{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,22]],"date-time":"2026-09-22T14:06:23Z","timestamp":1790085983587,"version":"4.0.1"},"reference-count":25,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2022,3,17]],"date-time":"2022-03-17T00:00:00Z","timestamp":1647475200000},"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>This investigation focuses on two novel Kadomtsev\u2013Petviashvili (KP) equations with time-dependent variable coefficients that describe the nonlinear wave propagation of small-amplitude surface waves in narrow channels or large straits with slowly varying width and depth and non-vanishing vorticity. These two variable coefficients, Kadomtsev\u2013Petviashvili (VCKP) equations in (2+1)-dimensions, are the main extensions of the KP equation. Applying the Lie symmetry technique, we carry out infinitesimal generators, potential vector fields, and various similarity reductions of the considered VCKP equations. These VCKP equations are converted into nonlinear ODEs via two similarity reductions. The closed-form analytic solutions are achieved, including in the shape of distinct complex wave structures of solitons, dark and bright soliton shapes, double W-shaped soliton shapes, multi-peakon shapes, curved-shaped multi-wave solitons, and novel solitary wave solitons. All the obtained solutions are verified and validated by using back substitution to the original equation through Wolfram Mathematica. We analyze the dynamical behaviors of these obtained solutions with some three-dimensional graphics via numerical simulation. The obtained variable coefficient solutions are more relevant and useful for understanding the dynamical structures of nonlinear KP equations and shallow water wave models.<\/jats:p>","DOI":"10.3390\/sym14030597","type":"journal-article","created":{"date-parts":[[2022,3,20]],"date-time":"2022-03-20T21:37:17Z","timestamp":1647812237000},"page":"597","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":105,"title":["Lie Symmetries, Closed-Form Solutions, and Various Dynamical Profiles of Solitons for the Variable Coefficient (2+1)-Dimensional KP Equations"],"prefix":"10.3390","volume":"14","author":[{"given":"Sachin","family":"Kumar","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Mathematical Sciences, University of Delhi, Delhi 110007, India"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Shubham K.","family":"Dhiman","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Mathematical Sciences, University of Delhi, Delhi 110007, India"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0286-7244","authenticated-orcid":false,"given":"Dumitru","family":"Baleanu","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Arts and Sciences, \u00c7ankaya University, Ankara 06530, Turkey"},{"name":"Institute of Space Sciences, Magurele-Bucharest, P.O. Box, MG-23, R 76900 M\u0103gurele, Romania"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5783-0940","authenticated-orcid":false,"given":"Mohamed S.","family":"Osman","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Applied Science, Umm Al-Qura University, Makkah 21955, Saudi Arabia"},{"name":"Department of Mathematics, Faculty of Science, Cairo University, Giza 12613, Egypt"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8325-7500","authenticated-orcid":false,"given":"Abdul-Majid","family":"Wazwaz","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Saint Xavier University, Chicago, IL 60655, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2022,3,17]]},"reference":[{"key":"ref_1","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_2","doi-asserted-by":"crossref","first-page":"2559","DOI":"10.1088\/0951-7715\/23\/10\/012","article-title":"Inverse scattering transform for the DegasperisProcesi equation","volume":"23","author":"Constantin","year":"2010","journal-title":"Nonlinearity"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"125255","DOI":"10.1088\/1402-4896\/ac3879","article-title":"A study of multi-soliton solutions, breather, lumps, and their interactions for kadomtsev\u2013Petviashvili equation with variable time coeffcient using hirota method","volume":"96","author":"Kumar","year":"2021","journal-title":"Phys. Scr."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"950","DOI":"10.1016\/j.camwa.2010.12.043","article-title":"Linear superposition principle applying to Hirota bilinear equations","volume":"61","author":"Ma","year":"2011","journal-title":"Comput. Math. Appl."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1217","DOI":"10.1016\/j.chaos.2004.09.109","article-title":"Simplest equation method to look for exact solutions of nonlinear differential equations","volume":"24","author":"Kudryashov","year":"2005","journal-title":"Chaos Solitons Fract."},{"key":"ref_6","first-page":"502","article-title":"Bell-polynomial approach applied to the seventh-order Sawada-Kotera-Ito equation","volume":"227","author":"Shen","year":"2014","journal-title":"Appl. Math. Comput."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"626","DOI":"10.1016\/j.ijleo.2019.03.112","article-title":"Optical solitons and other solutions to Biswas-Arshed equation using the extended simplest equation method","volume":"185","author":"Zayed","year":"2019","journal-title":"Optik"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"296","DOI":"10.1016\/j.wavemoti.2011.11.003","article-title":"Exact solutions for the transformed reduced Ostrovsky equation via the F-expansion method in terms of Weierstrass-elliptic and Jacobian-elliptic functions","volume":"49","author":"Ebaid","year":"2012","journal-title":"Wave Motion"},{"key":"ref_9","first-page":"155209","article-title":"Nonlocal symmetries related to B\u00e4cklund transformation and their applications","volume":"45","author":"Lou","year":"2012","journal-title":"J. Phys. A Math."