On astigmatic solutions of the wave and the Klein-Gordon-Fock equations with exponential fall--off

TL;DR

This paper presents highly localized explicit solutions to multidimensional wave and Klein-Gordon-Fock equations that exhibit astigmatic properties and Gaussian localization, with explicit Fourier transforms and asymptotic analysis.

Contribution

It introduces new explicit, localized solutions with astigmatic properties for wave and Klein-Gordon-Fock equations, including their Fourier transforms and asymptotic behavior.

Findings

Solutions are highly localized and depend on multiple parameters.

Fourier transforms of solutions are explicitly derived.

Solutions exhibit Gaussian localization near a moving point.

Abstract

Highly localized explicit solutions to multidimensional wave and Klein--Gordon--Fock equations are presented. Their Fourier transform is also found explicitly. Solutions depend on a set of parameters, and demonstrate astigmatic properties. Asymptotic analysis for large and moderate time shows that constructed solutions have Gaussian localisation near a point moving with the group speed.