Quasideterminant solutions of NC Painlevé II equation with the Toda solution at n= 1 as a seed solution in its Darboux transformation

Irfan Mahmood (1, 2)

1 LAREMA - Laboratoire Angevin de Recherche en Mathématiques
2 University of the Punjab

In this paper, I construct the Darboux transformations for the non-commutative Toda solutions at n = 1 with the help of linear systems whose compatibility condition yields zero curvature representation of associated systems of non-linear differential equations. I also derive the quasideterminant solutions of the non-commutative Painlevé II equation by taking the Toda solutions at n = 1 as a seed solution in its Darboux transformations. Further by iteration, I generalize the Darboux transformations of the seed solutions to N-th form. At the end I describe the zero curvature representation of quantum Painlevé II equation that involves Planck constant h explicitly and system reduces to the classical Painlevé II when h → 0.

Mots clés: Integrable systems; Painlevé II equation; Toda equations; zero curvature condition; Riccati equation; Darboux transformation.; Darboux transformation

Domaines: Physique mathématique [math-ph]

Irfan Mahmood. Quasideterminant solutions of NC Painlevé II equation with the Toda solution at n= 1 as a seed solution in its Darboux transformation. 2014. ⟨hal-00983782v2⟩