Coalescence, Deformation and Bäcklund Symmetries of Painlevé IV and II Equations

Abstract

We extend Painlevé IV model by adding quadratic terms to its Hamiltonian obtaining two classes of models (coalescence and deformation) that interpolate between Painlevé IV and II equations for special limits of the underlying parameters. We derive the underlying Bäcklund transformations, symmetry structure and requirements to satisfy Painlevé property. The structure of P IV model, definition and symmetriesThis section is devoted to a summary of relevant results on P IV equations, Bäcklund transformations and coalescence between P IV and P II available in the literature (e.g. [11,18]).