4 problems

Consider the repulsive one-dimensional Gross–Pitaevskii equation with cosine potential, parameters $\omega$ and $A$, and its associated Poincaré map $T$. Let $\Delta_n^\pm$ be the…

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 $B\in C(\mathbb R^2)$ 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…