Whitham modulation theory for dispersive shock waves in systems with nonlocal dispersion of Benjamin-Ono type

Dr Khiem Nguyen (James Watt School of Engineering)

Thursday 7th March 14:00-15:00 Maths 311B / Zoom (ID: 894 0173 1730)


Dispersive hydrodynamics is the domain concerned with fluid or fluid-like motion in which dissipation, e.g., viscosity, is negligible relative to wave dispersion. In conservative media such as superfluids, optical materials, and water waves, nonlinearity has the tendency to engender wave breaking that is modulated by dispersion. The result of these processes is a multiscale, unsteady, coherent wave structure called dispersive shock wave. Dispersive shock waves have been found in several physical applications: superfluids, nonlinear optics, geophysics, and fluid dynamics. In 1965, G.B. Whitham published a seminal paper that ushered in the mathematical study of nonlinear dispersive waves. It turns out the modulation theory proposed by Whitham is an excellent tool for studying dispersive shock waves. Thus, there has been an emerging interest in this research area. In Whitham modulation theory, the wave solution is assumed to be periodic in the phase variable θ and modulated by the wave parameters such as wavenumber k, wave frequency ω, and amplitude A which are functions of spatial and time coordinates (see Figure below). In this consideration, the phase variable θ and the wave parameters are the fast and the slow variables, respectively. By averaging the conservation laws of the wave equation over the fast variable, namely the phase variable, the modulation equations are derived in terms of the wave parameters. The modulation equations can also be derived by averaging the Lagrangian for the wave equation over the periodic phase variable θ.

In this talk, the Whitham modulation theory will be briefly introduced by deriving the modulation equations for a class of Klein-Gordon equations. Then, the Whitham modulation methods will be applied to study dispersive shock waves in systems with nonlocal dispersion of BenjaminOno type. These systems describe propagation of nonlinear dispersive waves in nonlocal media. The obtained modulation solution of dispersive shock waves will then be validated against the direct numerical solutions.

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