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We study J\'anossy densities of a randomly thinned Airy kernel determinantalpoint process. We prove that they can be expressed in terms of solutions to theStark and cylindrical Korteweg-de Vries equations; these solutions are Darbouxtranformations of the simpler ones related to the gap probability of the samethinned Airy point process. Moreover, we prove that the associated wavefunctions satisfy a variation of Amir-Corwin-Quastel's integro-differentialPainlev\'e II equation. Finally, we derive tail asymptotics for the relevantsolutions to the cylindrical Korteweg-de Vries equation and show that theydecompose asymptotically into a superposition of simpler solutions.

J\'anossy densities and Darboux transformations for the Stark and cylindrical KdV equations