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
[Attraction globale vers des ondes solitaires pour l'équation de Klein–Gordon couplé à un oscillateur non linéaire]

Présenté par : 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.

Résumé
Abstract

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 » $ψ (x) e^{- i ω t}$ .

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’ $ψ (x) e^{- i ω t}$ .

Reçu le : 2006-04-28
Accepté le : 2006-06-01
Publié le : 2006-07-11

DOI : 10.1016/j.crma.2006.06.009

Affiliations des auteurs :

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

@article{CRMATH_2006__343_2_111_0,
     author = {Alexander I. Komech and Andrew A. Komech},
     title = {On the global attraction to solitary waves for the {Klein{\textendash}Gordon} equation coupled to a nonlinear oscillator},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {111--114},
     publisher = {Elsevier},
     volume = {343},
     number = {2},
     year = {2006},
     doi = {10.1016/j.crma.2006.06.009},
     language = {en},
}

TY  - JOUR
AU  - Alexander I. Komech
AU  - Andrew A. Komech
TI  - On the global attraction to solitary waves for the Klein–Gordon equation coupled to a nonlinear oscillator
JO  - Comptes Rendus. Mathématique
PY  - 2006
SP  - 111
EP  - 114
VL  - 343
IS  - 2
PB  - Elsevier
DO  - 10.1016/j.crma.2006.06.009
LA  - en
ID  - CRMATH_2006__343_2_111_0
ER  -

%0 Journal Article
%A Alexander I. Komech
%A Andrew A. Komech
%T On the global attraction to solitary waves for the Klein–Gordon equation coupled to a nonlinear oscillator
%J Comptes Rendus. Mathématique
%D 2006
%P 111-114
%V 343
%N 2
%I Elsevier
%R 10.1016/j.crma.2006.06.009
%G en
%F CRMATH_2006__343_2_111_0

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/

Bibliographie
Cité par

[1] A.V. Babin; M.I. Vishik Attractors of Evolution Equations, Studies in Mathematics and its Applications, vol. 25, North-Holland Publishing Co., Amsterdam, 1992

[2] V.S. Buslaev; G.S. Perel'man Scattering for the nonlinear Schrödinger equation: states that are close to a soliton, Algebra i Analiz, Volume 4 (1992) no. 6, pp. 63-102

[3] S. Cuccagna Stabilization of solutions to nonlinear Schrödinger equations, Comm. Pure Appl. Math., Volume 54 (2001) no. 9, pp. 1110-1145

[4] J. Ginibre; G. Velo Time decay of finite energy solutions of the nonlinear Klein–Gordon and Schrödinger equations, Ann. Inst. H. Poincaré Phys. Théor., Volume 43 (1985) no. 4, pp. 399-442

[5] L. Hörmander The Analysis of Linear Partial Differential Operators. I, Springer-Verlag, Berlin, 1990 (Springer Study Edition)

[6] S. Klainerman Long-time behavior of solutions to nonlinear evolution equations, Arch. Rational Mech. Anal., Volume 78 (1982) no. 1, pp. 73-98

[7] A.I. Komech Stabilization of the interaction of a string with a nonlinear oscillator, Moscow Univ. Math. Bull., Volume 46 (1991) no. 6, pp. 34-39

[8] A. Komech On transitions to stationary states in one-dimensional nonlinear wave equations, Arch. Rational Mech. Anal., Volume 149 (1999) no. 3, pp. 213-228

[9] A. Komech; H. Spohn Long-time asymptotics for the coupled Maxwell–Lorentz equations, Comm. Partial Differential Equations, Volume 25 (2000) no. 3–4, pp. 559-584

[10] A. Komech; H. Spohn; M. Kunze Long-time asymptotics for a classical particle interacting with a scalar wave field, Comm. Partial Differential Equations, Volume 22 (1997) no. 1–2, pp. 307-335

[11] C.S. Morawetz; W.A. Strauss Decay and scattering of solutions of a nonlinear relativistic wave equation, Comm. Pure Appl. Math., Volume 25 (1972), pp. 1-31

[12] A. Soffer; M.I. Weinstein Multichannel nonlinear scattering for nonintegrable equations, Comm. Math. Phys., Volume 133 (1990) no. 1, pp. 119-146

[13] A. Soffer; M.I. Weinstein Resonances, radiation damping and instability in Hamiltonian nonlinear wave equations, Invent. Math., Volume 136 (1999) no. 1, pp. 9-74

[14] W.A. Strauss Decay and asymptotics for $□ u = f (u)$ , J. Funct. Anal., Volume 2 (1968), pp. 409-457

[15] R. Temam Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences, vol. 68, Springer-Verlag, New York, 1997

[16] E. Titchmarsh The zeros of certain integral functions, Proc. London Math. Soc., Volume 25 (1926), pp. 283-302

Cité par Sources :

Commentaires - Politique