{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:52:50Z","timestamp":1760151170144,"version":"build-2065373602"},"reference-count":22,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2022,2,17]],"date-time":"2022-02-17T00:00:00Z","timestamp":1645056000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11971475"],"award-info":[{"award-number":["11971475"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>By introducing a 3\u00d73 matrix Lie algebra and employing the generalized Tu scheme, a AKNS isospectral\u2013nonisospectral integrable hierarchy is generated by using a third-order matrix Lie algebra. Through a matrix transformation, we turn the 3\u00d73 matrix Lie algebra into a 2\u00d72 matrix case for which we conveniently enlarge it into two various expanding Lie algebras in order to obtain two different expanding integrable models of the isospectral\u2013nonisospectral AKNS hierarchy by employing the integrable coupling theory. Specially, we propose a method for generating nonlinear integrable couplings for the first time, and produce a generalized KdV-Schr\u00f6dinger integrable system and a nonlocal nonlinear Schr\u00f6dinger equation, which indicates that we unite the KdV equation and the nonlinear Schr\u00f6dinger equation as an integrable model by our method. This method presented in the paper could apply to investigate other integrable systems.<\/jats:p>","DOI":"10.3390\/sym14020402","type":"journal-article","created":{"date-parts":[[2022,2,17]],"date-time":"2022-02-17T20:26:41Z","timestamp":1645129601000},"page":"402","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Several Isospectral and Non-Isospectral Integrable Hierarchies of Evolution Equations"],"prefix":"10.3390","volume":"14","author":[{"given":"Shiyin","family":"Zhao","sequence":"first","affiliation":[{"name":"College of Mathematics, Suqian University, Suqian 223800, China"},{"name":"School of Mathematics, China University of Mining and Technology, Xuzhou 221116, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yufeng","family":"Zhang","sequence":"additional","affiliation":[{"name":"School of Mathematics, China University of Mining and Technology, Xuzhou 221116, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6537-2946","authenticated-orcid":false,"given":"Jian","family":"Zhou","sequence":"additional","affiliation":[{"name":"College of Mathematics, Suqian University, Suqian 223800, China"},{"name":"School of Mathematics, China University of Mining and Technology, Xuzhou 221116, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,2,17]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Newell, A.C. 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