New vortices in axisymmetric beams in an inhomogeneous magnetic field

We analyzed localized vortices in non-neutral electron beams with inhomogeneous density and velocity propagating in vacuum along an inhomogeneous external magnetic field. These vortices are different to vortices in earlier work because of the inhomogeneous external magnetic field. New types of vortices were also obtained using a new solution for non-linear equations, different to the standard Larichev–Reznik or Reznik method. A new axisymmetric expression was found for the electric field potential of a vortex in a wave frame. The new vortices represent new solitons in an inhomogeneous external magnetic field.