CHAPTER 1

Introduction

We begin in §1.1 with a grasshopper’s guide which contains a brief outline of

our results and the plan of the paper. In the rest of the introduction, starting

with §1.2, we explain our results and their proofs in more detail. We finish the

introduction in §1.6 with references to related work and a brief discussion of open

problems.

1.1. Grasshopper’s guide

The issue investigated in this paper is the dynamics of slow modulations of

nonlinear spatially-periodic travelling waves, in the following referred to as wave

trains, in reaction-diffusion equations

∂tu = D∂xxu + f(u), x ∈ R, u ∈

Rd.

Let u(x, t) = u0(ωt − kx; k) be such a wave train whose profile u0(θ; k) is 2π-

periodic in θ, and whose temporal frequency ω and spatial wave number k are

related through the nonlinear dispersion relation ω = ωnl(k). We define the group

velocity of the wave trains to be cg = ωnl(k) and denote their linear dispersion

relation by λlin(ν). If we modulate the wave number k of the wave trains over large

spatial scales, we are led to an ansatz of the form

u(x, t) = u0(ωt − kx − Φ(X, T ); k + δ∂X Φ(X, T )), (X, T ) =

(

δ(x − cgt),

δ2t

)

with 0 δ 1, which turns out to satisfy the underlying reaction-diffusion sys-

tem formally to leading order provided the wave-number modulation q(X, T ) =

∂X Φ(X, T ) is a solution of the (viscous) Burgers equation

∂T q =

1

2

λlin(0)∂XX q −

1

2

ωnl(k)∂X

(q2).

In this manuscript, we investigate the following issues:

• Validity results for the Burgers equation (setup: §4.1; results: §4.4; proofs:

§5):

We establish rigorous error estimates for the approximation of slowly-

varying modulated wave trains via the Burgers equation over the natural

time scale of order

δ−2.

The error estimates are uniform in the spatial

variable x provided the wave-number modulation q(X, T ) approaches lim-

its as X → ±∞. These results are formulated and proved separately

for the complex Ginzburg–Landau equation (§3) and for general reaction-

diffusion systems. For the latter case, we also present approximation

results for the inviscid Burgers equation over time scales of order

δ−1

(§6).

3