Appendix 1: The Extended Approximation
1.1 Derivation of the Extended TWI System
The water wave problem can be written as a first order system in the variables \( \eta \) and \( w \). The solutions of the water wave problem can be represented as infinite Fourier series for the two components. The linear system possesses solutions \( e^{i(jk_{*}x-\omega _{j,\pm }t)} \varphi _{j,\pm } \) with \( \varphi _{j,\pm } \in {\mathbb {C}}^2 \) for \( j \in {\mathbb {Z}} \). In principle there are infinitely many possibilities for the three resonant wave numbers. However, to keep the notation on a reasonable level we choose like in the previous counter example \( k_* ,\, 2 k_* \), and \(- 3 k_* \) as resonant wave numbers. Hence, the wave numbers and the eigenvectors are ordered in such a way that the resonant wave numbers are given by \( \omega _{j}=\omega _{j,+} \) for \( j = 1,2,-3\) and that the associated eigenvectors are given by \( \varphi _{j} =\varphi _{j,+} \) for \( j = 1,2,-3\).
For the derivation of the higher order TWI approximation we make the ansatz
$$\begin{aligned} \left( \begin{array}{c} \eta \\ w \end{array}\right) (x,t)&= \varepsilon {\mathcal {A}}_{1}(\varepsilon t)e^{i(k_*x-\omega _{1}t)}\varphi _1+\varepsilon {\mathcal {A}}_{2}(\varepsilon t)e^{i(2k_*x-\omega _{2}t)}\varphi _2+\varepsilon {\mathcal {A}}_{3}(\varepsilon t)e^{i(3k_*x-\omega _{3}t)}\varphi _3\\&\quad + \varepsilon ^{1+\theta } {\mathcal {A}}_{1,-}(\varepsilon t)e^{i(k_*x-\omega _{1,-}t)}\varphi _{1,-}+ \varepsilon ^{1+\theta } {\mathcal {A}}_{2,-}(\varepsilon t)e^{i(2k_*x-\omega _{2,-}t)}\varphi _{2,-}\\&\quad +\varepsilon ^{1+\theta } {\mathcal {A}}_{3,-}(\varepsilon t)e^{i(3k_*x-\omega _{3,-}t)}\varphi _{3,-}+ \sum \limits _{j=\pm } ( \varepsilon ^{1+\theta }{\mathcal {A}}_{4,j}(\varepsilon t)e^{i(4k_*x-\omega _{4,j}t)} \varphi _{4,j}\\&\quad +\varepsilon ^{1+\theta }{\mathcal {A}}_{5,j}(\varepsilon t)e^{i(5k_*x-\omega _{5,j}t)}\varphi _{4,j}+\varepsilon ^{1+\theta }{\mathcal {A}}_{6,j}(\varepsilon t)e^{i(6k_*x-\omega _{6,j}t)}\varphi _{6,j})\\&\quad + \sum \limits _{n=2}^N \sum \limits _{p=1}^3\sum \limits _{j=\pm }\varepsilon ^{1+n\theta }{\mathcal {A}}_{3n+p,j}(\varepsilon t)e^{i((3n+p)k_*x-\omega _{3n+p,j}t)}\varphi _{3n+p,j}+ c.c. \end{aligned}$$
where \( \theta \in [0,1/2) \) is chosen arbitrarily close to \( 1/2 \) and \( N \in {\mathbb {N}} \) is a later on suitably chosen number. If only estimates for all \( T \in [0,T_0] \) are necessary then \( \theta = 1 \) would be the optimal choice. By equating the coefficients in front of \( e^{i(l k_* x - \omega _{l,j} t)} \varphi _{l,j} \) to zero we find the following system of amplitude equations
$$\begin{aligned} \partial _{T}{\mathcal {A}}_{1}&= i \gamma _1 {\mathcal {A}}_{-2}{\mathcal {A}}_{3}+\sum \limits _{j=\pm }\varepsilon ^{\theta } e^{-i(\omega _{4,j}+\omega _{-3}-\omega _{1})T/\varepsilon }c_*{\mathcal {A}}_{-3}{\mathcal {A}}_{4,j}+\cdots ,\\ \partial _{T}{\mathcal {A}}_{2}&= i \gamma _2 {\mathcal {A}}_{3}{\mathcal {A}}_{-1}+\sum \limits _{j=\pm } e^{-i(\omega _{1}+\omega _{1}-\omega _{2})T/\varepsilon }c_*{\mathcal {A}}_{1}{\mathcal {A}}_{1}\\&\quad +\varepsilon ^{\theta }e^{-i(\omega _{4,j}+\omega _{-2}-\omega _{2})T/\varepsilon }c_*{\mathcal {A}}_{-2}{\mathcal {A}}_{4,j}+\cdots ,\\ \partial _{T}{\mathcal {A}}_{3}&= - i \gamma _3 {\mathcal {A}}_{1}{\mathcal {A}}_{2}+\sum \limits _{j=\pm }\varepsilon ^{\theta }e^{-i(\omega _{4,j}+\omega _{-1}-\omega _{3})T/\varepsilon }c_*{\mathcal {A}}_{-1}{\mathcal {A}}_{4,j}+\cdots ,\\ \partial _{T}{\mathcal {A}}_{4,j}&= \varepsilon ^{-\theta }e^{-i(\omega _{2}+\omega _{2}-\omega _{4,j})T/\varepsilon }c_*{\mathcal {A}}_{2}{\mathcal {A}}_{2}+\varepsilon ^{-\theta }e^{-i(\omega _{1}+\omega _{3}-\omega _{4,j})T/\varepsilon }c_*{\mathcal {A}}_{1}{\mathcal {A}}_{3}+\cdots ,\\&\vdots \\ \partial _{T}{\mathcal {A}}_{7,j}&= \sum \limits _{l=\pm } (\varepsilon ^{-\theta }e^{-i(\omega _{3}+\omega _{4,l}-\omega _{7,j})T/\varepsilon }c_*{\mathcal {A}}_{3}{\mathcal {A}}_{4,l} +\varepsilon ^{-\theta }e^{-i(\omega _{2}+\omega _{5,l}-\omega _{7,})T/\varepsilon }c_*{\mathcal {A}}_{2}{\mathcal {A}}_{5,l}\\&\qquad +\varepsilon ^{-\theta }e^{-i(\omega _{1}+\omega _{6,l}-\omega _{7,j})T/\varepsilon }c_*{\mathcal {A}}_{1}{\mathcal {A}}_{6,l})+\cdots ,\\&\vdots \end{aligned}$$
where the many occurring coefficients are simply denoted with the same symbol \( c_*\).
