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Astérisque - Parutions - 403 (2018)

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Large KAM tori for perturbations of the defocusing NLS equation
Massimiliano Berti, Thomas Kappeler, Riccardo Montalto
Astérisque 403 (2018), viii+148 pages
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Résumé :
Tores KAM de grande taille pour des perturbations de l’équation de Schrödinger nonlinéaire défocalisante
Dans ce travail on démontre que toutes les perturbations hamiltoniennes de l'équation de Schrödinger nonlinéaire défocalisante (dNLS), qui sont semi-linéaires et suffisamment petites, admettent un grand nombre de tores invariants de taille et de dimension finie arbitrairement grande. Aucune condition de symétrie n'est supposée pour la perturbation et il n'est pas nécessaire qu'elle soit analytique. La difficulté principale est la présence des paires de fréquences de l'équation dNLS qui sont presque résonnantes. La preuve est basée sur l'intégrabilité de l'équation dNLS et en particulier sur le fait, que la partie nonlinéaire des coordonnées de Birkhoff est régularisante. On applique une procédure d'itération de type Newton-Nash-Moser pour construire les tores invariants. Les éléments clé du schéma de la procédure d'itération sont la réduction de certains opérateurs linéaires à des opérateurs, qui sont 2 2 bloc-diagonaux à coéfficients constants, et des estimations asymptotiques précises de leurs valeurs propres.

Abstract:
We prove that small, semi-linear Hamiltonian perturbations of the defocusing nonlinear Schrödinger (dNLS) equation on the circle have an abundance of invariant tori of any size and (finite) dimension which support quasi-periodic solutions. When compared with previous results the novelty consists in considering perturbations which do not satisfy any symmetry condition (they may depend on x in an arbitrary way) and need not be analytic. The main difficulty is posed by pairs of almost resonant dNLS frequencies. The proof is based on the integrability of the dNLS equation, in particular the fact that the nonlinear part of the Birkhoff coordinates is one smoothing. We implement a Newton-Nash-Moser iteration scheme to construct the invariant tori. The key point is the reduction of linearized operators, coming up in the iteration scheme, to 2 2 block diagonal ones with constant coefficients together with sharp asymptotic estimates of their eigenvalues.

Keywords: Defocusing NLS equation, KAM for PDE, Nash-Moser theory, invariant tori.

Class. math. : 37K55, 35Q55.


ISBN : 978-2-85629-892-3
ISSN : 0303-1179
Publié avec le concours de : Centre National de la Recherche Scientifique

Bibliographie:

