{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,16]],"date-time":"2026-04-16T15:17:21Z","timestamp":1776352641710,"version":"3.51.2"},"reference-count":43,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2023,10,25]],"date-time":"2023-10-25T00:00:00Z","timestamp":1698192000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Deanship of Scientific Research at Najran University","award":["NU\/DRP\/SERC\/12\/24"],"award-info":[{"award-number":["NU\/DRP\/SERC\/12\/24"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this manuscript, we implement analytical multiple soliton wave and singular soliton wave solutions for coupled mKdV with a time-dependent variable coefficient. Based on the similarity transformation and Hirota bilinear technique, we construct both multiple wave kink and wave singular kink solutions for coupled mKdV with a time-dependent variable coefficient. We implement the Hirota bilinear technique to compute analytical solutions for the coupled mKdV system. Such calculations are made by using a software with symbolic computation software, for instance, Maple. Recently some researchers used Maple in order to show that the bilinear method of Hirota is a straightforward technique which can be used in the approach of differential, nonlinear models. We analyzed whether the experiments proved that the procedure is effective and can be successfully used for many other mathematical models used in physics and engineering. The results of this study display that the profiles of multiple-kink and singular-kink soliton types can be efficiently controlled by selecting the particular form of a similar time variable. The changes in the solitons based on the changes in the arbitrary function of time allows for more applications of them in applied sciences.<\/jats:p>","DOI":"10.3390\/sym15111972","type":"journal-article","created":{"date-parts":[[2023,10,25]],"date-time":"2023-10-25T09:54:08Z","timestamp":1698227648000},"page":"1972","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Multiple Soliton Solutions for Coupled Modified Korteweg\u2013de Vries (mkdV) with a Time-Dependent Variable Coefficient"],"prefix":"10.3390","volume":"15","author":[{"given":"Haroon D. S.","family":"Adam","sequence":"first","affiliation":[{"name":"Department of Basic Sciences, Najran University, P.O. Box 1988, Najran 61441, Saudi Arabia"}]},{"given":"Khalid I. A.","family":"Ahmed","sequence":"additional","affiliation":[{"name":"Department of Basic Sciences, Najran University, P.O. Box 1988, Najran 61441, Saudi Arabia"}]},{"given":"Mukhtar Yagoub","family":"Youssif","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, College of Science, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1552-3763","authenticated-orcid":false,"given":"Marin","family":"Marin","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Computer Science, Transilvania University of Brasov, 500036 Brasov, Romania"},{"name":"Academy of Romanian Scientists, Ilfov Street, No. 3, 050045 Bucharest, Romania"}]}],"member":"1968","published-online":{"date-parts":[[2023,10,25]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"2469","DOI":"10.1007\/s11071-018-4503-8","article-title":"Painlev\u00e9 analysis and invariant solutions of generalized fifth-order nonlinear integrable equation","volume":"94","author":"Kaur","year":"2018","journal-title":"Nonlinear Dyn."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"2209","DOI":"10.1007\/s11071-018-4686-z","article-title":"Complex simplified Hirota\u2019s forms and lie symmetry analysis for multiple real and complex soliton solutions","volume":"95","author":"Wazwaz","year":"2019","journal-title":"Nonlinear Dyn."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"2050167","DOI":"10.1142\/S0217984920501675","article-title":"Multiple lump solutions and dynamics of the generalized (3+1)-dimensional KP equation","volume":"34","author":"He","year":"2020","journal-title":"Mod. 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