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arXiv · 1402.3540

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

Irfan Mahmood
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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 $.

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Irfan Mahmood. 2014-02-14. Quasideterminant solutions of NC Painlev\'e II equation with the Toda solution at $ n=1 $ as a seed solution in its Darboux transformation. https://doi.org/10.1016/j.geomphys.2015.05.004

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