{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,28]],"date-time":"2026-03-28T03:04:48Z","timestamp":1774667088283,"version":"3.50.1"},"reference-count":45,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2023,8,29]],"date-time":"2023-08-29T00:00:00Z","timestamp":1693267200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Natural Science Foundation of China","award":["11971475"],"award-info":[{"award-number":["11971475"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>A new three-dimensional Lie algebra and its loop algebra are proposed by us, whose commutator is a vector product. Based on this, a positive flow and a negative flow are obtained by introducing a new kind of spectral problem expressed by the vector product, which reduces to a generalized KdV equation, a generalized Schr\u00f6dinger equation, a sine-Gordon equation, and a sinh-Gordon equation. Next, the well-known Tu scheme is generalized for generating isospectral integrable hierarchies and non-isospectral integrable hierarchies. It is important that we make use of the variational method to create a new vector-product trace identity for which the Hamiltonian structure of the isospectral integrable hierarchy presented in the paper is worded out. Finally, we further enlarge the three-dimensional loop algebra into a six-dimensional loop algebra so that a new isospectral integrable hierarchy which is a type of extended integrable model is produced whose bi-Hamiltonian structure is also derived from the vector-product trace identity. This new approach presented in the paper possesses extensive applications in the aspect of generating integrable hierarchies of evolution equations.<\/jats:p>","DOI":"10.3390\/axioms12090840","type":"journal-article","created":{"date-parts":[[2023,8,30]],"date-time":"2023-08-30T10:30:52Z","timestamp":1693391452000},"page":"840","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Using Vector-Product Loop Algebra to Generate Integrable Systems"],"prefix":"10.3390","volume":"12","author":[{"given":"Jian","family":"Zhang","sequence":"first","affiliation":[{"name":"School of Computer Science and Technology, China University of Mining and Technology, Xuzhou 221116, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Binlu","family":"Feng","sequence":"additional","affiliation":[{"name":"School of Mathematics and Information Sciences, Weifang University, Weifang 261061, 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"}]},{"given":"Long","family":"Ju","sequence":"additional","affiliation":[{"name":"School of Mathematics, China University of Mining and Technology, Xuzhou 221116, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,8,29]]},"reference":[{"key":"ref_1","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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