26 problems

Consider the cubic nonlinear Schrödinger equation referred to as, with initial data in the Sobolev space $H^s$. Local well-posedness conjecture. The cubic NLS equation is locally w…

Scattering conjecture for critical NLS. There exist functions

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…

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…

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…

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…

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…

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 $H^s$.

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…

In dimension $d=3$, let $\phi_\omega$ denote the ground state solution of the cubic-quintic Schrödinger equation at frequency $\omega$, with $0<\omega<\tfrac{3}{16}$, and let…

Absence-of-binding conjecture. For all $2\leq p<3$ and all $N\geq2$, $J(N)$ has no minimiser and

Binding conjecture. For every $N\geq2$ and every $1<p<\min\left(2,1+\frac{2}{d}\right)$, the binding inequalities

Stability threshold conjecture. For every $p$ satisfying $+\frac{4}{n}<p<p_n^*$, there exists $\omega_{p,n}$ such that $\phi_\omega$ is stable whenever $-n<\omega\leq\omega_{p,n}$…

Let $d\geq 3$, let $I$ be the lifespan interval, and let $u:I\times\mathbf{R}^{d}\to\mathbf{C}$ solve the focusing energy-critical nonlinear Schrödinger equation with…

Let $P_N$ denote the frequency projection onto frequencies $\lesssim N$, and let $e^{it\Delta_{\mathbb{T}^2}}$ be the Schrödinger propagator on $\mathbb{T}^2$. For all sufficiently…

Let the interaction parameter satisfy $\beta\in(0,1]$, and let $\{\Gamma_N\}_{N=1}^{\infty}$ be a sequence of quantum BBGKY hierarchy solutions satisfying the energy condition … fo…

Global well-posedness and scattering conjecture. For $d\geq 1$, the defocusing, mass-critical nonlinear Schrödinger initial value problem $$ is globally well-posed for $u0in L^2(ma…

Equal-rate blowup conjecture. The solution undergoes equal-rate collapse,

Shrinking-ring conjecture. The solution is quasi self-similar, $\psi\sim\psi_{Q_{\mathrm B}}$, for $r-r_{\rm \max}=\mathcal{O}(L)$; it is a shrinking ring, with…

Rigidity conjecture at the ground-state mass. If $M(u_0)=M(Q)$, then only the following cases can occur: if $u$ blows up in finite time, it coincides with $Pc(Q)$ up to symmetries…

Let $v$ solve the mass-critical nonlinear Schrödinger equation on ${\mathbb R}\times{\mathbb R}$, let $v_0\in L^2_x({\mathbb R})$ be its initial data, let $M(v_0)$ denote its mass,…

Let $S(t)$ be the linear propagator, let $\mathscr{A}f(t)=\int_{\mathbb{R}}S(t-\tau)f(\tau)\,d\tau$ be the associated full-time integral operator, and let $\Box_k$ be the frequency…