Literature Cited
I. M. Krichever, "Algebraic-geometrical 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 algebraic geometry methods," Funkts. Anal. Prilozhen.,11, No. 1, 15–31 (1977).
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).
V. E. Zakharov, S. V. Manakov, S. P. Novikov, and L. P. Pitaevskii, Theory of Solitons: the Inverse Problem Method [in Russian], Nauka, Moscow (1980).
I. M. Krichever, "Methods of algebraic geometry in the theory of nonlinear equations," Usp. Mat. Nauk,32, No. 6, 183–208 (1980).
I. M. Krichever and S. P. Novikov, "Holomorphic bundles over algebraic curves and nonlinear equations," Usp. Mat. Nauk,35, No. 6, 47–68 (1980).
B. A. Dubrovin, "Theta-functions and nonlinear equations," Usp. Mat. Nauk,36, No. 2, 11–80 (1981).
B. A. Dubrovin, I. M. Krichever, and S. P. Novikov, "Topological and algebraic geometry methods in contemporary mathematical physics. II," Soviet Scientific Reviews. Math. Phys. Reviews: OPA, Amsterdam, 1982, Vol. 3, pp. 1–151.
S. P. Novikov, "Two-dimensional Schrödinger operators in periodic fields," Contemporary Problems in Mathematics [in Russian], Vol. 23, Itogi Nauki i Tekhniki, VINITI Akad. Nauk SSSR, Moscow (1983), pp. 3–32.
B. A. Dubrovin, "Matrix finite-zone operators," Contemporary Problems in Mathematics [in Russian], Vol. 23, Itogi Nauki i Tekhniki, VINITI Akad. Nauk SSSR, Moscow (1983), pp. 33–78.
I. M. Krichever, "Nonlinear equations and elliptic curves," Contemporary Problems in Mathematics [in Russian], Vol. 23, Itogi Nauki i Tekhniki, VINITI Akad. Nauk SSSR, Moscow (1983), pp. 79–136.
S. A. Brazovskii, S. A. Gordyunin, and N. N. Kirova, "The exact solution of the Peierls model with an arbitrary number of electrons in a cell," Pis'ma Zh. Eksp. Teor. Fiz.,31, No. 8, 486–490 (1980).
E. D. Belokolos, "The Peierls—Frolich problems and finite-zone potentials. I," Teor. Mat. Fiz.,45, No. 2, 268–280 (1980).
E. D. Belokolos, "The Peierls—Frolich problems and finite-zone potentials. II," ibid.,48, No. 1, 60–69 (1981).
S. A. Brazovskii, I. E. Dzyaloshinskii, and I. M. Krichever, "Exactly solvable discrete Peierls models," Zh. Eksp. Teor. Fiz.,83, No. 1, 389–415 (1982).
I. E. Dzyaloshinskii and I. M. Krichever, "The sound and the wave of charge density in the discrete Peierls model," Zh. Eksp. Teor. Fiz.,85, No. 11, 1771–1789 (1983).
I. M. Krichever, "The Peierls model," Funkts. Anal. Prilozhen.,16, No. 4, 10–26 (1982).
I. E. Dzyaloshinskii and I. M. Krichever, "The effects of co-measurability in the discrete Peierls model," Zh. Eksp. Teor. Fiz.,83, No. 5, 1576–1581 (1982).
I. V. Cherednik, "Reality conditions in the ‘finite-zone’ integration," Dokl. Akad. Nauk SSSR,252, No. 5, 1104–1108 (1980).
V. E. Zakharov and A. B. Shabat, "The integration scheme for nonlinear equations of mathematical physics with the method of the inverse problem of the scattering theory. I," Funkts. Anal. Prilozhen.,8, No. 3, 43–53 (1974).
I. M. Krichever, "Rational solutions of the Kadomtsev—Petviashvili equation and integrable systems of particles on the direct line," Funkts. Anal. Prilozhen.,12, No. 1, 76–78 (1978).
L. A. Bordag, A. R. Its, V. B. Matveev, S. V. Manakov, and V. E. Zakharov, "Two-dimensional solitons of Kadomtsev—Petviashvili equation," Phys. Lett., 205–207 (1977).
I. M. Krichever, "The Laplace method, algebraic curves and nonlinear equations," Funkts. Anal. Prilozhen.,18, No. 3, 43–56 (1984).
J. Fay, "Theta-function on Riemann surfaces," Lect. Notes in Math.,352 (Springer-Verlag), Berlin (1973).
V. B. Matveev, "Darboux transformations and the solutions of KP equation depending on the functional parameters," Lett. Math. Phys.,3, 213–225 (1979).
E. A. Kuznetsov and A. B. Mikhailov, "Stability of stationary waves in nonlinear media with weak dispersion," Zh. Eksp. Teor. Fiz.,67, No. 11, 1019–1027 (1974).
B. A. Dubrovin, "The periodic problem for the Korteweg—de Vries equation," Funkts. Anal. Prilozhen.,9, No. 3, 41–51 (1975).
B. A. Dubrovin, "The inverse problem of scattering for periodic finite-zone potentials," Funkts. Anal. Prilozhen.,9, No. 1, 65–66 (1975).
I. V. Cherednik, "Differential equations for the Beiker—Akhiezer functions," Funkts. Anal. Prilozhen.,12, No. 3, 45–54 (1978).
N. Yajima and M. Oikawa, "Interactions between Langmure's and sonic waves," Prog. Theor. Phys.,56, 17–19 (1976).
V. G. Makhankov, "On the stationary solutions of Schrödinger equation with self-congruent potential satisfying Bussinesqu equation," Phys. Lett.,50, A–P, 42–44 (1974).
Y. Bogomolov, I. Kol'chugina, A. Litvak, and A. Sergeev, "Near-sonic Langmur solitons," Phys. Lett.,91, A–P, 9–14 (1982).
V. K. Mel'nikov, "Some new nonlinear evolution equations integrable by the inverse probblem method," Mat. Sb.,121, No. 4, 469–498 (1983).
H. Bateman and A. Erdelyi, Higher Transcendental Functions, McGraw-Hill, New York (1955).
E. Date, M. Jimbo, Kashivara, and T. Miva, "Transformation groups for soliton equations. I," Proc. Jpn. Acad.,57A, 342–347 (1981), and III, J. Jpn.,50, 3806–3812 (1981).
Additional information
G. M. Krzhizhanovskii Moscow State Institute of Energetics. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 20, No. 3, pp. 42–54, July–September, 1986.
Rights and permissions
About this article
Cite this article
Krichever, I.M. Spectral theory of finite-zone nonstationary Schrödinger operators. A nonstationary Peierls model. Funct Anal Its Appl 20, 203–214 (1986). https://doi.org/10.1007/BF01078472
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01078472