{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:24:59Z","timestamp":1760235899276,"version":"build-2065373602"},"reference-count":57,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2021,9,30]],"date-time":"2021-09-30T00:00:00Z","timestamp":1632960000000},"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>In recent years, symmetry in abstract partial differential equations has found wide application in the field of nonlinear integrable equations. The symmetries of the corresponding transformation groups for such equations make it possible to significantly simplify the procedure for establishing equivalence between nonlinear integrable equations from different areas of physics, which in turn open up opportunities to easily find their solutions. In this paper, we study the symmetry between differential geometry of surfaces\/curves and some integrable generalized spin systems. In particular, we investigate the gauge and geometrical equivalence between the local\/nonlocal nonlinear Schr\u00f6dinger type equations (NLSE) and the extended continuous Heisenberg ferromagnet equation (HFE) to investigate how nonlocality properties of one system are inherited by the other. First, we consider the space curves induced by the nonlinear Schr\u00f6dinger-type equations and its equivalent spin systems. Such space curves are governed by the Serret\u2013Frenet equation (SFE) for three basis vectors. We also show that the equation for the third of the basis vectors coincides with the well-known integrable HFE and its generalization. Two other equations for the remaining two vectors give new integrable spin systems. Finally, we investigated the relation between the differential geometry of surfaces and integrable spin systems for the three basis vectors.<\/jats:p>","DOI":"10.3390\/sym13101827","type":"journal-article","created":{"date-parts":[[2021,10,11]],"date-time":"2021-10-11T01:59:47Z","timestamp":1633917587000},"page":"1827","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["Surfaces and Curves Induced by Nonlinear Schr\u00f6dinger-Type Equations and Their Spin Systems"],"prefix":"10.3390","volume":"13","author":[{"given":"Akbota","family":"Myrzakul","sequence":"first","affiliation":[{"name":"Ratbay Myrzakulov Eurasian International Centre for Theoretical Physics, Nur-Sultan 010009, Kazakhstan"},{"name":"Eurasian International Center for Theoretical Physics and Department of General and Theoretical Physics, Eurasian National University, Astana 010008, Kazakhstan"}]},{"given":"Gulgassyl","family":"Nugmanova","sequence":"additional","affiliation":[{"name":"Ratbay Myrzakulov Eurasian International Centre for Theoretical Physics, Nur-Sultan 010009, Kazakhstan"},{"name":"Eurasian International Center for Theoretical Physics and Department of General and Theoretical Physics, Eurasian National University, Astana 010008, Kazakhstan"}]},{"given":"Nurzhan","family":"Serikbayev","sequence":"additional","affiliation":[{"name":"Ratbay Myrzakulov Eurasian International Centre for Theoretical Physics, Nur-Sultan 010009, Kazakhstan"},{"name":"Eurasian International Center for Theoretical Physics and Department of General and Theoretical Physics, Eurasian National University, Astana 010008, Kazakhstan"}]},{"given":"Ratbay","family":"Myrzakulov","sequence":"additional","affiliation":[{"name":"Ratbay Myrzakulov Eurasian International Centre for Theoretical Physics, Nur-Sultan 010009, Kazakhstan"},{"name":"Eurasian International Center for Theoretical Physics and Department of General and Theoretical Physics, Eurasian National University, Astana 010008, Kazakhstan"}]}],"member":"1968","published-online":{"date-parts":[[2021,9,30]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"354","DOI":"10.1016\/0375-9601(78)90264-5","article-title":"On the geometrical interpretation of solitons","volume":"64","author":"Lakshmanan","year":"1978","journal-title":"Phys. Lett. A"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"17","DOI":"10.1007\/BF01030253","article-title":"Equivalence of the nonlinear Schr\u00f6dinger equation and the equation of a Heisenberg ferromagnet","volume":"38","author":"Zakharov","year":"1979","journal-title":"Theor. Math. Phys."},{"key":"ref_3","unstructured":"Dubrovin, B., Novikov, S., and Fomenko, A. (1981). Modern Geometry (In Rush), Nauka."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"490","DOI":"10.1088\/0031-8949\/20\/3-4\/026","article-title":"Integrability of nonlinear Hamiltonian systems by inverse scattering method","volume":"20","author":"Chen","year":"1979","journal-title":"Phys. Scr."