{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,30]],"date-time":"2026-04-30T23:22:28Z","timestamp":1777591348165,"version":"3.51.4"},"reference-count":37,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2023,10,4]],"date-time":"2023-10-04T00:00:00Z","timestamp":1696377600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Ministry of Science and Higher Education of the Republic of Kazakhstan","award":["AP14870191"],"award-info":[{"award-number":["AP14870191"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>This article concerns new analytical wave solutions of the Kuralay-II equations (K-IIAE and K-IIBE) with exploration of a new definition of the derivative. This model is used in various fields, like nonlinear optics, ferromagnetic materials and optical fibers. For this purpose, the expa function, the extended sinh-Gordon equation expansion scheme, and the generalized Kudryashov schemes were utilized. The resulting solutions are dark, bright, dark-bright, periodic, singular and other kinds of solitons. These results are obtained and also verified by the Mathematica tool. Some of the solutions are explained with 2-D, 3-D and contour plots using the Mathematica tool. The solutions obtained succede the present solutions in the literature. For the first time, the effect of the fractional derivative on the solutions is also shown graphically for this model. The analytical wave solutions are highly desirable as they offer insights into the underlying physics or mathematics of a system and provide a framework for further analysis. The results obtained can also be fruitful for the development of models in the future. The schemes used in this research are effective, easy to apply, and reliably handle other fractional non-linear partial differential equations.<\/jats:p>","DOI":"10.3390\/sym15101862","type":"journal-article","created":{"date-parts":[[2023,10,4]],"date-time":"2023-10-04T07:47:52Z","timestamp":1696405672000},"page":"1862","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":32,"title":["Exact Solutions of M-Fractional Kuralay Equation via Three Analytical Schemes"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2242-8529","authenticated-orcid":false,"given":"Asim","family":"Zafar","sequence":"first","affiliation":[{"name":"Department of Mathematics, COMSATS University Islamabad, Vehari Campus, Vehari 61120, Pakistan"}]},{"given":"Muhammad","family":"Raheel","sequence":"additional","affiliation":[{"name":"Department of Mathematics, COMSATS University Islamabad, Vehari Campus, Vehari 61120, Pakistan"}]},{"given":"Mohamed R.","family":"Ali","sequence":"additional","affiliation":[{"name":"Faculty of Engineering and Technology, Future University in Egypt, New Cairo 11835, Egypt"},{"name":"Basic Engineering Science Department, Benha Faculty of Engineering, Benha University, Benha 13518, Egypt"}]},{"given":"Zhaidary","family":"Myrzakulova","sequence":"additional","affiliation":[{"name":"Ratbay Myrzakulov Eurasian International Centre for Theoretical Physics, Astana 010000, Kazakhstan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9394-4681","authenticated-orcid":false,"given":"Ahmet","family":"Bekir","sequence":"additional","affiliation":[{"name":"Neighbourhood of Akcaglan, Imarli Street, Number: 28\/4, Eskisehir 26030, Turkey"}]},{"given":"Ratbay","family":"Myrzakulov","sequence":"additional","affiliation":[{"name":"Ratbay Myrzakulov Eurasian International Centre for Theoretical Physics, Astana 010000, Kazakhstan"}]}],"member":"1968","published-online":{"date-parts":[[2023,10,4]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1950155","DOI":"10.1142\/S0217732319501554","article-title":"Generalized exponential rational function method for extended Zakharov\u2013Kuzetsov equation with conformable derivative","volume":"34","author":"Ghanbari","year":"2019","journal-title":"Mod. 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