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arXiv · 2306.05931

Damped nonlinear Schr\"odinger equation with Stark effect

Yi Hu·Yongki Lee·Shijun Zheng
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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 $\|Q\|_2$, 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.

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Yi Hu, Yongki Lee, Shijun Zheng. 2023-06-09. Damped nonlinear Schr\"odinger equation with Stark effect. https://arxiv.org/abs/2306.05931

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