1.2 Estimates for the Residual
The terms that remain at \( e^{i(lk_* x + \omega _{l,j} t)}\varphi _{l,j} \) with \( |l| > N \) form the so called residual. We choose \( N \) so large that formally \( \mathrm{Res }(\varepsilon \psi ) = {\mathcal {O}}(\varepsilon ^{\beta +\delta }) \). In order to prove the required estimates we have to show that the solutions of the system of amplitude equations stay \( {\mathcal {O}}(1) \) bounded for all \( T \in [0,\varepsilon ^{-\widetilde{\mu }}] \) and not only for \( T \in [0,T_0] \). In order to do so we define suitable energies, namely
$$\begin{aligned} E&= E_{twi}+E_{rest},\\ E_{twi}&= \sum \limits _{k\in I_{3}}\rho _{k}|{\mathcal {A}}_{k}|^{2},\\ E_{rest}&= \sum \limits _{k\notin I_{m}\setminus I_{3}} \sum \limits _{j=\pm } |{\mathcal {A}}_{k,j}|^{2}+ \sum \limits _{k\in I_{3}}|{\mathcal {A}}_{k,-}|^{2}, \end{aligned}$$
where \(I_{\nu }=\{-\nu ,-\nu +1,\ldots ,-1,1,\ldots ,\nu -1,\nu \}\) and \( m = 3(N+1) \). We find
$$\begin{aligned} \partial _{T}E_{twi}&= \sum \limits _{k\in I_{3}}\rho _{k}\left( {\mathcal {A}}_{k}\overline{\partial _{T}{\mathcal {A}}_{k}}+\overline{{\mathcal {A}}_{k}}\partial _{T}{\mathcal {A}}_{k}\right) \\&= \left( \rho _{1}\gamma _1+\rho _{2} \gamma _2 +\rho _{3}\gamma _3 \right) \left( {\mathcal {A}}_{1}{\mathcal {A}}_{2}{\mathcal {A}}_{-3}+\overline{{\mathcal {A}}_{1}{\mathcal {A}}_{2}{\mathcal {A}}_{-3}}\right) \nonumber \\&\quad + e^{-i(\omega _{1}+\omega _{1}-\omega _{2})T/\varepsilon }c_*{\mathcal {A}}_{1}{\mathcal {A}}_{1}{\mathcal {A}}_{-2}+ \cdots \\&\quad +\,\sum \limits _{l=\pm } \varepsilon ^{\theta } \left( e^{-i(\omega _{3}+\omega _{-4,l}- \omega _{-1})T/\varepsilon }c_*{\mathcal {A}}_{1}{\mathcal {A}}_{3}{\mathcal {A}}_{-4,l}\right. \nonumber \\&\left. \quad +e^{-i(\omega _{-3} +\omega _{4,l}-\omega _{1})T/\varepsilon }c_*{\mathcal {A}}_{-1}{\mathcal {A}}_{-3}{\mathcal {A}}_{4,l}\right) +\cdots ,\\ \partial _{T} E_{rest}&= \sum \limits _{k\notin I_{m}\setminus I_{3}} \sum \limits _{l=\pm } \left( {\mathcal {A}}_{k,l}\overline{\partial _{T}{\mathcal {A}}_{k,l}}+\overline{{\mathcal {A}}_{k,l}}\partial _{T}{\mathcal {A}}_{k,l}\right) +\cdots \\&= \sum \limits _{l=\pm } \left( {\mathcal {A}}_{4,l}\overline{\partial _{T}{\mathcal {A}}_{4,l}}+\overline{{\mathcal {A}}_{4,l}}\partial _{T}{\mathcal {A}}_{4,l}\right) +\cdots +\left( {\mathcal {A}}_{7,l}\overline{\partial _{T}{\mathcal {A}}_{7,l}}+\overline{{\mathcal {A}}_{7,l}}\partial _{T}{\mathcal {A}}_{7,l}\right) +\cdots \\&= \sum \limits _{l=\pm } \varepsilon ^{-\theta }\left( e^{-i(\omega _{-2}+\omega _{-2}-\omega _{4,l})T/\varepsilon }c_*{\mathcal {A}}_{-2}{\mathcal {A}}_{-2}{\mathcal {A}}_{-4,l}\right. \nonumber \\&\left. \quad +e^{-i(\omega _{2}+\omega _{2}-\omega _{-4,l})T/\varepsilon }c_*{\mathcal {A}}_{2}{\mathcal {A}}_{2}{\mathcal {A}}_{4,l}\right) +\cdots \end{aligned}$$
We choose \( \rho _j > 0 \) such that \( \rho _{1}\gamma _1+\rho _{2} \gamma _2 +\rho _{3}\gamma _3 =0 \). Integration w.r.t. time and using partial integration then yields
$$\begin{aligned} E_{twi}(T)&= E_{twi}(0) + c_* \int \limits _{0}^{T} \left( e^{-i(\omega _{1}+\omega _{1}-\omega _{2})T/\varepsilon }{\mathcal {A}}_{1}(s){\mathcal {A}}_{1}(s){\mathcal {A}}_{-2}(s) ds + \cdots \right) \\&\quad + \sum \limits _{l=\pm }\left( c_*\varepsilon ^{\theta }\int \limits _{0}^{T} e^{-i(\omega _{3}+\omega _{-4,l}-\omega _{-1})s/\varepsilon }{\mathcal {A}}_{1}(s){\mathcal {A}}_{3}(s){\mathcal {A}}_{-4,l}(s) ds \right. \\&\quad \left. +\, c_*\varepsilon ^{\theta }\int \limits _{0}^{T}e^{-i(\omega _{-3}+\omega _{4,l}-\omega _{1})s/\varepsilon }{\mathcal {A}}_{-1}(s){\mathcal {A}}_{-3}(s){\mathcal {A}}_{4,l}(s) ds +\cdots \right) \\&= E_{twi}(0) -c_*\varepsilon i({\omega _{1}+\omega _{1}}-\omega _{2})^{-1} e^{i(\omega _{1}+\omega _{1}- \omega _{2})s/\varepsilon }{\mathcal {A}}_{1}(s){\mathcal {A}}_{1}(s){\mathcal {A}}_{-2}(s)|_{0}^{T}\\&\quad +c_*\varepsilon i({\omega _{1}+\omega _{1}-\omega _{2}})^{-1}\nonumber \\&\times \int \limits _{0}^{T}\left( e^{-i(\omega _{1}+\omega _{1}-\omega _{2})s/\varepsilon }\partial _{s}({\mathcal {A}}_{1}(s){\mathcal {A}}_{1}(s){\mathcal {A}}_{-2}(s)) ds + \cdots \right) \\&\quad + \sum \limits _{l=\pm }\left( -c_*\varepsilon ^{1+\theta }i({\omega _{3}+\omega _{-4,l}}-\omega _{-1})^{-1}\right. \\&\quad \left. \times e^{i(\omega _{3}+\omega _{-4,l}- \omega _{-1})s/\varepsilon }{\mathcal {A}}_{1}(s){\mathcal {A}}_{3}(s){\mathcal {A}}_{-4,l}(s)|_{0}^{T}+ c_*\varepsilon ^{1+\theta }i\left( {\omega _{3}+\omega _{-4,l}-\omega _{-1}}\right) ^{-1} \right. \\&\quad \left. \times \int \limits _{0}^{T}\left( e^{-i(\omega _{3}+\omega _{-4,l}-\omega _{-1})s/\varepsilon }\partial _{s}({\mathcal {A}}_{1}(s){\mathcal {A}}_{3}(s){\mathcal {A}}_{-4,l}(s)) ds + \cdots \right) \right) \\&\le E_{twi}(0)+C \varepsilon \left( E_{twi}^{1/2}(T)E_{twi}^{1/2}(T)E_{rest}^{1/2}(T)+\cdots \right) \end{aligned}$$
with an \( \varepsilon \)-independent constant \( C \) since for instance \( \partial _s {\mathcal {A}}_{1,2} = {\mathcal {O}}(1) \) and \( \partial _s {\mathcal {A}}_{-4,l} = {\mathcal {O}}(\varepsilon ^{-\theta }) \). Similarly we obtain
$$\begin{aligned} E_{rest}(T)&= E_{rest}(0)+\sum \limits _{l=\pm }\left( \varepsilon ^{-\theta }\int \limits _{0}^{T}e^{-i(\omega _{2}+\omega _{2}-\omega _{4,l})s/\varepsilon }c_*{\mathcal {A}}_{2}(s){\mathcal {A}}_{2}(s){\mathcal {A}}_{-4,l}(s)) ds \right. \\&\left. \quad +\,\varepsilon ^{-\theta }\int \limits _{0}^{T} e^{-i(\omega _{-2}+\omega _{-2}-\omega _{-4,l})s/\varepsilon }c_*{\mathcal {A}}_{-2}(s){\mathcal {A}}_{-2}(s){\mathcal {A}}_{4,l}(s) ds +\cdots \right) +\cdots \\&= E_{rest}(0)+\sum \limits _{l=\pm }\left( -\varepsilon ^{1-\theta }i({\omega _{2}+\omega _{2}-\omega _{4,l}})^{-1}\right. \nonumber \\&\times \left. \quad e^{-i(\omega _{2}+\omega _{2}-\omega _{4,l})s/\varepsilon }c_*{\mathcal {A}}_{2}(s){\mathcal {A}}_{2}(s){\mathcal {A}}_{-4,l}(s)|_{0}^{T}+\varepsilon ^{1-\theta }i({\omega _{2}+\omega _{2}-\omega _{4,l}})^{-1}\right. \\&\quad \left. \times \int \limits _{0}^{T}\left( e^{-i(\omega _{2}+\omega _{2}-\omega _{4,l})s/\varepsilon }\partial _{s}(c_*{\mathcal {A}}_{2}(s){\mathcal {A}}_{2}(s){\mathcal {A}}_{-4,l}(s)\right) ds +\cdots \right) +\cdots \\&\le E_{rest}(0)+C (\varepsilon ^{1-2 \theta }E_{twi}^{1/2}(T)E_{twi}^{1/2}(T)E_{rest}^{1/2}(T)+\cdots ) \end{aligned}$$
with an \( \varepsilon \)-independent constant \( C \) since again \( \partial _s {\mathcal {A}}_{4,l} = {\mathcal {O}}(\varepsilon ^{-\theta }) \), etc. With
$$\begin{aligned} B_{twi}(T)=\sup _{T'\in [0,T]}E_{twi}(T'),\quad B_{rest}(T)=\sup _{T'\in [0,T]}E_{rest}(T') \end{aligned}$$
we obtain
$$\begin{aligned} B_{twi}(T)&\le E_{twi}(0)+C \varepsilon T(B_{twi}(T)+B_{rest}(T))^{3/2},\\ B_{rest}(T)&\le E_{rest}(0)+C \varepsilon ^{1-2\theta }T(B_{twi}(T)+B_{rest}(T))^{3/2}. \end{aligned}$$
Adding the two inequalities yields
$$\begin{aligned} B_{twi}(T)+B_{rest}(T)\le E_{twi}(0)+E_{rest}(0) + 2 C \varepsilon ^{1-2\theta }T (B_{twi}(T)+B_{rest}(T))^{3/2} \end{aligned}$$
and so
$$\begin{aligned} B_{twi}(T)+B_{rest}(T) \le 2(E_{twi}(0)+E_{rest}(0)) \end{aligned}$$
for \(0\le T\le \varepsilon ^{2 \theta -1+ \vartheta }\) for an arbitrary small, but fixed \( \vartheta > 0 \) if we choose \( \varepsilon _0 > 0 \) so small that
$$\begin{aligned} 2^{5/2}C\varepsilon _0^{\vartheta }\left( E_{twi}(0)+E_{rest}(0)\right) ^{1/2} < 1. \end{aligned}$$
Hence the solutions of the system of approximation equations stay \( {\mathcal {O}}(1) \) bounded for all \( T \in [0,\varepsilon ^{2 \theta -1+ \vartheta }] \) and not only for \( T \in [0,T_0] \). As a consequence all terms remaining in the residual can be estimated for all \( T \in [0,\varepsilon ^{2 \theta -1+ \vartheta }] \) by purely counting powers of \( \varepsilon \). Since we have chosen \( N \) so large that all terms up to order \( {\mathcal {O}}(\varepsilon ^{\beta + \delta } ) \) cancel, we are done.
Remark 8.1
Again we oversimplified the presentation in the sense that [12, Theorem 4.1] and its proof are formulated for the Lagrangian formulation of the water wave problem and the residual should have been made small in the Lagrangian formulation as already been said in the proof of Theorem 6.2. The transfer is straightforward and we leave it to the reader.
