\mathrm{i}\,\gamma_{12}\mu\epsilon^{2}+\mathrm{i}\,r_{9}(\mu\epsilon^{3},\mu^{2}\epsilon)&\tanh(\mathtt{h}\mu)+\gamma_{22}\mu\epsilon^{2}+r_{10}(\mu\epsilon^{3},\mu^{2}\epsilon)\end{pmatrix}$
(4.2b) $\displaystyle
F^{(1)}:=\begin{pmatrix}\mathtt{f}_{11}\mu\epsilon+\phi_{11}\mu\epsilon^{3}+r_{3}(\mu\epsilon^{4},\mu^{2}\epsilon^{2},\mu^{3}\epsilon)&\mathrm{i}\,\mu\epsilon{\mathtt{c}}_{\mathtt{h}}^{-\frac{1}{2}}+\mathrm{i}\,\phi_{12}\mu\epsilon^{3}+\mathrm{i}\,r_{4}({\mu\epsilon^{4}},\mu^{2}\epsilon^{2},\mu^{3}\epsilon)\\\
\mathrm{i}\,\phi_{21}\mu\epsilon+\mathrm{i}\,r_{6}(\mu\epsilon^{3},\mu^{2}\epsilon)&\phi_{22}\mu\epsilon+r_{7}(\mu\epsilon^{3},\mu^{2}\epsilon)\end{pmatrix}$
(4.2c)

where the coefficients appearing in the entries are the same of (3.3).

Note that the matrix $\mathtt{L}_{\mu,\epsilon}^{(1)}$, initially defined only
for $\mu\neq 0$, extends analytically to the zero matrix at $\mu=0$. For
$\mu\neq 0$ the spectrum of $\mathtt{L}_{\mu,\epsilon}^{(1)}$ coincides with
the spectrum of $\mathtt{L}_{\mu,\epsilon}$.

#### Non-perturbative step of block-decoupling.

The following lemma computes the first order Taylor expansions (4.4) of the
matrix entries in (4.3) and then the expansion (4.8) at a higher degree of
accuracy with respect to [10, Lemma 5.4].

######  Lemma 4.2.

(Step of block-decoupling) There exists a $2\times 2$ reversibility-preserving
matrix $X$, analytic in $(\mu,\epsilon)$, of the form

$X:=\begin{pmatrix}x_{11}&\mathrm{i}\,x_{12}\\\
\mathrm{i}\,x_{21}&x_{22}\end{pmatrix}\,,\qquad x_{ij}\in\mathbb{R}\,,\
i,j=1,2\,,$ (4.3)

with

$\displaystyle x_{11}$
$\displaystyle=x_{11}^{(1)}\epsilon+r(\epsilon^{3},\mu\epsilon)\,,\qquad
x_{12}=x_{12}^{(1)}\epsilon+r(\epsilon^{3},\mu\epsilon)$ (4.4a) $\displaystyle
x_{21}$
$\displaystyle=x_{21}^{(1)}\epsilon+x_{21}^{(3)}\epsilon^{3}+r(\epsilon^{4},\mu\epsilon^{2},\mu^{2}\epsilon)\,,\qquad
x_{22}=x_{22}^{(1)}\epsilon+x_{22}^{(3)}\epsilon^{3}+r(\epsilon^{4},\mu\epsilon^{2},\mu^{2}\epsilon)\,,$
where
$x_{21}^{(1)}:=-\tfrac{1}{2}\mathtt{D}_{\mathtt{h}}^{-1}\big{(}\mathtt{e}_{12}\mathtt{f}_{11}+2{\mathtt{c}}_{\mathtt{h}}^{-\frac{1}{2}}\big{)}\,,\quad
x_{22}^{(1)}:=\tfrac{1}{2}\mathtt{D}_{\mathtt{h}}^{-1}\big{(}{\mathtt{c}}_{\mathtt{h}}^{-\frac{1}{2}}\mathtt{e}_{12}+2\mathtt{h}\mathtt{f}_{11}\big{)}\,,$
(4.4b) and $\displaystyle x_{11}^{(1)}:$
$\displaystyle=\mathtt{D}_{\mathtt{h}}^{-1}\big{(}\tfrac{1}{16}\mathtt{e}_{12}\mathtt{e}_{22}x_{21}^{(1)}-\tfrac{1}{2}\mathtt{e}_{12}\phi_{21}+\phi_{22}-\tfrac{1}{8}\mathtt{e}_{22}x_{22}^{(1)}\big{)}\,,$
(4.4c) $\displaystyle x_{12}^{(1)}:$
$\displaystyle=\mathtt{D}_{\mathtt{h}}^{-1}\big{(}\tfrac{1}{8}\mathtt{h}\mathtt{e}_{22}x_{21}^{(1)}-\mathtt{h}\phi_{21}+\tfrac{1}{2}\mathtt{e}_{12}\phi_{22}-\tfrac{1}{16}\mathtt{e}_{12}\mathtt{e}_{22}x_{22}^{(1)}\big{)}\,,$
$\displaystyle x_{21}^{(3)}:$
$\displaystyle=\mathtt{D}_{\mathtt{h}}^{-1}\big{(}-\tfrac{1}{2}\mathtt{e}_{11}\mathtt{e}_{12}x_{11}^{(1)}+\tfrac{1}{2}(\gamma_{12}+\eta_{12})\mathtt{e}_{12}x_{21}^{(1)}+\tfrac{1}{2}\mathtt{e}_{12}\gamma_{11}x_{22}^{(1)}$
$\displaystyle\qquad-\tfrac{1}{2}\phi_{11}\mathtt{e}_{12}-\mathtt{e}_{11}x_{12}^{(1)}-\gamma_{22}x_{21}^{(1)}-(\gamma_{12}+\eta_{12})x_{22}^{(1)}-\phi_{12}\big{)}\,,$
$\displaystyle x_{22}^{(3)}:$
$\displaystyle={\mathtt{D}_{\mathtt{h}}^{-1}\big{(}\mathtt{h}\mathtt{e}_{11}x_{11}^{(1)}-\mathtt{h}(\gamma_{12}+\eta_{12})x_{21}^{(1)}-\mathtt{h}\gamma_{11}x_{22}^{(1)}+\mathtt{h}\phi_{11}}$
$\displaystyle\qquad+\tfrac{1}{2}\mathtt{e}_{11}\mathtt{e}_{12}x_{12}^{(1)}+\tfrac{1}{2}\mathtt{e}_{12}\gamma_{22}x_{21}^{(1)}+\tfrac{1}{2}\mathtt{e}_{12}(\gamma_{12}+\eta_{12})x_{22}^{(1)}+{\tfrac{1}{2}\mathtt{e}_{12}}\phi_{12}\big{)}\,,$

with $\mathtt{e}_{12}$, $\mathtt{e}_{22}$, $\mathtt{e}_{11}$,
$\phi_{21},\phi_{22},\gamma_{12},\eta_{12},\gamma_{11},\phi_{11},\gamma_{22},\phi_{12},\mathtt{f}_{11}$
computed in (3.4) and (cfr. [10, (5.7)])

$\mathtt{D}_{\mathtt{h}}:=\mathtt{h}-\frac{1}{4}\mathtt{e}_{12}^{2}>0\,,\quad\forall\mathtt{h}>0\,,$
(4.5)

such that the following holds true. By conjugating the Hamiltonian and
reversible matrix $\mathtt{L}_{\mu,\epsilon}^{(1)}$, defined in (4.1), with
the symplectic and reversibility-preserving $4\times 4$ matrix

$\exp\big{(}S^{(1)}\big{)}\,,\quad\text{ where }\qquad
S^{(1)}:=\mathtt{J}_{4}\begin{pmatrix}0&\Sigma\\\
\Sigma^{*}&0\end{pmatrix}\,,\qquad\Sigma:=\mathtt{J}_{2}X\,,$ (4.6)

we get the Hamiltonian and reversible matrix

$\mathtt{L}_{\mu,\epsilon}^{(2)}:=\exp\big{(}S^{(1)}\big{)}\mathtt{L}_{\mu,\epsilon}^{(1)}\exp\big{(}-S^{(1)}\big{)}=\mathtt{J}_{4}\mathtt{B}_{\mu,\epsilon}^{(2)}=\begin{pmatrix}\mathtt{J}_{2}E^{(2)}&\mathtt{J}_{2}F^{(2)}\\\
\mathtt{J}_{2}[F^{(2)}]^{*}&\mathtt{J}_{2}G^{(2)}\end{pmatrix}\,,$ (4.7)

where the reversibility-preserving $2\times 2$ self-adjoint matrix $E^{(2)}$
has the form

$\displaystyle
E^{(2)}={\begin{pmatrix}\mathtt{e}_{\scriptscriptstyle{\textsc{WB}}}\mu\epsilon^{2}+\eta_{\scriptscriptstyle{\textsc{WB}}}\mu\epsilon^{4}+r_{1}^{\prime}(\mu\epsilon^{5},\mu^{2}\epsilon^{3})-\mathtt{e}_{22}\frac{\mu^{3}}{8}(1+r_{1}^{\prime\prime}(\epsilon,\mu))&\mathrm{i}\,\big{(}\frac{1}{2}\mathtt{e}_{12}\mu+r_{2}(\mu\epsilon^{2},\mu^{2}\epsilon,\mu^{3})\big{)}\\\
-\mathrm{i}\,\big{(}\frac{1}{2}\mathtt{e}_{12}\mu+r_{2}(\mu\epsilon^{2},\mu^{2}\epsilon,\mu^{3})\big{)}&-\mathtt{e}_{22}\frac{\mu}{8}(1+r_{5}(\epsilon,\mu))\end{pmatrix}}\,,$
(4.8)

where $\mathtt{e}_{\scriptscriptstyle{\textsc{WB}}}$ is the Whitham-Benjamin
function in (1.3) and

$\displaystyle\eta_{\scriptscriptstyle{\textsc{WB}}}=\eta_{11}+{x_{21}^{(1)}\phi_{12}+x_{21}^{(3)}{\mathtt{c}}_{\mathtt{h}}^{-\frac{1}{2}}-x_{22}^{(1)}\phi_{11}-x_{22}^{(3)}\mathtt{f}_{11}+\frac{3}{2}(x_{21}^{(1)})^{2}x_{22}^{(1)}\phi_{22}+(x_{21}^{(1)})^{2}x_{12}^{(1)}{\mathtt{c}}_{\mathtt{h}}^{-\frac{1}{2}}}$
(4.9)
$\displaystyle\qquad\qquad{-\frac{3}{2}x_{21}^{(1)}x_{12}^{(1)}x_{22}^{(1)}\mathtt{f}_{11}+\frac{3}{2}(x_{22}^{(1)})^{2}x_{21}^{(1)}\phi_{21}-\frac{3}{2}x_{22}^{(1)}x_{11}^{(1)}x_{21}^{(1)}{\mathtt{c}}_{\mathtt{h}}^{-\frac{1}{2}}+(x_{22}^{(1)})^{2}{x_{11}^{(1)}\mathtt{f}_{11}}}$

with $x_{11}^{(1)}$, $x_{12}^{(1)}$, $x_{22}^{(1)}$, $x_{21}^{(1)}$,
$x_{21}^{(3)}$, $x_{22}^{(3)}$ in (4.4) and the remaining coefficients in
(3.4), whereas the reversibility-preserving $2\times 2$ self-adjoint matrix
$G^{(2)}$ has the form

$G^{(2)}=\begin{pmatrix}\mu+r_{8}(\mu\epsilon^{2},\mu^{3}\epsilon)&-\mathrm{i}\,r_{9}(\mu\epsilon^{2},\mu^{2}\epsilon)\\\
\mathrm{i}\,r_{9}(\mu\epsilon^{2},\mu^{2}\epsilon)&\tanh(\mathtt{h}\mu)+r_{10}(\mu\epsilon)\end{pmatrix}\,,$
(4.10)

and finally

$F^{(2)}=\begin{pmatrix}r_{3}(\mu\epsilon^{3})&\mathrm{i}\,r_{4}(\mu\epsilon^{3})\\\
\mathrm{i}\,r_{6}(\mu\epsilon^{3})&r_{7}(\mu\epsilon^{3})\end{pmatrix}\,.$
(4.11)

The rest of the section is devoted to the proof of Lemma 4.2. In Lemma 5.4 of
[10] we proved the existence of a matrix $X$ as in (4.3) such that we obtain
(4.7) with matrices $G^{(2)},F^{(2)}$ as in (4.10)-(4.11) and a $2\times
2$-self adjoint and reversibility preserving matrix $E^{(2)}$ whose first
entry has the form
$[E^{(2)}]_{11}=\mathtt{e}_{\scriptscriptstyle{\textsc{WB}}}\mu\epsilon^{2}+r_{1}(\mu\epsilon^{3},\mu^{2}\epsilon^{2})$.
The main result of Lemma 4.2 is that the first entry $[E^{(2)}]_{11}$ has the
better expansion

$[E^{(2)}]_{11}=\mathtt{e}_{\scriptscriptstyle{\textsc{WB}}}\mu\epsilon^{2}+r_{1}(\mu\epsilon^{3},\mu^{2}\epsilon^{2})=\mathtt{e}_{\scriptscriptstyle{\textsc{WB}}}\mu\epsilon^{2}+\eta_{\scriptscriptstyle{\textsc{WB}}}\mu\epsilon^{4}+r_{1}^{\prime}(\mu\epsilon^{5},\mu^{2}\epsilon^{3})\,$

with $\eta_{\scriptscriptstyle{\textsc{WB}}}$ computed in (4.9), which is
relevant to determine the stability/instablity of the Stokes wave at the
critical depth. Clearly we could compute explicitly also other Taylor
coefficients of the matrix entries of $E^{(2)},G^{(2)},F^{(2)}$, but it is not
needed.

The coefficients $x_{21}^{(1)}$ and $x_{22}^{(1)}$ in (4.4b) were already
computed in [10, Lemma 5.4].

We now expand in Lie series the Hamiltonian and reversible matrix
$\mathtt{L}_{\mu,\epsilon}^{(2)}=\exp(S)\mathtt{L}_{\mu,\epsilon}^{(1)}\exp(-S)$
where for simplicity we set $S:=S^{(1)}$. We split
$\mathtt{L}_{\mu,\epsilon}^{(1)}$ into its $2\times 2$-diagonal and off-
diagonal Hamiltonian and reversible matrices

$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\mathtt{L}_{\mu,\epsilon}^{(1)}=D^{(1)}+R^{(1)}\,,$
(4.12) $\displaystyle D^{(1)}:=\begin{pmatrix}D_{1}&0\\\
0&D_{0}\end{pmatrix}:=\begin{pmatrix}\mathtt{J}_{2}E^{(1)}&0\\\
0&\mathtt{J}_{2}G^{(1)}\end{pmatrix},\quad
R^{(1)}:=\begin{pmatrix}0&\mathtt{J}_{2}F^{(1)}\\\
\mathtt{J}_{2}[F^{(1)}]^{*}&0\end{pmatrix},$

and we perform the Lie expansion

$\displaystyle\mathtt{L}_{\mu,\epsilon}^{(2)}=\exp(S)\mathtt{L}_{\mu,\epsilon}^{(1)}\exp(-S)=D^{(1)}+[S,D^{(1)}]+\frac{1}{2}\big{[}S,[S,D^{(1)}]\big{]}+R^{(1)}+[S,R^{(1)}]$
(4.13) $\displaystyle+\frac{1}{2}\int_{0}^{1}(1-\tau)^{2}\exp(\tau
S)\text{ad}_{S}^{3}(D^{(1)})\exp(-\tau
S)\,\mathrm{d}\tau+\int_{0}^{1}(1-\tau)\,\exp(\tau
S)\,\text{ad}_{S}^{2}(R^{(1)})\,\exp(-\tau S)\,\mathrm{d}\tau$

where $\text{ad}_{A}(B):=[A,B]:=AB-BA$ denotes the commutator between the
linear operators $A,B$.

