Soliton-like structures on a water-ice interface. (December 2015)

• Record Type: Journal Article
• Authors: Il'ichev, A. T.
• Is Part Of: Russian mathematical surveys. Volume 70:Number 6(2015)
• Page Start: 1051
• Page End: 1103
• Publication Date: 2015-12
• Subjects: Primary 35J61 -- 74J35; ice cover -- solitary wave -- bifurcation -- closed operator -- normal forms -- centre manifold -- resolvent estimates.; Mathematics -- Soviet Union -- Periodicals; Mathematics -- Russia (Federation) -- Periodicals; Mathematics -- Periodicals; Mathematicians -- Soviet Union -- Periodicals; Mathematicians -- Russia (Federation) -- Periodicals; 510.5
• DOI: 10.1070/RM2015v070n06ABEH004974
• Languages: English
• ISSNs: 0036-0279

Abstract

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.