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Russian Mathematical Surveys, 2013, Volume 68, Issue 2, Pages 283–334
DOI: https://doi.org/10.1070/RM2013v068n02ABEH004830 (Mi rm9509) 		 
This article is cited in 5 scientific papers (total in 5 papers)

Asymptotic stability of solitons for nonlinear hyperbolic equations

E. A. Kopylovaab

a Kharkevich Institute for Information Transmission Problems, Russian Academy of Sciences
b University of Vienna, Austria
English version PDF (908 kB) Citations (5)
References:
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DOI: https://doi.org/10.1070/RM2013v068n02ABEH004830
Abstract: Fundamental results on asymptotic stability of solitons are surveyed, methods for proving asymptotic stability are illustrated based on the example of a nonlinear relativistic wave equation with Ginzburg–Landau potential. Asymptotic stability means that a solution of the equation with initial data close to one of the solitons can be asymptotically represented for large values of the time as a sum of a (possibly different) soliton and a dispersive wave solving the corresponding linear equation. The proof techniques depend on the spectral properties of the linearized equation and may be regarded as a modern extension of the Lyapunov stability theory. Examples of nonlinear equations with prescribed spectral properties of the linearized dynamics are constructed.
Bibliography: 45 titles.
Keywords: nonlinear hyperbolic equations, asymptotic stability, relativistic invariance, Hamiltonian structure, symplectic projection, invariant manifold, soliton, kink, Fermi's golden rule, scattering of solitons, asymptotic state.
Funding agency Grant number
Russian Foundation for Basic Research 12-01-00203-а
Austrian Science Fund M1329-N13
Received: 06.02.2013
Russian version:
Uspekhi Matematicheskikh Nauk, 2013, Volume 68, Issue 2(410), Pages 91–144
DOI: https://doi.org/10.4213/rm9509
Bibliographic databases:
Document Type: Article
UDC: 517.957
MSC: Primary 35C08; Secondary 35L05, 35Q56, 35L75, 37K40
Language: English
Original paper language: Russian
Citation: E. A. Kopylova, “Asymptotic stability of solitons for nonlinear hyperbolic equations”, Uspekhi Mat. Nauk, 68:2(410) (2013), 91–144; Russian Math. Surveys, 68:2 (2013), 283–334
Citation in format AMSBIB
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    • This publication is cited in the following 5 articles:
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    References:98
    First page:39
     
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