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Paraxial approximation of ultrarelativistic intense beams. (English) Zbl 0816.65119
A paraxial mathematical model of short beams (bunches) of highly relativistic noncollisional charged particles in the interior of a perfectly conducting cylindrical hollow tube in an external confining magnetic field is developed. The Vlasov-Maxwell equations in the beam frame, i.e. in a frame moving along the optical axis with the light velocity are presented. Simpler approximations than the Vlasov-Maxwell equations are sought. The equations are derived for the distribution function $$f$$ and electromagnetic force $$F$$ acting on the particles.
A scaling of the model is introduced. It leads to taking as a small parameter the ratio $$\eta = \overline{v}/c$$ ($$\overline{v}$$ is the characteristic transversal velocity, $$c$$ the light velocity). Asymptotic expansions are carried out for $$f$$ and $$F$$ in powers of $$\eta$$ and the first few terms of these expansions are characterized. The paraxial model retains the first four terms of the asymptotic expansion of $$f$$. The Vlasov-Maxwell model and the paraxial model coincide up to the third order of $$\eta$$.
Reviewer: V.Burjan (Praha)

##### MSC:
 65Z05 Applications to the sciences 78A35 Motion of charged particles 35Q60 PDEs in connection with optics and electromagnetic theory
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