Exact Solutions of Two-Dimensional Vortex Systems in Statistical Equilibrium
Statistical equilibrium states of two-dimensional vortex systems  are described by the solutions of a nonlinear elliptic partial differential equation, the sinh-Poisson equation ∇2 ø + λ2 sinh = ø, with (ø = 0 on a 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 solutions showing nonlinear superposition are displayed.
KeywordsImaginary Axis Elliptic Partial Differential Equation Statistical Equilibrium State General Analytic Solution Mirror Image Position
Unable to display preview. Download preview PDF.
- 7.V.O. Kozel, V.P. Kotlyrarov, Dokl. A. N. Ukr. SSR, Ser. A 10, 878–881 (1976)Google Scholar
- 8.M.J. Ablowitz, D.J. Kaup, A.C. Newell, H. Segur, Stud. Appl. Math., 53, 249–315 (1974)Google Scholar
- 9.For application of Riemann complex theory to periodic problems, see Theory of Nonlinear Lattices, by Morikazu Toda, Springer Verlag (1981)Google Scholar