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Uspekhi Mat. Nauk, 2015, Volume 70, Issue 6(426), Pages 85–138 (Mi umn9690)  

This article is cited in 16 scientific papers (total in 16 papers)

Soliton-like structures on a water-ice interface

A. T. Il'ichev

Steklov Mathematical Institute of Russian Academy of Sciences

Abstract: This paper contains a proof of the existence of soliton-like solutions of the complete system of equations describing wave propagation in a fluid of finite depth under an ice cover. These solutions correspond to solitary waves of various kinds propagating along the water-ice interface. The plane-parallel motion is considered in a layer of a perfect fluid of finite depth whose characteristics obey the complete two-dimensional Euler system of equations. The ice cover is modelled by an elastic Kirchhoff–Love plate and has significant thickness, so that the plate's inertia is taken into account in the formulation of the model. The Euler equations contain the additional pressure arising from the presence of the elastic plate floating freely on the fluid surface. The indicated families of solitary waves are parameterized by the speed of the waves, and their existence is proved for speeds lying in some neighbourhood of the critical value corresponding to the quiescent state. The solitary waves, in turn, bifurcate from the quiescent state and lie in some neighbourhood of it. In other words, it is proved that solitary waves of sufficiently small amplitude exist on the water-ice interface. The proof is conducted using the projection of the required system of equations on the centre manifold and a further analysis of the finite-dimensional reduced dynamical system on the centre manifold.
Bibliography: 84 titles.

Keywords: ice cover, solitary wave, bifurcation, closed operator, normal forms, centre manifold, resolvent estimates.

Funding Agency Grant Number
Russian Science Foundation 14-50-00005
This work is supported by the Russian Science Foundation under grant 14-50-00005.


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English version:
Russian Mathematical Surveys, 2015, 70:6, 1051–1103

Bibliographic databases:

UDC: 532.59
MSC: 35J61, 74J35
Received: 19.01.2015
Revised: 25.08.2015

Citation: A. T. Il'ichev, “Soliton-like structures on a water-ice interface”, Uspekhi Mat. Nauk, 70:6(426) (2015), 85–138; Russian Math. Surveys, 70:6 (2015), 1051–1103

Citation in format AMSBIB
\by A.~T.~Il'ichev
\paper Soliton-like structures on a water-ice interface
\jour Uspekhi Mat. Nauk
\yr 2015
\vol 70
\issue 6(426)
\pages 85--138
\jour Russian Math. Surveys
\yr 2015
\vol 70
\issue 6
\pages 1051--1103

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    This publication is cited in the following articles:
    1. 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  elib  elib  scopus
    2. V. V. Markov, G. B. Sizykh, “Exact solutions of the Euler equations for some two-dimensional incompressible flows”, Proc. Steklov Inst. Math., 294 (2016), 283–290  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    3. A. T. Il'ichev, A. P. Chugainova, “Spectral stability theory of heteroclinic solutions to the Korteweg–de Vries–Burgers equation with an arbitrary potential”, Proc. Steklov Inst. Math., 295 (2016), 148–157  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    4. A. T. Il'ichev, “Stability of solitary waves in membrane tubes: A weakly nonlinear analysis”, Theoret. and Math. Phys., 193:2 (2017), 1593–1601  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    5. A. T. Il'ichev, A. S. Savin, “Process of establishing a plane-wave system on ice cover over a dipole moving uniformly in an ideal fluid column”, Theoret. and Math. Phys., 193:3 (2017), 1801–1810  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    6. A. E. Bukatov, A. A. Bukatov, “Vibrations of a floating elastic plate upon nonlinear interaction of flexural-gravity waves”, J. Appl. Mech. Tech. Phys., 59:4 (2018), 662–672  crossref  crossref  mathscinet  zmath  isi  elib  elib  scopus
    7. A. Il'ichev, “Physical parameters of envelope solitary waves at a water-ice interface”, Mathematical Methods and Computational Techniques in Science and Engineering II, AIP Conf. Proc., 1982, ed. N. Bardis, Amer. Inst. Phys., 2018, 020036-1  crossref  isi  scopus
    8. A. E. Bukatov, A. A. Bukatov, “Phase structure of fluid fluctuations with a floating elastic ice plate under nonlinear interaction of progressive surface waves”, Phys. Oceanogr., 25:1 (2018), 3–17  crossref  isi
    9. E. B. Pavelyeva, A. S. Savin, “Establishment of waves generated by a pulsating source in a finite-depth fluid”, Fluid Dyn., 53:4 (2018), 461–470  crossref  crossref  zmath  isi  elib
    10. A. T. Il'ichev, “Envelope solitary waves at a water-ice interface: the case of positive initial tension”, Math. Montisnigri, 43 (2018), 49–57  mathscinet  isi
    11. A. T. Il'ichev, V. J. Tomashpolskii, “Characteristic parameters of nonlinear surface envelope waves beneath an ice cover under pre-stress”, Wave Motion, 86 (2019), 11–20  crossref  mathscinet  isi  scopus
    12. A. A. Bukatov, “Nonlinear vibrations of a floating longitudinally compressed elastic plate in the interaction of wave harmonics of finite amplitude”, Fluid Dyn., 54:2 (2019), 194–204  crossref  isi
    13. M. M. Bhatti, D.-Q. Lu, “An application of Nwogu's Boussinesq model to analyze the head-on collision process between hydroelastic solitary waves”, Open Phys., 17:1 (2019), 177–191  crossref  isi
    14. A. T. Il'ichev, “Physical parameters of solitary wave packets in shallow basins under ice cover”, Theoret. and Math. Phys., 201:3 (2019), 1710–1722  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    15. A. A. Bukatov, “Analiz fazovoi struktury kolebanii zhidkosti s plavayuschei prodolno szhatoi uprugoi plastinkoi pri nelineinom vzaimodeistvii poverkhnostnykh progressivnykh voln”, Vestn. S.-Peterburg. un-ta. Ser. 10. Prikl. matem. Inform. Prots. upr., 16:3 (2020), 226–237  mathnet  crossref
    16. A. T. Il'ichev, “Effective wavelength of envelope waves on the water surface beneath an ice sheet: small amplitudes and moderate depths”, Theoret. and Math. Phys., 208:3 (2021), 1182–1200  mathnet  crossref  crossref  mathscinet  isi  elib
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