# SĂ©minaire du LPTMS: Mark Hoefer (lecture nÂ°1)

## Lectures on Whitham Modulation Theory and Dispersive Hydrodynamics (1)

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

Nonlinear wave modulation theory, developed by G. B. Whitham over 50Â years ago, is a powerful mathematical tool to investigate dispersiveÂ hydrodynamics. Â Dispersive hydrodynamics encompass fluid and fluid-likeÂ media in which nonlinear, hydrodynamic phenomena (e.g., shock formationÂ and expansion waves) are influenced more prominently by wave dispersionÂ than by irreversible, dissipative processes. Examples includeÂ superfluids, intense light propagation through a nonlinear medium, andÂ the interface between two classical fluids. Â A familiar feature of suchÂ media includes the solitary wave or soliton, whose width represents theÂ characteristic coherence length of the medium, e.g., the so-calledÂ healing length of a Bose-Einstein condensate. Â Whitham theory is used to study modulations of nonlinear waves on a scale much larger than the

medium’s coherence length. Â It has been successfully used to describeÂ the most fundamental object in dispersive hydrodynamics, a dispersiveÂ shock wave.

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

**Date/Time : ** 16/06/2017 - *11:00 - 12:30*

**Location : ** LPTMS, salle 201, 2Ă¨me Ă©tage, BĂ˘t 100, Campus d'Orsay

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