New vortices in axisymmetric inhomogeneous beams

We analyzed localized vortices in non-neutral inhomogeneous beams by the density and velocity of electron beams propagating in vacuum along the external magnetic field. These vortices distinguish from well-known vortices of Larichev-Reznik or Reznik types, which were used in [Yu.Ya. Golub, et al., Nonlinear World: IV Intern. Workshop on Nonlin. and Turbul. Proc. in Phys., vol.2, p.857 (1990)]. New types of vortex are obtained by new solution method of nonlinear equations. The new method is development of a method described in [Yu.Ya. Golub, Proceedings of EPAC, p.1253 (2002)]. That method distinguish from standard Larichev-Reznik or Reznik method, which were used in [Yu.Ya. Golub, et al., Nonlinear World: IV Intern. Workshop on Nonlin. and Turbul. Proc. in Phys., vol.2, p.857 (1990)]. A new expression for the electric field potential of the vortex in a wave frame has been found. The expression is axisymmetric in a wave frame. New vortices are new solitons. New vortices are the result of external disturbances or the appearance and development of instabilities like for example a diocotron instability in hollow beams and a slipping-instability in solid beams