8 problems

Stability conjecture. Let $1\leq d\leq 6$. For $\alpha\leq 2/d$, the solution $\varphi_\omega$ minimizes $\mathcal{E}$ at mass $M=M(\omega)$ for all $\omega\in(0,\omega^*)$ and is…

Asymptotic stability conjecture. The line solitary waves of the Amick-Schonbek system are asymptotically stable.

Let $m\in\mathbb{N}$ and $\tilde{c}>0$. For sufficiently small $\varepsilon>0$, consider the FPU system and the speed $c=1-\varepsilon^2\tilde{c}$. A generalized moving pulse solut…

Let $u_0\in L^2(\mathbb{R})$ be smooth initial data with a single hump for the fractional Benjamin–Bona–Mahony equation, with parameter $0<\alpha$. Fractional Benjamin–Bona–Mahony…

Let $u_0\in L^2(\mathbb{R})$ be smooth initial data with a single hump for the focusing modified fractional Korteweg–de Vries equation, with dispersion parameter $\alpha$. Let $Q$…

Fix $q<0$ and $p>4$, and consider the linearized operator $\mathcal{L}(\omega)$ for $\omega\in(\tfrac12q^2,\infty)$. Let $csigma_c(\mathcal{L}(\omega))$ denote its continuous spect…

Fix $q<0$ and $p>4$. Let $\sigma=-1$ denote the focusing case and $\sigma=1$ the defocusing case, and let $\Omega=\Omega(q,p)$ be the minimal unstable frequency. The spectral condi…

Let $d\ge 1$, and consider the focusing mass-critical nonlinear Schrödinger equation. Let $Q$ denote its ground state, and suppose that a global solution with initial data…