Revivals in time evolution problems

Abstract

Subject to periodic boundary conditions, it is known that the solution to a certain family of linear dispersive partial differential equations, such as the free linear Schr¨odinger and Airy evolution, exhibits a dichotomy at rational and irrational times. At rational times, the solution is decomposed into a finite number of translated copies of the initial condition. Consequently, when the initial function has a jump discontinuity, then the solution also exhibits finitely many jump discontinuities. On the other hand, at irrational times the solution becomes a continuous, but nowhere differentiable function. These two effects form the revival and fractalisation phenomenon at rational and irrational times, respectively. The main aim of the thesis is to further investigate the phenomenon of revivals in time evolution problems posed under appropriate boundary conditions on a finite interval. We consider both first-order and second-order in time problems. For the former, we examine the influence of non-periodic boundary conditions on the revival effect. For the latter, we study the revivals under periodic and non-periodic boundary conditions. In terms of first-order in time evolution problems, we show that the revival phenomenon persists in the free linear Schr¨odinger equation under pseudo-periodic and Robin-type boundary conditions. Moreover, we prove that under quasi-periodic boundary conditions, the Airy equation does not in general exhibit revivals. With respect to second-order in time equations, we first formulate an abstract setting for the revival phenomenon, which we then apply to establish that the periodic, even-order poly-harmonic wave equation exhibits revivals. Finally, following the lack of revivals in Airy’s quasi-periodic problem, we characterise quasi-periodic and periodic problems, either of first-order or second-order in time, for which the revival effect breaks. In general, our approach relies on identifying the canonical periodic components of the generalised Fourier series representations of solutions, in order to utilise the classical periodic theory of revivals.