# The Dynamics of Modulated Wave Trains

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*Arjen Doelman; Björn Sandstede; Arnd Scheel; Guido Schneider*

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

# Table of Contents

## The Dynamics of Modulated Wave Trains

- Contents v6 free
- Notation 110 free
- Chapter 1. Introduction 312
- Chapter 2. The Burgers equation 1524
- Chapter 3. The complex cubic Ginzburg–Landau equation 1928
- 3.1. Set-up 1928
- 3.2. Slowly-varying modulations of the k = 0 wave train: Results 2029
- 3.3. Derivation of the Burgers equation 2332
- 3.4. The construction of higher-order approximations 2433
- 3.5. The approximation theorem for the wave numbers 2534
- 3.6. Mode filters, and separation into critical and noncritical modes 2534
- 3.7. Estimates of the linear semigroups 2938
- 3.8. Estimates of the residual 3039
- 3.9. Estimates of the errors 3140
- 3.10. Proofs of the theorems from §3.2 3443

- Chapter 4. Reaction-diffusion equations: Set-up and results 3948
- Chapter 5. Validity of the Burgers equation in reaction-diffusion equations 5362
- Chapter 6. Validity of the inviscid Burgers equation in reaction-diffusion systems 6574
- Chapter 7. Modulations of wave trains near sideband instabilities 7382
- Chapter 8. Existence and stability of weak shocks 8392
- Chapter 9. Existence of shocks in the long-wavelength limit 93102
- Chapter 10. Applications 99108
- Bibliography 103112