{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,17]],"date-time":"2026-03-17T16:59:55Z","timestamp":1773766795565,"version":"3.50.1"},"reference-count":28,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2015,5,22]],"date-time":"2015-05-22T00:00:00Z","timestamp":1432252800000},"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":["11371326"],"award-info":[{"award-number":["11371326"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>A direct approach is proposed for constructing conservation laws of discrete evolution equations, regardless of the existence of a Lagrangian. The approach utilizes pairs of symmetries and adjoint symmetries, in which adjoint symmetries make up for the disadvantage of non-Lagrangian structures in presenting a correspondence between symmetries and conservation laws. Applications are made for the construction of conservation laws of the Volterra lattice equation.<\/jats:p>","DOI":"10.3390\/sym7020714","type":"journal-article","created":{"date-parts":[[2015,5,26]],"date-time":"2015-05-26T04:16:36Z","timestamp":1432613796000},"page":"714-725","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":61,"title":["Conservation Laws of Discrete Evolution Equations by Symmetries and Adjoint Symmetries"],"prefix":"10.3390","volume":"7","author":[{"given":"Wen-Xiu","family":"Ma","sequence":"first","affiliation":[{"name":"College of Mathematics and Physics, Shanghai University of Electric Power, Shanghai 200090, China"},{"name":"Department of Mathematics and Statistics, University of South Florida, Tampa, FL 33620, USA"}]}],"member":"1968","published-online":{"date-parts":[[2015,5,22]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Bluman, G., and Kumei, S. (1989). 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Lett."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"311","DOI":"10.1016\/j.jmaa.2006.10.078","article-title":"A New Conservation Theorem","volume":"333","author":"Ibragimov","year":"2007","journal-title":"J. Math. Anal. Appl."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Ibragimov, N.H. (2011). Nonlinear Self-Adjointness and Conservation Laws. J. Phys. A Math. Theor., 44.","DOI":"10.1088\/1751-8113\/44\/43\/432002"},{"key":"ref_8","unstructured":"Volterra, V. (1931). Le\u00e7ons sur la Th\u00e9orie Math\u00e9matique de la Lutte pour la vie, Gauthier-Villars. In French."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"2400","DOI":"10.1063\/1.532872","article-title":"Algebraic Structure of Discrete Zero Curvature Equations and Master Symmetries of Discrete Evolution Equations","volume":"40","author":"Ma","year":"1999","journal-title":"J. Math. Phys."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Clarkson, P.A., and Nijhoff, F.W. 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Phys."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"2375","DOI":"10.1088\/0305-4470\/22\/13\/031","article-title":"On Liouville Integrability of Zero-Curvature Equations and the Yang Hierarchy","volume":"22","author":"Tu","year":"1989","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"10787","DOI":"10.1088\/0305-4470\/39\/34\/013","article-title":"Hamiltonian and Quasi-Hamiltonian Structures Associated with Semi-Direct Sums of Lie Algebras","volume":"39","author":"Ma","year":"2006","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1156","DOI":"10.1063\/1.523777","article-title":"A Simple Model of the Integrable Hamiltonian Equation","volume":"19","author":"Magri","year":"1978","journal-title":"J. Math. Phys."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"43","DOI":"10.1016\/0167-2789(82)90049-5","article-title":"Canonical Structure of Soliton Equations I","volume":"5","author":"Alberty","year":"1982","journal-title":"Phys. D Nonlinear Phenom."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"573","DOI":"10.1016\/S0960-0779(01)00238-7","article-title":"The Conservation Laws of Some Discrete Soliton Systems","volume":"14","author":"Zhang","year":"2002","journal-title":"Chaos Solitons Fractals"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"47","DOI":"10.1016\/0167-2789(81)90004-X","article-title":"Symplectic Structures, Their B\u00e4cklund Transformations and Hereditary Symmetries","volume":"4","author":"Fuchssteiner","year":"1981","journal-title":"Phys. D Nonlinear Phenom."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"913","DOI":"10.1360\/csb1979-24-20-913","article-title":"Relationship Between Symmetries and Conservation Laws of Nonlinear Evolution Equations","volume":"24","author":"Tu","year":"1979","journal-title":"Chin. Sci. Bull."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"3197","DOI":"10.1088\/0951-7715\/26\/12\/3197","article-title":"Integrability Properties of the Differential-Difference Kadomtsev-Petviashvili Hierarchy and Continuum Limits","volume":"26","author":"Fu","year":"2013","journal-title":"Nonlinearity"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1943","DOI":"10.1088\/0305-4470\/28\/7\/016","article-title":"The Symmetry in the Structure of Dynamical and Adjoint Symmetries of Second-Order Differential Equations","volume":"28","author":"Morando","year":"1995","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"2877","DOI":"10.1063\/1.533277","article-title":"Adjoint Symmetries, Separability, and Volume Forms","volume":"41","author":"Sarlet","year":"2000","journal-title":"J. Math. Phys."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Mohanasubha, R., Chandrasekar, V.K., Senthilvelan, M., and Lakshmanan, M. (2014). Interplay of Symmetries, Null Forms, Darboux Polynomials, Integrating Factors and Jacobi Multipliers in Integrable Second-Order Differential Equations. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 470.","DOI":"10.1098\/rspa.2013.0656"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"5329","DOI":"10.1088\/0305-4470\/25\/20\/014","article-title":"The Algebraic Structures of Isospectral Lax Operators and Applications to Integrable Equations","volume":"25","author":"Ma","year":"1992","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"2464","DOI":"10.1063\/1.529616","article-title":"Lax Representations and Lax Operator Algebras of Isospectral and Nonisospectral Hierarchies of Evolution Equations","volume":"33","author":"Ma","year":"1992","journal-title":"J. Math. Phys."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1336","DOI":"10.9734\/BJAST\/2014\/3094","article-title":"Lie Algebra Structures Associated with Zero Curvature Equations and Generalized Zero Curvature Equations","volume":"3","author":"Ma","year":"2013","journal-title":"Br. J. Appl. Sci. Tech."},{"key":"ref_28","unstructured":"Ge, M.L. (1993). Proceedings of the 21st International Conference on the Differential Geometry Methods in Theoretical Physics, World Scientific."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/7\/2\/714\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T20:46:47Z","timestamp":1760215607000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/7\/2\/714"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,5,22]]},"references-count":28,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2015,6]]}},"alternative-id":["sym7020714"],"URL":"https:\/\/doi.org\/10.3390\/sym7020714","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2015,5,22]]}}}