Uspekhi Matematicheskikh Nauk
RUS  ENG    JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB  
General information
Latest issue
Archive
Impact factor
Subscription
License agreement
Submit a manuscript

Search papers
Search references

RSS
Latest issue
Current issues
Archive issues
What is RSS



Uspekhi Mat. Nauk:
Year:
Volume:
Issue:
Page:
Find






Personal entry:
Login:
Password:
Save password
Enter
Forgotten password?
Register


Uspekhi Mat. Nauk, 2020, Volume 75, Issue 1(451), Pages 3–94 (Mi umn9900)  

This article is cited in 2 scientific papers (total in 2 papers)

Attractors of nonlinear Hamiltonian partial differential equations

A. I. Komech*, E. A. Kopylova

Institute for Information Transmission Problems of the Russian Academy of Sciences (Kharkevich Institute)

Abstract: This is a survey of the theory of attractors of nonlinear Hamiltonian partial differential equations since its appearance in 1990. Included are results on global attraction to stationary states, to solitons, and to stationary orbits, together with results on adiabatic effective dynamics of solitons and their asymptotic stability, and also results on numerical simulation. The results obtained are generalized in the formulation of a new general conjecture on attractors of $G$-invariant nonlinear Hamiltonian partial differential equations. This conjecture suggests a novel dynamical interpretation of basic quantum phenomena: Bohr transitions between quantum stationary states, de Broglie's wave-particle duality, and Born's probabilistic interpretation.
Bibliography: 212 titles.

Keywords: Hamiltonian equations, nonlinear partial differential equations, wave equation, Maxwell equations, Klein–Gordon equation, limiting amplitude principle, limiting absorption principle, attractor, steady states, soliton, stationary orbits, adiabatic effective dynamics, symmetry group, Lie group, Schrödinger equation, quantum transitions, wave-particle duality.

Funding Agency Grant Number
Austrian Science Fund P28152-N35
Russian Foundation for Basic Research 18-01-00524
The first author was supported by the Austrian Science Fund (FWF) under project no. P28152-N35, and the second author by the Russian Foundation for Basic Research (grant no. 18-01-00524).

* Author to whom correspondence should be addressed

DOI: https://doi.org/10.4213/rm9900

Full text: PDF file (1709 kB)
First page: PDF file
References: PDF file   HTML file

English version:
Russian Mathematical Surveys, 2020, 75:1, 1–87

Bibliographic databases:

UDC: 517.957
MSC: Primary 35B41; Secondary 35B40, 35C08
Received: 13.07.2019

Citation: A. I. Komech, E. A. Kopylova, “Attractors of nonlinear Hamiltonian partial differential equations”, Uspekhi Mat. Nauk, 75:1(451) (2020), 3–94; Russian Math. Surveys, 75:1 (2020), 1–87

Citation in format AMSBIB
\Bibitem{KomKop20}
\by A.~I.~Komech, E.~A.~Kopylova
\paper Attractors of nonlinear Hamiltonian partial differential equations
\jour Uspekhi Mat. Nauk
\yr 2020
\vol 75
\issue 1(451)
\pages 3--94
\mathnet{http://mi.mathnet.ru/umn9900}
\crossref{https://doi.org/10.4213/rm9900}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=4070019}
\elib{https://elibrary.ru/item.asp?id=43282202}
\transl
\jour Russian Math. Surveys
\yr 2020
\vol 75
\issue 1
\pages 1--87
\crossref{https://doi.org/10.1070/RM9900}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000548535900001}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85085053435}


Linking options:
  • http://mi.mathnet.ru/eng/umn9900
  • https://doi.org/10.4213/rm9900
  • http://mi.mathnet.ru/eng/umn/v75/i1/p3

    SHARE: VKontakte.ru FaceBook Twitter Mail.ru Livejournal Memori.ru


    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. V. I. Bogachev, “Non-uniform Kozlov–Treschev averagings in the ergodic theorem”, Russian Math. Surveys, 75:3 (2020), 393–425  mathnet  crossref  crossref  mathscinet  isi  elib
    2. A. R. Alimov, “Characterization of Sets with Continuous Metric Projection in the Space $\ell^\infty_n$”, Math. Notes, 108:3 (2020), 309–317  mathnet  crossref  crossref  mathscinet  isi  elib
  • Успехи математических наук Russian Mathematical Surveys
    Number of views:
    This page:402
    References:39
    First page:66

     
    Contact us:
     Terms of Use  Registration to the website  Logotypes © Steklov Mathematical Institute RAS, 2021