  We investigate the points of constancy in the piecewise constant solution
profiles of the periodic linearized Korteweg--deVries equation with step
function initial data at rational times. The solution formulas are given by
certain Weyl sums, and we employ number theoretic techniques, including Kummer
sums, in our analysis. These results constitute an initial attempt to
understand the phenomenon of "fractalization" observed at irrational times.
