J\'anossy densities and Darboux transformations for the Stark and cylindrical KdV equations
Abstract
We study J\'anossy densities of a randomly thinned Airy kernel determinantal point process. We prove that they can be expressed in terms of solutions to the Stark and cylindrical Korteweg-de Vries equations; these solutions are Darboux tranformations of the simpler ones related to the gap probability of the same thinned Airy point process. Moreover, we prove that the associated wave functions satisfy a variation of Amir-Corwin-Quastel's integro-differential Painlev\'e II equation. Finally, we derive tail asymptotics for the relevant solutions to the cylindrical Korteweg-de Vries equation and show that they decompose asymptotically into a superposition of simpler solutions.
Keywords
Cite
@article{arxiv.2303.09848,
title = {J\'anossy densities and Darboux transformations for the Stark and cylindrical KdV equations},
author = {Tom Claeys and Gabriel Glesner and Giulio Ruzza and Sofia Tarricone},
journal= {arXiv preprint arXiv:2303.09848},
year = {2024}
}
Comments
V1: 41 pages, 3 figures. V2: 45 pages, 3 figures; revised introduction, minor corrections. V3: 46 pages, 3 figures; minor corrections