All solutions of the n = 5 Lane-Emden equation

TL;DR

This paper derives all real solutions of the n=5 Lane-Emden equation using elliptic functions, introduces a new solution family, and discusses their properties and potential applications.

Contribution

It provides a complete set of solutions for the n=5 Lane-Emden equation, including a novel family expressed with simple elliptic functions.

Findings

All solutions expressed via Jacobian and Weierstrass elliptic functions.

Discovery of a new family of solutions with simple formulae.

Discussion of properties, symmetries, and applications of solutions.

Abstract

All real solutions of the Lane-Emden equation for n = 5 are obtained in terms of Jacobian and Weierstrass elliptic functions. A new family of solutions is found. It is expressed by remarkably simple formulae involving Jacobian elliptic functions only. The general properties and discrete scaling symmetries of these new solutions are discussed. We also comment on their possible applications.