{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,25]],"date-time":"2026-02-25T21:17:15Z","timestamp":1772054235028,"version":"3.50.1"},"reference-count":32,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2022,11,24]],"date-time":"2022-11-24T00:00:00Z","timestamp":1669248000000},"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"]}]},{"name":"National Natural Science Foundation of China","award":["Z202225"],"award-info":[{"award-number":["Z202225"]}]},{"name":"SuQian Sci&amp;Tech Program","award":["11971475"],"award-info":[{"award-number":["11971475"]}]},{"name":"SuQian Sci&amp;Tech Program","award":["Z202225"],"award-info":[{"award-number":["Z202225"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this paper, we first generalize the Dirac spectral problem to isospectral and non-isospectral problems and use the Tu scheme to derive the hierarchy of some new soliton evolution equations. Then, integrable coupling is obtained by solving the isospectral and non-isospectral zero curvature equations.We find that the obtained hierarchy has the bi-Hamiltonian structure of the combined form. In particular, one of the integrable soliton hierarchies is reduced to be similar to the coupled nonlinear Sch\u00f6rdinger system in the AKNS hierarchy. Next, the strict self-adjointness of the reduced equation system is verified, and conservation laws are constructed with the aid of the Ibragimov method. In addition, we apply the extended Kudryashov method to obtain some exact solutions of this reduced equation system.<\/jats:p>","DOI":"10.3390\/sym14122489","type":"journal-article","created":{"date-parts":[[2022,11,24]],"date-time":"2022-11-24T02:25:36Z","timestamp":1669256736000},"page":"2489","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Integrable Coupling of Expanded Isospectral and Non-Isospectral Dirac Hierarchy and Its Reduction"],"prefix":"10.3390","volume":"14","author":[{"given":"Cheng","family":"Chen","sequence":"first","affiliation":[{"name":"School of Mathematics, China University of Mining and Technology, Xuzhou 221116, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jian","family":"Zhou","sequence":"additional","affiliation":[{"name":"School of Mathematics, China University of Mining and Technology, Xuzhou 221116, China"},{"name":"College of Mathematics, Suqian University, Suqian 223800, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Shiyin","family":"Zhao","sequence":"additional","affiliation":[{"name":"School of Mathematics, China University of Mining and Technology, Xuzhou 221116, China"},{"name":"College of Mathematics, Suqian University, Suqian 223800, 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"}]}],"member":"1968","published-online":{"date-parts":[[2022,11,24]]},"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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