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.

This project concerns mathematical models of water waves, which are relevant for the description and understanding of the dynamics of wave propagation.The main focus of this project revolved around so-called traveling wave solutions, which have the property that they travel at constant speed in one direction without changing their shape. These type of solutions, for instance in the form of periodic waves or localised humps, are well-known and observable phenomena. To justify the validity and applicability of a mathematical model it is therefore desirable to prove that the involved equations allow for the existence and stability of such solutions - this was one of the main goals that we have achieved in this project. Moreover, we succeeded in obtaining a better understanding of the role of symmetric wave profiles in this context, as well as in obtaining a more profound understanding of the functional relationship which exists between the length and amplitude of periodic wave trains. We found that for model equations that describe waves of larger amplitude, one can obtain entirely new types of wave profiles. Such profiles, which may involve cusps and peaks at both the wave crests and wave troughs, do not occur in models for moderate amplitude waves. In order to obtain these results we developed new mathematical tools, drawing both from abstract theoretical results in the framework of dynamical systems as well as techniques developed in the context of water waves.A further development of such tools and the derivation of additional model equations, which take into account the complex interplay among various physical factors present in wave propagation (such as the interaction of waves with currents), is the subject of ongoing and future investigations.