Soliton asymptotics for systems of ‘field-particle’ type

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.