Polynomial dynamical systems and the Korteweg–de Vries equation

V. M. Buchstaber

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Modern problems of mathematics, mechanics, and mathematical physics. II, Tome 294 (2016), pp. 191-215

Résumé

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 $\partial z_4(u_1,u_3)/\partial u_1=\partial x_2(u_1,u_3)/\partial u_3$.

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