{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:16:02Z","timestamp":1760238962504,"version":"build-2065373602"},"reference-count":20,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2020,9,19]],"date-time":"2020-09-19T00:00:00Z","timestamp":1600473600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>A geometrical formulation for adjoint-symmetries as one-forms is studied for general partial differential equations (PDEs), which provides a dual counterpart of the geometrical meaning of symmetries as tangent vector fields on the solution space of a PDE. Two applications of this formulation are presented. Additionally, for systems of evolution equations, adjoint-symmetries are shown to have another geometrical formulation given by one-forms that are invariant under the flow generated by the system on the solution space. This result is generalized to systems of evolution equations with spatial constraints, where adjoint-symmetry one-forms are shown to be invariant up to a functional multiplier of a normal one-form associated with the constraint equations. All of the results are applicable to the PDE systems of interest in applied mathematics and mathematical physics.<\/jats:p>","DOI":"10.3390\/sym12091547","type":"journal-article","created":{"date-parts":[[2020,9,19]],"date-time":"2020-09-19T07:06:09Z","timestamp":1600499169000},"page":"1547","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Geometrical Formulation for Adjoint-Symmetries of Partial Differential Equations"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2125-3627","authenticated-orcid":false,"given":"Stephen C.","family":"Anco","sequence":"first","affiliation":[{"name":"Department of Mathematics and Statistics, Brock University, St. Catharines, ON L2S3A1, Canada"}]},{"given":"Bao","family":"Wang","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, Brock University, St. Catharines, ON L2S3A1, Canada"}]}],"member":"1968","published-online":{"date-parts":[[2020,9,19]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Bluman, G.W., Cheviakov, A., and Anco, S.C. 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