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International Journal of Differential Equations
Volume 2012 (2012), Article ID 412569, 9 pages
http://dx.doi.org/10.1155/2012/412569
Research Article

A Nonlinear Differential Equation Related to the Jacobi Elliptic Functions

Institute of Technology and Innovation, University of Southern Denmark, Niels Bohrs Allé 1, 5230 Odense M, Denmark

Received 3 May 2012; Accepted 6 August 2012

Academic Editor: Yuriy Rogovchenko

Copyright © 2012 Kim Johannessen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

A nonlinear differential equation for the polar angle of a point of an ellipse is derived. The solution of this differential equation can be expressed in terms of the Jacobi elliptic function dn(u,k). If the polar angle is extended to the complex plane, the Jacobi imaginary transformation properties and the dependence on the real and complex quarter periods can be described. From the differential equation of the polar angle, exact solutions of the Poisson Boltzmann and the sinh-Poisson equations are found in terms of the Jacobi elliptic functions.