Bi-differential calculus and the KdV equation

TL;DR

This paper introduces a gauged bi-differential calculus framework over associative algebras, applying it to the KdV equation to systematically derive conserved densities and currents.

Contribution

It develops a novel algebraic approach using bi-differential calculus to analyze integrable systems like the KdV equation, enabling the construction of conserved quantities.

Findings

Construction of generalized conserved currents

Explicit computation of conserved densities for KdV

Framework applicable to nonlinear differential equations

Abstract

A gauged bi-differential calculus over an associative (and not necessarily commutative) algebra A is an N-graded left A-module with two covariant derivatives acting on it which, as a consequence of certain (e.g., nonlinear differential) equations, are flat and anticommute. As a consequence, there is an iterative construction of generalized conserved currents. We associate a gauged bi-differential calculus with the Korteweg-de-Vries equation and use it to compute conserved densities of this equation.