1.3 Qualitative Behavior of Solutions of the Extended TWI System
The amplitudes \( {\mathcal {A}}_j \) for \( j \in \{1,2,3\} \) satisfy a perturbed TWI system, namely the first three equations of (19),
$$\begin{aligned} \partial _{T}{\mathcal {A}}_{1}&= i \gamma _1 {\mathcal {A}}_{-2}{\mathcal {A}}_{3}+\sum \limits _{j=\pm }\varepsilon ^{\theta } e^{-i(\omega _{4,j}+\omega _{-3}-\omega _{1})T/\varepsilon }c_*{\mathcal {A}}_{-3}{\mathcal {A}}_{4,j}+\cdots ,\\ \partial _{T}{\mathcal {A}}_{2}&= i \gamma _2 {\mathcal {A}}_{3}{\mathcal {A}}_{-1}+\sum \limits _{j=\pm } e^{-i(\omega _{1}+\omega _{1}-\omega _{2})T/\varepsilon }c_*{\mathcal {A}}_{1}{\mathcal {A}}_{1}\\&\quad +\varepsilon ^{\theta }e^{-i(\omega _{4,j}+\omega _{-2}-\omega _{2})T/\varepsilon }c_*{\mathcal {A}}_{-2}{\mathcal {A}}_{4,j}+\cdots ,\\ \partial _{T}{\mathcal {A}}_{3}&= - i \gamma _3 {\mathcal {A}}_{1}{\mathcal {A}}_{2}+\sum \limits _{j=\pm }\varepsilon ^{\theta }e^{-i(\omega _{4,j}+\omega _{-1}-\omega _{3})T/\varepsilon }c_*{\mathcal {A}}_{-1}{\mathcal {A}}_{4,j}+\cdots . \end{aligned}$$
In general the solutions of the classical TWI system (15) and the extended TWI system (19) will differ by an \( {\mathcal {O}}(1) \)-amount for \( T = \widetilde{T} \) and so quantitatively the classical TWI system (15) is not a good approximation of the extended TWI system (19). However, qualitatively for the solutions we are interested in, the classical TWI system (15) and the extended TWI system (19) show the same behavior. For showing this, we integrate the first two equations of (19) w.r.t. \( T \), and after some partial integration we obtain
$$\begin{aligned} {\mathcal {A}}_{1}(T)&= {\mathcal {A}}_{1}(0)+ i \gamma _1\int \limits _0^T {\mathcal {A}}_{-2}(s){\mathcal {A}}_{3}(s) ds\\&\quad +\sum \limits _{j=\pm }\int \limits _0^T \varepsilon ^{1+\theta }(\omega _{4,j}+\omega _{-3}-\omega _{1})^{-1} e^{-i(\omega _{4,j}+\omega _{-3}-\omega _{1})s/\varepsilon }c_* \partial _s ({\mathcal {A}}_{-3}(s){\mathcal {A}}_{4,j}(s)) \\&\quad +\cdots ds,\\&-\sum \limits _{j=\pm }\varepsilon ^{1+\theta }(\omega _{4,j}+\omega _{-3}-\omega _{1})^{-1} e^{-i(\omega _{4,j}+\omega _{-3}-\omega _{1})s/\varepsilon }c_* ({\mathcal {A}}_{-3}(s){\mathcal {A}}_{4,j}(s)) \\&\quad +\cdots |_{s=0}^T, \\ {\mathcal {A}}_{2}(T)&= {\mathcal {A}}_{2}(0)+ i \gamma _2 \int \limits _0^T {\mathcal {A}}_{3}(s){\mathcal {A}}_{-1}(s)ds\\&\quad +\sum \limits _{j=\pm } \int \limits _0^T\varepsilon (\omega _{1}+\omega _{1}-\omega _{2})^{-1} e^{-i(\omega _{1}+\omega _{1}-\omega _{2})s/\varepsilon }c_*\partial _s ({\mathcal {A}}_{1}(s){\mathcal {A}}_{1}(s)) ds\\&\quad - \sum \limits _{j=\pm }\varepsilon (\omega _{1}+\omega _{1}-\omega _{2})^{-1} e^{-i(\omega _{1}+\omega _{1}-\omega _{2})s/\varepsilon }c_* ({\mathcal {A}}_{1}(s){\mathcal {A}}_{1}(s)) |_{s=0}^T, \\&\quad +\sum \limits _{j=\pm } \int \limits _0^T \varepsilon ^{1+\theta }(\omega _{4,j}+\omega _{-2}-\omega _{2})^{-1} e^{-i(\omega _{4,j}+\omega _{-2}-\omega _{2})s/\varepsilon }c_*\partial _s ({\mathcal {A}}_{-2}(s){\mathcal {A}}_{4,j}(s)) \\&\quad +\cdots ds\\&\quad -\sum \limits _{j=\pm } \varepsilon ^{1+\theta }(\omega _{4,j} \!+\! \omega _{-2}-\omega _{2})^{-1} e^{-i(\omega _{4,j}+\omega _{-2}-\omega _{2})s/\varepsilon }c_*{\mathcal {A}}_{-2}(s){\mathcal {A}}_{4,j}(s) \!+\! \cdots |_{s=0}^T, \end{aligned}$$
Since for instance \( \partial _s {\mathcal {A}}_{1}(s) = {\mathcal {O}}(1) \) and \( \partial _s {\mathcal {A}}_{4}(s) = {\mathcal {O}}(\varepsilon ^{-\theta }) \) we have
$$\begin{aligned} {\mathcal {A}}_{1}(T)&= {\mathcal {A}}_{1}(0)+ i \gamma _1\int \limits _0^T {\mathcal {A}}_{-2}(s){\mathcal {A}}_{3}(s) ds + {\mathcal {O}}(\varepsilon ) + {\mathcal {O}}(\varepsilon T) , \\ {\mathcal {A}}_{2}(T)&= {\mathcal {A}}_{2}(0)+ i \gamma _2 \int \limits _0^T {\mathcal {A}}_{3}(s){\mathcal {A}}_{-1}(s)ds+ {\mathcal {O}}(\varepsilon ) + {\mathcal {O}}(\varepsilon T) , \end{aligned}$$
uniformly for all \( T \in [0,\varepsilon ^{2 \theta -1+\mu }] \). As before we introduce variables \( {\mathcal {B}}_j \) by
$$\begin{aligned} {\mathcal {A}}_1 = -i {\mathcal {B}}_1 /\sqrt{|\gamma _2 \gamma _3|}, \qquad {\mathcal {A}}_2 = -i {\mathcal {B}}_2 /\sqrt{|\gamma _1 \gamma _3|}, \quad \mathrm{and } \quad {\mathcal {A}}_3 = -i {\mathcal {B}}_3 /\sqrt{|\gamma _1 \gamma _2|} \end{aligned}$$
which satisfy
$$\begin{aligned} {\mathcal {B}}_{1}(T)&= {\mathcal {B}}_{1}(0)+ \int \limits _0^T {\mathcal {B}}_{-2}(s){\mathcal {B}}_{3}(s) ds + {\mathcal {O}}(\varepsilon ) + {\mathcal {O}}(\varepsilon T) , \end{aligned}$$
(21)
$$\begin{aligned} {\mathcal {B}}_{2}(T)&= {\mathcal {B}}_{2}(0)+ \int \limits _0^T {\mathcal {B}}_{3}(s){\mathcal {B}}_{-1}(s)ds+ {\mathcal {O}}(\varepsilon ) + {\mathcal {O}}(\varepsilon T) . \end{aligned}$$
(22)
To construct our counter example we start the extended TWI system (19) with
$$\begin{aligned} {\mathcal {B}}_1|_{T=0} = \varepsilon ^{1/2},\quad {\mathcal {B}}_2|_{T=0} = \varepsilon ^{1/2}, \quad \mathrm{and } \qquad {\mathcal {B}}_3|_{T=0} = 1 . \end{aligned}$$
All other initial conditions we assume to vanish.
