Calogero-Moser-Krichever system

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

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