{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,26]],"date-time":"2026-02-26T15:23:01Z","timestamp":1772119381600,"version":"3.50.1"},"reference-count":39,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2023,11,13]],"date-time":"2023-11-13T00:00:00Z","timestamp":1699833600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2023,11,13]],"date-time":"2023-11-13T00:00:00Z","timestamp":1699833600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/501100004561","name":"Ministry of Education and Science of the Republic of Kazakhstan","doi-asserted-by":"publisher","award":["AP08856912"],"award-info":[{"award-number":["AP08856912"]}],"id":[{"id":"10.13039\/501100004561","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Soft Comput"],"published-print":{"date-parts":[[2024,1]]},"DOI":"10.1007\/s00500-023-09375-3","type":"journal-article","created":{"date-parts":[[2023,11,13]],"date-time":"2023-11-13T11:02:33Z","timestamp":1699873353000},"page":"105-117","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Solutions of localized induction equation associated with involute\u2013evolute curve pair"],"prefix":"10.1007","volume":"28","author":[{"given":"Kemal","family":"Eren","sequence":"first","affiliation":[]},{"given":"Zhaidary","family":"Myrzakulova","sequence":"additional","affiliation":[]},{"given":"Soley","family":"Ersoy","sequence":"additional","affiliation":[]},{"given":"Ratbay","family":"Myrzakulov","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2023,11,13]]},"reference":[{"issue":"3","key":"9375_CR1","first-page":"556","volume":"9","author":"NH Abdel-All","year":"2012","unstructured":"Abdel-All NH, Hussien RA, Youssef T (2012) Hasimoto surfaces. Life Sci 9(3):556\u2013560","journal-title":"Life Sci"},{"issue":"4","key":"9375_CR2","first-page":"75","volume":"6","author":"M Akyi\u011fit","year":"2010","unstructured":"Akyi\u011fit M, Azak AZ, Ersoy S (2010) Involute-evolute curves in Galilean space $${G}^3$$. Scientia Magna 6(4):75\u201380","journal-title":"Scientia Magna"},{"key":"9375_CR3","doi-asserted-by":"publisher","unstructured":"Angelidis A, Neyret F (2005) Simulation of smoke based on vortex filament primitives. SCA 05, pp. 87\u201396. Association for Computing Machinery, New York, NY, USA. https:\/\/doi.org\/10.1145\/1073368.1073380","DOI":"10.1145\/1073368.1073380"},{"key":"9375_CR4","doi-asserted-by":"publisher","first-page":"553","DOI":"10.1063\/1.1761268","volume":"8","author":"RJ Arms","year":"1995","unstructured":"Arms RJ, Hama FR (1995) Localized-induction concept on a curved vortex and motion of an elliptic vortex ring. Phys Fluids 8:553\u2013559. https:\/\/doi.org\/10.1063\/1.1761268","journal-title":"Phys Fluids"},{"issue":"5","key":"9375_CR5","doi-asserted-by":"publisher","first-page":"1650062","DOI":"10.1142\/S0219887816500626","volume":"13","author":"E Bayram","year":"2016","unstructured":"Bayram E, Bilici M (2016) Surface family with a common involute asymptotic curve. Int J Geom Methods Mod Phys 13(5):1650062. https:\/\/doi.org\/10.1142\/S0219887816500626","journal-title":"Int J Geom Methods Mod Phys"},{"issue":"4","key":"9375_CR6","doi-asserted-by":"publisher","first-page":"685","DOI":"10.1002\/cpa.22091","volume":"76","author":"J Bedrossian","year":"2023","unstructured":"Bedrossian J, Germain P, Harrop-Griffiths B (2023) Vortex filament solutions of the Navier-Stokes equations. Commun Pure Appl Math 76(4):685\u2013787. https:\/\/doi.org\/10.1002\/cpa.22091","journal-title":"Commun Pure Appl Math"},{"key":"9375_CR7","doi-asserted-by":"publisher","unstructured":"Bilici M (2017) On the invariants of ruled surfaces generated by the dual involute frenet trihedron. Commun Fac Sci Univ Ank Series