We look for a $4\times 4$ matrix $S$ as in (4.6) that solves the homological
equation $R^{(1)}+\left[S\,,\,D^{(1)}\right]=0$, which, recalling (4.12),
reads

$\begin{pmatrix}0&\mathtt{J}_{2}F^{(1)}+\mathtt{J}_{2}\Sigma
D_{0}-D_{1}\mathtt{J}_{2}\Sigma\\\
\mathtt{J}_{2}{[F^{(1)}]}^{*}+\mathtt{J}_{2}\Sigma^{*}D_{1}-D_{0}\mathtt{J}_{2}\Sigma^{*}&0\end{pmatrix}=0\,.$
(4.14)

Writing $\Sigma=\mathtt{J}_{2}X$, namely $X=-\mathtt{J}_{2}\Sigma$, the
equation (4.14) amounts to solve the “Sylvester” equation

$D_{1}X-XD_{0}=-\mathtt{J}_{2}F^{(1)}\,.$ (4.15)

We write the matrices $E^{(1)},F^{(1)},G^{(1)}$ in (4.1) as

$E^{(1)}=\begin{pmatrix}E_{11}^{(1)}&\mathrm{i}\,E_{12}^{(1)}\\\
-\mathrm{i}\,E_{12}^{(1)}&E_{22}^{(1)}\end{pmatrix}\,,\quad
F^{(1)}=\begin{pmatrix}F_{11}^{(1)}&\mathrm{i}\,F_{12}^{(1)}\\\
\mathrm{i}\,F_{21}^{(1)}&F_{22}^{(1)}\end{pmatrix}\,,\quad
G^{(1)}=\begin{pmatrix}G_{11}^{(1)}&\mathrm{i}\,G_{12}^{(1)}\\\
-\mathrm{i}\,G_{12}^{(1)}&G_{22}^{(1)}\end{pmatrix}$ (4.16)

where the real numbers $E_{ij}^{(1)},F_{ij}^{(1)},G_{ij}^{(1)}$, $i,j=1,2$,
have the expansion in (4.2a), (4.2b), (4.2c). Thus, by (4.12), (4.3) and
(4.16), the equation (4.15) amounts to solve the $4\times 4$ real linear
system

$\displaystyle\underbrace{\begin{pmatrix}G_{12}^{(1)}-E_{12}^{(1)}&G_{11}^{(1)}&E_{22}^{(1)}&0\\\
G_{22}^{(1)}&G_{12}^{(1)}-E_{12}^{(1)}&0&-E_{22}^{(1)}\\\
E_{11}^{(1)}&0&G_{12}^{(1)}-E_{12}^{(1)}&-G_{11}^{(1)}\\\
0&-E_{11}^{(1)}&-G_{22}^{(1)}&G_{12}^{(1)}-E_{12}^{(1)}\end{pmatrix}}_{=:{\cal
A}}\underbrace{\begin{pmatrix}x_{11}\\\ x_{12}\\\ x_{21}\\\
x_{22}\end{pmatrix}}_{=:\vec{x}}=\underbrace{\begin{pmatrix}-F_{21}^{(1)}\\\
F_{22}^{(1)}\\\ -F_{11}^{(1)}\\\ F_{12}^{(1)}\end{pmatrix}}_{=:\vec{f}}.$
(4.17)

By [10] system (4.17) admits a unique solution. We now prove that it has the
form (4.4).

######  Lemma 4.3.

The vector $\vec{x}=(x_{11},x_{12},x_{21},x_{22})$ with entries in (4.4)
solves (4.17).

###### Proof.

Since $\tanh(\mathtt{h}\mu)=\mathtt{h}\mu+r(\mu^{3})$, we have

$\displaystyle
G_{12}^{(1)}-E_{12}^{(1)}=-\mathtt{e}_{12}\frac{\mu}{2}-(\gamma_{12}+\eta_{12})\mu\epsilon^{2}+r(\mu\epsilon^{3},\mu^{2}\epsilon,\mu^{3})\,,$
(4.18) $\displaystyle
G_{11}^{(1)}=\mu+\gamma_{11}\mu\epsilon^{2}+r_{8}(\mu\epsilon^{3},{\mu^{2}\epsilon^{2}},\mu^{3}\epsilon)\,,\
\ E_{22}^{(1)}=-\mathtt{e}_{22}\frac{\mu}{8}(1+r_{5}(\epsilon^{2},\mu))\,,$
$\displaystyle
G_{22}^{(1)}=\mu\mathtt{h}+\gamma_{22}\mu\epsilon^{2}+r(\mu\epsilon^{3},\mu^{2}\epsilon,\mu^{3})\,,\
\
E_{11}^{(1)}=\mathtt{e}_{11}\mu\epsilon^{2}+r(\mu\epsilon^{4},\mu^{2}\epsilon^{3},\mu^{3})\,,$

with coefficients $\mathtt{e}_{12}$, $\gamma_{12}$, $\eta_{12}$,
$\gamma_{11}$, $\mathtt{e}_{22}$, $\gamma_{22}$ and $\mathtt{e}_{11}$ computed
in (3.4). We exploit that the terms $x_{21}^{(1)}$ and $x_{22}^{(2)}$ have
been already computed in [10, Lemma 5.4], in order to get $x_{11}^{(1)}$ and
$x_{12}^{(2)}$ in (4.4) as solutions of the system

$\begin{pmatrix}-\tfrac{1}{2}\mathtt{e}_{12}&1\\\
\mathtt{h}&-\tfrac{1}{2}\mathtt{e}_{12}\end{pmatrix}\begin{pmatrix}x_{11}^{(1)}\\\
x_{12}^{(1)}\end{pmatrix}=\begin{pmatrix}\tfrac{1}{8}\mathtt{e}_{22}x_{21}^{(1)}-\phi_{21}\\\
\phi_{22}-\tfrac{1}{8}\mathtt{e}_{22}x_{22}^{(1)}\end{pmatrix}\,,\quad\det\begin{pmatrix}-\tfrac{1}{2}\mathtt{e}_{12}&1\\\
\mathtt{h}&-\tfrac{1}{2}\mathtt{e}_{12}\end{pmatrix}\stackrel{{\scriptstyle\eqref{defDh}}}{{=}}-\mathtt{D}_{\mathtt{h}}<0\,,$
(4.19)

given, using also (4.2c), by the first two lines in (4.17) at order
$\mu\epsilon$. Similarly, $x_{21}^{(3)}$ and $x_{22}^{(3)}$ in (4.4) solve the
system

$\begin{pmatrix}-\tfrac{1}{2}\mathtt{e}_{12}&-1\\\
-\mathtt{h}&-\tfrac{1}{2}\mathtt{e}_{12}\end{pmatrix}\begin{pmatrix}x_{21}^{(3)}\\\
x_{22}^{(3)}\end{pmatrix}=\begin{pmatrix}-\mathtt{e}_{11}x_{11}^{(1)}+(\gamma_{12}+\eta_{12})x_{21}^{(1)}+\gamma_{11}x_{22}^{(1)}-\phi_{11}\\\
\mathtt{e}_{11}x_{12}^{(1)}+\gamma_{22}x_{21}^{(1)}+(\gamma_{12}+\eta_{12})x_{22}^{(1)}+\phi_{12}\end{pmatrix}\,,$
(4.20)

which comes, also by (4.2c), from the last two lines of (4.17) at order
$\mu\epsilon^{3}$. The solutions of (4.19)-(4.20) are given in (4.4c). ∎

We now prove the expansion (4.8). Since the matrix $S$ solves the homological
equation $\left[S\,,\,D^{(1)}\right]+R^{(1)}=0$, identity (4.13) simplifies to

$\mathtt{L}_{\mu,\epsilon}^{(2)}=D^{(1)}+\frac{1}{2}\big{[}S,R^{(1)}\big{]}+\frac{1}{2}\int_{0}^{1}(1-\tau^{2})\,\exp(\tau
S)\,\text{ad}_{S}^{2}(R^{(1)})\,\exp(-\tau S)\mathrm{d}\tau\,.$ (4.21)

By plugging the Lie expansion

$\displaystyle\exp(\tau S)\,\text{ad}_{S}^{2}(R^{(1)})\,\exp(-\tau S)$
$\displaystyle\qquad=\text{ad}_{S}^{2}(R^{(1)})+\tau\text{ad}_{S}^{3}(R^{(1)})+\tau^{2}\int_{0}^{1}(1-\tau^{\prime})\exp(\tau^{\prime}\tau
S)\,\text{ad}_{S}^{4}(R^{(1)})\,\exp(-\tau^{\prime}\tau
S)\,\mathrm{d}\tau^{\prime}$

into (4.21) we get

$\displaystyle\mathtt{L}_{\mu,\epsilon}^{(2)}=D^{(1)}+\frac{1}{2}\big{[}S,R^{(1)}\big{]}+\frac{1}{3}\text{ad}_{S}^{2}(R^{(1)})+\frac{1}{8}\text{ad}_{S}^{3}(R^{(1)})$
(4.22a)
$\displaystyle+\frac{1}{2}\int_{0}^{1}(1-\tau^{2})\tau^{2}\int_{0}^{1}(1-\tau^{\prime})\exp(\tau\tau^{\prime}S)\text{ad}_{S}^{4}(R^{(1)})\exp(-\tau\tau^{\prime}S)\mathrm{d}\tau^{\prime}\mathrm{d}\tau\,.$
(4.22b)

Next we compute the commutators in the expansion (4.22a).

######  Lemma 4.4.

One has

$\frac{1}{2}\big{[}S,R^{(1)}\big{]}=\begin{pmatrix}\mathtt{J}_{2}\tilde{E}_{1}&0\\\
0&\mathtt{J}_{2}\tilde{G}_{1}\end{pmatrix}$ (4.23)

where $\tilde{E}_{1}$, $\tilde{G}_{1}$ are self-adjoint and reversibility-
preserving matrices of the form

$\displaystyle\tilde{E}_{1}=\begin{pmatrix}\tilde{\mathtt{e}}_{11}\mu\epsilon^{2}+\tilde{\eta}_{11}^{(a)}\mu\epsilon^{4}+\tilde{r}_{1}(\mu\epsilon^{5},\mu^{2}\epsilon^{3},\mu^{3}\epsilon^{2})&\mathrm{i}\,\big{(}\tilde{\mathtt{e}}_{12}\mu\epsilon^{2}+\tilde{r}_{2}(\mu\epsilon^{4},\mu^{2}\epsilon^{2})\big{)}\\\
-\mathrm{i}\,\big{(}\tilde{\mathtt{e}}_{12}\mu\epsilon^{2}+\tilde{r}_{2}(\mu\epsilon^{4},\mu^{2}\epsilon^{2})\big{)}&\tilde{r}_{5}(\mu\epsilon^{2})\end{pmatrix}\,,$
(4.24)
$\displaystyle\tilde{G}_{1}=\begin{pmatrix}\tilde{\mathtt{g}}_{11}\mu\epsilon^{2}+\tilde{r}_{8}(\mu\epsilon^{4},\mu^{2}\epsilon^{2})&\mathrm{i}\,\big{(}\tilde{\mathtt{g}}_{12}\mu\epsilon^{2}+\tilde{r}_{9}(\mu\epsilon^{4},\mu^{2}\epsilon^{2})\big{)}\\\
-\mathrm{i}\,\big{(}\tilde{\mathtt{g}}_{12}\mu\epsilon^{2}+\tilde{r}_{9}(\mu\epsilon^{4},\mu^{2}\epsilon^{2})\big{)}&\tilde{\mathtt{g}}_{22}\mu\epsilon^{2}+\tilde{r}_{10}(\mu\epsilon^{4},\mu^{2}\epsilon^{2})\end{pmatrix}\,,$

where

$\displaystyle\tilde{\mathtt{e}}_{11}:=x_{21}^{(1)}{\mathtt{c}}_{\mathtt{h}}^{-\frac{1}{2}}-x_{22}^{(1)}\mathtt{f}_{11}\,,\quad\tilde{\eta}_{11}^{(a)}:=x_{21}^{(1)}\phi_{12}+x_{21}^{(3)}{\mathtt{c}}_{\mathtt{h}}^{-\frac{1}{2}}-x_{22}^{(1)}\phi_{11}-x_{22}^{(3)}\mathtt{f}_{11}\,,\quad$
(4.25)
$\displaystyle\tilde{\mathtt{e}}_{12}:=-\tilde{\mathtt{g}}_{12}:={\frac{1}{2}\big{(}x_{21}^{(1)}\phi_{22}+x_{22}^{(1)}\phi_{21}-x_{11}^{(1)}{\mathtt{c}}_{\mathtt{h}}^{-\frac{1}{2}}-x_{12}^{(1)}\mathtt{f}_{11}\big{)}}\,,$
$\displaystyle\tilde{\mathtt{g}}_{11}:=x_{11}^{(1)}\mathtt{f}_{11}+x_{21}^{(1)}\phi_{21}\,,\quad\tilde{\mathtt{g}}_{22}:=x_{22}^{(1)}\phi_{22}+x_{12}^{(1)}{\mathtt{c}}_{\mathtt{h}}^{-\frac{1}{2}}\,.$

###### Proof.

By (4.6), (4.12), and since $\Sigma=\mathtt{J}_{2}X$, we have

$\frac{1}{2}\left[S\,,\,R^{(1)}\right]=\begin{pmatrix}\mathtt{J}_{2}\tilde{E}_{1}&0\\\
0&\mathtt{J}_{2}\tilde{G}_{1}\end{pmatrix},\quad\tilde{E}_{1}:=\textup{{Sym}}\big{[}\mathtt{J}_{2}X\mathtt{J}_{2}[F^{(1)}]^{*}\big{]}\,,\quad\tilde{G}_{1}:=\textup{{Sym}}\big{[}X^{*}F^{(1)}\big{]}\,,$
(4.26)

where $\textup{{Sym}}[A]:=\frac{1}{2}(A+A^{*})$, see [10, (5.28)-(5.29)]. By
(4.3), (4.16), setting $F=F^{(1)}$, we have

$\displaystyle\mathtt{J}_{2}X\mathtt{J}_{2}F^{*}$
$\displaystyle=\begin{pmatrix}x_{21}F_{12}-x_{22}F_{11}&\mathrm{i}\,(x_{21}F_{22}+x_{22}F_{21})\\\
\mathrm{i}\,(x_{11}F_{12}+x_{12}F_{11})&-x_{11}F_{22}+x_{12}F_{21}\end{pmatrix}\,,$
(4.27) $\displaystyle X^{*}F$
$\displaystyle=\begin{pmatrix}x_{11}F_{11}+x_{21}F_{21}&\mathrm{i}\,(x_{11}F_{12}-x_{21}F_{22})\\\
\mathrm{i}\,(x_{22}F_{21}-x_{12}F_{11})&x_{22}F_{22}+x_{12}F_{12}\end{pmatrix}\,,$
(4.28)

and the expansions in (4.24) with the coefficients given in (LABEL:primocomm)
follow by (4.27), (4.28), (4.4) and (4.2c). ∎

