A Non-Archimedean Wave Equation

Abstract

Let K be a non-Archimedean local field with the normalized absolute value $|\cdot |$. It is shown that a ``plane wave'' $f(t+{\omega }_1{x}_1+...+{\omega }_n{x}_n)$, where f is a Bruhat-Schwartz complex-valued test function on K, $(t,x_1,...,x_n)\in K^{n+1}$, $\underset{1\le j\le n}{\max }|{\omega }_j|=1$, 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.

17 pages; the final version, to appear in Pacif. J. Math