{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,26]],"date-time":"2026-03-26T13:07:25Z","timestamp":1774530445819,"version":"3.50.1"},"reference-count":45,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2022,6,13]],"date-time":"2022-06-13T00:00:00Z","timestamp":1655078400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Khalifa University"},{"name":"A. Abdeljabbar"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>This article reflects on the Klein\u2013Gordon model, which frequently arises in the fields of solid-state physics and quantum field theories. We analytically delve into solitons and composite rogue-type wave propagation solutions of the model via the generalized Kudryashov and the extended Sinh Gordon expansion approaches. We obtain a class of analytically exact solutions in the forms of exponential and hyperbolic functions involving some arbitrary parameters with the help of Maple, which included comparing symmetric and non-symmetric solutions with other methods. After analyzing the dynamical behaviors, we caught distinct conditions on the accessible parameters of the solutions for the model. By applying conditions to the existing parameters, we obtained various types of rogue waves, bright and dark bells, combing bright\u2013dark, combined dark\u2013bright bells, kink and anti-kink solitons, and multi-soliton solutions. The nature of the solitons is geometrically explained for particular choices of the arbitrary parameters. It is indicated that the nonlinear rogue-type wave packets are restricted in two dimensions that characterized the rogue-type wave envelopes.<\/jats:p>","DOI":"10.3390\/sym14061223","type":"journal-article","created":{"date-parts":[[2022,6,14]],"date-time":"2022-06-14T01:43:32Z","timestamp":1655171012000},"page":"1223","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":32,"title":["Bright, Dark, and Rogue Wave Soliton Solutions of the Quadratic Nonlinear Klein\u2013Gordon Equation"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5775-516X","authenticated-orcid":false,"given":"Alrazi","family":"Abdeljabbar","sequence":"first","affiliation":[{"name":"Department of Mathematics, Khalifa University, Abu Dhabi P.O. Box 127788, United Arab Emirates"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1687-623X","authenticated-orcid":false,"given":"Harun-Or","family":"Roshid","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Pabna University of Science and Technology, Pabna 6600, Bangladesh"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Abdullah","family":"Aldurayhim","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science and Humanities in Alkharj, Prince Sattam bin Abdulaziz University, Alkharj 11942, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,6,13]]},"reference":[{"key":"ref_1","unstructured":"Kharif, C., Pelinovsky, E., and Slunyaev, A. (2009). 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