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A non-Archimedean wave equation. (English) Zbl 1190.35235
Summary: Let K be a non-Archimedean local field with the normalized absolute value |·|. It is shown that a “plane wave” f(t+ω 1 ×1++ω n ×n), where f is a Bruhat-Schwartz complex-valued test function on K with (t,x 1 ,,x n )K n+1 and max 1jn |ω j |=1, satisfies, for any f, a certain homogeneous pseudodifferential equation, an analogue of the classical wave equation. A theory of the Cauchy problem for this equation is developed.

MSC:
35S10Initial value problems for pseudodifferential operators
35L99Hyperbolic equations and systems
46S10Functional analysis over fields (not , , or quaternions)
11S80Other analytic theory of local fields