{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,27]],"date-time":"2026-03-27T16:13:46Z","timestamp":1774628026806,"version":"3.50.1"},"reference-count":37,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2021,1,17]],"date-time":"2021-01-17T00:00:00Z","timestamp":1610841600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Isothermic surfaces are defined as immersions with the curvture lines admitting conformal parameterization. We present and discuss the reconstruction of the iterated Darboux transformation using Clifford numbers instead of matrices. In particulalr, we derive a symmetric formula for the two-fold Darboux transformation, explicitly showing Bianchi\u2019s permutability theorem. In algebraic calculations an important role is played by the main anti-automorphism (reversion) of the Clifford algebra C(4,1) and the spinorial norm in the corresponding Spin group.<\/jats:p>","DOI":"10.3390\/sym13010148","type":"journal-article","created":{"date-parts":[[2021,1,21]],"date-time":"2021-01-21T02:36:05Z","timestamp":1611196565000},"page":"148","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Iterated Darboux Transformation for Isothermic Surfaces in Terms of Clifford Numbers"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1730-0950","authenticated-orcid":false,"given":"Jan L.","family":"Cie\u015bli\u0144ski","sequence":"first","affiliation":[{"name":"Wydzia\u0142 Fizyki, Uniwersytet w Bia\u0142ymstoku, ul. Cio\u0142kowskiego 1L, 15-245 Bia\u0142ystok, Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4574-9536","authenticated-orcid":false,"given":"Zbigniew","family":"Hasiewicz","sequence":"additional","affiliation":[{"name":"Wydzia\u0142 Fizyki, Uniwersytet w Bia\u0142ymstoku, ul. Cio\u0142kowskiego 1L, 15-245 Bia\u0142ystok, Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,1,17]]},"reference":[{"key":"ref_1","first-page":"147","article-title":"M\u00e9moire sur les surfaces isothermes dans les corps solides homog\u00e8nes en \u00e9quilibre de temp\u00e9rature","volume":"2","year":"1837","journal-title":"J. Math. Pures Appl."},{"key":"ref_2","first-page":"117","article-title":"M\u00e9moire sur les surfaces isothermes orthogonales","volume":"9","author":"Bertrand","year":"1844","journal-title":"J. Math. Pures Appl."},{"key":"ref_3","unstructured":"Klimczewski, P., Nieszporski, M., and Sym, A. (2000). Luigi Bianchi, Pasquale Calapso and solitons. Rend. Sem. Mat. Messina (Atti del Congresso Internazionale in onore di Pasquale Calapso), 223\u2013240."},{"key":"ref_4","first-page":"1299","article-title":"Sur les surfaces isothermiques","volume":"128","author":"Darboux","year":"1899","journal-title":"C. R. Acad. Sci. Paris"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"93","DOI":"10.1007\/BF02419963","article-title":"Ricerche sulle superficie isoterme e sulle deformazione delle quadriche","volume":"11","author":"Bianchi","year":"1905","journal-title":"Ann. Matem."},{"key":"ref_6","unstructured":"Novikov, S., Manakov, S.V., Pitaevskii, L.P., and Zakharov, V.E. (1984). Theory of Solitons. The Inverse Sattering Method, Springer."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"37","DOI":"10.1016\/0375-9601(95)00504-V","article-title":"Isothermic surfaces in E3 as soliton surfaces","volume":"205","author":"Goldstein","year":"1995","journal-title":"Phys. Lett. A"},{"key":"ref_8","first-page":"187","article-title":"Discrete isothermic surfaces","volume":"475","author":"Bobenko","year":"1996","journal-title":"J. Reine Angew. Math."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"199","DOI":"10.1007\/PL00004308","article-title":"Curved flats and isothermic surfaces","volume":"225","author":"Burstall","year":"1997","journal-title":"Math. Z."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"313","DOI":"10.4171\/dm\/32","article-title":"Remarks on the Darboux transform of isothermic surfaces","volume":"2","author":"Pedit","year":"1997","journal-title":"Doc. Math."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"129","DOI":"10.1090\/conm\/288\/04822","article-title":"Special isothermic surfaces and solitons","volume":"288","author":"Musso","year":"2001","journal-title":"Contemp. Math."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1090\/amsip\/036\/01","article-title":"Isothermic surfaces: Conformal geometry, Clifford algebras and integrable systems","volume":"36","author":"Burstall","year":"2006","journal-title":"AMS\/IP Stud. Adv. Math."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"503","DOI":"10.1016\/j.difgeo.2006.04.007","article-title":"The Ribaucour transformation in Lie sphere geometry","volume":"24","author":"Burstall","year":"2006","journal-title":"Differ. Geom. Appl."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"43","DOI":"10.1007\/s10711-016-0143-7","article-title":"Semi-discrete isothermic surfaces","volume":"183","author":"Burstall","year":"2016","journal-title":"Geom. Dedicata"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"295","DOI":"10.1016\/S0034-4877(17)30010-1","article-title":"Covariant Description of Isothermic Surfaces","volume":"78","author":"Tafel","year":"2016","journal-title":"Rep. Math. Phys."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"81","DOI":"10.1007\/s13366-016-0301-y","article-title":"Minimal Darboux transformations","volume":"58","author":"Honda","year":"2017","journal-title":"Beitr. Algebra Geom."