraic
635:Yang–Baxter equation
596:fermionic Fock space
365:
328:
302:
180:Frobenius integrable
157:Lagrangian foliation
33:conserved quantities
2632:Clifford S. Gardner
2402:Quantum Sine-Gordon
2349:Topological soliton
2339:integrals of motion
2152:Harmonic oscillator
2025:2008SchpJ...3.7216C
1975:2016PhRvA..93f3859S
1884:2016JSMTE..06.4001C
1751:1986PhyD...18..161H
1673:"Integrable system"
1642:2004PhRvE..69e6501S
1511:Dynamical Systems V
1509:; Shil'nikov, L.P.
1507:Il'yashenko, Yu. S.
1447:2016dsi..book.....H
1336:Classical Mechanics
1089:Nicolai Reshetikhin
848:Three-wave equation
718:Harmonic oscillator
572:projection operator
317:{\displaystyle 2n,}
264:Hamiltonian systems
227:Hamiltonian systems
50:the existence of a
2729:Integrable systems
2601:Sofia Kovalevskaya
2499:Chiral Potts model
2494:Hard hexagon model
2489:Eight-vertex model
2103:Integrable systems
1611:10.1007/BFb0094792
1272:. Addison-Wesley.
1223:. Academic Press.
908:gravitational wave
731:Neumann oscillator
647:Lieb–Liniger model
586:as expressing the
564:Grassmann manifold
543:could be derived.
537:inverse scattering
398:superintegrability
371:
334:
314:
67:algebraic geometry
2734:Dynamical systems
2716:
2715:
2712:
2711:
2562:
2561:
2463:
2462:
2422:Toda field theory
2412:Quantum Liouville
2390:
2389:
2344:Soliton solutions
2302:Gross–Neveu model
2235:
2234:
2061:978-0-521-59699-2
1935:978-0-521-58646-7
1772:Sato, M. (1981).
1723:978-0-19-967677-4
1630:Physical Review E
1620:978-3-540-60542-3
1565:978-5-396-00687-4
1552:Sardanashvily, G.
1543:978-0-19-954758-6
1499:Afrajmovich, V.S.
1490:978-0-521-58646-7
1464:978-1-107-04272-8
1323:978-0-415-29805-6
1301:978-2-88124-901-3
1279:978-0-387-15579-1
1253:978-0-19-857063-9
1230:978-0-12-083180-7
1151:978-0-387-96890-2
1053:Igor M. Krichever
1018:Vladimir Drinfeld
865:Ishimori equation
588:PlĂĽcker relations
374:{\displaystyle 1}
346:totally isotropic
337:{\displaystyle n}
281:Poisson commuting
207:dynamical systems
195:special functions
188:dynamical systems
176:Frobenius theorem
147:method in 1967.
61:the existence of
29:integrable system
25:dynamical systems
2751:
2611:Joseph Liouville
2571:
2570:
2512:
2511:
2484:Square ice model
2433:
2432:
2337:Infinitely many
2270:
2269:
2167:Two body problem
2142:
2141:
2096:
2089:
2082:
2073:
2072:
2066:
2065:
2045:
2039:
2038:
2036:
2001:
1995:
1994:
1968:
1946:
1940:
1939:
1918:
1912:
1911:
1863:
1857:
1856:
1854:
1830:
1824:
1823:
1803:
1797:
1796:
1778:
1769:
1763:
1762:
1745:(1–3): 161–170.
1734:
1728:
1727:
1707:
1686:
1661:
1624:
1594:
1592:
1569:
1547:
1524:
1494:
1468:
1429:Hietarinta, J.;
1425:
1383:
1349:
1327:
1305:
1283:
1257:
1234:
1212:
1181:
1155:
1143:
1067:Vladimir Matveev
1033:Hermann Flaschka
758:Volterra lattice
655:Heisenberg model
584:Hirota equations
533:Hirota equations
380:
378:
377:
372:
343:
341:
340:
335:
323:
321:
320:
315:
285:Poisson brackets
270:sense. (See the
209:, the notion of
19:In mathematics,
2759:
2758:
2754:
2753:
2752:
2750:
2749:
2748:
2719:
2718:
2717:
2708:
2699:Elliott H. Lieb
2677:
2656:
2620:
2581:Vladimir Arnold
2558:
2531:
2503:
2459:
2436:Master theories
2431:
2386:
2382:Ward conjecture
2360:
2316:
2261:
2231:
2205:
2174:Integrable tops
2133:
2105:
2100:
2070:
2069:
2062:
2046:
2042:
2002:
1998:
1947:
1943:
1936:
1919:
1915:
1864:
1860:
1845:(3): 943–1001.
