{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,22]],"date-time":"2025-12-22T04:33:02Z","timestamp":1766377982807,"version":"build-2065373602"},"reference-count":41,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2023,6,6]],"date-time":"2023-06-06T00:00:00Z","timestamp":1686009600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Liaoning BaiQianWan Talents Program of China","award":["2020921037","LJ2020002","11547005","2020D01B01"],"award-info":[{"award-number":["2020921037","LJ2020002","11547005","2020D01B01"]}]},{"name":"Natural Science Foundation of Education Department of Liaoning Province of China","award":["2020921037","LJ2020002","11547005","2020D01B01"],"award-info":[{"award-number":["2020921037","LJ2020002","11547005","2020D01B01"]}]},{"DOI":"10.13039\/501100001809","name":"National Science Foundation of China","doi-asserted-by":"publisher","award":["2020921037","LJ2020002","11547005","2020D01B01"],"award-info":[{"award-number":["2020921037","LJ2020002","11547005","2020D01B01"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Natural Science Foundation of Xinjiang Autonomous Region of China","award":["2020921037","LJ2020002","11547005","2020D01B01"],"award-info":[{"award-number":["2020921037","LJ2020002","11547005","2020D01B01"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The existence of solutions with localized solitary wave structures is one of the significant characteristics of nonlinear integrable systems. Darboux transformation (DT) is a well-known method for constructing multi-soliton solutions, using a type of localized solitary wave, of integrable systems, but there are still no reports on extending DT techniques to construct such solitary wave solutions of fractional integrable models. This article takes the coupled nonlinear Schr\u00f6dinger (CNLS) equations with conformable fractional derivatives as an example to illustrate the feasibility of extending the DT and generalized DT (GDT) methods to construct symmetric and asymmetric solitary wave solutions for fractional integrable systems. Specifically, the traditional n-fold DT and the first-, second-, and third-step GDTs are extended for the fractional CNLS equations. Based on the extended GDTs, explicit solutions with symmetric\/asymmetric soliton and soliton\u2013rogon (solitrogon) spatial structures of the fractional CNLS equations are obtained. This study found that the symmetric solitary wave solutions of the integer-order CNLS equations exhibit asymmetry in the fractional order case.<\/jats:p>","DOI":"10.3390\/sym15061211","type":"journal-article","created":{"date-parts":[[2023,6,7]],"date-time":"2023-06-07T01:38:41Z","timestamp":1686101921000},"page":"1211","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":11,"title":["Localized Symmetric and Asymmetric Solitary Wave Solutions of Fractional Coupled Nonlinear Schr\u00f6dinger Equations"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4631-7033","authenticated-orcid":false,"given":"Sheng","family":"Zhang","sequence":"first","affiliation":[{"name":"School of Mathematical Sciences, Bohai University, Jinzhou 121013, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Feng","family":"Zhu","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences, Bohai University, Jinzhou 121013, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7956-4882","authenticated-orcid":false,"given":"Bo","family":"Xu","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences, Bohai University, Jinzhou 121013, China"},{"name":"School of Educational Sciences, Bohai University, Jinzhou 121013, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,6,6]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1095","DOI":"10.1103\/PhysRevLett.19.1095","article-title":"Method for solving the Korteweg-deVries equation","volume":"19","author":"Gardner","year":"1967","journal-title":"Phys. Rev. Lett."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Ablowitz, M.J., and Clarkson, P.A. (1991). Solitons, Nonlinear Evolution Equations and Inverse Scattering, Cambridge University Press.","DOI":"10.1017\/CBO9780511623998"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Matveev, V.B., and Salle, M.A. (1991). Darboux Transformation and Soliton, Springer.","DOI":"10.1007\/978-3-662-00922-2"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"805","DOI":"10.1063\/1.1666399","article-title":"Exact envelope-soliton solutions of a nonlinear wave equation","volume":"14","author":"Hirota","year":"1973","journal-title":"J. Math. Phys."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"279","DOI":"10.1016\/0375-9601(96)00103-X","article-title":"Exact solutions for a compound KdV-Burgers equation","volume":"213","author":"Wang","year":"1996","journal-title":"Phys. Lett. A"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"700","DOI":"10.1016\/j.chaos.2006.03.020","article-title":"Exp-function method for nonlinear wave equations","volume":"30","author":"He","year":"2006","journal-title":"Chaos Soliton. Fract."