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''Calogero–Moser–Sutherland–Krichever system''

A finite-dimensional [[Hamiltonian system|Hamiltonian system]] which is algebraically completely integrable (cf. [[Completely-integrable differential equation|Completely-integrable differential equation]]). It admits several generalizations, which will be indicated below. During the enormous activity on mathematical aspects of integrable systems and [[Soliton|soliton]] equations, starting in the late 1960{}s, some prototypes that represent the main features have emerged. For example, the [[Korteweg–de Vries equation|Korteweg–de Vries equation]] and the non-linear [[Schrödinger equation|Schrödinger equation]] for inverse scattering [[#References|[a8]]], rank- $  2 $
perturbations for algebraic complete integrability [[#References|[a19]]], [[#References|[a20]]], the Kadomtsev–Petviashvili equation (cf. [[Soliton|Soliton]]) for Grassmannians and Schur functions [[#References|[a30]]], and the modified Korteweg–de Vries equation for representation theory [[#References|[a6]]], [[#References|[a15]]].

As a prototype, the Calogero–Moser–Krichever system possesses all the standard features of algebraic complete integrability: 1) a Lax pair, spectral curve, and Abelian integrals; 2) a connection with soliton equations; 3) an  $  r $-
matrix; 4) a geodesic-motion interpretation. However, it has also more unique features: 5) a link with elliptic functions; 6) a more sophisticated interpretation of the Lax pair, giving a link with the Hitchin system, which is a more powerful source of algebraic complete integrability over moduli spaces of vector bundles; 7) the bispectral property.

A brief illustration of these terms follows.

1) The system has a Hamiltonian function

$$ 
H = {
\frac{1}{2}
 } \sum _ {i = 1 } ^ { n }  p _ {i}  ^ {2} + \sum _ {i < j } v ( q _ {i} - q _ {j} ) ,
$$

where  $  ( q _ {1} \dots q _ {n} ; p _ {1} \dots p _ {n} ) $
are position/momentum variables (all quantities are complex numbers) and  $  v $
is an even function. If  $  L $
and  $  M $
are the  $  ( n \times n ) $-
matrices with entries

$$ 
L _ {jk }  = p _ {j} \delta _ {jk }  + \sqrt {- 1 } ( 1 - \delta _ {jk }  ) u ( q _ {j} - q _ {k} ) ,
$$

$$ 
M _ {jk }  = \delta _ {jk }  \sum _ {l \neq j } w ( q _ {j} - q _ {l} ) - ( 1 - \delta _ {jk }  ) z ( q _ {j} - q _ {k} ) ,
$$

where  $  u $
is odd and  $  w $, 
$  z $
are even, then the equation  $  \sqrt {- 1 } {\dot{L} } = [ M,L ] $
implies  $  z = - {\dot{u} } $, 
$  v = u  ^ {2} + \textrm{ const  } $, 
as well as a functional equation for  $  u $
and  $  w $
with solutions

$$ 
u = {
\frac{1}{q}
 } , {
\frac{1}{ \sin  q }
 } , {
\frac{1}{ { \mathop{\rm sn} } q }
 } .
$$

These three cases are referred to as rational (Calogero–Moser system), trigonometric (Calogero–Sutherland system) and elliptic (Calogero–Moser–Krichever system), respectively; the first two can be viewed as limits of the third as one or both periods of the Weierstrass function  $  { \mathop{\rm sn} } $(
cf. [[Weierstrass elliptic functions|Weierstrass elliptic functions]]) tend to infinity. In each case, the invariants  $  F _ {k} = { \mathop{\rm tr} } ( L  ^ {k} ) /k $, 
$  k = 1 \dots n $, 
are (generically) functionally independent and in involution (cf. [[Integrals in involution|Integrals in involution]]), so that the system is completely integrable. This was the first example [[#References|[a21]]] of a Lax pair with as parameter a function over a curve of genus  $  > 0 $(
generalizations are still (1996) quite rare, cf. [[#References|[a10]]] for hyperelliptic parameters).

2) By interpolating an eigenvector of  $  L $
into a Baker–Akhiezer function, it was shown in [[#References|[a21]]] that the solutions correspond to elliptic (in  $  x $) 
solutions of the Kadomtsev–Petviashvili equation  $  u _ {yy }  = ( u _ {t} + u _ {xxx }  - 6uu _ {x} ) _ {x} $. 
The first breakthrough in this respect was made in [[#References|[a1]]], concerning Korteweg–de Vries solutions and Lamé equations (cf. also [[Lamé equation|Lamé equation]]); moduli spaces of the corresponding algebro-geometric configurations (tangential covers) were described in [[#References|[a31]]] and their density was detected in [[#References|[a3]]].

3) Several models of the  $  r $-
matrix, both dynamic and non-dynamic, recently (1990{}s) became available (for a most complete set of references see [[#References|[a11]]]); however a finite-dimensional geodesic motion interpretation 4) has only been achieved for the rational and trigonometric cases [[#References|[a17]]].