Remark 8.2
Before we go on we would like to remark that the estimates in the last section easily imply
$$\begin{aligned} | E_{twi}(T) - E_{twi}(0) | \le 2^{3/2} C \varepsilon T \left( E_{twi}(0) + E_{rest}(0)\right) ^{3/2}. \end{aligned}$$
(23)
As a consequence if we assume that \( {\mathcal {B}}_{1}(T) \) and \( {\mathcal {B}}_{2}(T) \) remain of order \( {\mathcal {O}}(\varepsilon ^{1/2}) \) for all \( T \in [0,\varepsilon ^{2 \theta - 1+ \mu }] \) then
$$\begin{aligned} {\mathcal {B}}_{3}(T) = 1 +{\mathcal {O}}(\varepsilon ) + {\mathcal {O}}(\varepsilon T) \end{aligned}$$
for all \( T \in [0,\varepsilon ^{2 \theta - 1+ \mu }] \). This immediately leads to a contradiction as can be seen as follows. The quantity \( r(T) = {\mathcal {B}}_{1}(T) +{\mathcal {B}}_{2}(T) + {\mathcal {B}}_{-1}(T) +{\mathcal {B}}_{-2}(T) \) then satisfies
$$\begin{aligned} r(T)&= 4 \varepsilon ^{1/2} + \int \limits _0^T r(s) ds + \int \limits _0^T ({\mathcal {O}}(\varepsilon ) + {\mathcal {O}}(\varepsilon T) ) ds + {\mathcal {O}}(\varepsilon ) + {\mathcal {O}}(\varepsilon T) \\&= 4 \varepsilon ^{1/2} + \int \limits _0^T r(s) ds + {\mathcal {O}}(\varepsilon ) + {\mathcal {O}}(\varepsilon T)+ {\mathcal {O}}(\varepsilon T^2) . \end{aligned}$$
The first two terms lead to the desired growth. The last three terms in the worst case will stop the solutions to grow. We find
$$\begin{aligned} r(T)&= 4 \varepsilon ^{1/2} e^{T} + \int \limits _0^T e^{T-s} ( {\mathcal {O}}(\varepsilon ) + {\mathcal {O}}(\varepsilon s)+{\mathcal {O}}(\varepsilon s^2) ) ds \end{aligned}$$
We recall that \( \varepsilon ^{1/2} e^{\widetilde{T}} = 1/2 \), that \( \widetilde{T} = {\mathcal {O}}(|\ln (\varepsilon ) |) \), and find
$$\begin{aligned} |r(\widetilde{T}) |&\ge 4 \varepsilon ^{1/2} e^{\widetilde{T}} - C \varepsilon e^{\widetilde{T}} \left( 1+\widetilde{T}+\widetilde{T}^2\right) \ge 2 - C \varepsilon ^{1/2} \left( 1+\widetilde{T}+\widetilde{T}^2 \right) \ge 1. \end{aligned}$$
Then either \( {\mathcal {B}}_{1}(\widetilde{T}) \) or \( {\mathcal {B}}_{2}(\widetilde{T}) \) are of order \( {\mathcal {O}}(1) \) contradicting our assumption.
The previous remark gives some first insight into the problem and allows to conclude by a contradiction argument that \( {\mathcal {B}}_{1,2}(\widetilde{T}) \gg {\mathcal {O}}(\varepsilon ^{1/2}) \), but it does not allow to conclude \( {\mathcal {B}}_{1,2}(\widetilde{T}) = {\mathcal {O}}(1) \). In order to prove this statement we proceed as follows.
We split \( {\mathcal {B}}_j (T) = \mathfrak {R}{\mathcal {B}}_j (T) +i \mathfrak {I}{\mathcal {B}}_j (T) \) in real and imaginary part. The quantity \( r(T) \) from the previous remark then satisfies
$$\begin{aligned} r(T)&= 4 \varepsilon ^{1/2} + \int \limits _0^T (\mathfrak {R}{\mathcal {B}}_{3}(s)) r(s) ds + q(T) + {\mathcal {O}}(\varepsilon ) + {\mathcal {O}}(\varepsilon T) \end{aligned}$$
with
$$\begin{aligned} q(T)&= i \int \limits _0^T (\mathfrak {I}{\mathcal {B}}_{3}(s))({\mathcal {B}}_{-2}(s)+ {\mathcal {B}}_{-1}(s)- {\mathcal {B}}_{1}(s)- {\mathcal {B}}_{2}(s)) ds . \end{aligned}$$
The factor \( \mathfrak {R}{\mathcal {B}}_{3}(s) \) will be close to \( 1 \) and will lead to exponential growth rates. The last three terms in the worst case will stop the solutions to grow. In order to bound \( q(t) \) we have to bound \( \mathfrak {I}{\mathcal {B}}_{3}(s)\) which together with all other \( \mathfrak {I}{\mathcal {B}}_{j}(s)\) is initially zero.
