{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:43:36Z","timestamp":1760060616308,"version":"build-2065373602"},"reference-count":28,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2025,9,4]],"date-time":"2025-09-04T00:00:00Z","timestamp":1756944000000},"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 research paper, the negative order Korteweg\u2013de Vries equation expressed as nonlinear partial differential equation, firstly introduced by Wazwaz, is solved for the exact Jacobi elliptic function solution. For this purpose, the Jacobi elliptic function scheme, one of the direct algebraic methods, was used. The obtained exact solutions of the negative-order Korteweg\u2013de Vries equation, a symmetry evolution equation, contains the combination of Jacobi elliptic functions and incomplete elliptic integral of second function. The three unique families of exact solutions are classified and presented. The degeneration of the obtained Jacobi elliptic function solutions into various solitons, periodic and rational solutions, is reported using the modulus transformation of Jacobi elliptic function solutions. The necessary condition existence of certain Jacobi elliptic function solutions is presented. The two-dimensional graphs for certain Jacobi elliptic function solutions are drawn to show the variation in wave propogation with respect to velocity and modulus. The non-existence of certain Jacobi elliptic function solutions for negative-order Korteweg\u2013de Vries equations is reported. Finally, the obtained solutions were compared with the previously obtained solutions of negative-order Korteweg\u2013de Vries equation.<\/jats:p>","DOI":"10.3390\/sym17091447","type":"journal-article","created":{"date-parts":[[2025,9,4]],"date-time":"2025-09-04T08:08:46Z","timestamp":1756973326000},"page":"1447","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["The Jacobi Elliptic Function and Incomplete Elliptic Integral of Second Kind Solutions of the Wazwaz Negative Order Korteweg\u2013de Vries Equation"],"prefix":"10.3390","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5096-5164","authenticated-orcid":false,"given":"Rathinavel","family":"Silambarasan","sequence":"first","affiliation":[{"name":"Department of Computer Science and Engineering, Sri Venkateswara College of Engineering and Technology, (SVCET Autonomous), Chittoor 517127, Andhra Pradesh, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1217-963X","authenticated-orcid":false,"given":"Adem","family":"Kilicman","sequence":"additional","affiliation":[{"name":"Faculty of Computer and Mathematical Sciences, Universiti Teknologi MARA, Shah Alam Selangor 40450, Malaysia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Purushotham","family":"Jyotheeswari","sequence":"additional","affiliation":[{"name":"Department of Computer Science and Engineering, Sri Venkateswara College of Engineering and Technology, (SVCET Autonomous), Chittoor 517127, Andhra Pradesh, India"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,9,4]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.aml.2018.08.004","article-title":"Multiple complex soliton solutions for integrable negative-order KdV and integrable nega-tive-order modified KdV equations","volume":"88","author":"Wazwaz","year":"2019","journal-title":"Appl. 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