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 Uspekhi Mat. Nauk, 2017, Volume 72, Issue 6(438), Pages 113–138 (Mi umn9806)

Singular solitons and spectral meromorphy

P. G. Grinevicha, S. P. Novikovb

a Landau Institute for Theoretical Physics of Russian Academy of Sciences, Moscow
b Steklov Mathematical Institute of Russian Academy of Sciences, Moscow

Abstract: Along with regular solutions of soliton equations one usually can also construct interesting classes of singular solutions. The conditions for the compatibility of their singularities with the dynamics prescribed by the equation impose stringent restrictions on the form of the singular points. For instance, the known meromorphic solutions of the Korteweg-de Vries equation have second-order poles with respect to the space variable, and the leading coefficient is always a triangular number. Singular finite-gap solutions are an important example of this type of solution. In the case of one space dimension the eigenfunctions of the auxiliary linear operators with pole singularities that are compatible with the dynamics turn out to be also locally meromorphic for all values of the spectral parameter. This property, which will be called spectral meromorphy, makes it possible to define a natural indefinite metric on the space spanned by the eigenfunctions, and the number of negative squares of this metric is a new integral of motion. Also discussed are analogues of these results for two-dimensional problems.
Bibliography: 50 titles.

Keywords: singular solitons, indefinite metrics, finite-gap potentials, finite-gap property on a single energy level, Moutard transformations.

 Funding Agency Grant Number Russian Foundation for Basic Research 17-01-00366 Russian Science Foundation 14-50-00005 The new result presented in this survey is contained in § 3. A programme for establishing this result was proposed by S. P. Novikov in an investigation supported by the Russian Science Foundation under grant no. 14-50-00005 at the Steklov Mathematical Institute of Russian Academy of Sciences. The idea of its proof was proposed by P. G. Grinevich in an investigation supported by the Russian Foundation for Basic Research (grant no. 17-01-00366).

DOI: https://doi.org/10.4213/rm9806

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English version:
Russian Mathematical Surveys, 2017, 72:6, 1083–1107

Bibliographic databases:

UDC: 517.984.4+517.927.25+517.984.52
MSC: Primary 34L40, 34M05; Secondary 47B50

Citation: P. G. Grinevich, S. P. Novikov, “Singular solitons and spectral meromorphy”, Uspekhi Mat. Nauk, 72:6(438) (2017), 113–138; Russian Math. Surveys, 72:6 (2017), 1083–1107

Citation in format AMSBIB
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