The Snapshot Problem for the Wave equation

TL;DR

This paper investigates the uniqueness and existence of wave solutions based on their snapshots at different times, revealing infinite solutions under certain conditions and connecting the problem to small denominators and Liouville numbers.

Contribution

It introduces new conditions for the uniqueness and existence of wave solutions given snapshots, and extends the analysis to shifted wave equations on symmetric spaces and spheres.

Findings

Infinitely many waves share the same snapshots at times 0 and 1.

Necessary and compatibility conditions for wave snapshots at multiple times.

Connection between the snapshot problem and small denominator issues.

Abstract

By definition, a wave is a $C^\infty$ solution $u(x,t)$ of the wave equation on $\mathbb R^n$, and a snapshot of the wave $u$ at time $t$ is the function $u_t$ on $\mathbb R^n$ given by $u_t(x)=u(x,t)$. We show that there are infinitely many waves with given $C^\infty$ snapshots $f_0$ and $f_1$ at times $t=0$ and $t=1$ respectively, but that all such waves have the same snapshots at integer times. We present a necessary condition for the uniqueness, and a compatibility condition for the existence, of a wave $u$ to have three given snapshots at three different times, and we show how this compatibility condition leads to the problem of small denominators and Liouville numbers. We extend our results to shifted wave equations on noncompact symmetric spaces. Finally, we consider the two-snapshot problem and corresponding small denominator results for the shifted wave equation on the $n$-sphere.