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Corpus ID: 233864907

Scattering of the three-dimensional cubic nonlinear Schr\"odinger equation with partial harmonic potentials

@inproceedings{Cheng2021ScatteringOT,
title={Scattering of the three-dimensional cubic nonlinear Schr\"odinger equation with partial harmonic potentials},
author={Xing Cheng and Chang-Yu Guo and Zihua Guo and Xian Liao and Jia Shen},
year={2021}
}

In this paper, we consider the following three dimensional defocusing cubic nonlinear Schr\"odinger equation (NLS) with partial harmonic potential \begin{equation*}\tag{NLS} i\partial_t u + \left(\Delta_{\mathbb{R}^3 }-x^2 \right) u = |u|^2 u, \quad u|_{t=0} = u_0. \end{equation*} Our main result shows that the solution $u$ scatters for any given initial data $u_0$ with finite mass and energy. The main new ingredient in our approach is to approximate (NLS) in the large-scale case by a relevant… Expand

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This work identifies Fermi-Pasta-Ulam-like recurrence phenomena, whereby the normal-mode spectrum passes in close proximity of the initial configuration, and two-mode states with time-independent mode amplitude spectra that translate into long-lived breathers of the original NLS equation.Expand