|
Tr. Mat. Inst. Steklova, 2016, Volume 294, Pages 191–215
(Mi tm3729)
|
|
|
|
This article is cited in 8 scientific papers (total in 8 papers)
Polynomial dynamical systems and the Korteweg–de Vries equation
V. M. Buchstaber Steklov Mathematical Institute of Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991 Russia
Abstract:
We explicitly construct polynomial vector fields $\mathcal L_k$, $k=0,1,2,3,4,6$, on the complex linear space $\mathbb C^6$ with coordinates $X=(x_2,x_3,x_4)$ and $Z=(z_4,z_5,z_6)$. The fields $\mathcal L_k$ are linearly independent outside their discriminant variety $\Delta\subset\mathbb C^6$ and are tangent to this variety. We describe a polynomial Lie algebra of the fields $\mathcal L_k$ and the structure of the polynomial ring $\mathbb C[X,Z]$ as a graded module with two generators $x_2$ and $z_4$ over this algebra. The fields $\mathcal L_1$ and $\mathcal L_3$ commute. Any polynomial $P(X,Z)\in\mathbb C[X,Z]$ determines a hyperelliptic function $P(X,Z)(u_1,u_3)$ of genus $2$, where $u_1$ and $u_3$ are the coordinates of trajectories of the fields $\mathcal L_1$ and $\mathcal L_3$. The function $2x_2(u_1,u_3)$ is a two-zone solution of the Korteweg–de Vries hierarchy, and $\partial z_4(u_1,u_3)/\partial u_1=\partial x_2(u_1,u_3)/\partial u_3$.
Funding Agency |
Grant Number |
Russian Science Foundation  |
14-50-00005 |
This work is supported by the Russian Science Foundation under grant 14-50-00005. |
DOI:
https://doi.org/10.1134/S0371968516030110
Full text:
PDF file (314 kB)
References:
PDF file
HTML file
English version:
Proceedings of the Steklov Institute of Mathematics, 2016, 294, 176–200
Bibliographic databases:
UDC:
515.178.2+517.958 Received: May 11, 2016
Citation:
V. M. Buchstaber, “Polynomial dynamical systems and the Korteweg–de Vries equation”, Modern problems of mathematics, mechanics, and mathematical physics. II, Collected papers, Tr. Mat. Inst. Steklova, 294, MAIK Nauka/Interperiodica, Moscow, 2016, 191–215; Proc. Steklov Inst. Math., 294 (2016), 176–200
Citation in format AMSBIB
\Bibitem{Buc16}
\by V.~M.~Buchstaber
\paper Polynomial dynamical systems and the Korteweg--de Vries equation
\inbook Modern problems of mathematics, mechanics, and mathematical physics.~II
\bookinfo Collected papers
\serial Tr. Mat. Inst. Steklova
\yr 2016
\vol 294
\pages 191--215
\publ MAIK Nauka/Interperiodica
\publaddr Moscow
\mathnet{http://mi.mathnet.ru/tm3729}
\crossref{https://doi.org/10.1134/S0371968516030110}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=3628500}
\elib{http://elibrary.ru/item.asp?id=26601058}
\transl
\jour Proc. Steklov Inst. Math.
\yr 2016
\vol 294
\pages 176--200
\crossref{https://doi.org/10.1134/S0081543816060110}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000386554900011}
\elib{http://elibrary.ru/item.asp?id=27421251}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84992135203}
Linking options:
http://mi.mathnet.ru/eng/tm3729https://doi.org/10.1134/S0371968516030110 http://mi.mathnet.ru/eng/tm/v294/p191
Citing articles on Google Scholar:
Russian citations,
English citations
Related articles on Google Scholar:
Russian articles,
English articles
Related presentations:
This publication is cited in the following articles:
-
V. M. Buchstaber, A. V. Mikhailov, “Infinite-Dimensional Lie Algebras Determined by the Space of Symmetric Squares of Hyperelliptic Curves”, Funct. Anal. Appl., 51:1 (2017), 2–21
-
O. K. Sheinman, “Almost graded current algebras on the symmetric square of a curve”, Russian Math. Surveys, 72:2 (2017), 384–386
-
T. Ayano, V. M. Buchstaber, “The field of meromorphic functions on a sigma divisor of a hyperelliptic curve of genus 3 and applications”, Funct. Anal. Appl., 51:3 (2017), 162–176
-
V. M. Buchstaber, “Polynomial Lie algebras and the Zelmanov–Shalev theorem”, Russian Math. Surveys, 72:6 (2017), 1168–1170
-
V. V. Zharinov, “Hamiltonian operators in differential algebras”, Theoret. and Math. Phys., 193:3 (2017), 1725–1736
-
O. K. Sheinman, “Certain reductions of Hitchin systems of rank 2 and genera 2 and 3”, Dokl. Math., 97:2 (2018), 144–146
-
E. Yu. Bunkova, “Differentiation of genus 3 hyperelliptic functions”, Eur. J. Math., 4:1 (2018), 93–112
-
D. V. Millionshchikov, “Polynomial Lie algebras and growth of their finitely generated Lie subalgebras”, Proc. Steklov Inst. Math., 302 (2018), 298–314
|
Number of views: |
This page: | 297 | Full text: | 13 | References: | 34 | First page: | 14 |
|