Break-up geometry conjecture for semiclassical DS II

Consider rapidly decreasing smooth initial data in $L_{2}(\mathbb{R}^{2})$ with a single maximum. Let the defocusing and focusing variants of the semiclassical DS II equation be the corresponding choices of sign, and let break-up points be finite-time gradient-catastrophe points. Semiclassical DS II break-up conjecture. For the defocusing variant, solutions show the same type of break-up as the corresponding semiclassical limit of the $1+1$-dimensional NLS equation: there are two break-up points in each spatial direction, generically of cubic type. For the focusing variant, solutions generally have two break-up points of the same type as focusing $1+1$-dimensional NLS, with a square-root cusp in each spatial direction; if the initial data are symmetric under interchange of the spatial coordinates, these cusps occur at the same time and location. These statements summarize the predicted local break-up geometry in the paper's numerical study; no proof or resolution is supplied.