10 problems

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

Let $v_0$, $(x_0,t_0)$, $a_0=a(v_0)$, $a_0'=a'(v_0)$, $c_0=c(v_0)$, and $\kappa=-(a”'(v_0)t_0+b”'(v_0))$ be the quantities associated with a generic solution, where $kappa eq0$ and…

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…

Let $r_\pm(x,t;\epsilon)$ solve the relevant Hamiltonian perturbation of a hyperbolic system, with an $\epsilon$-independent Cauchy problem, no $\mathcal O(\epsilon)$ terms, and a…

Consider a generalized KdV equation near a generic critical point of the corresponding dispersionless equation, and let the PI2 equation denote the second member of the Painlevé I…

Let $(x_0,t_0)$ be a point of gradient catastrophe for the focusing nonlinear Schrödinger equation, and consider a nongeneric gradient catastrophe, meaning that more than one new m…

Let the system referred to as … for generic initial data do not develop a point of gradient catastrophe in finite time. The source presents this as a task for investigation in nume…