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 h explicitly and system reduces to the classical Painlevé II when h → 0.
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Pré-publication, Document de travail
2014
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  • HAL Id : hal-00983782, version 2

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Irfan Mahmood. Quasideterminant solutions of NC Painlevé II equation with the Toda solution at n= 1 as a seed solution in its Darboux transformation. 2014. 〈hal-00983782v2〉

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