Russian Mathematical Surveys
RUS  ENG    JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB  
General information
Latest issue
Archive
Impact factor
Submit a manuscript

Search papers
Search references

RSS
Latest issue
Current issues
Archive issues
What is RSS



Uspekhi Mat. Nauk:
Year:
Volume:
Issue:
Page:
Find






Personal entry:
Login:
Password:
Save password
Enter
Forgotten password?
Register


Russian Mathematical Surveys, 2017, Volume 72, Issue 6, Pages 1083–1107
DOI: https://doi.org/10.1070/RM9806
(Mi rm9806)
 

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
References:
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).
Received: 27.10.2017
Russian version:
Uspekhi Matematicheskikh Nauk, 2017, Volume 72, Issue 6(438), Pages 113–138
DOI: https://doi.org/10.4213/rm9806
Bibliographic databases:
Document Type: Article
UDC: 517.984.4+517.927.25+517.984.52
MSC: Primary 34L40, 34M05; Secondary 47B50
Language: English
Original paper language: Russian
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
\Bibitem{GriNov17}
\by P.~G.~Grinevich, S.~P.~Novikov
\paper Singular solitons and spectral meromorphy
\jour Uspekhi Mat. Nauk
\yr 2017
\vol 72
\issue 6(438)
\pages 113--138
\mathnet{http://mi.mathnet.ru/rm9806}
\crossref{https://doi.org/10.4213/rm9806}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=3748690}
\zmath{https://zbmath.org/?q=an:1407.34123}
\adsnasa{https://adsabs.harvard.edu/cgi-bin/bib_query?2017RuMaS..72.1083G}
\elib{https://elibrary.ru/item.asp?id=30737982}
\transl
\jour Russian Math. Surveys
\yr 2017
\vol 72
\issue 6
\pages 1083--1107
\crossref{https://doi.org/10.1070/RM9806}
\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=000429465700002}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85045681926}
Linking options:
  • https://www.mathnet.ru/eng/rm9806
  • https://doi.org/10.1070/RM9806
  • https://www.mathnet.ru/eng/rm/v72/i6/p113
  • Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Успехи математических наук Russian Mathematical Surveys
    Statistics & downloads:
    Abstract page:848
    Russian version PDF:133
    English version PDF:39
    References:83
    First page:65
     
      Contact us:
     Terms of Use  Registration to the website  Logotypes © Steklov Mathematical Institute RAS, 2024