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The Dynamics of Modulated Wave Trains
Arjen Doelman, CWI, Amsterdam, The Netherlands, Björn Sandstede, University of Surrey, Guildford, United Kingdom, Arnd Scheel, University of Minnesota, Minneapolis, MN, and Guido Schneider, Universität Stuttgart, Germany

Memoirs of the American Mathematical Society
2009; 105 pp; softcover
Volume: 199
ISBN-10: 0-8218-4293-5
ISBN-13: 978-0-8218-4293-5
List Price: US$70
Individual Members: US$42
Institutional Members: US$56
Order Code: MEMO/199/934
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The authors investigate the dynamics of weakly-modulated nonlinear wave trains. For reaction-diffusion systems and for the complex Ginzburg-Landau equation, they establish rigorously that slowly varying modulations of wave trains are well approximated by solutions to the Burgers equation over the natural time scale. In addition to the validity of the Burgers equation, they show that the viscous shock profiles in the Burgers equation for the wave number can be found as genuine modulated waves in the underlying reaction-diffusion system. In other words, they establish the existence and stability of waves that are time-periodic in appropriately moving coordinate frames which separate regions in physical space that are occupied by wave trains of different, but almost identical, wave number.

The speed of these shocks is determined by the Rankine-Hugoniot condition where the flux is given by the nonlinear dispersion relation of the wave trains. The group velocities of the wave trains in a frame moving with the interface are directed toward the interface. Using pulse-interaction theory, the authors also consider similar shock profiles for wave trains with large wave number, that is, for an infinite sequence of widely separated pulses. The results presented here are applied to the FitzHugh-Nagumo equation and to hydrodynamic stability problems.

Table of Contents

  • Notation
  • Introduction
  • The Burgers equation
  • The complex cubic Ginzburg-Landau equation
  • Reaction-diffusion equations: Set-up and results
  • Validity of the Burgers equation in reaction-diffusion equations
  • Validity of the inviscid Burgers equation in reaction-diffusion systems
  • Modulations of wave trains near sideband instabilities
  • Existence and stability of weak shocks
  • Existence of shocks in the long-wavelength limit
  • Applications
  • Bibliography
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