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Peng, W., Pu, J., and Chen, Y. (2021). PINN Deep Learning for the Chen-Lee-Liu Equation: Rogue Wave on the Periodic Background. arXiv.","DOI":"10.1016\/j.cnsns.2021.106067"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Inui, T., Tanabe, Y., and Onodera, Y. (1990). Group Theory and Its Applications in Physics, Springer.","DOI":"10.1007\/978-3-642-80021-4"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"391","DOI":"10.1016\/S0375-9601(97)00457-X","article-title":"A (2+1)-dimensional integrable spin model: Geometrical and gauge equivalent counterpart, solitons and localized coherent structures","volume":"233","author":"Myrzakulov","year":"1997","journal-title":"Phys. Lett. A"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"441","DOI":"10.1007\/BF02557332","article-title":"Geometry and multidimensional soliton equations","volume":"118","author":"Myrzakulov","year":"1999","journal-title":"Theor. Math. Phys."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1352","DOI":"10.3390\/sym7031352","article-title":"Integrable (2+1)-dimensional spin models with self-consistent potentials","volume":"7","author":"Myrzakulov","year":"2015","journal-title":"Symmetry"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1550134","DOI":"10.1142\/S0219887815501340","article-title":"Darboux Transformation Exact Solutions of the Heisenberg Ferromagnetic Equation with Self-Consistent Potentials","volume":"13","author":"Yersultanova","year":"2016","journal-title":"Int. J. Geom. Methods Mod. Phys."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Rogers, C., and Schief, W.K. (2002). Backlund and Darboux Transfotmations. Geometry and Modern Applications in Soliton Theory, Cambridge University Press.","DOI":"10.1017\/CBO9780511606359"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1262","DOI":"10.1103\/PhysRevLett.30.1262","article-title":"Method for solving the Sine-Gordon equation","volume":"30","author":"Ablowitz","year":"1973","journal-title":"Phys. Rev. Lett."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"5","DOI":"10.1103\/PhysRevLett.110.064105","article-title":"Integrable nonlocal nonlinear Schr\u00f6dinger equation","volume":"110","author":"Ablowitz","year":"2013","journal-title":"Phys. Rev. Lett."},{"key":"ref_14","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_15","doi-asserted-by":"crossref","first-page":"590","DOI":"10.1002\/(SICI)1097-0312(200005)53:5<590::AID-CPA2>3.0.CO;2-R","article-title":"Schr\u00f6dinger maps","volume":"53","author":"Chang","year":"2000","journal-title":"Commun. Pure Appl. Math."},{"key":"ref_16","first-page":"117","article-title":"On the motion of an unbounded fluid with a vortex filament of any shape","volume":"22","year":"1906","journal-title":"Rend. Circ. Mat. Palermo"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"49","DOI":"10.1016\/S0375-9601(98)00697-5","article-title":"A note on the NLS and the Schr\u00f6dinger flow of maps","volume":"248","author":"Ding","year":"1998","journal-title":"Phys. Lett. A"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"669","DOI":"10.1016\/j.chaos.2003.12.092","article-title":"Schr?odinger flows, binormal motion of curves and the second AKNS hierarchies","volume":"21","author":"Ding","year":"2004","journal-title":"Chaos Solitons Fractals"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"746","DOI":"10.1007\/BF02901957","article-title":"Schr\u00f6dinger flows of maps into symplectic manifolds","volume":"41","author":"Ding","year":"1998","journal-title":"Sci. China A"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"3201","DOI":"10.1016\/j.physleta.2010.06.001","article-title":"A motion of spacelike curves in the Minkowski 3-space and the KdV equation","volume":"374","author":"Ding","year":"2010","journal-title":"Phys. Lett. A"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"777","DOI":"10.1007\/s11425-018-9350-0","article-title":"The complex 2-sphere in C3 and Schr\u00f6dinger flows","volume":"63","author":"Ding","year":"2021","journal-title":"Sci. China Math."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"373","DOI":"10.1016\/0375-9601(94)90170-8","article-title":"An elementary geometric characterization of the integrable motions of a curve","volume":"185","author":"Doliwa","year":"1994","journal-title":"Phys. Lett. A"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"369","DOI":"10.1017\/S0022112091001143","article-title":"Three-dimensional distortions of a vortex filament with axial velocity","volume":"22","author":"Fukumoto","year":"1991","journal-title":"J. Fluid. Mech."