Convergence of linear solutions through convergence of periodic initial data

TL;DR

This paper proves that solutions to linear PDEs with subharmonic initial data converge to solutions with localized initial data as the subharmonic period tends to infinity, formalizing the intuition about stability and localization.

Contribution

It establishes a rigorous link between subharmonic and localized initial data convergence for linear PDE solutions, extending understanding of stability in periodic systems.

Findings

Subharmonic initial data converges to localized data as period increases.

Linear solutions with subharmonic perturbations converge to localized solutions.

Formal proof of the convergence intuition for linear PDEs.

Abstract

When studying the stability of $T$-periodic solutions to partial differential equations, it is common to encounter subharmonic perturbations, i.e. perturbations which have a period that is an integer multiple (say $n$) of the background wave, and localized perturbations, i.e. perturbations that are integrable on the line. Formally, we expect solutions subjected to subharmonic perturbations to converge to solutions subjected to localized perturbations as $n$ tends to infinity since larger $n$ values force the subharmonic perturbation to become more localized. In this paper, we study the convergence of solutions to linear initial value problems when subjected to subharmonic and localized perturbations. In particular, we prove the formal intuition outlined above; namely, we prove that if the subharmonic initial data converges to some localized initial datum, then the linear solutions converge.