1 Erratum to: J Nonlinear Sci (2014) 24:117–144 DOI 10.1007/s00332-013-9186-1

The expansion given in the main result, Theorem 1, of Goh and Scheel (2014) is incorrect. The correct statement is as follows.

Theorem 1

Fix \(\alpha ,\gamma \in \mathbb {R}\) and assume that there exists a generic free front. Then there exist trigger fronts for \(c<c_\mathrm {lin}\), \(|c-c_\mathrm {lin}|\) sufficiently small. The frequency of the trigger front possesses the expansion

$$\begin{aligned} \omega _\mathrm {tf}( c) = \omega _\mathrm {abs}(c) + \frac{2}{\pi }(1+\alpha ^2)^{3/4} |\Delta Z_\mathrm {i}| ( c_\mathrm {lin}- c)^{3/2} + \mathcal {O}( ( c_\mathrm {lin}- c)^{2}). \end{aligned}$$
(0.1)

Here,

$$\begin{aligned} \omega _\mathrm {abs}(c)=-\alpha +\frac{\alpha c^2}{2(1+\alpha ^2)}, \end{aligned}$$

and \(\Delta Z_\mathrm {i}\) is defined in (3.22), below. Furthermore, for \(\alpha \ne \gamma \) the selected wavenumber has the expansion

$$\begin{aligned} k_\mathrm {tf} = k_\mathrm {lin}- g_1(\alpha ,\gamma ) (c_\mathrm {lin}- c) - \frac{(1+\alpha ^2)^{3/4} |\Delta Z_\mathrm {i}|}{\pi (1+\gamma ^2)^{1/2}} ( c_\mathrm {lin}- c)^{3/2} + \mathcal {O}(( c_\mathrm {lin}- c)^{2}) \end{aligned}$$

where \(g_1(\alpha ,\gamma ) =\frac{1}{2(\gamma - \alpha )}\left(1-\frac{1+2\alpha \gamma -\alpha ^2 }{\sqrt{(1+\alpha ^2)(1+\gamma ^2)}} \right).\) The distance between the trigger and the front interface is given by

$$\begin{aligned} {\xi }_* = \pi (1+\alpha ^2)^{1/4} ( c_\mathrm {lin}- c)^{-1/2} + (1+\alpha ^2)^{1/2}\, \Delta Z_\mathrm {r} + \mathcal {O}(( c_\mathrm {lin}- c)^{1/2}), \end{aligned}$$

where \(\Delta Z_\mathrm {r}\) 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 \(\omega \). A correct calculation traces the scalings as a nonlinear transformation \(\varUpsilon : (c, \omega ) \mapsto (\hat{c}, \hat{\omega })\), which can, for the purpose of the expansion, be approximated by its linearization near the linear spreading parameters \(c_\mathrm {lin},\omega _\mathrm {lin}\),

$$\begin{aligned} ( \hat{c}, \hat{\omega }) =(2,0)+ D \varUpsilon \Big |_{c_\mathrm {lin},\omega _\mathrm {lin}} (\Delta c, \Delta \omega ) +\mathcal {O}((\Delta \hat{c})^2, (\hat{\omega })^2), \end{aligned}$$

with \(\Delta c := ( c_\mathrm {lin}- c)\) and \(\Delta \omega := (\omega _\mathrm {lin}- \omega )\). The Inverse Function Theorem then gives \(\Delta \hat{c} \sim (1+\alpha ^2)^{1/2} \Delta c\) instead of our previous \(\Delta \hat{c} \sim (1+\alpha ^2)^{-1/2} \Delta c\). This leads to the change in the exponent of the \((1+\alpha ^2)\)–term throughout Theorem 1.

We also note a sign error of \(\hat{\omega }\) in (3.11) that propagates through Sections 3.2–3.5 such that the first expansion in (3.27) should read

$$\begin{aligned} \hat{\omega }_*(\Delta \hat{c}) = - \frac{2\Delta Z_\mathrm {i}}{\pi \Delta j}(\Delta \hat{c})^{3/2} + \mathcal {O}((\Delta \hat{c})^2). \end{aligned}$$

With this sign change and the correct scaling, (3.27) yields

$$\begin{aligned} \Delta \omega&= -\left( \partial _c\,\omega _\mathrm {abs}(\alpha ,c_\mathrm {lin})\right)\Delta c +\frac{2 \Delta Z_\mathrm {i}}{\pi \Delta j} (1+\alpha ^2)^{3/4}(\Delta c)^{3/2}+ \mathcal {O}((\Delta c)^2), \end{aligned}$$

from which we obtain the expansion as stated here by setting \(\Delta j = -1\):

$$\begin{aligned} \omega = \omega _\mathrm {abs}(\alpha , c) - \frac{2 \Delta Z_\mathrm {i}(1+\alpha ^2)^{3/4}}{\pi \Delta j} (c_\mathrm {lin}- c)^{3/2} + \mathcal {O}( ( c_\mathrm {lin}- c)^{2}). \end{aligned}$$

In the same manner, we obtain the expansions for the unscaled distance between the trigger and the invasion front, \(\xi _*\), and for the selected wavenumber \(k_\mathrm {tf}\) as stated in Theorem 1 above.