5) S.P. Novikov posed the following question: Which algebro-geometric Kadomtsev–Petviashvili solutions can be expressed in terms of elliptic functions (cf. also [[Elliptic function|Elliptic function]])? These non-trivially overlap with elliptic solitons and a state-of-the-art report can be found in [[#References|[a22]]].

6) is the deep theory that ensues if the moving data is viewed as a rank- $  n $
bundle over the elliptic curve, cf. [[#References|[a4]]], [[#References|[a9]]].

Lastly, 7) is an extra property of the Baker function in the rational case, defined by F.A. Grünbaum [[#References|[a7]]] in connection with computerized [[Tomography|tomography]] (roughly stated, the Baker function is both an  $  x $- 
and a  $  z $-
eigenfunction for a pair of operators with eigenvalues that are functions of  $  z $, 
$  x $, 
respectively); its manifestation for the matrix  $  L $
above (cf. [[#References|[a1]]]) is investigated in [[#References|[a16]]].

A final word about generalizations: [[#References|[a27]]] adapts the geodesic problem 4) to other groups and metrics; [[#References|[a28]]] defines a relativistic Calogero–Moser–Krichever system; [[#References|[a24]]] and [[#References|[a25]]] provide discretized versions of the Calogero–Moser–Krichever and Ruijsenaars system, respectively; an Euler Calogero–Moser–Krichever system is related to a multi-component Kadomtsev–Petviashvili equation in [[#References|[a23]]]; and the most recent application of the Calogero–Moser–Krichever system is that conjecturally it provides moduli for solutions to the Seiberg–Witten equations, [[#References|[a5]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Airault,   H.P. McKean,   J. Moser,   "Rational and elliptic solutions of the Korteweg–de Vries equation and a related many-body problem"  ''Commun. Pure Appl. Math.'' , '''30'''  (1977)  pp. 95–148</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  E.D. Belokolos,   V.Z. Enol'skii,   "Algebraically integrable nonlinear equations and Humbert surfaces, plasma theory and nonlinear and turbulent processes in physics" , '''1–2''' , World Sci.  (1988)  pp. 20–57</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  E. Colombo,   G.P. Pirola,   E. Previato,   "Density of elliptic solitons"  ''J. Reine Angew. Math.'' , '''451'''  (1994)  pp. 161–169</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  R. Donagi,   E. Markman,   "Spectral covers, algebraically completely integrable Hamiltonian systems, and moduli of bundles" , ''Integrable Systems and Quantum Groups'' , ''Lecture Notes in Mathematics'' , '''1620''' , Springer  (1996)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  R. Donagi,   E. Witten,   "Supersymmetric Yang–Mills systems and integrable systems"  ''hep-th/9510101''  (1995)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  V.G. Drinfeld,   V.V. Sokolov,   "Lie algebras and equations of Korteweg–de Vries type"  ''J. Soviet Math.'' , '''30'''  (1985)  pp. 1975–2005</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  J.J. Duistermaat,   F.A. Grünbaum,   "Differential equations in the spectral parameter"  ''Comm. Math. Phys.'' , '''103'''  (1986)  pp. 177–240</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  L.D. Faddeev,   L.A. Takhtajan,   "Hamiltonian methods in the theory of solitons" , Springer  (1987)</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  G. Faltings,   "Stable G-bundles and projective connections"  ''J. Algebraic Geom.'' , '''2'''  (1993)  pp. 507–568</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  Yu.N. Fedorov,   "Integrable systems, Lax representations, and confocal quadrics"  ''Amer. Math. Soc. Transl. Ser. 2'' , '''168'''  (1995)  pp. 173–199</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  K. Hasegawa,   "Ruijsenaars' commuting difference operators as commuting transfer matrices"  ''q-alg/9512029''  (1995)</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top">  N. Hitchin,   "Stable bundles and integrable systems"  ''Duke Math. J.'' , '''54'''  (1987)  pp. 91–114</TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top">  E.L. Ince,   "Further investigations into the periodic Lamé function"  ''Proc. Roy. Soc. Edinburgh'' , '''60'''  (1940)  