Nonlinear waves

Abstract

This project concerns the propagation of nonlinear waves: the equations describing them, their solutions and general properties; the physical systems where they show up; their relevance in the description of natural phenomena. The general context is the study of macroscopic extended systems, malnly hydrodynamics. Specifically, we will be interested is dispersive and dissipative systems, and we will focus on the study of wave propagation and formation of stable structures phenomena. There are two main lines of research. The first one consists in the study of general mathematical methods applicable to a wide range of physical systems. It deals with nonlinear partial differential equations of physics and correlated subjects, perturbation methods, solitons and integrable systems. The second one concerns specific physical systems where the mentioned methods may be applied: solitary-waves in inviscid fluids, convective hydrodynamic systems, geophysical fluid dynamics, porous media, solitons in condensed matter physics. In the background of all these studies, asymptotic rnethods, in particular the multiple scales method. Mathematical problems related to asymptotic modelling are also discussed. The project is divided in eleven research units, which are: short-wave dynamics; waves over topography; the renormalization group and perturbative series; nonlinear aspects of Bose-Einstein condensation; time series, solitons and oceanography; dissipative systems; the Green-Naghdi equations; systems under action of rapidly varying external forces; multiple-scales method in quantum mechanics; geophysical fluid dynamics; integrable differential equations. (AU)