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"227","DOI":"10.1088\/1751-8113\/40\/2\/003","article-title":"A generalized auxiliary equation method and its application to (2+1)-dimensional asymmetric Nizhnik-Novikov-Vesselov equations","volume":"40","author":"Zhang","year":"2007","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Zeng, X.H., Wu, X.L., Lang, C.Z., Yuan, C.P., and Cai, J.P. (2023). Exact solutions for coupled variable coefficient KdV equation via quadratic Jacobi\u2019s elliptic function expansion. Symmetry, 15.","DOI":"10.3390\/sym15051021"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"6925","DOI":"10.1088\/0305-4470\/33\/39\/308","article-title":"Darboux transformation and soliton-like solutions for the Gerdjikov-Ivanov equation","volume":"33","author":"Fan","year":"2000","journal-title":"J. Phys. A Gen. Phys."},{"key":"ref_10","first-page":"650","article-title":"Explicit N-Fold Darboux transformations and soliton solutions for nonlinear derivative Schr\u00f6dinger equations","volume":"35","author":"Fan","year":"2001","journal-title":"Commun. Theor. Phys."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1677","DOI":"10.1016\/j.chaos.2006.03.015","article-title":"Explicit N-fold Darboux transformation and multi-soliton solutions for the (1+1)-dimensional higher-order Broer-Kaup system","volume":"33","author":"Huang","year":"2007","journal-title":"Chaos Soliton. Fract."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"211","DOI":"10.1016\/S0034-4877(12)60005-6","article-title":"N-fold Darboux transformation and soliton solutions for Toda lattice equation","volume":"68","author":"Wen","year":"2011","journal-title":"Rep. Math. Phys."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"3243","DOI":"10.1088\/0951-7715\/28\/9\/3243","article-title":"Darboux transformation and multi-dark soliton for N-component nonlinear Schr\u00f6dinger equations","volume":"28","author":"Ling","year":"2015","journal-title":"Nonlinearity"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Sun, H.Q., and Zhu, Z.N. (2023). Darboux transformation and soliton solution of the nonlocal generalized Sasa\u2013Satsuma equation. Mathematics, 11.","DOI":"10.3390\/math11040865"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"026607","DOI":"10.1103\/PhysRevE.85.026607","article-title":"Nonlinear Schr\u00f6dinger equation: Generalized Darboux transformation and rogue wave solutions","volume":"85","author":"Guo","year":"2012","journal-title":"Phys. Rev. E"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"317","DOI":"10.1111\/j.1467-9590.2012.00568.x","article-title":"High-order solutions and generalized Darboux transformations of derivative nonlinear Schr\u00f6dinger equations","volume":"130","author":"Guo","year":"2013","journal-title":"Stud. Appl. Math."},{"key":"ref_17","unstructured":"Guo, B.L., Tian, L.X., Yan, Z.Y., and Lin, L.M. (2015). Rogue Wave and Its Mathematical Theory, Zhejiang Science and Technology Press."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"403","DOI":"10.1016\/j.cnsns.2018.02.008","article-title":"Three-component nonlinear Schrodinger equations: Modulational instability, Nth-order vector rational and semi-rational rogue waves, and dynamics","volume":"62","author":"Zhang","year":"2018","journal-title":"Commun. Nonlinear Sci. Numer. Simula."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"403","DOI":"10.1016\/j.cnsns.2018.07.023","article-title":"Some semirational solutions and their interactions on the zero-intensity background for the coupled nonlinear Schrdinger equations","volume":"67","author":"Jiang","year":"2019","journal-title":"Commun. Nonlinear Sci. Numer. Simula."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"459","DOI":"10.1140\/epjp\/s13360-021-01400-4","article-title":"Generalized Darboux transformation, semi-rational solutions and novel degenerate soliton solutions for a coupled nonlinear Schrdinger equation","volume":"136","author":"Zhang","year":"2021","journal-title":"Eur. Phys. J. Plus"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"170338","DOI":"10.1016\/j.ijleo.2022.170338","article-title":"Solitons, rogon-solitons and their propagations and reflections in three-component coupled nonlinear Schr\u00f6dinger equation","volume":"272","author":"Zhang","year":"2023","journal-title":"Optik"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"113902","DOI":"10.1103\/PhysRevLett.90.113902","article-title":"Exact self-similar solutions of the generalized nonlinear Schr\u00f6dinger equation with distributed coefficients","volume":"90","author":"Kruglov","year":"2003","journal-title":"Phys. Rev. Lett."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"074102","DOI":"10.1103\/PhysRevLett.98.074102","article-title":"Nonautonomous solitons in external potentials","volume":"98","author":"Serkin","year":"2007","journal-title":"Phys. Rev. Lett."