KAM theory for the hamiltonian derivative wave equation

Berti, Massimiliano; Biasco, Luca; Procesi, Michela

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 46 (2013) no. 2, pp. 301-373.

Nous prouvons un théorème KAM en dimension infinie, qui implique l'existence de familles de Cantor de tores invariants de petite amplitude, réductibles, elliptiques et analytiques, pour les équations des ondes hamiltoniennes avec dérivées.

We prove an infinite dimensional KAM theorem which implies the existence of Cantor families of small-amplitude, reducible, elliptic, analytic, invariant tori of Hamiltonian derivative wave equations.

Classification : 37K55, 35L05
Keywords: infinite dimensional KAM theorem, Cantor families, hamiltonian derivative wave equations
Mot clés : théorème KAM en dimension infinie, familles de Cantor, équation des ondes hamiltonienne avec dérivées

[1] D. Bambusi & B. Grébert, Birkhoff normal form for partial differential equations with tame modulus, Duke Math. J. 135 (2006), 507-567. | MR

[2] M. Berti & L. Biasco, Branching of Cantor manifolds of elliptic tori and applications to PDEs, Comm. Math. Phys. 305 (2011), 741-796. | MR

[3] M. Berti & P. Bolle, Sobolev quasi-periodic solutions of multidimensional wave equations with a multiplicative potential, Nonlinearity 25 (2012), 2579-2613. | MR

[4] M. Berti & P. Bolle, Quasi-periodic solutions with Sobolev regularity of NLS on $T^d$ with a multiplicative potential, Eur. Jour. Math. 15 (2013), 229-286. | MR

[5] J. Bourgain, Construction of quasi-periodic solutions for Hamiltonian perturbations of linear equations and applications to nonlinear PDE, Int. Math. Res. Not. 1994 (1994). | MR

[6] J. Bourgain, Quasi-periodic solutions of Hamiltonian perturbations of 2D linear Schrödinger equations, Ann. of Math. 148 (1998), 363-439. | MR

[7] J. Bourgain, Periodic solutions of nonlinear wave equations, in Harmonic analysis and partial differential equations (Chicago, IL, 1996), Chicago Lectures in Math., Univ. Chicago Press, 1999, 69-97. | MR

[8] J. Bourgain, Green's function estimates for lattice Schrödinger operators and applications, Annals of Math. Studies 158, Princeton Univ. Press, 2005. | MR

[9] L. Chierchia & J. You, KAM tori for 1D nonlinear wave equations with periodic boundary conditions, Comm. Math. Phys. 211 (2000), 497-525. | MR

[10] W. Craig, Problèmes de petits diviseurs dans les équations aux dérivées partielles, Panoramas et Synthèses 9, Soc. Math. France, 2000. | MR

[11] W. Craig & C. E. Wayne, Newton's method and periodic solutions of nonlinear wave equations, Comm. Pure Appl. Math. 46 (1993), 1409-1498. | MR

[12] J.-M. Delort, A quasi-linear Birkhoff normal forms method. Application to the quasi-linear Klein-Gordon equation on ${𝕊}^1$, Astérique 341 (2012). | Numdam | MR

[13] J.-M. Delort & J. Szeftel, Long-time existence for small data nonlinear Klein-Gordon equations on tori and spheres, Int. Math. Res. Not. 2004 (2004), 1897-1966. | MR

[14] L. H. Eliasson & S. B. Kuksin, KAM for the nonlinear Schrödinger equation, Ann. of Math. 172 (2010), 371-435. | MR

[15] J. Geng, X. Xu & J. You, An infinite dimensional KAM theorem and its application to the two dimensional cubic Schrödinger equation, Adv. Math. 226 (2011), 5361-5402. | MR

[16] J. Geng & J. You, A KAM theorem for Hamiltonian partial differential equations in higher dimensional spaces, Comm. Math. Phys. 262 (2006), 343-372. | MR

[17] P. Gérard & S. Grellier, Effective integrable dynamics for some nonlinear wave equation, preprint http://hal.archives-ouvertes.fr/hal-00635686/fr/.

[18] B. Grébert & L. Thomann, KAM for the quantum harmonic oscillator, Comm. Math. Phys. 307 (2011), 383-427. | MR

[19] T. Kappeler & J. Pöschel, KAM and KdV, Springer, 2003.

[20] S. B. Kuksin, Hamiltonian perturbations of infinite-dimensional linear systems with imaginary spectrum, Funktsional. Anal. i Prilozhen. 21 (1987), 22-37, 95. | MR

[21] S. B. Kuksin, A KAM-theorem for equations of the Korteweg-de Vries type, Rev. Math. Math. Phys. 10 (1998), 1-64. | MR

[22] S. B. Kuksin, Analysis of Hamiltonian PDEs, Oxford Lecture Series in Mathematics and its Applications 19, Oxford Univ. Press, 2000. | MR

[23] S. Kuksin & J. Pöschel, Invariant Cantor manifolds of quasi-periodic oscillations for a nonlinear Schrödinger equation, Ann. of Math. 143 (1996), 149-179. | MR

[24] J. Liu & X. Yuan, A KAM theorem for Hamiltonian partial differential equations with unbounded perturbations, Comm. Math. Phys. 307 (2011), 629-673. | MR

[25] N. V. Nikolenko, The method of Poincaré normal forms in problems of integrability of equations of evolution type, Uspekhi Mat. Nauk 41 (1986), 109-152, 263. | MR

[26] J. Pöschel, On elliptic lower-dimensional tori in Hamiltonian systems, Math. Z. 202 (1989), 559-608. | MR

[27] J. Pöschel, A KAM-theorem for some nonlinear partial differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 23 (1996), 119-148. | Numdam | MR

[28] J. Pöschel, Quasi-periodic solutions for a nonlinear wave equation, Comment. Math. Helv. 71 (1996), 269-296. | MR

[29] J. Pöschel & E. Trubowitz, Inverse spectral theory, Pure and Applied Mathematics 130, Academic Press Inc., 1987. | MR

[30] M. Procesi & X. Xu, Quasi-Töplitz functions in KAM theorem, to appear in SIAM Journal Math. Anal.

[31] W. M. Wang, Supercritical nonlinear Schrödinger equations I: quasi-periodic solutions, preprint arXiv:1007.0156.

[32] C. E. Wayne, Periodic and quasi-periodic solutions of nonlinear wave equations via KAM theory, Comm. Math. Phys. 127 (1990), 479-528. | MR