{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,23]],"date-time":"2026-01-23T07:12:44Z","timestamp":1769152364357,"version":"3.49.0"},"reference-count":39,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2021,1,29]],"date-time":"2021-01-29T00:00:00Z","timestamp":1611878400000},"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>A comprehensive study of the negative-order Kadomtsev\u2013Petviashvili (nKP) partial differential equation by Lie group method has been presented. Initially the infinitesimal generators and symmetry reduction, which were obtained by applying the Lie group method on the negative-order Kadomtsev\u2013Petviashvili equation, have been used for constructing the reduced equations. In particular, the traveling wave solutions for the negative-order KP equation have been derived from the reduced equations as an invariant solution. Finally, the extended improved (G\u2032\/G) method and the extended tanh method are described and applied in constructing new explicit expressions for the traveling wave solutions. Many new and more general exact solutions are obtained.<\/jats:p>","DOI":"10.3390\/sym13020224","type":"journal-article","created":{"date-parts":[[2021,1,29]],"date-time":"2021-01-29T09:25:22Z","timestamp":1611912322000},"page":"224","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":12,"title":["Lie Group Method for Solving the Negative-Order Kadomtsev\u2013Petviashvili Equation (nKP)"],"prefix":"10.3390","volume":"13","author":[{"given":"Ghaylen","family":"Laouini","sequence":"first","affiliation":[{"name":"College of Engineering and Technology, American University of the Middle East, Egaila 54200, Kuwait"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Amr M.","family":"Amin","sequence":"additional","affiliation":[{"name":"School of Sciences and Engineering, The American University in Cairo, New Cairo 11835, Egypt"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Mohamed","family":"Moustafa","sequence":"additional","affiliation":[{"name":"School of Sciences and Engineering, The American University in Cairo, New Cairo 11835, Egypt"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,1,29]]},"reference":[{"key":"ref_1","unstructured":"Clarkson, P.A., and Priestley, T.J. 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