Generalized Riemann Problems In Dispersive Hydrodynamics

Sprenger, Patrick

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Graduate Thesis Or Dissertation

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Citeable URL: https://scholar.colorado.edu/concern/graduate_thesis_or_dissertations/g445cf038

Abstract

Nonlinear, dispersive wave phenomena occur in a variety of physical contexts, both in nature and the laboratory. Mathematically, their dynamics can be modeled by a dispersive hydrodynamic system---a first order system of conservation laws modified by dispersion. In appropriate physical regimes, a multi-scale asymptotic expansion can by employed to derive a scalar equation from which we can infer approximate dynamics of the overarching system.

We first study scalar models of dispersive hydrodynamics when dispersion is of higher order. Higher order dispersion in nonlinear, real-valued, local scalar equations can manifest when spatial derivatives are higher than third order. The primary mathematical framework we utilize is Whitham modulation theory, an asymptotic method to describe the slow modulations of a periodic wave's parameters. We identify three new classes of DSWs solutions to the Kawahara equation---a weakly nonlinear model that contains both third and fifth order dispersive terms. Numerical and asymptotic studies of the DSW solutions to the Kawahara equation motivate a further comprehensive study of the Whitham modulation equations for the fifth order Korteweg-de Vries (KdV5) equation. We compute various heteroclinic traveling waves that are shown to correspond to weak, discontinuous shock solutions of the KdV5-Whitham modulation system in the zero dispersion limit. The discontinuous shock solutions are shown to arise from a so-called generalized Riemann problem. The existence of heteroclinic traveling waves of the governing equation allow us to define the admissibility of discontinuous, weak solutions of the Whitham modulation equations, which we term Whitham shocks. Furthermore, the structure of the modulation equations, e.g. hyperbolicity or ellipticity, determine the modulational (in)stability of the heteroclinic traveling wave corresponding to an admissible Whitham shock. We then revisit the DSW solution of the KdV5 equation and demonstrate that it can be described in terms of a shock-rarefaction solution of the KdV5-Whitham modulation system. We conclude this portion with a discussion of how our results can be applied to other model dispersive hydrodynamic systems.

We then investigate the interaction of a soliton and an evolving mean flow in bi-directional dispersive hydrodynamic media. The model equation is the defocusing nonlinear Schrödinger equation. Utilizing Whitham modulation theory and posing a generalized Riemann problem for an initial jump in the mean flow and the soliton amplitude, a simple wave solution of the diagonalized NLS-Whitham modulation equations is obtained. This yields algebraic relationships between far-field initial data and the solitary wave amplitude from which we may infer the long-time soliton dynamics including the hydrodynamic trapping or transmission of the soliton by a, possibly, oscillatory mean flow.

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