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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2512.19136 |
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| _version_ | 1866914214119997440 |
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| author | Gonzalez, Fulton Nebeker, Adelaide Hallett, Katie Sailstad, Andew |
| author_facet | Gonzalez, Fulton Nebeker, Adelaide Hallett, Katie Sailstad, Andew |
| contents | 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 $0,\,k,\,\ell$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_19136 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Snapshot Problem for Wave Equations on Homogeneous Trees Gonzalez, Fulton Nebeker, Adelaide Hallett, Katie Sailstad, Andew Combinatorics Primary: 43A85, Secondary: 39A14 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 $0,\,k,\,\ell$. |
| title | The Snapshot Problem for Wave Equations on Homogeneous Trees |
| topic | Combinatorics Primary: 43A85, Secondary: 39A14 |
| url | https://arxiv.org/abs/2512.19136 |