{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T15:40:37Z","timestamp":1760197237125,"version":"build-2065373602"},"reference-count":38,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2018,4,16]],"date-time":"2018-04-16T00:00:00Z","timestamp":1523836800000},"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>The compatibility method is used for a generalized variable-coefficient Gardner equation (GVGE) with a forcing term. By the compatibility of the considered equation and a non-classical symmetry of a given form, four types of symmetry are obtained. Then, by solving the characteristic equations of symmetry, the GVGE is reduced to variable coefficients ordinary differential equations, and rich varieties of new similarity solutions are presented. Our results show that the compatibility method can be employed for variable coefficients nonlinear evolution equations with forcing terms.<\/jats:p>","DOI":"10.3390\/sym10040112","type":"journal-article","created":{"date-parts":[[2018,4,16]],"date-time":"2018-04-16T12:40:26Z","timestamp":1523882426000},"page":"112","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["New Similarity Solutions of a Generalized Variable-Coefficient Gardner Equation with Forcing Term"],"prefix":"10.3390","volume":"10","author":[{"given":"Jianping","family":"Zhou","sequence":"first","affiliation":[{"name":"School of Computer Science &amp; Technology, Anhui University of Technology, Ma\u2019anshan 243032, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yuan","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Computer Science &amp; Technology, Anhui University of Technology, Ma\u2019anshan 243032, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yang","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Computer Science &amp; Technology, Anhui University of Technology, Ma\u2019anshan 243032, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zhilian","family":"Yan","sequence":"additional","affiliation":[{"name":"School of Mathematics &amp; Physics, Anhui University of Technology, Ma\u2019anshan 243032, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7188-5828","authenticated-orcid":false,"given":"Zhen","family":"Wang","sequence":"additional","affiliation":[{"name":"College of Mathematics &amp; Systems Science, Shandong University of Science &amp; Technology, Qingdao 266590, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2018,4,16]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"587","DOI":"10.1088\/0305-4470\/24\/11\/003","article-title":"Exact solitary waves in a convecting fluid","volume":"24","author":"Lou","year":"1991","journal-title":"J. Phys. A Gen. Phys."},{"doi-asserted-by":"crossref","unstructured":"Wang, H., Wang, Y., and Dong, H. (2018). Interaction solutions of a (2+1)-dimensional dispersive long wave system. Comput. Math. Appl.","key":"ref_2","DOI":"10.1016\/j.camwa.2017.12.032"},{"key":"ref_3","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_4","doi-asserted-by":"crossref","first-page":"67","DOI":"10.1016\/0375-9601(96)00283-6","article-title":"Applications of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics","volume":"216","author":"Wang","year":"1996","journal-title":"Phys. Lett. A"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"032116","DOI":"10.1063\/1.4794947","article-title":"New approach of (G\u2032\/G)-expansion method and new approach of generalized (G\u2032\/G)-expansion method for nonlinear evolution equation","volume":"3","author":"Naher","year":"2013","journal-title":"AIP Adv."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"256","DOI":"10.4208\/eajam.110215.010815a","article-title":"Generalised (2+1)-dimensional super MKdV hierarchy for integrable systems in soliton theory","volume":"5","author":"Dong","year":"2015","journal-title":"East Asian J. Appl. Math."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"499","DOI":"10.1016\/j.mcm.2003.12.010","article-title":"A sine-cosine method for handling nonlinear wave equations","volume":"40","author":"Wazwaz","year":"2004","journal-title":"Math. Comput. Model."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"435","DOI":"10.1016\/j.amc.2013.10.047","article-title":"Frobenius integrable decompositions of nonlinear evolution equations with modified term","volume":"226","author":"Fang","year":"2014","journal-title":"Appl. Math. Comput."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"3182","DOI":"10.3390\/e17053182","article-title":"Exact solutions of non-linear lattice equations by an improved exp-function method","volume":"17","author":"Zhang","year":"2015","journal-title":"Entropy"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"6025","DOI":"10.3390\/e17096025","article-title":"New exact solutions of the new Hamiltonian amplitude-equation and Fokas Lenells equation","volume":"17","author":"Demiray","year":"2015","journal-title":"Entropy"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"203","DOI":"10.1016\/j.camwa.2016.11.012","article-title":"Exact travelling wave solutions for the local fractional two-dimensional Burgers-type equations","volume":"73","author":"Yang","year":"2017","journal-title":"Comput. Math. Appl."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1192","DOI":"10.1103\/PhysRevLett.27.1192","article-title":"Exact solution of the Korteweg-de Vries equation for multiple collisions of solitons","volume":"27","author":"Hirota","year":"1971","journal-title":"Phys. Rev. Lett."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1399","DOI":"10.1016\/j.camwa.2017.06.034","article-title":"Mixed lump-kink solutions to the KP equation","volume":"74","author":"Zhao","year":"2017","journal-title":"Comput. Math. Appl."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"2653","DOI":"10.1016\/j.jde.2017.10.033","article-title":"Lump solutions to nonlinear partial differential equations via Hirota bilinear forms","volume":"264","author":"Ma","year":"2018","journal-title":"J. Differ. Equ."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"2693","DOI":"10.1016\/j.amc.2012.11.053","article-title":"Two integrable lattice hierarchies and their respective Darboux transformations","volume":"219","author":"Zhao","year":"2013","journal-title":"Appl. Math. Comput."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"3328","DOI":"10.22436\/jnsa.010.06.42","article-title":"An integrable coupling hierarchy of Dirac integrable hierarchy, its Liouville integrability and Darboux transformation","volume":"10","author":"Xu","year":"2017","journal-title":"J. Nonlinear Sci. Appl."