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 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 $\hslash$ explicitly and system reduces to the classical Painlevé II when $\hslash \to 0$.

17 pages. arXiv admin note: substantial text overlap with arXiv:1201.0900

References in corpus (3)

• On a direct approach to quasideterminant solutions of a noncommutative modified KP equation
• With a Cole-Hopf transformation to solutions of the noncommutative KP hierarchy in terms of Wronski matrices
• Quantum Painlevé systems of type $A_l^{(1)}$

Cited by in corpus (1)

• Quantitative analysis and simulation of Noncommutative Langmuir Oscillator solutions