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 j = jθ(ρ, z)eθ. Magnetic field of these sheets has two components B = Bz(ρ, z)ez + Bρ(ρ, z)eρ. From mathematical point of view, we find solutions of the nonlinear equation in partial derivatives for some function u(ρ, z): ∂2u/∂x2 + x−1∂2u/∂z2 = e−u, where x = ρ2. 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 ρ and for ρ ∼ 0. We discuss applications of these solutions for description of current sheets in magnetospheres of planets with magnetic dipoles located near the ecliptic plane.