{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,16]],"date-time":"2026-03-16T16:08:50Z","timestamp":1773677330185,"version":"3.50.1"},"reference-count":33,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2023,5,4]],"date-time":"2023-05-04T00:00:00Z","timestamp":1683158400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Guangdong Distinctive Key Disciplines construction","award":["F2017STSZD01"],"award-info":[{"award-number":["F2017STSZD01"]}]},{"name":"Guangdong Distinctive Key Disciplines construction","award":["2021ZDJS141"],"award-info":[{"award-number":["2021ZDJS141"]}]},{"name":"Guangdong Distinctive Key Disciplines construction","award":["2022ZDJS150"],"award-info":[{"award-number":["2022ZDJS150"]}]},{"name":"Guangdong Distinctive Key Disciplines construction","award":["2021JYS001"],"award-info":[{"award-number":["2021JYS001"]}]},{"name":"Guangdong Distinctive Key Disciplines construction","award":["2015S001"],"award-info":[{"award-number":["2015S001"]}]},{"name":"Guangdong Key construction Discipline Scientific Research Capacity Improvement","award":["F2017STSZD01"],"award-info":[{"award-number":["F2017STSZD01"]}]},{"name":"Guangdong Key construction Discipline Scientific Research Capacity Improvement","award":["2021ZDJS141"],"award-info":[{"award-number":["2021ZDJS141"]}]},{"name":"Guangdong Key construction Discipline Scientific Research Capacity Improvement","award":["2022ZDJS150"],"award-info":[{"award-number":["2022ZDJS150"]}]},{"name":"Guangdong Key construction Discipline Scientific Research Capacity Improvement","award":["2021JYS001"],"award-info":[{"award-number":["2021JYS001"]}]},{"name":"Guangdong Key construction Discipline Scientific Research Capacity Improvement","award":["2015S001"],"award-info":[{"award-number":["2015S001"]}]},{"name":"Doctoral guidance program of Guangzhou Xinhua University, the Finance Course Teaching and Research Section of Guangzhou Xinhua University","award":["F2017STSZD01"],"award-info":[{"award-number":["F2017STSZD01"]}]},{"name":"Doctoral guidance program of Guangzhou Xinhua University, the Finance Course Teaching and Research Section of Guangzhou Xinhua University","award":["2021ZDJS141"],"award-info":[{"award-number":["2021ZDJS141"]}]},{"name":"Doctoral guidance program of Guangzhou Xinhua University, the Finance Course Teaching and Research Section of Guangzhou Xinhua University","award":["2022ZDJS150"],"award-info":[{"award-number":["2022ZDJS150"]}]},{"name":"Doctoral guidance program of Guangzhou Xinhua University, the Finance Course Teaching and Research Section of Guangzhou Xinhua University","award":["2021JYS001"],"award-info":[{"award-number":["2021JYS001"]}]},{"name":"Doctoral guidance program of Guangzhou Xinhua University, the Finance Course Teaching and Research Section of Guangzhou Xinhua University","award":["2015S001"],"award-info":[{"award-number":["2015S001"]}]},{"name":"Financial Vocational Skills Training Experimental Teaching Demonstration Center","award":["F2017STSZD01"],"award-info":[{"award-number":["F2017STSZD01"]}]},{"name":"Financial Vocational Skills Training Experimental Teaching Demonstration Center","award":["2021ZDJS141"],"award-info":[{"award-number":["2021ZDJS141"]}]},{"name":"Financial Vocational Skills Training Experimental Teaching Demonstration Center","award":["2022ZDJS150"],"award-info":[{"award-number":["2022ZDJS150"]}]},{"name":"Financial Vocational Skills Training Experimental Teaching Demonstration Center","award":["2021JYS001"],"award-info":[{"award-number":["2021JYS001"]}]},{"name":"Financial Vocational Skills Training Experimental Teaching Demonstration Center","award":["2015S001"],"award-info":[{"award-number":["2015S001"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The exact traveling wave solutions to coupled KdV equations with variable coefficients are obtained via the use of quadratic Jacobi\u2019s elliptic function expansion. The presented coupled KdV equations have a more general form than those studied in the literature. Nine couples of quadratic Jacobi\u2019s elliptic function solutions are found. Each couple of traveling wave solutions is symmetric in mathematical form. In the limit cases m\u21921, these periodic solutions degenerate as the corresponding soliton solutions. After the simple parameter substitution, the trigonometric function solutions are also obtained.<\/jats:p>","DOI":"10.3390\/sym15051021","type":"journal-article","created":{"date-parts":[[2023,5,5]],"date-time":"2023-05-05T01:38:21Z","timestamp":1683250701000},"page":"1021","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["Exact Solutions for Coupled Variable Coefficient KdV Equation via Quadratic Jacobi\u2019s Elliptic Function Expansion"],"prefix":"10.3390","volume":"15","author":[{"given":"Xiaohua","family":"Zeng","sequence":"first","affiliation":[{"name":"School of Economics and Trade, Guangzhou Xinhua University, Dongguan 523133, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xiling","family":"Wu","sequence":"additional","affiliation":[{"name":"School of Economics and Trade, Guangzhou Xinhua University, Dongguan 523133, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Changzhou","family":"Liang","sequence":"additional","affiliation":[{"name":"School