},{"key":"ref_17","first-page":"1025","article-title":"The general similarity of the heat equation","volume":"18","author":"Bluman","year":"1969","journal-title":"J. Math. Mech."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"2201","DOI":"10.1063\/1.528613","article-title":"New similarity solutions of the Boussinesq equation","volume":"30","author":"Clarkson","year":"1989","journal-title":"J. Math. Phys."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"267","DOI":"10.1088\/0305-4470\/33\/2\/304","article-title":"Lie symmetries of nonlinear multidimensional reaction-diffusion systems: I","volume":"33","author":"Cherniha","year":"2000","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"405","DOI":"10.1088\/0305-4470\/36\/2\/309","article-title":"Lie symmetries of nonlinear multidimensional reaction-diffusion systems: II","volume":"36","author":"Cherniha","year":"2003","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_21","first-page":"132","article-title":"Symmetry reductions and exact solutions of a variable coefficient (2+1)-Zakharov-Kuznetsov equation","volume":"17","author":"Moleleki","year":"2012","journal-title":"Math. Comput. Appl."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1613","DOI":"10.3390\/sym7031613","article-title":"Symmetries, Lagrangians and conservation laws of an Easter Island population model","volume":"7","author":"Nucci","year":"2015","journal-title":"Symmetry"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"5107","DOI":"10.22436\/jnsa.009.07.13","article-title":"A new integrable symplectic map and the lie point symmetry associated with nonlinear lattice equations","volume":"9","author":"Dong","year":"2016","journal-title":"J. Nonlinear Sci. Appl."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"81","DOI":"10.1016\/j.cnsns.2016.09.019","article-title":"Lie symmetries of the shigesada-Kawasaki-Teramoto system","volume":"45","author":"Cherniha","year":"2017","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"doi-asserted-by":"crossref","unstructured":"Huang, D., Li, X., and Yu, S. (2017). Lie symmetry classification of the generalized nonlinear Beam equation. Symmetry, 9.","key":"ref_25","DOI":"10.3390\/sym9070115"},{"doi-asserted-by":"crossref","unstructured":"Hydon, P.E. (2000). Symmetry Methods for Differential Equations: A Beginner\u2019s Guide, Cambridge University Press.","key":"ref_26","DOI":"10.1017\/CBO9780511623967"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"288","DOI":"10.1016\/j.amc.2005.12.021","article-title":"Symmetry and similarity solutions of variable coefficient generalized Zakharov-Kuznetsov equation","volume":"180","author":"Yan","year":"2006","journal-title":"Appl. Math. Comput."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"333","DOI":"10.1016\/j.amc.2007.12.027","article-title":"Symmetry reductions and similarity solutions of the (3+1)-dimensional breaking soliton equation","volume":"201","author":"Yan","year":"2008","journal-title":"Appl. Math. Comput."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"965","DOI":"10.1088\/0253-6102\/54\/6\/03","article-title":"New explicit solutions of (1+1)-dimensional variable-coefficient Broer-Kaup system","volume":"54","author":"Yan","year":"2010","journal-title":"Commun. Theor. Phys."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"181","DOI":"10.1139\/p2012-002","article-title":"Exact solutions of Wick-type stochastic Korteweg-de Vries equation","volume":"90","author":"Zhang","year":"2012","journal-title":"Can. J. Phys."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"1543","DOI":"10.1007\/s11071-016-3132-3","article-title":"Nonclassical symmetries and similarity solutions of variable coefficient coupled KdV system using compatibility method","volume":"87","author":"Gupta","year":"2017","journal-title":"Nonlinear Dyn."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"2732","DOI":"10.1016\/j.amc.2012.08.104","article-title":"New exact solutions for the generalized variable-coefficient Gardner equation with forcing term","volume":"219","author":"Hong","year":"2012","journal-title":"Appl. Math. Comput."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"1443","DOI":"10.1016\/j.jmaa.2007.03.064","article-title":"Lax pair: B\u00e4cklund transformation and N-soliton-like solution for a variable-coefficient Gardner equation from nonlinear lattice, plasma physics and ocean dynamics with symbolic computation","volume":"336","author":"Li","year":"2007","journal-title":"J. Math. Anal. Appl."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"313","DOI":"10.1016\/j.amc.2008.10.049","article-title":"Analytic N-solitary-wave solution of a variable-coefficient Gardner equation from fluid dynamics and plasma physics","volume":"210","author":"Xu","year":"2009","journal-title":"Appl. Math. Comput."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"327","DOI":"10.1016\/j.ijleo.2017.01.071","article-title":"New solitary and optical wave structures to the (1+1)-dimensional combined KdV-mKdV equation","volume":"135","author":"Bulut","year":"2017","journal-title":"Optik"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"1783","DOI":"10.1016\/j.cnsns.2010.07.023","article-title":"Some exact solutions of KdV equation with variable coefficient","volume":"16","author":"Latif","year":"2009","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"3761","DOI":"10.1016\/j.amc.2010.05.043","article-title":"Group analysis of KdV equation with time dependent coefficients","volume":"216","author":"Johnpillai","year":"2010","journal-title":"Appl. Math. Comput."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"1207","DOI":"10.1016\/j.cnsns.2010.06.025","article-title":"Lie group classification and invariant solutions of mKdV equation with time-dependent coefficients","volume":"16","author":"Johnpillai","year":"2011","journal-title":"Commun. Nonlinear Sci. Numer. Simul."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/10\/4\/112\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T15:00:52Z","timestamp":1760194852000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/10\/4\/112"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,4,16]]},"references-count":38,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2018,4]]}},"alternative-id":["sym10040112"],"URL":"https:\/\/doi.org\/10.3390\/sym10040112","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2018,4,16]]}}}