The snapshot problem for the wave equation

Abstract

By definition, a wave is a C∞ solution u(x,t) of the wave equation on Rn, and a snapshot of the wave u at time t is the function ut on Rn given by ut(x)=u(x,t). We show that there are infinitely many waves with given C∞ snapshots f0 and f1 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 wave equations on noncompact symmetric spaces. Finally, we consider the two-snapshot problem and corresponding small denominator results for the wave equation on the n-sphere.