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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].
Cong, Hongzi 1 ; Yuan, Xiaoping 2
@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://geodesic.mathdoc.fr/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://geodesic.mathdoc.fr/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://geodesic.mathdoc.fr/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://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2020.09.006/
[1] Time quasi-periodic gravity water waves in finite depth, Invent. Math., Volume 214 (2018) no. 2, pp. 739-911 | Zbl | DOI
[2] KAM for quasi-linear and fully nonlinear forced perturbations of Airy equation, Math. Ann., Volume 359 (2014) no. 1–2, pp. 471-536 | Zbl | DOI
[3] KAM for autonomous quasi-linear perturbations of KdV, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 33 (2016) no. 6, pp. 1589-1638 | Zbl | mathdoc-id | DOI
[4] Sobolev quasi-periodic solutions of multidimensional wave equations with a multiplicative potential, Nonlinearity, Volume 25 (2012) no. 9, pp. 2579-2613 | Zbl | DOI
[5] Quasi-periodic solutions with Sobolev regularity of NLS on with a multiplicative potential, J. Eur. Math. Soc., Volume 15 (2013) no. 1, pp. 229-286 | Zbl | DOI
[6] 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 | Zbl | DOI
[7] Quasi-periodic solutions of Hamiltonian perturbations of 2D linear Schrödinger equations, Ann. Math. (2), Volume 148 (1998) no. 2, pp. 363-439 | Zbl | DOI
[8] Green's Function Estimates for Lattice Schrödinger Operators and Applications, Annals of Mathematics Studies, vol. 158, Princeton University Press, Princeton, NJ, 2005 | Zbl
[9] On invariant tori of full dimension for 1D periodic NLS, J. Funct. Anal., Volume 229 (2005) no. 1, pp. 62-94 | Zbl | DOI
[10] Anderson localization for two interacting quasiperiodic particles, Geom. Funct. Anal. (2019) | Zbl | DOI
[11] The stability of full dimensional KAM tori for nonlinear Schrödinger equation, J. Differ. Equ., Volume 264 (2018) no. 7, pp. 4504-4563 | Zbl | DOI
[12] Newton's method and periodic solutions of nonlinear wave equations, Commun. Pure Appl. Math., Volume 46 (1993) no. 11, pp. 1409-1498 | Zbl | DOI
[13] KAM for the nonlinear Schrödinger equation, Ann. Math. (2), Volume 172 (2010) no. 1, pp. 371-435 | Zbl | DOI
[14] Quasi-periodic solutions for fully nonlinear forced reversible Schrödinger equations, J. Differ. Equ., Volume 259 (2015) no. 7, pp. 3389-3447 | Zbl | DOI
[15] Invariant tori for 1D quintic nonlinear wave equation, J. Differ. Equ., Volume 263 ( Dec. 2017 ), pp. 8533-8564 | Zbl | DOI
[16] Almost periodic solutions of one dimensional Schrödinger equation with the external parameters, J. Dyn. Differ. Equ., Volume 25 (2013) no. 2, pp. 435-450 | Zbl | DOI
[17] KdV & KAM, Ergebnisse der Mathematik und Ihrer Grenzgebiete. 3. Folge, vol. 45, Springer-Verlag, Berlin, 2003 | Zbl
[18] Invariant Cantor manifolds of quasi-periodic oscillations for a nonlinear Schrödinger equation, Ann. Math. (2), Volume 143 (1996) no. 1, pp. 149-179 | Zbl | DOI
[19] Hamiltonian perturbations of infinite-dimensional linear systems with imaginary spectrum, Funkc. Anal. Prilozh., Volume 21 (1987) no. 3, pp. 22-37 (95) | Zbl
[20] Analysis of Hamiltonian PDEs, Oxford Lecture Series in Mathematics and Its Applications, vol. 19, Oxford University Press, Oxford, 2000 | Zbl
[21] 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] Almost periodic invariant tori for the nls on the circle, 2019 | arXiv | Zbl
[23] A KAM theorem for Hamiltonian partial differential equations with unbounded perturbations, Commun. Math. Phys., Volume 307 (2011) no. 3, pp. 629-673 | Zbl | DOI
[24] Quasi-periodic solutions for a nonlinear wave equation, Comment. Math. Helv., Volume 71 (1996) no. 2, pp. 269-296 | Zbl | DOI
[25] On the construction of almost periodic solutions for a nonlinear Schrödinger equation, Ergod. Theory Dyn. Syst., Volume 22 (2002) no. 5, pp. 1537-1549 | Zbl | DOI
[26] Energy supercritical nonlinear Schrödinger equations: quasiperiodic solutions, Duke Math. J., Volume 165 (2016) no. 6, pp. 1129-1192 | Zbl
[27] Periodic and quasi-periodic solutions of nonlinear wave equations via KAM theory, Commun. Math. Phys., Volume 127 (1990) no. 3, pp. 479-528 | Zbl | DOI
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