A Reciprocal Transformation for the Constant Astigmatism Equation

TL;DR

This paper presents a nonlocal reciprocal transformation linked to Bäcklund transformations of the sine-Gordon equation, enabling the generation of exact solutions for the constant astigmatism equation, a key nonlinear PDE in differential geometry.

Contribution

The paper introduces a novel nonlocal transformation related to Bäcklund transformations, providing a new method to find exact solutions of the constant astigmatism equation.

Findings

Derived explicit solutions for the constant astigmatism equation.

Connected the transformation to Bäcklund transformations of sine-Gordon.

Established the transformation as a nonlocal symmetry.

Abstract

We introduce a nonlocal transformation to generate exact solutions of the constant astigmatism equation $z_{yy} + (1/z)_{xx} + 2 = 0$. The transformation is related to the special case of the famous B\"acklund transformation of the sine-Gordon equation with the B\"acklund parameter $\lambda = \pm1$. It is also a nonlocal symmetry.