  We investigate the Cauchy problem for linear, constant-coefficient evolution
PDEs on the real line with discontinuous initial conditions (ICs) in the
small-time limit. The small-time behavior of the solution near discontinuities
is expressed in terms of universal, computable special functions. We show that
the leading-order behavior of the solution of dispersive PDEs near a
discontinuity of the ICs is characterized by Gibbs-type oscillations and gives
exactly the Wilbraham-Gibbs constant.