1
Baldi, Pietro and Berti, Massimiliano and Montalto, Riccardo
KAM for quasi-linear and fully nonlinear forced perturbations of Airy equation
Math. Ann. 359 (2014) 471–536
Math Reviews MR3201904
2
Baldi, Pietro and Berti, Massimiliano and Montalto, Riccardo
KAM for autonomous quasi-linear perturbations of KdV
Ann. Inst. H. Poincaré Anal. Non Linéaire 33 (2016) 1589–1638
Math Reviews MR3569244
3
Baldi, Pietro and Berti, Massimiliano and Montalto, Riccardo
KAM for autonomous quasi-linear perturbations of mKdV
Boll. Unione Mat. Ital. 9 (2016) 143–188
Math Reviews MR3502158
4
Baldi, Pietro and Haus, Emanuele
A Nash-Moser-Hörmander implicit function theorem with applications to control and Cauchy problems for PDEs
J. Funct. Anal. 273 (2017) 3875–3900
Math Reviews MR3711883
5
Berti, Massimiliano
Nonlinear oscillations of Hamiltonian PDEs
Progress in Nonlinear Differential Equations and their Applications, vol. 74, Birkhauser Boston, Inc., Boston, MA, 2007
Math Reviews MR2345400
6
Berti, Massimiliano and Bolle, Philippe
Cantor families of periodic solutions of wave equations with C^k nonlinearities
NoDEA Nonlinear Differential Equations Appl. 15 (2008) 247–276
Math Reviews MR2408354
7
Berti, Massimiliano and Bolle, Philippe
Quasi-periodic solutions with Sobolev regularity of NLS on T^d with a multiplicative potential
J. Eur. Math. Soc. (JEMS) 15 (2013) 229–286
Math Reviews MR2998835
8
Berti, Massimiliano and Bolle, Philippe
A Nash-Moser approach to KAM theory
in Hamiltonian partial differential equations and applications
Fields Inst. Commun. 75 (2015) 255–284
Math Reviews MR3445505
9
Berti, Massimiliano and Bolle, Philippe and Procesi, Michela
An abstract Nash-Moser theorem with parameters and applications to PDEs
Ann. Inst. H. Poincaré Anal. Non Linéaire 27 (2010) 377–399
Math Reviews MR2580515
10
Biasco, Luca and Coglitore, Federico
Periodic orbits accumulating onto elliptic tori for the (N+1)-body problem
Celestial Mech. Dynam. Astronom. 101 (2008) 349–373
Math Reviews MR2430357
11
Bourgain, Jean
Construction of quasi-periodic solutions for Hamiltonian perturbations of linear equations and applications to nonlinear PDE
Int. Math. Res. Not. 1994 (1994) 475–496
Math Reviews MR1316975
12
Bourgain, Jean
Quasi-periodic solutions of Hamiltonian perturbations of 2D linear Schrödinger equations
Ann. of Math. 148 (1998) 363–439
Math Reviews MR1668547
13
Bourgain, Jean
Green's function estimates for lattice Schrödinger operators and applications
Annals of Math. Studies, vol. 158, Princeton Univ. Press, 2005
Math Reviews MR2100420
14
Chierchia, Luigi and You, Jiangong
KAM tori for 1D nonlinear wave equations with periodic boundary conditions
Comm. Math. Phys. 211 (2000) 497–525
Math Reviews MR1754527
15
Craig, Walter and Wayne, C. Eugene
Periodic solutions of nonlinear Schrödinger equations and the Nash-Moser method
in Hamiltonian mechanics (Toruń, 1993)
NATO Adv. Sci. Inst. Ser. B Phys. 331 (1994) 103–122
Math Reviews MR1316671
16
Eliasson, L. Hakan and Kuksin, Sergei B.
KAM for the nonlinear Schrödinger equation
Ann. of Math. 172 (2010) 371–435
Math Reviews MR2680422
17
Feola, Roberto and Procesi, Michela
Quasi-periodic solutions for fully nonlinear forced reversible Schrödinger equations
J. Differential Equations 259 (2015) 3389–3447
Math Reviews MR3360677
18
Geng, Jiansheng and Xu, Xindong and You, Jiangong
An infinite dimensional KAM theorem and its application to the two dimensional cubic Schrödinger equation
Adv. Math. 226 (2011) 5361–5402
Math Reviews MR2775905
19
Geng, Jiansheng and Yi, Yingfei
Quasi-periodic solutions in a nonlinear Schrödinger equation
J. Differential Equations 233 (2007) 512–542
Math Reviews MR2292517
20
Geng, Jiansheng and You, Jiangong