######  Lemma 4.5.

One has

$\displaystyle\frac{1}{3}\mathrm{ad}_{S}^{2}(R^{(1)})=\begin{pmatrix}0&\mathtt{J}_{2}\tilde{F}\\\
\mathtt{J}_{2}\tilde{F}^{*}&0\end{pmatrix},$ (4.29)

where $\tilde{F}$ is a reversibility-preserving matrix of the form

$\tilde{F}=\begin{pmatrix}\tilde{\mathtt{f}}_{11}\mu\epsilon^{3}+\tilde{r}_{3}(\mu\epsilon^{5},\mu^{2}\epsilon^{3})&\mathrm{i}\,\tilde{\mathtt{f}}_{12}\mu\epsilon^{3}+\mathrm{i}\,\tilde{r}_{4}(\mu\epsilon^{5},\mu^{2}\epsilon^{3})\\\
\mathrm{i}\,\tilde{r}_{6}(\mu\epsilon^{3})&\tilde{r}_{7}(\mu\epsilon^{3})\end{pmatrix}\,,$
(4.30)

with

$\displaystyle\tilde{\mathtt{f}}_{11}$
$\displaystyle:={\frac{4}{3}x_{21}^{(1)}x_{11}^{(1)}{\mathtt{c}}_{\mathtt{h}}^{-\frac{1}{2}}-\frac{4}{3}x_{22}^{(1)}x_{11}^{(1)}\mathtt{f}_{11}-\frac{4}{3}x_{22}^{(1)}x_{21}^{(1)}\phi_{21}+\frac{2}{3}x_{21}^{(1)}x_{12}^{(1)}\mathtt{f}_{11}-\frac{2}{3}(x_{21}^{(1)})^{2}\phi_{22}}\,,$
(4.31) $\displaystyle\tilde{\mathtt{f}}_{12}$
$\displaystyle:={\frac{4}{3}x_{21}^{(1)}x_{22}^{(1)}\phi_{22}+\frac{4}{3}x_{12}^{(1)}x_{21}^{(1)}{\mathtt{c}}_{\mathtt{h}}^{-\frac{1}{2}}-\frac{4}{3}x_{12}^{(1)}x_{22}^{(1)}\mathtt{f}_{11}+\frac{2}{3}(x_{22}^{(1)})^{2}\phi_{21}-\frac{2}{3}x_{11}^{(1)}x_{22}^{(1)}{\mathtt{c}}_{\mathtt{h}}^{-\frac{1}{2}}}\,.$

###### Proof.

Using the form of $S$ in (4.6) and $[S,R^{(1)}]$ in (4.23) we deduce (4.29)
with

$\tilde{F}:=\frac{2}{3}\big{(}\mathtt{J}_{2}X\mathtt{J}_{2}\tilde{G}_{1}+\tilde{E}_{1}X\big{)}$
(4.32)

where $\tilde{E}_{1}$ and $\tilde{G}_{1}$ are the matrices in (4.24). Writing
$\tilde{E}_{1}={\footnotesize\begin{pmatrix}{[\tilde{E}_{1}]}_{11}&\mathrm{i}\,{[\tilde{E}_{1}]}_{12}\\\
-\mathrm{i}\,{[\tilde{E}_{1}]}_{12}&{[\tilde{E}_{1}]}_{22}\end{pmatrix}}$,
$\tilde{G}_{1}={\footnotesize\begin{pmatrix}{[\tilde{G}_{1}]}_{11}&\mathrm{i}\,{[\tilde{G}_{1}]}_{12}\\\
-\mathrm{i}\,{[\tilde{G}_{1}]}_{12}&{[\tilde{G}_{1}]}_{22}\end{pmatrix}}$ we
have, in view of (4.3),

$\displaystyle\mathtt{J}_{2}X\mathtt{J}_{2}\tilde{G}_{1}$
$\displaystyle=\begin{pmatrix}x_{21}{[\tilde{G}_{1}]}_{12}-x_{22}{[\tilde{G}_{1}]}_{11}&\mathrm{i}\,\big{(}x_{21}{[\tilde{G}_{1}]}_{22}-x_{22}{[\tilde{G}_{1}]}_{12}\big{)}\\\
\mathrm{i}\,\big{(}x_{11}{[\tilde{G}_{1}]}_{12}+x_{12}{[\tilde{G}_{1}]}_{11}\big{)}&-x_{11}{[\tilde{G}_{1}]}_{22}-x_{12}{[\tilde{G}_{1}]}_{12}\end{pmatrix}\,,$
(4.33) $\displaystyle\tilde{E}_{1}X$
$\displaystyle=\begin{pmatrix}x_{11}{[\tilde{E}_{1}]}_{11}-x_{21}{[\tilde{E}_{1}]}_{12}&\mathrm{i}\,\big{(}x_{12}{[\tilde{E}_{1}]}_{11}+x_{22}{[\tilde{E}_{1}]}_{12}\big{)}\\\
\mathrm{i}\,\big{(}x_{21}{[\tilde{E}_{1}]}_{22}-x_{11}{[\tilde{E}_{1}]}_{12}\big{)}&x_{12}{[\tilde{E}_{1}]}_{12}+x_{22}{[\tilde{E}_{1}]}_{22}\end{pmatrix}\,.$

By (4.32), (4.33), (4.4) and (4.24) we deduce that the matrix $\tilde{F}$ has
the expansion (4.30) with

$\displaystyle\tilde{\mathtt{f}}_{11}$
$\displaystyle=\frac{2}{3}\big{(}x_{21}^{(1)}\tilde{\mathtt{g}}_{12}-x_{22}^{(1)}\tilde{\mathtt{g}}_{11}+x_{11}^{(1)}\tilde{\mathtt{e}}_{11}-x_{21}^{(1)}\tilde{\mathtt{e}}_{12}\big{)}\,,$
$\displaystyle\tilde{\mathtt{f}}_{12}$
$\displaystyle=\frac{2}{3}\big{(}x_{21}^{(1)}\tilde{\mathtt{g}}_{22}-x_{22}^{(1)}\tilde{\mathtt{g}}_{12}+x_{12}^{(1)}\tilde{\mathtt{e}}_{11}+x_{22}^{(1)}\tilde{\mathtt{e}}_{12}\big{)}\,,$

which, by (LABEL:primocomm), gives (4.31). ∎

######  Lemma 4.6.

One has

$\displaystyle\frac{1}{8}\mathrm{ad}_{S}^{3}(R^{(1)})=\begin{pmatrix}\mathtt{J}_{2}\tilde{E}_{3}&0\\\
0&\mathtt{J}_{2}\tilde{G}_{3}\end{pmatrix},$ (4.34)

where the self-adjoint and reversibility-preserving matrices
$\tilde{E}_{3},\tilde{G}_{3}$ in (4.34) have entries of size
$\mathcal{O}(\mu\epsilon^{4})$. In particular the first entry of the matrix
$\tilde{E}_{3}$ has the expansion

${[\tilde{E}_{3}]}_{11}=\tilde{\eta}_{11}^{(b)}\mu\epsilon^{4}+r(\mu\epsilon^{5},\mu^{2}\epsilon^{4})$
(4.35)

with

$\displaystyle\tilde{\eta}_{11}^{(b)}$
$\displaystyle:={\frac{3}{2}(x_{21}^{(1)})^{2}x_{22}^{(1)}\phi_{22}+(x_{21}^{(1)})^{2}x_{12}^{(1)}{\mathtt{c}}_{\mathtt{h}}^{-\frac{1}{2}}-\frac{3}{2}x_{21}^{(1)}x_{12}^{(1)}x_{22}^{(1)}\mathtt{f}_{11}}$
(4.36) $\displaystyle\
\quad{+\frac{3}{2}(x_{22}^{(1)})^{2}x_{21}^{(1)}\phi_{21}-\frac{3}{2}x_{22}^{(1)}x_{11}^{(1)}x_{21}^{(1)}{\mathtt{c}}_{\mathtt{h}}^{-\frac{1}{2}}+(x_{22}^{(1)})^{2}{x_{11}^{(1)}\mathtt{f}_{11}}}\,.$

###### Proof.

Since
$\frac{1}{8}\mathrm{ad}_{S}^{3}(R^{(1)})=\frac{3}{8}[S,\frac{1}{3}\mathrm{ad}_{S}^{2}(R^{(1)})]$
and using (4.29), the identity (4.34) holds with

$\tilde{E}_{3}:=\frac{3}{4}\textup{{Sym}}\big{[}\mathtt{J}_{2}X\mathtt{J}_{2}[\tilde{F}]^{*}\big{]}\,,\qquad\tilde{G}_{3}:=\frac{3}{4}\textup{{Sym}}\big{[}X^{*}\tilde{F}\big{]}\,.$
(4.37)

Since, by (4.4) the matrix $X$ in (4.3) has entries of size
$\mathcal{O}(\epsilon)$ and the matrix $\tilde{F}$ in (4.30) has entries of
size $\mathcal{O}(\mu\epsilon^{3})$ we deduce that the matrices
$\tilde{E}_{3}$, $\tilde{G}_{3}$ in (4.37) have entries of size
$\mathcal{O}(\mu\epsilon^{4})$. By (4.37) and denoting
$\tilde{F}={\footnotesize\begin{pmatrix}\tilde{F}_{11}&\mathrm{i}\,\tilde{F}_{12}\\\
\mathrm{i}\,\tilde{F}_{21}&\tilde{F}_{22}\end{pmatrix}}$, we deduce, similarly
to (4.27), that
${[\tilde{E}_{3}]}_{11}=\frac{3}{4}(x_{21}\tilde{F}_{12}-x_{22}\tilde{F}_{11})$
which, by (4.4) and (4.30), gives (4.35) with
$\tilde{\eta}_{11}^{(b)}=\frac{3}{4}\big{(}x_{21}^{(1)}\tilde{\mathtt{f}}_{12}-x_{22}^{(1)}\tilde{\mathtt{f}}_{11}\big{)}$
which by (4.31) gives (4.36). ∎

Finally we show that the term in (4.22b) is small.

######  Lemma 4.7.

The $4\times 4$ Hamiltonian and reversible matrix
$\footnotesize\begin{pmatrix}\mathtt{J}_{2}\widehat{E}&\mathtt{J}_{2}\widehat{F}\\\
\mathtt{J}_{2}[\widehat{F}]^{*}&\mathtt{J}_{2}\widehat{G}\end{pmatrix}$ given
by (4.22b) has the $2\times 2$ self-adjoint and reversibility-preserving
blocks $\widehat{E}$, $\widehat{G}$ and the $2\times 2$ reversibility-
preserving block $\widehat{F}$ all with entries of size
$\mathcal{O}(\mu\epsilon^{5})$.

###### Proof.

By the Hamiltonian and reversibility properties of $S$ and $R^{(1)}$ the
matrix $\textup{ad}_{S}^{4}(R^{(1)})$ is Hamiltonian and reversible and the
same holds, for any $\tau,\tau^{\prime}\in[0,1]$, for

$\exp(\tau\tau^{\prime}S)\,\textup{ad}_{S}^{4}(R^{(1)})\,\exp(-\tau\tau^{\prime}S)=[S,\textup{ad}_{S}^{3}(R^{(1)})](1+\mathcal{O}(\mu,\epsilon))\,.$
(4.38)

The claimed estimate on the entries of the matrix given by (4.22b) follows by
(4.38) and because $S$ in (4.6) has entries of size $\mathcal{O}(\epsilon)$
and $\textup{ad}_{S}^{3}(R^{(1)})$ in (4.34) has entries of size
$\mathcal{O}(\mu\epsilon^{4})$. ∎

###### Proof of Lemma 4.2..

It follows by (4.22), (4.12) and Lemmata 4.4, 4.5, 4.6 and 4.7. The matrix
$E^{(2)}:=E^{(1)}+\tilde{E}_{1}+\tilde{E}_{3}+\widehat{E}$ has the expansion
in (4.8), with

$\mathtt{e}_{\scriptscriptstyle{\textsc{WB}}}:=\mathtt{e}_{11}+\tilde{\mathtt{e}}_{11}=\mathtt{e}_{11}-\mathtt{D}_{\mathtt{h}}^{-1}\big{(}{\mathtt{c}}_{\mathtt{h}}^{-1}+\mathtt{h}\mathtt{f}_{11}^{2}+\mathtt{e}_{12}\mathtt{f}_{11}{\mathtt{c}}_{\mathtt{h}}^{-\frac{1}{2}}\big{)}\,,\qquad\eta_{\scriptscriptstyle{\textsc{WB}}}:=\eta_{11}+\tilde{\eta}_{11}^{(a)}+\tilde{\eta}_{11}^{(b)}\,,$

as in (4.9). Furthermore
$G^{(2)}:=G^{(1)}+\tilde{G}_{1}+\tilde{G}_{3}+\widehat{G}$ has the expansion
in (4.10) and $F^{(2)}:=\tilde{F}+\widehat{F}$ has the expansion in (4.11). ∎

#### Complete block-decoupling and proof of the main result.

Finally Theorem 1.3 is proved as in [10] by block-diagonalizing the $4\times
4$ Hamiltonian and reversible matrix $\mathtt{L}_{\mu,\epsilon}^{(2)}$ in
(4.7),

$\displaystyle\mathtt{L}_{\mu,\epsilon}^{(2)}=D^{(2)}+R^{(2)}\,,\quad
D^{(2)}:=\begin{pmatrix}\mathtt{J}_{2}E^{(2)}&0\\\
0&\mathtt{J}_{2}G^{(2)}\end{pmatrix},\quad
R^{(2)}:=\begin{pmatrix}0&\mathtt{J}_{2}F^{(2)}\\\
\mathtt{J}_{2}[F^{(2)}]^{*}&0\end{pmatrix}.$ (4.39)

The next lemma is [10, Lemma 5.9].

######  Lemma 4.8.

There exist a $4\times 4$ reversibility-preserving Hamiltonian matrix
$S^{(2)}:=S^{(2)}(\mu,\epsilon)$ of the form (4.6), analytic in
$(\mu,\epsilon)$, of size $\mathcal{O}(\epsilon^{3})$, and a $4\times 4$
block-diagonal reversible Hamiltonian matrix $P:=P(\mu,\epsilon)$, analytic in
$(\mu,\epsilon)$, of size ${\mathcal{O}(\mu\epsilon^{6})}$ such that

$\exp(S^{(2)})(D^{(2)}+R^{(2)})\exp(-S^{(2)})=D^{(2)}+P\,.$ (4.40)

By (4.40), (4.8)-(4.10) and the fact that $P$ has size
${\mathcal{O}(\mu\epsilon^{6})}$ we deduce Theorem 1.3: there exists a
symplectic and reversibility-preserving linear map that conjugates the matrix
$\mathrm{i}\,{\mathtt{c}}_{\mathtt{h}}\mu+\mathtt{L}_{\mu,\epsilon}$ (which
represents ${\cal L}_{\mu,\epsilon}$) with $\mathtt{L}_{\mu,\epsilon}$ in
(2.8) into the Hamiltonian and reversible matrix (1.17) with $\mathtt{U}$ in
(1.18) and $\mathtt{S}$ in (1.19). The function
$\Delta_{\scriptscriptstyle{\textsc{BF}}}(\mathtt{h};\mu,\epsilon)$ expands as
in (1.2).