1831:
1827:
1814:(11): 3806–12.
1804:
1800:
1776:
1770:
1766:
1735:
1731:
1724:
1708:
1704:
1699:
1671:
1668:
1621:
1590:
1576:
1574:Further reading
1566:
1544:
1521:
1491:
1465:
1414:
1380:
1364:, eds. (2000).
1346:
1324:
1302:
1280:
1266:Takhtajan, L.A.
1254:
1231:
1209:
1178:
1152:
1129:
1124:
1107:Elliott H. Lieb
1097:Evgeny Sklyanin
1038:Israel Gel'fand
995:
963:
951:
898:, of which the
691:
663:
604:
521:
485:
441:
411:
403:superintegrable
366:
363:
362:
329:
326:
325:
303:
300:
299:
260:
203:
90:chaotic systems
37:first integrals
17:
12:
11:
5:
2757:
2747:
2746:
2741:
2736:
2731:
2714:
2713:
2710:
2709:
2707:
2706:
2704:Yang Chen-Ning
2701:
2696:
2694:Ludvig Faddeev
2691:
2685:
2683:
2679:
2678:
2676:
2675:
2670:
2664:
2662:
2658:
2657:
2655:
2654:
2649:
2644:
2639:
2637:John M. Greene
2634:
2628:
2626:
2622:
2621:
2619:
2618:
2613:
2608:
2603:
2598:
2593:
2588:
2586:Leonhard Euler
2583:
2577:
2575:
2568:
2564:
2563:
2560:
2559:
2557:
2556:
2551:
2546:
2539:
2537:
2533:
2532:
2530:
2529:
2524:
2518:
2516:
2509:
2505:
2504:
2502:
2501:
2496:
2491:
2486:
2481:
2474:
2472:
2470:lattice models
2465:
2464:
2461:
2460:
2458:
2457:
2452:
2446:
2439:
2437:
2430:
2429:
2424:
2419:
2417:Thirring model
2414:
2409:
2404:
2398:
2396:
2392:
2391:
2388:
2387:
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2384:
2379:
2374:
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2351:
2341:
2335:
2330:
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2318:
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2314:
2309:
2307:Thirring model
2304:
2299:
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2287:
2276:
2274:
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2260:
2259:
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2232:
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2207:
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2203:
2201:Hitchin system
2198:
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2181:
2171:
2170:
2169:
2164:
2154:
2148:
2146:
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2113:
2111:
2107:
2106:
2099:
2098:
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2068:
2067:
2060:
2040:
1996:
1941:
1934:
1913:
1858:
1825:
1798:
1764:
1729:
1722:
1701:
1700:
1698:
1695:
1694:
1693:
1687:
1667:
1666:External links
1664:
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1575:
1572:
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1548:
1542:
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1495:
1489:
1473:Korepin, V. E.
1469:
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1412:
1384:
1378:
1358:Winternitz, P.
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1177:978-0521779197
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1093:Aleksei Shabat
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1058:Martin Kruskal
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1028:Ludvig Faddeev
1025:
1023:Boris Dubrovin
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955:Hitchin system
950:
947:
946:
945:
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937:Ice-type model
934:
928:
923:
921:8-vertex model
917:
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911:
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843:Thirring model
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594:, viewed as a
592:exterior space
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252:chaotic motion
202:
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141:Norman Zabusky
137:Martin Kruskal
82:
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9:
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4:
3:
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2690:
2689:Rodney Baxter
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2596:Nigel Hitchin
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2285:KdV hierarchy
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2255:
2253:
2252:Hydrogen atom
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2162:Kepler system
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2000:
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1984:
1980:
1976:
1972:
1967:
1962:
1959:(6): 063859.
1958:
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1953:
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1937:
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1922:Korepin, V.E.
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1581:Beilinson, A.
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1379:0-8218-2093-1
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1362:Sabidussi, G.
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1345:0-201-02918-9
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1332:Goldstein, H.
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1311:
1310:Fomenko, A.T.
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1289:
1288:Fomenko, A.T.
