Abstract

Let K be a non-Archimedean local field with the normalized absolute value | • |. It is shown that a “plane wave” f(t + ω₁x₁ + ⋯ + ω_nx_n), where f is a Bruhat–Schwartz complex-valued test function on K with (t,x₁,…,x_n) in Kⁿ⁺¹ and max_1≤j≤n|ω_j| = 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.

Keywords

local field, plane wave, pseudodifferential equation, Cauchy problem

Mathematical Subject Classification

Primary: 11S80, 35S10

Secondary: 35L99

Authors