I will report on joint works with M. Cafasso, C. Charlier, T. Claeys, in which we study a new class of Korteweg-de Vries solutions. They are built out of certain multiplicative expectations of the Airy point process and they generalize the self-similar solution associated with the Hastings-McLeod Painlevé II transcendent; in general they are associated with a specific solution, again characterized by an Airy limiting behavior, of an "integro-differential" generalization of the Painlevé II equation. The solutions are unbounded and the classical scattering-inverse scattering theory cannot be applied; however, they can be characterized through a Riemann-Hilbert problem, which allows to study rigorously and precisely their small time asymptotics, which we do uniformly in the space variable. A special case of the construction provides refined tail asymptotics for a specific solution ("narrow-wedge solution") of the Kardar-Parisi-Zhang stochastic equation. Depending on time I will comment on some more recent generalizations.


A recoding of the talk can be found here. Speaker(s): Giulio Ruzza (Université Catholique de Louvain)