Exponentially localized solutions of the Klein-Gordon equation

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.

💡 Research Summary

The paper presents a family of exponentially localized solutions to the Klein‑Gordon equation in two and three spatial dimensions. The authors start by introducing a complex eikonal (phase) function Φ(x,t) that satisfies the complex eikonal equation (∂tΦ)² – (∇Φ)² = 1. An explicit exact solution of this eikonal equation is constructed using four free parameters (α, β, γ, δ) and a constant velocity vector v. In compact form the phase reads

 Φ(x,t) =