A1 66(2):62\u201370. https:\/\/doi.org\/10.1501\/Commua1_0000000801","DOI":"10.1501\/Commua1_0000000801"},{"key":"9375_CR8","doi-asserted-by":"publisher","unstructured":"Bilici M (2023) A survey on timelike-spacelike involute-evolute curve pair. Erzincan Uni J Sci Tech 6(1):49\u201357. https:\/\/doi.org\/10.18185\/erzifbed.1129800","DOI":"10.18185\/erzifbed.1129800"},{"key":"9375_CR9","doi-asserted-by":"publisher","first-page":"1","DOI":"10.5269\/bspm.51594","volume":"41","author":"M Bilici","year":"2023","unstructured":"Bilici M (2023) A new method for designing involute trajectory timelike ruled surfaces in Minkowski 3-space. Bol Soc Parana Mat 41:1\u201311. https:\/\/doi.org\/10.5269\/bspm.51594","journal-title":"Bol Soc Parana Mat"},{"issue":"2","key":"9375_CR10","first-page":"20","volume":"14","author":"M Bilici","year":"2021","unstructured":"Bilici M, Bayram E (2021) Surface construction with a common involute line of curvature. Int J Open Problems Compt Math 14(2):20\u201331","journal-title":"Int J Open Problems Compt Math"},{"issue":"31","key":"9375_CR11","first-page":"1497","volume":"4","author":"M Bilici","year":"2009","unstructured":"Bilici M, Caliskan M (2009) On the involutes of the spacelike curve with a timelike binormal in Minkowski 3-space. Int Math Forum 4(31):1497\u20131509","journal-title":"Int Math Forum"},{"issue":"44","key":"9375_CR12","first-page":"573","volume":"3","author":"M Bilici","year":"2018","unstructured":"Bilici M, Caliskan M (2018) A new perspective on the involutes of the spacelike curve with a spacelike binormal in Minkowski 3-space. J Sci Arts 3(44):573\u2013582","journal-title":"J Sci Arts"},{"key":"9375_CR13","unstructured":"Bilici M, Caliskan M (2011) Some new notes on the involutes of the timelike curves in Minkowski 3-space. Int J Contemp Math Sci 6(41):2019\u20132030"},{"key":"9375_CR14","doi-asserted-by":"crossref","unstructured":"Bilici M, Koseoglu G (2023) Tubular involutive surfaces with Frenet frame in Euclidean 3-space. Maejo Int J Sci Technol 17(2):96\u2013106","DOI":"10.1016\/j.heliyon.2023.e18822"},{"issue":"3","key":"9375_CR15","first-page":"199","volume":"16","author":"M Bilici","year":"2022","unstructured":"Bilici M, Palavar S (2022) New-type tangent indicatrix of involute and ruled surface according to Blaschke frame in dual space. Maejo Int J Sci Technol 16(3):199\u2013207","journal-title":"Maejo Int J Sci Technol"},{"key":"9375_CR16","volume-title":"A history of mathematics","author":"C Boyer","year":"1968","unstructured":"Boyer C (1968) A history of mathematics. Wiley, New York"},{"issue":"2","key":"9375_CR17","first-page":"289","volume":"21","author":"M Caliskan","year":"2002","unstructured":"Caliskan M, Bilici M (2002) Some characterizations for the pair of involute-evolute curves in Euclidean space $${E}^3$$. Bull Pure Appl 21(2):289\u2013294","journal-title":"Bull Pure Appl"},{"key":"#cr-split#-9375_CR18.1","unstructured":"Commons W (2023) File:F5 tornado Elie Manitoba 2007.jpg-Wikimedia Commons, the free media repository. [Online"},{"key":"#cr-split#-9375_CR18.2","unstructured":"Accessed 23 Aug 2023]. https:\/\/commons.wikimedia.org\/w\/index.php?title=File:F5_tornado_Elie_Manitoba_2007.jpg &oldid=792557327"},{"key":"9375_CR19","doi-asserted-by":"publisher","first-page":"1331","DOI":"10.1007\/s11425-020-1839-5","volume":"64","author":"Q Ding","year":"2021","unstructured":"Ding Q, Zhong S (2021) On the vortex filament in 3-spaces and its generalizations. Sci China Math 64:1331\u20131348. https:\/\/doi.org\/10.1007\/s11425-020-1839-5","journal-title":"Sci China Math"},{"key":"9375_CR20","volume-title":"Differential geometry of curves and surfaces","author":"M Do