A non-Archimedean wave equation

Anatoly N. Kochubei

Vol. 235 (2008), No. 2, 245–261

DOI: 10.2140/pjm.2008.235.245

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) ∈ 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 2000

Primary: 11S80, 35S10

Secondary: 35L99

Milestones

Received: 18 July 2007

Revised: 5 November 2007

Accepted: 5 December 2007

Published: 1 April 2008