Some notes on elliptic equation method

TL;DR

This paper analyzes solutions to a specific elliptic equation used in nonlinear differential equations, showing that claimed new solutions are actually re-expressions of known solutions, and deriving new elliptic function identities.

Contribution

It demonstrates that purported new solutions are just re-expressions of existing solutions and derives new identities of elliptic functions.

Findings

New solutions are re-expressions of known solutions.

Derived new identities of elliptic functions.

Provided detailed analysis of the elliptic equation solutions.

Abstract

Elliptic equation $(y')^2=a_0+a_2y^2+a_4y^4$ is the foundation of the elliptic function expansion method of finding exact solutions to nonlinear differential equation. In some references, some new form solutions to the elliptic equation have been claimed. In the paper, we discuss its solutions in detail. By detailed computation, we prove that those new form solutions can be derived from a very few known solutions. This means that those new form solutions are just new representations of old solutions. From our discussion, some new identities of the elliptic function can be obtained. In the course of discussion, we give an example of this kind of formula.