Points of constancy of the periodic linearized Korteweg--deVries equation

Abstract

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.