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Solitary waves in the nolinear wave equation and in gauge theories. (English) Zbl 1122.35121

Summary: Roughly speaking a solitary wave is a solution of a field equation whose energy travels as a localized packet and which preserves this localization in time. This paper is an introduction to the study of solitary waves relative to the nonlinear wave equation and to the Abelian gauge theories. Abelian gauge theories consist of a class of field equations obtained by coupling in a suitable way the nonlinear wave equation with the Maxwell equations. They provide a model for the interaction of matter with the electromagnetic field. One of the motivations of this study lies in the fact that the nonlinear wave equation and the Abelian gauge theories are the simplest equations which satisfy the basic principles of modern physics.

MSC:

35Q51 Soliton equations
58E30 Variational principles in infinite-dimensional spaces
83C05 Einstein’s equations (general structure, canonical formalism, Cauchy problems)
70S15 Yang-Mills and other gauge theories in mechanics of particles and systems
37K30 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with infinite-dimensional Lie algebras and other algebraic structures
35Q75 PDEs in connection with relativity and gravitational theory
83C22 Einstein-Maxwell equations
49S05 Variational principles of physics
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