Jumps, cusps and fractals in time-evolution models

The phenomenon of revivals in linear dispersive equations was discovered first experimentally in optics, in around 1834, then rediscovered several times by theoretical and experimental investigations. While the concept has been used systematically and consistently by many authors, there is no consensus on a rigorous definition.
Several have described it by stating that a given periodic time-dependent boundaryvalue problem exhibits revivals at rational times, if the solution evaluated at a certain dense subset of times, is given by finite superpositions of translated copies of the initial condition. When this initial condition has jump discontinuities at time zero, these discontinuities are propagated and remain present in the solution at each rational time.
The complementary phenomenon to revivals is fractalisation. One instance of this is that, for initial conditions with jump discontinuities, the solution is continuous in space for every irrational time, but its graph is a curve of high fractal dimension.
In this talk I will report on the presence of revivals and fractalisation, defined in the context of spectral theory, for three models of parabolic differential equations.

• Nonlocal equations that arise in water wave theory and are defined by convolution kernels [1] and [5].
• Schrödinger equations with different types of boundary conditions [2] or with complex potentials [3].
• Indefinite Laplacian time-evolution equations [4].