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Uspekhi Mat. Nauk, 2013, Volume 68, Issue 2(410), Pages 33–90 (Mi umn9510)  

This article is cited in 3 scientific papers (total in 3 papers)

Soliton asymptotics for systems of ‘field-particle’ type

V. M. Imaykin

Moscow Institute for the Development of Education

Abstract: This survey is devoted to the recent mathematical progress in the study of interaction between particles and fields. It covers a series of papers from 2000 till now. Three systems describing the interaction of a field and a charged particle are considered: the scalar Klein–Gordon field or the wave field coupled to a particle, and the Maxwell–Lorentz system describing a charged particle in the Maxwell field. The Wiener condition on the charge density of the particle was introduced in the first papers on long-time convergence to solitons in the absence of external potentials (the 1990s) and turned out to play an important role in the investigations reflected here of soliton asymptotics for solutions with initial data sufficiently close to invariant solitary manifolds. Our approach is based on using the Hamiltonian structure of the systems and the Buslaev–Perelman method of symplectic projection.
Bibliography: 49 titles.

Keywords: non-linear system of ‘field-particle’ type, soliton, solitary manifold, symplectic projection, linearization around a soliton, modulation equations, decay of the transversal component, Wiener condition.

DOI: https://doi.org/10.4213/rm9510

Full text: PDF file (1008 kB)
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English version:
Russian Mathematical Surveys, 2013, 68:2, 227–281

Bibliographic databases:

UDC: 517.955.8
MSC: Primary 34C08, 37K40, 35Q61; Secondary 35P25, 35Q60
Received: 06.02.2013

Citation: V. M. Imaykin, “Soliton asymptotics for systems of ‘field-particle’ type”, Uspekhi Mat. Nauk, 68:2(410) (2013), 33–90; Russian Math. Surveys, 68:2 (2013), 227–281

Citation in format AMSBIB
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    This publication is cited in the following articles:
    1. E. .A. Kopylova, A. I. Komech, “Asymptotic stability of stationary states in the wave equation coupled to a nonrelativistic particle”, Russ. J. Math.Phys., 23:1 (2016), 93–100  crossref  mathscinet  zmath  isi  scopus
    2. Komech A., “Attractors of Hamilton nonlinear PDEs”, Discret. Contin. Dyn. Syst., 36:11 (2016), 6201–6256  crossref  mathscinet  zmath  isi  elib  scopus
    3. A. I. Komech, E. A. Kopylova, “Attraktory nelineinykh gamiltonovykh uravnenii v chastnykh proizvodnykh”, UMN, 75:1(451) (2020), 3–94  mathnet  crossref
  • Успехи математических наук Russian Mathematical Surveys
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