On Global Attraction to Solitary Waves for the Klein-Gordon Equation Coupled to Nonlinear Oscillator

Abstract

The long-time asymptotics is analyzed for all finite energy solutions to a model U(1)-invariant nonlinear Klein-Gordon equation in one dimension, with the nonlinearity concentrated at a point. Our main result is that each finite energy solution converges as $t\to\pm\infty$ to the set of ``nonlinear eigenfunctions'' $\psi(x)e\sp{-i\omega t}$.