Beyond periodic revivals for linear dispersive PDEs

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.

The Time-fractional Airy Equation on the Metric Graph

Initial boundary value problem for the time-fractional Airy equation on a graph with finite bonds is considered in the paper. Properties of potentials for this equation are studied. Using these properties the solutions of the considered problem were found. The uniqueness theorem is proved using the analogue of Gr¨onwall-Bellman inequality and a-priory estimate

Almost-Periodic Response Solutions for a Forced Quasi-Linear Airy Equation

We prove the existence of almost-periodic solutions for quasi-linear perturbations of the Airy equation. This is the first result about the existence of this type of solutions for a quasi-linear PDE. The solutions turn out to be analytic in time and space. To prove our result we use a Craig–Wayne approach combined with a KAM reducibility scheme and pseudo-differential calculus on $$\mathbb {T}^\infty $$ T ∞ .