##  Appendix A Fourth-order expansion of the Stokes waves

In this Appendix we compute the Taylor coefficients (A.1) of the fourth order
expansions (1.9) of the Stokes waves. We also compute the fourth order
expansion (A.58) of the $2\pi$-periodic functions $p_{\epsilon}(x)$ and
$a_{\epsilon}(x)$ and the expansion (A.47) of the constant
$\mathtt{f}_{\epsilon}$ in (1.12).

###  A.1 Expansion of Stokes waves

######  Proposition A.1.

(Expansion of Stokes waves) The Stokes waves $\eta_{\epsilon}(x)$,
$\psi_{\epsilon}(x)$ and the speed $c_{\epsilon}$ in Theorem 1.2 have the
expansions (1.9) with coefficients

$\displaystyle\eta_{2}^{[0]}:=\frac{{\mathtt{c}}_{\mathtt{h}}^{4}-1}{4{\mathtt{c}}_{\mathtt{h}}^{2}}\,,\qquad\eta_{2}^{[2]}:=\frac{3-{\mathtt{c}}_{\mathtt{h}}^{4}}{4{\mathtt{c}}_{\mathtt{h}}^{6}}\,,\qquad\psi_{2}^{[2]}:=\frac{3+{\mathtt{c}}_{\mathtt{h}}^{8}}{8{\mathtt{c}}_{\mathtt{h}}^{7}}\,,\qquad$
(A.1a) $\displaystyle
c_{2}:=\frac{9-10{\mathtt{c}}_{\mathtt{h}}^{4}+9{\mathtt{c}}_{\mathtt{h}}^{8}}{16{\mathtt{c}}_{\mathtt{h}}^{7}}+{\frac{(1-{\mathtt{c}}_{\mathtt{h}}^{4})}{2{\mathtt{c}}_{\mathtt{h}}}}\eta_{2}^{[0]}=\frac{-2{\mathtt{c}}_{\mathtt{h}}^{12}+13{\mathtt{c}}_{\mathtt{h}}^{8}-12{\mathtt{c}}_{\mathtt{h}}^{4}+9}{16{\mathtt{c}}_{\mathtt{h}}^{7}}\,,$
(A.1b) $\displaystyle\begin{aligned}
&\eta_{3}^{[1]}:=\frac{-2{\mathtt{c}}_{\mathtt{h}}^{12}+3{\mathtt{c}}_{\mathtt{h}}^{8}+3}{16{\mathtt{c}}_{\mathtt{h}}^{8}(1+{\mathtt{c}}_{\mathtt{h}}^{2})}\,,\qquad\eta_{3}^{[3]}:=\frac{-3{\mathtt{c}}_{\mathtt{h}}^{12}+9{\mathtt{c}}_{\mathtt{h}}^{8}-9{\mathtt{c}}_{\mathtt{h}}^{4}+27}{64{\mathtt{c}}_{\mathtt{h}}^{12}}\,,\\\
&\psi_{3}^{[1]}:=\frac{2{\mathtt{c}}_{\mathtt{h}}^{12}-3{\mathtt{c}}_{\mathtt{h}}^{8}-3}{16{\mathtt{c}}_{\mathtt{h}}^{7}(1+{\mathtt{c}}_{\mathtt{h}}^{2})}\,,\qquad\psi_{3}^{[3]}:=\frac{-9{\mathtt{c}}_{\mathtt{h}}^{12}+19{\mathtt{c}}_{\mathtt{h}}^{8}+5{\mathtt{c}}_{\mathtt{h}}^{4}+9}{64{\mathtt{c}}_{\mathtt{h}}^{13}}\,,\end{aligned}$
(A.1c) $\displaystyle\begin{aligned}
&\eta_{4}^{[0]}:=\frac{-4{\mathtt{c}}_{\mathtt{h}}^{20}-4{\mathtt{c}}_{\mathtt{h}}^{18}+17{\mathtt{c}}_{\mathtt{h}}^{16}+6{\mathtt{c}}_{\mathtt{h}}^{14}-48{\mathtt{c}}_{\mathtt{h}}^{8}+6{\mathtt{c}}_{\mathtt{h}}^{6}+36{\mathtt{c}}_{\mathtt{h}}^{4}-9}{64{\mathtt{c}}_{\mathtt{h}}^{14}}\,,\\\
&\eta_{4}^{[2]}:=\frac{1}{384{\mathtt{c}}_{\mathtt{h}}^{18}({\mathtt{c}}_{\mathtt{h}}^{2}+1)}\Big{(}-24{\mathtt{c}}_{\mathtt{h}}^{22}+285{\mathtt{c}}_{\mathtt{h}}^{18}+177{\mathtt{c}}_{\mathtt{h}}^{16}-862{\mathtt{c}}_{\mathtt{h}}^{14}-754{\mathtt{c}}_{\mathtt{h}}^{12}\\\
&\qquad+1116{\mathtt{c}}_{\mathtt{h}}^{10}+1080{\mathtt{c}}_{\mathtt{h}}^{8}-162{\mathtt{c}}_{\mathtt{h}}^{6}-54{\mathtt{c}}_{\mathtt{h}}^{4}-81{\mathtt{c}}_{\mathtt{h}}^{2}-81\Big{)}\,,\\\
&\eta_{4}^{[4]}:=\frac{21{\mathtt{c}}_{\mathtt{h}}^{20}+{\mathtt{c}}_{\mathtt{h}}^{16}-262{\mathtt{c}}_{\mathtt{h}}^{12}+522{\mathtt{c}}_{\mathtt{h}}^{8}+81{\mathtt{c}}_{\mathtt{h}}^{4}+405}{384{\mathtt{c}}_{\mathtt{h}}^{18}({\mathtt{c}}_{\mathtt{h}}^{4}+5)}\,,\\\
&\psi_{4}^{[2]}:=\frac{1}{768{\mathtt{c}}_{\mathtt{h}}^{19}({\mathtt{c}}_{\mathtt{h}}^{2}+1)}\Big{(}-12{\mathtt{c}}_{\mathtt{h}}^{26}-36{\mathtt{c}}_{\mathtt{h}}^{24}+57{\mathtt{c}}_{\mathtt{h}}^{22}+93{\mathtt{c}}_{\mathtt{h}}^{20}+51{\mathtt{c}}_{\mathtt{h}}^{18}-21{\mathtt{c}}_{\mathtt{h}}^{16}-646{\mathtt{c}}_{\mathtt{h}}^{14}\,,\\\
&\qquad-502{\mathtt{c}}_{\mathtt{h}}^{12}+1098{\mathtt{c}}_{\mathtt{h}}^{10}+1098{\mathtt{c}}_{\mathtt{h}}^{8}-243{\mathtt{c}}_{\mathtt{h}}^{6}-135{\mathtt{c}}_{\mathtt{h}}^{4}-81{\mathtt{c}}_{\mathtt{h}}^{2}-81\Big{)}\,,\\\
&\psi_{4}^{[4]}:=\frac{-21{\mathtt{c}}_{\mathtt{h}}^{24}+60{\mathtt{c}}_{\mathtt{h}}^{20}+343{\mathtt{c}}_{\mathtt{h}}^{16}-1648{\mathtt{c}}_{\mathtt{h}}^{12}+3177{\mathtt{c}}_{\mathtt{h}}^{8}+756{\mathtt{c}}_{\mathtt{h}}^{4}+405}{1536{\mathtt{c}}_{\mathtt{h}}^{19}({\mathtt{c}}_{\mathtt{h}}^{4}+5)}\,,\end{aligned}$
(A.1d) $\displaystyle
c_{4}=\frac{1}{1024{\mathtt{c}}_{\mathtt{h}}^{19}({\mathtt{c}}_{\mathtt{h}}^{2}+1)}\Big{(}56{\mathtt{c}}_{\mathtt{h}}^{30}+88{\mathtt{c}}_{\mathtt{h}}^{28}-272{\mathtt{c}}_{\mathtt{h}}^{26}-528{\mathtt{c}}_{\mathtt{h}}^{24}-7{\mathtt{c}}_{\mathtt{h}}^{22}+497{\mathtt{c}}_{\mathtt{h}}^{20}+1917{\mathtt{c}}_{\mathtt{h}}^{18}$
(A.1e)
$\displaystyle\quad+1437{\mathtt{c}}_{\mathtt{h}}^{16}-4566{\mathtt{c}}_{\mathtt{h}}^{14}-4038{\mathtt{c}}_{\mathtt{h}}^{12}+4194{\mathtt{c}}_{\mathtt{h}}^{10}+3906{\mathtt{c}}_{\mathtt{h}}^{8}-891{\mathtt{c}}_{\mathtt{h}}^{6}-675{\mathtt{c}}_{\mathtt{h}}^{4}+81{\mathtt{c}}_{\mathtt{h}}^{2}+81\Big{)}\,.$

The rest of this section is devoted to the proof of Proposition A.1.

In [10, Theorem 2.1] we have yet computed the second order expansion of the
Stokes waves in (1.9) and proved that the coefficients $\eta_{2}^{[0]}$,
$\eta_{2}^{[2]}$, $\psi_{2}^{[2]}$ are (A.1a) and $c_{2}$ is (A.1b). Then we
write

$\displaystyle\eta_{\epsilon}(x)=\epsilon\cos(x)+\epsilon^{2}(\eta_{2}^{[0]}+\eta_{2}^{[2]}\cos(2x))+\epsilon^{3}\eta_{3}(x)+\epsilon^{4}\eta_{4}(x)+\mathcal{O}(\epsilon^{5})\,,$
(A.2)
$\displaystyle\psi_{\epsilon}(x)=\epsilon{\mathtt{c}}_{\mathtt{h}}^{-1}\sin(x)+\epsilon^{2}\psi_{2}^{[2]}\sin(2x)+\epsilon^{3}\psi_{3}(x)+\epsilon^{4}\psi_{4}(x)+\mathcal{O}(\epsilon^{5})\,,$
$\displaystyle
c_{\epsilon}={\mathtt{c}}_{\mathtt{h}}+\epsilon^{2}c_{2}+\epsilon^{3}c_{3}+\epsilon^{4}c_{4}+\mathcal{O}(\epsilon^{5})\,,$

where the functions $\eta_{3}(x)$, $\eta_{4}(x)$ are even and $\psi_{3}(x)$,
$\psi_{4}(x)$ are odd. We rewrite the equations (1.8) as the system

$\begin{cases}\eta-c\,\psi_{x}+\dfrac{\psi_{x}^{2}}{2}-\dfrac{\eta_{x}^{2}}{2(1+\eta_{x}^{2})}(c-\psi_{x})^{2}=0\\\
c\,\eta_{x}+G(\eta)\psi=0\,,\end{cases}$ (A.3)

where in the first equation we have replaced $G(\eta)\psi$ with
$-c\,\eta_{x}$.

We Taylor expand the Dirichlet-Neumann operator $G(\eta)$ as

$G(\eta)=G_{0}+G_{1}(\eta)+G_{2}(\eta)+G_{3}(\eta)+\mathcal{O}(\eta^{4})$

where, by [16, formulae (39)-(40)],

$\displaystyle G_{0}:=D\tanh(\mathtt{h}D)=|D|\tanh(\mathtt{h}|D|)\,,$ (A.4)
$\displaystyle G_{1}(\eta):=-\partial_{x}\eta\partial_{x}-G_{0}\eta G_{0}\,,$
$\displaystyle
G_{2}(\eta):=-\frac{1}{2}G_{0}\partial_{x}\eta^{2}\partial_{x}+\frac{1}{2}\partial_{x}^{2}\eta^{2}G_{0}-G_{0}\eta
G_{1}(\eta)\,,$ $\displaystyle
G_{3}(\eta):=\frac{1}{6}\partial_{x}^{3}\eta^{3}\partial_{x}+\frac{1}{6}G_{0}\partial_{x}^{2}\eta^{3}G_{0}-G_{0}\eta
G_{2}(\eta)+\frac{1}{2}\partial_{x}^{2}\eta^{2}G_{1}(\eta)$ $\displaystyle
G_{4}(\eta):=\frac{1}{24}G_{0}\partial_{x}^{3}\eta^{4}\partial_{x}-\frac{1}{24}\partial_{x}^{4}\eta^{4}G_{0}+\frac{1}{2}\partial_{x}^{2}\eta^{2}G_{2}(\eta)+\frac{1}{6}G_{0}\partial_{x}^{2}\eta^{3}G_{1}(\eta)-G_{0}\eta
G_{3}(\eta)\,.$

######  Remark A.2.

In order to check that (LABEL:expDiriNeu) coincides with [16, formulae
(39)-(40)] use the identity $D^{2}=-\partial_{x}^{2}$. We point out that
(LABEL:expDiriNeu) coincides with [17, formulae (2.13)-(2.14)] and the
recursion formulae of [50, p. 24].

The ‘unperturbed" linear part of the system (A.3) is associated with the self-
adjoint closed operator

${\cal B}_{0}=\begin{bmatrix}1&-{\mathtt{c}}_{\mathtt{h}}\partial_{x}\\\
{\mathtt{c}}_{\mathtt{h}}\partial_{x}&G_{0}\end{bmatrix}:\mathcal{D}\subset
H^{\sigma,s}_{ev}\times H^{\sigma,s}_{odd}\longrightarrow
H^{\sigma,s}_{ev}\times H^{\sigma,s}_{odd}\,,$ (A.5)

with domain $\mathcal{D}:=H^{\sigma,s+1}_{ev}\times H^{\sigma,s+1}_{odd}$. To
compute the higher-order expansions one needs the following

######  Lemma A.3.

The kernel of the operator ${\cal B}_{0}$ in (A.5) is

$K:=\text{Ker}\;{\cal B}_{0}=\text{span}\,\Big{\\{}\begin{bmatrix}\cos(x)\\\
{\mathtt{c}}_{\mathtt{h}}^{-1}\sin(x)\end{bmatrix}\Big{\\}}$ (A.6a) and its
range $R:=\text{Rn}\;{\cal B}_{0}=K^{\perp_{\mathcal{D}}}$ is given by
$R=R_{0}\oplus R_{1}\oplus R_{\varnothing}$, where $\displaystyle\quad
R_{0}:=\text{span}\,\Big{\\{}\begin{bmatrix}1\\\ 0\end{bmatrix}\Big{\\}}\,,\ \
R_{1}:=\text{span}\,\Big{\\{}\begin{bmatrix}-\cos(x)\\\
{\mathtt{c}}_{\mathtt{h}}\sin(x)\end{bmatrix}\Big{\\}}\,,$ (A.6b)
$\displaystyle R_{\varnothing}:=\overline{\bigoplus_{k=2}^{\infty}R_{k}}\,,\ \
R_{k}:=\text{span}\,\Big{\\{}\begin{bmatrix}\cos(kx)\\\
0\end{bmatrix}\,,\;\begin{bmatrix}0\\\ \sin(kx)\end{bmatrix}\Big{\\}}\,.$

Consequently there exists a unique self-adjoint bounded linear operator ${\cal
B}_{0}^{-1}:R\to R$, given by

$\displaystyle{\cal B}_{0}^{-1}\begin{bmatrix}1\\\
0\end{bmatrix}=\begin{bmatrix}1\\\ 0\end{bmatrix}\,,\quad{\cal
B}_{0}^{-1}\begin{bmatrix}-\cos(x)\\\
{\mathtt{c}}_{\mathtt{h}}\sin(x)\end{bmatrix}=\frac{1}{1+{\mathtt{c}}_{\mathtt{h}}^{2}}\begin{bmatrix}-\cos(x)\\\
{\mathtt{c}}_{\mathtt{h}}\sin(x)\end{bmatrix}\,,$ (A.7) $\displaystyle{\cal
B}_{0}^{-1}\begin{bmatrix}f(x)\\\
g(x)\end{bmatrix}=\big{(}|D|\tanh(\mathtt{h}|D|)+{\mathtt{c}}_{\mathtt{h}}^{2}\partial_{x}^{2}\big{)}^{-1}\begin{bmatrix}|D|\tanh(\mathtt{h}|D|)&{\mathtt{c}}_{\mathtt{h}}\partial_{x}\\\
-{\mathtt{c}}_{\mathtt{h}}\partial_{x}&1\end{bmatrix}\begin{bmatrix}f(x)\\\
g(x)\end{bmatrix}\,,\ \forall\begin{bmatrix}f\\\ g\end{bmatrix}\in
R_{\varnothing}\,,$

such that ${\cal B}_{0}{\cal B}_{0}^{-1}={\cal
B}_{0}^{-1}{\left.\kern-1.2pt{\cal
B}_{0}\vphantom{\big{|}}\right|_{R}}=\mathrm{Id}_{R}$.

