{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,13]],"date-time":"2026-03-13T06:37:25Z","timestamp":1773383845759,"version":"3.50.1"},"reference-count":21,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2025,5,22]],"date-time":"2025-05-22T00:00:00Z","timestamp":1747872000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100004329","name":"Slovenian Research Agency","doi-asserted-by":"publisher","award":["P2-0405"],"award-info":[{"award-number":["P2-0405"]}],"id":[{"id":"10.13039\/501100004329","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004329","name":"Slovenian Research Agency","doi-asserted-by":"publisher","award":["J2-60053"],"award-info":[{"award-number":["J2-60053"]}],"id":[{"id":"10.13039\/501100004329","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004329","name":"Slovenian Research Agency","doi-asserted-by":"publisher","award":["633053"],"award-info":[{"award-number":["633053"]}],"id":[{"id":"10.13039\/501100004329","id-type":"DOI","asserted-by":"publisher"}]},{"name":"EUROfusion Consortium","award":["P2-0405"],"award-info":[{"award-number":["P2-0405"]}]},{"name":"EUROfusion Consortium","award":["J2-60053"],"award-info":[{"award-number":["J2-60053"]}]},{"name":"EUROfusion Consortium","award":["633053"],"award-info":[{"award-number":["633053"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>An exact solution of the collisionless time-dependent Vlasov equation is found. For the first time in a century, an analytical solution to the one-dimensional time-dependent Vlasov\u2013Boltzmann equation has been found. It has been found that instead of the widely discussed damping, waves are subject to instability. By means of this solution, the behavior of the Langmuir waves in the nonlinear stage is considered. A symmetry method is found that allows us to establish the dependence on time of the desired quantity based on the dependence on the previous time. The analysis is restricted by the consideration of the first nonlinear approximation\u2014keeping the second power of the electric strength. It is shown that in general the waves with finite amplitudes are not subjected to the damping. Conditions have been found under which waves can be unstable.<\/jats:p>","DOI":"10.3390\/sym17060809","type":"journal-article","created":{"date-parts":[[2025,5,22]],"date-time":"2025-05-22T12:15:43Z","timestamp":1747916143000},"page":"809","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["On the Theory of Nonlinear Landau Damping"],"prefix":"10.3390","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1790-7093","authenticated-orcid":false,"given":"Leon","family":"Kos","sequence":"first","affiliation":[{"name":"LECAD Laboratory, Faculty of Mechanical Engineering, University of Ljubljana, Askerceva Cesta 6, 1000 Ljubljana, Slovenia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5710-2070","authenticated-orcid":false,"given":"Ivona","family":"Vasileska","sequence":"additional","affiliation":[{"name":"Faculty of Science, Technology and Medicine, University of Luxembourg, Maison du Nombre 6, Avenue de la Fonte, L-4364 Esch, Luxembourg"}]},{"given":"Davy D.","family":"Tskhakaya","sequence":"additional","affiliation":[{"name":"Andronikashvili Institute of Physics, Georgian Academy of Sciences, Mikheil Tamarashvili 6, 0177 Tbilisi, Georgia"}]}],"member":"1968","published-online":{"date-parts":[[2025,5,22]]},"reference":[{"key":"ref_1","first-page":"25","article-title":"On the vibrations of the electronic plasma","volume":"10","author":"Landau","year":"1946","journal-title":"J. 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A Course of Modern Analysis, Cambridge University Press."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/6\/809\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T17:38:43Z","timestamp":1760031523000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/6\/809"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,5,22]]},"references-count":21,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2025,6]]}},"alternative-id":["sym17060809"],"URL":"https:\/\/doi.org\/10.3390\/sym17060809","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,5,22]]}}}