The Snapshot Problem for Wave Equations on Homogeneous Trees

Abstract

By definition, a wave on a homogeneous tree $\mathfrak{X}$ is a solution to the discrete wave equation on $\mathfrak{X}$; that is, a family $\{f_k{\}}_{k\in \mathbb{Z}}$ of complex-valued functions on $\mathfrak{X}$ satisfying the partial difference equation $\hat{I}{¼}_1{f}_k=(f_{k+1}+f_{k-1})/2$ for all $k$, where $\hat{I}{¼}_1$ is the mean value operator on $\mathfrak{X}$ of radius $1$. The function $f_k$ is called the snapshot of the wave at time $k$. For $k\ge 2$, we will show that there exist infinitely many waves having given snapshots at times $0$ and $k$, but that all such waves have the same snapshots at times which are multiples of $k$. For integers $0<k<\ell$, we then consider necessary and sufficient conditions for the existence and uniqueness of a wave with given snapshots at times $0,k,\ell$.