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Uspekhi Mat. Nauk, 2004, Volume 59, Issue 2(356), Pages 37–52 (Mi umn716)  

This article is cited in 21 scientific papers (total in 21 papers)

Recent progress on quasi-periodic lattice Schrödinger operators and Hamiltonian PDEs

J. Bourgain

Institute for Advanced Study, School of Mathematics

Abstract: This is a survey of recent investigations of quasi-periodic localization on lattices (of both methods based on perturbation theory and non-perturbative methods) and of applications of KAM theories in connection with infinite-dimensional Hamiltonian systems. The focus is on applications of these investigations to the Schrödinger equation and the wave equation with periodic boundary conditions, and to non-linear random Schrödinger equations with short-range potentials.


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English version:
Russian Mathematical Surveys, 2004, 59:2, 231–246

Bibliographic databases:

UDC: 517.984+517.958
MSC: Primary 35Q55; Secondary 35J10, 35L05, 37K99, 82B44, 37N20, 47B39
Received: 23.01.2004

Citation: J. Bourgain, “Recent progress on quasi-periodic lattice Schrödinger operators and Hamiltonian PDEs”, Uspekhi Mat. Nauk, 59:2(356) (2004), 37–52; Russian Math. Surveys, 59:2 (2004), 231–246

Citation in format AMSBIB
\by J.~Bourgain
\paper Recent progress on quasi-periodic lattice Schr\"odinger operators and Hamiltonian PDEs
\jour Uspekhi Mat. Nauk
\yr 2004
\vol 59
\issue 2(356)
\pages 37--52
\jour Russian Math. Surveys
\yr 2004
\vol 59
\issue 2
\pages 231--246

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    This publication is cited in the following articles:
    1. Yuan X., “Kamtheorem With Normal Frequencies of Finitelimit-Pointsfor Some Shallow Water Equations”, Commun. Pure Appl. Math.  crossref  isi
    2. Bourgain, J, “Anderson localization for quasi-periodic lattice Schrödinger operators on z(d), d arbitrary”, Geometric and Functional Analysis, 17:3 (2007), 682  crossref  mathscinet  zmath  isi
    3. Jing Zhang, Meina Gao, Xiaoping Yuan, “KAM tori for reversible partial differential equations”, Nonlinearity, 24:4 (2011), 1189  crossref  mathscinet  zmath  isi
    4. Jianjun Liu, Xiaoping Yuan, “A KAM Theorem for Hamiltonian Partial Differential Equations with Unbounded Perturbations”, Commun. Math. Phys, 2011  crossref  mathscinet  isi
    5. Helge Krüger, “Multiscale analysis for ergodic Schrödinger operators and positivity of Lyapunov exponents”, Jama, 115:1 (2011), 343  crossref  mathscinet  zmath  isi
    6. Lufang Mi, “Quasi-periodic solutions of derivative nonlinear Schrödinger equations with a given potential”, Journal of Mathematical Analysis and Applications, 2012  crossref  mathscinet  zmath  isi
    7. Jiansheng Geng, Jian Wu, “Real analytic quasi-periodic solutions for the derivative nonlinear Schrödinger equations”, J. Math. Phys, 53:10 (2012), 102702  crossref  mathscinet  zmath  isi
    8. Lufang Mi, Kangkang Zhang, “Invariant Tori for Benjamin-Ono Equation with Unbounded quasi-periodically forced Perturbation”, DCDS-A, 34:2 (2013), 689  crossref  mathscinet  isi
    9. Xiaoping Yuan, Kangkang Zhang, “A reduction theorem for time dependent Schrödinger operator with finite differentiable unbounded perturbation”, J. Math. Phys, 54:5 (2013), 052701  crossref  mathscinet  zmath  isi
    10. Berti M., Bolle Ph., “Quasi-Periodic Solutions with Sobolev Regularity of NLS on T-D with a Multiplicative Potential”, J. Eur. Math. Soc., 15:1 (2013), 229–286  crossref  mathscinet  zmath  isi
    11. XiaoPing Yuan, “KAM theorems and open problems for infinite-dimensional Hamiltonian with short range”, Sci. China Math, 2014  crossref  mathscinet  isi
    12. Mi L., Zhang K., “Quasi-periodic solutions for perturbed generalized KdV equation”, Nonlinear Anal.-Real World Appl., 32 (2016), 314–337  crossref  mathscinet  zmath  isi  scopus
    13. Cong H., Liu J., Yuan X., “Introduction and Main Results”, Mem. Am. Math. Soc., 239:1134 (2016), 1+  mathscinet  isi  elib
    14. Cong H., Liu J., Shi Yu., Yuan X., “The Stability of Full Dimensional Kam Tori For Nonlinear Schrodinger Equation”, J. Differ. Equ., 264:7 (2018), 4504–4563  crossref  mathscinet  zmath  isi
    15. Cui W., Mi L., Yin L., “Quasi-Periodic Solutions For Non-Autonomous Mkdv Equation”, Indian J. Pure Appl. Math., 49:2 (2018), 313–337  crossref  mathscinet  isi
    16. Cui W., Mi L., Zhang J., Yin L., “Invariant Tori For a Fifth Order Nonlinear Partial Differential Equation With Unbounded Perturbation”, Dyn. Partial Differ. Equ., 15:3 (2018), 183–199  crossref  zmath  isi
    17. Cui W., Mi L., Yin L., “Kam Tori For Defocusing Kdv-Mkdv Equation”, Acta Math. Sci., 39:1 (2019), 243–258  crossref  isi
    18. Cong H., Mi L., Shi Yu., Wu Yu., “on the Existence of Full Dimensional Kam Torus For Nonlinear Schrodinger Equation”, Discret. Contin. Dyn. Syst., 39:11 (2019), 6599–6630  crossref  isi
    19. Mi L., Cui W., “Invariant Tori For the Schrodinger Equation in the Heisenberg Ferromagnetic Chain”, Appl. Anal., 98:13 (2019), 2440–2453  crossref  isi
    20. Mi L. Cui W. You H., “Periodic and Quasi-Periodic Solutions For the Complex Swift-Hohenberg Equation”, J. Appl. Anal. Comput., 10:1 (2020), 297–313  crossref  isi
    21. Wu Yu., Yuan X., “On the Existence of Full Dimensional Kam Torus For Fractional Nonlinear Schrodinger Equation”, J. Appl. Anal. Comput., 10:2 (2020), 771–794  crossref  mathscinet  isi
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