A Non-Archimedean Wave Equation

TL;DR

This paper introduces a non-Archimedean analog of the classical wave equation, demonstrating that certain plane waves satisfy a pseudo-differential equation and developing a corresponding Cauchy problem theory.

Contribution

It formulates a non-Archimedean wave equation and establishes a theory for solving the associated Cauchy problem, extending classical wave analysis to non-Archimedean fields.

Findings

Plane waves satisfy a non-Archimedean wave equation.

A pseudo-differential equation analogous to the classical wave equation is derived.

A Cauchy problem theory for this non-Archimedean wave equation is developed.

Abstract

Let K be a non-Archimedean local field with the normalized absolute value $|\cdot |$. It is shown that a ``plane wave'' $f(t+\omega_1 x_1+... +\omega_nx_n)$, where f is a Bruhat-Schwartz complex-valued test function on K, $(t,x_1,..., x_n)\in K^{n+1}$, $\max\limits_{1\le j\le n}|\omega_j|=1$, 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.