258 problems

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 $u$ solve the one-dimensional fourth-order mixed-dispersion nonlinear Schrödinger equation, with power $alpha$ and ground-state solutions of the associated stationary equation.…

Let $u$ solve the one-dimensional $L^2$-critical nonlinear Schrödinger equation with repulsive delta potential $u<0$, with initial data $u0in H^1(mathbb{R})$ satisfying … where…

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…

Sirakov's equal-frequency uniqueness conjecture. In this equal-frequency setting, this is the unique positive solution.

Global well-posedness and scattering conjecture. Then $u$ is global and scatters, and

Fermi golden rule formula. The identity

Consider the defocusing cubic nonlinear Schrödinger initial value problem … with initial data $\phi(x,0)=\phi_0(x)\in H^s(\mathbb{R}^3)$, where $H^s(\mathbb{R}^3)$ is the usual inh…

Let $Q$ be the ground state solution of the focusing cubic nonlinear Schrödinger equation, and consider the deterministic Zakharov system in the energy space…

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…

Nondegeneracy conjecture. The kernel of the real linearized operator is trivial:

Let $Q$ be the unique positive radial solution of … Consider solutions $v$ of the $L^2$-critical nonlinear Schrödinger equation with critical mass $\|v\|_{L^2}=\|Q\|_{L^2}$, where…

The intensity-dependent dispersion nonlinear Schrödinger model is given by equation, with $\Sigma$ the energy space in and $\mathcal{H}$ the exponentially weighted…

Consider the complex Ginzburg–Landau equation with parameter $\theta=\pi/2$, namely the nonlinear Schrödinger equation. A solution is unbounded if its norm becomes arbitrarily larg…

Global scattering theorem. In the four sign regimes, the following conclusions hold: if $\mu_1=\mu_2=-1$, then $u$ is global and scatters; if $\mu_1=1$ and $\mu_2=-1$, then $u$ is…

Consider the nonlinear Schrödinger equation in the setting of the paper, and let $Q$ denote the ground state with critical mass $\|Q\|_{L^2}^2$. Let the symmetries of the equation…

Minimal-mass classification conjecture. Up to the symmetries of the equation, the only non-scattering solutions are the soliton $e^{it}Q$ and its pseudoconformal transformation.

Let $L_+$ be the linearized operator in the Hartree problem, and let $gKer[L_+]$ denote its generalized kernel. At the mass-critical exponent … the linearized operator has an extra…

Small-data asymptotic-freedom conjecture. If

Global wellposedness conjecture. When $\mu=1$, the equation is globally wellposed: for each $u_0\in\Sigma$ there is a unique global solution…

Let $u(t)$ be a solution of the cubic nonlinear Schrödinger equation on $\mathbb{T}^2$, and let $s>0$ with $s\neq 1$. For every $\epsilon>0$, let $C_\epsilon$ be a constant. Bourga…

Sharp-threshold conjecture. The sharp threshold for the existence of collapsing ring solutions is $p<5$ in any dimension $N\geq 2$.

For $\beta<0$, let $(u(r),f(r)e^{\sqrt{-1}d\theta})$ be a radially symmetric solution of the two-component nonlinear Schrödinger system described above, with $d\in\mathbb{N}$ and…

For the one-dimensional linearized NLS operator, let $\mathcal{L}$ be the operator whose continuous-spectrum endpoints are $\lambda=\pm i$, and let $\kappa=\sqrt{\mu_2}$ denote the…

Translated rigidity conjecture. Theorem 5.1 remains true if