Gronwall’s inequality then applied to (21) and (22) and to the imaginary part of (21) and (22) yields
$$\begin{aligned} | {\mathcal {B}}_1(T) | + | {\mathcal {B}}_2(T) |&\le C (\varepsilon ^{1/2}+ \varepsilon T+ \varepsilon T^2 ) e^T \le C \varepsilon ^{1/2} e^T, \\ |\mathfrak {I}{\mathcal {B}}_1(T) | + | \mathfrak {I}{\mathcal {B}}_2(T) |&\le C ( \varepsilon T + \varepsilon T^2 )e^T, \end{aligned}$$
where we used \( | {\mathcal {B}}_{3}(T) | \approx 1 \le 2 \). Here and in the following all constants which can be chosen independently of the small bifurcation parameter \( 0 < \varepsilon \ll 1 \) are denoted with the same symbol \( C \). Similarly as in Remark 8.2, we find
$$\begin{aligned} {\mathcal {B}}_{3}(T) = {\mathcal {B}}_{3}(0)- \int \limits _0^T {\mathcal {B}}_{1}(s){\mathcal {B}}_{2}(s) ds + {\mathcal {O}}(\varepsilon ) + {\mathcal {O}}(\varepsilon T) . \end{aligned}$$
As a consequence the imaginary part satisfies
$$\begin{aligned} \mathfrak {I}{\mathcal {B}}_{3}(T) =- \int \limits _0^T (\mathfrak {I}{\mathcal {B}}_{1}(s))(\mathfrak {R}{\mathcal {B}}_{2}(s)) ds - \int \limits _0^T (\mathfrak {R}{\mathcal {B}}_{1}(s))(\mathfrak {I}{\mathcal {B}}_{2}(s)) ds + {\mathcal {O}}(\varepsilon ) + {\mathcal {O}}(\varepsilon T) . \end{aligned}$$
Therefore, it obeys the estimates
$$\begin{aligned} | \mathfrak {I}{\mathcal {B}}_{3}(T) | \le C \left( \varepsilon ^{3/2} (1+T^2) e^{2 T} +\varepsilon (1+T^2)\right) . \end{aligned}$$
Assume now that \( | {\mathcal {B}}_{3}(T) - 1| \le 1/10 \) for all \( T \in [0,2 \tilde{T}] \). If this is not satisfied due to (23) we would be done. Under this assumption we obtain
$$\begin{aligned} |q(T)| \le C T \times \varepsilon ^{1/2} e^T \times \varepsilon ^{3/2} (1+T^2) e^{2 T} . \end{aligned}$$
Hence finally
$$\begin{aligned} r(T)&= 4 \varepsilon ^{1/2} + \int \limits _0^T (\mathfrak {R}{\mathcal {B}}_{3}(s)) r(s) ds + {\mathcal {O}}\left( \varepsilon ^{2}(T+T^3) e^{3 T} \right) + {\mathcal {O}}(\varepsilon ) + {\mathcal {O}}(\varepsilon T) . \end{aligned}$$
The first two terms lead to the desired growth. The last three terms in the worst case will stop the solutions to grow. We find
$$\begin{aligned} r(T)&= 4 \varepsilon ^{1/2} e^{T} + \int \limits _0^T e^{T-s} {\mathcal {O}}\left( \varepsilon ^{2}(s+s^3) e^{3 s} \right) + {\mathcal {O}}(\varepsilon ) + {\mathcal {O}}(\varepsilon s) ) ds \end{aligned}$$
and so since \( \mathfrak {R}{\mathcal {B}}_{3}(T) > 0.9 \) finally
$$\begin{aligned} |r(10\widetilde{T}/9) |&\ge 4 \varepsilon ^{1/2} e^{\widetilde{ T}} - 2 \varepsilon ^2 e^{30 \widetilde{T}/9} \left( 1+\widetilde{T}^2+\widetilde{T}^4\right) - 2 \varepsilon e^{10 \widetilde{T}/9} \left( 1+\widetilde{T}+\widetilde{T}^2\right) \\&\ge 2 -2 \varepsilon ^{3/9} \left( 1+\widetilde{T}^2+\widetilde{T}^4 \right) - 2 \varepsilon ^{4/9} \left( 1+\widetilde{T}+\widetilde{T}^2 \right) \ge 1 \end{aligned}$$
for \( \varepsilon > 0 \) sufficiently small. This contradicts the assumption \( | {\mathcal {B}}_{3}(T) - 1| \le 1/10 \) and so there exists a \( \widehat{T} \in [0,2 \tilde{T}] \) where either \( {\mathcal {B}}_{1}(\widehat{T}) \) or \( {\mathcal {B}}_{2}(\widehat{T}) \) are of order \( {\mathcal {O}}(1) \).
Appendix 2: The TWI Coefficients
1.1 The Case of Infinite Depth
The water wave problem in infinite depth differs from the one in finite depth \( h \) only by replacing the symbol of the linearized Dirichlet–Neumann operator \( \mathrm{tanh}(h k) \) by \( \mathrm{sign}(k) \), cf. [12]. This simplifies the equations of water wave problem at various points and so we expect that it is rather straightforward to transfer [12, Theorem 1.1] respectively Theorem 6.2 from finite to infinite depth.
The linear dispersion relation of the water wave problems with infinite depth is given by
$$\begin{aligned} \omega ^2 = (k+\sigma k^3)\mathrm{sign}(k) \end{aligned}$$
(24)
and so the function \( f \) in Sect. 4 has to be replaced by
$$\begin{aligned} f(k_*,\sigma )&= \sqrt{\left( n_1 k_* + \sigma (n_1 k_*)^3\right) \mathrm{sign}(n_1 k_*)} + \sqrt{\left( n_2 k_* + \sigma (n_2 k_*)^3\right) \mathrm{sign}(n_2 k_*)} \\&\quad -\sqrt{\left( n_3 k_* + \sigma (n_3 k_*)^3 \right) \mathrm{sign}(n_3 k_*)}. \end{aligned}$$
Again as an example we have for \( n_1 = 1 ,\, n_2 = 2 \), and \( n_3 = -3 \) that
$$\begin{aligned} f(10,0) \approx \sqrt{10}+ \sqrt{20} - \sqrt{30} > 0 \end{aligned}$$
and
$$\begin{aligned} f(10,1) \approx \sqrt{1000}+ \sqrt{8000} - \sqrt{27000} < 0 \end{aligned}$$
such that a zero exists due to the intermediate value theorem. In Fig. 4 we have plotted the zeroes of \( f(k_0,\sigma ) \) for two different cases of \( n_1 ,\, n_2 \), and \( n_3 \).
These curves are typical and in infinite depth such zeroes exist for all \( \sigma > 0 \). In case of finite depth qualitatively the same behavior occurs. Since in the definition of \( f \) we have either \( \mathrm{sign}(\cdot ) \) functions or \( \tanh (h \cdot ) \) functions, for \( h k_* \) large the functions in finite and infinite depth almost show no difference. For \( h k_* \) small the difference is visible.
