19 problems

Let $n$-vortex states $\phi$ be considered for a harmonic potential, with rotation frequency $\Omega\geq 0$ and interaction parameter $a>0$. Instability conjecture. An $n$-vortex w…

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…

Let $w$ be a positive-definite interaction potential, and let $\mu$ be a sufficiently small positive density parameter. Consider infinite ground states of the Gross–Pitaevskii equa…

In the energy-supercritical case $p=4$ and dimension $d\geq 5$, consider the singular radial solution $u_{\infty}$ of … for some $\omega_{\infty}\in(d-4,d)$ satisfying … Define the…

Nonexistence conjecture. There is no nontrivial finite-energy solution of $$ when $c(\mathcal{W})\geq c_*(\mathcal{W})$.

Let $N=2$ or $3$, let $c\in(0,\sqrt{2})$, and for sufficiently large periods $T$ let $\tilde{\psi}_T$ be the non-constant $T$-periodic mountain-pass solution of … with … where…

Consider the Gross–Pitaevskii equation … on $\mathbb{R}^N\times\mathbb{R}$, where $N=2$ or $3$, and traveling-wave profiles $\psi:\mathbb{R}^N\to\mathbb{C}$ satisfying … for speed…

Nonexistence conjecture. When $m-n>1$, there will be no balancing configurations satisfying those requirements.

Let $\mathcal{W}$ be a potential satisfying hypotheses … . Let $E_{\min}(\mathfrak{q})$ denote the minimizing curve and let $\mathcal{S}_{\mathfrak{q}}$ denote the corresponding se…

Let $\mathcal{W}$ be a potential with speed of sound … Consider traveling-wave solutions of the nonlocal Gross–Pitaevskii equation, and call a solution nontrivial if it is not the…

Initial-state dependence conjecture. Clever choices of such initial density-matrix sequences lead, in the large $N$ limit, to one-particle effective dynamics different from the Gro…

Dipolar Gross–Pitaevskii convergence conjecture. Similar results to the paper's derivation of the dipolar Gross–Pitaevskii energy should hold for

Let $\psi$ solve the three-dimensional Gross–Pitaevskii equation, and write $u=\psi-1$. For traveling waves of the form $\psi(t,x)=v(x-ct)$, define … where $E$ is the conserved Gro…

Shadow-vortex profile conjecture. Minimizers of the Gross–Pitaevski energy in the intermediate regime may also have a similar profile near the zero level set of $a(x)$.

Holmer–Zworski higher-Sobolev-norm conjecture. The error bounds

Radiation-damped oscillation conjecture. In the infinite-time limit, the soliton executes a radiation-damped oscillation to some stable nonlinear bound state, corresponding to the…

Ground-state uniqueness conjecture. The ground state solution is unique, up to the invariances of the problem.

Let $B\in C(\mathbb R^2)$ be a continuous magnetic field, and let $\mathcal E_B$ denote the energy space associated with $B$. The notation $B=\operatorname{curl} A'$ means that the…

Let $\psi=1+u$ be a global solution of the Gross–Pitaevskii equation in $C(\mathbb{R};1+F_1)$, and let $E_1$ denote its renormalized energy. Fix the threshold $\mathcal{E}_0>0$ fro…