Beyond periodic revivals for linear dispersive PDEs

We study the phenomenon of revivals for the linear Schrödinger and Airy equations over a finite interval, by considering several types of non-periodic boundary conditions. In contrast to the case of the linear Schrödinger equation examined recently (which we develop further), we prove that, remarkably, the Airy equation does not generally exhibit revivals even for boundary conditions very close to periodic. We also describe a new, weaker form of revival phenomena, present in the case of certain Robin-type boundary conditions for the linear Schrödinger equation. In this weak revival, the dichotomy between the behaviour of the solution at rational and irrational times persists, but in contrast to the classical periodic case, the solution is not given by a finite superposition of copies of the initial condition.