Dynamical Systems Methods in Hydrodynamics

This research project is concerned with mathematical models for describing the propagation of gravity water waves. More specifically, I will focus on nonlinear dispersive equations which arise as approximations of the Euler equations for homogeneous, inviscid and incompressible fluids with a free boundary. The main goals of this project are: i. to close the gap between two well-known model equations for waves propagating in shallow water, the Korteweg-de Vries and the Camassa-Holm equation. ii. to investigate the traveling wave solutions of an equation for surface waves corresponding to the latter equation. Concerning research objective (i), I plan to derive a new model equation by means of asymptotic expansions in terms of two parameters, relating the wave amplitude, length and water depth, in a physical regime characterized by these non dimensional quantities. Promising further insight into the propagation of shallow water waves, this equation attempts to extend the range of applicability compared to the KdV equation on the one hand, while reducing the technical complexity associated with the CH equation on the other hand. Regarding objective (ii), I will refine the results obtained during my PhD studies to continue and extend the investigation of solitary and periodic traveling wave solutions of equations modeling surface waves of moderate amplitude in shallow water. I will investigate their dependence on the wave speed, qualitative features concerning their shape, as well as their orbital stability. To prove the existence of such solutions I will apply techniques from the theory of dynamical systems, building on an approach that has already been initiated in a joint project with Prof. Gasull. Therefore, my research is planned to be carried out at his home institution, the Dynamical Systems Research Group (GSD) at the faculty of mathematics of the Universitat Autonoma de Barcelona. I am convinced that the collaboration with Prof. Gasull, a leading expert in the field of dynamical systems and a key member of the GSD research group, will be beneficial for my scientific progress, my future research output, and ultimately, for successfully completing my habilitation, planned to be initiated upon my return to Vienna.

No additional funding sources recorded.