Geodesic
On a~$p$-adic Wave Equation
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Selected topics of mathematical physics and $p$-adic analysis, Tome 265 (2009), pp. 154-158

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It is shown that a "$p$-adic plane wave" $f(t+\omega_1x_1+\dots+\omega_nx_n)$, $(t,x_1,\dots,x_n)\in\mathbb Q_p^{n+1}$, where $f$ is a Bruhat–Schwartz complex-valued test function and $\max_{1\le j\le n}|\omega_j|_p=1$, satisfies, for any $f$, a certain homogeneous pseudodifferential equation, an analog of the classical wave equation. A theory of the Cauchy problem for this equation is developed. 
@article{TM_2009_265_a12,
     author = {A. N. Kochubei},
     title = {On a~$p$-adic {Wave} {Equation}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {154--158},
     publisher = {mathdoc},
     volume = {265},
     year = {2009},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TM_2009_265_a12/}
}
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A. N. Kochubei. On a~$p$-adic Wave Equation. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Selected topics of mathematical physics and $p$-adic analysis, Tome 265 (2009), pp. 154-158. http://geodesic.mathdoc.fr/item/TM_2009_265_a12/