###### Proof.

This is the content of [10, Lemma B.1]. Then (A.7) follows by inspection. ∎

From the second-order expansion of the Stokes waves in [10, Appendix B] we
recover

$G_{0}\psi_{1}={\mathtt{c}}_{\mathtt{h}}\sin(x)\,,\quad
G_{1}(\eta_{1})\psi_{1}=\frac{1-{\mathtt{c}}_{\mathtt{h}}^{4}}{{\mathtt{c}}_{\mathtt{h}}(1+{\mathtt{c}}_{\mathtt{h}}^{4})}\sin(2x)\,.$
(A.8)

Third order in $\epsilon$. By plugging the expansion (A.2) in system (A.3) and
discarding quartic terms we find the following linear system (we drop the
dependence w.r.t. $x$)

${\cal B}_{0}\begin{bmatrix}\eta_{3}\\\
\psi_{3}\end{bmatrix}=\begin{bmatrix}c_{2}(\psi_{1})_{x}-(\psi_{1})_{x}(\psi_{2})_{x}-(\eta_{1})_{x}^{2}(\psi_{1})_{x}{\mathtt{c}}_{\mathtt{h}}+(\eta_{1})_{x}(\eta_{2})_{x}{\mathtt{c}}_{\mathtt{h}}^{2}\\\
-c_{2}(\eta_{1})_{x}-G_{1}(\eta_{1})\psi_{2}-G_{1}(\eta_{2})\psi_{1}-G_{2}(\eta_{1})\psi_{1}\end{bmatrix}=:\begin{bmatrix}f_{3}\\\
g_{3}\end{bmatrix}\,.$ (A.9)

In view of (1.9) we have

$\begin{aligned}
&(\psi_{1})_{x}(\psi_{2})_{x}=\frac{\psi_{2}^{[2]}}{{\mathtt{c}}_{\mathtt{h}}}(\cos(x)+\cos(3x))\,,\\\
&(\eta_{1})_{x}^{2}(\psi_{1})_{x}{\mathtt{c}}_{\mathtt{h}}=\frac{1}{4}(\cos(x)-\cos(3x))\,,\end{aligned}\quad{\mathtt{c}}_{\mathtt{h}}^{2}(\eta_{1})_{x}(\eta_{2})_{x}={\mathtt{c}}_{\mathtt{h}}^{2}\eta_{2}^{[2]}(\cos(x)-\cos(3x))\,.$
(A.10)

By means of (LABEL:expDiriNeu) and since

$\tanh(\mathtt{h})={\mathtt{c}}_{\mathtt{h}}^{2}\,,\quad\tanh(2\mathtt{h})=\frac{2{\mathtt{c}}_{\mathtt{h}}^{2}}{1+{\mathtt{c}}_{\mathtt{h}}^{4}}\,,\quad\tanh(3\mathtt{h})=\frac{3{\mathtt{c}}_{\mathtt{h}}^{2}+{\mathtt{c}}_{\mathtt{h}}^{6}}{1+3{\mathtt{c}}_{\mathtt{h}}^{4}}\,,$
(A.11)

whereby

$G_{0}\psi_{2}=\frac{4{\mathtt{c}}_{\mathtt{h}}^{2}}{1+{\mathtt{c}}_{\mathtt{h}}^{4}}\psi_{2}^{[2]}\sin(2x)\,,\quad$
(A.12)

we have, in view of (A.8) too,

$\displaystyle
G_{1}(\eta_{1})\psi_{2}=\psi_{2}^{[2]}\frac{1-{\mathtt{c}}_{\mathtt{h}}^{4}}{1+{\mathtt{c}}_{\mathtt{h}}^{4}}\sin(x)+3\psi_{2}^{[2]}\frac{1-2{\mathtt{c}}_{\mathtt{h}}^{4}+{\mathtt{c}}_{\mathtt{h}}^{8}}{1+4{\mathtt{c}}_{\mathtt{h}}^{4}+3{\mathtt{c}}_{\mathtt{h}}^{8}}\sin(3x)\,,$
(A.13) $\displaystyle
G_{2}(\eta_{1})\psi_{1}=\frac{{\mathtt{c}}_{\mathtt{h}}}{4}\frac{3{\mathtt{c}}_{\mathtt{h}}^{4}-1}{1+{\mathtt{c}}_{\mathtt{h}}^{4}}\sin(x)-\frac{3}{4}{\mathtt{c}}_{\mathtt{h}}\frac{{\mathtt{c}}_{\mathtt{h}}^{8}-4{\mathtt{c}}_{\mathtt{h}}^{4}+3}{1+4{\mathtt{c}}_{\mathtt{h}}^{4}+3{\mathtt{c}}_{\mathtt{h}}^{8}}\sin(3x)\,,$
$\displaystyle
G_{1}(\eta_{2})\psi_{1}=\frac{1}{{\mathtt{c}}_{\mathtt{h}}}\big{(}\eta_{2}^{[0]}(1-{\mathtt{c}}_{\mathtt{h}}^{4})+\frac{1}{2}\eta_{2}^{[2]}(1+{\mathtt{c}}_{\mathtt{h}}^{4})\big{)}\sin(x)+\frac{3}{2{\mathtt{c}}_{\mathtt{h}}}\eta_{2}^{[2]}\frac{1-{\mathtt{c}}_{\mathtt{h}}^{8}}{1+3{\mathtt{c}}_{\mathtt{h}}^{4}}\sin(3x)\,.$

By (A.1b), (A.10), (A.13) and (A.1a) the right-hand side of system (A.9) is
given by

$\begin{bmatrix}f_{3}(x)\\\\[5.69054pt]
g_{3}(x)\end{bmatrix}=\begin{bmatrix}f_{3}^{[1]}\cos(x)+f_{3}^{[3]}\cos(3x)\\\\[5.69054pt]
g_{3}^{[1]}\sin(x)+g_{3}^{[3]}\sin(3x)\end{bmatrix}\,,$ (A.14)

with

$\displaystyle
f_{3}^{[1]}:=\frac{-2{\mathtt{c}}_{\mathtt{h}}^{12}+3{\mathtt{c}}_{\mathtt{h}}^{8}+3}{16{\mathtt{c}}_{\mathtt{h}}^{8}}\,,\quad
f_{3}^{[3]}:=\frac{3{\mathtt{c}}_{\mathtt{h}}^{8}-6{\mathtt{c}}_{\mathtt{h}}^{4}-3}{8{\mathtt{c}}_{\mathtt{h}}^{8}}\,,$
(A.15) $\displaystyle
g_{3}^{[1]}:=\frac{2{\mathtt{c}}_{\mathtt{h}}^{12}-3{\mathtt{c}}_{\mathtt{h}}^{8}-3}{16{\mathtt{c}}_{\mathtt{h}}^{7}}\,,\quad
g_{3}^{[3]}:=\frac{-6{\mathtt{c}}_{\mathtt{h}}^{8}+15{\mathtt{c}}_{\mathtt{h}}^{4}-9}{4{\mathtt{c}}_{\mathtt{h}}^{7}(1+3{\mathtt{c}}_{\mathtt{h}}^{4})}\,.$

We recall that the term $c_{2}$ in (A.1b) is determined (cfr. [10, Appendix
B]) to make the vector $(f(x),g(x))$ in (A.14) orthogonal to the kernel of the
operator ${\cal B}_{0}$, therefore ensuring the existence of a solution

$\begin{bmatrix}\eta_{3}(x)\\\
\psi_{3}(x)\end{bmatrix}:=\begin{bmatrix}\eta_{3}^{[1]}\cos(x)+\eta_{3}^{[3]}\cos(3x)\\\
\psi_{3}^{[1]}\sin(x)+\psi_{3}^{[3]}\sin(3x)\end{bmatrix}:={\cal
B}_{0}^{-1}\begin{bmatrix}f_{3}(x)\\\ g_{3}(x)\end{bmatrix}\,.$

By Lemma A.3 and (A.11) we have

$\displaystyle\begin{bmatrix}\eta_{3}^{[3]}\cos(3x)\\\
\psi_{3}^{[3]}\sin(3x)\end{bmatrix}={\cal
B}_{0}^{-1}\begin{bmatrix}\beta\cos(3x)\\\ \delta\sin(3x)\end{bmatrix}$
$\displaystyle=-\frac{1+3{\mathtt{c}}_{\mathtt{h}}^{4}}{24{\mathtt{c}}_{\mathtt{h}}^{6}}\begin{bmatrix}|D|\tanh(\mathtt{h}|D|)&{\mathtt{c}}_{\mathtt{h}}\partial_{x}\\\
-{\mathtt{c}}_{\mathtt{h}}\partial_{x}&1\end{bmatrix}\begin{bmatrix}f_{3}^{[3]}\cos(3x)\\\
g_{3}^{[3]}\sin(3x)\end{bmatrix}$ (A.16)
$\displaystyle=-\frac{1+3{\mathtt{c}}_{\mathtt{h}}^{4}}{24{\mathtt{c}}_{\mathtt{h}}^{6}}\begin{bmatrix}3\big{(}\frac{3{\mathtt{c}}_{\mathtt{h}}^{2}+{\mathtt{c}}_{\mathtt{h}}^{6}}{1+3{\mathtt{c}}_{\mathtt{h}}^{4}}f_{3}^{[3]}+{\mathtt{c}}_{\mathtt{h}}g_{3}^{[3]}\big{)}\cos(3x)\\\
(3{\mathtt{c}}_{\mathtt{h}}f_{3}^{[3]}+g_{3}^{[3]})\sin(3x)\end{bmatrix}\,,$
(A.17)

and $\begin{bmatrix}\eta_{3}^{[1]}\cos(x)\\\
\psi_{3}^{[1]}\sin(x)\end{bmatrix}={\cal
B}_{0}^{-1}\begin{bmatrix}f_{3}^{[1]}\cos(x)\\\
g_{3}^{[1]}\sin(x)\end{bmatrix}=\frac{1}{1+{\mathtt{c}}_{\mathtt{h}}^{2}}\begin{bmatrix}f_{3}^{[1]}\cos(x)\\\
g_{3}^{[1]}\sin(x)\end{bmatrix}$. The coefficients in (A.1c) follow.
Fourth order in $\epsilon$. By plugging the expansion (A.2) in system (A.3)
and discarding quintic terms we find the linear system

${\cal B}_{0}\begin{bmatrix}\eta_{4}\\\
\psi_{4}\end{bmatrix}=\begin{bmatrix}f_{4}\\\ g_{4}\end{bmatrix}\,,$ (A.18)

with

$\displaystyle f_{4}$
$\displaystyle:=c_{3}(\psi_{1})_{x}+c_{2}(\psi_{2})_{x}-(\psi_{1})_{x}(\psi_{3})_{x}-\frac{1}{2}(\psi_{2})_{x}^{2}+{\mathtt{c}}_{\mathtt{h}}(\eta_{1})_{x}^{2}\big{(}c_{2}-(\psi_{2})_{x}\big{)}$
(A.19a) $\displaystyle\ \
+(\eta_{1})_{x}(\eta_{3})_{x}{\mathtt{c}}_{\mathtt{h}}^{2}+\frac{1}{2}(\eta_{2})_{x}^{2}{\mathtt{c}}_{\mathtt{h}}^{2}+\frac{1}{2}(\eta_{1})_{x}^{2}(\psi_{1})_{x}^{2}-\frac{1}{2}{\mathtt{c}}_{\mathtt{h}}^{2}(\eta_{1})_{x}^{4}-2{\mathtt{c}}_{\mathtt{h}}(\eta_{1})_{x}(\eta_{2})_{x}(\psi_{1})_{x}\,,$
$\displaystyle g_{4}$
$\displaystyle:=-c_{3}(\eta_{1})_{x}-c_{2}(\eta_{2})_{x}-G_{1}(\eta_{1})\psi_{3}-G_{1}(\eta_{2})\psi_{2}-G_{1}(\eta_{3})\psi_{1}$
(A.19b) $\displaystyle\ \
-G_{2}^{\prime}(\eta_{1})[\eta_{2}]\psi_{1}-G_{2}(\eta_{1})\psi_{2}-G_{3}(\eta_{1})\psi_{1},$

where, in view of (LABEL:expDiriNeu),

$G_{2}^{\prime}(\eta)[\hat{\eta}]:=-G_{0}\partial_{x}\eta\hat{\eta}\partial_{x}+\partial_{x}^{2}\eta\hat{\eta}G_{0}-G_{0}\hat{\eta}G_{1}(\eta)-G_{0}\eta
G_{1}(\hat{\eta})\,.$ (A.20)

