Symmetry-Preserving Finite-Difference Schemes and Auto-B\"acklund Transformations for the Schwarz Equation
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 ) can be related through a B\"acklund-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 .
Keywords
Cite
@article{arxiv.2503.03369,
title = {Symmetry-Preserving Finite-Difference Schemes and Auto-B\"acklund Transformations for the Schwarz Equation},
author = {E. I. Kaptsov and V. A. Dorodnitsyn},
journal= {arXiv preprint arXiv:2503.03369},
year = {2025}
}
Comments
Corrected a large number of misprints related to the scheme for the second order ODE. Different symbols are used for constants that are part of the general solutions of the ODE and the scheme