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
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