Uspekhi Matematicheskikh Nauk
 RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Archive Impact factor Subscription License agreement Submit a manuscript Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Uspekhi Mat. Nauk: Year: Volume: Issue: Page: Find

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

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.

DOI: https://doi.org/10.4213/rm9690

Full text: PDF file (989 kB)
References: PDF file   HTML file

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

Bibliographic databases:

UDC: 532.59
MSC: 35J61, 74J35
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
\Bibitem{Ili15} \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 \mathnet{http://mi.mathnet.ru/umn9690} \crossref{https://doi.org/10.4213/rm9690} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3462716} \zmath{https://zbmath.org/?q=an:06608782} \elib{https://elibrary.ru/item.asp?id=25707780} \transl \jour Russian Math. Surveys \yr 2015 \vol 70 \issue 6 \pages 1051--1103 \crossref{https://doi.org/10.1070/RM2015v070n06ABEH004974} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000372362900003} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84962876030} 

• http://mi.mathnet.ru/eng/umn9690
• https://doi.org/10.4213/rm9690
• http://mi.mathnet.ru/eng/umn/v70/i6/p85

 SHARE:

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, “Solitary wave packets beneath a compressed ice cover”, Fluid Dyn., 51:3 (2016), 327–337
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
•  Number of views: This page: 513 Full text: 122 References: 63 First page: 39