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

TL;DR

This paper analyzes the points of constancy in solutions of the periodic linearized Korteweg--deVries equation at rational times, using number theory to understand fractalization phenomena at irrational times.

Contribution

It introduces a number theoretic approach to identify points of constancy in solutions of the linearized KdV equation at rational times, exploring fractalization effects.

Findings

Identifies points of constancy using Weyl sums and Kummer sums.

Provides initial insights into fractalization at irrational times.

Establishes a link between number theory and PDE solution structures.

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.