{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,16]],"date-time":"2026-07-16T18:14:25Z","timestamp":1784225665834,"version":"3.55.0"},"reference-count":69,"publisher":"Wiley","license":[{"start":{"date-parts":[[2022,3,25]],"date-time":"2022-03-25T00:00:00Z","timestamp":1648166400000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Computational and Mathematical Methods"],"published-print":{"date-parts":[[2022,3,25]]},"abstract":"<jats:p>The purpose of this article is to explore different types of solutions for the Kadomtsev-Petviashvili-modified Kadomtsev-Petviashvili (KP-mKP) equation which is termed as KP-Gardner equation, extensively used to model strong nonlinear internal waves in (<jats:inline-formula>\n                     <a:math xmlns:a=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M1\">\n                        <a:mn>1<\/a:mn>\n                        <a:mo>+<\/a:mo>\n                        <a:mn>2<\/a:mn>\n                     <\/a:math>\n                  <\/jats:inline-formula>)-dimensions on the stratified ocean shelf. This evolution equation is also used to describe weakly nonlinear shallow-water wave and dispersive interracial waves traveling in a mildly rotating channel with slowly varying topography. Introducing Liu\u2019s approach regarding the complete discrimination system for polynomial and the trial equation technique, a set of new solutions to the KP-mKP equation containing Jacobi elliptic function have been derived. It is found that these analytical solutions numerically exhibit different nonlinear structures such as solitary waves, shock waves, and periodic wave profiles. The reliability and effectiveness are confirmed from the numerical graphs of the solutions. Finally, the existence and validity of the various topological structures of the solutions are confirmed from the phase portrait of the dynamical system. Based on this investigation, it is confirmed that the method is not only suited for obtaining the classification of the solutions but also for qualitative analysis, which means that it can also be extended to other fields of application.<\/jats:p>","DOI":"10.1155\/2022\/3844031","type":"journal-article","created":{"date-parts":[[2022,3,25]],"date-time":"2022-03-25T17:50:40Z","timestamp":1648230640000},"page":"1-14","source":"Crossref","is-referenced-by-count":15,"title":["The Classification of the Exact Single Travelling Wave Solutions to the Constant Coefficient KP-mKP Equation Employing Complete Discrimination System for Polynomial Method"],"prefix":"10.1155","volume":"2022","author":[{"given":"Tanay","family":"Sarkar","sequence":"first","affiliation":[{"name":"Department of Mathematics, Jadavpur University, Kolkata 700032, India"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0619-5577","authenticated-orcid":true,"given":"Santanu","family":"Raut","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Mathabhanga College, Cooch Behar 736146, India"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Prakash Chandra","family":"Mali","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Jadavpur University, Kolkata 700032, India"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"311","reference":[{"key":"1","volume-title":"B\u00c3\u00a4Cklund Transformations and Their Application Mathematics in Science and Engineering","author":"C. Rogers","year":"1982"},{"key":"2","volume-title":"Solitons, Nonlinear Evolution Equations and Inverse