A new approach to deriving Bäcklund transformations
Resumen
We give a new, surprisingly simple approach to the derivation of Bäcklund transformations. Motivated by the use of integrating factors to solve linear ordinary differential equations, for the nonlinear case this new technique leads to differential relations between equations. Although our interest here is in Painlevé equations, our approach is applicable to nonlinear equations more widely. As a completely new result we obtain a matrix version of a classical mapping between solutions of special cases of the second Painlevé equation. This involves the derivation of a new matrix second Painlevé equation, for which we also present a Lax pair. In addition, we give a matrix version of the Schwarzian second Painlevé equation, again a completely new result. In this way we also discover a new definition of matrix Schwarzian derivative.
Descripción
The author is grateful to the Ministerio de Ciencia e Innovación/Agencia Estatal de Investigación for financial support: Project PID2020-115273GB-I00 funded by MCIN/AEI/10.13039/501100011033; and Grant RED2022-134301-T funded by MCIN/AEI/10.13039/501100011033. He also thanks the Universidad Rey Juan Carlos for funding as a member of the Grupo de Investigación de alto rendimiento DELFO.
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