no. 6
KAM for autonomous quasi-linear perturbations of KdV
Annales de l'I.H.P. Analyse non linéaire, Tome 33 (2016) no. 6, pp. 1589-1638.

We prove the existence and the stability of Cantor families of quasi-periodic, small amplitude solutions of quasi-linear (i.e. strongly nonlinear) autonomous Hamiltonian differentiable perturbations of KdV. This is the first result that extends KAM theory to quasi-linear autonomous and parameter independent PDEs. The core of the proof is to find an approximate inverse of the linearized operators at each approximate solution and to prove that it satisfies tame estimates in Sobolev spaces. A symplectic decoupling procedure reduces the problem to the one of inverting the linearized operator restricted to the normal directions. For this aim we use pseudo-differential operator techniques to transform such linear PDE into an equation with constant coefficients up to smoothing remainders. Then a linear KAM reducibility technique completely diagonalizes such operator. We introduce the “initial conditions” as parameters by performing a “weak” Birkhoff normal form analysis, which is well adapted for quasi-linear perturbations.

DOI : 10.1016/j.anihpc.2015.07.003
Classification : 37K55, 35Q53
Mots clés : KdV, KAM for PDEs, Quasi-linear PDEs, Nash–Moser theory, Quasi-periodic solutions 
@article{AIHPC_2016__33_6_1589_0,
     author = {Baldi, Pietro and Berti, Massimiliano and Montalto, Riccardo},
     title = {KAM for autonomous quasi-linear perturbations of {KdV}},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {1589--1638},
     publisher = {Elsevier},
     volume = {33},
     number = {6},
     year = {2016},
     doi = {10.1016/j.anihpc.2015.07.003},
     mrnumber = {3569244},
     zbl = {1370.37134},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1016/j.anihpc.2015.07.003/}
}
TY  - JOUR
AU  - Baldi, Pietro
AU  - Berti, Massimiliano
AU  - Montalto, Riccardo
TI  - KAM for autonomous quasi-linear perturbations of KdV
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2016
SP  - 1589
EP  - 1638
VL  - 33
IS  - 6
PB  - Elsevier
UR  - http://archive.numdam.org/articles/10.1016/j.anihpc.2015.07.003/
DO  - 10.1016/j.anihpc.2015.07.003
LA  - en
ID  - AIHPC_2016__33_6_1589_0
ER  - 
%0 Journal Article
%A Baldi, Pietro
%A Berti, Massimiliano
%A Montalto, Riccardo
%T KAM for autonomous quasi-linear perturbations of KdV
%J Annales de l'I.H.P. Analyse non linéaire
%D 2016
%P 1589-1638
%V 33
%N 6
%I Elsevier
%U http://archive.numdam.org/articles/10.1016/j.anihpc.2015.07.003/
%R 10.1016/j.anihpc.2015.07.003
%G en
%F AIHPC_2016__33_6_1589_0
Baldi, Pietro; Berti, Massimiliano; Montalto, Riccardo. KAM for autonomous quasi-linear perturbations of KdV. Annales de l'I.H.P. Analyse non linéaire, Tome 33 (2016) no. 6, pp. 1589-1638. doi : 10.1016/j.anihpc.2015.07.003. http://archive.numdam.org/articles/10.1016/j.anihpc.2015.07.003/

[1] Baldi, P. Periodic solutions of fully nonlinear autonomous equations of Benjamin–Ono type, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 30 (2013), pp. 33–77 | DOI | Numdam | MR | 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), pp. 471–536 | DOI | MR | Zbl

[3] Baldi, P.; Berti, M.; Montalto, R. KAM for quasi-linear KdV, C. R. Acad. Sci. Paris, Ser. I, Volume 352 (2014), pp. 603–607 | DOI | MR | Zbl

[4] Berti, M.; Biasco, P.; Procesi, M. KAM theory for the Hamiltonian DNLW, Ann. Sci. Éc. Norm. Supér. (4), Volume 46 (2013) no. 2, pp. 301–373 | MR | Zbl

[5] Berti, M.; Biasco, P.; Procesi, M. KAM theory for the reversible derivative wave equation, Arch. Ration. Mech. Anal., Volume 212 (2014), pp. 905–955 | DOI | MR | Zbl

[6] Berti, M.; Bolle, P. Quasi-periodic solutions with Sobolev regularity of NLS on Td with a multiplicative potential, Eur. J. Math., Volume 15 (2013), pp. 229–286 | MR | Zbl

[7] Berti, M.; Bolle, P.; Guyenne, P.; Nicholls, D.; Sulem, C. A Nash–Moser approach to KAM theory, Special Volume “Hamiltonian PDEs and Applications”, Fields Institute Communications, vol. 75, 2015 | DOI | MR

[8] M. Berti, P. Bolle, Quasi-periodic solutions for autonomous NLW on Td with a multiplicative potential, in preparation.

