  We study the (1+1) focussing nonlinear Schr\"{o}dinger equation for an
initial condition with compactly-supported parabolic profile and phase
depending quadratically on the spatial coordinate. In the absence of
dispersion, using the natural class of self-similar solutions, we provide a
criterion for blow-up in finite time, generalising a result by Talanov et al.
In the presence of dispersion, we numerically show that the same criterion
determines, even beyond the semi-classical regime, whether the solution relaxes
or develops a high-order rogue wave, whose onset time is predicted by the
corresponding dispersionless catastrophe time. The sign of the chirp appears to
determine the prevailing scenario among two competing mechanisms for rogue wave
formation. For negative values, the numerical simulations are suggestive of the
dispersive regularisation of a gradient catastrophe described by Bertola and
Tovbis for a different class of smooth, bell-shaped initial data. As the chirp
becomes positive, the rogue wave seems to result from the interaction of
counter-propagating dispersive dam break flows, as in the box problem recently
studied by El, Khamis and Tovbis. As the chirp and amplitude of the initial
profile are relatively easy to manipulate in optical devices and water tank
wave generators, we expect our observation to be relevant for experiments in
nonlinear optics and fluid dynamics.
