Bispectral KP solutions and linearization of Calogero-Moser particle systems
Article Received: 06 June 1994 Revised: 21 November 1994 Abstract
Rational and soliton solutions of the KP hierarchy in the subgrassmannian
Gr 1 are studied within the context of finite dimensional dual grassmannians. In the rational case, properties of the tau function, τ, which are equivalent to bispectrality of the associated wave function, ψ, are identified. In particular, it is shown that there exists a bound on the degree of all time, variables in τ if and only if ψ is a rank one bispectral wave function. The action of the bispectral involution, β, in the generic rational case is determined explicitly in terms of dual grassmannian parameters. Using the correspondence between rational solutions, and particle systems, it is demonstrated that β is a linearizing map of the Calogero-Moser particle system and is essentially the map σ introduced by Airault, McKean and Moser in 1977 .
Research supported by NSA Grant MDA904-92-H-3032
Adler, M., Moser, J.: On a Class of Polynomials Connected with the Korteweg-deVries Equation. Commun. Math. Phys.
, 1–30 (1978)
Airault, H., McKean, H.P., Moser, J.: Rational and Elliptic Solutions of the Korteweg-DeVries Equation and a Related Many-Body Problem. Commun. Pure and Applied Math.
, 95–148 (1977)
Duistermaat, J.J., Grünbaum, F.A.: Differential Equations in the Spectral Parameter. Commun. Math. Phys.
, 177–240 (1986)
Grünbaum, F.A.: The Kadomtsev-Petviashvili Equation: An Alternative Approach to the ‘Rank Two’ Solutions of Krichever and Novikov. Phy., Lett. A
, 146–150 (1989)
Hodge, W.V.D., Pedoe, D.: Methods of Algebraic Geometry Volume II. Cambridge: Cambridge University Press, 1952
Krichever, I.M.: Rational Solutions of the Kadomtsev-Petviashvili Equation and Integrable Systems of
Particles on, a Line. Funct. Anal. Appl.
, 59–61 (1978)
Krichever, I.M.: Rational Solutions of the Zakharov-Shabat Equations and Completely Intergrable Systems of
Particles on a Line. J. Soviet Math.
, 335–345 (1983)
Krichever, I.M.: Methods of Algebraic Geometry in the Theory of Non-linear Equations. Russ. Math. Surv.
:6 185–213 (1977)
Mulase, M.: Cohomological Structure in Soliton Equations and Jacobian Varieties. J. Diff. Geom.
, 403–430 (1984)
Mulase, M.: Algebraic Theory of the KP Equations. In Perspectives in Mathematical Physics. Hong Kong: International Press, 1994
Orlov, A.Yu. Schulman, E.I.: Additional symmetries for 2D integrable systems. Lett. Math. Phy.
, 171–179 (1986)
Previato, E.: Seventy Years of Spectral Curves: 1923–1993. To appear in Proceedings of CIME 1993, Springer-Verlag, Lecture Notes in Physics
Segal, G., Wilson, G.: Loop Groups and Equations of KdV Type. Publications Mathematiques No.
de l'Institut des Hautes Etudes Scientifiques, 5–65 (1985)
Shiota, T.: Calogero-Moser hierarchy and KP hierarchy. J. Math. Phys.
, 5844–5849 (1994)
Veselov, A.P.: Rational Solutions of the KP Equation and Hamiltonian Systems. Commun. Moscow Math. Soc., Russ. Math. Surv
:1 239–240 (1980)
Wilson, G.: Bispectral Commutative Ordinary Differential Operators. J. reine angew. Math.
, 177–204 (1993)
Zubelli, J.: On the Polynomial Tau Function for the KP Hierarchy and the Bispectral Property. Lett. Math. Phys.
, 41–48 (1992)