Differential Calculi on Associative Algebras and Integrable Systems

TL;DR

This paper explores the use of bidifferential calculus on associative algebras to derive transformations like Bäcklund and Darboux, and applies these methods to integrable systems, including those with sources.

Contribution

It introduces a framework for symmetries of zero curvature equations and extends Darboux transformations, including a deformation approach for integrable equations with sources.

Findings

Derived Bäcklund and Darboux transformations from symmetry considerations.

Presented a matrix version of the binary Darboux transformation.

Developed a deformation of the matrix binary Darboux transformation for integrable equations with sources.

Abstract

After an introduction to some aspects of bidifferential calculus on associative algebras, we focus on the notion of a "symmetry" of a generalized zero curvature equation and derive Backlund and (forward, backward and binary) Darboux transformations from it. We also recall a matrix version of the binary Darboux transformation and, inspired by the so-called Cauchy matrix approach, present an infinite system of equations solved by it. Finally, we sketch recent work on a deformation of the matrix binary Darboux transformation in bidifferential calculus, leading to a treatment of integrable equations with sources.