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Erratum to: Triggered Fronts in the Complex Ginzburg Landau Equation
Erratum to: J Nonlinear Sci
Erratum to: Triggered Fronts in the Complex Ginzburg Landau Equation
Ryan Goh 0
Arnd Scheel 0
Here 0
B Ryan Goh gohxx 0
@umn.edu 0
0 School of Mathematics, University of Minnesota-Twin Cities , Vincent Hall, 206 Church St. SE, Minneapolis, MN 55455 , USA
Theorem 1 Fix α, γ ∈ R and assume that there exists a generic free front. Then there exist trigger fronts for c < clin, |c − clin| sufficiently small. The frequency of the trigger front possesses the expansion
-
Zi|(clin − c)3/2 + O((clin − c)2).
and Zi is defined in (3.22), below. Furthermore, for α = γ the selected wavenumber
has the expansion
π(1 + γ 2)|1/2Zi| (clin − c)3/2 + O((clin − c)2)
. The distance between the trigger
and the front interface is given by
ξ∗ = π(1 + α2)1/4(clin − c)−1/2 + (1 + α2)1/2
Zr + O((clin − c)1/2),
Zr is defined in (3.22) as well.
The error in the original statement is caused by an incorrect transformation of the
expansion from “hat” variables back to original variables. In particular, we neglected
the dependence of m on c and ω. A correct calculation traces the scalings as a nonlinear
transformation Υ : (c, ω) → (cˆ, ωˆ ), which can, for the purpose of the expansion, be
approximated by its linearization near the linear spreading parameters clin, ωlin,
with c := (clin − c) and ω := (ωlin − ω). The Inverse Function Theorem then
gives cˆ ∼ (1 + α2)1/2 c instead of our previous cˆ ∼ (1 + α2)−1/2 c. This leads
to the change in the exponent of the (1 + α2)–term throughout Theorem 1.
We also note a sign error of ωˆ in (3.11) that propagates through Sections 3.2–3.5
such that the first expansion in (3.27) should read
2 Zi
With this sign change and the correct scaling, (3.27) yields
from which we obtain the expansion as stated here by setting
j = −1:
2 Zi(1 + α2)3/4
(clin − c)3/2 + O((clin − c)2).
In the same manner, we obtain the expansions for the unscaled distance between the
trigger and the invasion front, ξ∗, and for the selected wavenumber ktf as stated in
Theorem 1 above.
Goh , R. , Scheel , A. : Triggered fronts in the complex Ginzburg-Landau equation . J. Nonlinear Sci . 24 , 117 - 144 ( 2014 ) (...truncated)