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