10 problems

Let the spectral problem be the linearized spectral problem denoted by (third), with parameter $\omega\in(-1,1)$. Its isolated eigenvalues and end-point resonances are considered a…

Let $\alpha$ be sufficiently close to zero, let $\mathfrak{M}_\alpha$ be a blow-up of the compactification of Minkowski space, and let $W$ denote the soliton profile. A function is…

Let $\sigma=-1$. Consider the stationary nonlinear Schrödinger equation … where $H$ has lowest eigenvalue $e_0<0$, and let $S$ denote the set of solitons, while $M$ is the mass,…

Global uniform smoothness conjecture. Identifying $\mathbf S_U^N$ and $\mathbf M_U^N$, the soliton addition and removal maps are uniformly smooth globally,

Uniform foliation conjecture. The fibers $\mathbf M_{\mathbf z}$ provide a uniformly smooth foliation of $\mathbf M_U^N$.

Uniform smoothness conjecture. The maps $B^N_+$ and $B^N_-$ are uniformly smooth on bounded sets in $H^s$.

Consider the discrete nonlinear Schrödinger evolution and its soliton family $e^{-i(-\mu t+\nu)}\alpha_\mu$, parametrized by $(\mu,\nu)$. The associated soliton manifold is the set…

Injectivity conjecture. The mapping $\omega\mapsto\beta(\omega)$ is injective; equivalently, ground state solitons are uniquely identified by the ratio $\beta(\omega)$. The claim i…

Let $P_\omega$ denote the ground state soliton for the cubic-quintic nonlinear Schrödinger equation on $\mathbb{R}^3$, and let $M(P_\omega)$ be its mass. Mass-curve conjecture. The…

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…