122 problems

Let $d\geq 3$ and consider the mass-critical nonlinear Schrödinger equation with nonlinearity $F(z)=+|z|^{4/d}z$. For initial data $u_0\in L^2_x(\mathbf{R}^d)$ and initial time…

Global well-posedness and scattering conjecture. For every $E\in(0,\infty)$ and every smooth initial datum $\vec{u}_0$ with $E_{ls}(\vec{u}_0)\leq E$, there exists a global solutio…

White-noise evolution-system conjecture. The family $\{\mu_{1+t}\}_{t\in\mathbb{R}_+}$ of white-noise measures with variance $1+t$ is an evolution system of measures for the stocha…

Global well-posedness and scattering conjecture. Global well-posedness and scattering hold for $(1.1)$ when $0<s_c<1$. The mass-critical case $s_c=0$ and energy-critical case…

A soliton solution is a solution constructed using Bäcklund transformations, and an attached Akhmediev breather is an Akhmediev breather attached to such a soliton solution. Instab…

Spectral mapping conjecture. For the operator $\boldsymbol{A}_k$, the spectral mapping theorem should hold. This would imply estimate (6.2) in the cited stability analysis. The con…

Negative-slope instability conjecture. The availability of a section of the stability curve with negative slope is a sufficient condition for instability of the soliton. The paper…

Stability conjecture. Contracting sphere blow-up solutions are stable under small radial perturbations but unstable under small nonradial (symmetry-breaking) perturbations.

Hessian determinant conjecture. The determinant satisfies

Scattering conjecture for critical NLS. There exist functions

Let $u_0\in L^2$ be initial data for the one-dimensional quintic nonlinear Schrödinger equation, and consider a solution $u$ that develops mass concentration near a blow-up time…

Consider the critical focusing nonlinear Schrödinger equation in one spatial dimension and its standing-wave blow-up solutions. The generalized root space of the linearized operato…

Quintic NLS well-posedness conjecture. Local and global well-posedness should hold in $H^s(\mathbb T)$ for all $s\geq 0$.

Semiclassical mass conjecture. In semiclassical quantization, the classical formula for $M$ should be modified in the same way as for the conventional nonlinear Schrödinger equatio…

Let $U_0$ be the nonlinear interaction coefficient, let $\mu$ be the chemical potential, and let $(\mathcal{E}_n,\phi_n)$ be an eigenvalue and eigenfunction of the associated linea…

Renormalised density $L^3$ estimate conjecture. If $b1 in [3/2,2)$ and $c3 > 2/3 - 1/b1$, then

Quadratic-phase conjecture. For the combined nonlinear Schrödinger equation, if $b>0$, the solution with initial condition $v_0$ exists globally and scatters, whereas if $b<0$ and…

Fix $(X,T)$ in a compact set, with $zeta>0$ and $mu_{cD_1}>0$. Let $R(Z)$ be the unique solution of the specified Riemann–Hilbert problem, with asymptotic expansion … where…

Let $B_1$ be the unit ball, let $\beta$ denote the first Dirichlet eigenvalue parameter, and let $Q_b$ be the least-energy (ground state) solution of the stationary nonlinear Schrö…

A repulsive $\delta'$ interaction in one dimension can be approximated by smooth potentials with a negative well, and the nonlinear energy admits ground states. Ground-state inheri…

Adaptability conjecture. The results of this work can be adapted to more general classes of metric graphs, such as single-knot metric graphs.

Critical-norm global well-posedness and scattering conjecture. If $s_c>0$, then $u$ is global and scatters as $t\to\pm\infty$.

Shape conjecture. The action ground state $u=(u_c,u_h)$ has one of the shapes given by Proposition, moving from case $(iv)$ to case $(i)$ as $L$ increases from $0$ to $\infty$.

Branch-counting conjecture. The supercritical CGL equation has a countable number of branches of nontrivial radial self-similar singular solutions. These branches can be labeled by…

Consider the cubic nonlinear Schrödinger equation in the mass-critical case $d=2$, where the $L^2$-norm is invariant under the equation's natural scaling. Generic blowup conjecture…