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''Calogero–Moser–Sutherland–Krichever system''
 
''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-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110030/c1100301.png" /> 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]]].
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110030/c1100303.png" />-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.
+
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.
 
A brief illustration of these terms follows.
Line 9: Line 23:
 
1) The system has a Hamiltonian function
 
1) The system has a Hamiltonian function
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110030/c1100304.png" /></td> </tr></table>
+
$$
 +
H = {
 +
\frac{1}{2}
 +
} \sum _ {i = 1 } ^ { n }  p _ {i}  ^ {2} + \sum _ {i < j } v ( q _ {i} - q _ {j} ) ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110030/c1100305.png" /> are position/momentum variables (all quantities are complex numbers) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110030/c1100306.png" /> is an even function. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110030/c1100307.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110030/c1100308.png" /> are the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110030/c1100309.png" />-matrices with entries
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110030/c11003010.png" /></td> </tr></table>
+
$$
 +
L _ {jk }  = p _ {j} \delta _ {jk }  + \sqrt {- 1 } ( 1 - \delta _ {jk }  ) u ( q _ {j} - q _ {k} ) ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110030/c11003011.png" /></td> </tr></table>
+
$$
 +
M _ {jk }  = \delta _ {jk }  \sum _ {l \neq j } w ( q _ {j} - q _ {l} ) - ( 1 - \delta _ {jk }  ) z ( q _ {j} - q _ {k} ) ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110030/c11003012.png" /> is odd and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110030/c11003013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110030/c11003014.png" /> are even, then the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110030/c11003015.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110030/c11003016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110030/c11003017.png" />, as well as a functional equation for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110030/c11003018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110030/c11003019.png" /> with solutions
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110030/c11003020.png" /></td> </tr></table>
+
$$
 +
u = {
 +
\frac{1}{q}
 +
} , {
 +
\frac{1}{ \sin  q }
 +
} , {
 +
\frac{1}{ { \mathop{\rm sn} } q }
 +
} .
 +
$$
  
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110030/c11003021.png" /> (cf. [[Weierstrass elliptic functions|Weierstrass elliptic functions]]) tend to infinity. In each case, the invariants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110030/c11003022.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110030/c11003023.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110030/c11003024.png" /> (generalizations are still (1996) quite rare, cf. [[#References|[a10]]] for hyperelliptic parameters).
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110030/c11003025.png" /> into a Baker–Akhiezer function, it was shown in [[#References|[a21]]] that the solutions correspond to elliptic (in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110030/c11003026.png" />) solutions of the Kadomtsev–Petviashvili equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110030/c11003027.png" />. 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]]].
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110030/c11003028.png" />-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]]].
+
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]]].
 
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-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110030/c11003029.png" /> bundle over the elliptic curve, cf. [[#References|[a4]]], [[#References|[a9]]].
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110030/c11003030.png" />- and a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110030/c11003031.png" />-eigenfunction for a pair of operators with eigenvalues that are functions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110030/c11003032.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110030/c11003033.png" />, respectively); its manifestation for the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110030/c11003034.png" /> above (cf. [[#References|[a1]]]) is investigated in [[#References|[a16]]].
+
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]]].
 
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====
 
====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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110030/c11003035.png" />-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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110030/c11003036.png" />-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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110030/c11003037.png" /> 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>
+
<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>

Latest revision as of 09:09, 26 March 2023


Calogero–Moser–Sutherland–Krichever system

A finite-dimensional Hamiltonian system which is algebraically completely integrable (cf. 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 equations, starting in the late 1960{}s, some prototypes that represent the main features have emerged. For example, the Korteweg–de Vries equation and the non-linear Schrödinger equation for inverse scattering [a8], rank- $ 2 $ perturbations for algebraic complete integrability [a19], [a20], the Kadomtsev–Petviashvili equation (cf. Soliton) for Grassmannians and Schur functions [a30], and the modified Korteweg–de Vries equation for representation theory [a6], [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) 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), so that the system is completely integrable. This was the first example [a21] of a Lax pair with as parameter a function over a curve of genus $ > 0 $( generalizations are still (1996) quite rare, cf. [a10] for hyperelliptic parameters).

2) By interpolating an eigenvector of $ L $ into a Baker–Akhiezer function, it was shown in [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 [a1], concerning Korteweg–de Vries solutions and Lamé equations (cf. also Lamé equation); moduli spaces of the corresponding algebro-geometric configurations (tangential covers) were described in [a31] and their density was detected in [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 [a11]); however a finite-dimensional geodesic motion interpretation 4) has only been achieved for the rational and trigonometric cases [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)? These non-trivially overlap with elliptic solitons and a state-of-the-art report can be found in [a22].