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"062124","DOI":"10.1103\/PhysRevA.93.062124","article-title":"Towards a gauge-equivalent magnetic structure of the nonlocal nonlinear Schr\u00f6dinger equation","volume":"93","author":"Gadzhimuradov","year":"2016","journal-title":"Phys. Rev. A"},{"key":"ref_25","unstructured":"Gollek, H. (2021, July 08). Deformations of Minimal Curves in C3, in Proc: 1-st NOSONGE Conference. Available online: http:\/\/www.cns.gatech.edu\/~danek\/preprints\/gollek.ps.gz."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"53","DOI":"10.1007\/s00025-006-0235-z","article-title":"Duals of vector fields and of null curves","volume":"50","author":"Gollek","year":"2007","journal-title":"Result. Math."},{"key":"ref_27","unstructured":"Gray, A. (1995). Modern Differential Geometry of Curves and Surfaces, CRC Press."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"329","DOI":"10.1016\/S0375-9601(98)00151-0","article-title":"Motion of curves on two-dimensional surfaces and soliton equations","volume":"241","author":"Gurses","year":"1998","journal-title":"Phys. Lett. A"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"17","DOI":"10.1063\/1.4997835","article-title":"Nonlocal nonlinear Schr\u00f6dinger equations and their soliton solutions","volume":"59","author":"Gurses","year":"2018","journal-title":"J. Math. Phys."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"477","DOI":"10.1017\/S0022112072002307","article-title":"A soliton on a vortex filament","volume":"51","author":"Hasimoto","year":"1972","journal-title":"J. Fluid Mech."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"127","DOI":"10.1007\/BF00382623","article-title":"Motion of strings, embedding problem and soliton equations","volume":"37","author":"Lakshmanan","year":"1981","journal-title":"Appl. Sci. Res."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"1654","DOI":"10.1063\/1.523453","article-title":"Solitons on moving space curves","volume":"18","author":"Lamb","year":"1977","journal-title":"J. Math. Phys."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"95","DOI":"10.1016\/0375-9601(83)90149-4","article-title":"On the gauge equivalence of the Landau-Lifshitz and the nonlinear Schr\u00f6dinger equations on symmetric spaces","volume":"95","author":"Makhankov","year":"1983","journal-title":"Phys. Lett. A"},{"key":"ref_34","first-page":"193","article-title":"New geometries associated with the nonlinear Schr\u00f6dinger equation","volume":"29","author":"Murugesh","year":"2002","journal-title":"Eur. Phys. J. B Condens. Matter Phys."},{"key":"ref_35","unstructured":"Novikov, S.P., Manakov, S.V., Pitaevskii, L.P., and Zakharov, V.E. (1984). Theory of Solitons: The Inverse Scattering Method, Plenum."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"163","DOI":"10.1016\/j.geomphys.2016.02.009","article-title":"On holomorphic Riemannian geometry and submanifolds of Wick-related spaces","volume":"104","author":"Pessers","year":"2016","journal-title":"J. Geom. Phys."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"7","DOI":"10.1103\/PhysRevE.89.052918","article-title":"Continuous and discrete Schr\u00f6dinger systems with parity-time-symmetric nonlinearities","volume":"89","author":"Sarma","year":"2014","journal-title":"Phys. Rev. E"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"235","DOI":"10.1090\/amsip\/036\/06","article-title":"Schr\u00f6dinger flows on Grassmannians","volume":"36","author":"Terng","year":"2006","journal-title":"AMS\/IP Stud. Adv. Math."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"195201","DOI":"10.1088\/1751-8121\/ab81d9","article-title":"Nonlocal gauge equivalence: Hirota versus extended continuous Heisenberg and Landau-Lifschitz equation","volume":"53","author":"Cen","year":"2020","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_40","unstructured":"de Laire, A. (2021, June 08). The Landau-Lifshitz Equation and Related Models. Analysis of PDEs [math.AP]. Available online: https:\/\/hal.archives-ouvertes.fr\/tel-02985356\/document."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"7","DOI":"10.1111\/sapm.12153","article-title":"Integrable nonlocal nonlinear equations","volume":"139","author":"Ablowitz","year":"2017","journal-title":"Stud. Appl. Math."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"915","DOI":"10.1088\/0951-7715\/29\/3\/915","article-title":"Inverse scattering transform for the integrable nonlocal nonlinear Schr\u00f6dinger equation","volume":"29","author":"Ablowitz","year":"2016","journal-title":"Nonlinearity"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"127516","DOI":"10.1016\/j.physleta.2021.127516","article-title":"Integrable space-time shifted nonlocal nonlinear equations","volume":"409","author":"Ablowitz","year":"2021","journal-title":"Phys. Lett. A"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"106170","DOI":"10.1016\/j.aml.2019.106170","article-title":"Water-wave symbolic computation for the Earth, Enceladus and Titan: The higher-order Boussinesq-Burgers system, auto- and non-auto-B\u00e4cklund transformations","volume":"104","author":"Gao","year":"2020","journal-title":"Appl. Math. Lett."