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"28","DOI":"10.1016\/j.geomphys.2016.11.028","article-title":"Group interpretation of the spectral parameter. The case of isothermic surfaces","volume":"113","author":"Kobus","year":"2017","journal-title":"J. Geom. Phys."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"529","DOI":"10.1007\/s10455-018-9639-8","article-title":"Transformations and singularities of polarized curves","volume":"55","author":"Fuchs","year":"2019","journal-title":"Ann. Glob. Anal. Geom."},{"key":"ref_19","unstructured":"Corro, A.M.V., and Ferro, M.L. (2020). New Isothermic surfaces. arXiv."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"251","DOI":"10.1016\/S0375-9601(00)00086-4","article-title":"A class of linear spectral problems in Clifford algebras","volume":"267","year":"2000","journal-title":"Phys. Lett. A"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"L363","DOI":"10.1088\/0305-4470\/33\/41\/101","article-title":"The Darboux-B\u00e4cklund transformation without using a matrix representation","volume":"33","year":"2000","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_22","unstructured":"Bobenko, A.I., and Hertrich-Jeromin, U.J. (1998). Orthogonal nets and Clifford algebras. arXiv."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/S0926-2245(97)00002-8","article-title":"The Darboux-Bianchi transformation for isothermic surfaces. Classical results versus the soliton approach","volume":"7","year":"1997","journal-title":"Diff. Geom. Appl."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"154","DOI":"10.1007\/3-540-16039-6_6","article-title":"Soliton surfaces and their applications","volume":"Volume 239","author":"Martini","year":"1985","journal-title":"Geometric Aspects of the Einstein Equations and Integrable Systems"},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Lounesto, P. (1997). Clifford Algebras and Spinors, Cambridge University Press.","DOI":"10.1007\/978-1-4612-4104-1_2"},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Vaz, J., and da Rocha, R. (2016). An Introduction to Clifford Algebras and Spinors, Oxford University Press.","DOI":"10.1093\/acprof:oso\/9780198782926.001.0001"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"5670","DOI":"10.1063\/1.531282","article-title":"An algebraic method to construct the Darboux matrix","volume":"36","year":"1995","journal-title":"J. Math. Phys."},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Gu, C.H. (1995). B\u00e4cklund Transformations and Darboux Transformations. Soliton Theory and Its Applications, Springer.","DOI":"10.1007\/978-3-662-03102-5_3"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"73","DOI":"10.1016\/0167-2789(81)90120-2","article-title":"The reduction problem and the inverse scattering method","volume":"3","author":"Mikhailov","year":"1981","journal-title":"Phys. D Nonlinear Phenom."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"404003","DOI":"10.1088\/1751-8113\/42\/40\/404003","article-title":"Algebraic construction of the Darboux matrix revisited","volume":"42","year":"2009","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1016\/S0375-9601(01)00551-5","article-title":"A compact form of the Darboux-B\u00e4cklund transformation for some spectral problems in Clifford algebras","volume":"288","author":"Biernacki","year":"2001","journal-title":"Phys. Lett. A"},{"key":"ref_32","first-page":"1394","article-title":"Geometry of submanifolds derived from Spin-valued spectral problems","volume":"137","year":"2003","journal-title":"Theor. Math. Phys."},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Matveev, V.B., and Salle, M.A. (1991). Darboux Transformations and Solitons, Springer.","DOI":"10.1007\/978-3-662-00922-2"},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Rogers, C., and Schief, W.K. (2002). B\u00e4cklund and Darboux Transformations. Geometry and Modern Applications in Soliton Theory, Cambridge University Press.","DOI":"10.1017\/CBO9780511606359"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"274","DOI":"10.1006\/jmaa.2001.7577","article-title":"Generalized B\u00e4cklund\u2013Darboux Transformation: Spectral Properties and Nonlinear Equations","volume":"262","author":"Sakhnovich","year":"2001","journal-title":"J. Math. Anal. Appl."},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Sakhnovich, A.L., Sakhnovich, L.A., and Roitberg, I.Y. (2013). Inverse Problems and Nonlinear Evolution Equations, De Gruyter.","DOI":"10.1515\/9783110258615"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"9491","DOI":"10.1088\/0305-4470\/38\/43\/006","article-title":"A new approach to the Darboux-B\u00e4cklund transformation versus the standard dressing method","volume":"38","author":"Biernacki","year":"2005","journal-title":"J. Phys. A Math. Gen."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/1\/148\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T05:12:09Z","timestamp":1760159529000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/1\/148"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,1,17]]},"references-count":37,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2021,1]]}},"alternative-id":["sym13010148"],"URL":"https:\/\/doi.org\/10.3390\/sym13010148","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,1,17]]}}}