[9] Bourgain, J. Gibbs measures and quasi-periodic solutions for nonlinear Hamiltonian partial differential equations, Gelfand Math. Sem., Birkhäuser Boston, Boston, MA, 1996, pp. 23–43 | DOI | MR | Zbl

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

[11] Craig, W. Problèmes de petits diviseurs dans les équations aux dérivées partielles, Panoramas et Synthèses, vol. 9, Société Mathématique de France, Paris, 2000 | MR | Zbl

[12] Craig, W.; Wayne, C.E. Newton's method and periodic solutions of nonlinear wave equation, Commun. Pure Appl. Math., Volume 46 (1993), pp. 1409–1498 | DOI | MR | Zbl

[13] Eliasson, L.H.; Kuksin, S. KAM for non-linear Schrödinger equation, Ann. Math., Volume 172 (2010), pp. 371–435 | DOI | MR | Zbl

[14] 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), pp. 5361–5402 | DOI | MR | Zbl

[15] Guan, H.; Kuksin, S. The KdV equation under periodic boundary conditions and its perturbations, Nonlinearity, Volume 27 (2014) no. 9, pp. R61–R88 | DOI | MR | Zbl

[16] Iooss, G.; Plotnikov, P.I. Small divisor problem in the theory of three-dimensional water gravity waves, Mem. Am. Math. Soc., Volume 200 (2009) no. 940 | MR | Zbl

[17] Iooss, G.; Plotnikov, P.I. Asymmetrical three-dimensional travelling gravity waves, Arch. Ration. Mech. Anal., Volume 200 (2011) no. 3, pp. 789–880 | DOI | MR | Zbl

[18] Iooss, G.; Plotnikov, P.I.; Toland, J.F. Standing waves on an infinitely deep perfect fluid under gravity, Arch. Ration. Mech. Anal., Volume 177 (2005) no. 3, pp. 367–478 | DOI | MR | Zbl

[19] Lax, P. Development of singularities of solutions of nonlinear hyperbolic partial differential equations, J. Math. Phys., Volume 5 (1964), pp. 611–613 | MR | Zbl

[20] 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 | MR | Zbl

[21] Kappeler, T.; Pöschel, J. KAM and KdV, Springer, 2003 | DOI | MR | Zbl

[22] Klainerman, S.; Majda, A. Formation of singularities for wave equations including the nonlinear vibrating string, Commun. Pure Appl. Math., Volume 33 (1980), pp. 241–263 | DOI | MR | Zbl

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

[24] Kuksin, S. A KAM theorem for equations of the Korteweg–de Vries type, Rev. Math. Phys., Volume 10 (1998) no. 3, pp. 1–64 | MR | Zbl

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

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

[27] Pöschel, J. A KAM-theorem for some nonlinear PDEs, Ann. Sc. Norm. Pisa, Volume 23 (1996), pp. 119–148 | Numdam | MR | Zbl

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

[29] Procesi, M.; Procesi, C. A normal form for the Schrödinger equation with analytic non-linearities, Commun. Math. Phys., Volume 312 (2012), pp. 501–557 | DOI | MR | Zbl

[30] Procesi, C.; Procesi, M. A KAM algorithm for the completely resonant nonlinear Schrödinger equation, Adv. Math., Volume 272 (2015), pp. 399–470 | DOI | MR | Zbl

[31] Taylor, M.E. Pseudodifferential Operators and Nonlinear PDEs, Progress in Mathematics, Birkhäuser, 1991 | MR | Zbl

[32] W.M. Wang, Supercritical nonlinear Schrödinger equations I: quasi-periodic solutions, preprint. | MR | Zbl

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

[34] Zhang, J.; Gao, M.; Yuan, X. KAM tori for reversible partial differential equations, Nonlinearity, Volume 24 (2011), pp. 1189–1228 | DOI | MR | Zbl

[35] Zehnder, E. Generalized implicit function theorems with applications to some small divisors problems I, Commun. Pure Appl. Math., Volume 28 (1975), pp. 91–140 | MR | Zbl

[35] Zehnder, E. Generalized implicit function theorems with applications to some small divisors problems II, Commun. Pure Appl. Math., Volume 29 (1976), pp. 49–113 | DOI | MR | Zbl

Cité par Sources :