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Uspekhi Mat. Nauk, 1976, Volume 31, Issue 1(187), Pages 55–136 (Mi umn3642)  

This article is cited in 373 scientific papers (total in 373 papers)

Non-linear equations of Korteweg–de Vries type, finite-zone linear operators, and Abelian varieties

B. A. Dubrovin, V. B. Matveev, S. P. Novikov

Abstract: The basic content of this survey is an exposition of a recently developed method of constructing a broad class of periodic and almost-periodic solutions of non-linear equations of mathematical physics to which (in the rapidly decreasing case) the method of the inverse scattering problem is applicable. These solutions are such that the spectrum of their associated linear differential operators has a finite-zone structure. The set of linear operators with a given finite-zone spectrum is the Jacobian variety of a Riemann surface, which is determined by the structure of the spectrum. We give an explicit solution of the corresponding non-linear equations in the language of the theory of Abelian functions.

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English version:
Russian Mathematical Surveys, 1976, 31:1, 59–146

Bibliographic databases:

UDC: 517.9+517.4
MSC: 35Q53, 47A10, 35B10, 35B15, 14Kxx, 35Q55
Received: 02.06.1975

Citation: B. A. Dubrovin, V. B. Matveev, S. P. Novikov, “Non-linear equations of Korteweg–de Vries type, finite-zone linear operators, and Abelian varieties”, Uspekhi Mat. Nauk, 31:1(187) (1976), 55–136; Russian Math. Surveys, 31:1 (1976), 59–146

Citation in format AMSBIB
\by B.~A.~Dubrovin, V.~B.~Matveev, S.~P.~Novikov
\paper Non-linear equations of Korteweg--de Vries type, finite-zone linear
operators, and Abelian varieties
\jour Uspekhi Mat. Nauk
\yr 1976
\vol 31
\issue 1(187)
\pages 55--136
\jour Russian Math. Surveys
\yr 1976
\vol 31
\issue 1
\pages 59--146

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