We overview recent existence results and techniques about KAM theory for PDEs.
@article{JEDP_2011____A2_0, author = {Berti, Massimiliano}, title = {Quasi-periodic solutions of Hamiltonian PDEs}, journal = {Journ\'ees \'equations aux d\'eriv\'ees partielles}, year = {2011}, pages = {1-13}, doi = {10.5802/jedp.74}, language = {en}, url = {http://dml.mathdoc.fr/item/JEDP_2011____A2_0} }
Berti, Massimiliano. Quasi-periodic solutions of Hamiltonian PDEs. Journées équations aux dérivées partielles, (2011), pp. 1-13. doi : 10.5802/jedp.74. http://gdmltest.u-ga.fr/item/JEDP_2011____A2_0/
[1] Bambusi D., Berti M., Magistrelli E., Degenerate KAM theory for partial differential equations, J. Differential Equations 250, 3379-3397, 2011. | MR 2772395 | Zbl 1213.37103
[2] Bambusi D., Delort J.M., Grebért B., Szeftel J., Almost global existence for Hamiltonian semilinear Klein-Gordon equations with small Cauchy data on Zoll manifolds, Comm. Pure Appl. Math. 60, 11, 1665-1690, 2007. | MR 2349351 | Zbl 1170.35481
[3] Berti M., Nonlinear Oscillations of Hamiltonian PDEs, Progr. Nonlinear Differential Equations Appl. 74, H. Brézis, ed., Birkhäuser, Boston, 1-181, 2008. | MR 2345400 | Zbl 1146.35002
[4] Berti M., Biasco L., Branching of Cantor manifolds of elliptic tori and applications to PDEs, Comm. Math. Phys, 305, 3, 741-796, 2011. | MR 2819413 | Zbl pre05942049
[5] Berti M., Bolle P., Sobolev Periodic solutions of nonlinear wave equations in higher spatial dimension, Archive for Rational Mechanics and Analysis, 195, 609-642, 2010. | MR 2592290 | Zbl 1186.35113
[6] Berti M., Bolle P., Quasi-periodic solutions with Sobolev regularity of NLS on with a multiplicative potential, to appear on the Journal European Math. Society.
[7] Berti M., Bolle P., Quasi-periodic solutions of nonlinear Schrödinger equations on , Rend. Lincei Mat. Appl. 22, 223-236, 2011. | MR 2813578 | Zbl pre05935225
[8] Berti M., Bolle P., Procesi M., An abstract Nash-Moser theorem with parameters and applications to PDEs, Ann. I. H. Poincaré, 27, 377-399, 2010. | Numdam | MR 2580515 | Zbl 1203.47038
[9] Berti M., Procesi M., Nonlinear wave and Schrödinger equations on compact Lie groups and homogeneous spaces, Duke Math. J., 159, 3, 479-538, 2011. | MR 2831876
[10] Bourgain J., Construction of quasi-periodic solutions for Hamiltonian perturbations of linear equations and applications to nonlinear PDE, Internat. Math. Res. Notices, no. 11, 1994. | MR 1316975 | Zbl 0817.35102
[11] Bourgain J., Construction of periodic solutions of nonlinear wave equations in higher dimension, Geom. Funct. Anal. 5, no. 4, 629-639, 1995. | MR 1345016 | Zbl 0834.35083
[12] Bourgain J., On Melnikov’s persistency problem, Internat. Math. Res. Letters, 4, 445 - 458, 1997. | MR 1470416 | Zbl 0897.58020
[13] Bourgain J., Quasi-periodic solutions of Hamiltonian perturbations of linear Schrödinger equations, Annals of Math. 148, 363-439, 1998. | MR 1668547 | Zbl 0928.35161
[14] Bourgain J., Green’s function estimates for lattice Schrödinger operators and applications, Annals of Mathematics Studies 158, Princeton University Press, Princeton, 2005. | MR 2100420 | Zbl 1137.35001
[15] Burq N., Gérard P., Tzvetkov N., Bilinear eigenfunction estimates and the nonlinear Schrödinger equation on surfaces, Invent. Math., 159, 187-223, 2005. | MR 2142336 | Zbl 1092.35099
[16] Colliander J., Keel M., Staffilani G., Takaoka H., Tao T., Weakly turbolent solutions for the cubic defocusing nonlinear Schrödinger equation, 181, 1, 39-113, Inventiones Math., 2010. | MR 2651381 | Zbl 1197.35265
[17] Craig W., Wayne C. E., Newton’s method and periodic solutions of nonlinear wave equation, Comm. Pure Appl. Math. 46, 1409-1498, 1993. | MR 1239318 | Zbl 0794.35104
[18] Delort J.M., Periodic solutions of nonlinear Schrödinger equations: a para-differential approach, to appear in Analysis and PDEs.
[19] Eliasson L.H., Perturbations of stable invariant tori for Hamiltonian systems, Ann. Sc. Norm. Sup. Pisa., 15, 115-147, 1988. | Numdam | MR 1001032 | Zbl 0685.58024
[20] Eliasson L. H., Kuksin S., On reducibility of Schrödinger equations with quasiperiodic in time potentials, Comm. Math. Phys, 286, 125-135, 2009. | MR 2470926 | Zbl 1176.35141
[21] Eliasson L. H., Kuksin S., KAM for nonlinear Schrödinger equation, Annals of Math., 172, 371-435, 2010. | MR 2680422 | Zbl 1201.35177
[22] Kuksin S., Hamiltonian perturbations of infinite-dimensional linear systems with imaginary spectrum, Funktsional Anal. i Prilozhen. 2, 22-37, 95, 1987. | MR 911772 | Zbl 0631.34069
[23] Kuksin S., Analysis of Hamiltonian PDEs, Oxford Lecture series in Math. and its applications, 19, Oxford University Press, 2000. | MR 1857574 | Zbl 0960.35001
[24] Lojasiewicz S., Zehnder E., An inverse function theorem in Fréchet-spaces, J. Funct. Anal. 33, 165-174, 1979. | MR 546504 | Zbl 0431.46032
[25] Procesi C., Procesi M., A normal form for the Schrödinger equation with analytic non-linearities, to appear on Comm. Math. Phys. | MR 2727802
[26] Wang W. M., Supercritical nonlinear Schrödinger equations I: quasi-periodic solutions, preprint 2010.
[27] Wayne E., Periodic and quasi-periodic solutions of nonlinear wave equations via KAM theory, Comm. Math. Phys. 127, 479-528, 1990. | MR 1040892 | Zbl 0708.35087