This article is cited in 11 scientific papers (total in 11 papers)
Soliton-like structures on a water-ice interface
A. T. Il'ichev
Steklov Mathematical Institute of Russian Academy of Sciences
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.
ice cover, solitary wave, bifurcation, closed operator, normal forms, centre manifold, resolvent estimates.
|Russian Science Foundation
|This work is supported by the Russian Science Foundation under grant 14-50-00005.
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Russian Mathematical Surveys, 2015, 70:6, 1051–1103
MSC: 35J61, 74J35
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
\paper Soliton-like structures on a water-ice interface
\jour Uspekhi Mat. Nauk
\jour Russian Math. Surveys
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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
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
A. T. Il'ichev, “Stability of solitary waves in membrane tubes: A weakly nonlinear analysis”, Theoret. and Math. Phys., 193:2 (2017), 1593–1601
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
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
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
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
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
A. T. Il'ichev, “Envelope solitary waves at a water-ice interface: the case of positive initial tension”, Math. Montisnigri, 43 (2018), 49–57
Il'ichev A.T., Tomashpolskii V.J., “Characteristic Parameters of Nonlinear Surface Envelope Waves Beneath An Ice Cover Under Pre-Stress”, Wave Motion, 86 (2019), 11–20
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