Painlevé equations and deformations of rational surfaces with rational double points. (Q2751981)

The authors study the Painlevé equations. This family of 6 equations are part of the Painlevé classification of all second-order differential equations \(y''=R(t,y,y')\) analytic in \(t\), polynomial in \(y\) and algebraic in \(y'\) whose general solution has no movable critical point. The Painlevé equations play a fundamental role in integrability theory as they admit symmetries, Bäcklund transformations, a Hamiltonian description and many other fundamental properties. The authors further explore these properties by performing a geometric analysis of the second Painlevé equation \(P_{II}:\;y''=2y^{3+t y+\alpha}\) based on the projective surface constructed by \textit{K. Okamoto} [Jap. J. Math., New Ser. 5, 1--79 (1979; Zbl 0426.58017)]. They show explicitly that the Bäcklund transformations arise from the rational double points and give an explicit form of the Hamiltonian function.NEWLINENEWLINEFor the entire collection see [Zbl 0964.00054].