A non-Archimedean wave equation

Let K be a non-Archimedean local field with the normalized absolute value | � | . It is shown that a “plane wave” f(t + !1x1 + � � � + !nxn), where f is a Bruhat-Schwartz complex-valued test function on K, (t, x1, . . . , xn) ∈ K n+1 , max 1≤j≤n |!j| = 1, satisfies, for any f, a certain homogeneous pseudo-differential equation, an analog of the classical wave equation. A theory of the Cauchy problem for this equation is developed.