Cherenkov Radiation versus X-shaped Localized Waves

The Localized Waves (LW) are nondiffracting ("soliton-like") solutions to the wave equations, and are known to exist with subluminal, luminal and superluminal peak-velocities V. For mathematical and experimental reasons, the ones that called more attention were the "X-shaped" superluminal waves. Such waves are associated with a cone, so that one may be tempted --let us confine ourselves to electromagnetism-- to look for links between them and the Cherenkov radiation[1]. However, the X-shaped waves belong to a very different realm: For instance, they can be shown to exist, independently of any media, even in the vacuum, as localized non-diffracting pulses propagating rigidly with a peak-velocity[2] V>c. In this paper we dissect the whole question on the basis of a rigorous formalism, and of clear physical considerations. [In particular we show, by explicit calculations based on Maxwell equations only, that, at variance with what was previously assumed by some authors: (i) the "X-waves" exist in all space, and in particular inside both the front and the rear part of their double cone (which has nothing to do with Cherenkov's); (ii) they are to be found not heuristically, but by use of strict mathematical (or experimental) procedures, without any ad hoc assumptions; (iii) the ideal X-waves, as well as the plane waves, are endowed with infinite energy, but finite-energy X-waves may be easily constructed (even without recourse to space-time truncations): And at the end of this article, by following a new technique, we construct finite-energy exact solutions, totally free from backward-traveling waves; (iv) the X-waves' most interesting property lies in the circumstance that they are LWs, endowed with a characteristic self-reconstruction property, which promises important practical applications (in part already realized, starting with 1992), quite independently of the superluminality ---or not--- of their peak-velocity; (v) insistence in attempting a comparison of Cherenkov radiation with X-waves would lead one to an unconventional sphere: that of considering the rather different situation of the (X-shaped, too) field generated by a superluminal point-charge, a non-orthodox question actually exploited in previous papers[3]: We show here explicitly that in such a case the point-charge would not lose energy in the vacuum, and that its field would not need to be continuously fed by incoming side-waves (as it is the case, on the contrary, for an ordinary X-wave).]