Polynomial dynamical systems and Korteweg--de Vries equation

TL;DR

This paper constructs polynomial vector fields related to hyperelliptic functions and demonstrates their connection to solutions of the Korteweg--de Vries (KdV) hierarchy, revealing algebraic structures underlying integrable systems.

Contribution

It explicitly constructs polynomial vector fields on complex space, describes their algebraic relations, and links these to hyperelliptic functions and KdV solutions, advancing understanding of integrable systems.

Findings

Polynomial vector fields form a Lie algebra with specific properties.

Hyperelliptic functions of genus 2 are linked to solutions of the KdV hierarchy.

The constructed functions satisfy particular differential relations.

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 $2 x_2(u_1, u_3)$ is a 2-zone solution of the KdV hierarchy and $\frac{\partial}{\partial u_1}z_4(u_1, u_3)=\frac{\partial}{\partial u_3}x_2(u_1, u_3)$.