11 problems

Weak asymptotic decomposition conjecture. The solution admits a decomposition

Let $\rho>0$ be a mass, let the initial datum have mass $\rho$ and momentum $0$, and let $g_t$ be the associated solution in rescaled variables. Let $G_\rho$ be the steady state of…

For a nonlinear dispersive equation, let $u(t)$ denote a solution at time $t$. A soliton is a localized coherent wave, and a radiation term is the dispersive remainder of the solut…

Let the Sawada–Kotera (SK) equation and modified Sawada–Kotera (mSK) equation be considered in their long-time asymptotic regimes, and let $x=0$ denote the central spatial region.…

Let the Sawada–Kotera (SK) equation be considered in its long-time asymptotic decomposition into a dispersive wave region and a Painlevé region. A reflection coefficient $r(k)$ des…

Let $q(x,t)$ be the solution considered in Theorem 1, and let $\xi=x/(4t)$. The asymptotic sectors are separated by the rays $\xi=\pm\operatorname{Re}p_m$, $m=\overline{1,n}$, by t…

Let ${\bar\rho}_{k,1}$ be the stationary profile and let ${\mathcal J}_\infty$ denote the limiting value of the second-moment-related quantity in the two-case analysis. Admissibili…

Consider solutions of the coagulation equation with integrable initial data, and let the transition region denote the spatial region connecting the parts of the asymptotic $N$-wave…

The Novikov equation is formulated through a Riemann–Hilbert problem whose jump matrix has the same form and $k$-dependence as the jump matrix for the Degasperis–Procesi equation.…

Let $(G(z,t))_{z\in \mathbb{H},t\geqslant 0}$ be the solution of the evolution equation governing the continuum Stieltjes transform. Here $z$ lies in the upper half-plane and $t$ d…

The control takes values $\a$ and $A$, where $a$ and $A$ are the minimal and maximal admissible control rates, respectively. The constant control $\alpha^*$ is the optimal constant…