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On weakly singular and fully nonlinear travelling shallow capillary–gravity waves in the critical regime

Mitsotakis, Dimitrios ; Dutykh, Denys ; Assylbekuly, Aydar ; Zhakebayev, Dauren

Physics Letters A, 25 May 2017, Vol.381(20), pp.1719-1726 [Tạp chí có phản biện]

ISSN: 0375-9601 ; E-ISSN: 1873-2429 ; DOI: 10.1016/j.physleta.2017.03.041

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  • Nhan đề:
    On weakly singular and fully nonlinear travelling shallow capillary–gravity waves in the critical regime
  • Tác giả: Mitsotakis, Dimitrios ; Dutykh, Denys ; Assylbekuly, Aydar ; Zhakebayev, Dauren
  • Chủ đề: Capillary–Gravity Waves ; Peakons ; Nonlinear Dispersive Waves ; Physics
  • Là 1 phần của: Physics Letters A, 25 May 2017, Vol.381(20), pp.1719-1726
  • Mô tả: To access, purchase, authenticate, or subscribe to the full-text of this article, please visit this link: http://dx.doi.org/10.1016/j.physleta.2017.03.041 Byline: Dimitrios Mitsotakis [dmitsot@gmail.com] (a), Denys Dutykh [Denys.Dutykh@univ-savoie.fr] (b,*), Aydar Assylbekuly [asylbekuly@mail.ru] (c), Dauren Zhakebayev [daurjaz@mail.ru] (d) Keywords Capillary--gravity waves; Peakons; Nonlinear dispersive waves Highlights * A model for long capillary--gravity weakly dispersive and fully nonlinear water waves is derived. * Shallow capillary--gravity waves are classified using phase plane analysis. * Peaked travelling waves are found in the critical regime. * The dynamics of peakons in Serre--Green--Naghdi equations is studied numerically. Abstract In this Letter we consider long capillary--gravity waves described by a fully nonlinear weakly dispersive model. First, using the phase space analysis methods we describe all possible types of localized travelling waves. Then, we especially focus on the critical regime, where the surface tension is exactly balanced by the gravity force. We show that our long wave model with a critical Bond number admits stable travelling wave solutions with a singular crest. These solutions are usually referred to in the literature as peakons or peaked solitary waves. They satisfy the usual speed-amplitude relation, which coincides with Scott--Russel's empirical formula for solitary waves, while their decay rate is the same regardless their amplitude. Moreover, they can be of depression or elevation type independent of their speed. The dynamics of these solutions are studied as well. Author Affiliation: (a) Victoria University of Wellington, School of Mathematics, Statistics and Operations Research, PO Box 600, Wellington 6140, New Zealand (b) LAMA, UMR 5127 CNRS, UniversitAaAaAeA@ Savoie Mont Blanc, Campus Scientifiqu F-73376 Le Bourget-du-Lac Cedex, France (c) Khoja Akhmet Yassawi International Kazakh--Turkish University, Faculty of Natural Science, Department of Mathematics, 161200 Turkestan, Kazakhstan (d) Al-Farabi Kazakh National University, Faculty of Mechanics and Mathematics, Department of Mathematical and Computer Modelling, 050000 Almaty, Kazakhstan * Corresponding author. Article History: Received 7 November 2016; Accepted 23 March 2017 (miscellaneous) Communicated by A.P. Fordy
  • Ngôn ngữ: English
  • Số nhận dạng: ISSN: 0375-9601 ; E-ISSN: 1873-2429 ; DOI: 10.1016/j.physleta.2017.03.041

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