1.2 The TWI Coefficients in Case of Infinite Depth
Throughout the next two sections assume \(k_j,\omega _j\in {\mathbb {R}}\) for \(j=1,2,3\) so that the linear dispersion relation (24) and
$$\begin{aligned} k_1+k_2+k_3=0,\quad \omega _1+\omega _2+\omega _3=0 \end{aligned}$$
(25)
are satisfied with \( 0 < k_1 \le k_2 < |k_3 | \) and \( \omega _1 > 0 ,\, \omega _2 > 0 ,\, \omega _3 < 0 \). We start with the water wave problem in case of infinite depth where in [2] the following explicit formulas for the coefficients \( \gamma _j \) appearing in the TWI system can be found. We have
$$\begin{aligned} \gamma _1&= \dfrac{1}{2}\left\{ |k_1|\omega _1^{-1} (\omega _2^{2}+\omega _3^{2}) + \left( \omega _3 |k_3| + \omega _2 |k_2| + |k_1| \omega _2 \omega _3 \omega _1^{-1}\right) \right. \\&\quad \left. + k_2 k_3 \left[ \omega _2 |k_2|^{-1} +\omega _3 |k_3|^{-1} -\omega _2 \omega _3 |k_1| (|k_2| |k_3| \omega _1)^{-1}\right] \right\} , \\ \gamma _2&= \dfrac{1}{2}\left\{ |k_2| \omega _2^{-1} (\omega _1^{2} + \omega _3^{2}) +\left( \omega _1 |k_1| + \omega _3 |k_3| + |k_2| \omega _1 \omega _3 \omega _2^{-1}\right) \right. \\&\quad \left. + k_1 k_3 \left[ \omega _1 |k_1|^{-1} + \omega _3 |k_3|^{-1} -\omega _1 \omega _3 |k_2| (|k_1| |k_3| \omega _2)^{-1}\right] \right\} , \\ \gamma _3&= \dfrac{1}{2}\left\{ |k_3| \omega _3^{-1} (\omega _1^{2} + \omega _2^{2}) + \left( \omega _1 |k_1| +\omega _2 |k_2| + |k_3| \omega _1 \omega _2 \omega _3^{-1}\right) \right. \\&\quad \left. + k_1 k_2\left[ \omega _2 |k_2|^{-1} + \omega _1 |k_1|^{-1} -\omega _1 \omega _2 |k_3| (|k_1| |k_2| \omega _3)^{-1}\right] \right\} . \end{aligned}$$
As pointed out in Sect. 5 these coefficients determine the stability of the \( A_3 ,\, A_2 \), and \( A_1 \) subspace in the TWI system, respectively. The formulas from [2] for the coefficients \( \gamma _j \) can be simplified strongly in the two-dimensional case. To our knowledge these simplifications are not documented in the literature so far, and they will allow us to analyse the signs of the \( \gamma _j \)s immediately.
Rather than working with the \(\gamma _j\) we consider \( \delta _j=2|k_1||k_2||k_3|\frac{\omega _j}{|k_j|}\gamma _j\) which allows us to eliminate the denominators in the above expressions. We find
$$\begin{aligned} \delta _1&=|k_1||k_2||k_3|(\omega _2^2+\omega _3^2)+|k_2||k_3|^2\omega _1\omega _3+|k_2|^2|k_3|\omega _1\omega _2+|k_1||k_2||k_3|\omega _2\omega _3\\&\quad +k_2\cdot k_3|k_3|\omega _1\omega _2+k_2\cdot k_3|k_2|\omega _1\omega _3-|k_1|k_2\cdot k_3\omega _2\omega _3,\\ \delta _2&=|k_1||k_2||k_3|(\omega _1^2+\omega _3^2)+|k_1||k_3|^2\omega _2\omega _3+|k_1|^2|k_3|\omega _1\omega _2+|k_1||k_2||k_3|\omega _1\omega _3\\&\quad +k_1\cdot k_3|k_3|\omega _1\omega _2+k_1\cdot k_3|k_1|\omega _2\omega _3-|k_2|k_1\cdot k_3\omega _1\omega _3,\\ \delta _3&=|k_1||k_2||k_3|(\omega _1^2+\omega _2^2)+|k_1||k_2|^2\omega _2\omega _3+|k_1|^2|k_2|\omega _1\omega _3+|k_1||k_2||k_3|\omega _1\omega _2\\&\quad +k_1\cdot k_2|k_1|\omega _2\omega _3+k_1\cdot k_2|k_2|\omega _1\omega _3-|k_3|k_1\cdot k_2\omega _1\omega _2. \end{aligned}$$
Recall that \(k_1,k_2>0>k_3\) and \(\omega _1,\omega _2>0>\omega _3\). Set \(K_j=|k_j|\) and \(\Omega _j=|\omega _j|\) for \(j=1,2,3\). The resonance condition becomes
$$\begin{aligned} K_1+K_2 =K_3 \quad \mathrm{and } \quad \Omega _1+\Omega _2 =\Omega _3. \end{aligned}$$
Then we can write
$$\begin{aligned} \delta _1&=K_1K_2K_3\left( \Omega _2^2+\Omega _3^2\right) -K_2K_3^2\Omega _1\Omega _3+K_2^2K_3\Omega _1\Omega _2-K_1K_2K_3\Omega _2\Omega _3\\&\quad -K_2K_3^2\Omega _1\Omega _2+K_2^2K_3\Omega _1\Omega _3-K_1K_2K_3\Omega _2\Omega _3\\&=K_1K_2K_3\left( \Omega _2^2-2\Omega _2\Omega _3+\Omega _3^2\right) +K_2K_3\Omega _1\left( K_2\Omega _2-K_3\Omega _2+K_2\Omega _3-K_3\Omega _3\right) \\&=K_1K_2K_3\left( \Omega _2-\Omega _3\right) ^2+(K_2-K_3)K_2K_3\Omega _1\left( \Omega _2+\Omega _3\right) \\&=K_1K_2K_3\Omega _1\left( \Omega _1-\Omega _3-\Omega _2\right) \\&=-2K_1K_2K_3\Omega _1\Omega _2. \end{aligned}$$
So that finally we arrive at
$$\begin{aligned} \gamma _1=-2|k_1||k_2||k_3||\omega _1||\omega _2|\cdot \frac{1}{2|k_2||k_3|\omega _1}=-|k_1|\mathrm{sign}(\omega _1)|\omega _2|. \end{aligned}$$
Since for \( \delta _2\) only the roles of \(k_1\) and \(k_2\) as well as those of \(\omega _1\) and \(\omega _2\) have to be exchanged and \(\mathrm{sign}(k_1)=\mathrm{sign}(k_2)\) as well as \(\mathrm{sign}(\omega _1)=\mathrm{sign}(\omega _2)\), we have after the same computation that
