Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 4 (2008), 010, 23 pages      arXiv:0711.0041      https://doi.org/10.3842/SIGMA.2008.010
Contribution to the Proceedings of the Seventh International Conference Symmetry in Nonlinear Mathematical Physics

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

Alexander I. Komech a, c and Andrew A. Komech b, c
a) Faculty of Mathematics, University of Vienna, Wien A-1090, Austria
b) Mathematics Department, Texas A&M University, College Station, TX 77843, USA
c) Institute for Information Transmission Problems, B. Karetny 19, Moscow 101447, Russia

Received November 01, 2007, in final form January 22, 2008; Published online January 31, 2008

Abstract
We review recent results on global attractors of U(1)-invariant dispersive Hamiltonian systems. We study several models based on the Klein-Gordon equation and sketch the proof that in these models, under certain generic assumptions, the weak global attractor is represented by the set of all solitary waves. In general, the attractors may also contain multifrequency solitary waves; we give examples of systems which contain such solutions.

Key words: global attractors; solitary waves; solitary asymptotics; nonlinear Klein-Gordon equation; dispersive Hamiltonian systems; unitary invariance.

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