# Darboux solutions of non-abelian quantum Painlev\'e II equation in terms of quasideterminants

## TL;DR

This paper derives Darboux solutions for a non-abelian quantum Painlevé II equation using quasideterminants, extending previous work to a noncommutative setting with new commutation relations.

## Contribution

It introduces a non-abelian quantum Painlevé II equation with new commutation relations and derives its Darboux solutions in quasideterminant form, extending existing frameworks.

## Key findings

- Derived non-abelian quantum Painlevé II equation with new commutation relations.
- Obtained quasideterminant solutions using Darboux transformations.
- Generalized solutions to the N-th form.

## Abstract

In this article non-abelian version of quantum Painlev\'e II equation is presented with Its quasideterminant solutions has been derived by using the Darboux transformations. This non-abelian quantum Painlev\'e II equation may be considered as a specific case of its purely noncommutatie analogue presented by V. Retakh and V. Rubtsov . In these computations the quantum Painlev\'e II symmetric form with commutation relations presented by H. Nagoya are applied to derive Nonabelian quantum Painlev\'e II equation and a new commutation relation between variable $z$ and the solution $ f(z)$ such as $ z f - f z = \frac{1}{2} i \hbar f $ is presented. Finally, the Darboux solutions of that system are generalized to the $N$-th form in terms of quasideterminants.