{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:37:01Z","timestamp":1760150221201,"version":"build-2065373602"},"reference-count":22,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2023,11,3]],"date-time":"2023-11-03T00:00:00Z","timestamp":1698969600000},"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":["11805114","1197050803","2018TDJH101"],"award-info":[{"award-number":["11805114","1197050803","2018TDJH101"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"SDUST Research Fund","award":["11805114","1197050803","2018TDJH101"],"award-info":[{"award-number":["11805114","1197050803","2018TDJH101"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>The 4\u00d74 trace-free complex matrix set is introduced in this study. By using it, we are able to create a novel soliton hierarchy that is reduced to demonstrate its bi-Hamiltonian structure. Additionally, we give the Darboux matrix T, whose elements are connected to the spectral parameter in accordance with the various positions and numbers of the spectral parameter \u03bb. The Darboux transformation approach has also been successfully applicated to superintegrable systems.<\/jats:p>","DOI":"10.3390\/axioms12111032","type":"journal-article","created":{"date-parts":[[2023,11,3]],"date-time":"2023-11-03T02:41:56Z","timestamp":1698979316000},"page":"1032","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Solution of High-Order Nonlinear Integrable Systems Using Darboux Transformation"],"prefix":"10.3390","volume":"12","author":[{"given":"Xinhui","family":"Wu","sequence":"first","affiliation":[{"name":"College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China"},{"name":"Department of Fundamental Course, Shandong University of Science and Technology, Taian 271019, China"}]},{"given":"Jiawei","family":"Hu","sequence":"additional","affiliation":[{"name":"College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China"},{"name":"Department of Fundamental Course, Shandong University of Science and Technology, Taian 271019, China"}]},{"given":"Ning","family":"Zhang","sequence":"additional","affiliation":[{"name":"College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China"},{"name":"Department of Fundamental Course, Shandong University of Science and Technology, Taian 271019, China"}]}],"member":"1968","published-online":{"date-parts":[[2023,11,3]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"3903","DOI":"10.1088\/0305-4470\/23\/17\/020","article-title":"A trace identity and its applications to the theory of discrete integrable systems","volume":"23","author":"Tu","year":"1990","journal-title":"J. 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