Nonlinear theta-pinch equilibria

Analytic solutions for cylindrical, rigid-drift equilibria were searched for by checking the invariance of the second-order, nonlinear differential equations of the line density and particle density under one-parameter Lie groups. No invariance was found for the cylindrical screw-pinch for the polytrope index γ ╪ 2 of the pressure equation of state. For the Z-pinch, the differential equation for the line density for arbitrary γ is invariant under one group and reduces by Euler's method to an intractable first-order, nonlinear differential equation. In the case of a θ-pinch, the nonlinear differential equations for the particle density and line density are invariant at least under the translation group. The particle density is found as an implicit function of the radial distance r, involving incomplete beta functions of the γth power of the particle density.