{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,15]],"date-time":"2026-05-15T09:19:58Z","timestamp":1778836798294,"version":"3.51.4"},"reference-count":24,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2018,8,15]],"date-time":"2018-08-15T00:00:00Z","timestamp":1534291200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this work, we study the completely integrable sixth-order nonlinear Ramani equation. By applying the Lie symmetry analysis technique, the Lie point symmetries and the optimal system of one-dimensional sub-algebras of the equation are derived. The optimal system is further used to derive the symmetry reductions and exact solutions. In conjunction with the Riccati Bernoulli sub-ODE (RBSO), we construct the travelling wave solutions of the equation by solving the ordinary differential equations (ODEs) obtained from the symmetry reduction. We show that the equation is nonlinearly self-adjoint and construct the conservation laws (CL) associated with the Lie symmetries by invoking the conservation theorem due to Ibragimov. Some figures are shown to show the physical interpretations of the acquired results.<\/jats:p>","DOI":"10.3390\/sym10080341","type":"journal-article","created":{"date-parts":[[2018,8,15]],"date-time":"2018-08-15T10:40:07Z","timestamp":1534329607000},"page":"341","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":39,"title":["Symmetry Analysis, Explicit Solutions, and Conservation Laws of a Sixth-Order Nonlinear Ramani Equation"],"prefix":"10.3390","volume":"10","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9756-7374","authenticated-orcid":false,"given":"Aliyu","family":"Aliyu","sequence":"first","affiliation":[{"name":"Department of Mathematics, Science Faculty, Firat University, Elazig 23119, Turkey"},{"name":"Department of Mathematics, Faculty of Science, Federal University Dutse, Dutse 71560, Nigeria"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4996-8373","authenticated-orcid":false,"given":"Mustafa","family":"Inc","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Science Faculty, Firat University, Elazig 23119, Turkey"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8308-7943","authenticated-orcid":false,"given":"Abdullahi","family":"Yusuf","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Science Faculty, Firat University, Elazig 23119, Turkey"},{"name":"Department of Mathematics, Faculty of Science, Federal University Dutse, Dutse 71560, Nigeria"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0286-7244","authenticated-orcid":false,"given":"Dumitru","family":"Baleanu","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Cankaya University, Ankara 06790, Turkey"},{"name":"Institute of Space Sciences, Magurele 077126, Romania"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2018,8,15]]},"reference":[{"key":"ref_1","unstructured":"Whitham, G.B. 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