A new explicit expression for the Korteweg-De Vries hierarchy

TL;DR

This paper presents a new explicit formula for the Korteweg-De Vries hierarchy's right-hand sides, utilizing heat kernel relations to improve understanding and computation of higher-order operators.

Contribution

The paper introduces a novel explicit expression for the KdV hierarchy's right-hand sides, linking it to heat kernel methods and providing formulas for powers of Schrödinger-like operators.

Findings

Derived an explicit formula for the KdV hierarchy

Connected heat kernel methods to hierarchy computation

Provided formulas for powers of differential operators

Abstract

We derive an improved fully explicit expression for the right-hand sides of the matrix KdV hierarchy using the relation to the heat kernel of the one-dimensional Schr\"odinger operator. Our method of "matrix elements" produces, moreover, an explicit expression for the powers of a Schr\"odinger-like differential operator of any order.