  Motivated by the recent discovery of a dispersive-to-nondispersive transition
for linear waves in shear flows, we accurately explored the wavenumber-Reynolds
number parameter map of the plane Poiseuille flow, in the limit of least-damped
waves. We have discovered the existence of regions of the map where the
dispersion and propagation features vary significantly from their surroundings.
These regions are nested in the dispersive, low-wavenumber part of the map.
This complex dispersion scenario demonstrates the existence of linear
dispersive focusing in wave envelopes evolving out of an initial, spatially
localized, three-dimensional perturbation. An asymptotic wave packet's
representation, based on the saddle-point method, allows to enlighten the
nature of the packet's morphology, in particular the arrow-shaped structure and
spatial spreading rates. A correlation is also highlighted between the regions
of largest dispersive focusing and the regions which are most subject to strong
nonlinear coupling in observations.
