Abstract
We present a Lax pair for the field elliptic Calogero-Moser system and establish a connection between this system and the Kadomtsev-Petviashvili equation. Namely, we consider elliptic families of solutions of the KP equation such that their poles satisfy a constraint of being balanced. We show that the dynamics of these poles is described by a reduction of the field elliptic CM system.
We construct a wide class of solutions to the field elliptic CM system by showing that any N-fold branched cover of an elliptic curve gives rise to an elliptic family of solutions of the KP equation with balanced poles.
Similar content being viewed by others
References
H. Airault, H. McKean, and J. Moser, “Rational and elliptic solutions of the Korteweg-de Vries equation and related many-body problem,” Commun. Pure Appl. Math., 30, No.1, 95–148 (1977).
O. Babelon, E. Billey, I. Krichever, and M. Talon, “Spin generalisation of the Calogero-Moser system and the matrix KP equation,” In: Topics in Topology and Mathematical Physics, Amer. Math. Soc. Transl., Ser. 2, Vol. 170, Amer. Math. Soc., Providence, 1995, pp. 83–119.
H. Bateman and A. Erdelyi, Higher Transcendental Functions, Vol. I I, McGraw-Hill, 1953.
F. Calogero, “Exactly solvable one-dimensional many-body systems,” Lett. Nuo vo Cimento (2), 13, No.11, 411–416 (1975).
I. Krichever, “An algebraic-geometric construction of the Zakharov-Shabat equation and their periodic solutions,” Dokl. Ak ad. Nauk USSR, 227, No.2, 291–294 (1976).
I. Krichever, “The integration of nonlinear equation with the help of algebraic-geometrical methods,” Funkts. Anal. Prilozhen., 11, No.1, 15–31 (1977).
I. Krichever, “Elliptic solutions of Kadomtsev-Petviashvili equations and integrable systems of particles,” Funkts. Anal. Prilozhen., 14, No.1, 45–54 (1980).
I. Krichever, “Elliptic solutions to difference non-linear equations and nested Bethe ansatz equations,” In: Calogero-Moser-Sutherland models (Montreal, QC, 1997), CRM Ser. Math. Phys., Springer-Verlag, New York, 2000, pp. 249–271.
I. Krichever, “Elliptic analog of the Toda lattice,” Internat. Math. Res. Notices, No. 8, 383–412 (2000).
I. Krichever, Vector Bundles and Lax Equations on Algebraic Curves, hep-th/0108110 (2001).
I. Krichever, O. Lipan, P. Wiegmann, and A. Zabrodin, “Quantum integrable models and discrete classical Hirota equations,” Comm. Math. Phys., 188, No.2, 267–304 (1997).
I. Krichever and A. Zabrodin, “Spin generalisation of the Ruijsenaars-Schneider model, the nonabelian two-dimensionalized Toda lattice, and representations of the Sklyanin algebra,” Usp. Mat. Nauk, 50, No.6, 3–56 (1995).
A. Levin, M. Olshanetsky, and A. Zotov, Hitchin Systems - Symplectic Maps and Two-Dimensional Version, arXiv:nlin. SI/0110045 (2001).
A. Gorsky and N. Nekrasov, Elliptic Calogero-Moser System from Two-Dimensional Current Algebra, hep-th/9401021.
N. Nekrasov, “Holomorphic bundles and many-body systems,” Comm. Math. Ph ys., 180, No.3, 587–603 (1996).
A. M. Perelomov, Integrable Systems of ClassicalMechanics and Lie Algebras, Vol. I, Birkhauser Verlag, Basel, 1990.
E. Markman, “Spectral curves and integrable systems,” Compositio Math., 93, 255–290 (1994).
Rights and permissions
About this article
Cite this article
Akhmetshin, A.A., Krichever, I.M. & Volvovski, Y.S. Elliptic Families of Solutions of the Kadomtsev--Petviashvili Equation and the Field Elliptic Calogero--Moser System. Functional Analysis and Its Applications 36, 253–266 (2002). https://doi.org/10.1023/A:1021706525301
Issue Date:
DOI: https://doi.org/10.1023/A:1021706525301