Beyond periodic revivals for linear dispersive PDEs

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Abstract

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.

Original languageEnglish
Article number20210241
JournalProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Volume477
Issue number2251
Early online date7 Jul 2021
DOIs
Publication statusPublished - 28 Jul 2021

Keywords

  • Airy equation
  • Talbot effect
  • boundary value problems
  • linear Schrödinger equation
  • revivals

ASJC Scopus subject areas

  • Mathematics(all)
  • Engineering(all)
  • Physics and Astronomy(all)

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