{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,8]],"date-time":"2026-07-08T07:43:19Z","timestamp":1783496599447,"version":"3.55.0"},"reference-count":62,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2022,8,31]],"date-time":"2022-08-31T00:00:00Z","timestamp":1661904000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Key Programs of Universities in Henan Province of China","award":["22A140006"],"award-info":[{"award-number":["22A140006"]}]},{"name":"Key Programs of Universities in Henan Province of China","award":["NSFRF210324"],"award-info":[{"award-number":["NSFRF210324"]}]},{"name":"Key Programs of Universities in Henan Province of China","award":["B2018-40"],"award-info":[{"award-number":["B2018-40"]}]},{"name":"Fundamental Research Funds for the Universities of Henan Province","award":["22A140006"],"award-info":[{"award-number":["22A140006"]}]},{"name":"Fundamental Research Funds for the Universities of Henan Province","award":["NSFRF210324"],"award-info":[{"award-number":["NSFRF210324"]}]},{"name":"Fundamental Research Funds for the Universities of Henan Province","award":["B2018-40"],"award-info":[{"award-number":["B2018-40"]}]},{"name":"Program of Henan Polytechnic University","award":["22A140006"],"award-info":[{"award-number":["22A140006"]}]},{"name":"Program of Henan Polytechnic University","award":["NSFRF210324"],"award-info":[{"award-number":["NSFRF210324"]}]},{"name":"Program of Henan Polytechnic University","award":["B2018-40"],"award-info":[{"award-number":["B2018-40"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>This study focuses on investigating the modified Benjamin-Bona-Mahony equation that is used to model the long wave in nonlinear dispersive media of the optical illusion field. Two effective techniques, the variational direct method and He\u2019s frequency formulation method, are employed to seek the travelling wave solutions. Using these two techniques, abundant exact solutions such as the bright wave, bright-dark wave, bright-like wave, kinky-bright wave and periodic wave solutions, are obtained. The 3-D contours and 2-D curves are drawn to present the dynamic physical behaviors of the solutions by assigning the proper parameters. It shows that the proposed methods are effective but simple and only need one or two steps to construct the exact solutions, which are expected to provide some new insights to study the travelling wave solutions of the PDEs arising in physics.<\/jats:p>","DOI":"10.3390\/axioms11090445","type":"journal-article","created":{"date-parts":[[2022,8,31]],"date-time":"2022-08-31T23:53:21Z","timestamp":1661990001000},"page":"445","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":42,"title":["Variational Principle and Diverse Wave Structures of the Modified Benjamin-Bona-Mahony Equation Arising in the Optical Illusions Field"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3905-0844","authenticated-orcid":false,"given":"Kang-Jia","family":"Wang","sequence":"first","affiliation":[{"name":"School of Physics and Electronic Information Engineering, Henan Polytechnic University, Jiaozuo 454003, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2022,8,31]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"085208","DOI":"10.1088\/1402-4896\/ac00e5","article-title":"Contribution of joule heating and viscous dissipation on three dimensional flow of Casson model comprising temperature dependent conductance utilizing shooting method","volume":"96","author":"Sohail","year":"2021","journal-title":"Phys. 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