of Economics and Trade, Guangzhou Xinhua University, Dongguan 523133, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Chiping","family":"Yuan","sequence":"additional","affiliation":[{"name":"Institute of Guangdong, Hong Kong and Macao Development Studies, Sun Yat-sen University, Guangzhou 510275, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jieping","family":"Cai","sequence":"additional","affiliation":[{"name":"School of Economics and Trade, Guangzhou Xinhua University, Dongguan 523133, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,5,4]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"306","DOI":"10.1016\/j.aml.2019.06.014","article-title":"Inverse scattering transformation for generalized nonlinear Schr\u00f6dinger equation","volume":"98","author":"Zhang","year":"2019","journal-title":"Appl. Math. Lett."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"102787","DOI":"10.1016\/j.wavemoti.2021.102787","article-title":"Darboux-Backlund transformation and localized excitation on the periodic wave background for the nonlinear Schr\u00f6dinger equation","volume":"106","author":"Yang","year":"2021","journal-title":"Wave Motion"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1491","DOI":"10.1007\/s11071-019-05275-0","article-title":"Darboux transformation and analytic solutions for a generalized super-NLS-mKdV equation","volume":"98","author":"Guan","year":"2019","journal-title":"Nonlinear Dyn."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"22","DOI":"10.1088\/0253-6102\/49\/1\/04","article-title":"Soliton solutions of coupled KdV system from Hirota\u2019s bilinear direct method","volume":"49","author":"Yang","year":"2008","journal-title":"Commun. Theor. Phys."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"801","DOI":"10.1007\/s11071-019-04822-z","article-title":"Periodic attenuating oscillation between soliton interactions for higher-order variable coefficient nonlinear Schr\u00f6dinger equation","volume":"96","author":"Liu","year":"2019","journal-title":"Nonlinear Dyn."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"684","DOI":"10.1016\/j.jmaa.2012.05.066","article-title":"The first integral method for some time fractional differential equations","volume":"395","author":"Lu","year":"2012","journal-title":"J. Math. Anal. Appl."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"2766","DOI":"10.1016\/j.amc.2010.03.100","article-title":"Exact solutions for a Wick-type stochastic 2D KdV equation","volume":"216","author":"Zhu","year":"2010","journal-title":"Appl. Math. Comput."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"11","DOI":"10.1016\/j.cnsns.2016.07.022","article-title":"Backlund transformation classification, integrability and exact solutions to the generalized Burgers\u2019-KdV equation","volume":"44","author":"Liu","year":"2017","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"128970","DOI":"10.1155\/2013\/128970","article-title":"New Exact Solutions for a Higher-Order Wave Equation of KdV Type Using Extended F-Expansion Method","volume":"2013","author":"He","year":"2013","journal-title":"Math. Probl. Eng."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"438","DOI":"10.1016\/j.ijleo.2016.11.027","article-title":"Exact analytical solutions of higher-order nonlinear Schrodinger equation","volume":"131","author":"Wang","year":"2017","journal-title":"Optik"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"215","DOI":"10.1007\/s11082-022-04476-z","article-title":"On solitary wave solutions for the extended nonlinear Schrodinger equation via the modified F-expansion method","volume":"55","author":"Ozisik","year":"2023","journal-title":"Opt. Quantum Electron."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"468","DOI":"10.1088\/0253-6102\/52\/3\/17","article-title":"Exact Solutions to a (3+1)-Dimensional Variable-Coefficient Kadomtsev-Petviashvilli Equation via the Bilinear Method and Wronskian Technique","volume":"52","author":"Zhang","year":"2009","journal-title":"Commun. Theor. Phys."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"164457","DOI":"10.1016\/j.ijleo.2020.164457","article-title":"Optical bright and dark soliton solutions for coupled nonlinear Schrodinger (CNLS) equations by the variational iteration method","volume":"207","author":"Wazwaz","year":"2020","journal-title":"Optik"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"108","DOI":"10.2306\/scienceasia1513-1874.2022.133","article-title":"Meromorphic solutions of the seventh-order KdV equation by using an extended complex method and Painleve analysis","volume":"49","author":"Dang","year":"2023","journal-title":"ScienceAsia"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"181","DOI":"10.1016\/j.joes.2016.06.003","article-title":"The simplified Hirota\u2019s method for studying three extended higher-order KdV-type equations","volume":"1","author":"Wazwaz","year":"2016","journal-title":"J. Ocean Eng. Sci."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.joes.2016.11.001","article-title":"A new integrable nonlocal modified KdV equation: Abundant solutions with distinct physical structures","volume":"2","author":"Wazwaz","year":"2017","journal-title":"J. Ocean Eng. Sci."