Amplitude Equation Models for the Interaction of Shocks with Nonlinear Dispersive Wave Envelopes

  • P. K. Newton
  • R. M. Axel


Mode coupling between hyperbolic and dispersive waves is discussed in a general context. A class of amplitude equations is then introduced and used to study the interaction of a shock wave with a dispersive wave envelope. In particular, we describe and analyze a system similar in structure to the well-known Zakharov equations from weak plasma turbulence theory, but with a nonlinear conservation equation that allows for finite time shock formation. Special attention is focused on the incompressible limit in which the shock speed is large compared to the underlying group velocity of the dispersive wave. This limit is also an integrable limit in which the model reduces to the nonlinear Schrödinger equation. In this limit, we study the interaction of a ‘fast’ shock wave and a single hump soliton. First we analyze the weakly coupled problem in which the shock evolves independently of the dispersive wave. We show that the shock formation rives a finite time blow-up in the generalized frequency, —θ t , of the dispersive wave. We then analyze the fully coupled problem and develop a multi-scale, near-integrable asymptotic method to derive the effective leading order shock equations and the leading order modulation equations governing the phase and amplitude of the dispersive wave envelope. A sequence of numerical experiments is then performed on the leading order shock equation to depict some of the interesting interactions predicted by the analysis.


Shock Wave Dispersive Wave Shock Strength Shock Speed Linear Limit 
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Copyright information

© Springer Science+Business Media New York  2002

Authors and Affiliations

  • P. K. Newton
    • 1
  • R. M. Axel
    • 2
  1. 1.Department of Aerospace Engineering, and Center for Applied Mathematical SciencesUniversity of Southern CaliforniaLos AngelesUSA
  2. 2.Department of MathematicsUniversity of IllinoisUrbanaUSA

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