52 problems

Let $b<0$, and consider the minimization problem in $A_{T/2}$ with fixed mass $m$, denoted by … operatorname{sn}{alpha,beta,k}(x)=frac{1}{alpha}operatorname{sn}left(frac{x}{beta},k…

Spherically symmetric scattering conjecture. In the defocusing case ($\mu=+1$) we have $m_{0,\operatorname{rad}}=+\infty$, while in the focusing case ($\mu=-1$) we have…

Scattering conjecture. In the defocusing case ($\mu=+1$) we have $m_0=+\infty$, while in the focusing case ($\mu=-1$) we have $m_0=M(Q)$. This predicts global well-posedness and sc…

Let $L^2_{+}(\mathbb{T})$ and $L^2_{+}(\mathbb{R})$ denote the positive-frequency Hardy subspaces of $L^2(\mathbb{T})$ and $L^2(\mathbb{R})$, respectively. Consider the defocusing…

Let $\Gamma$ be a $k$-dimensional submanifold in $\mathbb R^\mathcal{N}$ and a nondegenerate critical point of the functional … Assume $1<p<\frac{n+2-k}{n-2+k}$ for…

Classification of minimal-mass blowup solutions. Only the following two cases occur:

Non-scattering-data dispersion-manageability conjecture. Given such $u_0$, both of the following assertions hold: (i) for fixed $t_0,t_1$, there exist $\gamma_+=\gamma_+(u_0)$ and…

Dispersion-manageability conjecture. There exists a Schwartz function $u_0$ such that, with $u(0,x)=u_0(x)$, either $(NLS)_\gamma$ or $(NLS)^\gamma$ is dispersion-manageable.

Delayed-blowup conjecture. The corresponding solution of either managed model does not blow up in $[0,T]$; thus, blowup can be delayed by dispersion or nonlinearity management.

Consider weakly localized solutions and their stability analysis by the method of modulation equations. Let the halo part denote the component outside the core or localized part of…

Consider a time-dependent potential that is localized in space. A weakly localized solution (WLS) is a solution with a component exhibiting weak localization, while a localized sol…

A localized solution is a spatially localized solution, and regularity in space refers to the spatial regularity assumed in the surrounding discussion. A weakly localized solution…

Let $\Omega\subset\mathbb{R}^3$ be the exterior domain of a smooth, compact, strictly convex obstacle, and consider the defocusing nonlinear Schrödinger equation with Dirichlet bou…

Consider the defocusing nonlinear Schrödinger equation … in $\mathbb{R}^d$, with $d\geq 1$ and $\alpha\geq 4/d$. Set $s_c=d/2-2/\alpha$, and let…

Global existence and scattering conjecture. All maximal-lifespan solutions are globally well-posed and scatter.

Kenig–Merle critical norm conjecture. If $u$ has bounded critical Sobolev norm in this sense, then $u$ is global and scatters to a free solution.

Stable-manifold conjecture. Such initial data are globally well-posed under the nonlinear Schrödinger dynamics if and only if $u_0\notin\mathcal{W}^s(0)$.

Let $p\in(2,6)$ and let $\nu>0$ be a mass. An energy ground state is a positive solution of the NLS variational problem on the $\mathcal{T}$-metric graph that minimizes the energy…

Let $m>0$, and let $\mathcal{S}_m$ denote the set of stationary solutions with total mass $m$. Let $Q_{gs}$ be a ground state, and let $E[Q]$ denote the total energy of a stationar…

Global scattering conjecture. Such initial data should give rise to a solution that exists globally in time and scatters.

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,…

Perturbation-determinant equicontinuity conjecture. For any $L^2$-bounded and equicontinuous $Q\subseteq\mathcal{S}$, the set $Q_*$ is also $L^2$-equicontinuous.

Equicontinuity conjecture. For any $Q\subseteq\mathcal{S}$ that is $L^2$-bounded and equicontinuous, the set $Q_*$ is also $L^2$-equicontinuous.

Let $\mathcal{PT}$-symmetric vector rational rogue waves be solutions of the multi-component nonlinear Schrödinger equations, and let their asymptotic structures refer to the patte…

For $\omega<\omega_c$, consider initial data of the form … Here $Q_\omega$ is a cubic-quintic ground state, $M$ denotes mass, and…