On classification of the solutions for general elliptic equation

TL;DR

This paper classifies solutions of the general elliptic equation using equivalence relations, constructs transformations from Riccati equations, and derives new solutions for a modified Camassa-Holm equation.

Contribution

It introduces a classification scheme for solutions of the general elliptic equation and applies it to find new exact solutions for a related nonlinear PDE.

Findings

36 known solutions are shown to be equivalent to 10 solutions.

A new classification of solutions for the elliptic equation is established.

New exact traveling wave solutions for the modified Camassa-Holm equation are obtained.

Abstract

The B\"acklund transformations and the superposition formulas for two sub--equations of the general elliptic equation are constructed from the Riccati equation by using an indirect mapping method.The thirty-six previously known solutions of the general elliptic equation are proved to be equivalent to the another ten solutions.The classification of solutions for the general elliptic equation is obtained based on the equivalence relations.The general elliptic equation expansion method with new classified solutions are used to obtain abundant new exact traveling wave solutions of a modified Camassa-Holm equation.