Symmetry-Preserving Finite-Difference Schemes and Auto-Bäcklund Transformations for the Schwarz Equation
E. I. Kaptsov, V. A. Dorodnitsyn
TL;DR
This work examines symmetry‑preserving finite‑difference schemes for the Schwarz equation and a related second‑order ODE from Lie classifications, showing that in the continuous setting one equation forms a first integral of Schwarz. It compares invariant discretizations, including the Winternitz Schwarz scheme and an exact invariant scheme for the second‑order ODE, and derives a discrete Bäcklund‑type transformation (in the case $ K=C^2=4 $) that links these two schemes via their first integrals. The transformation, not realizable by a point change, reduces to an auto‑Bäcklund map for Schwarz in the continuous limit, illustrating that multiple exact schemes for the same ODE can be interconnected by discrete transformations. The results advance the understanding of how symmetry and integrals persist under discretization and suggest a broader framework for relating invariant schemes beyond point transformations.
Abstract
It is demonstrated that one of the equations from the Lie classification list of second-order ODEs is a first integral of the Schwarz equation. As symmetry-preserving finite-difference schemes have been previously constructed for both equations, the preservation of a similar connection between these schemes is studied. It is shown that the schemes for the Schwarz equation and the second-order ODE (with an arbitrary constant $C$) can be related through a Bäcklund-type difference transformation. In addition, previously unexamined aspects of the difference scheme for the second-order ODE are discussed, including its singular solution and the complete set of difference first integrals for the case $C^2=4$.