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

Rogue wave formation scenarios for the focusing nonlinear Schr\"odinger equation with parabolic-profile initial data on a compact support