The kinetic model of the two dimensional cylindrical current sheet

In this paper we develop a one class of solutions of the steady Vlasov-Maxwell equations, which describes two dimensional cylindrical current sheets with current directed azimuthally $\textbf{j}=j_\theta(\rho,z)\textbf{e}_\theta$. Magnetic field of these sheets has two components $\textbf{B}=B_z(\rho,z)\textbf{e}_z+B_\rho(\rho,z)\textbf{e}_\rho$. From mathematical point of view, we find solutions of the nonlinear equation in partial derivatives for some function $u(\rho,z)$: $\partial^2u/\partial x^2+x^{-1}\partial^2u/\partial z^2=e^{-u}$, where ${x=\rho^2}$. We apply methods of group theory to develop three-parameter class of solutions. We also derive asymptotic behavior of these solutions for large values of $\rho$ and for ${\rho\sim0}$. We discuss applications of these solutions for description of current sheets in magnetospheres of planets with magnetic dipoles located near the ecliptic plane.