On Transformations of the Rabelo Equations

TL;DR

This paper investigates four Rabelo equations describing pseudospherical surfaces, transforming them into well-known integrable equations like sine-Gordon, Liouville, and linear equations to derive their general solutions.

Contribution

It introduces transformations of Rabelo equations into classical integrable equations, revealing their solutions and connections to established models.

Findings

Two Rabelo equations relate to sine-Gordon.

Remaining equations transform into linear and Liouville equations.

General solutions are explicitly obtained for all four equations.

Abstract

We study four distinct second-order nonlinear equations of Rabelo which describe pseudospherical surfaces. By transforming these equations to the constant-characteristic form we relate them to some well-studied integrable equations. Two of the Rabelo equations are found to be related to the sine-Gordon equation. The other two are transformed into a linear equation and the Liouville equation, and in this way their general solutions are obtained.