{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,13]],"date-time":"2025-10-13T14:11:51Z","timestamp":1760364711406,"version":"build-2065373602"},"reference-count":28,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2020,1,28]],"date-time":"2020-01-28T00:00:00Z","timestamp":1580169600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The known three-component reaction\u2013diffusion system modeling competition and co-existence of different language speakers is under study. A modification of this system is proposed, which is examined by the Lie symmetry method; furthermore, exact solutions in the form of traveling fronts are constructed and their properties are identified. Plots of the traveling fronts are presented and the relevant interpretation describing the language shift that has occurred in Ukraine during the Soviet times is suggested.<\/jats:p>","DOI":"10.3390\/e22020154","type":"journal-article","created":{"date-parts":[[2020,1,29]],"date-time":"2020-01-29T10:51:07Z","timestamp":1580295067000},"page":"154","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Exact Solutions of a Mathematical Model Describing Competition and Co-Existence of Different Language Speakers"],"prefix":"10.3390","volume":"22","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1733-5240","authenticated-orcid":false,"given":"Roman","family":"Cherniha","sequence":"first","affiliation":[{"name":"Institute of Mathematics, National Academy of Sciences of Ukraine, 3, Tereshchenkivs\u2019ka Street, 01004 Kyiv, Ukraine"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Vasyl\u2019","family":"Davydovych","sequence":"additional","affiliation":[{"name":"Institute of Mathematics, National Academy of Sciences of Ukraine, 3, Tereshchenkivs\u2019ka Street, 01004 Kyiv, Ukraine"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,1,28]]},"reference":[{"doi-asserted-by":"crossref","unstructured":"Cherniha, R., Serov, M., and Pliukhin, O. 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