{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,10]],"date-time":"2026-04-10T11:21:59Z","timestamp":1775820119108,"version":"3.50.1"},"reference-count":50,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2025,7,29]],"date-time":"2025-07-29T00:00:00Z","timestamp":1753747200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Imam Mohammad Ibn Saud Islamic University","award":["IMSIU-DDRSP2502"],"award-info":[{"award-number":["IMSIU-DDRSP2502"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>This study explores the (1+1)-dimensional Klein\u2013Fock\u2013Gordon equation, a distinct third-order nonlinear differential equation of significant theoretical interest. The Klein\u2013Fock\u2013Gordon equation (KFGE) plays a pivotal role in theoretical physics, modeling high-energy particles and providing a fundamental framework for simulating relativistic wave phenomena. To find the exact solution of the proposed model, for this purpose, we utilized two effective techniques, including the sine-Gordon equation method and a new extended direct algebraic method. The novelty of these approaches lies in the form of different solutions such as hyperbolic, trigonometric, and rational functions, and their graphical representations demonstrate the different form of solitons like kink solitons, bright solitons, dark solitons, and periodic waves. To illustrate the characteristics of these solutions, we provide two-dimensional, three-dimensional, and contour plots that visualize the magnitude of the (1+1)-dimensional Klein\u2013Fock\u2013Gordon equation. By selecting suitable values for physical parameters, we demonstrate the diversity of soliton structures and their behaviors. The results highlighted the effectiveness and versatility of the sine-Gordon equation method and a new extended direct algebraic method, providing analytical solutions that deepen our insight into the dynamics of nonlinear models. These results contribute to the advancement of soliton theory in nonlinear optics and mathematical physics.<\/jats:p>","DOI":"10.3390\/axioms14080590","type":"journal-article","created":{"date-parts":[[2025,7,29]],"date-time":"2025-07-29T16:16:10Z","timestamp":1753805770000},"page":"590","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Exact Solutions and Soliton Transmission in Relativistic Wave Phenomena of Klein\u2013Fock\u2013Gordon Equation via Subsequent Sine-Gordon Equation Method"],"prefix":"10.3390","volume":"14","author":[{"given":"Muhammad","family":"Uzair","sequence":"first","affiliation":[{"name":"Center for High Energy Physics, University of the Punjab, Quaid-e-Azam Campus, Lahore 54590, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0009-0003-6122-0779","authenticated-orcid":false,"given":"Ali H.","family":"Tedjani","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11564, Saudi Arabia"}]},{"given":"Irfan","family":"Mahmood","sequence":"additional","affiliation":[{"name":"Center for High Energy Physics, University of the Punjab, Quaid-e-Azam Campus, Lahore 54590, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0009-0002-7410-3267","authenticated-orcid":false,"given":"Ejaz","family":"Hussain","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of the Punjab, Quaid-e-Azam Campus, Lahore 54590, Pakistan"}]}],"member":"1968","published-online":{"date-parts":[[2025,7,29]]},"reference":[{"key":"ref_1","first-page":"827","article-title":"New Soliton Wave Solutions to a Nonlinear Equation Arising in Plasma Physics","volume":"137","author":"Almatrafi","year":"2023","journal-title":"CMES-Comput. 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