Dubrovin has conjectured that the local behavior near a breaking point can generically be modeled using solutions of the Painlevé-I equation for a broad class of Hamiltonian PDEs. This conjecture has been proven for the one-dimensional focusing nonlinear Schrodinger equation by Bertola and Tovbis.

In 2017, Suleimanov formally showed that self-focused solutions of the nonlinear geometric optics equations are generically described in the semiclassical limit by a function satisfying (for each fixed time) the second member of Sakka's Painlevé-III hierarchy. This same function has since been discovered in several different contexts, including rogue waves, multiple-pole solitons, and semiclassical soliton ensembles, indicating it is a new type of universal, albeit nongeneric, focusing. We will discuss the universal nature of Painlevé-III focusing and its relation to Painlevé-I focusing with an emphasis on multiple-pole solitons of the focusing nonlinear Schrodinger equation.

This is joint work with Deniz Bilman, Bob Jenkins, and Peter Miller.