{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,31]],"date-time":"2026-01-31T18:09:58Z","timestamp":1769882998199,"version":"3.49.0"},"reference-count":32,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2024,7,18]],"date-time":"2024-07-18T00:00:00Z","timestamp":1721260800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Ministry of Science and Technology of China","award":["G2021016032L"],"award-info":[{"award-number":["G2021016032L"]}]},{"name":"Ministry of Science and Technology of China","award":["G2023016011L"],"award-info":[{"award-number":["G2023016011L"]}]},{"name":"Ministry of Science and Technology of China","award":["12271488"],"award-info":[{"award-number":["12271488"]}]},{"name":"Ministry of Science and Technology of China","award":["11975145"],"award-info":[{"award-number":["11975145"]}]},{"name":"Ministry of Science and Technology of China","award":["11972291"],"award-info":[{"award-number":["11972291"]}]},{"name":"National Natural Science Foundation of China","award":["G2021016032L"],"award-info":[{"award-number":["G2021016032L"]}]},{"name":"National Natural Science Foundation of China","award":["G2023016011L"],"award-info":[{"award-number":["G2023016011L"]}]},{"name":"National Natural Science Foundation of China","award":["12271488"],"award-info":[{"award-number":["12271488"]}]},{"name":"National Natural Science Foundation of China","award":["11975145"],"award-info":[{"award-number":["11975145"]}]},{"name":"National Natural Science Foundation of China","award":["11972291"],"award-info":[{"award-number":["11972291"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>The paper aims to demonstrate that a linear expansion in a unital two-dimensional algebra can generate integrable couplings, proposing a novel approach for their construction. The integrable couplings presented encompass a range of perturbation equations and nonlinear integrable couplings. Their corresponding Lax pairs and hereditary recursion operators are explicitly detailed. Concrete applications to the KdV equation and the AKNS system of nonlinear Schr\u00f6dinger equations are extensively explored.<\/jats:p>","DOI":"10.3390\/axioms13070481","type":"journal-article","created":{"date-parts":[[2024,7,18]],"date-time":"2024-07-18T10:39:26Z","timestamp":1721299166000},"page":"481","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":34,"title":["Integrable Couplings and Two-Dimensional Unital Algebras"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5309-1493","authenticated-orcid":false,"given":"Wen-Xiu","family":"Ma","sequence":"first","affiliation":[{"name":"Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China"},{"name":"Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia"},{"name":"Department of Mathematics and Statistics, University of South Florida, Tampa, FL 33620-5700, USA"},{"name":"Material Science Innovation and Modelling, Department of Mathematical Sciences, North-West University, Mafikeng Campus, Mmabatho 2735, South Africa"}]}],"member":"1968","published-online":{"date-parts":[[2024,7,18]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1227","DOI":"10.1016\/0960-0779(95)00104-2","article-title":"Integrable theory of the perturbation equations","volume":"7","author":"Ma","year":"1996","journal-title":"Chaos Solitons Fractals"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"125","DOI":"10.1016\/j.physleta.2005.09.087","article-title":"Semi-direct sums of Lie algebras and continuous integrable couplings","volume":"351","author":"Ma","year":"2006","journal-title":"Phys. Lett. A"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"2643","DOI":"10.1016\/j.camwa.2007.10.012","article-title":"Two integrable couplings of the Tu hierarchy and their Hamiltonian structures","volume":"55","author":"Li","year":"2008","journal-title":"Comput. Math. Appl."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"395201","DOI":"10.1088\/1751-8113\/42\/39\/395201","article-title":"Integrable couplings of relativistic Toda lattice systems in polynomial form and rational form, their hierarchies and bi-Hamiltonian structures","volume":"42","author":"Xu","year":"2009","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1109","DOI":"10.1016\/j.chaos.2007.04.027","article-title":"Coupling commutator pairs and integrable systems","volume":"39","author":"Zhang","year":"2009","journal-title":"Chaos Solitons Fractals"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"123510","DOI":"10.1063\/1.3669484","article-title":"Nonlinear super integrable Hamiltonian couplings","volume":"52","author":"You","year":"2011","journal-title":"J. Math. Phys."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"3925","DOI":"10.1002\/mma.3618","article-title":"Integrable couplings of fractional L-hierarchy and its Hamiltonian structures","volume":"39","author":"Wu","year":"2006","journal-title":"Math. Methods Appl. Sci."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"106075","DOI":"10.1016\/j.cnsns.2021.106075","article-title":"A new multi-component integrable coupling and its application to isospectral and nonisospectral problems","volume":"105","author":"Wang","year":"2022","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"2250160","DOI":"10.1142\/S0219887822501602","article-title":"Integrable couplings of two expanded non-isospectral soliton hierarchies and their bi-Hamiltonian structures","volume":"19","author":"Wang","year":"2022","journal-title":"Int. J. Geom. Methods Mod. Phys."},{"key":"ref_10","first-page":"7238","article-title":"Nonlinear continuous integrable Hamiltonian couplings","volume":"217","author":"Ma","year":"2011","journal-title":"Appl. Math. Comput."