Center Manifold for Nonintegrable Nonlinear Schrödinger Equations on the Line
Center Manifold for Nonintegrable Nonlinear Schrödinger Equations on the Line
复制标题
在线不可积非线性薛定谔方程的中心流形
DOI：
10.1007/s002200000298
复制
发表时间：
2000
影响因子：
2.4
通讯作者：
R. Weder
中科院分区：
文献类型：
--
作者：
R. Weder
In this paper we study the following nonlinear Schrödinger equation on the line,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\end{document}wherefis real-valued, and it satisfies suitable conditions on regularity, on growth as a function ofuand on decay asx→±∞. Thegenericpotential,V, is real-valued and it is chosen so that the spectrum of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\end{document} consists of one simple negative eigenvalue and absolutely-continuous spectrum filling [0, ∞). The solutions to this equation have, in general, a localized and a dispersive component. The nonlinear bound states, that bifurcate from the zero solution at the energy of the eigenvalue ofH, define an invariant center manifold that consists of the orbits of time-periodic localized solutions. We prove that all small solutions approach a particular periodic orbit in the center manifold ast→±∞. In general, the periodic orbits are different fort→±∞. Our result implies also that the nonlinear bound states are asymptotically stable, in the sense that each solution with initial data near a nonlinear bound state is asymptotic ast→±∞ to the periodic orbits of nearby nonlinear bound states that are, in general, different fort→±∞.
In this paper we study the following nonlinear Schrödinger equation on the line,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\end{document}wherefis real-valued, and it satisfies suitable conditions on regularity, on growth as a function ofuand on decay asx→±∞. Thegenericpotential,V, is real-valued and it is chosen so that the spectrum of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\end{document} consists of one simple negative eigenvalue and absolutely-continuous spectrum filling [0, ∞). The solutions to this equation have, in general, a localized and a dispersive component. The nonlinear bound states, that bifurcate from the zero solution at the energy of the eigenvalue ofH, define an invariant center manifold that consists of the orbits of time-periodic localized solutions. We prove that all small solutions approach a particular periodic orbit in the center manifold ast→±∞. In general, the periodic orbits are different fort→±∞. Our result implies also that the nonlinear bound states are asymptotically stable, in the sense that each solution with initial data near a nonlinear bound state is asymptotic ast→±∞ to the periodic orbits of nearby nonlinear bound states that are, in general, different fort→±∞.