5 problems

Gradient-catastrophe conjecture. Initial data violating the non-cavitation condition in one point in 1D lead to a gradient catastrophe in finite time.

Consider initial data that are the $x$-derivative of a rapidly decreasing smooth function in $L_{2}(\mathbb{R}^{2})$ with a single maximum. Write the two-dimensional dispersionless…

Consider generic rapidly decreasing initial data with a single hump for the focusing semiclassical DS II system, and let $x_c$ and $y_c$ denote the spatial coordinates of gradient-…

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 th…

Vortex-filament tritronquée conjecture. The generic behavior near the gradient-catastrophe point is