The existence of full dimensional invariant tori for 1-dimensional nonlinear wave equation
Cong, Hongzi1 ; Yuan, Xiaoping2
1 a School of Mathematical Sciences, Dalian University of Technology, Dalian, Liaoning 116024, PR China
2 b School of Mathematical Sciences, Fudan University, Shanghai 200433, PR China
Annales de l'I.H.P. Analyse non linéaire, mai – juin 2021, Tome 38 (2021) no. 3, pp. 759-786.
Le texte intégral des articles récents est réservé aux abonnés de la revue. Consultez l'article sur le site de la revue.

In this paper we prove the existence and linear stability of full dimensional tori with subexponential decay for 1-dimensional nonlinear wave equation with external parameters, which relies on the method of KAM theory and the idea proposed by Bourgain [9].

Reçu le :
Révisé le :
Accepté le :
DOI : 10.1016/j.anihpc.2020.09.006
Mots-clés : KAM theory, Almost periodic solution, Nonlinear wave equation, Gevrey space
Cong, Hongzi 1 ; Yuan, Xiaoping 2

1 a School of Mathematical Sciences, Dalian University of Technology, Dalian, Liaoning 116024, PR China
2 b School of Mathematical Sciences, Fudan University, Shanghai 200433, PR China 
@article{AIHPC_2021__38_3_759_0,
     author = {Cong, Hongzi and Yuan, Xiaoping},
     title = {The existence of full dimensional invariant tori for 1-dimensional nonlinear wave equation},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {759--786},
     publisher = {Elsevier},
     volume = {38},
     number = {3},
     year = {2021},
     doi = {10.1016/j.anihpc.2020.09.006},
     zbl = {1466.35314},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1016/j.anihpc.2020.09.006/}
}
TY  - JOUR
AU  - Cong, Hongzi
AU  - Yuan, Xiaoping
TI  - The existence of full dimensional invariant tori for 1-dimensional nonlinear wave equation
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2021
SP  - 759
EP  - 786
VL  - 38
IS  - 3
PB  - Elsevier
UR  - http://archive.numdam.org/articles/10.1016/j.anihpc.2020.09.006/
DO  - 10.1016/j.anihpc.2020.09.006
LA  - en
ID  - AIHPC_2021__38_3_759_0
ER  - 
%0 Journal Article
%A Cong, Hongzi
%A Yuan, Xiaoping
%T The existence of full dimensional invariant tori for 1-dimensional nonlinear wave equation
%J Annales de l'I.H.P. Analyse non linéaire
%D 2021
%P 759-786
%V 38
%N 3
%I Elsevier
%U http://archive.numdam.org/articles/10.1016/j.anihpc.2020.09.006/
%R 10.1016/j.anihpc.2020.09.006
%G en
%F AIHPC_2021__38_3_759_0
Cong, Hongzi; Yuan, Xiaoping. The existence of full dimensional invariant tori for 1-dimensional nonlinear wave equation. Annales de l'I.H.P. Analyse non linéaire, mai – juin 2021, Tome 38 (2021) no. 3, pp. 759-786. doi : 10.1016/j.anihpc.2020.09.006. http://archive.numdam.org/articles/10.1016/j.anihpc.2020.09.006/

[1] Baldi, P.; Berti, M.; Haus, E.; Montalto, R. Time quasi-periodic gravity water waves in finite depth, Invent. Math., Volume 214 (2018) no. 2, pp. 739-911 | DOI | Zbl

[2] Baldi, P.; Berti, M.; Montalto, R. KAM for quasi-linear and fully nonlinear forced perturbations of Airy equation, Math. Ann., Volume 359 (2014) no. 1–2, pp. 471-536 | DOI | Zbl

[3] Baldi, P.; Berti, M.; Montalto, R. KAM for autonomous quasi-linear perturbations of KdV, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 33 (2016) no. 6, pp. 1589-1638 | DOI | Numdam | Zbl

[4] Berti, M.; Bolle, P. Sobolev quasi-periodic solutions of multidimensional wave equations with a multiplicative potential, Nonlinearity, Volume 25 (2012) no. 9, pp. 2579-2613 | DOI | Zbl

[5] Berti, M.; Bolle, P. Quasi-periodic solutions with Sobolev regularity of NLS on Td with a multiplicative potential, J. Eur. Math. Soc., Volume 15 (2013) no. 1, pp. 229-286 | DOI | Zbl

[6] Bourgain, J. Construction of approximative and almost periodic solutions of perturbed linear Schrödinger and wave equations, Geom. Funct. Anal., Volume 6 (1996) no. 2, pp. 201-230 | DOI | Zbl