Let us inspect the terms in (A.19a). In view of (1.9) we have

$\displaystyle
c_{3}(\psi_{1})_{x}={\mathtt{c}}_{\mathtt{h}}^{-1}c_{3}\cos(x)\,,\quad
c_{3}(\eta_{1})_{x}=-c_{3}\sin(x)\,,\quad
c_{2}(\psi_{2})_{x}=2c_{2}\psi_{2}^{[2]}\cos(2x)\,,$ (A.21) $\displaystyle
c_{2}(\eta_{2})_{x}=-2c_{2}\eta_{2}^{[2]}\sin(2x)\,,\quad\frac{1}{2}(\psi_{2})_{x}^{2}=(\psi_{2}^{[2]})^{2}+(\psi_{2}^{[2]})^{2}\cos(4x)\,,$
$\displaystyle(\psi_{1})_{x}(\psi_{3})_{x}=\frac{1}{2}{\mathtt{c}}_{\mathtt{h}}^{-1}\psi_{3}^{[1]}+\frac{1}{2}{\mathtt{c}}_{\mathtt{h}}^{-1}\big{(}\psi_{3}^{[1]}+3\psi_{3}^{[3]}\big{)}\cos(2x)+\frac{3}{2}{\mathtt{c}}_{\mathtt{h}}^{-1}\psi_{3}^{[3]}\cos(4x)\,,$
$\displaystyle{\mathtt{c}}_{\mathtt{h}}(\eta_{1})^{2}_{x}(\psi_{2})_{x}=-\frac{1}{2}{\mathtt{c}}_{\mathtt{h}}\psi_{2}^{[2]}+{\mathtt{c}}_{\mathtt{h}}\psi_{2}^{[2]}\cos(2x)-\frac{1}{2}{\mathtt{c}}_{\mathtt{h}}\psi_{2}^{[2]}\cos(4x)\,,$
$\displaystyle\frac{1}{2}{\mathtt{c}}_{\mathtt{h}}^{2}(\eta_{2})_{x}^{2}={\mathtt{c}}_{\mathtt{h}}^{2}(\eta_{2}^{[2]})^{2}-{\mathtt{c}}_{\mathtt{h}}^{2}(\eta_{2}^{[2]})^{2}\cos(4x)\,,\quad{\mathtt{c}}_{\mathtt{h}}c_{2}(\eta_{1})_{x}^{2}=\frac{1}{2}{\mathtt{c}}_{\mathtt{h}}c_{2}\big{(}1-\cos(2x)\big{)}\,,$
$\displaystyle{\mathtt{c}}_{\mathtt{h}}^{2}(\eta_{1})_{x}(\eta_{3})_{x}=\frac{1}{2}{\mathtt{c}}_{\mathtt{h}}^{2}\eta_{3}^{[1]}+\frac{1}{2}{\mathtt{c}}_{\mathtt{h}}^{2}\big{(}-\eta_{3}^{[1]}+3\eta_{3}^{[3]}\big{)}\cos(2x)-\frac{3}{2}{\mathtt{c}}_{\mathtt{h}}^{2}\eta_{3}^{[3]}\cos(4x)\,,$
$\displaystyle\frac{1}{2}(\eta_{1})^{2}_{x}(\psi_{1})^{2}_{x}=\frac{{\mathtt{c}}_{\mathtt{h}}^{-2}}{16}\big{(}1-\cos(4x)\big{)}\,,\quad\frac{1}{2}{\mathtt{c}}_{\mathtt{h}}^{2}(\eta_{1})_{x}^{4}=\frac{{\mathtt{c}}_{\mathtt{h}}^{2}}{16}\big{(}3-4\cos(2x)+\cos(4x)\big{)}\,,$
$\displaystyle
2{\mathtt{c}}_{\mathtt{h}}(\eta_{1})_{x}(\eta_{2})_{x}(\psi_{1})_{x}=\eta_{2}^{[2]}-\eta_{2}^{[2]}\cos(4x)\,.$

Let us inspect the terms in (A.19b). In view of (LABEL:expDiriNeu), (A.2),
(A.11), whereby

$G_{0}\psi_{3}={\mathtt{c}}_{\mathtt{h}}^{2}\psi_{3}^{[1]}\sin(x)+3{\mathtt{c}}_{\mathtt{h}}^{2}\frac{3+{\mathtt{c}}_{\mathtt{h}}^{4}}{1+3{\mathtt{c}}_{\mathtt{h}}^{4}}\psi_{3}^{[3]}\sin(3x)\,,$
(A.22)

and since

$\tanh(4\mathtt{h})=\frac{4{\mathtt{c}}_{\mathtt{h}}^{2}+4{\mathtt{c}}_{\mathtt{h}}^{6}}{1+6{\mathtt{c}}_{\mathtt{h}}^{4}+{\mathtt{c}}_{\mathtt{h}}^{8}}\,,$
(A.23)

we have

$\displaystyle G_{1}(\eta_{1})\psi_{3}$
$\displaystyle=\Big{(}\frac{1-{\mathtt{c}}_{\mathtt{h}}^{4}}{1+{\mathtt{c}}_{\mathtt{h}}^{4}}\psi_{3}^{[1]}+\frac{3(1-{\mathtt{c}}_{\mathtt{h}}^{4})^{2}\psi_{3}^{[3]}}{(1+{\mathtt{c}}_{\mathtt{h}}^{4})(1+3{\mathtt{c}}_{\mathtt{h}}^{4})}\Big{)}\sin(2x)+6\frac{(1-{\mathtt{c}}_{\mathtt{h}}^{4})^{3}\psi_{3}^{[3]}\sin(4x)}{(1+3{\mathtt{c}}_{\mathtt{h}}^{4})(1+6{\mathtt{c}}_{\mathtt{h}}^{4}+{\mathtt{c}}_{\mathtt{h}}^{8})}\,,$
(A.24) $\displaystyle G_{1}(\eta_{2})\psi_{2}$
$\displaystyle=4\frac{(1-{\mathtt{c}}_{\mathtt{h}}^{4})^{2}}{(1+{\mathtt{c}}_{\mathtt{h}}^{4})^{2}}\psi_{2}^{[2]}\eta_{2}^{[0]}\sin(2x)+4\frac{(1-{\mathtt{c}}_{\mathtt{h}}^{4})^{2}\psi_{2}^{[2]}\eta_{2}^{[2]}}{1+6{\mathtt{c}}_{\mathtt{h}}^{4}+{\mathtt{c}}_{\mathtt{h}}^{8}}\sin(4x)\,,$
$\displaystyle G_{1}(\eta_{3})\psi_{1}$
$\displaystyle=\Big{(}\frac{1-{\mathtt{c}}_{\mathtt{h}}^{4}}{{\mathtt{c}}_{\mathtt{h}}(1+{\mathtt{c}}_{\mathtt{h}}^{4})}\eta_{3}^{[1]}+\frac{1+3{\mathtt{c}}_{\mathtt{h}}^{4}}{{\mathtt{c}}_{\mathtt{h}}(1+{\mathtt{c}}_{\mathtt{h}}^{4})}\eta_{3}^{[3]}\Big{)}\sin(2x)+2\frac{(1-{\mathtt{c}}_{\mathtt{h}}^{4})(1+3{\mathtt{c}}_{\mathtt{h}}^{4})}{{\mathtt{c}}_{\mathtt{h}}(1+6{\mathtt{c}}_{\mathtt{h}}^{4}+{\mathtt{c}}_{\mathtt{h}}^{8})}\eta_{3}^{[3]}\sin(4x)\,.$

In view of (1.9), (A.20), (A.11), (A.8), (A.13) and (A.23), we have

$\displaystyle
G_{2}^{\prime}(\eta_{1})[\eta_{2}]\psi_{1}=\Big{(}\frac{8{\mathtt{c}}_{\mathtt{h}}^{5}\eta_{2}^{[2]}}{(1+3{\mathtt{c}}_{\mathtt{h}}^{4})(1+{\mathtt{c}}_{\mathtt{h}}^{4})}-\frac{4{\mathtt{c}}_{\mathtt{h}}(1-{\mathtt{c}}_{\mathtt{h}}^{4})}{(1+{\mathtt{c}}_{\mathtt{h}}^{4})^{2}}\eta_{2}^{[0]}\Big{)}\sin(2x)-\frac{8{\mathtt{c}}_{\mathtt{h}}(1-{\mathtt{c}}_{\mathtt{h}}^{8})\eta_{2}^{[2]}}{(1+3{\mathtt{c}}_{\mathtt{h}}^{4})(1+6{\mathtt{c}}_{\mathtt{h}}^{4}+{\mathtt{c}}_{\mathtt{h}}^{8})}\sin(4x)\,.$
(A.25)

By (1.9), (LABEL:expDiriNeu), (A.13), (A.11) and (A.23) we have

$G_{2}(\eta_{1})\psi_{2}=-\frac{8{\mathtt{c}}_{\mathtt{h}}^{2}(1-{\mathtt{c}}_{\mathtt{h}}^{4})\psi_{2}^{[2]}}{(1+{\mathtt{c}}_{\mathtt{h}}^{4})^{2}(1+3{\mathtt{c}}_{\mathtt{h}}^{4})}\sin(2x)-\frac{16{\mathtt{c}}_{\mathtt{h}}^{2}(1-{\mathtt{c}}_{\mathtt{h}}^{4})^{2}\psi_{2}^{[2]}}{(1+{\mathtt{c}}_{\mathtt{h}}^{4})(1+3{\mathtt{c}}_{\mathtt{h}}^{4})(1+6{\mathtt{c}}_{\mathtt{h}}^{4}+{\mathtt{c}}_{\mathtt{h}}^{8})}\sin(4x)\,.$
(A.26)

Finally, by (1.9), (LABEL:expDiriNeu), (A.13), (A.8), (A.11) and (A.23), we
have

$G_{3}(\eta_{1})\psi_{1}=\frac{-1+14{\mathtt{c}}_{\mathtt{h}}^{4}-9{\mathtt{c}}_{\mathtt{h}}^{8}}{3{\mathtt{c}}_{\mathtt{h}}(1+{\mathtt{c}}_{\mathtt{h}}^{4})^{2}(1+3{\mathtt{c}}_{\mathtt{h}}^{4})}\sin(2x)+2\frac{-1+15{\mathtt{c}}_{\mathtt{h}}^{4}-23{\mathtt{c}}_{\mathtt{h}}^{8}+9{\mathtt{c}}_{\mathtt{h}}^{12}}{3{\mathtt{c}}_{\mathtt{h}}(1+{\mathtt{c}}_{\mathtt{h}}^{4})(1+3{\mathtt{c}}_{\mathtt{h}}^{4})(1+6{\mathtt{c}}_{\mathtt{h}}^{4}+{\mathtt{c}}_{\mathtt{h}}^{8})}\sin(4x)\,.$
(A.27)

By (A.19), (A.21), (A.24), (A.25), (A.26), (A.27) and (A.1a)-(A.1c) system
(A.18) reads as

$\begin{bmatrix}f_{4}(x)\\\\[5.69054pt]
g_{4}(x)\end{bmatrix}=\begin{bmatrix}f_{4}^{[0]}+{\mathtt{c}}_{\mathtt{h}}^{-1}c_{3}\cos(x)+f_{4}^{[2]}\cos(2x)+f_{4}^{[4]}\cos(4x)\\\\[5.69054pt]
c_{3}\sin(x)+g_{4}^{[2]}\sin(2x)+g_{4}^{[4]}\sin(4x)\end{bmatrix}\,,$ (A.28)

with

$\displaystyle f_{4}^{[0]}$
$\displaystyle=\frac{-4{\mathtt{c}}_{\mathtt{h}}^{20}-4{\mathtt{c}}_{\mathtt{h}}^{18}+17{\mathtt{c}}_{\mathtt{h}}^{16}+6{\mathtt{c}}_{\mathtt{h}}^{14}-48{\mathtt{c}}_{\mathtt{h}}^{8}+6{\mathtt{c}}_{\mathtt{h}}^{6}+36{\mathtt{c}}_{\mathtt{h}}^{4}-9}{64{\mathtt{c}}_{\mathtt{h}}^{14}}\,,\
\
f_{4}^{[4]}=\frac{7{\mathtt{c}}_{\mathtt{h}}^{16}-48{\mathtt{c}}_{\mathtt{h}}^{12}+126{\mathtt{c}}_{\mathtt{h}}^{8}-168{\mathtt{c}}_{\mathtt{h}}^{4}-45}{128{\mathtt{c}}_{\mathtt{h}}^{14}}\,,$
$\displaystyle f_{4}^{[2]}$
$\displaystyle=\frac{4{\mathtt{c}}_{\mathtt{h}}^{22}+12{\mathtt{c}}_{\mathtt{h}}^{20}-27{\mathtt{c}}_{\mathtt{h}}^{18}-31{\mathtt{c}}_{\mathtt{h}}^{16}+78{\mathtt{c}}_{\mathtt{h}}^{14}+66{\mathtt{c}}_{\mathtt{h}}^{12}-72{\mathtt{c}}_{\mathtt{h}}^{10}-84{\mathtt{c}}_{\mathtt{h}}^{8}+6{\mathtt{c}}_{\mathtt{h}}^{6}-6{\mathtt{c}}_{\mathtt{h}}^{4}+27{\mathtt{c}}_{\mathtt{h}}^{2}+27}{128{\mathtt{c}}_{\mathtt{h}}^{14}(1+{\mathtt{c}}_{\mathtt{h}}^{2})}\,,$
$\displaystyle g_{4}^{[2]}$
$\displaystyle=\frac{3{\mathtt{c}}_{\mathtt{h}}^{16}-12{\mathtt{c}}_{\mathtt{h}}^{14}-39{\mathtt{c}}_{\mathtt{h}}^{12}+18{\mathtt{c}}_{\mathtt{h}}^{10}+139{\mathtt{c}}_{\mathtt{h}}^{8}-225{\mathtt{c}}_{\mathtt{h}}^{4}+18{\mathtt{c}}_{\mathtt{h}}^{2}+18}{48{\mathtt{c}}_{\mathtt{h}}^{9}(1+{\mathtt{c}}_{\mathtt{h}}^{4})}\,,$
(A.29) $\displaystyle g_{4}^{[4]}$
$\displaystyle=\frac{-21{\mathtt{c}}_{\mathtt{h}}^{20}+61{\mathtt{c}}_{\mathtt{h}}^{16}+14{\mathtt{c}}_{\mathtt{h}}^{12}-198{\mathtt{c}}_{\mathtt{h}}^{8}+279{\mathtt{c}}_{\mathtt{h}}^{4}-135}{48{\mathtt{c}}_{\mathtt{h}}^{13}({\mathtt{c}}_{\mathtt{h}}^{8}+6{\mathtt{c}}_{\mathtt{h}}^{4}+1)}\,.$

As a consequence of (A.28) we have

$\Big{(}\begin{bmatrix}f_{4}(x)\\\
g_{4}(x)\end{bmatrix}\,,\,\begin{bmatrix}\cos(x)\\\
{\mathtt{c}}_{\mathtt{h}}^{-1}\sin(x)\end{bmatrix}\Big{)}={\mathtt{c}}_{\mathtt{h}}^{-1}c_{3}\,.$

By Lemma A.3, with $c_{3}=0$ as stated in (1.9c), one ensures that the system
(A.18) is solved by

