Almost periodic invariant tori for the NLS on the circle
Biasco, Luca1 ; Massetti, Jessica Elisa1 ; Procesi, Michela1
1 Università degli Studi Roma Tre, Italy
Annales de l'I.H.P. Analyse non linéaire, mai – juin 2021, Tome 38 (2021) no. 3, pp. 711-758.
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 study the existence and linear stability of almost periodic solutions for a NLS equation on the circle with external parameters. Starting from the seminal result of Bourgain in [15] on the quintic NLS, we propose a novel approach allowing to prove in a unified framework the persistence of finite and infinite dimensional invariant tori, which are the support of the desired solutions. The persistence result is given through a rather abstract “counter-term theorem” à la Herman, directly in the original elliptic variables without passing to action-angle ones. Our framework allows us to find “many more” almost periodic solutions with respect to the existing literature and consider also non-translation invariant PDEs.

Reçu le :
Révisé le :
Accepté le :
DOI : 10.1016/j.anihpc.2020.09.003
Mots-clés : Almost periodic solutions, Nonlinear Schrodinger equation, KAM for PDEs
Biasco, Luca 1 ; Massetti, Jessica Elisa 1 ; Procesi, Michela 1

1 Università degli Studi Roma Tre, Italy 
@article{AIHPC_2021__38_3_711_0,
     author = {Biasco, Luca and Massetti, Jessica Elisa and Procesi, Michela},
     title = {Almost periodic invariant tori for the {NLS} on the circle},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {711--758},
     publisher = {Elsevier},
     volume = {38},
     number = {3},
     year = {2021},
     doi = {10.1016/j.anihpc.2020.09.003},
     mrnumber = {4227050},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1016/j.anihpc.2020.09.003/}
}
TY  - JOUR
AU  - Biasco, Luca
AU  - Massetti, Jessica Elisa
AU  - Procesi, Michela
TI  - Almost periodic invariant tori for the NLS on the circle
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2021
SP  - 711
EP  - 758
VL  - 38
IS  - 3
PB  - Elsevier
UR  - http://archive.numdam.org/articles/10.1016/j.anihpc.2020.09.003/
DO  - 10.1016/j.anihpc.2020.09.003
LA  - en
ID  - AIHPC_2021__38_3_711_0
ER  - 
%0 Journal Article
%A Biasco, Luca
%A Massetti, Jessica Elisa
%A Procesi, Michela
%T Almost periodic invariant tori for the NLS on the circle
%J Annales de l'I.H.P. Analyse non linéaire
%D 2021
%P 711-758
%V 38
%N 3
%I Elsevier
%U http://archive.numdam.org/articles/10.1016/j.anihpc.2020.09.003/
%R 10.1016/j.anihpc.2020.09.003
%G en
%F AIHPC_2021__38_3_711_0
Biasco, Luca; Massetti, Jessica Elisa; Procesi, Michela. Almost periodic invariant tori for the NLS on the circle. Annales de l'I.H.P. Analyse non linéaire, mai – juin 2021, Tome 38 (2021) no. 3, pp. 711-758. doi : 10.1016/j.anihpc.2020.09.003. http://archive.numdam.org/articles/10.1016/j.anihpc.2020.09.003/

[1] Berti, M.; Biasco, L. Branching of Cantor manifolds of elliptic tori and applications to PDEs, Commun. Math. Phys., Volume 305 (2011) no. 3, pp. 741-796 | DOI | MR | Zbl

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

[3] Berti, M.; Bolle, Ph. 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 | MR | Zbl

[4] Berti, M.; Bolle, Ph. A Nash-Moser approach to KAM theory, Hamiltonian Partial Differential Equations and Applications, Fields Inst. Commun., vol. 75, Fields Inst. Res. Math. Sci., Toronto, ON, 2015, pp. 255-284 | DOI | MR

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

[6] Baldi, Pietro; Berti, M.; Montalto, Riccardo 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 | MR

[7] Berti, M.; Corsi, L.; Procesi, M. 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) no. 3, pp. 1413-1454 | DOI | MR | Zbl

[8] Berti, M.; Delort, J.-M. Almost Global Solutions of Capillary-Gravity Water Waves Equations on the Circle, Lecture Notes of the Unione Matematica Italiana., vol. 24, Springer, Cham, 2018 (Unione Matematica Italiana, Bologna) | MR

[9] Bambusi, D.; Grébert, B. Birkhoff normal form for partial differential equations with tame modulus, Duke Math. J., Volume 135 (2006) no. 3, pp. 507-567 | DOI | MR | Zbl

[10] Broer, H.W.; Huitema, G.B.; Takens, F.; Braaksma, B.L.J. Unfoldings and bifurcations of quasi-periodic tori, Mem. Am. Math. Soc., Volume 83 (1990) no. 421 (viii+175) | MR | Zbl

[11] Berti, M.; Montalto, R.; Kappeler, T. Large KAM tori for quasi-linear perturbations of KdV (preprint) | arXiv | DOI | MR

[12] Biasco, L.; Massetti, J.E.; Procesi, M. An Abstract Brkhoff Normal Form Theorem and exponential type stability of the 1d NLS, Commun. Math. Phys., Volume 375 (2020), pp. 2089-2153 | DOI | MR

[13] Biasco, L.; Massetti, J.E.; Procesi, M. On the construction of Sobolev almost periodic invariant tori for the 1d NLS, 2020 (preprint) | arXiv | MR

[14] Bourgain, J. Construction of quasi-periodic solutions for Hamiltonian perturbations of linear equations and applications to nonlinear PDE, Int. Math. Res. Not., Volume 1994 (1994) no. 11, pp. 475-497 | DOI | MR | Zbl

