Rational solutions of the Zakharov-Shabat equations and completely integrable systems of N particles on a line
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.
KeywordsIntegrable System Hamiltonian System Rational Solution
Unable to display preview. Download preview PDF.
- 1.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).Google Scholar
- 2.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).Google Scholar
- 3.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).Google Scholar
- 4.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).Google Scholar
- 5.I. M. Krichever, “Integration of nonlinear equations by the methods of algebraic geometry,” Funkts. Anal. Prilozhen.,11, No. 1, 15–31 (1977).Google Scholar
- 6.I. M. Krichever, “Methods of algebraic geometry in the theory of nonlinear equations,” Usp. Mat. Nauk,32, No. 6, 183–208 (1977).Google Scholar
- 7.S. P. Novikov, “A periodic problem for the Korteweg-de Vries equation. I,” Funkts. Anal. Prilozhen.,8, No. 3, 54–66 (1974).Google Scholar
- 8.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).Google Scholar
- 9.J. Moser, “Three integrable Hamiltonian systems connected with isospectral deformations,” Adv. Math.,16, 197–220 (1975).Google Scholar
- 10.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).Google Scholar