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Yang, J.K. (2010). Nonlinear Waves in Integrable and Nonintegrable Systems, SIAM.","DOI":"10.1137\/1.9780898719680"},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Liu, W.M., and Kengne, E. (2019). Schr\u00f6dinger Equations in Nonlinear Systems, Springer Nature Singapore Pte Ltd.","DOI":"10.1007\/978-981-13-6581-2"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"100201","DOI":"10.1088\/0256-307X\/39\/10\/100201","article-title":"Matrix integrable fourth-order nonlinear Schr\u00f6dinger equations and their exact soliton solutions","volume":"39","author":"Ma","year":"2022","journal-title":"Chin. Phys. Lett."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"020201","DOI":"10.1088\/1674-1056\/ac7dc1","article-title":"Matrix integrable fifth-order mKdV equations and their soliton solutions","volume":"32","author":"Ma","year":"2023","journal-title":"Chin. Phys. B"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"44","DOI":"10.1007\/s13324-021-00477-5","article-title":"A binary Darboux transformation for multicomponent NLS equations and their reductions","volume":"11","author":"Ma","year":"2021","journal-title":"Anal. Math. Phys."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"110824","DOI":"10.1016\/j.chaos.2021.110824","article-title":"Binary Darboux transformation for general matrix mKdV equations and reduced counterparts","volume":"146","author":"Ma","year":"2021","journal-title":"Chaos Soliton. Fract."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"100515","DOI":"10.1016\/j.padiff.2023.100515","article-title":"Soliton hierarchies and soliton solutions of type (\u2212\u03bb*, \u2212\u03bb) reduced nonlocal nonlinear Schr\u00f6dinger equations of arbitrary even order","volume":"7","author":"Ma","year":"2023","journal-title":"Partial Differ. Equ. Appl. Math."},{"key":"ref_31","unstructured":"Podlubny, I. (1999). Fractional Differential Equations, Academic Press."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"1126","DOI":"10.1016\/j.physleta.2009.12.051","article-title":"Fractional optical solitons","volume":"374","author":"Fujioka","year":"2010","journal-title":"Phys. Lett. A"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"S145","DOI":"10.2298\/TSCI11S1145H","article-title":"A new fractal derivation","volume":"15","author":"He","year":"2011","journal-title":"Therm. Sci."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"1069","DOI":"10.1016\/j.physleta.2011.01.029","article-title":"Fractional sub-equation method and its applications to nonlinear fractional PDEs","volume":"375","author":"Zhang","year":"2011","journal-title":"Phys. Lett. A"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"3698","DOI":"10.1007\/s10773-014-2123-8","article-title":"A tutorial review on fractal spacetime and fractional calculus","volume":"53","author":"He","year":"2014","journal-title":"Int. J. Theor. Phys."},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Yang, X.J., Baleanu, D., and Srivastava, H.M. (2015). Local Fractional Integral Transforms and Their Applications, Academic Press.","DOI":"10.1016\/B978-0-12-804002-7.00002-4"},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Xu, B., and Zhang, S. (2021). Riemann-Hilbert approach for constructing analytical solutions and conservation laws of a local time-fractional nonlinear Schr\u00f6dinger equation. Symmetry, 13.","DOI":"10.3390\/sym13091593"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"184101","DOI":"10.1103\/PhysRevLett.128.184101","article-title":"Fractional integrable nonlinear soliton equation","volume":"128","author":"Ablowitz","year":"2022","journal-title":"Phys. Rev. Lett."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"110202","DOI":"10.1088\/0256-307X\/28\/11\/110202","article-title":"Rogue wave, breathers and bright-dark-rogue solutions for the coupled Schr\u00f6dinger equations","volume":"28","author":"Guo","year":"2011","journal-title":"Chin. Phys. Lett."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1016\/j.cam.2014.01.002","article-title":"A new definition of fractional derivative","volume":"264","author":"Khalil","year":"2014","journal-title":"J. Comput. Appl. Math."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"6664039","DOI":"10.1155\/2021\/6664039","article-title":"Line soliton interactions for shallow ocean-waves and novel solutions with peakon, ring, conical, columnar and lump structures based on fractional KP equation","volume":"2021","author":"Xu","year":"2021","journal-title":"Adv. Math. Phys."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/6\/1211\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T19:49:28Z","timestamp":1760125768000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/6\/1211"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,6,6]]},"references-count":41,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2023,6]]}},"alternative-id":["sym15061211"],"URL":"https:\/\/doi.org\/10.3390\/sym15061211","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2023,6,6]]}}}