The dynamics of modulated wave trains.

We investigate the dynamics of weakly-modulated nonlinear wave trains. For reaction-diffusion systems and for the complex Ginzburg--Landau equation, we establish rigorously that slowly varying modulations of wave trains are well approximated by solutions to Burgers equation over the natural time scale. In addition to the validity of Burgers equation, we show that the viscous shock profiles in Burgers equation for the wave number can be found as genuine modulated waves in the underlying reaction-diffusion system. In other words, we 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, we also consider similar shock profiles for wave trains with large wave number, that is, for an infinite sequence of widely separated pulses. The results represented here are then applied to the FitzHugh--Nagumo equation and to hydrodynamic stability problems