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Uspekhi Mat. Nauk, 2000, Volume 55, Issue 1(331), Pages 45–98 (Mi umn249)  

This article is cited in 7 scientific papers (total in 7 papers)

Attractors of non-linear Hamiltonian one-dimensional wave equations

A. I. Komech

M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: A theory is constructed for attractors of all finite-energy solutions of conservative one-dimensional wave equations on the whole real line. The attractor of a non-degenerate (that is, generic) equation is the set of all stationary solutions. Each finite-energy solution converges as $t\to\pm\infty$ to this attractor in the Frechet topology determined by local energy seminorms. The attraction is caused by energy dissipation at infinity. Our results provide a mathematical model of Bohr transitions (“quantum jumps”) between stationary states in quantum systems.

DOI: https://doi.org/10.4213/rm249

Full text: PDF file (541 kB)
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English version:
Russian Mathematical Surveys, 2000, 55:1, 43–92

Bibliographic databases:

UDC: 517.9
MSC: Primary 35L10, 35L70; Secondary 35B40, 35B45, 34C15, 58F05, 34D45, 35Q55
Received: 19.08.1998

Citation: A. I. Komech, “Attractors of non-linear Hamiltonian one-dimensional wave equations”, Uspekhi Mat. Nauk, 55:1(331) (2000), 45–98; Russian Math. Surveys, 55:1 (2000), 43–92

Citation in format AMSBIB
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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. O. Yu. Dinariev, “On dissipative phenomena of the interaction of Hamiltonian systems”, Siberian Math. J., 44:1 (2003), 61–72  mathnet  crossref  mathscinet  zmath  isi
    2. Komech A.I., “On attractor of a singular nonlinear U(I)-invariant Klein-Gordon equation”, Progress in Analysis, I–II (2003), 599–611  crossref  mathscinet  zmath  isi
    3. Komech A.I., Mauser N.J., Vinnichenko A.P., “Attraction to solitons in relativistic nonlinear wave equations”, Russ. J. Math. Phys., 11:3 (2004), 289–307  mathscinet  zmath  isi  elib
    4. Bertini M., Noja D., Posilicano A., “Dynamics and Lax–Phillips scattering for generalized Lamb models”, J. Phys. A, 39:49 (2006), 15173–15195  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    5. Merzon A.E., Taneco-Hernández M.A., “Scattering in the zero-mass Lamb system”, Phys. Lett. A, 372:27-28 (2008), 4761–4767  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus
    6. Komech A.I., Merzon A.E., “Scattering in the nonlinear Lamb system”, Phys. Lett. A, 373:11 (2009), 1005–1010  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus
    7. Komech A., “Attractors of Hamilton nonlinear PDEs”, Discret. Contin. Dyn. Syst., 36:11 (2016), 6201–6256  crossref  mathscinet  zmath  isi  elib  scopus
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