Scattering of the threedimensional cubic nonlinear Schrödinger equation with partial harmonic potentials
Abstract
In this paper, we consider the following three dimensional defocusing cubic nonlinear Schrödinger 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 largescale case by a relevant dispersive continuous resonant (DCR) system. The proof of global wellposedness and scattering of the new (DCR) system is greatly inspired by the fundamental works of Dodson \cite{D3,D1,D2} in his study of scattering for the masscritical nonlinear Schrödinger equation. The analysis of (DCR) system allows us to utilize the additional regularity of the smooth nonlinear profile so that the celebrated concentrationcompactness/rigidity argument of Kenig and Merle applies.
 Publication:

arXiv eprints
 Pub Date:
 May 2021
 arXiv:
 arXiv:2105.02515
 Bibcode:
 2021arXiv210502515C
 Keywords:

 Mathematics  Analysis of PDEs;
 Mathematical Physics
 EPrint:
 71 pages