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

We consider a U(1)-invariant nonlinear Klein-Gordon equation in dimension one or larger, 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 time goes to infinity) to the two-dimensional set of all ``nonlinear eigenfunctions'' of the form \phi(x)e\sp{-i\omega t}. This global attraction is caused by the nonlinear energy transfer from lower harmonics to the continuous spectrum and subsequent dispersive radiation

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

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.Comment: This is a contribution to the Proc. of the Seventh International Conference ''Symmetry in Nonlinear Mathematical Physics'' (June 24-30, 2007, Kyiv, Ukraine), published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA

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

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.Comment: This is a contribution to the Proc. of the Seventh International Conference ''Symmetry in Nonlinear Mathematical Physics'' (June 24-30, 2007, Kyiv, Ukraine), published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA

MSC: 35B41, 37K40, 37L30

Abstract The global attraction is established for the U(1)-invariant Klein-Gordon equation in one dimension coupled to a finite number of nonlinear oscillators. Each finite energy solution is shown to converge as t → ±∞ to the set of all solitary waves which are the "nonlinear eigenfunctions" of the form φ(x)e −iωt , under the conditions that all oscillators are strictly nonlinear and polynomial and the distances between neighboring oscillators are small. Our approach is based on the spectral analysis of omega-limit trajectories. We apply the Titchmarsh convolution theorem to prove that the time spectrum of each omega-limit trajectory consists of one point. Physically, the convergence to solitary waves is caused by the nonlinear energy transfer from lower harmonics to the continuous spectrum and subsequent dispersive radiation. The Titchmarsh theorem implies that such radiation is absent only for the solitary waves. To demonstrate the sharpness of our conditions, we construct counterexamples showing that the global attractor can contain "multifrequency solitary waves" if the distance between oscillators is large or if some oscillators are linear. Résumé. Onétablit l'attraction globale pour l'équation de Klein-Gordon U(1)-invariante monodimensionnelle couplée au nombre fini d'oscillateurs non linéaires. On démontre que chaque solution d'énergie finie converge vers un ensemble de toutes ondes solitaires qui sont des "fonctions propres non-linéaires" de la forme φ(x)e −iωt , sous la condition que tous les oscillateurs sont polynomiaux strictement non-linéaires et que les distances entre les oscillateurs voisins sont suffisament petites. Notre approche est fondé sur l'analyse spectrale des trajectoires omega-limite. Nous utilisons le théorème de convolution de Titchmarsh pour reduire le spectre temporel de chaque trajectoire omega-limiteà un seul point. Du point de vue physique, la raison de cette attraction globale est le transfert non linéair d'energie des modes inférieures vers les modes suprêmes, suivies par la radiation dispersive. Le théorème de Titchmarsh implique l'absence du transfert et de la radiation exclusivement pour les ondes solitaires. Pour démontrer l'optimalité de nos conditions, nous construisons des contre-exemples montrant que l'attracteur global peut contenir des "ondes solitaires multi-frequence" si la distance entre les oscillateurs est grande ou si certains oscillateurs sont linéaires

Global attractor for a nonlinear oscillator coupled to the Klein-Gordon field

The long-time asymptotics is 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 single point: each finite energy solution converges as time goes to plus or minus infinity to the set of all ``nonlinear eigenfunctions'' of the form \psi(x)e\sp{-i\omega t}. The global attraction is caused by the nonlinear energy transfer from lower harmonics to the continuous spectrum and subsequent dispersive radiation. We justify this mechanism by the following novel strategy based on inflation of spectrum by the nonlinearity. We show that any omega-limit trajectory has the time-spectrum in the spectral gap [-m,m] and satisfies the original equation. This equation implies the key spectral inclusion for spectrum of the nonlinear term. Then the application of the Titchmarsh Convolution Theorem reduces the spectrum of each omega-limit trajectory to a single harmonic in [-m,m]. The research is inspired by Bohr's postulate on quantum transitions and Schroedinger's identification of the quantum stationary states to the nonlinear eigenfunctions of the coupled U(1)-invariant Maxwell-Schroedinger and Maxwell-Dirac equations.Comment: 29 pages, 1 figur