Four-dimensional conformal transformations due originally to Bateman have been used in the past by Hillion as alternative approaches to focus wave mode solutions to the 3D scalar wave equation. More recently, more extended families of focus wave mode solutions to the 3D scalar wave equation have been derived by Borisov and Utkin, as well as Kiselev, based on Bateman transformations together with a dimension-reduction approach, whereby the wave function is separated incompletely into a product of two functions. One particular goal in this exposition is to comment on and extend the work of Borisov and Utkin and simplify and extend the method used by Kiselev. More generally, however, the aim is to show that an already existing method, known as the bidirectional spectral representation, when examined in conjunction with Bateman conformal transformations, encompasses the Borisov-Utkin-Kiselev theories as special cases and allows a systematic derivation of extended families of FWM-type localized waves beyond the ranges of their applicability.
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