Séminaire du LPTMS: Mark Hoefer (lecture n°2)


11:00 - 12:30

LPTMS, salle 201, 2ème étage, Bât 100, Campus d'Orsay
15 Rue Georges Clemenceau, Orsay, 91405

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Lectures on Whitham Modulation Theory and Dispersive Hydrodynamics (2)

Mark Hoefer (Department of Applied Mathematics, University of Colorado,  Boulder, USA)

Nonlinear wave modulation theory, developed by G. B. Whitham over 50  years ago, is a powerful mathematical tool to investigate dispersive  hydrodynamics.  Dispersive hydrodynamics encompass fluid and fluid-like  media in which nonlinear, hydrodynamic phenomena (e.g., shock formation  and expansion waves) are influenced more prominently by wave dispersion  than by irreversible, dissipative processes. Examples include  superfluids, intense light propagation through a nonlinear medium, and  the interface between two classical fluids.  A familiar feature of such  media includes the solitary wave or soliton, whose width represents the  characteristic coherence length of the medium, e.g., the so-called  healing length of a Bose-Einstein condensate.  Whitham theory is used to study modulations of nonlinear waves on a scale much larger than the
medium’s coherence length.  It has been successfully used to describe  the most fundamental object in dispersive hydrodynamics, a dispersive  shock wave.

These lectures will introduce the listener to the basic theory of  Whitham with applications to several modern examples.  Mathematically,  the Whitham modulation equations are a system of first order,  quasi-linear partial differential equations.  Properties of these  hydrodynamic type systems such as (strict) hyperbolicity, ellipticity,  genuine nonlinearity, and simple wave solutions will be elucidated with  a view toward understanding the physical implications of these abstract  concepts.  Fundamental problems in dispersive hydrodynamics such as the  Riemann problem and the piston problem will be described. The theory  will be sufficiently developed to describe a new type of hydrodynamic  interaction where a soliton coherently interacts with a hydrodynamic  flow, termed hydrodynamic soliton tunneling.

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