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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