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

Abstract

We consider a $\mathbf{U}(1)$-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 $t\to \pm \infty$ to the two-dimensional set of all “nonlinear eigenfunctions” of the form $\varphi (x)e^{-i\omega t}$. This global attraction is caused by the nonlinear energy transfer from lower harmonics to the continuous spectrum and subsequent dispersive radiation.