Nonhomogeneous three-dimensional structures described by the nonlinear Klein–Gordon equation

Abstract

We present exact solutions to the nonlinear Klein–Gordon equation in (3 + 1) dimensions. Vortex rings, spirals, and other solutions are demonstrated. On the basis of our solutions we discuss various nonhomogeneous patterns, which can be formed at the critical point of systems described by the Ginzburg–Landau model.