  The formation of extreme localizations in nonlinear dispersive media can be
explained and described within the framework of nonlinear evolution equations,
such as the nonlinear Schr\"odinger equation (NLS). Within the class of exact
NLS breather solutions on finite background, which describe the modulational
instability of monochromatic wave trains, the hierarchy of both in time and
space localized rational solutions are considered to be appropriate prototypes
to model rogue wave dynamics. Here, we use the time-reversal invariance of the
NLS to propose and experimentally demonstrate a new approach to construct
strongly nonlinear localized waves focused both in time and space. The
potential areas of applications of this time-reversal approach range from
remote sensing to motivated analogous experimental analysis in other nonlinear
dispersive media, such as optics, Bose-Einstein condensates and plasma, where
the wave motion dynamics is governed by the NLS.
