Scalar potential calculations for relativistic charged-particle beam in unbounded coaxial drift tube

We receive the first order analytical expression for the scalar potential transversal distribution of finite width uniform density axisymmetric relativistic charged-particle beam propagating in the unbounded in the longitudinal direction coaxial drift tube in the strong guide magnetic field, which exhibits nonlinear dependence on the beam injection current. Ranges of the injection current values that show non-uniqueness of the solutions to the nonlinear ordinary differential equation describing the transverse distribution of the scalar potential in unbounded coaxial drift tube are also found numerically.