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This article is cited in 20 scientific papers (total in 20 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 $\partial z_4(u_1,u_3)/\partial u_1=\partial x_2(u_1,u_3)/\partial u_3$.
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
Linking options:
https://www.mathnet.ru/eng/tm3729https://doi.org/10.1134/S0371968516030110 https://www.mathnet.ru/eng/tm/v294/p191
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