Quasideterminant solutions of NC Painlev\'e II equation with the Toda solution at $ n=1 $ as a seed solution in its Darboux transformation

Abstract

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 Painlev\'e 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 Painlev\'e II equation that involves Planck constant $ \hbar $ explicitly and system reduces to the classical Painlev\'e II when $ \hbar \rightarrow 0 $.