Such a fluid model is amenable to direct analysis by transforming to Lagrangian variables following the motion of a fluid element. The resulting model is shown to be exactly equivalent to a (truncated) warm-fluid description with zero heat flow and triple-adiabatic equation of state with scalar pressure Pb(x,s) = const3. Coupled nonlinear equations are derived describing the self-consistent evolution of the boundary curves, x+′(x,s) and x-′(x,s), and the self-field potential ψ(x,s) = ebφ(x,s)/γbmbβb2c2.
The analysis considers the special class of distribution functions with uniform phase-space density fb(x,x′,s) = A = const inside of the simply connected boundary curves, x+′(x,s) and x-′(x,s), in the two-dimensional phase space (x,x′).
The Vlasov-Maxwell equations are used to investigate the nonlinear evolution of an intense sheet beam with distribution function fb(x,x′,s) propagating through a periodic focusing lattice κx(s+S) = κx(s), where S = const is the lattice period.
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