eBook ISBN:  9781470405403 
Product Code:  MEMO/199/934.E 
List Price:  $70.00 
MAA Member Price:  $63.00 
AMS Member Price:  $42.00 
eBook ISBN:  9781470405403 
Product Code:  MEMO/199/934.E 
List Price:  $70.00 
MAA Member Price:  $63.00 
AMS Member Price:  $42.00 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 199; 2009; 105 ppMSC: Primary 35; 37
The authors investigate the dynamics of weaklymodulated nonlinear wave trains. For reactiondiffusion 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 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 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 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 FitzHugh–Nagumo equation and to hydrodynamic stability problems.

Table of Contents

Chapters

Chapter 1. Introduction

Chapter 2. The Burgers equation

Chapter 3. The complex cubic GinzburgLandau equation

Chapter 4. Reactiondiffusion equations: Setup and results

Chapter 5. Validity of the Burgers equation in reactiondiffusion equations

Chapter 6. Validity of the inviscid Burgers equation in reactiondiffusion systems

Chapter 7. Modulations of wave trains near sideband instabilities

Chapter 8. Existence and stability of weak shocks

Chapter 9. Existence of shocks in the longwavelength limit

Chapter 10. Applications


RequestsReview Copy – for publishers of book reviewsPermission – for use of book, eBook, or Journal contentAccessibility – to request an alternate format of an AMS title
 Book Details
 Table of Contents
 Requests
The authors investigate the dynamics of weaklymodulated nonlinear wave trains. For reactiondiffusion 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 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 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 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 FitzHugh–Nagumo equation and to hydrodynamic stability problems.

Chapters

Chapter 1. Introduction

Chapter 2. The Burgers equation

Chapter 3. The complex cubic GinzburgLandau equation

Chapter 4. Reactiondiffusion equations: Setup and results

Chapter 5. Validity of the Burgers equation in reactiondiffusion equations

Chapter 6. Validity of the inviscid Burgers equation in reactiondiffusion systems

Chapter 7. Modulations of wave trains near sideband instabilities

Chapter 8. Existence and stability of weak shocks

Chapter 9. Existence of shocks in the longwavelength limit

Chapter 10. Applications