7 problems

Painlevé I hierarchy asymptotics conjecture. A special solution to the second equation in the Painlevé I hierarchy provides an asymptotic description of the break-up of Camassa–Hol…

The Amick-Schonbek system is the dispersive system obtained by setting $\epsilon=1$ in … Its zero-dispersion limit is the corresponding dispersionless Saint-Venant system, and the…

Let $u(x,t)$ denote the real pole-free solution of the second member of the Painlevé I hierarchy, … with $t\in\mathbb{R}$. Consider solutions of generalized Kadomtsev–Petviashvili…

A dispersive shock wave (DSW) is a region of rapid modulated oscillations that forms near the shock of the solution of a dispersionless equation under dispersive regularization. Th…

Let the radial nonlinear Schrödinger equation generate a dispersive shock wave (DSW) from an initial density jump of size $\rho_b$, and let the cavitation threshold for the one-dim…

Let $\epsilon$ be the small dispersion parameter, and let $a$, $b$, and $c_0$ be constants. Let $(x_0,y_0)$ be the position where a singularity of the Whitham system is expected to…

The Benjamin–Ono equation is … In the small-dispersion analysis of a wider class of integrable dispersive equations that includes the Benjamin–Ono equation, the function $U(X,T)$ a…