Speaker: Alexander Stokes

Title: On the zeroes of generalised Okamoto polynomials and singularity structure of real solutions of Painlevé-IV

Abstract: At special parameter values the fourth Painlevé equation admits rational special solutions, including a hierarchy of those expressed in terms of generalised Okamoto polynomials. Recently B. Yang and J. Yang showed that the number of real zeroes of these polynomials govern partial-rogue waves in the Sasa-Satsuma equation, but there has been no proof of the number of these or their distribution. We consider the Bäcklund transformations generating the hierarchy of rational solutions and use topological arguments based on their action on the Sakai surfaces forming the space of initial conditions to obtain detailed information about the sequence in which poles and zeroes of these rational solutions occur on the real line. As a corollary we obtain a closed formula for the number of real zeroes of the generalised Okamoto polynomials as well as various interlacing properties. Based on joint work with Pieter Roffelsen (The University of Sydney) .

This seminar is a part of the first TSVP Thematic Program "Exact Asymptotics: From Fluid Dynamics to Quantum Geometry" (https://groups.oist.jp/tsvp/exact-asymptotics). OIST students and researchers are welcome to participate in all scientific sessions of the program without registration.

List and profiles of program participants can be found here: https://groups.oist.jp/tsvp/23ea-program-participants