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

A. T. Il'ichev ; V. Ya. Tomashpolskii

Teoretičeskaâ i matematičeskaâ fizika, Tome 182 (2015) no. 2, pp. 277-293

Résumé

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.

Détail

Keywords: ice cover, solitary wave, center manifold, resolvent estimate.
Mots-clés : bifurcation