Cherenkov radiation versus X-shaped localized waves

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, those that have attracted more attention are 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 [Phys. Rev. Lett.99, 244802 (2007)] for links between them and the Cherenkov radiation. 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 vacuum, as localized non-diffracting pulses propagating rigidly with a peak-velocity V>c [Hernández et al., eds., Localized Waves (Wiley, 2008)]. We dissect the whole question on the basis of a rigorous formalism and clear physical considerations.