Carmo","year":"1976","unstructured":"Do Carmo M (1976) Differential geometry of curves and surfaces. Prentice-Hall Inc., Englewood Cliffs, New Jersey"},{"key":"9375_CR21","doi-asserted-by":"publisher","DOI":"10.1016\/j.ijleo.2022.169915","volume":"270","author":"K Eren","year":"2022","unstructured":"Eren K, Yesmakhanova K, Ersoy S, Myrzakulov R (2022) Involute evolute curve family induced by the coupled dispersionless equations. Optik 270:169915. https:\/\/doi.org\/10.1016\/j.ijleo.2022.169915","journal-title":"Optik"},{"issue":"12","key":"9375_CR22","doi-asserted-by":"publisher","first-page":"4152","DOI":"10.1143\/JPSJ.55.4152","volume":"55","author":"Y Fukumoto","year":"1986","unstructured":"Fukumoto Y, Miyazaki T (1986) N-solitons on a curved vortex filament. J Phys Soc Jpn 55(12):4152\u20134155. https:\/\/doi.org\/10.1143\/JPSJ.55.4152","journal-title":"J Phys Soc Jpn"},{"key":"9375_CR23","unstructured":"Ginibre J, Velo G (1978) On a class of non linear Schr\u00f6dinger equations . III. Special theories in dimensions 1, 2 and 3. Annales de I\u2019I.H.P. Ann I\u2019I.H.P. Phys Th\u00e9or 28(3):287\u2013316"},{"key":"9375_CR24","doi-asserted-by":"publisher","DOI":"10.1007\/s00028-023-00897-9","author":"A Gu\u00e9rin","year":"2023","unstructured":"Gu\u00e9rin A (2023) On self-similar singularity formation for the binormal flow. J Evol Equ. https:\/\/doi.org\/10.1007\/s00028-023-00897-9","journal-title":"J Evol Equ"},{"issue":"10","key":"9375_CR25","doi-asserted-by":"publisher","first-page":"1156","DOI":"10.1063\/1.1706500","volume":"5","author":"FR Hama","year":"1962","unstructured":"Hama FR (1962) Progressive deformation of a curved vortex filament by its own induction. Phys Fluids 5(10):1156\u20131162","journal-title":"Phys Fluids"},{"issue":"1","key":"9375_CR26","doi-asserted-by":"publisher","first-page":"293","DOI":"10.1143\/JPSJ.31.293","volume":"31","author":"H Hasimoto","year":"1971","unstructured":"Hasimoto H (1971) Motion of a vortex filament and its relation to elastica. J Phys Soc Jpn 31(1):293\u2013294. https:\/\/doi.org\/10.1143\/JPSJ.31.293","journal-title":"J Phys Soc Jpn"},{"key":"9375_CR27","doi-asserted-by":"publisher","unstructured":"Hasimoto H (1972) A soliton on a vortex filament. J Fluid Mech 51(3):477\u2013485. https:\/\/doi.org\/10.1017\/S0022112072002307","DOI":"10.1017\/S0022112072002307"},{"key":"9375_CR28","doi-asserted-by":"publisher","unstructured":"Khalifa Saad M (2023) On the harmonic evolute of time-like Hasimoto surfaces in Lorentz-Minkowski space. Int J Geom Methods Mod Phys 2350206. https:\/\/doi.org\/10.1142\/S0219887823502067","DOI":"10.1142\/S0219887823502067"},{"issue":"8","key":"9375_CR29","doi-asserted-by":"publisher","first-page":"18822","DOI":"10.1016\/j.heliyon.2023.e18822","volume":"9","author":"G Koseoglu","year":"2023","unstructured":"Koseoglu G, Bilici M (2023) Involutive sweeping surfaces with Frenet frame in Euclidean 3-space. Heliyon 9(8):18822. https:\/\/doi.org\/10.1016\/j.heliyon.2023.e18822","journal-title":"Heliyon"},{"key":"9375_CR30","unstructured":"Millman RS, Parker GD (1977) Elements of differential geometry. Prentice-Hall Inc., Englewoodd Cliffs, New Jersey"},{"issue":"30","key":"9375_CR31","doi-asserted-by":"publisher","first-page":"2118","DOI":"10.1016\/j.physleta.2014.05.010","volume":"378","author":"R Myrzakulov","year":"2014","unstructured":"Myrzakulov R, Mamyrbekova GK, Nugmanova GN, Yesmakhanova KR, Lakshmanan M (2014) Integrable motion of curves in self-consistent potentials: relation to spin systems and soliton equations. Phys Lett A 378(30):2118\u20132123. https:\/\/doi.org\/10.1016\/j.physleta.2014.05.010","journal-title":"Phys Lett