International Journal of Differential Equations

A Nonlinear Differential Equation Related to the Jacobi Elliptic Functions

Kim Johannessen

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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.

Article information

Int. J. Differ. Equ., Volume 2012 (2012), Article ID 412569, 9 pages.

Received: 3 May 2012
Accepted: 6 August 2012
First available in Project Euclid: 24 January 2017

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Johannessen, Kim. A Nonlinear Differential Equation Related to the Jacobi Elliptic Functions. Int. J. Differ. Equ. 2012 (2012), Article ID 412569, 9 pages. doi:10.1155/2012/412569.

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