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"238","DOI":"10.1016\/j.physa.2004.06.070","article-title":"The multi-component coupled Burgers hierarchy of soliton equations and its multi-component integrable couplings system with two arbitrary functions","volume":"343","author":"Xia","year":"2004","journal-title":"Phys. A"},{"key":"ref_12","first-page":"344","article-title":"An integrable coupling hierarchy of the Mkdv_integrable systems, its Hamiltonian structure and corresponding nonisospectral integrable hierarchy","volume":"216","author":"Xu","year":"2010","journal-title":"Appl. Math. Comput."},{"key":"ref_13","first-page":"71","article-title":"Integrable nonlinear perturbed hierarchies of NLS-mKDV equation and soliton solutions","volume":"2022","author":"Zhao","year":"2022","journal-title":"Electr. J. Differ. Equ."},{"key":"ref_14","first-page":"105","article-title":"Integrable couplings and matrix loop algebras","volume":"Volume 1562","author":"Ma","year":"2013","journal-title":"Nonlinear and Modern Mathematical Physics"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"2601","DOI":"10.1016\/j.camwa.2010.08.076","article-title":"Constructing nonlinear discrete integrable Hamiltonian couplings","volume":"60","author":"Ma","year":"2010","journal-title":"Comput. Math. Appl."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"283","DOI":"10.1007\/BF01692479","article-title":"\u00dcber Systeme complexer Zahlen und ihre Anwendung in der Theorie der Transformationsgruppen","volume":"1","author":"Study","year":"1890","journal-title":"Monatshefte Math. Phys."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"849","DOI":"10.1016\/0362-546X(79)90052-X","article-title":"Application of hereditary symmetries to nonlinear evolution equations","volume":"3","author":"Fuchssteiner","year":"1979","journal-title":"Nonlinear Anal."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"957","DOI":"10.1088\/0305-4470\/33\/5\/311","article-title":"Bi-differential calculi and integrable models","volume":"33","author":"Dimakis","year":"2000","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"511","DOI":"10.1016\/S0375-9601(98)00555-6","article-title":"Extending Hamiltonian operators to get bi-Hamiltonian coupled KdV systems","volume":"246","author":"Ma","year":"1998","journal-title":"Phys. Lett. A"},{"key":"ref_20","unstructured":"G\u00fcrses, M., and Pekcan, A. (2024). On SK and KK integrable systems. arXiv."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1149","DOI":"10.1088\/0253-6102\/71\/10\/1149","article-title":"Fifth-order Alice-Bob systems and their abundant periodic and solitary wave solutions","volume":"71","author":"Zhao","year":"2019","journal-title":"Commun. Theor. Phys."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"249","DOI":"10.1002\/sapm1974534249","article-title":"The inverse scattering transform-Fourier analysis for nonlinear problems","volume":"53","author":"Ablowitz","year":"1974","journal-title":"Stud. Appl. Math."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"075001","DOI":"10.1088\/1572-9494\/ad3dd9","article-title":"A combined Liouville integrable hierarchy associated with a fourth-order matrix spectral problem","volume":"76","author":"Ma","year":"2024","journal-title":"Commun. Theor. Phys."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"101","DOI":"10.59277\/RomJPhys.2024.69.101","article-title":"Four-component Liouville integrable models and their bi-Hamiltonian formulations","volume":"69","author":"Yang","year":"2024","journal-title":"Rom. J. Phys."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1763","DOI":"10.1142\/S0217984909019922","article-title":"Applications of the Lie algebra gl(2)","volume":"23","author":"Zhang","year":"2009","journal-title":"Mod. Phys. Lett. B"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"441","DOI":"10.1016\/j.physleta.2005.05.013","article-title":"Bilinear forms and B\u00e4cklund transformations of the perturbation systems","volume":"341","author":"Ma","year":"2005","journal-title":"Phys. Lett. A"},{"key":"ref_27","first-page":"163","article-title":"New non-isospectral integrable couplings of the AKNS system","volume":"203","author":"Sun","year":"2008","journal-title":"Appl. Math. Comput."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1028","DOI":"10.1016\/j.jmaa.2006.10.084","article-title":"Exact controllability of the nonlinear third-order dispersion equation","volume":"332","author":"George","year":"2007","journal-title":"J. Math. Anal. Appl."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"100713","DOI":"10.1016\/j.padiff.2024.100713","article-title":"Fractional stochastic Schr\u00f6dinger evolution system with complex potential and poisson jumps: Qualitative behavior and T-controllability","volume":"10","author":"Sandrasekaran","year":"2024","journal-title":"Partial Differ. Equ. Appl. Math."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"207","DOI":"10.1007\/s11401-012-0702-7","article-title":"Loop algebras and bi-integrable couplings","volume":"33","author":"Ma","year":"2012","journal-title":"Chin. Ann. Math. Ser. B"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"377","DOI":"10.1515\/ijnsns-2013-0011","article-title":"Tri-integrable couplings by matrix loop algebras","volume":"14","author":"Ma","year":"2013","journal-title":"Int. J. Nonlinear Sci. Numer. Simul."},{"key":"ref_32","first-page":"627924","article-title":"Tri-integrable couplings of the Giachetti-Johnson soliton hierarchy as well as their Hamiltonian structure","volume":"2014","author":"Wang","year":"2014","journal-title":"Abstr. Appl. Anal."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/7\/481\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T15:18:54Z","timestamp":1760109534000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/7\/481"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,7,18]]},"references-count":32,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2024,7]]}},"alternative-id":["axioms13070481"],"URL":"https:\/\/doi.org\/10.3390\/axioms13070481","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,7,18]]}}}