{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,14]],"date-time":"2025-11-14T10:41:51Z","timestamp":1763116911408,"version":"3.45.0"},"reference-count":43,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2025,11,14]],"date-time":"2025-11-14T00:00:00Z","timestamp":1763078400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Science Committee of the Ministry of Science and Higher Education of the Republic of Kazakhstan","award":["AP25794905"],"award-info":[{"award-number":["AP25794905"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Nonlinear evolution equations play a key role in modeling various physical processes, such as wave propagation in nonlinear optical and hydrodynamic media, as well as in the dynamics of plasma and quantum systems. In this paper, we study an integrable generalization of the nonlinear Schr\u00f6dinger equation: the Fokas\u2013Lenells (FL) equation. We derive a new (1+1)-dimensional FL equation with self-consistent sources, which enables modeling the interaction of solitons with external disturbances within the framework of integrable systems. For the frist time, we obtain, two distinct types of solutions for the spin system of the FL equation, namely, a traveling wave and a one-soliton solution, derived using the Darboux transformation (DT). We also construct exact one-soliton and two-soliton solutions for the (2+1)-dimensional FL equation using the DT. These results advance analytical methods in the theory of integrable nonlinear systems, including spin models widely used to describe magnetic, quantum, and soliton phenomena. We illustrate the dynamics of the solutions graphically.<\/jats:p>","DOI":"10.3390\/sym17111961","type":"journal-article","created":{"date-parts":[[2025,11,14]],"date-time":"2025-11-14T10:07:03Z","timestamp":1763114823000},"page":"1961","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Modified Fokas\u2013Lenells Equation: Self-Consistent Sources and Soliton Solutions of the Spin and (2+1)-Dimensional Models"],"prefix":"10.3390","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8491-0893","authenticated-orcid":false,"given":"Meruyert","family":"Zhassybayeva","sequence":"first","affiliation":[{"name":"Department of General and Theoretical Physics, L.N. Gumilyov Eurasian National University, Astana 010000, Kazakhstan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7351-0472","authenticated-orcid":false,"given":"Kuralay","family":"Yesmakhanova","sequence":"additional","affiliation":[{"name":"Department of Mathematical and Computer Modeling, L.N. Gumilyov Eurasian National University, Astana 010000, Kazakhstan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4047-4484","authenticated-orcid":false,"given":"Zhaidary","family":"Myrzakulova","sequence":"additional","affiliation":[{"name":"Department of Algebra and Geometry, L.N. Gumilyov Eurasian National University, Astana 010000, Kazakhstan"}]}],"member":"1968","published-online":{"date-parts":[[2025,11,14]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Wazwaz, A.M. (2009). Partial Differential Equations and Solitary Waves Theory, Springer.","DOI":"10.1007\/978-3-642-00251-9"},{"key":"ref_2","unstructured":"Dodd, R.K., Eilbeck, J.C., Gibbon, J.D., and Morris, H.C. 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