&
\widehat W_{-}(\eta,k)=\widehat W_{+}(\eta,-k),\quad \widehat Y_{-}(\eta,k)=\widehat
  Y_{+}^\star(-\eta,k).\end{aligned}$$ The duality pairing between ${\cal A}'$ and ${\cal A}$ is determined by the relation $$\begin{aligned}
\label{pairing}
&
\left\langle {\bf F},{\bf
    W}\right\rangle:=\int_{{{\mathbb T}}_{\epsilon}\times{{\mathbb T}}}\widehat {\bf
  F}(\eta,k)\cdot  {\bf
    W}(\eta,k)d\eta dk\\
&
=2\int_{{{\mathbb T}}_{\epsilon}\times{{\mathbb T}}}\left\{\widehat F_{+}(\eta,k)\widehat
  W_{+}^\star(\eta,k)+{\rm Re}\left(\widehat H_{+}(\eta,k)\widehat
  Y_{+}^\star(\eta,k)\right)\right\},\nonumber\end{aligned}$$ wher the scalar product of two matrices is given by $$\label{scalar}
\widehat {\bf
  F}\cdot \widehat  {\bf
    W}=\sum_{\iota=\pm}\left(\widehat {
  F}_\iota \widehat {
  W}^\star_\iota+\widehat {
  H}_\iota \widehat {
  Y}^\star_\iota\right).$$ The norm $\|{\bf
  W}\|_{{\cal A}'}$ is therefore the sum of the norm of its entries.

Thanks to and we conclude that $$\label{030406-19}
\sup_{{\epsilon}\in(0,1]}\sup_{t\ge0}\|{\bf  W}_{{\epsilon}}(t)\|_{{\cal A}'}=:A_*'<+\infty.$$ Therefore $\left({\bf  W}_{{\epsilon}}(\cdot)\right)$ is bounded in $L^\infty([0,+\infty),{\cal A}')$. In consequence, from any sequence ${\epsilon}_n\to0$ we can choose a subsequence, that we still denote by the same symbol, such that $\left({\bf  W}_{{\epsilon}_n}(\cdot)\right)$ is $\vphantom{1}^\star$-weakly convergent in $\left(L^1([0,+\infty),{\cal A})\right)'$.

In what follows we shall also consider the Hilbert spaces ${\cal
  L}_{2,{\epsilon}}$ with the scalar product $\langle\cdot,\cdot\rangle_{{\cal L}_{2,{\epsilon}}}$ given by the formula . The respective Hilbert space norms are $$\|{\bf W}\|_{{\cal L}_{2,{\epsilon}}}:=\left\{2\left(\| W_{+}\|_{{\cal
        L}_{2,{\epsilon}}}^2+\| Y_{+}\|_{{\cal
        L}_{2,{\epsilon}}}^2\right)\right\}^{1/2}. $$ We introduce the following notation, given a function $f:{{\mathbb T}}\to\mathbb C$, we let $$\label{bar-f}
\bar f(k,\eta):=\frac{1}{2}\left[f\left(k+\frac{\eta}{2}\right)+f\left(k-\frac{ \eta}{2}\right)\right]$$ and the difference quotient for the dispersion relation $$\label{d-om}
\delta_{{\epsilon}}{\omega}(k,\eta):=\frac{1}{{\epsilon}}\left[{\omega}\left(k+\frac{{\epsilon}\eta}{2}\right)-{\omega}\left(k-\frac{{\epsilon}\eta}{2}\right)\right].$$ Equipped with this 