Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 10 (2014), 091, 20 pages      arXiv:1111.2027      https://doi.org/10.3842/SIGMA.2014.091

A Reciprocal Transformation for the Constant Astigmatism Equation

Adam Hlaváč and Michal Marvan
Mathematical Institute in Opava, Silesian University in Opava, Na Rybníčku 1, 746 01 Opava, Czech Republic

Received May 07, 2014, in final form August 14, 2014; Published online August 25, 2014

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 Bäcklund transformation of the sine-Gordon equation with the Bäcklund parameter $\lambda = \pm1$. It is also a nonlocal symmetry.

Key words: constant astigmatism equation; exact solution; constant astigmatism surface; orthogonal equiareal pattern; reciprocal transformation; sine-Gordon equation.

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