A KAM theorem for one dimensional Schrödinger equation with periodic boundary conditions
J. Differential Equations 209 (2005) 1–56
Math Reviews MR2107467
21
Geng, Jiansheng and You, Jiangong
A KAM theorem for Hamiltonian partial differential equations in higher dimensional spaces
Comm. Math. Phys. 262 (2006) 343–372
Math Reviews MR2200264
22
Grébert, Benoît and Kappeler, Thomas
Perturbations of the defocusing nonlinear Schrödinger equation
Milan J. Math. 71 (2003) 141–174
Math Reviews MR2120919
23
Grébert, Benoît and Kappeler, Thomas
The defocusing NLS equation and its normal form
EMS Series of Lectures in Mathematics, European Mathematical Society, 2014
Math Reviews MR3203027
24
Hamilton, Richard S.
The inverse function theorem of Nash and Moser
Bull. Amer. Math. Soc. (N.S.) 7 (1982) 65–222
Math Reviews MR656198
25
Hörmander, Lars
The boundary problems of physical geodesy
Arch. Rational Mech. Anal. 62 (1976) 1–52
Math Reviews MR0602181
26
Iooss, G. and Plotnikov, P. I. and Toland, J. F.
Standing waves on an infinitely deep perfect fluid under gravity
Arch. Ration. Mech. Anal. 177 (2005) 367–478
Math Reviews MR2187619
27
Kappeler, Thomas and Liang, Zhenguo
A KAM theorem for the defocusing NLS equation
J. Differential Equations 252 (2012) 4068–4113
Math Reviews MR2875612
28
Kappeler, Thomas and Pöschel, Jürgen
KdV KAM
Ergebn. Math. Grenzg., vol. 45, Springer, 2003
Math Reviews MR1997070
29
Kappeler, Thomas and Schaad, Beat and Topalov, Peter
Semi-linearity of the non-linear Fourier transform of the defocusing NLS equation
Int. Math. Res. Not. 2016 (2016) 7212–7229
Math Reviews MR3632080
30
Kappeler, Thomas and Schaad, Beat and Topalov, Peter
Scattering-like phenomena of the periodic defocusing NLS equation
Math. Res. Lett. 24 (2017) 803–826
Math Reviews MR3696604
31
Kuksin, Sergei B.
Analysis of Hamiltonian PDEs
Oxford Lecture Series in Mathematics and its Applications, vol. 19, Oxford Univ. Press, 2000
Math Reviews MR1857574
32
Kuksin, Sergej and Pöschel, Jürgen
Invariant Cantor manifolds of quasi-periodic oscillations for a nonlinear Schrödinger equation
Ann. of Math. 143 (1996) 149–179
Math Reviews MR1370761
33
Liang, Zhenguo and You, Jiangong
Quasi-periodic solutions for 1D Schrödinger equations with higher order nonlinearity
SIAM J. Math. Anal. 36 (2005) 1965–1990
Math Reviews MR2178228
34
Moser, Jürgen
A new technique for the construction of solutions of nonlinear differential equations
Proc. Nat. Acad. Sci. U.S.A. 47 (1961) 1824–1831
Math Reviews MR0132859
35
Moser, Jürgen
A rapidly convergent iteration method and non-linear partial differential equations. I
Ann. Scuola Norm. Sup. Pisa 20 (1966) 265–315
Math Reviews MR0199523
36
Nirenberg, Louis
Topics in nonlinear functional analysis
Courant Lecture Notes in Math., vol. 6, New York University, 2001
Math Reviews MR1850453
37
Pöschel, Jürgen
Integrability of Hamiltonian systems on Cantor sets
Comm. Pure Appl. Math. 35 (1982) 653–696
Math Reviews MR668410
38
Pöschel, Jürgen
A KAM-theorem for some nonlinear partial differential equations
Ann. Scuola Norm. Sup. Pisa Cl. Sci. 23 (1996) 119–148
Math Reviews MR1401420
39
Procesi, C. and Procesi, Michela
A KAM algorithm for the resonant non-linear Schrödinger equation
Adv. Math. 272 (2015) 399–470
Math Reviews MR3303238
40
Salamon, Dietmar and Zehnder, Eduard
KAM theory in configuration space
Comment. Math. Helv. 64 (1989) 84–132
Math Reviews MR982563
41
Wang, W.-M.
Energy supercritical nonlinear Schrödinger equations: quasiperiodic solutions
Duke Math. J. 165 (2016) 1129–1192
Math Reviews MR3486416
42
Zehnder, Eduard
Generalized implicit function theorems with applications to some small divisor problems. I
Comm. Pure Appl. Math. 28 (1975) 91–140
Math Reviews MR0380867