Seminarium "Nieliniowość i Geometria"

While solutions of the fourth Painlevé equation are in general transcendental, at special parameter values it admits rational solutions, including a hierarchy of those expressed in terms of generalised Okamoto polynomials.Recently, B. Yang and J. Yang derived a family of rational solutions to the Sasa-Satsuma equation, and showed that any of its members constitutes a "partial-rogue wave" provided that an associated generalised Okamoto polynomial has no real roots or no imaginary roots.Roots of these polynomials correspond to singularities of the rational solutions, and we develop an algorithmic procedure to derive the qualitative distribution of singularities on the real line for real solutions of Painlevé equations using the Okamoto-Sakai theory of algebraic surfaces associated with Painlevé equations.We apply this procedure to the rational solutions expressed in terms of generalised Okamoto polynomials and as a corollary obtain exact formulas for the number of real and the number of imaginary roots.Based on joint work with Pieter Roffelsen (The University of Sydney).