5 problems

Real-line Green's function conjecture. For every $c>0$ and every $\alpha\in(2,4]$, $G_{\mathbb{R}}$ is not strictly positive on $\mathbb{R}$ and is not monotonically decreasing on…

Periodic Green's function conjecture. For each $\alpha\in(2,4]$, there exists $c_0>0$ such that, for $c\in(0,c_0)$, $G_{\mathbb{T}}$ is even, strictly positive on $\mathbb{T}$, and…

Let $\psi_0 \in H^{\alpha}_{\rm per}(\mathbb{T})$ be the solution to the boundary-value problem with $c=c_0$ obtained from the main existence theorem, and let $X_0$ and…

Let $u_0\in L_2(\mathbb{R})$ be smooth initial data with a single negative hump. For $-1<\alpha<0$, solutions to the Whitham equation and to the fKdV equations with sufficiently sm…

Let $u0 in L2(\mathbb{R})$ be smooth initial data with a single hump. The fKdV equation is conjectured to have the following behavior. For $\alpha0.5$, solutions remain smooth for al…