A"},{"key":"9375_CR32","doi-asserted-by":"publisher","first-page":"433","DOI":"10.1007\/BF02101741","volume":"162","author":"T Nishiyama","year":"1994","unstructured":"Nishiyama T, Tani A (1994) Solvability of the localized induction equation for vortex motion. Commun Math Phys 162:433\u2013445. https:\/\/doi.org\/10.1007\/BF02101741","journal-title":"Commun Math Phys"},{"issue":"1","key":"9375_CR33","doi-asserted-by":"publisher","first-page":"117","DOI":"10.5666\/KMJ.2018.58.1.117","volume":"58","author":"G Ozturk","year":"2018","unstructured":"Ozturk G, Arslan K, Bulca B (2018) A characterization of involutes and evolutes of a given curve in $${E}^n$$. Kyungpook Math J 58(1):117\u2013135. https:\/\/doi.org\/10.5666\/KMJ.2018.58.1.117","journal-title":"Kyungpook Math J"},{"issue":"1","key":"9375_CR34","first-page":"39","volume":"15","author":"S Palavar","year":"2022","unstructured":"Palavar S, Bilici M (2022) Dual ruled surface constructed by the pole curve of the involute curve. Int J Open Problems Compt Math 15(1):39\u201353","journal-title":"Int J Open Problems Compt Math"},{"key":"9375_CR35","doi-asserted-by":"publisher","DOI":"10.1007\/s00500-023-08848-9","author":"J Qian","year":"2023","unstructured":"Qian J, Sun M, Zhang B (2023) Involutes of null Cartan curves and their representations in Minkowski 3-space. Soft Comput. https:\/\/doi.org\/10.1007\/s00500-023-08848-9","journal-title":"Soft Comput"},{"key":"9375_CR36","doi-asserted-by":"publisher","first-page":"910","DOI":"10.1145\/1073204.1073282","volume":"24","author":"A Selle","year":"2005","unstructured":"Selle A, Rasmussen N, Fedkiw R (2005) A Vortex Particle Method for Smoke. Water and Explosions 24:910\u2013914. https:\/\/doi.org\/10.1145\/1073204.1073282","journal-title":"Water and Explosions"},{"key":"9375_CR37","doi-asserted-by":"publisher","unstructured":"Senoussi B, Akbay Kelleci A (2022) Characterizations of Hasimoto surfaces in Euclidean 3-spaces $${E}^3$$. Appl Math Inf Sci 16(5):689\u2013694. https:\/\/doi.org\/10.18576\/amis\/160504","DOI":"10.18576\/amis\/160504"},{"key":"9375_CR38","first-page":"113","volume":"1","author":"S Senyurt","year":"2015","unstructured":"Senyurt S, Bilici M, Caliskan M (2015) Some characterizations for the involute curves in dual space. Int J Math Comput Sci 1:113\u2013125","journal-title":"Int J Math Comput Sci"}],"container-title":["Soft Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00500-023-09375-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00500-023-09375-3\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00500-023-09375-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,1,4]],"date-time":"2024-01-04T15:12:21Z","timestamp":1704381141000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00500-023-09375-3"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,11,13]]},"references-count":39,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2024,1]]}},"alternative-id":["9375"],"URL":"https:\/\/doi.org\/10.1007\/s00500-023-09375-3","relation":{"has-preprint":[{"id-type":"doi","id":"10.21203\/rs.3.rs-3134305\/v1","asserted-by":"object"}]},"ISSN":["1432-7643","1433-7479"],"issn-type":[{"value":"1432-7643","type":"print"},{"value":"1433-7479","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,11,13]]},"assertion":[{"value":"10 October 2023","order":1,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"13 November 2023","order":2,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare that they have no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}},{"value":"This article does not contain any studies with human participants performed by any of the authors.","order":3,"name":"Ethics","group":{"name":"EthicsHeading","label":"Ethical approval"}}]}}