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

Lastly, 7) is an extra property of the Baker function in the rational case, defined by F.A. Grünbaum [a7] in connection with computerized 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. [a1]) is investigated in [a16].

A final word about generalizations: [a27] adapts the geodesic problem 4) to other groups and metrics; [a28] defines a relativistic Calogero–Moser–Krichever system; [a24] and [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 [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, [a5].

References

[a1] 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
[a2] 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
[a3] E. Colombo, G.P. Pirola, E. Previato, "Density of elliptic solitons" J. Reine Angew. Math. , 451 (1994) pp. 161–169
[a4] 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)
[a5] R. Donagi, E. Witten, "Supersymmetric Yang–Mills systems and integrable systems" hep-th/9510101 (1995)
[a6] V.G. Drinfeld, V.V. Sokolov, "Lie algebras and equations of Korteweg–de Vries type" J. Soviet Math. , 30 (1985) pp. 1975–2005
[a7] J.J. Duistermaat, F.A. Grünbaum, "Differential equations in the spectral parameter" Comm. Math. Phys. , 103 (1986) pp. 177–240
[a8] L.D. Faddeev, L.A. Takhtajan, "Hamiltonian methods in the theory of solitons" , Springer (1987)
[a9] G. Faltings, "Stable G-bundles and projective connections" J. Algebraic Geom. , 2 (1993) pp. 507–568
[a10] Yu.N. Fedorov, "Integrable systems, Lax representations, and confocal quadrics" Amer. Math. Soc. Transl. Ser. 2 , 168 (1995) pp. 173–199
[a11] K. Hasegawa, "Ruijsenaars' commuting difference operators as commuting transfer matrices" q-alg/9512029 (1995)
[a12] N. Hitchin, "Stable bundles and integrable systems" Duke Math. J. , 54 (1987) pp. 91–114
[a13] E.L. Ince, "Further investigations into the periodic Lamé function" Proc. Roy. Soc. Edinburgh , 60 (1940) pp. 83–99
[a14] 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
[a15] 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
[a16] A. Kasman, "Bispectral KP solutions and linearization of Calogero–Moser particle systems" Comm. Math. Phys. , 172 (1995) pp. 427–448
[a17] D. Kazhdan, B. Kostant, S. Sternberg, "Hamiltonian group actions and dynamical systems of Calogero type" Comm. Pure Appl. Math. , 31 (1978) pp. 481–507
[a18] 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
[a19] M. Adler, P. van Moerbeke, "Completely integrable systems, Euclidean Lie algebras, and curves" Adv. in Math. , 38 (1980) pp. 267–317
[a20] M. Adler, P. van Moerbeke, "Linearization of Hamiltonian systems, Jacobi varieties and representation theory" Adv. in Math. , 38 (1980) pp. 318–379
[a21] I.M. Krichever, "Elliptic solutions of the KP equation and integrable systems of particles" Funct. Anal. Appl. , 14 (1980) pp. 282–290
[a22] I.M. (ed.) Krichever, "Special issue on elliptic solitons, dedicated to the memory of J.-L. Verdier" Acta Applic. Math. , 36 : 1–2 (1994)
[a23] 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
[a24] F.W. Nijhoff, G.-D. Pang, "A time-discretized version of the Calogero–Moser model" Phys. Lett. A , 191 (1994) pp. 101–107
[a25] F.W. Nijhoff, O. Ragnisco, V.B. Kuznetsov, "Integrable time-discretisation of the Ruijsenaars–Schneider model" Comm. Math. Phys. , 176 (1996) pp. 681–700
[a26] S.P. Novikov, "A periodic problem for the KdV equation" Funct. Anal. Appl. , 8 (1974) pp. 236–246
[a27] M.A. Olshanetsky, A.M. Perelomov, "Classical integrable finite-dimensional systems related to Lie algebras" Phys. Rep. , 71 (1981) pp. 313–400
[a28] S.N.M. Ruijsenaars, "Complete integrability of relativistic Calogero–Moser systems and elliptic function identities" Comm. Math. Phys. , 20 (1987) pp. 191–213
[a29] G. Segal, G. Wilson, "Loop groups and equations of KdV type" IHES Publ. Math. , 61 (1985) pp. 5–65
[a30] 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
[a31] A. Treibich, J.-L. Verdier, "Solitons elliptiques" , The Grothendieck Festschrift , III , Birkhäuser (1990) pp. 437–480
[a32] A.P. Veselov, "Rational solutions of the KP equation and hamiltonian systems" Russian Math. Surveys , 35 : 1 (1980) pp. 239–240
How to Cite This Entry:
Calogero-Moser-Krichever system. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Calogero-Moser-Krichever_system&oldid=22233
This article was adapted from an original article by E. Previato (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article