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Russian Mathematical Surveys, 2020, Volume 75, Issue 1, Pages 1–87
DOI: https://doi.org/10.1070/RM9900 (Mi rm9900) 		 
This article is cited in 8 scientific papers (total in 8 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)
English version PDF (1616 kB) Citations (8) Russian version article
References:
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DOI: https://doi.org/10.1070/RM9900
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).
Received: 13.07.2019
Bibliographic databases:
Document Type: Article
UDC: 517.957
MSC: Primary 35B41; Secondary 35B40, 35C08
Language: English
Original paper language: Russian
Citation: A. I. Komech, E. A. Kopylova, “Attractors of nonlinear Hamiltonian partial differential equations”, Russian Math. Surveys, 75:1 (2020), 1–87
Citation in format AMSBIB
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\pages 1--87
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Linking options:
  • https://www.mathnet.ru/eng/rm9900
  • https://doi.org/10.1070/RM9900
  • https://www.mathnet.ru/eng/rm/v75/i1/p3
    • This publication is cited in the following 8 articles:
    1. Lawrence Frolov, Samuel Leigh, Shadi Tahvildar-Zadeh, “Joint evolution of a Lorentz-covariant massless scalar field and its point-charge source in one space dimension”, Journal of Mathematical Physics, 65:6 (2024)  crossref
    2. A. I. Komech, E. A. Kopylova, “On the Hamilton–Poisson structure and solitons for the Maxwell–Lorentz equations with spinning particle”, Journal of Mathematical Analysis and Applications, 522:2 (2023), 126976  crossref  mathscinet
    3. A. I. Komech, E. A. Kopylova, “On the stability of solitons for the Maxwell-Lorentz equations with rotating particle”, Milan J. Math., 91:1 (2023), 155  crossref  mathscinet
    4. St. Petersburg Math. J., 35:5 (2024), 827–838  mathnet  crossref
    5. A. I. Komech, “On quantum jumps and attractors of the Maxwell-Schrödinger equations”, Ann. Math. Qué., 46:1 (2022), 139–159  crossref  mathscinet  isi  scopus
    6. A. Komech, E. Kopylova, Attractors of Hamiltonian Nonlinear Partial Differential Equations, Cambridge Tracts in Mathematics, 224, Cambridge University Press, 2021  crossref  mathscinet
    7. V. I. Bogachev, “Non-uniform Kozlov–Treschev averagings in the ergodic theorem”, Russian Math. Surveys, 75:3 (2020), 393–425  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    8. A. R. Alimov, “Characterization of Sets with Continuous Metric Projection in the Space ℓn∞”, Math. Notes, 108:3 (2020), 309–317  mathnet  crossref  crossref  mathscinet  isi  elib
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    Abstract page:873
    Russian version PDF:226
    English version PDF:81
    References:132
    First page:76
     
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