Trajectories of Liquid Particles in a Dark Soliton Field in a Fluid Beneath an Ice Cover

Abstract

A fluid layer of finite depth described by Euler equations is considered. The ice cover is modeled by a geometrically non-linear elastic Kirchhoff–Love plate. The trajectories of liquid particles under the ice cover are found in the field of a nonlinear surface traveling wave tending to a periodic wave at infinity: the so-called “dark soliton” (which is a nonlinear product of a step-wave and a periodic wave) of small but finite amplitude. The presence of dark solitons in the system is an indicator of the modulation stability of the carrier periodic wave (defocusing). The analysis uses explicit asymptotic expressions for solutions describing wave structures on the water–ice interface such as dark solitons, as well as asymptotic solutions for the velocity field in the liquid column generated by these waves.