Differential Calculi on Associative Algebras and Integrable Systems

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 Bäcklund 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.

Obituary: Aristophanes Dimakis

“…In the framework of Noncommutative Geometry, we therefore started to explore the case of a linear dependence on such a parameter, which lead to a structure which we called "bidifferential calculus" and which can be thought of as a generalization of Frölicher-Nijenhuis theory from Differential to Noncommutative Geometry. It turned out that indeed many integrable models (though probably not all) can in fact be treated in this framework [47][48][49][50][51]56,57,59,82,86,87,91,93,97,101]. A crucial point was that integrability features could be formulated in an extremely general way, only using simple calculation rules of bidifferential calculus.…”

Двухкомпонентное Комплексное Модифицированное Уравнение Кортевега-Де Фриза: Новые Солитонные Решения, Получающиеся Из Нового Бинарного Преобразования Дарбу

“…В теории солитонов существуют различные эффективные методы исследования интегрируемых систем [16]- [21]. Одним из мощных инструментов построения солитонных решений является бинарное ПД, которое сохраняет инвариантными две пары спектральных задач и сопряженных спектральных задач, связанных с изучаемыми уравнениями [17], [22], [23].…”

Lax Pairs of Integrable Systems in Bidifferential Graded Algebras*

“…It is interesting that similar to the Cauchy matrix approach, [12−14] an infinite system of equations in bidifferential calculus with solutions is presented in Ref. [15], generated by the binary Darboux transformation. As a whole, much work in this framework has been carried out recently and satisfactory results are derived to investigate integrable systems.…”