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"103769","DOI":"10.1016\/j.rinp.2020.103769","article-title":"Novel multiple soliton solutions for some nonlinear PDEs via multiple Exp-function method","volume":"21","author":"Nisar","year":"2021","journal-title":"Result Phys."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"164132","DOI":"10.1016\/j.ijleo.2019.164132","article-title":"Optical soliton solutions to the generalized nonautonomous nonlinear Schr\u00f6dinger equations in optical fibers via the sine-Gordon expansion method","volume":"208","author":"Ali","year":"2020","journal-title":"Optik"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"103604","DOI":"10.1016\/j.rinp.2020.103604","article-title":"Nonautonomous complex wave solutions to the (2+1)-dimensional variable-coefficients nonlinear Chiral Schr\u00f6dinger equation","volume":"19","author":"Sulaiman","year":"2020","journal-title":"Result Phys."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Niwas, M., Kumar, S., and Kharbanda, H. (2021). Symmetry analysis, closed-form invariant solutions and dynamical wave structures of the generalized (3+1)-dimensional breaking soliton equation using optimal system of Lie subalgebra. J. Ocean Eng. Sci.","DOI":"10.1016\/j.joes.2021.08.002"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"776","DOI":"10.1080\/17455030.2018.1559962","article-title":"Two new integrable Kadomtsev\u2013Petviashvili equations with time-dependent coefficients: Multiple real and complex soliton solutions","volume":"30","author":"Wazwaz","year":"2020","journal-title":"Waves Random Complex Media"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"212","DOI":"10.1111\/sapm.12026","article-title":"On the integrability of a generalized variable-coefficient forced Korteweg-de Vries equation in fluids","volume":"132","author":"Tian","year":"2014","journal-title":"Stud. Appl. Math."},{"key":"ref_16","first-page":"102","article-title":"Special types of elastic resonant soliton solutions of the Kadomtsev\u2013Petviashvili II equation","volume":"70","author":"Chen","year":"2018","journal-title":"Rom. Rep. Phys."},{"key":"ref_17","first-page":"539","article-title":"On the stability of solitary waves in weakly dispersive media","volume":"15","author":"Kadomtsev","year":"1970","journal-title":"Sov. Phys. Dokl."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"13","DOI":"10.1016\/S0378-4754(96)00053-5","article-title":"Symbolic methods to construct exact solutions of nonlinear partial differential equations","volume":"43","author":"Hereman","year":"1997","journal-title":"Math. Comput. Simul."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1841","DOI":"10.1007\/s11071-020-06186-1","article-title":"Multiple rogue wave, breather wave and interaction solutions of a generalized (3+1)-dimensional variable-coefficient nonlinear wave equation","volume":"103","author":"Liu","year":"2021","journal-title":"Nonlinear Dyn."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1094","DOI":"10.1080\/00207160903111592","article-title":"A symbolic algorithm for computing recursion operators of nonlinear partial differential equations","volume":"87","author":"Baldwin","year":"2010","journal-title":"Int. J. Comput. Math."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"257","DOI":"10.1016\/0167-2789(91)90148-3","article-title":"Nonclassical symmetry reductions for the Kadomtsev\u2013Petviashvili equation","volume":"49","author":"Clarkson","year":"1991","journal-title":"Physica D"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"2547","DOI":"10.1007\/s11071-018-4509-2","article-title":"On Lie symmetries and soliton solutions of (2+1)-dimensional Bogoyavlenskii equations","volume":"94","author":"Kumar","year":"2018","journal-title":"Nonlinear Dyn."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"90","DOI":"10.1016\/j.cjph.2018.12.007","article-title":"N-solitons and cuspon waves solutions of (2+1)-dimensional Broer-Kaup-Kupershmidt equations via hidden symmetries of Lie optimal system","volume":"57","author":"Kassem","year":"2018","journal-title":"Chin. J. Phys."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"065215","DOI":"10.1088\/1402-4896\/abf00a","article-title":"Optimal system, symmetry reductions and group-invariant solutions of (2+1)-dimensional ZK-BBM equation","volume":"96","author":"Tanwar","year":"2021","journal-title":"Phys. Scr."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Laouini, G., Amin, A.M., and Moustafa, M. (2021). Lie Group Method for Solving the Negative-Order Kadomtsev-Petviashvili Equation (nKP). Symmetry, 13.","DOI":"10.3390\/sym13020224"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/3\/597\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T22:38:02Z","timestamp":1760135882000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/3\/597"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,3,17]]},"references-count":25,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2022,3]]}},"alternative-id":["sym14030597"],"URL":"https:\/\/doi.org\/10.3390\/sym14030597","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,3,17]]}}}