{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,10]],"date-time":"2026-01-10T03:06:11Z","timestamp":1768014371047,"version":"3.49.0"},"reference-count":45,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2025,8,1]],"date-time":"2025-08-01T00:00:00Z","timestamp":1754006400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"European Union under the REFRESH\u2014Research Excellence For Region Sustainability and High-tech Industries","award":["CZ.10.03.01\/00\/22_003\/0000048"],"award-info":[{"award-number":["CZ.10.03.01\/00\/22_003\/0000048"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In the present paper, a mathematical analysis of the Gardner equation with varying coefficients has been performed to give a more realistic model of physical phenomena, especially in regards to plasma physics. First, a Lie symmetry analysis was carried out, as a result of which a symmetry classification following the different representations of the variable coefficients was systematically derived. The reduced ordinary differential equation obtained is solved using the power-series method and solutions to the equation are represented graphically to give an idea of their dynamical behavior. Moreover, a fully connected neural network has been included as an efficient computation method to deal with the complexity of the reduced equation, by using traveling-wave transformation. The validity and correctness of the solutions provided by the neural networks have been rigorously tested and the solutions provided by the neural networks have been thoroughly compared with those generated by the Runge\u2013Kutta method, which is a conventional and well-recognized numerical method. The impact of a variation in the loss function of different coefficients has also been discussed, and it has also been found that the dispersive coefficient affects the convergence rate of the loss contribution considerably compared to the other coefficients. The results of the current work can be used to improve knowledge on the nonlinear dynamics of waves in plasma physics. They also show how efficient it is to combine the approaches, which consists in the use of analytical and semi-analytical methods and methods based on neural networks, to solve nonlinear differential equations with variable coefficients of a complex nature.<\/jats:p>","DOI":"10.3390\/sym17081218","type":"journal-article","created":{"date-parts":[[2025,8,4]],"date-time":"2025-08-04T09:41:17Z","timestamp":1754300477000},"page":"1218","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Comparative Analysis of the Gardner Equation in Plasma Physics Using Analytical and Neural Network Methods"],"prefix":"10.3390","volume":"17","author":[{"given":"Zain","family":"Majeed","sequence":"first","affiliation":[{"name":"Abdus Salam School of Mathematical Sciences, Government College University, Lahore 54600, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6747-425X","authenticated-orcid":false,"given":"Adil","family":"Jhangeer","sequence":"additional","affiliation":[{"name":"IT4Innovations, VSB\u2014Technical University of Ostrava, Poruba, 708 00 Ostrava, Czech Republic"},{"name":"Center for Theoretical Physics, Khazar University, 41 Mehseti Str., Baku AZ1096, Azerbaijan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6995-5820","authenticated-orcid":false,"given":"F. M.","family":"Mahomed","sequence":"additional","affiliation":[{"name":"Abdus Salam School of Mathematical Sciences, Government College University, Lahore 54600, Pakistan"},{"name":"School of Computer Science and Applied Mathematics, University of the Witwatersrand, Johannesburg 2050, South Africa"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5024-866X","authenticated-orcid":false,"given":"Hassan","family":"Almusawa","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Sciences, Jazan University, Jazan 45142, Saudi Arabia"}]},{"given":"F. D.","family":"Zaman","sequence":"additional","affiliation":[{"name":"School of Mathematics, University of the Witwatersrand, Johannesburg 2050, South Africa"}]}],"member":"1968","published-online":{"date-parts":[[2025,8,1]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Tajima, T. (2018). Computational Plasma Physics: With Applications to Fusion and Astrophysics, CRC Press.","DOI":"10.1201\/9780429501470"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"8647","DOI":"10.1007\/s11071-023-08260-w","article-title":"Auto-B\u00e4cklund transformations and soliton solutions on the nonzero background for a (3 + 1)-dimensional Korteweg-de Vries-Calogero-Bogoyavlenskii-Schif equation in a fluid","volume":"111","author":"Zhou","year":"2023","journal-title":"Nonlinear Dyn."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"128970","DOI":"10.1016\/j.physleta.2023.128970","article-title":"Line-rogue waves, transformed nonlinear waves and their interactions for a (3 + 1)-dimensional Korteweg-de Vries equation in a fluid","volume":"480","author":"Cheng","year":"2023","journal-title":"Phys. Lett. A"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"2127","DOI":"10.1007\/s11071-019-05110-6","article-title":"Multi-waves, breather wave and lump\u2013stripe interaction solutions in a (2 + 1)-dimensional variable-coefficient Korteweg\u2013de Vries equation","volume":"97","author":"Liu","year":"2019","journal-title":"Nonlinear Dyn."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"075002","DOI":"10.1088\/1572-9494\/acd99c","article-title":"Higher-dimensional integrable deformations of the modified KdV equation","volume":"75","author":"Hao","year":"2023","journal-title":"Commun. Theor. Phys."