On the global attraction to solitary waves for the Klein–Gordon equation coupled to a nonlinear oscillator

Alexander I. Komech; Andrew A. Komech

doi:10.1016/j.crma.2006.06.009

Partial Differential Equations

On the global attraction to solitary waves for the Klein–Gordon equation coupled to a nonlinear oscillator

Presented by: Pierre-Louis Lions

Alexander I. Komech ¹; Andrew A. Komech ²

¹ Faculty of Mathematics, Vienna University, A-1090 Vienna, Austria
² Mathematics Department, Texas A&M University, College Station, TX, USA

Comptes Rendus. Mathématique, Volume 343 (2006) no. 2, pp. 111-114.

Abstract
Résumé

The long-time asymptotics are analyzed for all finite energy solutions to a model $U (1)$ -invariant nonlinear Klein–Gordon equation in one dimension, with the nonlinearity concentrated at a point. Our main result is that each finite energy solution converges as $t \to \pm \infty$ to the set of ‘nonlinear eigenfunctions’ $ψ (x) e^{- i ω t}$ .

On s'intéresse aux solutions d'énergie finie d'une équation non linéaire de Klein–Gordon $U (1)$ -invariante monodimensionnelle, avec une non linéarité ponctuelle, et on analyse leur comportement asymptotique aux temps longs. Le principal résultat que nous avons obtenu est que toute solution d'énergie finie converge pour $t \to \pm \infty$ vers un ensemble de « fonctions propres non linéaires » $ψ (x) e^{- i ω t}$ .

Received: 2006-04-28
Accepted: 2006-06-01
Published online: 2006-07-11

DOI: 10.1016/j.crma.2006.06.009

Author's affiliations:

Alexander I. Komech ¹; Andrew A. Komech ²

¹ Faculty of Mathematics, Vienna University, A-1090 Vienna, Austria
² Mathematics Department, Texas A&M University, College Station, TX, USA

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Alexander I. Komech; Andrew A. Komech. On the global attraction to solitary waves for the Klein–Gordon equation coupled to a nonlinear oscillator. Comptes Rendus. Mathématique, Volume 343 (2006) no. 2, pp. 111-114. doi : 10.1016/j.crma.2006.06.009. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2006.06.009/

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