The role of periodicity in the solution of third order boundary value problems

TL;DR

This paper investigates how periodicity influences solutions of third order boundary value problems, especially in linear dispersive equations, highlighting the impact of boundary conditions on solution behavior and phenomena like revivals.

Contribution

It demonstrates that solutions to certain boundary value problems can be viewed as perturbations of purely periodic solutions, clarifying boundary condition effects.

Findings

Solutions can be expressed as perturbations of periodic solutions

Boundary conditions significantly affect solution properties

Insights into revivals and fractalization phenomena

Abstract

In this short paper, we elucidate how the solution of certain illustrative boundary value problems for the Airy equation $u_t+u_{xxx}=0$ on $[0,1]$ can be expressed as a perturbation of the solution of the purely periodic problem. The motivation is to understand the role boundary conditions play in the properties of the solution. This is particularly important in related work on the solution of linear dispersive problems with discontinuous initial data and the phenomena of revivals and fractalization.