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"109018","DOI":"10.1016\/j.aml.2024.109018","article-title":"Two-layer-liquid and lattice considerations through a (3 + 1)-dimensional generalized Yu-Toda-Sasa-Fukuyama system","volume":"152","author":"Gao","year":"2024","journal-title":"Appl. Math. Lett."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1455","DOI":"10.1007\/s11071-016-2971-2","article-title":"An extended modified KdV equation and its Painlev\u00e9 integrability","volume":"86","author":"Wazwaz","year":"2016","journal-title":"Nonlinear Dyn."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"121","DOI":"10.1515\/zna-2020-0263","article-title":"Dust-acoustic Gardner solitons in cryogenic plasma with the effect of polarization in the presence of a quantizing magnetic field","volume":"76","author":"Atteya","year":"2021","journal-title":"Z. Naturforschung A"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"110578","DOI":"10.1016\/j.chaos.2020.110578","article-title":"Quasi-periodic, chaotic and travelling wave structures of modified Gardner equation","volume":"143","author":"Jhangeer","year":"2021","journal-title":"Chaos Solitons Fractals"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"207","DOI":"10.5194\/npg-29-207-2022","article-title":"Effects of rotation and topography on internal solitary waves governed by the rotating Gardner equation","volume":"29","author":"Helfrich","year":"2022","journal-title":"Nonlinear Process. Geophys."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"12345","DOI":"10.1007\/s11071-024-09680-y","article-title":"Integrability, bilinearization, B\u00e4cklund transformations and solutions for a generalized variable-coefficient Gardner equation with an external-force term in a fluid or plasma","volume":"112","author":"Liu","year":"2024","journal-title":"Nonlinear Dyn."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"575","DOI":"10.1007\/s11071-010-9936-7","article-title":"Multi-soliton solutions of the forced variable-coefficient extended Korteweg\u2013de Vries equation arisen in fluid dynamics of internal solitary waves","volume":"66","author":"Liu","year":"2011","journal-title":"Nonlinear Dyn."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"053204","DOI":"10.1103\/PhysRevE.88.053204","article-title":"Integrable aspects and soliton interaction for a generalized inhomogeneous Gardner model with external force in plasmas and fluids","volume":"88","author":"Liu","year":"2013","journal-title":"Phys. Rev. E\u2014Stat. Nonlinear Soft Matter Phys."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"3655","DOI":"10.1007\/s11071-024-10397-1","article-title":"N-soliton, H th-order breather, hybrid and multi-pole solutions for a generalized variable-coefficient Gardner equation with an external force in a plasma or fluid","volume":"113","author":"Liu","year":"2025","journal-title":"Nonlinear Dyn."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"827","DOI":"10.1007\/s11071-018-4093-5","article-title":"Exact similarity and traveling wave solutions to an integrable evolution equation for surface waves in deep water","volume":"92","author":"Meng","year":"2018","journal-title":"Nonlinear Dyn."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"55","DOI":"10.1007\/s00021-019-0460-3","article-title":"A novel Lie group classification method for generalized cylindrical KdV type of equation: Exact solutions and conservation laws","volume":"21","author":"Liu","year":"2019","journal-title":"J. Math. Fluid Mech."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Olver, P.J. (1993). Applications of Lie Groups to Differential Equations, Springer Science & Business Media.","DOI":"10.1007\/978-1-4612-4350-2"},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Bluman, G.W. (2010). Applications of Symmetry Methods to Partial Differential Equations, Springer.","DOI":"10.1007\/978-0-387-68028-6"},{"key":"ref_19","unstructured":"Ovsiannikov, L.V.E. (2014). Group Analysis of Differential Equations, Academic Press."},{"key":"ref_20","unstructured":"Bluman, G.W., and Kumei, S. (2013). Symmetries and Differential Equations, Springer Science & Business Media."},{"key":"ref_21","unstructured":"Ibragimov, N.H. (2006). Selected Works, ALGA Publications BTH."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"3887","DOI":"10.1016\/j.cnsns.2010.01.037","article-title":"More common errors in finding exact solutions of nonlinear differential equations: Part I","volume":"15","author":"Popovych","year":"2010","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_23","first-page":"125","article-title":"On symmetries and conservation laws of a Gardner equation involving arbitrary functions","volume":"290","author":"Gandarias","year":"2016","journal-title":"Appl. Math. Comput."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1243","DOI":"10.1016\/j.cnsns.2014.09.016","article-title":"Enhanced group classification of Gardner equations with time-dependent coefficients","volume":"22","author":"Vaneeva","year":"2015","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"377","DOI":"10.1016\/j.aej.2022.08.008","article-title":"Modeling of the vibration and stability of a dynamical system coupled with an energy harvesting device","volume":"63","author":"Abohamer","year":"2023","journal-title":"Alex. Eng. J."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"011001","DOI":"10.1115\/1.4054039","article-title":"Surrogate modeling of nonlinear dynamic systems: A comparative study","volume":"23","author":"Zhao","year":"2023","journal-title":"J. Comput. Inf. Sci. Eng."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"3325","DOI":"10.1016\/j.aej.2020.04.051","article-title":"Design of a hybrid NAR-RBFs neural network for nonlinear dusty plasma system","volume":"59","author":"Bukhari","year":"2020","journal-title":"Alex. Eng. J."