Polynomial dynamical systems and Korteweg--de Vries equation

Abstract

In this work 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 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 coordinates of trajectories of the fields ${\mathcal{L}}_1$ and ${\mathcal{L}}_3$. The function $2x_2(u_1,u_3)$ is a 2-zone solution of the KdV hierarchy and $\frac{\partial }{\partial u_1}{z}_4(u_1,u_3)=\frac{\partial }{\partial u_3}{x}_2(u_1,u_3)$.

Cited by in corpus (2)

• Differentiation of Genus 3 Hyperelliptic Functions
• Some reductions of rank 2 and genera 2 and 3 Hitchin systems