{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,13]],"date-time":"2026-01-13T07:16:27Z","timestamp":1768288587580,"version":"3.49.0"},"reference-count":56,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2024,8,19]],"date-time":"2024-08-19T00:00:00Z","timestamp":1724025600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Ministry of Science and Technology of China","award":["G2023016011L"],"award-info":[{"award-number":["G2023016011L"]}]},{"name":"Ministry of Science and Technology of China","award":["G2021016032L"],"award-info":[{"award-number":["G2021016032L"]}]},{"name":"Ministry of Science and Technology of China","award":["12271488"],"award-info":[{"award-number":["12271488"]}]},{"name":"Ministry of Science and Technology of China","award":["11975145"],"award-info":[{"award-number":["11975145"]}]},{"name":"National Natural Science Foundation of China","award":["G2023016011L"],"award-info":[{"award-number":["G2023016011L"]}]},{"name":"National Natural Science Foundation of China","award":["G2021016032L"],"award-info":[{"award-number":["G2021016032L"]}]},{"name":"National Natural Science Foundation of China","award":["12271488"],"award-info":[{"award-number":["12271488"]}]},{"name":"National Natural Science Foundation of China","award":["11975145"],"award-info":[{"award-number":["11975145"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>This study introduces a 4\u00d74 matrix eigenvalue problem and develops an integrable hierarchy with a bi-Hamiltonian structure. Integrability is ensured by the zero-curvature condition, while the Hamiltonian structure is supported by the trace identity. Explicit derivations yield second-order and third-order integrable equations, illustrating the integrable hierarchy.<\/jats:p>","DOI":"10.3390\/axioms13080563","type":"journal-article","created":{"date-parts":[[2024,8,19]],"date-time":"2024-08-19T10:11:28Z","timestamp":1724062288000},"page":"563","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":17,"title":["An Integrated Integrable Hierarchy Arising from a Broadened Ablowitz\u2013Kaup\u2013Newell\u2013Segur Scenario"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5309-1493","authenticated-orcid":false,"given":"Wen-Xiu","family":"Ma","sequence":"first","affiliation":[{"name":"Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China"},{"name":"Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia"},{"name":"Department of Mathematics and Statistics, University of South Florida, Tampa, FL 33620-5700, USA"},{"name":"Material Science Innovation and Modelling, Department of Mathematical Sciences, North-West University, Mafikeng Campus, Mmabatho 2735, South Africa"}]}],"member":"1968","published-online":{"date-parts":[[2024,8,19]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"467","DOI":"10.1002\/cpa.3160210503","article-title":"Integrals of nonlinear equations of evolution and solitary waves","volume":"21","author":"Lax","year":"1968","journal-title":"Comm. 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