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"2450066","DOI":"10.1142\/S0217979224500668","article-title":"Mathematical modeling and simulation of biophysics systems using neural network","volume":"38","author":"Makhdoom","year":"2024","journal-title":"Int. J. Mod. Phys. B"},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Ul Rahman, J., Danish, S., and Lu, D. (2023). Deep neural network-based simulation of Sel\u2019kov model in glycolysis: A comprehensive analysis. Mathematics, 11.","DOI":"10.3390\/math11143216"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"168761","DOI":"10.1016\/j.ijleo.2022.168761","article-title":"On the optical soliton solutions of Kundu\u2013Mukherjee\u2013Naskar equation via two different analytical methods","volume":"257","author":"Onder","year":"2022","journal-title":"Optik"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"85","DOI":"10.1016\/j.joes.2019.01.003","article-title":"An efficient analytical technique for fractional partial differential equations occurring in ion acoustic waves in plasma","volume":"4","author":"Goswami","year":"2019","journal-title":"J. Ocean. Eng. Sci."},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Iqbal, N., Chughtai, M.T., and Ullah, R. (2023). Fractional study of the non-linear Burgers\u2019 equations via a semi-analytical technique. Fractal Fract., 7.","DOI":"10.3390\/fractalfract7020103"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"1133","DOI":"10.12693\/APhysPolA.133.1133","article-title":"Exact solutions and conservation laws of the bogoyavlenskii equation","volume":"133","author":"Mustafa","year":"2018","journal-title":"Acta Phys. Pol. A"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"94","DOI":"10.1007\/s11082-018-1373-8","article-title":"Lie symmetry analysis and explicit solutions for the time fractional generalized Burgers\u2013Huxley equation","volume":"50","author":"Inc","year":"2018","journal-title":"Opt. Quantum Electron."},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Huang, Y., Sun, S., Duan, X., and Chen, Z. (2016, January 3\u20135). A study on deep neural networks framework. Proceedings of the 2016 IEEE Advanced Information Management, Communicates, Electronic and Automation Control Conference (IMCEC), Xi\u2019an, China.","DOI":"10.1109\/IMCEC.2016.7867471"},{"key":"ref_36","first-page":"5416722","article-title":"Analysis of artificial neural network: Architecture, types, and forecasting applications","volume":"2022","author":"Madhiarasan","year":"2022","journal-title":"J. Electr. Comput. Eng."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"100682","DOI":"10.1016\/j.padiff.2024.100682","article-title":"The bilinear neural network method for solving Benney-Luke equation","volume":"10","author":"Tuan","year":"2024","journal-title":"Partial. Differ. Equ. Appl. Math."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"521","DOI":"10.1007\/s11071-022-07207-x","article-title":"Bilinear residual network method for solving the exactly explicit solutions of nonlinear evolution equations","volume":"108","author":"Zhang","year":"2022","journal-title":"Nonlinear Dyn."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"116232","DOI":"10.1016\/j.chaos.2025.116232","article-title":"Neural network-based symbolic calculation approach for solving the Korteweg\u2013de Vries equation","volume":"194","author":"Xie","year":"2025","journal-title":"Chaos Solitons Fractals"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"686","DOI":"10.1016\/j.jcp.2018.10.045","article-title":"Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations","volume":"378","author":"Raissi","year":"2019","journal-title":"J. Comput. Phys."},{"key":"ref_41","doi-asserted-by":"crossref","unstructured":"Kumar, R., Kumar, A., and Kumar, A. (2025). Dynamic Behavior of Coupled mKdV\u2013Calogero\u2013Bogoyavlenskii\u2013Schiff Equations in Fluid Dynamics. Differ. Equ. Dyn. Syst., 1\u201320.","DOI":"10.1007\/s12591-025-00717-1"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"075223","DOI":"10.1088\/1402-4896\/ad51b6","article-title":"Kinks and soliton solutions to the coupled Burgers equation by Lie symmetry approach","volume":"99","author":"Tanwar","year":"2024","journal-title":"Phys. Scr."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"045240","DOI":"10.1088\/1402-4896\/ad32fd","article-title":"Some more variety of analytical solutions to (2+1)-Bogoyavlensky-Konopelchenko equation","volume":"99","author":"Kumar","year":"2024","journal-title":"Phys. Scr."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"125615","DOI":"10.1016\/j.jmaa.2021.125615","article-title":"Optimal system and classification of invariant solutions of nonlinear class of wave equations and their conservation laws","volume":"505","author":"Raza","year":"2022","journal-title":"J. Math. Anal. Appl."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"1600118","DOI":"10.1002\/minf.201600118","article-title":"Performance of deep and shallow neural networks, the universal approximation theorem, activity cliffs, and QSAR","volume":"36","author":"Winkler","year":"2017","journal-title":"Mol. Inform."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/8\/1218\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T18:21:20Z","timestamp":1760034080000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/8\/1218"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,8,1]]},"references-count":45,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2025,8]]}},"alternative-id":["sym17081218"],"URL":"https:\/\/doi.org\/10.3390\/sym17081218","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,8,1]]}}}