{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:10:32Z","timestamp":1760145032507,"version":"build-2065373602"},"reference-count":42,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2024,6,14]],"date-time":"2024-06-14T00:00:00Z","timestamp":1718323200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"NSF of China","doi-asserted-by":"publisher","award":["12271334"],"award-info":[{"award-number":["12271334"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In the paper, we describe a method for deriving generalized symmetries for a generic discrete quadrilateral equation that allows a Lax pair. Its symmetry can be interpreted as a flow along the tangent direction of its solution evolving with a Lie group parameter t. Starting from the spectral problem of the quadrilateral equation and assuming the eigenfunction evolves with the parameter t, one can obtain a differential-difference equation hierarchy, of which the flows are proved to be commuting symmetries of the quadrilateral equation. We prove this result by using the zero-curvature representations of these flows. As an example, we apply this method to derive symmetries for the lattice potential Korteweg\u2013de Vries equation.<\/jats:p>","DOI":"10.3390\/sym16060744","type":"journal-article","created":{"date-parts":[[2024,6,14]],"date-time":"2024-06-14T10:42:34Z","timestamp":1718361754000},"page":"744","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["On Symmetries of Integrable Quadrilateral Equations"],"prefix":"10.3390","volume":"16","author":[{"given":"Junwei","family":"Cheng","sequence":"first","affiliation":[{"name":"School of Information Science and Engineering, Shandong Agricultural University, Taian 271018, China"}]},{"given":"Jin","family":"Liu","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Shanghai University, Shanghai 200444, China"},{"name":"Newtouch Center for Mathematics, Shanghai University, Shanghai 200444, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3691-4165","authenticated-orcid":false,"given":"Da-jun","family":"Zhang","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Shanghai University, Shanghai 200444, China"},{"name":"Newtouch Center for Mathematics, Shanghai University, Shanghai 200444, China"}]}],"member":"1968","published-online":{"date-parts":[[2024,6,14]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"337","DOI":"10.1016\/0375-9601(91)90955-8","article-title":"Similarity reductions of integrable lattices and discrete analogues of the Painlev\u00e9 II equation","volume":"153","author":"Nijhoff","year":"1991","journal-title":"Phys. Lett. A"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"235201","DOI":"10.1088\/1751-8121\/ab8b36","article-title":"Symmetries of ZN graded discrete integrable systems","volume":"53","author":"Fordy","year":"2020","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"10357","DOI":"10.1088\/0305-4470\/34\/48\/302","article-title":"Integrable hierarchies of nonlinear difference-difference equations and symmetries","volume":"34","author":"Levi","year":"2001","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"4141","DOI":"10.1088\/1751-8113\/40\/15\/006","article-title":"Continuous symmetries of the lattice potential KdV equation","volume":"40","author":"Levi","year":"2007","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"325202","DOI":"10.1088\/1751-8113\/44\/32\/325202","article-title":"Method for searching higher symmetries for quad-graph equations","volume":"44","author":"Garifullin","year":"2011","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"12753","DOI":"10.1088\/1751-8113\/40\/42\/S18","article-title":"The lattice Schwarzian KdV equation and its symmetries","volume":"40","author":"Levi","year":"2007","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"454012","DOI":"10.1088\/1751-8113\/42\/45\/454012","article-title":"The generalized symmetry method for discrete equations","volume":"42","author":"Levi","year":"2009","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Levi, D., Winternitz, P., and Yamilov, R.I. (2023). Continuous Symmetries and Integrability of Discrete Equations, AMS.","DOI":"10.1090\/crmm\/038"},{"key":"ref_9","unstructured":"Mikhailov, A.V. (2023, December 01). Formal Diagonalisation of the Lax-Darboux Scheme and Conservation Laws of Integrable Partial Differential, Differential Difference and Partial Difference. Available online: https:\/\/www.newton.ac.uk\/event\/disw05\/."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"795","DOI":"10.18255\/1818-1015-2015-6-795-817","article-title":"Formal diagonalisation of Lax-Darboux schemes","volume":"22","author":"Mikhailov","year":"2015","journal-title":"Model. Anal. Inform. Syst."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"421","DOI":"10.1007\/s11232-011-0033-y","article-title":"Recursion operators, conservation laws and integrability conditions for difference equations","volume":"167","author":"Mikhailov","year":"2011","journal-title":"Theor. Math. Phys."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"235201","DOI":"10.1088\/1751-8113\/43\/23\/235201","article-title":"Infinitely many symmetries and conservation laws for quad-graph equations via the Gardner method","volume":"43","author":"Rasin","year":"2010","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"253","DOI":"10.1111\/j.1467-9590.2007.00385.x","article-title":"Symmetries of integrable difference equations on the quad-graph","volume":"119","author":"Rasin","year":"2007","journal-title":"Stud. Appl. Math."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"13353","DOI":"10.1088\/1751-8113\/40\/44\/015","article-title":"Affine linear and D4 symmetric lattice equations: Symmetry analysis and reductions","volume":"40","author":"Tongas","year":"2007","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"807","DOI":"10.1007\/s00220-019-03548-8","article-title":"Rational recursion operators for integrable differential-difference equations","volume":"370","author":"Carpentier","year":"2019","journal-title":"Commun. Math. Phys."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"5524","DOI":"10.1063\/1.531742","article-title":"Lie algebraic structures of (1+1)-dimensional Lax integrable systems","volume":"37","author":"Chen","year":"1996","journal-title":"J. Math. Phys."