Lax pair representation and Darboux transformation of NC Painlev\'e-II equation

TL;DR

This paper explores the noncommutative Painlevé-II equation by developing its Lax pair, Darboux transformation, and quasideterminant solutions, advancing the understanding of integrable systems in noncommutative spaces.

Contribution

It introduces the Lax formulation, Darboux transformation, and quasideterminant solutions for the recently defined noncommutative Painlevé-II equation.

Findings

Established Lax pair for NC Painlevé-II

Constructed Darboux transformation for the equation

Derived quasideterminant solutions

Abstract

The extension of Painlev\'e equations to noncommutative spaces has been considering extensively in the theory of integrable systems and it is also interesting to explore some remarkable aspects of these equations such as Painlev\'e property, Lax representation, Darboux transformation and their connection to well know integrable equations. This paper is devoted to the Lax formulation, Darboux transformation and Quasideterminant solution of noncommutative Painlev\'e second equation which is recently introduced by V. Retakh and V. Rubtsov.