{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,11]],"date-time":"2026-06-11T10:38:57Z","timestamp":1781174337296,"version":"3.54.1"},"reference-count":33,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2025,10,21]],"date-time":"2025-10-21T00:00:00Z","timestamp":1761004800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Deanship of Scientific Research at Imam Mohammad Ibn Saud Islamic University","award":["IMSIU-DDRSP2502"],"award-info":[{"award-number":["IMSIU-DDRSP2502"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>We studied traveling-wave solutions of the Dullin\u2013Gottwald\u2013Holm (DGH) equation via a sub-ODE construction. Under explicit algebraic constraints, the approach yielded closed-form families\u2014bell-shaped, hyperbolic (sech\/tanh), Jacobi-elliptic function (JEF), Weierstrass-elliptic function (WEF), periodic, and rational\u2014and classified their symmetry properties. Optical solitons (bright and dark) arose as limiting cases of the elliptic solutions. We specified the parameter regimes that produced each profile and illustrated representative solutions with 2D\/3D plots to highlight symmetry. The results provide a unified, reproducible procedure for generating solitary and periodic DGH waves and expand the catalog of exact solutions for this model.<\/jats:p>","DOI":"10.3390\/sym17101773","type":"journal-article","created":{"date-parts":[[2025,10,21]],"date-time":"2025-10-21T14:17:35Z","timestamp":1761056255000},"page":"1773","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["Elliptic Functions and Advanced Analysis of Soliton Solutions for the Dullin\u2013Gottwald\u2013Holm Dynamical Equation with Applications of Mathematical Methods"],"prefix":"10.3390","volume":"17","author":[{"given":"Syed T. R.","family":"Rizvi","sequence":"first","affiliation":[{"name":"Department of Mathematics, COMSATS University Islamabad, Lahore Campus, Lahore 54000, Pakistan"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6215-5305","authenticated-orcid":false,"given":"Ibtehal","family":"Alazman","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11566, Saudi Arabia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"family":"Nimra","sequence":"additional","affiliation":[{"name":"Department of Mathematics, COMSATS University Islamabad, Lahore Campus, Lahore 54000, Pakistan"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7412-4773","authenticated-orcid":false,"given":"Aly R.","family":"Seadawy","sequence":"additional","affiliation":[{"name":"Mathematics Department, Faculty of Science, Taibah University, Al-Madinah Al-Munawarah 41411, Saudi Arabia"},{"name":"Basic Sciences Research Center (BSRC), Deanship of Scientific Research, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 13318, Saudi Arabia"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2025,10,21]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"3557","DOI":"10.1016\/j.camwa.2012.09.006","article-title":"Benjamin-Feir-instability in nonlinear dispersive waves","volume":"64","author":"Helal","year":"2012","journal-title":"Comput. 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