Abstract
One constructs all the decreasing rational solutions of the Kadomtsev-Petviashvili equations. The presented method allows us to identify the motion of the poles of the obtained functions with the motion of a system of N particles on a line with a Hamiltonian of the Calogero-Moser type. Thus, this Hamiltonian system is imbedded in the theory of the algebraic-geometric solutions of the Zakharov-Shabat equations.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
Literature cited
B. B. Kadomtsev and V. I. Petviashvili, “On the stability of solitary waves in weakly dispersing media,” Dokl. Akad. Nauk SSSR,192, No. 4, 753–756 (1970).
V. E. Zakharov and A. B. Shabat, “A plan for integrating the nonlinear equations of mathematical physics by the method of the inverse scattering problem. I,” Funkts. Anal. Prilozhen.,8, No. 3, 43–53 (1974).
V. S. Dryuma, “On the analytic solution of the two-dimensional Korteweg-de Vries equation,” Pis'ma Zh. Eksp. Teor. Fiz.,19, No. 753–755 (1974).
I. M. Krichever, “An algebraic-geometric construction of the Zakharov-Shabat equations and their periodic solutions,” Dokl. Akad. Nauk SSSR,227, No. 2, 291–294 (1976).
I. M. Krichever, “Integration of nonlinear equations by the methods of algebraic geometry,” Funkts. Anal. Prilozhen.,11, No. 1, 15–31 (1977).
I. M. Krichever, “Methods of algebraic geometry in the theory of nonlinear equations,” Usp. Mat. Nauk,32, No. 6, 183–208 (1977).
S. P. Novikov, “A periodic problem for the Korteweg-de Vries equation. I,” Funkts. Anal. Prilozhen.,8, No. 3, 54–66 (1974).
L. A. Bordag, A. R. Its, V. B. Matveev, S. V. Manacov, and V. E. Zakharov, “Two-dimensional solitons of the Kadomtsev-Petviashvili equation and their interaction,” Preprint KMU (1977).
J. Moser, “Three integrable Hamiltonian systems connected with isospectral deformations,” Adv. Math.,16, 197–220 (1975).
H. Airault, H. P. McKean, and J. Moser, “Rational and elliptic solutions of the Korteweg-De Vries Equation and a related many-body problem,” Commun. Pure Appl. Math.,30, 95–148 (1977).
Additional information
Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akad. Nauk SSSR, Vol. 84, pp. 117–130, 1979.
Rights and permissions
About this article
Cite this article
Krichever, I.M. Rational solutions of the Zakharov-Shabat equations and completely integrable systems of N particles on a line. J Math Sci 21, 335–345 (1983). https://doi.org/10.1007/BF01660590
Issue Date:
DOI: https://doi.org/10.1007/BF01660590