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

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.

💡 Research Summary

The paper introduces a novel construction called “conjugate Hamiltonian systems” and applies it to the classical Painlevé equations, thereby generating new integrable ordinary differential equations (ODEs). The authors start by recalling that each Painlevé equation can be written in Hamiltonian form H(p,q,t), where t appears explicitly. By solving the relation h = H(p,q,t) for t, they obtain a new function t = T(p,q,h). This defines a new Hamiltonian system with Hamiltonian T and independent variable h, which they call the conjugate system.

Using this framework, the authors focus on Painlevé II (PII). The standard Hamiltonian for PII is
H_II = ½ p² – (q² + ½ t) p – (α + ½) q.
Setting h = H_II and solving for t yields T(p,q,h). The resulting Hamiltonian equations in the variable h are
\dot q = ∂T/∂p, \dot p = –∂T/∂q.
Eliminating p leads to a second‑order ODE for q(h) that the authors term the “conjugate Painlevé II” equation. This new equation retains the nonlinear character of PII but contains explicit dependence on the new independent variable h.

A major part of the work is the demonstration that the conjugate equation remains integrable. The authors construct a Lax pair (L(λ), M(λ)) adapted to the conjugate variables and the new independent variable h. They verify the compatibility condition ∂_h L – ∂_λ M +