This project is focussed on evolution problems in which dispersion is predominant compared to dissipative or diffusive mechanisms. It is motivated by physical applications in which the total energy is - to some extent - conserved, and also by numerical issues regarding the approximation, with the least possible amount of numerical viscosity, of hyperbolic systems of conservation laws. The model, dispersive equations that are being considered include the Korteweg-de Vries, Nonlinear Schrödinger, Kadomtsev– Petviashvili, Kawahara, and the Davey–Stewartson equations, but also the more complicated systems of Euler–Korteweg (for capillary fluids), and of Green–Naghdi (for water waves). All these equations and systems are taken in their most general form, which means that their nonlinearities are not predefined, and that integrability arguments should not have a preponderant importance in the proposed work.
Furthermore, the project aims at dealing with problems in which boundaries play an important role, be they ‘physical’, fixed boundaries such as walls or pipe extremities in fluid flows, artificial boundaries introduced for numerical purposes, moving boundaries such as shocks in compressible fluids or free surface for incompressible fluids submitted to gravity. If the project does not involve any numerical analysis in the usual sense - e.g. proving the convergence of numerical methods -, the numerics is nevertheless ubiquitous, with the will of elaborating innovative numerical schemes and of understanding their qualitative properties.
One of its originalities is that, for various reasons, it is situated at the ‘interface’ between the hyperbolic and dispersive partial differential equations. A first task concerns the analysis of continuous, and also discrete, dispersive Initial Boundary Value Problems (IBVP), the latter coming from the discretization of hyperbolic IBVP and requiring a generalization of the Gustafsson-Kreiss-Sundström stability theory. Within this task, the design and analysis of Artificial Boundary Conditions (ABC) for the models mentioned above is a challenging issue. Another issue is the qualitative analysis of various remarkable solutions of those model equations, which we refer to as dispersive patterns. These include periodic waves, which are special, traveling waves with a usually large number of degrees of freedom, as well as dispersive shocks, which are complicated, unsteady patterns. Their understanding is closely related to the modulated theory initiated by Whitham in the 1970s. In order to go further, some asymptotic analysis is needed, which is also the case for the analysis of small amplitude wave trains, another topic of interest in this project. Finally, a more applied purpose is to derive new models that take into account multidimensional effects in surface waves such as tidal bores and roll waves.