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 n⩾1, 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 t→±∞ to the two-dimensional set of all “nonlinear eigenfunctions” of the form ϕ(x)e−iωt. This global attraction is caused by the nonlinear energy transfer from lower harmonics to the continuous spectrum and subsequent dispersive radiation.