Global Attractor for a Nonlinear Oscillator Coupled to the Klein–Gordon Field

The long-time asymptotics is analyzed for all finite energy solutions to a model \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{U}(1)$$\end{document}-invariant nonlinear Klein–Gordon equation in one dimension, with the nonlinearity concentrated at a single point: each finite energy solution converges as t→ ± ∞ to the set of all “nonlinear eigenfunctions” of the form ψ(x)e−iω t. The global attraction is caused by the nonlinear energy transfer from lower harmonics to the continuous spectrum and subsequent dispersive radiation.