On the heat kernel and the Korteweg-de Vries hierarchy

TL;DR

This paper derives explicit formulas for Hadamard's coefficients using the tau-function of the KdV hierarchy, revealing their symmetry and relation to the hierarchy's equations through Sato theory and Gegenbauer polynomials.

Contribution

It provides a novel explicit formula connecting Hadamard's coefficients with the tau-function of the KdV hierarchy, simplifying their analysis.

Findings

Hadamard's coefficients are symmetric about the diagonal.

On the diagonal, these coefficients determine the KdV hierarchy equations.

Explicit formulas are derived using Sato theory and Gegenbauer polynomials.

Abstract

We give explicit formulas for Hadamard's coefficients in terms of the tau-function of the KdV hierarchy. We show that some of the basic properties of these coefficients can be easily derived from these formulas. The first immediate corollary is the symmetry of Hadamard's coefficients about the diagonal. Another well known fact, which follows from this approach, is that on the diagonal Hadamard's coefficients determine the right-hand sides of the equations of the KdV hierarchy. The proof of the main result uses Sato theory and simple properties of Gegenbauer polynomials.