Elliptic solutions of the Kadomtsev-Petviashvili equation and integrable systems of particles. (English) Zbl 0473.35071

Krichever, I. M.

Funct. Anal. Appl. 14, 282-290 (1981).

Translation from Funkts. Anal. Prilozh. 14, No. 4, 45–54 (1980; Zbl 0462.35080).

MSC:

35Q53	KdV equations (Korteweg-de Vries equations)
35Q58	Other completely integrable PDE (MSC2000)
37K10	Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
37K20	Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions
37K15	Inverse spectral and scattering methods for infinite-dimensional Hamiltonian and Lagrangian systems

Keywords:

elliptic solutions; Kadomtsev-Petviashvili equation; integrable systems of particles; Lax-type representation; inverse problem method; Hamiltonian

Citations:

Zbl 0462.35080

References:

[1]	H. Bateman and A. Erdelyi, Higher Transcendental Functions. Elliptic and Automorphic Functions, Lamé and Mathieu Functions [Russian translation], Nauka, Moscow (1967).
[2]	F. Calogero, ”Exactly solvable one-dimensional many-body systems,” Lett. Nuovo Cimento,13, 411-415 (1975).
[3]	A. M. Perelomov, ”Completely integrable classical systems connected with semisimple Lie algebras,” Lett. Math. Phys.,1, 531-540 (1977).
[4]	H. Airault, H. McKean, and J. Moser, ”Rational and elliptic solutions of the KdV equation and related many-body problem,” Commun. Pure Appl. Math.,30, 95-125 (1977). · Zbl 0338.35024
[5]	I. M. Krichever, ”Rational solutions of the Kadomtsev?Petviashvili equation and integrable systems of N particles on a line,” Funkts. Anal. Prilozhen.,12, No. 1, 76-78 (1978). · Zbl 0374.70008
[6]	D. V. Choodnovsky and G. V. Choodnovsky, ”Pole expansions of nonlinear partial differential equations,” Nuovo Cimento,40B, 339-350 (1977). · Zbl 0324.02066
[7]	F. Calogero, ”Integrable many-body problem,” Preprint, Univ. di Roma, No. 89 (1978). · Zbl 0399.70022
[8]	K. M. Case, ”The N-soliton solutions of the Bendgamine?Ono equation,” Proc. Nat. Acad. Sci. USA,75, 3562-3563 (1978). · Zbl 0449.35085
[9]	M. A. Olshanetsky and A. M. Perelomov, ”Explicit solutions of some completely integrable systems,” Lett. Nuovo Cimento,17, 97-133 (1976). · Zbl 0342.58017
[10]	M. A. Olshanetsky and N. V. Rogov, ”Bound states in completely integrable systems with two types of particles,” Ann. Inst. H. Poincare,29, 169-177 (1978). · Zbl 0416.58014
[11]	D. V. Choodnovsky, ”Meromorphic solutions of nonlinear partial differential equations and particle integrable systems,” J. Math. Phys.,20, No. 12, 2416-2424 (1979). · Zbl 0455.35096
[12]	V. S. Dryuma, ”An analytic solution of the two-dimensional Korteweg?de Vries equation,” Pis’ma Zh. Eksp. Teor. Fiz.,19, No. 12, 219-225 (1973).
[13]	V. E. Zakharov and A. B. Shabat, ”A scheme for the integration of nonlinear equations of mathematical physics by an inverse problem of scattering theory. I,” Funkts. Anal. Prilozhen.,8, No. 3, 43-53 (1974). · Zbl 0303.35024
[14]	E. Kamke, Handbook on Ordinary Differential Equations [in German], Chelsea Publ.
[15]	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). · Zbl 0361.35007
[16]	I. M. Krichever, ”The integration of nonlinear equations by the methods of algebraic geometry,” Funkts. Anal. Prilozhen.,11, No. 1, 15-31 (1977). · Zbl 0346.35028
[17]	H. Baker, ”Note on the foregoing paper, ?Commutative ordinary differential equations,?” Proc. R. Soc. London, Ser. A,118, 570-576 (1928).
[18]	B. A. Dubrovin and S. P. Novikov, ”Periodic and conditionally periodic analogues of many-soliton solutions of the Korteweg?de Vries equation,” Zh. Eksp. Teor. Fiz.,67, No. 12, 2131-2143 (1974).