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TMF, 2015, Volume 182, Number 2, Pages 277–293 (Mi tmf8708)  

This article is cited in 5 scientific papers (total in 5 papers)

Soliton-like structures on a liquid surface under an ice cover

A. T. Il'icheva, V. Ya. Tomashpolskiib

a Steklov Institute of Mathematics, RAS, Moscow, Russia
b Bauman Moscow State Technical University, Moscow, Russia

Abstract: For a complete system of equations describing wave propagation in a fluid of finite depth under an ice cover, we prove the existence of soliton-like solutions corresponding to a family of solitary waves of surface level depression. The ice cover is modeled as a Kirchhoff–Love elastic plate and has a significant thickness such that the plate inertia is taken into account in the model formulation. The family of solitary waves is parameterized by the wave propagation velocity, and its existence is proved for velocities that bifurcate from the characteristic velocity of linear waves and are rather close to this velocity. In turn, the solitary waves bifurcate from the rest state and are located in its neighborhood. In other words, we prove the existence of small-amplitude solitary waves of water–ice interface level depression. The proof uses the projection of the sought system of equations onto the center manifold {(}whose dimensionality is two in this case{\rm)} and a further analysis of a finite-dimensional reduced dynamical system on the center manifold.

Keywords: ice cover, solitary wave, bifurcation, center manifold, resolvent estimate

Funding Agency Grant Number
Russian Foundation for Basic Research 12-05-00889
14-01-00466


DOI: https://doi.org/10.4213/tmf8708

Full text: PDF file (517 kB)
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English version:
Theoretical and Mathematical Physics, 2015, 182:2, 231–245

Bibliographic databases:

Document Type: Article
Received: 13.05.2014
Revised: 08.07.2014

Citation: A. T. Il'ichev, V. Ya. Tomashpolskii, “Soliton-like structures on a liquid surface under an ice cover”, TMF, 182:2 (2015), 277–293; Theoret. and Math. Phys., 182:2 (2015), 231–245

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. A. T. Il'ichev, “Envelope solitary waves and dark solitons at a water–ice interface”, Proc. Steklov Inst. Math., 289 (2015), 152–166  mathnet  crossref  crossref  isi  elib
    2. A. T. Il'ichev, “Soliton-like structures on a water-ice interface”, Russian Math. Surveys, 70:6 (2015), 1051–1103  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    3. A. T. Il'ichev, “Solitary wave packets beneath a compressed ice cover”, Fluid Dyn., 51:3 (2016), 327–337  mathnet  crossref  crossref  mathscinet  zmath  isi  scopus
    4. P. Sprenger, M. A. Hoefer, “Shock waves in dispersive hydrodynamics with nonconvex dispersion”, SIAM J. Appl. Math., 77:1 (2017), 26–50  crossref  mathscinet  zmath  isi  scopus
    5. O. Trichtchenko, E. I. Parau, J.-M. Vanden-Broeck, P. Milewski, “Solitary flexural-gravity waves in three dimensions”, Philos. Trans. R. Soc. A-Math. Phys. Eng. Sci., 376:2129 (2018), 20170345  crossref  mathscinet  isi  scopus
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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