Memoirs of the American Mathematical Society 2009; 105 pp; softcover Volume: 199 ISBN10: 0821842935 ISBN13: 9780821842935 List Price: US$70 Individual Members: US$42 Institutional Members: US$56 Order Code: MEMO/199/934
 The authors investigate the dynamics of weaklymodulated nonlinear wave trains. For reactiondiffusion systems and for the complex GinzburgLandau 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 reactiondiffusion system. In other words, they establish the existence and stability of waves that are timeperiodic 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 RankineHugoniot 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 pulseinteraction 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 FitzHughNagumo equation and to hydrodynamic stability problems. Table of Contents  Notation
 Introduction
 The Burgers equation
 The complex cubic GinzburgLandau equation
 Reactiondiffusion equations: Setup and results
 Validity of the Burgers equation in reactiondiffusion equations
 Validity of the inviscid Burgers equation in reactiondiffusion systems
 Modulations of wave trains near sideband instabilities
 Existence and stability of weak shocks
 Existence of shocks in the longwavelength limit
 Applications
 Bibliography
