Literature cited
B. A. Dubrovin, V. B. Matveev, and S. P. Novikov, “Nonlinear equations of Korteweg-de Vries type, finite-zone linear operators, and Abelian manifolds,” Usp. Mat. Nauk,31, No. 1, 55–136 (1976).
A. R. Its and V. B. Matveev, “Hill operators with a finite number of lacuna and multisoliton solutions of the Korteweg-de Vries equation,” Teor. Mat. Fiz.,23, No. 1, 51–64 (1975).
V. A. Marchenko, Sturm-Liouville Operators and Their Applications [in Russian], Naukova Dumka Kiev (1977).
S. Yu. Dobrokhotov, “Resonance correction to the adiabatically perturbed finite-zone almost periodic solution of the Korteweg-de Vries equation,” Mat. Zametki,44, No. 4, 551–554 (1988).
S. Yu. Dobrokhotov and V. P. Maslov, “Finite-zone almost periodic solutions in WKB- approximations,” Itogi Nauki i Tekh., Ser. Sov. Probl. Mat.,15, 3–94 (1980).
H. Flashka, F. Forest, and D. W. McLaughlin, “Multiphase averaging and the inverse spectral solution of KdV,” Commun. Pure Appl. Math.,33, No. 6, 739–794 (1980).
I. M. Krichever, “Spectral theory of two-dimensional periodic operators and its applications,” Usp. Mat. Nauk,44, No. 2, 121–184 (1989).
D. J. Kaup, “A perturbation theory for inverse scattering transforms,” SIAM J. Appl. Math.,31, No. 1, 121–132 (1976).
V. E. Zakharov, S. V. Manakov, L. P. Pitaevskii, and S. P. Novikov, Theory of Solitons [in Russian], Nauka, Moscow (1979).
Yu. M. Vorob'ev and S. Yu. Dobrokhotov, “Completeness of the system of eigenfunctions of a nonelliptic operator on the torus generated by the Hill operator with finite-zone potential,” Funkts. Analiz Prilozhen.,22, No. 2, 65–66 (1988).
I. M. Krichever, “‘Hessians’ of integrals of the Korteweg-de Vries equation and perturbation of finite-zone solutions,” Dokl. Akad. Nauk SSSR,270, No. 6, 1312–1317 (1983).
B. A. Dubrovin, “Theta-functions and nonlinear equations,” Usp. Mat. Nauk,36, No. 2, 12–80 (1981).
B. S. Kashin and A. A. Saakyan, Orthogonal Series [in Russian], Nauka, Moscow (1984).
S. Yu. Dobrokhotov and V. P. Maslov, “Multiphase asymptotics of nonlinear participating differential equations,” Sov. Sci. Rev. Math. Phys. Rev.,3, 211–311 (1982).
Author information
Authors and Affiliations
Additional information
Translated from Matematicheskie Zametki, Vol. 47, No. 1, pp. 47–61, January, 1990.
Rights and permissions
About this article
Cite this article
Vorob'ev, Y.M., Dobrokhotov, S.Y. Basis systems on the torus generated by finite-zone integration of the Korteweg-de Vries equation. Mathematical Notes of the Academy of Sciences of the USSR 47, 32–41 (1990). https://doi.org/10.1007/BF01157281
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01157281