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)
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.
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.
|Austrian Science Fund
|Russian Foundation for Basic Research
|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
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Russian Mathematical Surveys, 2020, 75:1, 1–87
MSC: Primary 35B41; Secondary 35B40, 35C08
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
\by A.~I.~Komech, E.~A.~Kopylova
\paper Attractors of nonlinear Hamiltonian partial differential equations
\jour Uspekhi Mat. Nauk
\jour Russian Math. Surveys
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