Global attraction to solitary waves for Klein-Gordon equation with mean field interaction

We consider a U(1)-invariant nonlinear Klein-Gordon equation in dimension one or larger, self-interacting via the mean field mechanism. We analyze the long-time asymptotics of finite energy solutions and prove that, under certain generic assumptions, each solution converges (as time goes to infinity) to the two-dimensional set of all ``nonlinear eigenfunctions'' of the form $\phi(x)e\sp{-i\omega t}$. This global attraction is caused by the nonlinear energy transfer from lower harmonics to the continuous spectrum and subsequent dispersive radiation.