TY - JOUR

T1 - Beyond periodic revivals for linear dispersive PDEs

AU - Boulton, Lyonell

AU - Farmakis, George

AU - Pelloni, Beatrice

N1 - Funding Information:
Data accessibility. This article has no additional data. Authors’ contributions. L.B. and B.P. conceived the ideas of considering pseudo-periodic and Robin boundary conditions to study revivals in the Airy and Schrodinger equations. G.F. conceived the idea of decomposing revivals for the different boundary conditions into simple periodic components. All the authors participated in the writing of the manuscript at different stages, revised the manuscript, gave final approval for publication and agree to be held accountable for the work performed therein. Competing interests. We declare we have no competing interests. Funding. B.P. and L.B. are also grateful for the invitation to Yale-NUS College in January 2020 for a workshop funded by grant IG18-CW003, in which discussions leading to part of this work began. G.F. is being supported by The Maxwell Institute Graduate School in Analysis and its Applications, a Centre for Doctoral Training funded by EPSRC (grant EP/L016508/01), the Scottish Funding Council, Heriot-Watt University and the University of Edinburgh. Acknowledgements. We thank David Smith for his useful comments and suggestions on the contents of this paper.
Publisher Copyright:
© 2021 The Author(s).

PY - 2021/7/28

Y1 - 2021/7/28

N2 - 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.

AB - 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.

KW - Airy equation

KW - Talbot effect

KW - boundary value problems

KW - linear Schrödinger equation

KW - revivals

UR - http://www.scopus.com/inward/record.url?scp=85113431137&partnerID=8YFLogxK

U2 - 10.1098/rspa.2021.0241

DO - 10.1098/rspa.2021.0241

M3 - Article

SN - 1364-5021

VL - 477

JO - Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

JF - Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

IS - 2251

M1 - 20210241

ER -