Ordinary differential equations associated with the heat equation

TL;DR

This paper explores the connection between the one-dimensional heat equation and associated non-linear ordinary differential equations, revealing how solutions relate through polynomial dynamical systems and group actions, including classical special functions.

Contribution

It introduces an algorithm linking solutions of polynomial dynamical systems to the heat equation and describes the group action of SL(2, C) on these solutions, including the Chazy-3 equation.

Findings

The fundamental solution corresponds to n=0 case.

Solutions involving elliptic theta-functions relate to Chazy-3.

SL(2, C) acts on solution spaces, connecting different equations.

Abstract

This paper is devoted to the one-dimensional heat equation and the non-linear ordinary differential equations associated to it. We consider homogeneous polynomial dynamical systems in the n-dimensional space, n = 0, 1, 2, .... For any such system our construction matches a non-linear ordinary differential equation. We describe the algorithm that brings the solution of such an equation to a solution of the heat equation. The classical fundamental solution of the heat equation corresponds to the case n=0 in terms of our construction. Solutions of the heat equation defined by the elliptic theta-function lead to the Chazy-3 equation and correspond to the case n=2. The group SL(2, C) acts on the space of solutions of the heat equation. We show this action for each n induces the action of SL(2, C) on the space of solutions of the corresponding ordinary differential equations. In the case n=2 this leads to the well-known action of this group on the space of solutions of the Chazy-3 equation. An explicit description of the family of ordinary differential equations arising in our approach is given.