The Dynamics of Modulated Wave Trains
by
Arjen Doelman
Korteweg-de Vries Instituut, Universiteit van Amsterdam

In this talk, we consider the dynamics of nonlinear wave trains in reaction-diffusion equations. We prove that slowly varying modulations of wave trains are well approximated by solutions of the Burgers equation over the natural time scale. In addition to the validity of Burgers equation over long but finite time intervals, we show that the viscous shock profiles in Burgers equation can be found as genuine modulated waves, or defects, in the underlying reaction-diffusion system. Moreover, we study the stability of these defects. Since the group velocities of the wave trains in a frame moving with the defect are directed towards the defect, the modulated waves can be considered as sinks. Finally, the results are applied some explicit systems as the complex Ginzburg-Landau equation, the FitzHugh-Nagumo system and Taylor-Couette flow. This talk is based on joint work with Bjorn Sandstede, Arnd Scheel and Guido Schneider.

Date received: September 26, 2003