{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:26:58Z","timestamp":1760149618778,"version":"build-2065373602"},"reference-count":23,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2023,8,23]],"date-time":"2023-08-23T00:00:00Z","timestamp":1692748800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Natural Science Foundation of China","award":["11971475"],"award-info":[{"award-number":["11971475"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Based on work related to the R-matrix theory, we first abstract the Lax pairs proposed by Blaszak and Sergyeyev into a unified form. Then, a generalized zero-curvature equation expressed by the Poisson bracket is exhibited. As an application of this theory, a generalized (2+1)-dimensional integrable system is obtained, from which a resulting generalized Davey\u2013Stewartson (DS) equation and a generalized Pavlov equation (gPe) are further obtained. Via the use of a nonisospectral zero-curvature-type equation, some (3+1) -dimensional integrable systems are produced. Next, we investigate the recursion operator of the gPe using an approach under the framework of the R-matrix theory. Furthermore, a type of solution for the resulting linearized equation of the gPe is produced by using its conserved densities. In addition, by applying a nonisospectral Lax pair, a (3+1)-dimensional integrable system is generated and reduced to a Boussinesq-type equation in which the recursion operators and the linearization are produced by using a Lie symmetry analysis; the resulting invertible mappings are presented as well. Finally, a B\u00e4cklund transformation of the Boussinesq-type equation is constructed, which can be used to generate some exact solutions.<\/jats:p>","DOI":"10.3390\/sym15091623","type":"journal-article","created":{"date-parts":[[2023,8,23]],"date-time":"2023-08-23T07:25:24Z","timestamp":1692775524000},"page":"1623","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Applications of the R-Matrix Method in Integrable Systems"],"prefix":"10.3390","volume":"15","author":[{"given":"Binlu","family":"Feng","sequence":"first","affiliation":[{"name":"School of Mathematics and Information Sciences, Weifang University, Weifang 261061, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yufeng","family":"Zhang","sequence":"additional","affiliation":[{"name":"School of Mathematics and Information Sciences, Weifang University, Weifang 261061, China"},{"name":"School of Mathematics, China University of Mining and Technology, Xuzhou 221116, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hongyi","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":[[2023,8,23]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"191","DOI":"10.1016\/S0375-9601(02)00421-8","article-title":"Classical R-matrices on Poisson algebras and related dispersionless systems","volume":"297","author":"Blaszak","year":"2002","journal-title":"Phys. Lett. A"},{"key":"ref_2","first-page":"225","article-title":"Classical R-matrix theory of dispersionless systems I","volume":"35","author":"Blaszak","year":"2001","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"10325","DOI":"10.1088\/0305-4470\/35\/48\/308","article-title":"Classical R-matrix theory of dispersionless system II","volume":"35","author":"Blaszak","year":"2002","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Blaszak, M., and Sergyeyev, A. (2019). Contact Lax pairs and associated (3+1)-dimensional integrable dispersionless systems. arXiv.","DOI":"10.1201\/9780429263743-2"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Sergyeyev, A. (2017). New integrable (3+1)-dimensional systems and contact geometry. arXiv.","DOI":"10.1007\/s11005-017-1013-4"},{"key":"ref_6","unstructured":"Li, Y.S. (1999). Soliton and Integrable System, Shanghai Scientific and Technological Education Publishing House. (In Chinese)."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Sergyeyev, A. (2017). A simple construction of recursion operators for multidimensional dispersionless integrable systems. arXiv.","DOI":"10.1016\/j.jmaa.2017.04.050"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Bluman, G.W., and Kumei, S. (1989). Symmetries and Differential Equations, Springer.","DOI":"10.1007\/978-1-4757-4307-4"},{"key":"ref_9","first-page":"63","article-title":"The B\u00e4cklund transformations and conservation laws of the Boussinesq equation","volume":"4","author":"Tu","year":"1981","journal-title":"Acta Math. Appl. Sin."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"103538","DOI":"10.1016\/j.geomphys.2019.103538","article-title":"A method for generating isospectral and nonisospectral hierarchies of equations as well as symmetries","volume":"147","author":"Zhang","year":"2020","journal-title":"Geom. Phys."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"707","DOI":"10.1007\/s10114-021-0392-8","article-title":"A scheme for generating nonisospectral integrable hierarchies and its related applications","volume":"37","author":"Zhang","year":"2021","journal-title":"Acta Math. Sin. Engl. Ser."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"579","DOI":"10.1007\/s10255-022-1099-z","article-title":"Schemes for generating different nonlinear Schr\u00f6dinger integrable equations and their some properties","volume":"38","author":"Zhang","year":"2022","journal-title":"Acta Math. Appl. Engl. Ser."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"106622","DOI":"10.1016\/j.cnsns.2022.106622","article-title":"Coverings and nonlocal symmetries as well as fundamental solutions of nonlinear equations derived from the nonisospectral AKNS hierarchy","volume":"14","author":"Zhao","year":"2022","journal-title":"Commun. Nonlinear Sci. Numer. Simulat."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"L719","DOI":"10.1088\/0305-4470\/25\/12\/003","article-title":"An approach for constructing nonisospectral hierarchies of evolution equations","volume":"25","author":"Ma","year":"1992","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"377","DOI":"10.1016\/S0378-4371(97)00587-6","article-title":"New hierarchies of isospectral and non-isospectral integrable NLEEs derived from the Harry-Dym spectral problem","volume":"252","author":"Qiao","year":"1998","journal-title":"Physica A"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Ablowitz, M.J., and Segur, H. (1981). Solitons and the Inverse Scattering Transform, SIAM.","DOI":"10.1137\/1.9781611970883"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"3147","DOI":"10.1063\/1.1586967","article-title":"Integrable systems and reductions of the self-dual Yang\u2014CMills equations","volume":"44","author":"Ablowitz","year":"2003","journal-title":"J. Math. Phys."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Newell, A.C. (1985). Solitons in Mathematics and Physics, SIAM.","DOI":"10.1137\/1.9781611970227"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"330","DOI":"10.1063\/1.528449","article-title":"The trace identity, a powerful tool for constructing the Hamiltonian structure of integrable systems","volume":"30","author":"Tu","year":"1989","journal-title":"J. Math. Phys."},{"key":"ref_20","first-page":"79","article-title":"A new hierarchy of Liouville integrable generalized Hamiltonian equations and its reduction","volume":"13","author":"Ma","year":"1992","journal-title":"Chin. J. Contemp. Math."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"369","DOI":"10.1007\/BF02451423","article-title":"A hierarchy of Liouville integrable finite-dimensional Hamiltonian systems","volume":"13","author":"Ma","year":"1992","journal-title":"Appl. Math. Mech."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"2497","DOI":"10.1088\/0305-4470\/27\/7\/026","article-title":"A powerful approach to generate new integrable systems","volume":"27","author":"Hu","year":"1994","journal-title":"J. Phys. A"},{"key":"ref_23","unstructured":"Zhang, Y.F., Liu, Y.Y., Liu, J.G., and Feng, B.L. (2023). A New Non-isospectral Integrable Hierarchy and Some Associated Symmetries. J. Math. Res. Appl."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/9\/1623\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T20:40:02Z","timestamp":1760128802000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/9\/1623"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,8,23]]},"references-count":23,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2023,9]]}},"alternative-id":["sym15091623"],"URL":"https:\/\/doi.org\/10.3390\/sym15091623","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2023,8,23]]}}}