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"109950","DOI":"10.1016\/j.chaos.2020.109950","article-title":"Shallow water in an open sea or a wide channel: Auto-and non-auto-B\u00e4cklund transformations with solitons for a generalized (2+1)-dimensional dispersive long-wave system","volume":"138","author":"Gao","year":"2020","journal-title":"Chaos Solitons Fractals"},{"key":"ref_46","doi-asserted-by":"crossref","unstructured":"Gao, X.-Y., Guo, Y.-J., and Shan, W.-R. (2021). Cosmic dusty plasmas via a (3+1)-dimensional generalized variable-coefficient Kadomtsev-Petviashvili-Burgers-type equation: Auto-B\u00e4cklund transformations, solitons and similarity reductions plus observational\/experimental supports. Waves Random Complex Media.","DOI":"10.1080\/17455030.2021.1942308"},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"2122","DOI":"10.1063\/1.532279","article-title":"On the simplest (2+1) dimensional integrable spin systems and their equivalent nonlinear Schr\u00f6dinger equations","volume":"39","author":"Myrzakulov","year":"1998","journal-title":"J. Math. Phys."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"3765","DOI":"10.1063\/1.532466","article-title":"Motion of curves and surfaces and nonlinear evolution equations in (2+1) dimensions","volume":"39","author":"Myrzakulov","year":"1998","journal-title":"J. Math. Phys."},{"key":"ref_49","first-page":"9535","article-title":"Gauge equivalence between (2+1) - dimensional continuous Heisenberg ferromagnetic models and nonlinear Schr\u00f6dinger-type equations","volume":"31","author":"Myrzakulov","year":"1998","journal-title":"J. Phys. Math. Theor."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"715","DOI":"10.1016\/S0378-4371(96)00300-7","article-title":"Nonlinear spin-phonon excitations in an inhomogeneous compressible biquadratic Heisenberg spin chain","volume":"234","author":"Myrzakulov","year":"1997","journal-title":"Physica A"},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"1576","DOI":"10.1016\/j.geomphys.2010.05.013","article-title":"Integrable generalizations of Schr\u00f6dinger maps and Heisenberg spin models from Hamiltonian flows of curves and surfaces","volume":"60","author":"Anco","year":"2010","journal-title":"J. Geom. Phys."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"2118","DOI":"10.1016\/j.physleta.2014.05.010","article-title":"Integrable Motion of Curves in Self-Consistent Potentials: Relation to Spin Systems and Soliton Equations","volume":"378","author":"Myrzakulov","year":"2014","journal-title":"Phys. Lett. A"},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"1397","DOI":"10.1063\/1.1339831","article-title":"Deformation of surfaces, integrable systems, and Chern\u2013Simons theory","volume":"42","author":"Martina","year":"2001","journal-title":"J. Math. Phys."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"1750136","DOI":"10.1142\/S0219887817501365","article-title":"Integrable geometric flows of interacting curves\/surfaces, multilayer spin systems and the vector nonlinear Schr\u00f6dinger equation","volume":"14","author":"Myrzakul","year":"2017","journal-title":"Int. J. Geom. Methods Mod. Phys."},{"key":"ref_55","unstructured":"Ma, L., Shen, S., and Zhu, Z. (2017). From discrete nonlocal nonlinear Schr\u00f6dinger equation to coupled discrete Heisenberg ferromagnet equation. arXiv."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"083507","DOI":"10.1063\/1.4960818","article-title":"Nonlocal nonlinear Schr\u00f6dinger equation and its discrete version: Soliton solutions and gauge equivalence","volume":"57","author":"Ma","year":"2016","journal-title":"J. Math. Phys."},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"75","DOI":"10.22436\/jnsa.012.02.02","article-title":"A motion of complex curves in C3 and the nonlocal nonlinear Schr\u00f6dinger equation","volume":"12","author":"Zhong","year":"2019","journal-title":"J. Nonlinear Sci. Appl."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/10\/1827\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T07:08:09Z","timestamp":1760166489000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/10\/1827"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,9,30]]},"references-count":57,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2021,10]]}},"alternative-id":["sym13101827"],"URL":"https:\/\/doi.org\/10.3390\/sym13101827","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2021,9,30]]}}}