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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.

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

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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