$\begin{bmatrix}\eta_{4}(x)\\\ \psi_{4}(x)\end{bmatrix}={\cal
B}_{0}^{-1}\begin{bmatrix}f_{4}(x)\\\ g_{4}(x)\end{bmatrix}\,,$ (A.30)

where, in view of (A.7) and (A.28),

$\begin{bmatrix}\eta_{4}(x)\\\\[5.69054pt]
\psi_{4}(x)\end{bmatrix}=\begin{bmatrix}\eta_{4}^{[0]}+\eta_{4}^{[2]}\cos(2x)+\eta_{4}^{[4]}\cos(4x)\\\\[5.69054pt]
\psi_{4}^{[2]}\sin(2x)+\psi_{4}^{[4]}\sin(4x)\end{bmatrix}\,,$ (A.31)

with, in view of (A.11) and (A.23) too,

$\displaystyle\eta_{4}^{[0]}:=f_{4}^{[0]}\,,\quad\begin{bmatrix}\eta_{4}^{[2]}\\\\[5.69054pt]
\psi_{4}^{[2]}\end{bmatrix}=-\frac{1+{\mathtt{c}}_{\mathtt{h}}^{4}}{4{\mathtt{c}}_{\mathtt{h}}^{6}}\begin{bmatrix}\frac{4{\mathtt{c}}_{\mathtt{h}}^{2}}{1+{\mathtt{c}}_{\mathtt{h}}^{4}}&2{\mathtt{c}}_{\mathtt{h}}\\\
2{\mathtt{c}}_{\mathtt{h}}&1\end{bmatrix}\begin{bmatrix}f_{4}^{[2]}\\\\[5.69054pt]
g_{4}^{[2]}\end{bmatrix}=-\frac{1+{\mathtt{c}}_{\mathtt{h}}^{4}}{4{\mathtt{c}}_{\mathtt{h}}^{6}}\begin{bmatrix}\frac{4{\mathtt{c}}_{\mathtt{h}}^{2}}{1+{\mathtt{c}}_{\mathtt{h}}^{4}}f_{4}^{[2]}+2{\mathtt{c}}_{\mathtt{h}}g_{4}^{[2]}\\\\[5.69054pt]
2{\mathtt{c}}_{\mathtt{h}}f_{4}^{[2]}+g_{4}^{[2]}\end{bmatrix}\,,$ (A.32)
$\displaystyle\begin{bmatrix}\eta_{4}^{[4]}\\\\[5.69054pt]
\psi_{4}^{[4]}\end{bmatrix}=-\frac{1+6{\mathtt{c}}_{\mathtt{h}}^{4}+{\mathtt{c}}_{\mathtt{h}}^{8}}{16{\mathtt{c}}_{\mathtt{h}}^{6}(5+{\mathtt{c}}_{\mathtt{h}}^{4})}\begin{bmatrix}16{\mathtt{c}}_{\mathtt{h}}^{2}\frac{1+{\mathtt{c}}_{\mathtt{h}}^{4}}{1+6{\mathtt{c}}_{\mathtt{h}}^{4}+{\mathtt{c}}_{\mathtt{h}}^{8}}&4{\mathtt{c}}_{\mathtt{h}}\\\
4{\mathtt{c}}_{\mathtt{h}}&1\end{bmatrix}\begin{bmatrix}f_{4}^{[4]}\\\\[5.69054pt]
g_{4}^{[4]}\end{bmatrix}=-\frac{1+6{\mathtt{c}}_{\mathtt{h}}^{4}+{\mathtt{c}}_{\mathtt{h}}^{8}}{16{\mathtt{c}}_{\mathtt{h}}^{6}(5+{\mathtt{c}}_{\mathtt{h}}^{4})}\begin{bmatrix}16{\mathtt{c}}_{\mathtt{h}}^{2}\frac{1+{\mathtt{c}}_{\mathtt{h}}^{4}}{1+6{\mathtt{c}}_{\mathtt{h}}^{4}+{\mathtt{c}}_{\mathtt{h}}^{8}}f_{4}^{[4]}+4{\mathtt{c}}_{\mathtt{h}}g_{4}^{[4]}\\\\[5.69054pt]
4{\mathtt{c}}_{\mathtt{h}}f_{4}^{[4]}+g_{4}^{[4]}\end{bmatrix}\,.$

By (A.31) we conclude the proof of (1.9a)-(LABEL:psiexp) and, in view of
(A.32) and (A.29), of (A.1d). The proof of (1.9), (A.1) needs only the
computation of $c_{4}$.
Fifth order in $\epsilon$. By plugging the expansion (A.2) in system (A.3) and
discarding sextic terms we find the linear system

${\cal B}_{0}\begin{bmatrix}\eta_{5}\\\
\psi_{5}\end{bmatrix}=\begin{bmatrix}{\mathtt{f}}_{5}\\\
{\mathtt{g}}_{5}\end{bmatrix}+c_{4}\begin{bmatrix}(\psi_{1})_{x}\\\
-(\eta_{1})_{x}\end{bmatrix}\,,$ (A.33)

with

$\displaystyle{\mathtt{f}}_{5}$
$\displaystyle:=c_{2}(\psi_{3})_{x}-(\psi_{1})_{x}(\psi_{4})_{x}-(\psi_{2})_{x}(\psi_{3})_{x}+(\eta_{1})_{x}^{2}\big{(}-{\mathtt{c}}_{\mathtt{h}}(\psi_{3})_{x}-c_{2}(\psi_{1})_{x}+(\psi_{1})_{x}(\psi_{2})_{x}\big{)}$
(A.34a) $\displaystyle\ \
+(\eta_{1})_{x}(\eta_{2})_{x}\big{(}(\psi_{1})_{x}^{2}+2{\mathtt{c}}_{\mathtt{h}}c_{2}-2{\mathtt{c}}_{\mathtt{h}}(\psi_{2})_{x}\big{)}-{\mathtt{c}}_{\mathtt{h}}(\psi_{1})_{x}\big{(}(\eta_{2})_{x}^{2}-(\eta_{1})_{x}^{4}+2(\eta_{1})_{x}(\eta_{3})_{x}\big{)}$
$\displaystyle\ \
+{\mathtt{c}}_{\mathtt{h}}^{2}\big{(}(\eta_{1})_{x}(\eta_{4})_{x}+(\eta_{2})_{x}(\eta_{3})_{x}-2(\eta_{1})_{x}^{3}(\eta_{2})_{x}\big{)}\,,$
$\displaystyle{\mathtt{g}}_{5}$
$\displaystyle:=-c_{2}(\eta_{3})_{x}-G_{1}(\eta_{1})\psi_{4}-G_{1}(\eta_{2})\psi_{3}-G_{1}(\eta_{3})\psi_{2}-G_{1}(\eta_{4})\psi_{1}-G_{2}(\eta_{1})\psi_{3}$
(A.34b) $\displaystyle\ \
-G_{2}(\eta_{2})\psi_{1}-G_{2}^{\prime}(\eta_{1})[\eta_{2},\psi_{2}]-G_{2}^{\prime}(\eta_{1})[\eta_{3},\psi_{1}]-G_{3}(\eta_{1})\psi_{2}-G_{3}^{\prime}(\eta_{1})[\eta_{2},\psi_{1}]-G_{4}(\eta_{1})\psi_{1}\,,$

where $G_{2}^{\prime}$ is in (A.20) and, by (LABEL:expDiriNeu),

$G_{3}^{\prime}(\eta)[\hat{\eta}]=\frac{1}{2}\partial_{x}^{3}\eta^{2}\hat{\eta}\partial_{x}+\frac{1}{2}G_{0}\partial_{x}^{2}\eta^{2}\hat{\eta}G_{0}-G_{0}\hat{\eta}G_{2}(\eta)-G_{0}\eta
G_{2}^{\prime}(\eta)[\hat{\eta}]+\partial_{x}^{2}\eta\hat{\eta}G_{1}(\eta)+\frac{1}{2}\partial_{x}^{2}\eta^{2}G_{1}(\hat{\eta})\,.$
(A.35)

The term $c_{4}$ is obtained by imposing the expression on the right-hand side
of (A.33) to be orthogonal to the kernel of the operator ${\cal B}_{0}$ in
(A.6a), obtaining, in view of (1.14),

$c_{4}=-\dfrac{\big{(}{\mathtt{f}}_{5},\eta_{1}\big{)}+\big{(}{\mathtt{g}}_{5},\psi_{1}\big{)}}{\big{(}(\psi_{1})_{x},\eta_{1}\big{)}-\big{(}(\eta_{1})_{x},\psi_{1}\big{)}}=-{\mathtt{c}}_{\mathtt{h}}\Big{(}\big{(}{\mathtt{f}}_{5},\eta_{1}\big{)}+\big{(}{\mathtt{g}}_{5},\psi_{1}\big{)}\Big{)}\,.$
(A.36)

By (A.34) and (1.9) we find that $c_{4}$ in (A.36) has the explicit expression
in (A.1e).

######  Remark A.4.

Expansion (1.9), (A.1) coincides with that in [23, formulae (12)-(14)],
provided one rescales properly their amplitude $\varepsilon_{\rm
Fen}=\epsilon+f(\mathtt{h})\epsilon^{3}$ with a suitable $f(\mathtt{h})$,
translates their bottom to
$d:=\mathtt{h}+\epsilon^{2}\eta_{2}^{[0]}+\epsilon^{4}\eta_{4}^{[0]}+\mathcal{O}(\epsilon^{5})$
(in [23] the water surface $\eta$ has zero average) and removes from the
velocity potential a shear term $-\overline{u}x$ (which corresponds to a
Galilean reference frame).

###  A.2 Expansions of $a_{\epsilon}(x)$ and $p_{\epsilon}(x)$

In this section we compute the fourth order expansion of the functions
$a_{\epsilon}(x)$ and $p_{\epsilon}(x)$ and of the constant
$\mathtt{f}_{\epsilon}$ in (1.12).

By [10, formula (2.18)]) the functions $a_{\epsilon}(x)$ and $p_{\epsilon}(x)$
in (1.12) are given by

$\displaystyle{\mathtt{c}}_{\mathtt{h}}+p_{\epsilon}(x):$
$\displaystyle=\displaystyle{\frac{c_{\epsilon}-V(x+\mathfrak{p}(x))}{1+\mathfrak{p}_{x}(x)}}\,,$
(A.37) $\displaystyle 1+a_{\epsilon}(x):$
$\displaystyle=\displaystyle{\frac{1+(V(x+\mathfrak{p}(x))-c_{\epsilon})B_{x}(x+\mathfrak{p}(x))}{1+\mathfrak{p}_{x}(x)}}=\dfrac{1}{1+\mathfrak{p}_{x}(x)}-({\mathtt{c}}_{\mathtt{h}}+p_{\epsilon}(x))B_{x}(x+\mathfrak{p}(x))\,,$

and by [10, formula (2.14)]) the constant $\mathtt{f}_{\epsilon}$ is

$\mathtt{f}_{\epsilon}=\frac{1}{2\pi}\int_{\mathbb{T}}\eta_{\epsilon}(x+\mathfrak{p}(x))\mathrm{d}x\,.$
(A.38)

Here the function $\mathfrak{p}(x)$ is determined (cfr. [10, formula (2.14)],
[2, formula (A.15)]) by the fixed point equation

$\mathfrak{p}(x)=\frac{\mathcal{H}}{\tanh\big{(}(\mathtt{h}+\mathtt{f}_{\epsilon})|D|\big{)}}[\eta_{\epsilon}(x+\mathfrak{p}(x))]\,,$
(A.39)

where ${\mathcal{H}}:=-\mathrm{i}\,\operatorname*{sgn}(D)$ is the Hilbert
transform222$\operatorname*{sgn}(D)$ is the Fourier multiplier operator with
symbol $\operatorname*{sgn}(k):=1\ \forall k>0$, $\operatorname*{sgn}(0):=0$,
$\operatorname*{sgn}(k):=-1\ \forall k<0$., whereas the functions $V$ and $B$
(cfr. [10, formula (2.10)]) are given by

$V(x):=-B(x)\eta^{\prime}_{\epsilon}(x)+\psi^{\prime}_{\epsilon}(x)\,,\quad
B(x):=\frac{\psi^{\prime}_{\epsilon}(x)-c_{\epsilon}}{1+(\eta^{\prime}_{\epsilon}(x))^{2}}\eta^{\prime}_{\epsilon}(x)\,.$
(A.40)

To provide the Taylor expansion of the functions in (A.37) we need some
preparatory results.

######  Lemma A.5.

(Fourth order expansion of $\mathfrak{p}(x)$ and $\mathtt{f}_{\epsilon}$) The
function $\mathfrak{p}(x)$ in (A.39) admits the following Taylor expansion

$\displaystyle\mathfrak{p}(x)$
$\displaystyle=\epsilon{\mathtt{c}}_{\mathtt{h}}^{-2}\sin(x)+\epsilon^{2}\mathfrak{p}_{2}^{[2]}\sin(2x)+\epsilon^{3}\big{(}\mathfrak{p}_{3}^{[1]}\sin(x)+\mathfrak{p}_{3}^{[3]}\sin(3x)\big{)}$
(A.41)
$\displaystyle\quad+\epsilon^{4}\big{(}\mathfrak{p}_{4}^{[2]}\sin(2x)+\mathfrak{p}_{4}^{[4]}\sin(4x)\big{)}+\mathcal{O}(\epsilon^{5})$

with coefficients

$\displaystyle\mathfrak{p}_{2}^{[2]}:=\frac{3+4{\mathtt{c}}_{\mathtt{h}}^{4}+{\mathtt{c}}_{\mathtt{h}}^{8}}{8{\mathtt{c}}_{\mathtt{h}}^{8}}\,,$
(A.42)
$\displaystyle\mathfrak{p}_{3}^{[1]}:=\frac{4{\mathtt{c}}_{\mathtt{h}}^{14}+2{\mathtt{c}}_{\mathtt{h}}^{12}-17{\mathtt{c}}_{\mathtt{h}}^{10}-14{\mathtt{c}}_{\mathtt{h}}^{8}+10{\mathtt{c}}_{\mathtt{h}}^{6}+10{\mathtt{c}}_{\mathtt{h}}^{4}-15{\mathtt{c}}_{\mathtt{h}}^{2}-12}{16{\mathtt{c}}_{\mathtt{h}}^{10}({\mathtt{c}}_{\mathtt{h}}^{2}+1)}\,,$
(A.43)
$\displaystyle\mathfrak{p}_{3}^{[3]}:=\frac{9+41{\mathtt{c}}_{\mathtt{h}}^{4}+43{\mathtt{c}}_{\mathtt{h}}^{8}+3{\mathtt{c}}_{\mathtt{h}}^{12}}{64{\mathtt{c}}_{\mathtt{h}}^{14}}\,,$
(A.44)
$\displaystyle\mathfrak{p}_{4}^{[2]}:=-\frac{1}{256{\mathtt{c}}_{\mathtt{h}}^{20}({\mathtt{c}}_{\mathtt{h}}^{2}+1)}\big{(}8{\mathtt{c}}_{\mathtt{h}}^{24}-57{\mathtt{c}}_{\mathtt{h}}^{22}-37{\mathtt{c}}_{\mathtt{h}}^{20}+199{\mathtt{c}}_{\mathtt{h}}^{18}+175{\mathtt{c}}_{\mathtt{h}}^{16}+238{\mathtt{c}}_{\mathtt{h}}^{14}$
(A.45)
$\displaystyle\qquad\qquad+190{\mathtt{c}}_{\mathtt{h}}^{12}-130{\mathtt{c}}_{\mathtt{h}}^{10}-178{\mathtt{c}}_{\mathtt{h}}^{8}+171{\mathtt{c}}_{\mathtt{h}}^{6}+135{\mathtt{c}}_{\mathtt{h}}^{4}+27{\mathtt{c}}_{\mathtt{h}}^{2}+27\big{)}\,,$
$\displaystyle\mathfrak{p}_{4}^{[4]}:=\frac{{\mathtt{c}}_{\mathtt{h}}^{24}+44{\mathtt{c}}_{\mathtt{h}}^{20}+557{\mathtt{c}}_{\mathtt{h}}^{16}+2528{\mathtt{c}}_{\mathtt{h}}^{12}+3595{\mathtt{c}}_{\mathtt{h}}^{8}+1332{\mathtt{c}}_{\mathtt{h}}^{4}+135}{512{\mathtt{c}}_{\mathtt{h}}^{20}({\mathtt{c}}_{\mathtt{h}}^{4}+5)}\,.$
(A.46)

The real constant $\mathtt{f}_{\epsilon}$ in (1.12) has the Taylor expansion

$\mathtt{f}_{\epsilon}=\epsilon^{2}\mathtt{f}_{2}+\epsilon^{4}\mathtt{f}_{4}+\mathcal{O}(\epsilon^{5})$
(A.47)

with coefficients

$\displaystyle\mathtt{f}_{2}:=\frac{{\mathtt{c}}_{\mathtt{h}}^{4}-3}{4{\mathtt{c}}_{\mathtt{h}}^{2}}\,,$
(A.48)
$\displaystyle\mathtt{f}_{4}:=\frac{1}{64{\mathtt{c}}_{\mathtt{h}}^{14}({\mathtt{c}}_{\mathtt{h}}^{2}+1)}\big{(}-4{\mathtt{c}}_{\mathtt{h}}^{22}-8{\mathtt{c}}_{\mathtt{h}}^{20}+5{\mathtt{c}}_{\mathtt{h}}^{18}+23{\mathtt{c}}_{\mathtt{h}}^{16}+40{\mathtt{c}}_{\mathtt{h}}^{14}+22{\mathtt{c}}_{\mathtt{h}}^{12}-78{\mathtt{c}}_{\mathtt{h}}^{10}-72{\mathtt{c}}_{\mathtt{h}}^{8}+72{\mathtt{c}}_{\mathtt{h}}^{6}+54{\mathtt{c}}_{\mathtt{h}}^{4}-27{\mathtt{c}}_{\mathtt{h}}^{2}-27\big{)}\,.$

###### Proof.

We expand

$\displaystyle\mathfrak{p}(x)=\epsilon\mathfrak{p}_{1}(x)+\epsilon^{2}\mathfrak{p}_{2}(x)+\epsilon^{3}\mathfrak{p}_{3}(x)+\epsilon^{4}\mathfrak{p}_{4}(x)+\mathcal{O}(\epsilon^{5})\,,$
(A.49)
$\displaystyle\mathtt{f}_{\epsilon}=\epsilon^{2}\mathtt{f}_{2}+\epsilon^{3}\mathtt{f}_{3}+\epsilon^{4}\mathtt{f}_{4}+\mathcal{O}(\epsilon^{5})\,,$

where, by [10, formula (2.15)], $\mathfrak{p}_{1}(x)$, $\mathfrak{p}_{2}(x)$
are given in (A.42) and $\mathtt{f}_{2}$ in (A.48).

