Notation

Throughout this paper, we denote possibly different constants by the same

symbol C. We denote by Hul m the space of locally square-integrable functions on R

whose first m weak derivatives exist and are uniformly bounded in local L2 spaces

and for which the spatial translation y → u(· + y) is continuous with respect to the

Hul m-norm. Their norm is defined by

u

Hul

m

= sup

x∈R

u

H m(x,x+1)

where the Sobolev norm ·

H m(x,x+1)

is, for each fixed x ∈ R, given by

u

H m(x,x+1)

=

m

j=0

∂xu

j

L2(x,x+1)

.

We also use the weighted Sobolev spaces

Hm(n)

which we equip with the norm

u

H m(n)

= u ρw

n

H m

where ρw(x) = 1 + |x|2.

u(x, t) solution to reaction-diffusion system

u0(ωt − kx; k) wave train (2π-periodic in argument θ)

θ = ωt − kx travelling-wave coordinate (wave train)

k wave number

ω temporal frequency

ωnl(k) nonlinear dispersion relation

cp = ωnl(k)/k phase velocity

cg = dωnl(k)/dk group velocity

λ temporal eigenvalue

λlin(ν) linear dispersion relation in comoving frame

ν complex spatial Floquet exponent

imaginary part of spatial Floquet exponent ν = i

Φ(X, T ) slowly varying phase

q(X, T ) slowly varying wave number

0 δ 1 multi-scale expansion parameter

(X, T ) = (δ(x − cgt),

δ2t)

slow space and time variables

c∗ speed of viscous shock

ω∗ frequency of viscous shock

ξ = x − c∗t travelling-wave coordinate (shock)

τ = ω∗t rescaled time (2π-periodic)

σ = τ − k0ξ co-rotating coordinate (wave train)

ˆ( u ) = [Fu]( ) Fourier transform of u(x)

ˇ(x, u ) = [J u](x, ) Bloch transform of u(x)

When we fix a wave number k0, we will denote the associated frequency, group

and phase velocities evaluated at k0 by ω0 = ωnl(k0), cp

0

and cg,

0

respectively. When

confusion is unlikely, we will drop the index 0.

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