Lax pair representation and Darboux transformation of noncommutative Painlevé’s second equation

Abstract

Extension of the Painlevé equations to noncommutative spaces has been extensively investigated in the theory of integrable systems. An interesting topic is the exploration of some remarkable aspects of these equations, such as the Painlevé property, the Lax representation and the Darboux transformation, and their connection to well-known integrable equations. This paper addresses the Lax formulation, the Darboux transformation and a quasideterminant solution of the noncommutative form of Painlevé’s second equation introduced by Retakh and Rubtsov [V. Retakh, V. Rubtsov, Noncommutative Toda chain, Hankel quasideterminants and Painlevé II equation, J. Phys. A Math. 43 (2010) 505204].