{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,16]],"date-time":"2026-03-16T10:20:13Z","timestamp":1773656413055,"version":"3.50.1"},"reference-count":12,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2021,1,29]],"date-time":"2021-01-29T00:00:00Z","timestamp":1611878400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100002261","name":"Russian Foundation for Basic Research","doi-asserted-by":"publisher","award":["18-29-10013."],"award-info":[{"award-number":["18-29-10013."]}],"id":[{"id":"10.13039\/501100002261","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>We studied, for the Kortweg\u2013de Vries\u2013Burgers equations on cylindrical and spherical waves, the development of a regular profile starting from an equilibrium under a periodic perturbation at the boundary. The regular profile at the vicinity of perturbation looks like a periodical chain of shock fronts with decreasing amplitudes. Further on, shock fronts become decaying smooth quasi-periodic oscillations. After the oscillations cease, the wave develops as a monotonic convex wave, terminated by a head shock of a constant height and equal velocity. This velocity depends on integral characteristics of a boundary condition and on spatial dimensions. In this paper the explicit asymptotic formulas for the monotonic part, the head shock and a median of the oscillating part are found.<\/jats:p>","DOI":"10.3390\/sym13020220","type":"journal-article","created":{"date-parts":[[2021,1,29]],"date-time":"2021-01-29T02:58:21Z","timestamp":1611889101000},"page":"220","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["On Monotonic Pattern in Periodic Boundary Solutions of Cylindrical and Spherical Kortweg\u2013De Vries\u2013Burgers Equations"],"prefix":"10.3390","volume":"13","author":[{"given":"Alexey","family":"Samokhin","sequence":"first","affiliation":[{"name":"Trapeznikov Institute of Control Scienties, Russian Academy of Sciences, 65 Profsoyuznaya Street, 117997 Moscow, Russia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,1,29]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Leibovich, S., and Seebass, R. (1974). Nonlinear Waves, Cornell University Press. Chapter 4.","DOI":"10.1115\/1.3423757"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1973","DOI":"10.1017\/S0022112073002235","article-title":"Propagation of spherical and cylindrical N-waves","volume":"58","author":"Sachdev","year":"1973","journal-title":"J. Fluid. Mech."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Grava, T. (2016). Whitham modulation equations and application to small dispersion asymptotics and long time asymptotics of nonlinear dispersive equations. arXiv.","DOI":"10.1007\/978-3-319-39214-1_10"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"635","DOI":"10.1140\/epjp\/s13360-020-00659-3","article-title":"Traveling waves and undercompressive shocks in solutions of the generalized Korteweg\u2013de Vries\u2013Burgers equation with a time-dependent dissipation coefficient distribution","volume":"135","author":"Chugainova","year":"2020","journal-title":"Eur. Phys. J. 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Physics"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"33","DOI":"10.1016\/j.geomphys.2018.03.016","article-title":"Nonlinear waves in layered media: Solutions of the KdV\u2014Burgers equation","volume":"130","author":"Samokhin","year":"2018","journal-title":"J. Geom. Physics"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"91","DOI":"10.1016\/j.difgeo.2017.03.001","article-title":"On nonlinear superposition of the KdV-Burgers shock waves and the behavior of solitons in a layered medium","volume":"54","author":"Samokhin","year":"2017","journal-title":"J. Diff. Geom. Its Appl."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Whitham, G.B. (1999). Linear and Nonlinear Waves, John Wiley & Sons.","DOI":"10.1002\/9781118032954"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Krasil\u2019shchik, I.S., and Vinogradov, A.M. (1999). Symmetries and Conservation Laws for Differential Equations of Mathematical Physics, American Mathematical Society.","DOI":"10.1007\/978-94-017-3196-6_2"},{"key":"ref_12","unstructured":"Bronstein, M., Corless, R.M., Davenport, J.H., and Jeffrey, D.J. Algebraic properties of the Lambert W function from a result of Rosenlicht and of Liouville. Integral Transforms and Special Functions, Taylor & Francis."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/2\/220\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T05:16:56Z","timestamp":1760159816000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/2\/220"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,1,29]]},"references-count":12,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2021,2]]}},"alternative-id":["sym13020220"],"URL":"https:\/\/doi.org\/10.3390\/sym13020220","relation":{"has-preprint":[{"id-type":"doi","id":"10.20944\/preprints202012.0579.v1","asserted-by":"object"}]},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,1,29]]}}}