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.
\Bibitem{Ima13}
\by V.~M.~Imaykin
\paper Soliton asymptotics for systems of `field-particle' type
\jour Russian Math. Surveys
\yr 2013
\vol 68
\issue 2
\pages 227--281
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Linking options:
https://www.mathnet.ru/eng/rm9510
https://doi.org/10.1070/RM2013v068n02ABEH004829
https://www.mathnet.ru/eng/rm/v68/i2/p33
This publication is cited in the following 4 articles:
Alexander Komech, Elena Kopylova, Attractors of Hamiltonian Nonlinear Partial Differential Equations, 2021
A. I. Komech, E. A. Kopylova, “Attractors of nonlinear Hamiltonian partial differential equations”, Russian Math. Surveys, 75:1 (2020), 1–87
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
Komech A., “Attractors of Hamilton nonlinear PDEs”, Discret. Contin. Dyn. Syst., 36:11 (2016), 6201–6256
Citation in format AMSBIB
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