A Note On Dimension Reduction and Finite Energy Localized Wave Solutions To the 3-d kLein-Gordon and Scalar Wave Equations. Part Ii. x wAve-Type - Abstract

Abstract

Extending previous work [1], a novel ansatz is formulated, whereby a class of X wave-type finite energy localized wave (LW) solutions ψ(p, z, t) to the axisymmetric 3-D Klein-Gordon equation is obtained by means of a dimension-reduction approach. Each of these solutions consists of a product of the zero-order X wave solution to the 3-D scalar wave equation and an analytic solution ψ_1D (Z, T) to the 1-D Klein-Gordon equation, with variables Z and T appropriately defined in terms of z, t and the polar radial coordinate p. In the absence of dispersion, the same formalism, but with a different definition of the coordinates Z and T, can be used to obtain X wave-type finite energy LW solutions to the 3-D scalar wave equation.