Solutions to the Calapso, Zoomeron, and Davey-Stewartson III Equations using Ribaucour Transformations
DOI:
https://doi.org/10.17268/sel.mat.2023.02.06Keywords:
Isothermic surfaces, Ribaucour transformations, Calapso equation, Zoomeron equation, Davey-Stewartson III equation.Abstract
In this paper, we present a method for obtaining solutions to the Calapso and Zoomeron equations using Ribaucour transformations. We provide explicit formulas for the solutions of these equations. By applying this method to the generalized cylinder, we provide new solutions of these equations, which are determined by a free function and by two other functions, each one defined in a given variable. As a consequence, we provide new solutions for the Davey-Stewartson III equation.
References
Bianchi L. Lezioni di geometria Differenziale. Terza Edicione, Nicola Zanichelli Editore; 1927.
Calapso P. Sulle superficie a linee di curvatura isoterme. Palermo Rend. 1903; 17:275-286.
Changyan S, Heming F, Chengfa W. Soliton solutions to the reverse-time nonlocal Davey-Stewartson III equation. Wave Motion. 2021; 104: 102744. DOI: https://doi.org/10.1016/j.wavemoti.2021.102744.
Corro AMV, Ferreira WP, Tenenblat K. On Ribaucour transformations for hypersurfaces. Mat.
Contemp. 1999; 17:137-160.
Corro AMV, Ferreira WP, Tenenblat K. Ribaucour transformations for Constant mean curvature and linear weingarten surfaces. Pacific Journal of Mathematics. 2003; 212(2):265-296.
Corro AMV, Ferreira WP, Tenenblat K. Minimal surfaces obtained by Ribaucour transformations. Geometriae Dedicata, Nettherlands. 2003; 96(1):117-150.
Corro AMV, Tenenblat K. Ribaucour transformation revisited, Comum. Geom. 2004; 12(5):1055-1082.
Corro AMV, Fernandes KV, Riveros CMC. Isothermic surfaces and solutions of the Calapso equation. Serdica Math. J. 2018; 44:341-364.
Corro AMV, Ferro ML. New Isothermic Surfaces. arXiv. 2021; Available from: https://doi.org/10.48550/arXiv.2103.04679.
Corro, AMV, Ferro ML. New Isothermic Surfaces in S^3. Selecciones Matemáticas. 2023; 10(1):114-128.
Corro, AMV, Ferro ML, Riveros CMC. $epsilon$-isothermic surfaces in pseudo-Euclidean 3-space. Selecciones Matemáticas. 2023; 10(1):129-146.
Davey A, Stewartson K. On Three-dimensional Packets of Surface Waves. Proc. R. Soc. Lond. A. 1974; 338;101-110.
Ferreira WP, Tenenblat K. Hypersurfaces with flat r-mean curvature and Ribaucour transformations. International Journal of Applied Mathematics and Statistics. 2007; 11:38-51.
Heming F, Chenzhen R, Weiying H. Soliton soltuions to the nonlocal Davey-Stewartson III equation. Mod. Phys. Lett. B. 2021;35(1) 2150026. DOI: 10.1142/S0217984921500263
Lemes MV, Tenenblat K. On Ribaucour transformations and minimal surfaces. Matemática Contemporânea. 2005; 29:13-40.
Rogers R, Schief WK. Bäcklund and Darboux transformations. Cambridge University Press; 2002.
Tenenblat K. On Ribaucour transformations and applications to linear Weingarten surfaces. Anais da Academia Brasileira de Ciências (Impresso), ABC. 2002; 74:559-575.
Tenenblat K, Wang Q. Ribaucour Transformations for Hypersurfaces in Space Forms. Annals of Global Analysis and Geometry. 2006; 29:157-185.
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2023 Selecciones Matemáticas
This work is licensed under a Creative Commons Attribution 4.0 International License.
The authors who publish in this journal accept the following conditions:
1. The authors retain the copyright and assign to the journal the right of the first publication, with the work registered with the Creative Commons Attribution License,Atribución 4.0 Internacional (CC BY 4.0) which allows third parties to use what is published whenever they mention the authorship of the work And to the first publication in this magazine.
2. Authors may make other independent and additional contractual arrangements for non-exclusive distribution of the version of the article published in this journal (eg, include it in an institutional repository or publish it in a book) provided they clearly state that The paper was first published in this journal.
3. Authors are encouraged to publish their work on the Internet (for example, on institutional or personal pages) before and during the review and publication process, as it can lead to productive exchanges and to a greater and more rapid dissemination Of the published work.
