Rogue wave formation scenarios for the focusing nonlinear Schr\'odinger equation with parabolic-profile initial data on a compact support
| Autor: | Demontis, F., Ortenzi, G., Roberti, G., Sommacal, M. |
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| Rok vydání: | 2023 |
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| Druh dokumentu: | Working Paper |
| Popis: | 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. Comment: 17 pages, 7 figures, 1 table, typos corrected, references added, figures improved |
| Databáze: | arXiv |
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