Ablowitz–Ramani–Segur (ARS) conjecture on Painlevé property for ODE reductions of integrable PDEs

Prove that any ordinary differential equation obtained as a reduction of an integrable partial differential equation possesses the Painlevé property, possibly after a transformation of variables.

Background

In the discussion of Lie symmetries and integrability, the paper recalls a classical conjecture attributed to Ablowitz, Ramani, and Segur (ARS) that connects reductions of integrable partial differential equations to the Painlevé property for ordinary differential equations. This conjecture is widely cited in the integrable systems literature and remains a guiding principle for assessing the integrability of reduced ODEs.

The authors note that the generalized seventh-order KdV equation considered in the paper passes the Painlevé test only in three specific parameter regimes. By invoking the ARS conjecture, they infer that the corresponding reduced ODEs (obtained from the integrable PDE cases) possess the Painlevé property, highlighting the broader unresolved status of the conjecture.