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Title:The interaction of shock waves and dispersive waves
Author(s):Axel, Ralph Martin
Doctoral Committee Chair(s):Newton, Paul
Department / Program:Mathematics
Degree Granting Institution:University of Illinois at Urbana-Champaign
Engineering, Mechanical
Abstract:We introduce and analyze a coupled system of partial differential equations which model the interaction of shocks with a dispersive wave envelope. The system mimics the Zakharov equations from weak plasma turbulence theory but replaces the linear wave equation in that system by a nonlinear equation allowing shock formation. This nonlinear equation is a hyperbolic conservation law forced by the dispersive wave.
Chapter 1 considers a weak coupling in which the nonlinear wave evolves independently but appears as the potential in the time dependent Schrodinger equation governing the dispersive wave. We solve the Riemann problem in a frame of reference moving with the shock. A viscous diffusion term is then added to the shock equation and by constructing asymptotic expansions in the small diffusion coefficient, we show that the Riemann problem steady states are zero diffusion limits. A family of time dependent solutions is derived in the case that the initial data evolves to a steady-state shock in finite time. We prove that shock formation drives a finite time blow-up in the phase gradient of the dispersive wave and identify a family of transients of the Riemann problem steady states.
In Chapter 2, the incompressible limit of the fully coupled equations is considered. After presenting some exact solutions of the full system, a multi-time-scale perturbation method is used to resolve the interaction of solutions of Schrodinger's equation and a rapidly propagating shock wave. The leading order interaction equations are analyzed by the method of characteristics. The influence of the shock on the dispersive wave is manifested in the generalized frequency of the dispersive wave: the fast-time part of the frequency is the shock wave itself. Numerical experiments demonstrate rigorous results describing how, to leading order, the shock wave is affected by the dispersive wave.
Issue Date:1996
Rights Information:Copyright 1996 Axel, Ralph Martin
Date Available in IDEALS:2011-05-07
Identifier in Online Catalog:AAI9702452
OCLC Identifier:(UMI)AAI9702452

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