Geodesic
A KAM theorem for space-multidimensional Hamiltonian PDEs
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Modern problems of mechanics, Tome 295 (2016), pp. 142-162.

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We present an abstract KAM theorem adapted to space-multidimensional Hamiltonian PDEs with smoothing nonlinearities. The main novelties of this theorem are the following: (i) the integrable part of the Hamiltonian may contain a hyperbolic part and, as a consequence, the constructed invariant tori may be unstable; (ii) it applies to singular perturbation problems. In this paper we state the KAM theorem and comment on it, give the main ingredients of the proof, and present three applications of the theorem. 
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L. H. Eliasson; B. Grébert; S. B. Kuksin. A KAM theorem for space-multidimensional Hamiltonian PDEs. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Modern problems of mechanics, Tome 295 (2016), pp. 142-162. http://geodesic.mathdoc.fr/item/TM_2016_295_a6/

[1] Berti M., Bolle P., “Sobolev quasi periodic solutions of multidimensional wave equations with a multiplicative potential”, Nonlinearity, 25:9 (2012), 2579–2613 | DOI | MR | Zbl

[2] Berti M., Bolle P., “Quasi-periodic solutions with Sobolev regularity of NLS on $\mathbb T^d$ with a multiplicative potential”, J. Eur. Math. Soc., 15:1 (2013), 229–286 | DOI | MR | Zbl

[3] 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., 334:3 (2015), 1413–1454 | DOI | MR | Zbl

[4] Bourgain J., “Quasi-periodic solutions of Hamiltonian perturbations of 2D linear Schrödinger equations”, Ann. Math. Ser. 2, 148:2 (1998), 363–439 | DOI | MR | Zbl

[5] Bourgain J., Green's function estimates for lattice Schrödinger operators and applications, Ann. Math. Stud., 158, Princeton Univ. Press, Princeton, NJ, 2005 | MR | Zbl

[6] Eliasson L. H., “Perturbations of stable invariant tori for Hamiltonian systems”, Ann. Sc. Norm. Super. Pisa. Cl. Sci. Ser. 4, 15:1 (1988), 115–147 | MR | Zbl

[7] Eliasson L. H., Grébert B., Kuksin S. B., KAM for the non-linear beam equation 2: A normal form theorem, E-print, 2015, arXiv: 1502.02262[math.AP]

[8] Eliasson L. H., Grébert B., Kuksin S. B., KAM for the nonlinear beam equation, E-print, 2016, arXiv: 1604.01657[math.AP] | MR

[9] Eliasson L. H., Kuksin S. B., “On reducibility of Schrödinger equations with quasiperiodic in time potentials”, Commun. Math. Phys., 286:1 (2009), 125–135 | DOI | MR | Zbl

[10] Eliasson L. H., Kuksin S. B., “KAM for the nonlinear Schrödinger equation”, Ann. Math. Ser. 2, 172:1 (2010), 371–435 | DOI | MR | Zbl

[11] Geng J., You J., “A KAM theorem for Hamiltonian partial differential equations in higher dimensional spaces”, Commun. Math. Phys., 262:2 (2006), 343–372 | DOI | MR | Zbl

[12] Geng J., You J., “KAM tori for higher dimensional beam equations with constant potentials”, Nonlinearity, 19:10 (2006), 2405–2423 | DOI | MR | Zbl

[13] Grébert B., Paturel E., “KAM for the Klein–Gordon equation on $\mathbb S^d$”, Boll. Unione Mat. Ital. Ser. 9, 9:2 (2016), 237–288 | DOI | MR | Zbl

[14] Grébert B., Paturel E., On reducibility of quantum harmonic oscillator on $\mathbb R^d$ with quasiperiodic in time potential, E-print, 2016, arXiv: 1603.07455[math.AP]

[15] Grébert B., Thomann L., “KAM for the quantum harmonic oscillator”, Commun. Math. Phys., 307:2 (2011), 383–427 | DOI | MR | Zbl

[16] Kuksin S. B., “Gamiltonovy vozmuscheniya beskonechnomernykh lineinykh sistem s mnimym spektrom”, Funkts. analiz i ego pril., 21:3 (1987), 22–37 | MR | Zbl

[17] Kuksin S. B., Nearly integrable infinite-dimensional Hamiltonian systems, Lect. Notes Math., 1556, Springer, Berlin, 1993 | DOI | MR | Zbl

[18] Kuksin S. B., Analysis of Hamiltonian PDEs, Oxford Lect. Ser. Math. Appl., 19, Oxford Univ. Press, Oxford, 2000 | MR | Zbl

[19] Kuksin S., Pöschel J., “Invariant Cantor manifolds of quasi-periodic oscillations for a nonlinear Schrödinger equation”, Ann. Math. Ser. 2, 143:1 (1996), 149–179 | DOI | MR | Zbl

[20] Pöschel J., “A KAM-theorem for some nonlinear partial differential equations”, Ann. Sc. Norm. Super. Pisa. Cl. Sci. Ser. 4, 23:1 (1996), 119–148 | MR | Zbl

[21] Procesi C., Procesi M., “A KAM algorithm for the resonant non-linear Schrödinger equation”, Adv. Math., 272 (2015), 399–470 | DOI | MR | Zbl

[22] Procesi M., Procesi C., “A normal form for the Schrödinger equation with analytic non-linearities”, Commun. Math. Phys., 312:2 (2012), 501–557 | DOI | MR | Zbl

[23] Wang W.-M., “Energy supercritical nonlinear Schrödinger equations: Quasiperiodic solutions”, Duke Math J., 165:6 (2016), 1129–1192 | MR | Zbl

[24] You J., “Perturbations of lower dimensional tori for Hamiltonian systems”, J. Diff. Eqns., 152:1 (1999), 1–29 | DOI | MR | Zbl