Scattering","author":"M. J. Ablowitz","year":"2010"},{"key":"3","doi-asserted-by":"publisher","DOI":"10.1016\/0375-9601(90)90449-X"},{"key":"4","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.27.1192"},{"key":"5","doi-asserted-by":"publisher","DOI":"10.1016\/j.amc.2006.11.013"},{"issue":"3-4","key":"6","first-page":"263","article-title":"Shallow water waves and Korteweg de Vries equation (Oceanographical application)","volume":"7","author":"M. A. Helal","year":"1996","journal-title":"Pure Mathematics and Applications"},{"key":"7","volume-title":"Mathematical and Numerical Aspects of Wave Propagation","author":"J. A. Desanto","year":"1998"},{"key":"8","doi-asserted-by":"publisher","DOI":"10.1063\/1.1664700"},{"key":"9","doi-asserted-by":"publisher","DOI":"10.1063\/1.1664701"},{"key":"10","doi-asserted-by":"publisher","DOI":"10.1063\/1.1521391"},{"key":"11","doi-asserted-by":"publisher","DOI":"10.1016\/S0165-2125(02)00093-8"},{"key":"12","doi-asserted-by":"publisher","DOI":"10.1071\/MF05016"},{"key":"13","doi-asserted-by":"publisher","DOI":"10.1175\/1520-0485(1997)027<0871:ANMOIT>2.0.CO;2"},{"key":"14","doi-asserted-by":"publisher","DOI":"10.1029\/1999JC900144"},{"key":"15","doi-asserted-by":"publisher","DOI":"10.5506\/APhysPolB.45.1135"},{"key":"16","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611975529"},{"key":"17","doi-asserted-by":"publisher","DOI":"10.1007\/s12043-017-1446-4"},{"key":"18","doi-asserted-by":"publisher","DOI":"10.1007\/s40324-021-00242-5"},{"key":"19","doi-asserted-by":"publisher","DOI":"10.3934\/dcdss.2015.8.1155"},{"key":"20","doi-asserted-by":"publisher","DOI":"10.1016\/S0960-0779(04)00342-X"},{"key":"21","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.102.052215"},{"key":"22","doi-asserted-by":"publisher","DOI":"10.1002\/mma.6131"},{"key":"23","doi-asserted-by":"publisher","DOI":"10.1016\/j.asej.2014.03.004"},{"key":"24","doi-asserted-by":"publisher","DOI":"10.1016\/j.amc.2013.12.005"},{"key":"25","doi-asserted-by":"publisher","DOI":"10.1016\/j.amc.2008.06.011"},{"key":"26","doi-asserted-by":"publisher","DOI":"10.1016\/j.physa.2019.121320"},{"key":"27","doi-asserted-by":"publisher","DOI":"10.1140\/epjp\/i2019-12871-y"},{"key":"28","doi-asserted-by":"publisher","DOI":"10.1016\/j.chaos.2021.110921"},{"key":"29","doi-asserted-by":"publisher","DOI":"10.3934\/math.2019.5.1416"},{"key":"30","doi-asserted-by":"publisher","DOI":"10.1002\/mma.6996"},{"key":"31","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-81448-8_7"},{"key":"32","doi-asserted-by":"publisher","DOI":"10.1016\/j.chaos.2005.04.071"},{"key":"33","doi-asserted-by":"publisher","DOI":"10.1016\/j.camwa.2011.10.015"},{"key":"34","doi-asserted-by":"publisher","DOI":"10.1088\/1674-1056\/20\/1\/010205"},{"key":"35","doi-asserted-by":"publisher","DOI":"10.1016\/j.amc.2007.01.056"},{"key":"36","doi-asserted-by":"publisher","DOI":"10.1088\/0031-8949\/54\/6\/003"},{"key":"37","doi-asserted-by":"publisher","DOI":"10.1016\/S0375-9601(00)00725-8"},{"key":"38","doi-asserted-by":"publisher","DOI":"10.1016\/j.amc.2007.01.070"},{"key":"39","doi-asserted-by":"publisher","DOI":"10.1016\/S0096-3003(02)00806-8"},{"key":"40","doi-asserted-by":"publisher","DOI":"10.1080\/00207160412331286815"},{"key":"41","doi-asserted-by":"publisher","DOI":"10.1016\/j.amc.2005.11.025"},{"key":"42","doi-asserted-by":"publisher","DOI":"10.1143\/JPSJ.24.941"},{"key":"43","doi-asserted-by":"publisher","DOI":"10.1088\/0953-4075\/41\/4\/043001"},{"key":"44","doi-asserted-by":"publisher","DOI":"10.1016\/S0096-3003(01)00312-5"},{"key":"45","doi-asserted-by":"publisher","DOI":"10.1016\/j.chaos.2005.03.006"},{"key":"46","doi-asserted-by":"publisher","DOI":"10.1016\/j.mcm.2003.12.010"},{"key":"47","doi-asserted-