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

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.

DOI: https://doi.org/10.4213/rm716

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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

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
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• http://mi.mathnet.ru/eng/umn716
• https://doi.org/10.4213/rm716
• http://mi.mathnet.ru/eng/umn/v59/i2/p37

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Bourgain, J, “Anderson localization for quasi-periodic lattice Schrödinger operators on z(d), d arbitrary”, Geometric and Functional Analysis, 17:3 (2007), 682
2. Jing Zhang, Meina Gao, Xiaoping Yuan, “KAM tori for reversible partial differential equations”, Nonlinearity, 24:4 (2011), 1189
3. Jianjun Liu, Xiaoping Yuan, “A KAM Theorem for Hamiltonian Partial Differential Equations with Unbounded Perturbations”, Commun. Math. Phys, 2011
4. Helge Krüger, “Multiscale analysis for ergodic Schrödinger operators and positivity of Lyapunov exponents”, Jama, 115:1 (2011), 343
5. Lufang Mi, “Quasi-periodic solutions of derivative nonlinear Schrödinger equations with a given potential”, Journal of Mathematical Analysis and Applications, 2012
6. Jiansheng Geng, Jian Wu, “Real analytic quasi-periodic solutions for the derivative nonlinear Schrödinger equations”, J. Math. Phys, 53:10 (2012), 102702
7. Lufang Mi, Kangkang Zhang, “Invariant Tori for Benjamin-Ono Equation with Unbounded quasi-periodically forced Perturbation”, DCDS-A, 34:2 (2013), 689
8. Xiaoping Yuan, Kangkang Zhang, “A reduction theorem for time dependent Schrödinger operator with finite differentiable unbounded perturbation”, J. Math. Phys, 54:5 (2013), 052701
9. 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
10. XiaoPing Yuan, “KAM theorems and open problems for infinite-dimensional Hamiltonian with short range”, Sci. China Math, 2014
11. Mi L., Zhang K., “Quasi-periodic solutions for perturbed generalized KdV equation”, Nonlinear Anal.-Real World Appl., 32 (2016), 314–337
12. Cong H., Liu J., Yuan X., “Introduction and Main Results”, Mem. Am. Math. Soc., 239:1134 (2016), 1+
13. 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
14. Cui W., Mi L., Yin L., “Quasi-Periodic Solutions For Non-Autonomous Mkdv Equation”, Indian J. Pure Appl. Math., 49:2 (2018), 313–337
15. 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
16. Cui W., Mi L., Yin L., “Kam Tori For Defocusing Kdv-Mkdv Equation”, Acta Math. Sci., 39:1 (2019), 243–258
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