Ermakov-Pinney and Emden-Fowler equations: new solutions from novel B\"acklund transformations

TL;DR

This paper introduces novel Bäcklund transformations for a class of nonlinear ODEs including Emden-Fowler and Ermakov-Pinney equations, enabling the generation of new solutions from trivial ones, which is significant due to the equations' nonlinear complexity.

Contribution

The paper constructs new Bäcklund and auto Bäcklund transformations for a broad class of nonlinear ODEs, providing a systematic way to generate solutions from trivial solutions.

Findings

Constructed Bäcklund transformations for Emden-Fowler and Ermakov-Pinney equations.

Developed a ladder of solutions from trivial solutions using auto Bäcklund transformations.

Addressed the difficulty of numerical methods due to the nonlinear structure.

Abstract

The class of nonlinear ordinary differential equations $y^{\prime\prime}y = F(z,y^2)$, where F is a smooth function, is studied. Various nonlinear ordinary differential equations, whose applicative importance is well known, belong to such a class of nonlinear ordinary differential equations. Indeed, the Emden-Fowler equation, the Ermakov-Pinney equation and the generalized Ermakov equations are among them. B\"acklund transformations and auto B\"acklund transformations are constructed: these last transformations induce the construction of a ladder of new solutions adimitted by the given differential equations starting from a trivial solutions. Notably, the highly nonlinear structure of this class of nonlinear ordinary differential equations implies that numerical methods are very difficulty to apply.