pp. 83–99</TD></TR><TR><TD valign="top">[a14]</TD> <TD valign="top">  A.R. Its,   V.Z. Enol'skii,   "Dynamics of the Calogero–Moser system and the reduction of hyperelliptic integrals to elliptic integrals"  ''Funct. Anal. Appl.'' , '''20'''  (1986)  pp. 62–64</TD></TR><TR><TD valign="top">[a15]</TD> <TD valign="top">  V.G. Kac,   J.W. van de Leur,   "The $n$-component KP hierarchy and representation theory" , ''Important Developments in Soliton Theory'' , Springer  (1993)  pp. 302–343</TD></TR><TR><TD valign="top">[a16]</TD> <TD valign="top">  A. Kasman,   "Bispectral KP solutions and linearization of Calogero–Moser particle systems"  ''Comm. Math. Phys.'' , '''172'''  (1995)  pp. 427–448</TD></TR><TR><TD valign="top">[a17]</TD> <TD valign="top">  D. Kazhdan,   B. Kostant,   S. Sternberg,   "Hamiltonian group actions and dynamical systems of Calogero type"  ''Comm. Pure Appl. Math.'' , '''31'''  (1978)  pp. 481–507</TD></TR><TR><TD valign="top">[a18]</TD> <TD valign="top">  I.M. Krichever,   "Rational solutions of the Kadomtsev–Petviashvili equation and integrable systems of $n$ particles on a line"  ''Funct. Anal. Appl.'' , '''12'''  (1978)  pp. 59–61</TD></TR><TR><TD valign="top">[a19]</TD> <TD valign="top">  M. Adler,   P. van Moerbeke,   "Completely integrable systems, Euclidean Lie algebras, and curves"  ''Adv. in Math.'' , '''38'''  (1980)  pp. 267–317</TD></TR><TR><TD valign="top">[a20]</TD> <TD valign="top">  M. Adler,   P. van Moerbeke,   "Linearization of Hamiltonian systems, Jacobi varieties and representation theory"  ''Adv. in Math.'' , '''38'''  (1980)  pp. 318–379</TD></TR><TR><TD valign="top">[a21]</TD> <TD valign="top">  I.M. Krichever,   "Elliptic solutions of the KP equation and integrable systems of particles"  ''Funct. Anal. Appl.'' , '''14'''  (1980)  pp. 282–290</TD></TR><TR><TD valign="top">[a22]</TD> <TD valign="top">  I.M. (ed.) Krichever,   "Special issue on elliptic solitons, dedicated to the memory of J.-L. Verdier"  ''Acta Applic. Math.'' , '''36''' :  1–2  (1994)</TD></TR><TR><TD valign="top">[a23]</TD> <TD valign="top">  I.M. Krichever,   O. Babelon,   E. Billey,   M. Talon,   "Spin generalizations of the Calogero–Moser system and the matrix KP equation"  ''Amer. Math. Soc. Transl. Ser. 2'' , '''170'''  (1995)  pp. 83–119</TD></TR><TR><TD valign="top">[a24]</TD> <TD valign="top">  F.W. Nijhoff,   G.-D. Pang,   "A time-discretized version of the Calogero–Moser model"  ''Phys. Lett. A'' , '''191'''  (1994)  pp. 101–107</TD></TR><TR><TD valign="top">[a25]</TD> <TD valign="top">  F.W. Nijhoff,   O. Ragnisco,   V.B. Kuznetsov,   "Integrable time-discretisation of the Ruijsenaars–Schneider model"  ''Comm. Math. Phys.'' , '''176'''  (1996)  pp. 681–700</TD></TR><TR><TD valign="top">[a26]</TD> <TD valign="top">  S.P. Novikov,   "A periodic problem for the KdV equation"  ''Funct. Anal. Appl.'' , '''8'''  (1974)  pp. 236–246</TD></TR><TR><TD valign="top">[a27]</TD> <TD valign="top">  M.A. Olshanetsky,   A.M. Perelomov,   "Classical integrable finite-dimensional systems related to Lie algebras"  ''Phys. Rep.'' , '''71'''  (1981)  pp. 313–400</TD></TR><TR><TD valign="top">[a28]</TD> <TD valign="top">  S.N.M. Ruijsenaars,   "Complete integrability of relativistic Calogero–Moser systems and elliptic function identities"  ''Comm. Math. Phys.'' , '''20'''  (1987)  pp. 191–213</TD></TR><TR><TD valign="top">[a29]</TD> <TD valign="top">  G. Segal,   G. Wilson,   "Loop groups and equations of KdV type"  ''IHES Publ. Math.'' , '''61'''  (1985)  pp. 5–65</TD></TR><TR><TD valign="top">[a30]</TD> <TD valign="top">  M. Sato,   Y. Sato,   "Soliton equations as dynamical systems on infinite-dimensional Grassmann manifold" , ''Nonlinear Partial Differential Equations in Applied Science'' , ''Math. Stud.'' , '''81''' , North-Holland  (1983)  pp. 259–271</TD></TR>
<TR><TD valign="top">[a31]</TD> <TD valign="top">  A. Treibich,   J.-L. Verdier,   "Solitons elliptiques" , ''The Grothendieck Festschrift'' , '''III''' , Birkhäuser  (1990)  pp. 437–480</TD></TR><TR><TD valign="top">[a32]</TD> <TD valign="top">  A.P. Veselov,   "Rational solutions of the KP equation and hamiltonian systems"  ''Russian Math. Surveys'' , '''35''' :  1  (1980)  pp. 239–240</TD></TR></table>