Beyond periodic revivals for linear dispersive PDEs

TL;DR

This paper investigates revival phenomena in linear dispersive PDEs, revealing that the Airy equation generally lacks revivals and introducing a weaker form of revival under specific boundary conditions for the Schr"odinger equation.

Contribution

It extends understanding of revivals beyond periodic cases, showing the absence of revivals in the Airy equation and identifying a new weak revival phenomenon for certain boundary conditions.

Findings

Airy equation does not exhibit revivals under typical boundary conditions.

A weaker revival phenomenon exists for Schr"odinger with Robin boundary conditions.

Solution behavior at rational and irrational times differs, but not via finite superpositions.

Abstract

We study the phenomenon of revivals for the linear Schr\"odinger and Airy equations over a finite interval, by considering several types of non-periodic boundary conditions. In contrast with the case of the linear Schr\"odinger 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 Schr\"odinger equation. In this weak revival, the dichotomy between the behaviour of the solution at rational and irrational times persists, but in contrast with the classical periodic case, the solution is not given by a finite superposition of copies of the initial condition.