{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:10:12Z","timestamp":1760058612891,"version":"build-2065373602"},"reference-count":34,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2025,4,15]],"date-time":"2025-04-15T00:00:00Z","timestamp":1744675200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>This paper focuses on the investigation of the Yu\u2013Toda\u2013Sasa\u2013Fukuyama (YTSF) equation in its three-dimensional form. Based on the well-known Euler operator, a set of seven non-singular local multipliers is explored. Using these seven non-singular multipliers, the corresponding local conservation laws are derived. Additionally, an auxiliary potential-related system of partial differential equations (PDEs) is constructed. This study delves into nonlocal systems, which reveal numerous intriguing exact solutions of the YTSF equation. The nonlinear systems exhibit stable structures such as kink solitons, representing transitions, and breather or multi-solitons, modeling localized energy packets and complex interactions. These are employed in materials science, optics, communications, and plasma. Additionally, patterns such as parabolic backgrounds with ripples inform designs involving structured or varying media such as waveguides.<\/jats:p>","DOI":"10.3390\/axioms14040298","type":"journal-article","created":{"date-parts":[[2025,4,15]],"date-time":"2025-04-15T06:50:27Z","timestamp":1744699827000},"page":"298","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Unveiling the Transformative Power: Exploring the Nonlocal Potential Approach in the (3 + 1)-Dimensional Yu\u2013Toda\u2013Sasa\u2013Fukuyama Equation"],"prefix":"10.3390","volume":"14","author":[{"given":"Enas Y.","family":"Abu El Seoud","sequence":"first","affiliation":[{"name":"Department of Physics and Engineering Mathematics, Faculty of Engineering, Zagazig University, Zagazig 44519, Egypt"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7800-2768","authenticated-orcid":false,"given":"Ahmed S.","family":"Rashed","sequence":"additional","affiliation":[{"name":"Department of Physics and Engineering Mathematics, Faculty of Engineering, Zagazig University, Zagazig 44519, Egypt"},{"name":"Department of Mathematical Sciences, Saveetha School of Engineering, SIMATS, Chennai 602105, Tamilnadu, India"},{"name":"Department of Basic Science, Faculty of Engineering, Delta University for Science and Technology, Gamasa 11152, Egypt"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1877-3944","authenticated-orcid":false,"given":"Samah M.","family":"Mabrouk","sequence":"additional","affiliation":[{"name":"Department of Physics and Engineering Mathematics, Faculty of Engineering, Zagazig University, Zagazig 44519, Egypt"}]}],"member":"1968","published-online":{"date-parts":[[2025,4,15]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"3337","DOI":"10.1088\/0305-4470\/31\/14\/018","article-title":"N soliton solutions to the bogoyavlenskii-schiff equation and a quest for the soliton solution in (3 + 1) dimensions","volume":"31","author":"Yu","year":"1998","journal-title":"J. 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