The Snapshot Problem for Wave Equations on Homogeneous Trees

TL;DR

This paper investigates the snapshot problem for wave equations on homogeneous trees, establishing conditions for the existence and uniqueness of waves with prescribed snapshots at specific times.

Contribution

It introduces the snapshot problem for wave equations on homogeneous trees and provides necessary and sufficient conditions for wave reconstruction from given snapshots.

Findings

Existence of infinitely many waves with given snapshots at times 0 and k.

All such waves share the same snapshots at multiples of k.

Conditions for existence and uniqueness of waves with snapshots at 0, k, and l.

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 $\mu_1 f_k=(f_{k+1}+f_{k-1})/2$ for all $k$, where $\mu_1$ 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 $k\geq 2$, 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 $0,\,k,\,\ell$.