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"055003","DOI":"10.1088\/1572-9494\/abe9ad","article-title":"Infinitely many nonlocal symmetries and nonlocal conservation laws of the integrable modified KdV-sine-Gordon equation","volume":"73","author":"Liang","year":"2021","journal-title":"Commun. Theor. Phys."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"309","DOI":"10.1007\/s11082-022-03640-9","article-title":"Adequate soliton solutions to the space\u2013time fractional telegraph equation and modified third-order KdV equation through a reliable technique","volume":"54","author":"Arefin","year":"2022","journal-title":"Opt. Quant. Electron."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"17","DOI":"10.1007\/s12559-009-9031-x","article-title":"Adaptive-Wave Alternative for the Black-Scholes Option Pricing Model","volume":"2","author":"Ivancevic","year":"2010","journal-title":"Cogn. Comput."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"124380","DOI":"10.1016\/j.physa.2020.124380","article-title":"A novel analytical technique for the solution of time-fractional Ivancevic option pricing model","volume":"550","author":"Jena","year":"2020","journal-title":"Phys. A"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"2539","DOI":"10.1007\/s11071-021-06642-6","article-title":"Dark wave, rogue wave and perturbation solutions of Ivancevic option pricing model","volume":"105","author":"Chen","year":"2021","journal-title":"Nonlinear Dyn."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"4599194","DOI":"10.1155\/2022\/4599194","article-title":"Solitary Wave and Singular Wave Solutions for Ivancevic Option Pricing Model","volume":"2022","author":"Zeng","year":"2022","journal-title":"Math. Probl. Eng."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1250072","DOI":"10.1142\/S0217979212500725","article-title":"Exact Soliton Solutions and Quasi-Periodic Wave Solutions to The Forced Variable-Coefficient KdV Equation","volume":"26","author":"Zhang","year":"2012","journal-title":"Int. J. Mod. Phys. B"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1593","DOI":"10.1007\/s11071-014-1539-2","article-title":"Multi-soliton solutions of a variable-coefficient KdV hierarchy","volume":"78","author":"Zhang","year":"2014","journal-title":"Nonlinear Dyn."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"404","DOI":"10.1016\/j.cnsns.2013.07.023","article-title":"Traveling wave solutions for fifth-order KdV type equations with time-dependent coefficients","volume":"19","author":"Triki","year":"2014","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"2250181","DOI":"10.1142\/S0217979222501818","article-title":"Bifurcation; bilinear forms, conservation laws and soliton solutions of the temporal-second-order KdV equation","volume":"36","author":"Ahmed","year":"2022","journal-title":"Int. J. Mod. Phys. B"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"126894","DOI":"10.1016\/j.physleta.2020.126894","article-title":"Nonlocal KdV equations","volume":"384","author":"Gurses","year":"2020","journal-title":"Phys. Lett. A"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"69","DOI":"10.1515\/phys-2016-0002","article-title":"Bilinearization and new multi-soliton solutions of mKdV hierarchy with time-dependent coefficients","volume":"14","author":"Zhang","year":"2016","journal-title":"Open Phys."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1950234","DOI":"10.1142\/S0217984919502348","article-title":"Periodic and decay mode solutions of the generalized variable-coefficient Korteweg-de Vries equation","volume":"33","author":"Zhang","year":"2019","journal-title":"Mod. Phys. Lett. B"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"S677","DOI":"10.2298\/TSCI180512081Z","article-title":"Inverse scattering transform for a supersymmetric Korteweg-de Vries equation","volume":"23","author":"Zhang","year":"2019","journal-title":"Therm. Sci."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"409","DOI":"10.1016\/j.joes.2021.09.014","article-title":"Various exact wave solutions for KdV equation with time-variable coefficients","volume":"7","author":"Ismael","year":"2022","journal-title":"J. Ocean Eng. Sci."},{"key":"ref_32","first-page":"955","article-title":"The exact solutions to coupled KdV equations with variable coefficients by auxiliary equation method, Journal of Northwest University","volume":"50","author":"Liu","year":"2020","journal-title":"Nat. Sci. Ed."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"106249","DOI":"10.1016\/j.rinp.2023.106249","article-title":"Novel solutions to the coupled KdV equations and the coupled system of variant Boussinesq equations","volume":"45","author":"Yao","year":"2023","journal-title":"Results Phys."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/5\/1021\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T19:29:03Z","timestamp":1760124543000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/5\/1021"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,5,4]]},"references-count":33,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2023,5]]}},"alternative-id":["sym15051021"],"URL":"https:\/\/doi.org\/10.3390\/sym15051021","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,5,4]]}}}