On a Novel Class of Integrable ODEs Related to the Painlev\'e Equations

TL;DR

This paper introduces a new class of integrable ordinary differential equations derived from Painlevé equations using conjugate Hamiltonian systems, providing Lax pairs and implicit solutions.

Contribution

It develops conjugate Painlevé II equations and constructs associated equations for Painlevé I and IV, expanding the integrable systems framework.

Findings

Lax pair formulation for conjugate Painlevé II equations

Implicit solutions for the new integrable ODEs

Construction of conjugate equations for Painlevé I and IV

Abstract

One of the authors has recently introduced the concept of conjugate Hamiltonian systems: the solution of the equation $h=H(p,q,t),$ where $H$ is a given Hamiltonian containing $t$ explicitly, yields the function $t=T(p,q,h)$, which defines a new Hamiltonian system with Hamiltonian $T$ and independent variable $h.$ By employing this construction and by using the fact that the classical Painlev\'e equations are Hamiltonian systems, it is straightforward to associate with each Painlev\'e equation two new integrable ODEs. Here, we investigate the conjugate Painlev\'e II equations. In particular, for these novel integrable ODEs, we present a Lax pair formulation, as well as a class of implicit solutions. We also construct conjugate equations associated with Painlev\'e I and Painlev\'e IV equations.