[7] Bourgain, J. Quasi-periodic solutions of Hamiltonian perturbations of 2D linear Schrödinger equations, Ann. Math. (2), Volume 148 (1998) no. 2, pp. 363-439 | DOI | Zbl

[8] Bourgain, J. Green's Function Estimates for Lattice Schrödinger Operators and Applications, Annals of Mathematics Studies, vol. 158, Princeton University Press, Princeton, NJ, 2005 | Zbl

[9] Bourgain, J. On invariant tori of full dimension for 1D periodic NLS, J. Funct. Anal., Volume 229 (2005) no. 1, pp. 62-94 | DOI | Zbl

[10] Bourgain, J.; Kachkovskiy, I. Anderson localization for two interacting quasiperiodic particles, Geom. Funct. Anal. (2019) | DOI | Zbl

[11] Cong, H.; Liu, J.; Shi, Y.; Yuan, X. The stability of full dimensional KAM tori for nonlinear Schrödinger equation, J. Differ. Equ., Volume 264 (2018) no. 7, pp. 4504-4563 | DOI | Zbl

[12] Craig, W.; Wayne, C.E. Newton's method and periodic solutions of nonlinear wave equations, Commun. Pure Appl. Math., Volume 46 (1993) no. 11, pp. 1409-1498 | DOI | Zbl

[13] Eliasson, L.H.; Kuksin, S.B. KAM for the nonlinear Schrödinger equation, Ann. Math. (2), Volume 172 (2010) no. 1, pp. 371-435 | DOI | Zbl

[14] Feola, R.; Procesi, M. Quasi-periodic solutions for fully nonlinear forced reversible Schrödinger equations, J. Differ. Equ., Volume 259 (2015) no. 7, pp. 3389-3447 | DOI | Zbl

[15] Gao, M.; Liu, J. Invariant tori for 1D quintic nonlinear wave equation, J. Differ. Equ., Volume 263 ( Dec. 2017 ), pp. 8533-8564 | DOI | Zbl

[16] Geng, J.; Xu, X. Almost periodic solutions of one dimensional Schrödinger equation with the external parameters, J. Dyn. Differ. Equ., Volume 25 (2013) no. 2, pp. 435-450 | DOI | Zbl

[17] Kappeler, T.; Pöschel, J. KdV & KAM, Ergebnisse der Mathematik und Ihrer Grenzgebiete. 3. Folge, vol. 45, Springer-Verlag, Berlin, 2003 | Zbl

[18] Kuksin, S.; Pöschel, J. Invariant Cantor manifolds of quasi-periodic oscillations for a nonlinear Schrödinger equation, Ann. Math. (2), Volume 143 (1996) no. 1, pp. 149-179 | DOI | Zbl

[19] Kuksin, S.B. Hamiltonian perturbations of infinite-dimensional linear systems with imaginary spectrum, Funkc. Anal. Prilozh., Volume 21 (1987) no. 3, pp. 22-37 (95) | Zbl

[20] Kuksin, S.B. Analysis of Hamiltonian PDEs, Oxford Lecture Series in Mathematics and Its Applications, vol. 19, Oxford University Press, Oxford, 2000 | Zbl

[21] Kuksin, S.B. Fifteen years of KAM for PDE, Geometry, Topology, and Mathematical Physics, Amer. Math. Soc. Transl. Ser. 2, vol. 212, Amer. Math. Soc., Providence, RI, 2004, pp. 237-258 | Zbl

[22] Biasco, J.E.M.L.; Procesi, M. Almost periodic invariant tori for the nls on the circle, 2019 | arXiv | Zbl

[23] Liu, J.; Yuan, X. A KAM theorem for Hamiltonian partial differential equations with unbounded perturbations, Commun. Math. Phys., Volume 307 (2011) no. 3, pp. 629-673 | DOI | Zbl

[24] Pöschel, J. Quasi-periodic solutions for a nonlinear wave equation, Comment. Math. Helv., Volume 71 (1996) no. 2, pp. 269-296 | DOI | Zbl

[25] Pöschel, J. On the construction of almost periodic solutions for a nonlinear Schrödinger equation, Ergod. Theory Dyn. Syst., Volume 22 (2002) no. 5, pp. 1537-1549 | DOI | Zbl

[26] Wang, W.-M. Energy supercritical nonlinear Schrödinger equations: quasiperiodic solutions, Duke Math. J., Volume 165 (2016) no. 6, pp. 1129-1192 | Zbl

[27] Wayne, C.E. Periodic and quasi-periodic solutions of nonlinear wave equations via KAM theory, Commun. Math. Phys., Volume 127 (1990) no. 3, pp. 479-528 | DOI | Zbl

Cité par Sources :