Damped nonlinear Schrödinger equation with Stark effect

Abstract

We study the $L^2$-critical damped NLS with a Stark potential. We prove that the threshold for global existence and finite time blowup of this equation is given by $\Vert Q{\Vert }_2$, where $Q$ is the unique positive radial solution of $\Delta Q+|Q{|}^{4/d}Q=Q$ in $H^1({\mathbb{R}}^d)$. Moreover, in any small neighborhood of $Q$, there exists an initial data $u_0$ above the ground state such that the solution flow admits the log-log blowup speed. This verifies the structural stability for the ``$\log$-$\log$ law'' associated to the NLS mechanism under the perturbation by a damping term and a Stark potential. The proof of our main theorem is based on the Avron-Herbst formula and the analogous result for the unperturbed damped NLS.

13 pages