Coalescence, Deformation and B\"acklund Symmetries of Painlev\'e IV and II Equations

TL;DR

This paper extends Painlevé IV by adding quadratic terms, creating models that interpolate between Painlevé IV and II, and analyzes their symmetries, transformations, and Painlevé property.

Contribution

It introduces new interpolating models between Painlevé IV and II and explores their Bäcklund symmetries and Hamiltonian structures.

Findings

Derived Bäcklund transformations for the models

Identified symmetry structures and Painlevé property conditions

Established interpolation between Painlevé IV and II equations

Abstract

We extend Painlev\'e IV model by adding quadratic terms to its Hamiltonian obtaining two classes of models (coalescence and deformation) that interpolate between Painlev\'e IV and II equations for special limits of the underlying parameters. We derive the underlying B\"acklund transformations, symmetry structure and requirements to satisfy Painlev\'e property.