Quasideterminant solutions of NC Painlevé 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 noncommuting elements ϕ and ψ of noncommutative Toda system at n = 1 with the help of zero curvature representation to the associated systems of non-linear differential equations. I also derive the quasideterminant solutions to the noncommutative 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 these solutions to the N th form.