Soliton-like structures on a water-ice interface. (English. Russian original) Zbl 1353.35228

Il’ichev, A. T.

Russ. Math. Surv. 70, No. 6, 1051-1103 (2015); translation from Usp. Mat. Nauk 70, No. 6, 85-138 (2015).

Some results concerning the existence of solitary waves which emerge as bifurcations from quiescent state on the surface of a perfect fluid under an ice cover are presented. The geometrically non-linear elastic Kirchoff-Love theory is used to describe the ice cover. The Euler system is used to describe the wave process in the fluid. “The existence of solitary waves is proved for small speeds near critical values corresponding to the quiescent state”. The finite-dimensional reduced dynamical system on the center manifold (the invariant manifold of the considered dynamical system) is used. The semigroup theory is used. A survey of mathematical tools used for the study of soliton-like structures is given. The main result is following: “there can be three types of bifurcations of traveling waves from the quiescent state, each of them determined by its own set of critical values of the physical parameters” (see Conclusion).

Reviewer: Gelu Paşa (Bucureşti)

MSC:

35Q31	Euler equations
35B41	Attractors
35C08	Soliton solutions
74B20	Nonlinear elasticity

Keywords:

ice cover; solitary wave; bifurcation; close operator; normal forms; center manifold; resolvent estimates