Exact Vortex Solutions of Two-Dimensional Guiding-Center Plasmas

Statistical equilibrium states of a two-dimensional guiding-center plasma are described by the solutions of a nonlinear elliptic partial differential equation, the sinh-Poisson equation ${\nabla }^2\Phi +{\lambda }^{2}sinh\Phi =0\mathrm{}$, with conducting rectangular boundary. We present here the first general analytic solutions to such a nonlinear boundary-value problem. Some of these solutions were obtained previously only through numerical means. Explicit construction of the solutions shows nonlinear superposition among the solutions.