On a $p$-adic Wave Equation

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.