Higher-order nonlinear special functions: Painlevé hierarchies, a survey

Abstract

The six Painlevé transcendents are widely accepted as nonlinear special functions. Over the last quarter of a century or so, there has been a surge of interest in higher-order analogues of the Painlevé equations, most often defined as members of hierarchies of equations of increasing order, i.e., of so-called Painlevé hierarchies. We give here a survey of such Painlevé hierarchies, including of their derivation and the derivation of their properties. Amongst other aspects, we discuss the relationships between the properties of completely integrable hierarchies, e.g., Hamiltonian structures and Miura maps, nonisospectral scattering problems, and those of Painlevé hierarchies, e.g., Lax pairs, B¨acklund and auto-B¨acklund transformations, and sequences of special solutions. Given the large number of papers published on Painlevé hierarchies, we hope this review will serve as a useful future reference.