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Uspekhi Mat. Nauk, 2011, Volume 66, Issue 1(397), Pages 111–150 (Mi umn9403)  

This article is cited in 5 scientific papers (total in 5 papers)

Singular spectral curves in finite-gap integration

I. A. Taimanov

Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences

Abstract: A number of examples of applications of the method of finite-gap integration of non-linear equations are considered in which singular spectral curves occur. In particular, constructions of orthogonal curvilinear coordinate systems for which a reducible spectral curve consists of rational components are discussed, along with constructions of finite-gap Frobenius manifolds, and a soliton deformation of spectral curves is demonstrated which consists in the creation and annihilation of singular points and corresponds to equations with self-consistent sources.
Bibliography: 52 titles.

Keywords: finite-gap integration, non-linear equations, Riemann surfaces, singular algebraic curves.

DOI: https://doi.org/10.4213/rm9403

Full text: PDF file (955 kB)
References: PDF file   HTML file

English version:
Russian Mathematical Surveys, 2011, 66:1, 107–144

Bibliographic databases:

UDC: 517.957+517.22+514.83
MSC: Primary 37K20, 37K25; Secondary 35Q53
Received: 01.12.2010

Citation: I. A. Taimanov, “Singular spectral curves in finite-gap integration”, Uspekhi Mat. Nauk, 66:1(397) (2011), 111–150; Russian Math. Surveys, 66:1 (2011), 107–144

Citation in format AMSBIB
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    This publication is cited in the following articles:
    1. E. I. Shamaev, “O reshetkakh Darbu–Egorova v ${\mathbb R}^n$”, Sib. elektron. matem. izv., 10 (2013), 113–122  mathnet
    2. O. A. Bogoyavlenskaya, “Ob odnom klasse konechnozonnykh krivolineinykh ortogonalnykh koordinat”, Sib. elektron. matem. izv., 12 (2015), 947–954  mathnet  crossref
    3. Burban I. Zheglov A., “Fourier-Mukai Transform on Weierstrass Cubics and Commuting Differential Operators”, Int. J. Math., 29:10 (2018), 1850064  crossref  mathscinet  zmath  isi  scopus
    4. Simonetta Abenda, Petr G. Grinevich, “Real soliton lattices of the Kadomtsev–Petviashvili II equation and desingularization of spectral curves: the $\mathrm {Gr^{ \scriptscriptstyle TP}}(2,4)$ case”, Proc. Steklov Inst. Math., 302 (2018), 1–15  mathnet  crossref  crossref  isi  elib
    5. Abenda S. Grinevich P.G., “Reducible M-Curves For Le-Networks in the Totally-Nonnegative Grassmannian and Kp-II Multiline Solitons”, Sel. Math.-New Ser., 25:3 (2019), UNSP 43  crossref  isi
  • Успехи математических наук Russian Mathematical Surveys
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