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MiS Preprint
66/2007
On global attraction to quantum stationary states III. Klein-Gordon equation with mean field interaction
Alexander Komech and Andrew Komech
Abstract
We consider a $\mathbf{U}(1)$-invariant nonlinear Klein-Gordon equation in dimension $n\ge 1$, 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 $\phi(x)e^{-i\omega t}$. This global attraction is caused by the nonlinear energy transfer from lower harmonics to the continuous spectrum and subsequent dispersive radiation.
