Beyond periodic revivals for linear dispersive PDEs

We study the phenomenon of revivals for the linear Schrodinger and Airy equations over a finite interval, by considering several types of non-periodic boundary conditions. In contrast to the case of the linear Schrodinger 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 Schrodinger 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.

Automated summary

The study investigates the phenomenon of revivals for the linear Schrodinger and Airy equations over a finite interval with various non-periodic boundary conditions. It is found that the Airy equation does not generally exhibit revivals even for boundary conditions very close to periodic, in contrast to the linear Schrodinger equation. A new, weaker form of revival phenomena is also described in the case of certain Robin-type boundary conditions for the linear Schrodinger equation.