학술논문
Global well-posedness for the cubic nonlinear Schrödinger equation with initial data lying in $L^p$-based Sobolev spaces.
Document Type
Journal
Author
Dodson, Benjamin (1-JHOP) AMS Author Profile; Soffer, Avraham (1-RTG) AMS Author Profile; Spencer, Thomas (1-IASP-SM) AMS Author Profile
Source
Subject
81 Quantum theory -- 81Q General mathematical topics and methods in quantum theory
81Q80Special quantum systems, such as solvable systems
81Q80
Language
English
Abstract
Summary: ``In this paper, we continue our study [B. Dodson, A. Soffer, and T. Spencer, J. Stat. Phys. {\bf 180}, 910 (2020)] [MR4131020] of the nonlinear Schrödinger equation (NLS) with bounded initial data which do not vanish at infinity. Local well-posedness on $\Bbb R$ was proved for real analytic data. Here, we prove global well-posedness for the 1D NLS with initial data lying in $L^p$ for any $2 < p < \infty$, provided that the initial data are sufficiently smooth. We do not use the complete integrability of the cubic NLS.''