14 problems

Let $u$ be a multi-soliton solution of the generalized KdV equation, defined at a given time, together with its spatial derivatives. Pointwise exponential decay conjecture. For eac…

Scattering conjecture for critical gKdV. There exist functions

Let $u$ be a solution of the generalized Korteweg–de Vries initial-value problem referenced in the source, with nonlinearity exponent $k\in[1,2]$. Rammaha's conjecture. Such soluti…

Below-ground-state almost-periodic-solution conjecture. There does not exist a nonzero almost periodic solution $u$ to $(1.1)$ satisfying

Let $u(t)$ be a solution to the generalized Korteweg–de Vries equation in the mass-subcritical setting, let $\mathcal{S}_+$ denote the set of initial data whose solutions scatter f…

Arbitrary-speed conjecture. The assumption on the speeds $c_1,\ldots,c_N$ is not optimal, and the result should hold for any choice of speeds.

Nonexistence conjecture. For all $0<c_2<c_1$, the corresponding ingoing 2-solitons of the quartic gKdV equation are not pure 2-solitons.

Let $u_0\in L_2(\mathbb R)$ be smooth initial data with a single maximum, let $\epsilon\ll1$ be the dispersion parameter, and let $t_c$ be the break-up time of the corresponding ge…

Let $P$ be the solution of the ODE defining the proposed blow-up profile, and let $u(x,t)$ solve the generalized Korteweg–de Vries equation for $n=5$ with smooth initial data…

Let $Q$ be the soliton of the quartic generalized Korteweg–de Vries equation, and let $u(x,t)$ solve that equation for $n=4$ with smooth initial data $u_0\in L_2(\mathbb{R})$ of su…

Let $u(x,t)$) be a solution of a generalized Korteweg–de Vries equation with dispersion parameter $\epsilon$, and let $t_c$ denote the break-up point of the corresponding generaliz…

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 $u$ solve the mass-critical generalized KdV equation on ${\mathbb R}\times{\mathbb R}$, let $u_0\in L^2_x({\mathbb R})$ be its initial data, let $M(u_0)$ denote its mass, and l…

Let $(a,E,c)\in\Omega$ parameterize a periodic traveling wave of the generalized Korteweg–de Vries equation with a power-law nonlinearity. Write $T_E$ for the derivative of the per…