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Trudy Mat. Inst. Steklova, 2016, Volume 294, Pages 191–215 (Mi tm3729)  

This article is cited in 15 scientific papers (total in 15 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.


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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, Trudy Mat. Inst. Steklova, 294, MAIK Nauka/Interperiodica, Moscow, 2016, 191–215; Proc. Steklov Inst. Math., 294 (2016), 176–200

Citation in format AMSBIB
\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 Trudy Mat. Inst. Steklova
\yr 2016
\vol 294
\pages 191--215
\publ MAIK Nauka/Interperiodica
\publaddr Moscow
\jour Proc. Steklov Inst. Math.
\yr 2016
\vol 294
\pages 176--200

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    This publication is cited in the following articles:
    1. 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  mathnet  crossref  crossref  mathscinet  isi  elib
    2. O. K. Sheinman, “Almost graded current algebras on the symmetric square of a curve”, Russian Math. Surveys, 72:2 (2017), 384–386  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    3. 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  mathnet  crossref  crossref  mathscinet  isi  elib
    4. V. M. Buchstaber, “Polynomial Lie algebras and the Zelmanov–Shalev theorem”, Russian Math. Surveys, 72:6 (2017), 1168–1170  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    5. V. V. Zharinov, “Hamiltonian operators in differential algebras”, Theoret. and Math. Phys., 193:3 (2017), 1725–1736  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    6. O. K. Sheinman, “Certain reductions of Hitchin systems of rank 2 and genera 2 and 3”, Dokl. Math., 97:2 (2018), 144–146  mathnet  crossref  crossref  zmath  isi  elib  scopus
    7. E. Yu. Bunkova, “Differentiation of genus 3 hyperelliptic functions”, Eur. J. Math., 4:1 (2018), 93–112  crossref  mathscinet  zmath  isi  scopus
    8. D. V. Millionshchikov, “Polynomial Lie algebras and growth of their finitely generated Lie subalgebras”, Proc. Steklov Inst. Math., 302 (2018), 298–314  mathnet  crossref  crossref  mathscinet  isi  elib
    9. T. Ayano, V. M. Buchstaber, “Ultraelliptic integrals and two-dimensional sigma-functions”, Funct. Anal. Appl., 53:3 (2019), 157–173  mathnet  crossref  crossref  mathscinet  isi  elib
    10. Takanori Ayano, Victor M. Buchstaber, “Construction of Two Parametric Deformation of KdV-Hierarchy and Solution in Terms of Meromorphic Functions on the Sigma Divisor of a Hyperelliptic Curve of Genus 3”, SIGMA, 15 (2019), 032, 15 pp.  mathnet  crossref  mathscinet
    11. E. Yu. Bunkova, “On the problem of differentiation of hyperelliptic functions”, Eur. J. Math., 5:3, SI (2019), 712–719  crossref  mathscinet  zmath  isi
    12. V. M. Buchstaber, E. Yu. Bunkova, “Lie Algebras of Heat Operators in a Nonholonomic Frame”, Math. Notes, 108:1 (2020), 15–28  mathnet  crossref  crossref  mathscinet  isi  elib
    13. V. M. Buchstaber, E. Yu. Bunkova, “Sigma Functions and Lie Algebras of Schrödinger Operators”, Funct. Anal. Appl., 54:4 (2020), 229–240  mathnet  crossref  crossref  mathscinet  isi  elib
    14. T. Ayano, V. M. Buchstaber, “Analytical and number-theoretical properties of the two-dimensional sigma function”, Chebyshevskii sb., 21:1 (2020), 9–50  mathnet  crossref  mathscinet
    15. V. M. Buchstaber, E. Yu. Bunkova, “Hyperelliptic Sigma Functions and Adler–Moser Polynomials”, Funct. Anal. Appl., 55:3 (2021), 179–197  mathnet  crossref  crossref
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