On a Novel Class of Integrable ODEs Related to the Painlevé Equations

Details On a Novel Class of Integrable ODEs Related to the Painlevé Equations On a Novel Class of Integrable ODEs Related to the Painlevé Equations

2010-09-26

arxiv.org/abs/1009.5125v1

Nonlinear Sciences

Exactly Solvable and Integrable Systems

This paper is dedicated to Professor T. Bountis on the occasion of his 60th birthday with appreciation of his important contri

Scientific paper

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.

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