by":"publisher","DOI":"10.1016\/j.jaubas.2013.03.006"},{"key":"48","doi-asserted-by":"publisher","DOI":"10.1016\/j.physd.2005.08.002"},{"key":"49","doi-asserted-by":"publisher","DOI":"10.1016\/j.camwa.2006.12.083"},{"key":"50","doi-asserted-by":"publisher","DOI":"10.1016\/S0375-9601(98)00547-7"},{"key":"51","doi-asserted-by":"publisher","DOI":"10.1016\/0375-9601(96)00283-6"},{"key":"52","doi-asserted-by":"publisher","DOI":"10.1088\/0031-8949\/82\/06\/065003"},{"key":"53","doi-asserted-by":"publisher","DOI":"10.1155\/2020\/8018064"},{"key":"54","doi-asserted-by":"publisher","DOI":"10.1143\/JPSJ.71.2663"},{"key":"55","doi-asserted-by":"publisher","DOI":"10.1016\/S0960-0779(02)00472-1"},{"key":"56","doi-asserted-by":"publisher","DOI":"10.1088\/0253-6102\/45\/6\/006"},{"key":"57","doi-asserted-by":"publisher","DOI":"10.1088\/1009-1963\/16\/7\/004"},{"issue":"9","key":"58","doi-asserted-by":"crossref","first-page":"1710","DOI":"10.1088\/1009-1963\/14\/9\/005","article-title":"Exact travelling wave solutions for (1+1)-dimensional dispersive long wave equation","volume":"14","author":"C. S. Liu","year":"2005","journal-title":"Chinese Physics"},{"key":"59","doi-asserted-by":"publisher","DOI":"10.1088\/0253-6102\/49\/1\/33"},{"key":"60","first-page":"759","article-title":"The classification of single travelling wave solutions to the Camassa-Holm-Degasperis-Procesi equation for some values of the convective parameter","author":"C.-Y. Wang","year":"2011"},{"key":"61","doi-asserted-by":"publisher","DOI":"10.1007\/s12043-016-1249-z"},{"key":"62","doi-asserted-by":"publisher","DOI":"10.1016\/j.cjph.2019.03.003"},{"key":"63","doi-asserted-by":"publisher","DOI":"10.1007\/s12043-013-0626-0"},{"key":"64","article-title":"The classification of the single traveling wave solutions to (1+1) dimensional Gardner equation with variable coefficients","volume":"2019","author":"D. Cao","year":"2019","journal-title":"Advances in Difference Equations"},{"issue":"6","key":"65","first-page":"628","article-title":"A complete discrimination system for polynomials, science in China","volume":"39","author":"L. Yang","year":"1996","journal-title":"Series E"},{"key":"66","doi-asserted-by":"publisher","DOI":"10.1016\/j.chaos.2020.110314"},{"key":"67","unstructured":"ByrdP. F.FriedmanM. D.Handbook of Elliptic Integrals for Engineers and Scientists1971SpringerBerlin Heidelberg New York"},{"key":"68","unstructured":"GuckenheimerJ.HolmesP.Nonlinear oscillations, dynamical systems, and bifurcations of vector fields201342Springer Science & Business Media"},{"key":"69","doi-asserted-by":"publisher","DOI":"10.1016\/j.chaos.2020.110578"}],"container-title":["Computational and Mathematical Methods"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/downloads.hindawi.com\/journals\/cmm\/2022\/3844031.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/cmm\/2022\/3844031.xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/cmm\/2022\/3844031.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,3,25]],"date-time":"2022-03-25T17:50:54Z","timestamp":1648230654000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.hindawi.com\/journals\/cmm\/2022\/3844031\/"}},"subtitle":[],"editor":[{"given":"Qichun","family":"Zhang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"editor"}]}],"short-title":[],"issued":{"date-parts":[[2022,3,25]]},"references-count":69,"alternative-id":["3844031","3844031"],"URL":"https:\/\/doi.org\/10.1155\/2022\/3844031","relation":{},"ISSN":["2577-7408"],"issn-type":[{"value":"2577-7408","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,3,25]]}}}