Revivals, or the Talbot effect, for the Airy equation

Abstract

We study Dirichlet-type problems for the simplest third-order linear dispersive partial differential equations (PDE), often referred to as the Airy equation. Such problems have not been extensively studied, perhaps due to the complexity of the spectral structure of the spatial operator. Our specific interest is to determine whether the peculiar phenomenon of revivals, also known as Talbot effect, is supported by these boundary conditions, which for third-order problems are not reducible to periodic ones. We prove that this is the case only for a very special choice of the boundary conditions, for which a new type of weak cusp revival phenomenon has been recently discovered. We also give some new results on the functional class of the solution for other cases.