Communications in Mathematical Physics

, Volume 172, Issue 2, pp 427–448 | Cite as

Bispectral KP solutions and linearization of Calogero-Moser particle systems

  • Alex Kasman
Article

Abstract

Rational and soliton solutions of the KP hierarchy in the subgrassmannianGr1 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 [2].

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Adler, M., Moser, J.: On a Class of Polynomials Connected with the Korteweg-deVries Equation. Commun. Math. Phys.61, 1–30 (1978)Google Scholar
  2. 2.
    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.30, 95–148 (1977)Google Scholar
  3. 3.
    Duistermaat, J.J., Grünbaum, F.A.: Differential Equations in the Spectral Parameter. Commun. Math. Phys.103, 177–240 (1986)Google Scholar
  4. 4.
    Grünbaum, F.A.: The Kadomtsev-Petviashvili Equation: An Alternative Approach to the ‘Rank Two’ Solutions of Krichever and Novikov. Phy., Lett. A139, 146–150 (1989)Google Scholar
  5. 5.
    Hodge, W.V.D., Pedoe, D.: Methods of Algebraic Geometry Volume II. Cambridge: Cambridge University Press, 1952Google Scholar
  6. 6.
    Krichever, I.M.: Rational Solutions of the Kadomtsev-Petviashvili Equation and Integrable Systems ofN Particles on, a Line. Funct. Anal. Appl.12, 59–61 (1978)Google Scholar
  7. 7.
    Krichever, I.M.: Rational Solutions of the Zakharov-Shabat Equations and Completely Intergrable Systems ofN Particles on a Line. J. Soviet Math.21, 335–345 (1983)Google Scholar
  8. 8.
    Krichever, I.M.: Methods of Algebraic Geometry in the Theory of Non-linear Equations. Russ. Math. Surv.32:6 185–213 (1977)Google Scholar
  9. 9.
    Mulase, M.: Cohomological Structure in Soliton Equations and Jacobian Varieties. J. Diff. Geom.19, 403–430 (1984)Google Scholar
  10. 10.
    Mulase, M.: Algebraic Theory of the KP Equations. In Perspectives in Mathematical Physics. Hong Kong: International Press, 1994Google Scholar
  11. 11.
    Orlov, A.Yu. Schulman, E.I.: Additional symmetries for 2D integrable systems. Lett. Math. Phy.12, 171–179 (1986)Google Scholar
  12. 12.
    Previato, E.: Seventy Years of Spectral Curves: 1923–1993. To appear in Proceedings of CIME 1993, Springer-Verlag, Lecture Notes in PhysicsGoogle Scholar
  13. 13.
    Segal, G., Wilson, G.: Loop Groups and Equations of KdV Type. Publications Mathematiques No.61 de l'Institut des Hautes Etudes Scientifiques, 5–65 (1985)Google Scholar
  14. 14.
    Shiota, T.: Calogero-Moser hierarchy and KP hierarchy. J. Math. Phys.35, 5844–5849 (1994)Google Scholar
  15. 15.
    Veselov, A.P.: Rational Solutions of the KP Equation and Hamiltonian Systems. Commun. Moscow Math. Soc., Russ. Math. Surv35:1 239–240 (1980)Google Scholar
  16. 16.
    Wilson, G.: Bispectral Commutative Ordinary Differential Operators. J. reine angew. Math.442, 177–204 (1993)Google Scholar
  17. 17.
    Zubelli, J.: On the Polynomial Tau Function for the KP Hierarchy and the Bispectral Property. Lett. Math. Phys.24, 41–48 (1992)Google Scholar

Copyright information

© Springer-Verlag 1995

Authors and Affiliations

  • Alex Kasman
    • 1
  1. 1.Department of MathematicsBoston UniversityBostonUSA

Personalised recommendations