On ringing effects near jump discontinuities for periodic solutions to dispersive partial differential equations

TL;DR

This paper investigates the ringing effects near jump discontinuities in solutions to dispersive PDEs with periodic boundary conditions, revealing that solutions exhibit overshoot phenomena near rational times, regardless of the sequence of times approaching these values.

Contribution

It demonstrates that solutions to dispersive PDEs show a universal ringing effect near rational times, extending understanding of solution behavior at discontinuities.

Findings

Solutions are continuous at irrational times and piecewise constant at rational times.

Ringing effects with fixed amplitude occur near jump discontinuities as time approaches rational values.

The ringing phenomenon is consistent whether approaching rational times via rational or irrational sequences.

Abstract

We consider weak solutions to dispersive partial differential equations with periodic boundary conditions and initial data with jump discontinuities. These are already known to be continuous at irrational times and piecewise constant at rational times; we show that as time approaches a rational value the solution exhibits a ringing effect, with the characteristic overshoot of fixed amplitude near the discontinuities. Furthermore this effect is the same whether the sequence of times follows rational or irrational values.