  We study the presence of a non-trivial revival effect in the solution of
linear dispersive boundary value problems for two benchmark models which arise
in applications: the Airy equation and the dislocated Laplacian Schr{\"o}dinger
equation. In both cases, we consider boundary conditions of Dirichlet-type. We
prove that, at suitable times, jump discontinuities in the initial profile are
revived in the solution not only as jump discontinuities but also as
logarithmic cusp singularities. We explicitly describe these singularities and
show that their formation is due to interactions between the symmetries of the
underlying spatial operators with the periodic Hilbert transform.
