The Snapshot Problem for Wave Equations on Homogeneous Trees
Fulton Gonzalez, Adelaide Nebeker, Katie Hallett, Andew Sailstad
TL;DR
The paper analyzes the discrete wave equation on a homogeneous tree, focusing on snapshot problems that fix wave values at a few time steps. It derives a fundamental iterated mean-value formula and shows every radius-$k$ mean is a polynomial in the basic mean $ μ_1$, yielding surjectivity and a flexible mechanism to construct waves from partial data. A closed-form wave solution via Chebyshev polynomials is established, enabling precise two- and three-snapshot analyses: two snapshots yield infinitely many waves with fixed endpoints, while three snapshots determine the wave uniquely when the two time steps are relatively prime; compatibility conditions govern existence in general. The work further connects mean-value operators to the Euler–Poisson–Darboux equation on trees, giving a degree-$k$ polynomial expression for $ μ_k$ in terms of $ μ_1$ and situating these results within the Dickson-Cohen-Picardello framework. Finally, it discusses the Two-Circle Pompeiu problem, establishing existence criteria that mirror the uniqueness conditions, thereby unifying several classical results in a discrete-tree setting.
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 $μ_1 f_k=(f_{k+1}+f_{k-1})/2$ for all $k$, where $μ_1$ 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 $0,\,k,\,\ell$.