Exponentially localized solutions of the Klein-Gordon equation

TL;DR

This paper presents new exponentially localized solutions to the multidimensional Klein-Gordon equation, describing wave packets with Gaussian decay and analyzing their properties using complex eikonal solutions.

Contribution

It introduces explicit localized solutions for the Klein-Gordon equation in multiple dimensions, constructed via exact complex eikonal solutions, and links the nonlinear equation to an ODE.

Findings

Solutions describe wave packets with Gaussian decay

Solutions depend on four free parameters

Reduction of nonlinear Klein-Gordon to an ODE

Abstract

Exponentially localized solutions of the Klein-Gordon equation for two and three space variables are presented. The solutions depend on four free parameters. For some relations between the parameters, the solutions describe wave packets filled with oscillations whose amplitudes decrease in the Gaussian way with distance from a point running with group velocity along a straight line. The solutions are constructed using exact complex solutions of the eikonal equation and may be regarded as ray solutions with amplitudes involving one term. It is also shown that the multidimensional nonlinear Klein-Gordon equation can be reduced to an ordinary differential equation with respect to the complex eikonal.