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

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.