Novel luminal spatiotemporally localized waves

Summary form only given. Most of the well-known luminal spatiotemporally localized waves, such as the focus wave mode, the splash mode and the modified power spectrum pulse, are based on the Courant-Hilbert ansatz, whereby a solution to the scalar wave equation is given as a product of an appropriate “attenuation” factor and a function obeying a nonlinear characteristic (eikonal) equation. However, the derivation of more general finite-energy localized waves, e.g., the Bessel-Gauss focus wave mode, requires a superposition of more elementary Courant-Hilbert-type ones. The bidirectional spectral representation has been developed in order to address this problem. But, even within its framework, one has to choose among a very large number of spectra those that can lead to interesting solutions. The purpose in this talk is to use two old techniques in order to derive more directly novel classes of luminal spatiotemporally localized waves.In 1909 and 1910, Bateman and, independently, Cunningham discovered a class of transformations, more general than conformal changes of the metric, which could be used to transform solutions of the scalar wave equation and, more generally, of Maxwell's equations into similar ones. This technique will be illustrated by the derivation of a novel luminal finite-energy spatiotemporally localized wave, the Airy splash mode, which cannot easily be motivated by any other known method. Comparison will be also made with the simple splash mode based on the Courant-Hilbert ansatz. The second old method is due to Whittaker and Bateman, The derivation of electromagnetic fields in this case is based on two scalar potential functions obeying the nonlinear characteristic equation. Emphasis in this presentation is placed on appropriate choices of Whittaker-Bateman scalar potentials which, with an additional constraint, give rise to null spatiotemporally localized electromagnetic fields carrying angular momentum and characterized by vortical structures. The local energy velocity for these fields evolves in space-time without any deformation. Such an invariant structure, known as a Robinson congruence, results in a linked and knotted topology of the associated electromagnetic fields.