$$\begin{aligned} \gamma _2=-|k_2|\mathrm{sign}(\omega _2)|\omega _1|. \end{aligned}$$
The computations for \(\delta _3\) are different to the above since the distribution of the signs is different. We have
$$\begin{aligned} \delta _3&=K_1K_2K_3\left( \Omega _1^2+\Omega _2^2\right) -K_1^2K_2\Omega _1\Omega _3-K_1K_2^2\Omega _2\Omega _3+K_1K_2K_3\Omega _1\Omega _2\\&\quad -K_1^2K_2\Omega _2\Omega _3-K_1K_2^2\Omega _1\Omega _3-K_1K_2K_3\Omega _1\Omega _2\\&=K_1K_2K_3\left( \Omega _1^2+\Omega _2^2\right) -K_1K_2\Omega _3\left( K_1\Omega _1+K_2\Omega _2+K_1\Omega _2+K_2 \Omega _1\right) \\&=K_1K_2K_3\left( \Omega _1^2+\Omega _2^2\right) -K_1K_2(K_1+K_2)\left( \Omega _1+\Omega _2\right) \Omega _3\\&=K_1K_2K_3\left( \Omega _1^2+\Omega _2^2-\Omega _3^2\right) \\&=-2K_1K_2K_3\Omega _1\Omega _2 \end{aligned}$$
and so \( \delta _1=\delta _2 = \delta _3 \). This leads to
$$\begin{aligned} \gamma _3=-\frac{|k_3||\omega _1||\omega _2|}{|\omega _3|}\mathrm{sign}(\omega _3). \end{aligned}$$
So, in addition to these vastly simplified expressions we get the result that \(\mathrm{sign}(\gamma _j)=-\hbox {sign}(\omega _j)\). This guarantees that the \(\gamma _j\) cannot all have the same sign when the resonance condition \(\omega _1+\omega _2+\omega _3=0\) is satisfied. Moreover, the \( A_3 \) subspace associated to \( k_3 \) is unstable.
1.3 The TWI Coefficients in Case of Finite Depth
Since the water wave problem in infinite depth approximates the water wave problem with large, but finite depth \( h \), the coefficients \( \gamma _j \) that occur in finite depth \( h \gg 1\) and infinite depth are close together if \( h k_* \) is large. Similar to [2] the coefficients \( \gamma _j \) appearing in the TWI system can be computed in case of finite depth \( h \) in a straightforward manner, too. Make the ansatz
$$\begin{aligned} \phi (x,y,t)&= \sum \limits _{j=1}^3 \omega _j|k_j|^{-1}\left( i P_j(t)e^{\xi _j}-i\overline{P_j(t)}e^{-\xi _j}\right) \psi _j(y),\\ \eta (x,t)&= \sum \limits _{j=1}^3 \left( P_j(t)e^{\xi _j}+\overline{P_j(t)}e^{-\xi _j}\right) , \end{aligned}$$
where \(\xi _j=i (k_j x+\omega _jt)\) and
$$\begin{aligned} \psi _j(y)=\frac{\cosh (|k_j|(z+h))}{\sinh (|k_j|h)}. \end{aligned}$$
After some lengthy, but straightforward calculation, we obtain
$$\begin{aligned} \gamma _j=\frac{|k_j|}{\omega _j}\cdot \frac{1}{2|k_1||k_2||k_3|}\delta ^h_j, \end{aligned}$$
with
$$\begin{aligned} \delta _1^h&=-|k_1|\omega _2\omega _3(|k_2||k_3|+\coth (|k_2|h)\coth (|k_3|h)k_2\cdot k_3)\\&\quad -|k_2|\omega _1\omega _3(|k_1||k_3|+\coth (|k_3|h)k_1\cdot k_3)\\&\quad -|k_3|\omega _1\omega _2(|k_1||k_2|+\coth (|k_2|h)k_1\cdot k_2),\\ \delta _2^h&=-|k_1|\omega _2\omega _3(|k_2||k_3|+\coth (|k_3|h)k_2\cdot k_3)\\&\quad -|k_2|\omega _1\omega _3(|k_1||k_3|+\coth (|k_1|h)\coth (|k_3|h)k_1\cdot k_3)\\&\quad -|k_3|\omega _1\omega _2(|k_1||k_2|+\coth (|k_1|h)k_1\cdot k_2),\\ \delta _3^h&=-|k_1|\omega _2\omega _3(|k_2||k_3|+\coth (|k_2|h)k_2\cdot k_3)\\&\quad -|k_2|\omega _1\omega _3(|k_1||k_3|+\coth (|k_1|h)k_1\cdot k_3)\\&\quad -|k_3|\omega _1\omega _2(|k_1||k_2|+\coth (|k_1|h)\coth (|k_2|h)k_1\cdot k_2). \end{aligned}$$
As pointed out in Sect. 5 these coefficients determine the stability of the \( A_3 ,\, A_2 \), and \( A_1 \) subspace in the TWI system, respectively. In the limit \(h\rightarrow \infty \) we have \(\delta _1^\infty =\delta _2^\infty =\delta _3^\infty =\delta \) with
$$\begin{aligned} \delta&=-|k_1|\omega _2\omega _3\left( |k_2||k_3|+k_2\cdot k_3\right) \\&\quad -|k_2|\omega _1\omega _3\left( |k_1||k_3|+k_1\cdot k_3\right) \\&\quad -|k_3|\omega _1\omega _2\left( |k_1||k_2|+k_1\cdot k_2\right) \end{aligned}$$
In the scalar case we have \(k_1\cdot k_2=|k_1||k_2|>0\). Furthermore, it holds
$$\begin{aligned} \omega _1\omega _2>0,\quad \omega _1\omega _3<0,\quad \omega _2\omega _3<0, \quad k_1\cdot k_3=-|k_1||k_3|,\quad k_2\cdot k_3=-|k_2||k_3|, \end{aligned}$$
so that
$$\begin{aligned} \delta =-2|k_1||k_2||k_3|\omega _1\omega _2<0. \end{aligned}$$
Since \(\coth (x)>1\) for all \(x>0\) and since \(k_1\cdot k_2>0\), then we have for \(j=1,2,3\) and all \(h>0\) that \(\delta _j^h<0\). As a consequence, the \(\gamma _j\) cannot all have the same sign in the scalar case. Moreover, the \( A_3 \) subspace associated to \( k_3 \) is unstable.
