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Integration of nonlinear equations by the methods of algebraic geometry. (English) Zbl 0368.35022

35G20 Nonlinear higher-order PDEs
35A25 Other special methods applied to PDEs
47E05 General theory of ordinary differential operators
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[1] C. Gardner, J. Greene, M. Kruskal, and R. Miura, ”Method for solving the Korteweg?de Vries equation,” Phys. Rev. Lett.,19, 1095-1098 (1967). · Zbl 1103.35360
[2] V. E. Zakharov and A. B. Shabat, ”A scheme for integrating the nonlinear equations of mathematical physics by the method of the inverse scattering problem. I,” Funkts. Anal. Prilozhen.,8, 43-53 (1974). · Zbl 0303.35024
[3] B. A. Dubrovin, V. B. Matveev, and S. P. Novikov, ”Nonlinear equations of the Korteweg?de Vries type, finite-zone linear operators, and Abelian manifolds,” Usp. Mat. Nauk,31, No. 1, 55-136 (1976). · Zbl 0326.35011
[4] I. M. Krichever, ”Algebrogeometric construction of the Zakharov?Shabat equations and their periodic solutions,” Dokl. Akad. Nauk SSSR,227, No. 2, 291-294 (1976). · Zbl 0361.35007
[5] I. M. Krichever, ”Algebraic curves and commutative matrix differential operators,” Funkts. Anal. Prilozhen.,10, No. 2, 75-77 (1976). · Zbl 0338.35082
[6] B. A. Dubrovin, I. M. Krichever, and S. P. Novikov, ”The Schrödinger equation in a periodic field and Riemann surfaces,” Dokl. Akad. Nauk SSSR,229, No. 1, 15-19 (1976). · Zbl 0441.35021
[7] S. P. Novikov, ”The periodic problem for the Korteweg?de Vries equation. I,” Funkts. Anal. Prilozhen.,8, No. 3, 54-66 (1974). · Zbl 0301.54027
[8] I. M. Gel’fand and L. A. Dikii, ”Asymptotic behavior of the resolvent of Sturm?Liouville operators and the algebra of Korteweg?de Vries equations,” Usp. Mat. Nauk,30, No. 5, 67-101 (1975).
[9] O. I. Bogoyavlenskii, ”Integrals of higher stationary Korteweg?de Vries equations and eigenvalues of Hill’s operator,” Funkts. Anal. Prilozhen.,10, No. 2, 9-13 (1976).
[10] I. R. Shafarevich, Foundations of Algebraic Geometry [in Russian], Nauka, Moscow (1972). · Zbl 0204.21301
[11] I. M. Krichever, ”Potentials with zero coefficient of reflection on a background of finite-zone potentials,” Funkts. Anal. Prilozhen.,9, No. 2, 77-78 (1975). · Zbl 0333.34022
[12] G. Springer, Introduction to Riemann Surfaces, Addison-Wesley (1957). · Zbl 0078.06602
[13] N. I. Akhiezer, ”A continuous analog of orthogonal polynomials on a system of intervals,” Dokl. Akad. Nauk SSSR,141, No. 2, 263-266 (1961). · Zbl 0109.29602
[14] B. A. Dubrovin, ”Periodic problems for the Korteweg?de Vries equation in the class of finite-band potentials,” Funkts. Anal. Prilozhen.,9, No. 3, 41-51 (1975). · Zbl 0316.30019
[15] A. R. Its and V. B. Matveev, ”The Schrödinger operator in a finite-zone spectrum and N-soliton solutions of the Korteweg?de Vries equation,” Teor. Mat. Fiz.,23, No. 1, 51-67 (1975).
[16] B. A. Dubrovin and S. P. Novikov, ”A periodicity problem for the Korteweg?de Vries and Sturm?Liouville equations. Their connection with algebraic geometry,” Dokl. Akad. Nauk SSSR,219, No. 3, 19-22 (1974). · Zbl 0312.35015
[17] N. G. Chebotarev, Theory of Algebraic Functions [in Russian], Gostekhizdat, Moscow (1948). · Zbl 0038.15201
[18] E. I. Zverovich, ”Boundary-value problems in the theory of analytic functions,” Usp. Mat. Nauk,26, No. 1, 113-181 (1971). · Zbl 0217.10201
[19] A. R. Its and V. B. Matveev, ”A class of solutions of the Korteweg?de Vries equation,” in: Problems of Mathematical Physics [in Russian], No. 8, LGU (1976).
[20] B. B. Kadomtsev and V. I. Petviashvili, ”Stability of combined waves in weakly dispersing media,” Dokl. Akad. Nauk SSSR,192, No. 4, 753-756 (1970). · Zbl 0217.25004
[21] V. E. Zakharov, ”On the problem of stochastization of one-dimensional chains of nonlinear oscillators,” Zh. Eksp. Teor. Fiz.,65, No. 1, 219-225 (1973).
[22] O. I. Bogoyavlenskii, On Perturbations of the Toda Lattice, Preprint, Chernogolovka (1976).
[23] P. D. Lax, ”Periodic solutions of the KdV equations,” Comm. Pure Appl. Math.,28, 141-188 (1975). · Zbl 0302.35008
[24] I. M. Gel’fand and L. A. Dikii, ”Fractional powers of operators and Hamiltonian systems,” Funkts. Anal. Prilozhen.,10, No. 4, 13-29 (1976).
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