Revivals in time-evolution quasi-periodic problems

Abstract

We examine the influence of quasi-periodic boundary conditions on the phenomenon of revivals in linear dispersive PDEs. We show that, in general, quasi-periodic problems do not support the revival effect at rational times. Our method is based on a correspondence between quasi-periodic and periodic problems. We prove that the solution to a quasi-periodic problem is expressed via the solution to a corresponding periodic problem, and vice-versa. Then, our main results follow by deriving a representation of the periodic problem solution in terms of a composition of solutions for a particular class of periodic problems, where the latter supports the classical revival and fractalisation dichotomy at rational and irrational times.