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We review recent results on global attractors of U(1)-invariant dispersiveHamiltonian systems. We study several models based on the Klein-Gordon equationand sketch the proof that in these models, under certain generic assumptions,the weak global attractor is represented by the set of all solitary waves. Ingeneral, the attractors may also contain multifrequency solitary waves; we giveexamples of systems which contain such solutions.

Global Attraction to Solitary Waves in Models Based on the Klein-Gordon Equation