Comptes Rendus
no. 7
Mathematical analysis/Partial differential equations
Quasi-periodic solutions for nonlinear wave equations
[Solutions quasi périodiques pour l'équation des ondes non linéaire]

Présenté par : Gilles Lebeau

Wei-Min Wang1
1 CNRS and Department of Mathematics, Université Cergy-Pontoise, 95302 Cergy-Pontoise cedex, France
Comptes Rendus. Mathématique, Volume 353 (2015) no. 7, pp. 601-604.

On construit des solutions quasi-périodiques en temps pour l'équation des ondes non linéaire sur le tore en dimension quelconque. Tous les résultats précédents se limitent au cercle. Cet article étend la méthode développée pour le cas limite elliptique dans [12] au cas hyperbolique. Le nouvel ingrédient est une propriété diophantienne des nombres algébriques.

We construct time quasi-periodic solutions to nonlinear wave equations on the torus in arbitrary dimensions. All previously known results (in the case of zero or a multiplicative potential) seem to be limited to the circle. This extends the method developed in the limit-elliptic setting in [12] to the hyperbolic setting. The additional ingredient is a Diophantine property of algebraic numbers.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2015.04.014
Wei-Min Wang 1

1 CNRS and Department of Mathematics, Université Cergy-Pontoise, 95302 Cergy-Pontoise cedex, France 
@article{CRMATH_2015__353_7_601_0,
     author = {Wei-Min Wang},
     title = {Quasi-periodic solutions for nonlinear wave equations},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {601--604},
     publisher = {Elsevier},
     volume = {353},
     number = {7},
     year = {2015},
     doi = {10.1016/j.crma.2015.04.014},
     language = {en},
}
TY  - JOUR
AU  - Wei-Min Wang
TI  - Quasi-periodic solutions for nonlinear wave equations
JO  - Comptes Rendus. Mathématique
PY  - 2015
SP  - 601
EP  - 604
VL  - 353
IS  - 7
PB  - Elsevier
DO  - 10.1016/j.crma.2015.04.014
LA  - en
ID  - CRMATH_2015__353_7_601_0
ER  - 
%0 Journal Article
%A Wei-Min Wang
%T Quasi-periodic solutions for nonlinear wave equations
%J Comptes Rendus. Mathématique
%D 2015
%P 601-604
%V 353
%N 7
%I Elsevier
%R 10.1016/j.crma.2015.04.014
%G en
%F CRMATH_2015__353_7_601_0
Wei-Min Wang. Quasi-periodic solutions for nonlinear wave equations. Comptes Rendus. Mathématique, Volume 353 (2015) no. 7, pp. 601-604. doi : 10.1016/j.crma.2015.04.014. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2015.04.014/

[1] M. Berti; P. Bolle Sobolev quasi-periodic solutions of multidimensionalal wave equations with a multiplicative potential, Nonlinearity, Volume 25 (2012), pp. 2579-2613

[2] M. Berti; L. Corsi; M. Procesi An abstract Nash–Moser theorem and quasi-periodic solutions for NLW and NLS on compact Lie groups and homogeneous manifolds, Commun. Math. Phys., Volume 334 (2015), pp. 1413-1454

[3] J. Bourgain Construction of periodic solutions of nonlinear wave equations in higher dimensions, Geom. Funct. Anal., Volume 5 (1995), pp. 363-439

[4] J. Bourgain Construction of approximative and almost periodic solutions of perturbed linear Schrödinger and wave equations, Geom. Funct. Anal., Volume 6 (1996), pp. 629-639

[5] J. Bourgain Green's Function Estimates for Latttice Schrödinger Operators and Applications, Ann. of Math. Studies, vol. 158, Princeton University Press, 2005

[6] L. Chierchia; J. You KAM tori for 1D nonlinear wave equations with periodic boundary conditions, Commun. Math. Phys., Volume 211 (2000), pp. 497-525

[7] V. Guillemin Lectures on spectral theory of elliptic operators, Duke Math. J., Volume 44 (1977), pp. 485-517

[8] S. Kuksin Hamiltonian perturbation of infinite-dimensional linear systems with imaginary spectrum, Funkc. Anal. Prilozh., Volume 21 (1987), pp. 22-37

[9] J. Pöschel Quasi-periodic solutions for a nonlinear wave equation, Comment. Math. Helv., Volume 71 (1996), pp. 269-296

[10] W. Schmidt Norm form equations, Ann. Math., Volume 96 (1972), pp. 526-551

[11] W.-M. Wang, Quasi-periodic solutions for nonlinear wave equations, Preprint (2014), 50 p.

[12] W.-M. Wang Energy supercritical nonlinear Schrödinger equations: quasi-periodic solutions (2014, submitted for publication, 61 p) | arXiv

[13] E. Wayne Periodic and quasi-periodic solutions of nonlinear wave equations via KAM theory, Commun. Math. Phys., Volume 127 (1990), pp. 479-528

Cité par Sources :

Commentaires - Politique