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"377","DOI":"10.1088\/0305-4470\/24\/2\/010","article-title":"Lie algebraic structure for the AKNS system","volume":"24","author":"Chen","year":"1991","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"761","DOI":"10.1016\/S0960-0779(02)00178-9","article-title":"Lie algebraic structures of some (1+2)-dimensional Lax integrable systems","volume":"15","author":"Chen","year":"2003","journal-title":"Chaos Solitons Fractals"},{"key":"ref_19","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_20","doi-asserted-by":"crossref","first-page":"404","DOI":"10.1111\/sapm.12647","article-title":"Symmetries of the D\u0394mKP hierarchy and their continuum limits","volume":"152","author":"Liu","year":"2024","journal-title":"Stud. Appl. Math."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"2707","DOI":"10.1088\/0305-4470\/23\/13\/011","article-title":"K symmetries and \u03c4 symmetries of evolution equations and their Lie algebras","volume":"23","author":"Ma","year":"1990","journal-title":"J. Phys. A: Math. Gen."},{"key":"ref_22","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_23","doi-asserted-by":"crossref","first-page":"393","DOI":"10.1111\/j.1467-9590.2010.00493.x","article-title":"Symmetries for the Ablowitz-Ladik hierarchy: Part I. Four-potential case","volume":"125","author":"Zhang","year":"2010","journal-title":"Stud. Appl. Math."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"419","DOI":"10.1111\/j.1467-9590.2010.00494.x","article-title":"Symmetries for the Ablowitz-Ladik hierarchy: Part II. Integrable discrete nonlinear Schr\u00f6dinger equations and discrete AKNS hierarchy","volume":"125","author":"Zhang","year":"2010","journal-title":"Stud. Appl. Math."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"458","DOI":"10.1016\/j.physleta.2006.06.077","article-title":"New symmetries for the Ablowitz-Ladik hierarchies","volume":"359","author":"Zhang","year":"2006","journal-title":"Phys. Lett. A"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"8160","DOI":"10.1002\/mma.10008","article-title":"The \u03c4-symmetries and Lie algebra structure of the Blaszak-Marciniak lattice equation","volume":"47","author":"Zhang","year":"2024","journal-title":"Math. Method Appl. Sci."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"211","DOI":"10.1016\/S0076-5392(08)62801-5","article-title":"Hamiltonian structure and integrability","volume":"185","author":"Fuchssteiner","year":"1991","journal-title":"Math. Sci. Eng."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"7225","DOI":"10.1088\/0305-4470\/35\/33\/316","article-title":"Hamiltonian structure of discrete soliton systems","volume":"35","author":"Zhang","year":"2002","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Olver, P.J. (1993). Applications of Lie Groups to Differential Equations, Springer.","DOI":"10.1007\/978-1-4612-4350-2"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"133","DOI":"10.1007\/BF00994631","article-title":"The discrete Korteweg-de Vries equation","volume":"39","author":"Nijhoff","year":"1995","journal-title":"Acta Appl. Math."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"125","DOI":"10.1016\/0375-9601(83)90192-5","article-title":"Direct linearization of nonlinear difference-difference equations","volume":"97","author":"Nijhoff","year":"1983","journal-title":"Phys. Lett. A"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"513","DOI":"10.1007\/s00220-002-0762-8","article-title":"Classification of integrable equationson quad-graphs. The consistency approach","volume":"233","author":"Adler","year":"2003","journal-title":"Commun. Math. Phys."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"1386","DOI":"10.1103\/PhysRevLett.31.1386","article-title":"B\u00e4cklund transformation for solutions of the Korteweg-de Vries equation","volume":"31","author":"Wahlquist","year":"1973","journal-title":"Phys. Rev. Lett."},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Hietarinta, J., Joshi, N., and Nijhoff, F.W. (2016). Discrete Systems and Integrability, Cambridge University Press.","DOI":"10.1017\/CBO9781107337411"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"050202","DOI":"10.1088\/0256-307X\/29\/5\/050202","article-title":"Lax pairs for discrete integrable equations via Darboux transformations","volume":"29","author":"Cao","year":"2012","journal-title":"Chin. Phys. Lett."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"34","DOI":"10.2991\/jnmp.2000.7.1.4","article-title":"On the structure of the B\u00e4cklund transformations for the relativistic lattices","volume":"7","author":"Adler","year":"2000","journal-title":"J. Nonlinear Math. Phys."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"10453","DOI":"10.1088\/0305-4470\/34\/48\/310","article-title":"Discrete equations on planar graphs","volume":"34","author":"Adler","year":"2001","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_38","first-page":"060","article-title":"Multi-component extension of CAC systems","volume":"16","author":"Zhang","year":"2020","journal-title":"Symmetry Integr. Geom. Meth. Appl."},{"key":"ref_39","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_40","unstructured":"Miwa, T., Jimbo, M., and Date, E. (2000). Solitons: Differential Equations, Symmetries and Infinite Dimensional Algebras, Cambridge University Press."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"152","DOI":"10.13108\/2021-13-2-152","article-title":"Yamilov\u2019s theorem for differential and difference equations","volume":"13","author":"Levi","year":"2021","journal-title":"Ufa Math. J."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"199","DOI":"10.1016\/0378-4371(87)90024-0","article-title":"Lattice equations, hierarchies and Hamiltonian structures","volume":"142","author":"Wiersma","year":"1987","journal-title":"Phys. A"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/6\/744\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T14:59:07Z","timestamp":1760108347000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/6\/744"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,6,14]]},"references-count":42,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2024,6]]}},"alternative-id":["sym16060744"],"URL":"https:\/\/doi.org\/10.3390\/sym16060744","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2024,6,14]]}}}