Hydroelastic waves and related problems

Jean-Marc Vanden-Broeck
University College London
Nonlinear waves propagating at the surface of a fluid bounded above by an elastic sheet are considered. The fluid is assumed to be incompressible and inviscid and the flow to be irrotational. Gravity is included in the dynamic boundary condition. This configuration can be viewed as a model for waves propagating under an ice sheet. The understanding of the properties of these waves is important in polar regions in assuring the safety of human activities, such as transportation over ice sheets. The mathematical formulation is similar to that of the classical problem of gravity capillary waves. The main difference is that the curvature term in the dynamic boundary condition is now replaced by a nonlinear term involving higher derivatives of the curvature of the free surface. This allows for the existence of new types of waves. We will show how to obtain fully nonlinear solutions by boundary integral equation methods. Both two-dimensional and three-dimensional waves will be studied. Periodic waves, solitary waves, generalised solitary waves and dark solitons are among the solutions to be presented.