{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,13]],"date-time":"2026-04-13T02:37:19Z","timestamp":1776047839122,"version":"3.50.1"},"reference-count":36,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2025,9,12]],"date-time":"2025-09-12T00:00:00Z","timestamp":1757635200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Deanship of Scientific Research at Imam Mohammad Ibn Saud Islamic University (IMSIU)","award":["IMSIU-DDRSP2502"],"award-info":[{"award-number":["IMSIU-DDRSP2502"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this study, the complete discrimination system for the polynomial method (CDSPM) is employed to analyze the integrable Gardner Equation (IGE). Through a traveling wave transformation, the model is reduced to a nonlinear ordinary differential equation, enabling the derivation of a wide class of exact solutions, including trigonometric, hyperbolic, rational, and Jacobi elliptic functions. For example, a bright soliton solution is obtained for parameters A=1.3, \u03b2=\u22120.1, and \u03b3=0.8. Qualitative analysis reveals diverse phase portraits, indicating the presence of saddle points, centers, and cuspidal points depending on parameter values. Chaos and quasi-periodic dynamics are investigated via Poincar\u00e9 maps and time-series analysis, where chaotic patterns emerge for values like \u03bd1=1.45, \u03bd2=2.18, \u039e0=4, and \u03bb=2\u03c0. Sensitivity analysis confirms the model\u2019s sensitivity to initial conditions \u03c7=2.2,2.4,2.6, reflecting real-world unpredictability. Additionally, the energy balance method (EBM) is applied to approximate periodic solutions by conserving kinetic and potential energies. These results highlight the IGE\u2019s ability to capture complex nonlinear behaviors relevant to fluid dynamics, plasma waves, and nonlinear optics.<\/jats:p>","DOI":"10.3390\/sym17091529","type":"journal-article","created":{"date-parts":[[2025,9,12]],"date-time":"2025-09-12T13:43:02Z","timestamp":1757684582000},"page":"1529","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Energy Analysis, Soliton Dynamics, Chaos, and Sensitivity Analysis for a Forced Damped Gardner Model"],"prefix":"10.3390","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7206-2508","authenticated-orcid":false,"given":"Syed T. R.","family":"Rizvi","sequence":"first","affiliation":[{"name":"Department of Mathematics, COMSATS University Islamabad, Lahore Campus, Lahore 54000, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7676-1020","authenticated-orcid":false,"given":"Atef F.","family":"Hashem","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11432, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Aiman","family":"Shahbaz","sequence":"additional","affiliation":[{"name":"Department of Mathematics, COMSATS University Islamabad, Lahore Campus, Lahore 54000, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zunaira","family":"Iqbal","sequence":"additional","affiliation":[{"name":"Department of Mathematics, COMSATS University Islamabad, Lahore Campus, Lahore 54000, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ijaz","family":"Ali","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Sargodha, Sargodha 40100, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"A. S.","family":"Al-Moisheer","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11432, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7412-4773","authenticated-orcid":false,"given":"Aly R.","family":"Seadawy","sequence":"additional","affiliation":[{"name":"Mathematics Department, Faculty of Science, Taibah University, Al-Madinah Al-Munawarah 41411, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,9,12]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Drazin, P.G., and Johnson, R.S. (1989). Solitons: An Introduction, Cambridge University Press.","DOI":"10.1017\/CBO9781139172059"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Ablowitz, M.J., and Segur, H. (1981). 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