{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:00:49Z","timestamp":1760144449833,"version":"build-2065373602"},"reference-count":29,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2024,4,23]],"date-time":"2024-04-23T00:00:00Z","timestamp":1713830400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Scientific Research Start-Up Foundation of Jimei University","award":["C622125","JAT220172"],"award-info":[{"award-number":["C622125","JAT220172"]}]},{"name":"Fujian Provincial Education Department","award":["C622125","JAT220172"],"award-info":[{"award-number":["C622125","JAT220172"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>By using the classical Lie algebra, the stationary zero curvature equation, and the Lenard recursion equations, we obtain the non-isospectral TD hierarchy. Two kinds of expanding higher-dimensional Lie algebras are presented by extending the classical Lie algebra. By solving the expanded non-isospectral zero curvature equations, the multi-component non-isospectral TD hierarchies are derived. The Hamiltonian structure for one of them is obtained by using the trace identity.<\/jats:p>","DOI":"10.3390\/axioms13050282","type":"journal-article","created":{"date-parts":[[2024,4,23]],"date-time":"2024-04-23T08:08:27Z","timestamp":1713859707000},"page":"282","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["A Class of Multi-Component Non-Isospectral TD Hierarchies and Their Bi-Hamiltonian Structures"],"prefix":"10.3390","volume":"13","author":[{"given":"Jianduo","family":"Yu","sequence":"first","affiliation":[{"name":"School of Science, Jimei University, Xiamen 361021, China"},{"name":"School of Mathematics and Statistics, Ningbo University, Ningbo 315211, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1058-7111","authenticated-orcid":false,"given":"Haifeng","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Science, Jimei University, Xiamen 361021, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,4,23]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"233","DOI":"10.1007\/3-540-09971-9_40","article-title":"Nonlinear Evolution Equations and Dynamical Systems","volume":"Volume 120","author":"Magri","year":"1980","journal-title":"Springer Lecture Notes in Physics"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"330","DOI":"10.1063\/1.528449","article-title":"The trace identity, a powerful tool for constructing the Hamiltonian structure of integrable systems","volume":"30","author":"Tu","year":"1989","journal-title":"J. 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