Continuous-discrete integrable equations and Darboux transformations as deformations of associative algebras

TL;DR

This paper explores how deformations of associative algebra structure constants, governed by a Central System, relate to integrable equations like Boussinesq and WDVV, unifying Darboux transformations within this framework.

Contribution

It introduces a novel deformation scheme for associative algebras governed by a Central System, linking algebraic deformations to integrable equations and Darboux transformations.

Findings

Concrete deformations for 3D algebra linked to BSQ and WDVV equations.

Darboux transformations are fully incorporated into the deformation scheme.

The framework unifies algebraic deformations with integrable systems theory.

Abstract

Deformations of the structure constants for a class of associative noncommutative algebras generated by Deformation Driving Algebras (DDA's) are defined and studied. These deformations are governed by the Central System (CS). Such a CS is studied for the case of DDA being the algebra of shifts. Concrete examples of deformations for the three-dimensional algebra governed by discrete and mixed continuous-discrete Boussinesq (BSQ) and WDVV equations are presented. It is shown that the theory of the Darboux transformations, at least for the BSQ case, is completely incorporated into the proposed scheme of deformations.