Bäcklund transformations of the higher order Painlevé equations (Q2781671)

This paper presents rational or classical solutions to the second Painlevé equation \((P_2): w''=2w^3+zw+\alpha\), and of its higher analogue. For \((P_2)\), the author discusses the Bäcklund transformations, and proves the following: NEWLINENEWLINENEWLINE(1) Equation \((P_2)\) admits a rational solution, if and only if \(\alpha\in\mathbb{Z}\), and all the rational solutions are derived from \(w\equiv 0,\) a solution to \((P_2)\) with \(\alpha=0,\) by successive application of the Bäcklund transformations; NEWLINENEWLINENEWLINE(2) Equation \((P_2)\) admits a solution satisfying a first-order algebraic differential equation simultaneously, if and only if \(\alpha-1/2\in\mathbb{Z};\) for each \(\alpha \in\mathbb{Z} +1/2,\) such solutions are expressible in terms of the Airy function, and constitute a one-parameter family. NEWLINENEWLINENEWLINEA higher-order analogue \(({}_{2n}P_2)\) of \((P_2)\) (resp. \(({}_{2n}P_1)\) of \((P_1):w''=6w^2+z\)) is explained in relation to the (KdV)-hierarchy. For the existence of a rational solution to \(({}_{2n}P_2)\), a necessary and sufficient condition is obtained; and, in particular for \(({}_4P_2),\) rational solutions are concretely computed by the use of Bäcklund transformations.NEWLINENEWLINEFor the entire collection see [Zbl 0974.00040].