MERCREDI 18 MARS 2009
11 heures
Salle Séminaire 5
Centre de Physique Théorique
Marseille-Luminy

Sergey Dobrokhotov
A. Ishlinski Institute for Problems in Mechanics
of Russian Academy of Sciences

Title: Semiclassical asymptotics for fast decaying solutions of linear
hyperbolic systems and their applications to geostrophic flows

Abstract: We suggest a new asymptotic representation for the solutions to the
multidimensional linear hyperbolic systems with variable coefficients
with localized initial data. This representation is the generalization
of the Maslov canonical operator. It is based also on a simple
relationship between fast decaying and fast oscillating solutions and
on boundary layer ideas. Our main result is the explicit formula which
establishes the connection between initial localized perturbations and
wave profiles near the wave fronts including the neighborhood of
backtracking (focal or turning) and self intersection points. Also we
show that in the case when the original system possesses the vortical
solutions, the solitary vortices correspond to the focal points
propagated in the phase space. We show that wave profiles and the
structure of vortices are related with a form of initial sources and
also with the Lagrangian manifolds organized by the rays and
wavefronts in the phase space. In particular we discuss the influence
of such topological characteristics like the Maslov and Morse indices
to metamorphosis of the profiles after crossing the focal points. We
apply these formulas to the problem of a propagation of tsunami waves
in the frame of so-called "piston model" and also to the problem of
propagation of mesoscale vortices (typhoons and hurricanes) in the
atmosphere.