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We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

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.

Received:
Jul 30, 2007
Published:
Jul 30, 2007

Related publications

inJournal
2009 Repository Open Access
Alexander Komech and Andrew Komech

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

In: Annales de l'Institut Henri Poincaré / C, 26 (2009) 3, pp. 855-868