{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:25:09Z","timestamp":1760239509214,"version":"build-2065373602"},"reference-count":28,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2020,12,1]],"date-time":"2020-12-01T00:00:00Z","timestamp":1606780800000},"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":["No.11971475"],"award-info":[{"award-number":["No.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>The Frobenius KDV equation and the Frobenius KP equation are introduced, and the Frobenius Kompaneets equation, Frobenius Burgers equation and Frobenius Harry Dym equation are constructed by taking values in a commutative subalgebra Z2\u03b5 in the paper. The five equations are selected as examples to help us study the self-adjointness of Frobenius type equations, and we show that the first two equations are quasi self-adjoint and the last three equations are nonlinear self-adjointness. It follows that we give the symmetries of the Frobenius KDV and the Frobenius KP equation in order to construct the corresponding conservation laws.<\/jats:p>","DOI":"10.3390\/sym12121987","type":"journal-article","created":{"date-parts":[[2020,12,1]],"date-time":"2020-12-01T13:08:51Z","timestamp":1606828131000},"page":"1987","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Self-Adjointness and Conservation Laws of Frobenius Type Equations"],"prefix":"10.3390","volume":"12","author":[{"given":"Haifeng","family":"Wang","sequence":"first","affiliation":[{"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"}]}],"member":"1968","published-online":{"date-parts":[[2020,12,1]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1095","DOI":"10.1103\/PhysRevLett.19.1095","article-title":"Method for solving the Korteweg-de Vries equation","volume":"19","author":"Gardner","year":"1967","journal-title":"Phys. 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