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This article is cited in 8 scientific papers (total in 8 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
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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
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This publication is cited in the following articles:
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O. Yu. Dinariev, “On dissipative phenomena of the interaction of Hamiltonian systems”, Siberian Math. J., 44:1 (2003), 61–72
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Komech A.I., “On attractor of a singular nonlinear U(I)-invariant Klein-Gordon equation”, Progress in Analysis, I–II (2003), 599–611
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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
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Bertini M., Noja D., Posilicano A., “Dynamics and Lax–Phillips scattering for generalized Lamb models”, J. Phys. A, 39:49 (2006), 15173–15195
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Merzon A.E., Taneco-Hernández M.A., “Scattering in the zero-mass Lamb system”, Phys. Lett. A, 372:27-28 (2008), 4761–4767
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Komech A.I., Merzon A.E., “Scattering in the nonlinear Lamb system”, Phys. Lett. A, 373:11 (2009), 1005–1010
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Komech A., “Attractors of Hamilton nonlinear PDEs”, Discret. Contin. Dyn. Syst., 36:11 (2016), 6201–6256
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A. I. Komech, E. A. Kopylova, “Attractors of nonlinear Hamiltonian partial differential equations”, Russian Math. Surveys, 75:1 (2020), 1–87
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