{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,14]],"date-time":"2026-01-14T00:31:24Z","timestamp":1768350684328,"version":"3.49.0"},"reference-count":30,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2021,8,1]],"date-time":"2021-08-01T00:00:00Z","timestamp":1627776000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100003725","name":"National Research Foundation of Korea","doi-asserted-by":"publisher","award":["NRF_2019R1A6A1A10073079"],"award-info":[{"award-number":["NRF_2019R1A6A1A10073079"]}],"id":[{"id":"10.13039\/501100003725","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this paper, we propose the extended Boussinesq\u2013Whitham\u2013Broer\u2013Kaup (BWBK)-type equations with variable coefficients and fractional order. We consider the fractional BWBK equations, the fractional Whitham\u2013Broer\u2013Kaup (WBK) equations and the fractional Boussinesq equations with variable coefficients by setting proper smooth functions that are derived from the proposed equation. We obtain uniformly coupled fractional traveling wave solutions of the considered equations by employing the improved system method, and subsequently their asymmetric behaviors are visualized graphically. The result shows that the improved system method is effective and powerful to find explicit traveling wave solutions of the fractional nonlinear evolution equations.<\/jats:p>","DOI":"10.3390\/sym13081396","type":"journal-article","created":{"date-parts":[[2021,8,1]],"date-time":"2021-08-01T21:46:44Z","timestamp":1627854404000},"page":"1396","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Coupled Fractional Traveling Wave Solutions of the Extended Boussinesq\u2013Whitham\u2013Broer\u2013Kaup-Type Equations with Variable Coefficients and Fractional Order"],"prefix":"10.3390","volume":"13","author":[{"given":"Jin Hyuk","family":"Choi","sequence":"first","affiliation":[{"name":"Humanitas College, Kyung Hee University, Yongin 17104, Korea"}]},{"given":"Hyunsoo","family":"Kim","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Sungkyunkwan University, Suwon 16419, Korea"}]}],"member":"1968","published-online":{"date-parts":[[2021,8,1]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"36","DOI":"10.1017\/S0022112010001345","article-title":"A variational approach to Boussinesq modelling of fully nonlinear water waves","volume":"657","author":"Klopman","year":"2010","journal-title":"J. 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