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

[16] Corsi, L.; Feola, R.; Procesi, M. Finite dimensional invariant KAM tori for tame vector fields, Trans. Am. Math. Soc., Volume 372 (2019) no. 3, pp. 1913-1983 | DOI | MR

[17] Corsi, L.; Gentile, G.; Procesi, M. KAM theory in configuration space and cancellations in the Lindstedt series, Commun. Math. Phys., Volume 302 (2011) no. 2, pp. 359-402 | DOI | MR | Zbl

[18] Chenciner, A. Bifurcations de points fixes elliptiques. I. Courbes invariantes, Publ. Math. IHÉS, Volume 61 (1985), pp. 67-127 | DOI | Numdam | MR | Zbl

[19] 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 | MR

[20] Corsi, L.; Montalto, R.; Procesi, M. Almost-periodic Response Solutions for a forced quasi-linear Airy equation (preprint) | arXiv | DOI | MR

[21] 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 | MR | Zbl

[22] de la Llave, R.; Gonzalez, A.; Jorba, A.; Villanueva, J. KAM theory without action-angles, Nonlinearity, Volume 18 (2005) no. 2, pp. 855-895 | DOI | MR | Zbl

[23] Eliasson, L.H.; Fayad, B.; Krikorian, R. KAM-tori near an analytic elliptic fixed point, Regul. Chaotic Dyn., Volume 18 (2013) no. 6, pp. 801-831 | DOI | MR | Zbl

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

[25] Féjoz, J. Démonstration du “théorème d'Arnold” sur la stabilité du système planétaire (d'après Michael Herman), Michael Herman Memorial Issue, Ergod. Theory Dyn. Syst., Volume 24 (2004) no. 5, pp. 1521-1582 | DOI | MR | Zbl

[26] Faou, E.; Grébert, B. A Nekhoroshev-type theorem for the nonlinear Schrödinger equation on the torus, Anal. PDE, Volume 6 (2013) no. 6, pp. 1243-1262 | DOI | MR | Zbl

[27] Feola, R.; Iandoli, F. Local well-posedness for quasi-linear NLS with large Cauchy data on the circle, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 36 (2019) no. 1, pp. 119-164 | DOI | Numdam | MR

[28] Feola, R.; Iandoli, F. Long time existence for fully nonlinear NLS with small Cauchy data on the circle, Ann. Sc. Norm. Super. Pisa (2019) | DOI | MR

[29] Fayad, B.; Krikorian, R. Herman's last geometric theorem, Ann. Sci. Éc. Norm. Supér. (4), Volume 42 (2009) no. 2, pp. 193-219 | DOI | Numdam | MR | Zbl

[30] 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 | MR | Zbl

[31] Geng, J.; Xu, X.; You, J. An infinite dimensional KAM theorem and its application to the two dimensional cubic Schrödinger equation, Adv. Math., Volume 226 (2011) no. 6, pp. 5361-5402 | DOI | MR | Zbl

[32] Herman, M.-R. Sur les courbes invariantes par les difféomorphismes de l'anneau. Vol. 1, Astérisque, vol. 103, Société Mathématique de France, Paris, 1983 (with an appendix by Albert Fathi, with an English summary) | Numdam | MR | Zbl

[33] Herman, M.; Sergeraert, F. Sur un théorème d'Arnold et Kolmogorov, C. R. Acad. Sci. Paris Sér. A-B, Volume 273 (1971), p. A409-A411 | MR | Zbl

[34] 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 | MR | Zbl

[35] Li, X.; Liu, S. The relation between the size of perturbations and the dimension of tori in an infinite-dimensional KAM theorem of Pöschel, Nonlinear Anal., Volume 197 (2020) | MR

[36] Massetti, J.E. A normal form à la Moser for diffeomorphisms and a generalization of Rüssmann's translated curve theorem to higher dimensions, Anal. PDE, Volume 11 (2018) no. 1, pp. 149-170 | DOI | MR

[37] Massetti, J.E. Normal forms for perturbations of systems possessing a Diophantine invariant torus, Ergod. Theory Dyn. Syst., Volume 39 (2019) no. 8, pp. 2176-2222 | DOI | MR

[38] Moser, J. Convergent series expansions for quasi-periodic motions, Math. Ann., Volume 169 (1967), pp. 136-176 | DOI | MR | Zbl

[39] Montalto, R.; Procesi, M. Linear Schrödinger equation with an almost periodic potential (preprint) | arXiv | DOI | MR

[40] Maspero, A.; Procesi, M. Long time stability of small finite gap solutions of the cubic nonlinear Schrödinger equation on T2 , J. Differ. Equ., Volume 265 (2018) no. 7, pp. 3212-3309 | DOI | MR

[41] Pöschel, J. On elliptic lower-dimensional tori in Hamiltonian systems, Math. Z., Volume 202 (1989) no. 4, pp. 559-608 | DOI | MR | Zbl

[42] Pöschel, J. A KAM-theorem for some nonlinear partial differential equations, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (4), Volume 23 (1996) no. 1, pp. 119-148 | Numdam | MR | Zbl

[43] Pöschel, J. A lecture on the classical KAM theorem, Seattle, WA, 1999 (Proc. Sympos. Pure Math.), Volume vol. 69, Amer. Math. Soc., Providence, RI (2001), pp. 707-732 | DOI | MR | Zbl

[44] 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 | MR | Zbl

[45] Rüssmann, H. On the existence of invariant curves of twist mappings of an annulus, Rio de Janeiro, 1981 (Lecture Notes in Math.), Volume vol. 1007, Springer, Berlin (1983), pp. 677-718 | DOI | MR | Zbl

[46] Stolovitch, L. Singular complete integrability, Publ. Math. IHÉS, Volume 91 (2001), pp. 133-210 | DOI | MR | Zbl

[47] 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 | MR | Zbl

Cité par Sources :