1285:
1281:
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1267:
1263:
1262:Faddeev, L.D.
1259:
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961:Related areas
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22:
21:integrability
2652:Robert Miura
2567:Contributors
2544:Bethe ansatz
2527:Gaudin model
2445:(Lagrangian)
2280:KdV equation
2184:Kovalevskaya
2102:
2050:
2043:
2016:
2013:Scholarpedia
2012:
2005:Calogero, F.
1999:
1956:
1952:Phys. Rev. A
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1944:
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1585:Drinfeld, V.
1555:
1529:
1513:. Springer.
1510:
1503:Arnold, V.I.
1476:
1434:
1391:
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1335:
1313:
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1217:Baxter, R.J.
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1134:Arnold, V.I.
1112:Graeme Segal
1079:Tetsuji Miwa
1074:Robert Miura
1070:Henry McKean
1048:Michio Jimbo
926:Gaudin model
800:Dym equation
753:Toda lattice
683:
671:Bethe ansatz
668:
664:
643:Bethe ansatz
631:
608:
605:
583:
579:
560:group action
555:
545:
532:
525:Ryogo Hirota
524:
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129:Toda lattice
112:
110:
106:Lagrange top
94:
85:
83:
77:
70:
62:
55:
51:
45:
28:
20:
18:
2478:Ising model
2407:Quantum KdV
2019:(8): 7216.
1793:2433/102800
1084:Alan Newell
1010:Percy Deift
931:Ising model
904:Kerr metric
770:AKNS system
735:Two center
627:eigenstates
623:Hamiltonian
576:group orbit
568:determinant
424:phase space
292:phase space
229:, known as
169:Hamiltonian
121:Kerr effect
97:Hamiltonian
78:solvability
41:phase space
2723:Categories
2542:Algebraic
1966:1602.03136
1599:Donagi, R.
1388:Harnad, J.
1354:Harnad, J.
1127:References
1102:Mikio Sato
548:Mikio Sato
529:Ď„-function
354:autonomous
350:Lagrangian
296:symplectic
215:foliations
2647:Peter Lax
2333:Lax pairs
2122:Foliation
1991:119331736
1908:124170507
1900:1742-5468
1787:: 30–46.
1683:EMS Press
1431:Joshi, N.
1422:222379146
1160:Audin, M.
1063:Peter Lax
906:and some
426:known as
394:Liouville
392:, in the
268:Liouville
233:Liouville
184:foliation
102:Euler top
63:algebraic
2515:Examples
2273:Examples
2189:Lagrange
2145:Examples
2007:(2008).
1658:15244955
1558:. URSS.
1554:(2015).
1334:(1980).
1290:(1995).
1268:(1987).
1219:(1982).
1162:(1996).
1136:(1997).
949:See also
705:motion (
677:and the
578:to some
541:solitons
509:Lax pair
489:solitons
133:solitons
2021:Bibcode
1971:Bibcode
1880:Bibcode
1747:Bibcode
1685:, 2001
1638:Bibcode
1443:Bibcode
972:Soliton
939:of Lieb
422:on the
155:of the
119:), the
86:generic
52:maximal
2536:Theory
2321:Theory
2210:Theory
2058:
1989:
1932:
1906:
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1148:
902:, the
739:motion
649:, the
580:origin
432:action
352:. All
153:leaves
2661:IQFTs
2179:Euler
1987:S2CID
1961:arXiv
1904:S2CID
1777:(PDF)
1697:Notes
1591:(PDF)
1418:S2CID
619:local
613:on a
570:of a
35:, or
2625:PDEs
2056:ISBN
1930:ISBN
1896:ISSN
1876:2016
1718:ISBN
1654:PMID
1615:ISBN
1560:ISBN
1538:ISBN
1515:ISBN
1485:ISBN
1459:ISBN
1408:ISBN
1374:ISBN
1340:ISBN
1318:ISBN
1296:ISBN
1274:ISBN
1248:ISBN
1225:ISBN
1203:ISBN
1172:ISBN
1146:ISBN
886:The
416:tori
359:tori
223:flow
139:and
2029:doi
1979:doi
1888:doi
1847:doi
1816:doi
1789:hdl
1785:439
1755:doi
1646:doi
1607:doi
1451:doi
1400:doi
1195:doi
443:In
294:is
274:.)
243:or
135:by
108:).
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