Let us denote derivatives w.r.t $x$ with a prime ′. By (A.38), (1.9) and
(A.49), we get

$\mathtt{f}_{3}=\frac{1}{2\pi}\int_{\mathbb{T}}\big{(}\eta_{3}(x)+\eta_{2}^{\prime}(x)\mathfrak{p}_{1}(x)+\eta_{1}^{\prime}(x)\mathfrak{p}_{2}(x)+\frac{1}{2}\eta_{1}^{\prime\prime}(x)\mathfrak{p}_{1}^{2}(x)\big{)}\mathrm{d}x=0$

as stated in the expansion (A.47). In view of (A.39) and (A.48) we have

$\displaystyle\mathfrak{p}_{3}(x)$
$\displaystyle=-\mathrm{i}\,\frac{\operatorname*{sgn}(D)}{\tanh(\mathtt{h}|D|)}\Big{(}\eta_{3}(x)+\eta_{2}^{\prime}(x)\mathfrak{p}_{1}(x)+\eta_{1}^{\prime}(x)\mathfrak{p}_{2}(x)+\frac{1}{2}\eta_{1}^{\prime\prime}(x)\mathfrak{p}_{1}^{2}(x)\Big{)}$
(A.50)
$\displaystyle\qquad+\mathtt{f}_{2}\frac{\partial_{x}}{\tanh^{2}(\mathtt{h}|D|)}\big{(}1-\tanh^{2}(\mathtt{h}|D|)\big{)}\eta_{1}(x)\,.$

In view of (A.50), (1.9), (A.41)-(A.42) and (A.11), we have

$\displaystyle\mathfrak{p}_{3}(x)$
$\displaystyle=\frac{16\eta_{3}^{[1]}{\mathtt{c}}_{\mathtt{h}}^{8}-16\eta_{2}^{[2]}{\mathtt{c}}_{\mathtt{h}}^{6}-3-6{\mathtt{c}}_{\mathtt{h}}^{4}-{\mathtt{c}}_{\mathtt{h}}^{8}-16\mathtt{f}_{2}{\mathtt{c}}_{\mathtt{h}}^{6}+16\mathtt{f}_{2}{\mathtt{c}}_{\mathtt{h}}^{10}{16{\mathtt{c}}_{\mathtt{h}}^{10}}}{\sin}(x)$
(A.51)
$\displaystyle\qquad+\frac{(1+3{\mathtt{c}}_{\mathtt{h}}^{4})(8\eta_{3}^{[3]}{\mathtt{c}}_{\mathtt{h}}^{4}+8\eta_{2}^{[2]}{\mathtt{c}}_{\mathtt{h}}^{2}+4{\mathtt{c}}_{\mathtt{h}}^{4}\mathfrak{p}_{2}^{[2]}+1)}{8{\mathtt{c}}_{\mathtt{h}}^{6}(3+{\mathtt{c}}_{\mathtt{h}}^{4})}\sin(3x)\,,$

which, by (A.1a)-(A.1c), is (A.44). By (A.38), (1.9), (A.42) and (A.44), we
have

$\displaystyle\mathtt{f}_{4}$
$\displaystyle=\frac{1}{2\pi}\int_{\mathbb{T}}\big{(}\eta_{4}(x)+\eta_{3}^{\prime}(x)\mathfrak{p}_{1}(x)+\eta_{2}^{\prime}(x)\mathfrak{p}_{2}(x)+\eta_{1}^{\prime}(x)\mathfrak{p}_{3}(x)$
(A.52)
$\displaystyle\quad+\frac{1}{2}\eta_{2}^{\prime\prime}(x)\mathfrak{p}_{1}^{2}(x)+\eta_{1}^{\prime\prime}(x)\mathfrak{p}_{2}(x)\mathfrak{p}_{1}(x)+\frac{1}{6}\eta_{1}^{\prime\prime\prime}(x)\mathfrak{p}_{1}^{3}(x)\big{)}\mathrm{d}x$
$\displaystyle=\eta_{4}^{[0]}-\frac{\eta_{3}^{[1]}}{2{\mathtt{c}}_{\mathtt{h}}^{2}}-\mathfrak{p}_{2}^{[2]}\eta_{2}^{[2]}-\frac{1}{2}\mathfrak{p}_{3}^{[1]}+\frac{\eta_{2}^{[2]}}{2{\mathtt{c}}_{\mathtt{h}}^{4}}-\frac{\mathfrak{p}_{2}^{[2]}}{4{\mathtt{c}}_{\mathtt{h}}^{2}}+\frac{1}{16{\mathtt{c}}_{\mathtt{h}}^{6}}\,,$

which gives the fourth-order coefficient in (A.48). Finally, by (A.39) and
(A.48),

$\displaystyle\mathfrak{p}_{4}(x)$
$\displaystyle=-\mathrm{i}\,\frac{\operatorname*{sgn}(D)}{\tanh(\mathtt{h}|D|)}\big{(}\eta_{4}(x)+\eta_{3}^{\prime}(x)\mathfrak{p}_{1}(x)+\eta_{2}^{\prime}(x)\mathfrak{p}_{2}(x)+\eta_{1}^{\prime}(x)\mathfrak{p}_{3}(x)$
$\displaystyle\qquad+\frac{1}{2}\eta_{2}^{\prime\prime}(x)\mathfrak{p}_{1}^{2}(x)+\eta_{1}^{\prime\prime}(x)\mathfrak{p}_{2}(x)\mathfrak{p}_{1}(x)+\frac{1}{6}\eta_{1}^{\prime\prime\prime}(x)\mathfrak{p}_{1}^{3}(x)\big{)}$
$\displaystyle\qquad+\mathtt{f}_{2}\frac{\partial_{x}}{\tanh^{2}(\mathtt{h}|D|)}\big{(}1-\tanh^{2}(\mathtt{h}|D|)\big{)}\big{(}\eta_{2}(x)+\eta_{1}^{\prime}(x)\mathfrak{p}_{1}(x)\big{)}$
$\displaystyle=\frac{1+{\mathtt{c}}_{\mathtt{h}}^{4}}{2{\mathtt{c}}_{\mathtt{h}}^{2}}\big{(}\frac{1}{2}\mathfrak{p}_{3}^{[1]}-\frac{1}{2}\mathfrak{p}_{3}^{[3]}-\frac{1}{12{\mathtt{c}}_{\mathtt{h}}^{6}}-\frac{\eta_{2}^{[2]}}{{\mathtt{c}}_{\mathtt{h}}^{4}}+\frac{\eta_{3}^{[1]}}{2{\mathtt{c}}_{\mathtt{h}}^{2}}-\frac{3\eta_{3}^{[3]}}{2{\mathtt{c}}_{\mathtt{h}}^{2}}+\eta_{4}^{[2]}\big{)}\sin(2x)$
$\displaystyle\qquad+\frac{1+6{\mathtt{c}}_{\mathtt{h}}^{4}+{\mathtt{c}}_{\mathtt{h}}^{8}}{4{\mathtt{c}}_{\mathtt{h}}^{2}(1+{\mathtt{c}}_{\mathtt{h}}^{4})}\big{(}\frac{\mathfrak{p}_{2}^{[2]}}{4{\mathtt{c}}_{\mathtt{h}}^{2}}+\frac{3\eta_{3}^{[3]}}{2{\mathtt{c}}_{\mathtt{h}}^{2}}+\mathfrak{p}_{2}^{[2]}\eta_{2}^{[2]}+\frac{1}{2}\mathfrak{p}_{3}^{[3]}+\frac{1}{48{\mathtt{c}}_{\mathtt{h}}^{6}}+\frac{\eta_{2}^{[2]}}{2{\mathtt{c}}_{\mathtt{h}}^{4}}+\eta_{4}^{[4]}\big{)}\sin(4x)$
$\displaystyle\qquad-\mathtt{f}_{2}\frac{(1-{\mathtt{c}}_{\mathtt{h}}^{4})^{2}(1+2{\mathtt{c}}_{\mathtt{h}}^{2}\eta_{2}^{[2]})}{4{\mathtt{c}}_{\mathtt{h}}^{6}}\sin(2x)\,,$

which, by (A.1a)-(A.1c), gives (A.41) with the coefficients computed in
(A.45)-(A.46). ∎

The second preparatory result is given by the following

######  Lemma A.6 (Expansion of $B(x)$ and $V(x)$).

The functions $B(x)$ and $V(x)$ in (A.40) admit the following Taylor expansion

$\displaystyle B(x)$ $\displaystyle=\epsilon
B_{1}(x)+\epsilon^{2}B_{2}(x)+\epsilon^{3}B_{3}(x)+\epsilon^{4}B_{4}(x)+\mathcal{O}(\epsilon^{5})\,,$
(A.53) $\displaystyle V(x)$ $\displaystyle=\epsilon
V_{1}(x)+\epsilon^{2}V_{2}(x)+\epsilon^{3}V_{3}(x)+\epsilon^{4}V_{4}(x)+\mathcal{O}(\epsilon^{5})\,,$

where

$\displaystyle B_{1}(x)$
$\displaystyle={\mathtt{c}}_{\mathtt{h}}\sin(x)\,,\quad$ $\displaystyle
B_{2}(x)=B_{2}^{[2]}\sin(2x)\,,\quad$ $\displaystyle
B_{2}^{[2]}:=\frac{3-2{\mathtt{c}}_{\mathtt{h}}^{4}}{2{\mathtt{c}}_{\mathtt{h}}^{5}}\,,$
(A.54) $\displaystyle V_{1}(x)$
$\displaystyle={\mathtt{c}}_{\mathtt{h}}^{-1}\cos(x)\,,\quad$ $\displaystyle
V_{2}(x)=\frac{{\mathtt{c}}_{\mathtt{h}}}{2}+V_{2}^{[2]}\cos(2x)\,,\quad$
$\displaystyle
V_{2}^{[2]}:=\frac{3-{\mathtt{c}}_{\mathtt{h}}^{8}}{4{\mathtt{c}}_{\mathtt{h}}^{7}}\,,$

and

$\displaystyle B_{3}(x)=B_{3}^{[1]}\sin(x)+B_{3}^{[3]}\sin(3x)\,,\quad$
$\displaystyle B_{4}(x)=B_{4}^{[2]}\sin(2x)+B_{4}^{[4]}\sin(4x)\,,$ (A.55)
$\displaystyle V_{3}(x)=V_{3}^{[1]}\cos(x)+V_{3}^{[3]}\cos(3x)\,,\quad$
$\displaystyle
V_{4}(x)=V_{4}^{[0]}+V_{4}^{[2]}\cos(2x)+V_{4}^{[4]}\cos(4x)\,,$

with

$\displaystyle
B_{3}^{[1]}:=\frac{6+3{\mathtt{c}}_{\mathtt{h}}^{2}-8{\mathtt{c}}_{\mathtt{h}}^{4}-8{\mathtt{c}}_{\mathtt{h}}^{6}+6{\mathtt{c}}_{\mathtt{h}}^{8}+3{\mathtt{c}}_{\mathtt{h}}^{10}-4{\mathtt{c}}_{\mathtt{h}}^{12}-2{\mathtt{c}}_{\mathtt{h}}^{14}}{16{\mathtt{c}}_{\mathtt{h}}^{7}(1+{\mathtt{c}}_{\mathtt{h}}^{2})}\,,\
\
B_{3}^{[3]}:=\frac{81-99{\mathtt{c}}_{\mathtt{h}}^{4}+43{\mathtt{c}}_{\mathtt{h}}^{8}-{\mathtt{c}}_{\mathtt{h}}^{12}}{64{\mathtt{c}}_{\mathtt{h}}^{11}}\,,$
$\displaystyle
V_{3}^{[1]}:=\frac{2{\mathtt{c}}_{\mathtt{h}}^{12}-15{\mathtt{c}}_{\mathtt{h}}^{8}-12{\mathtt{c}}_{\mathtt{h}}^{6}+24{\mathtt{c}}_{\mathtt{h}}^{4}+24{\mathtt{c}}_{\mathtt{h}}^{2}-3}{16{\mathtt{c}}_{\mathtt{h}}^{7}(1+{\mathtt{c}}_{\mathtt{h}}^{2})}\,,\
\
V_{3}^{[3]}:=\frac{21{\mathtt{c}}_{\mathtt{h}}^{12}-39{\mathtt{c}}_{\mathtt{h}}^{8}+15{\mathtt{c}}_{\mathtt{h}}^{4}+27}{64{\mathtt{c}}_{\mathtt{h}}^{13}}\,,$
(A.56a) and $\displaystyle V_{4}^{[0]}:$
$\displaystyle=\frac{-2{\mathtt{c}}_{\mathtt{h}}^{18}-6{\mathtt{c}}_{\mathtt{h}}^{16}+3{\mathtt{c}}_{\mathtt{h}}^{14}+9{\mathtt{c}}_{\mathtt{h}}^{12}-33{\mathtt{c}}_{\mathtt{h}}^{6}-27{\mathtt{c}}_{\mathtt{h}}^{4}+36{\mathtt{c}}_{\mathtt{h}}^{2}+36}{32{\mathtt{c}}_{\mathtt{h}}^{11}({\mathtt{c}}_{\mathtt{h}}^{2}+1)}\,,$
(A.56b) $\displaystyle B_{4}^{[2]}:$
$\displaystyle=\frac{1}{192{\mathtt{c}}_{\mathtt{h}}^{17}({\mathtt{c}}_{\mathtt{h}}^{2}+1)}\big{(}-24{\mathtt{c}}_{\mathtt{h}}^{22}+24{\mathtt{c}}_{\mathtt{h}}^{20}+354{\mathtt{c}}_{\mathtt{h}}^{18}+210{\mathtt{c}}_{\mathtt{h}}^{16}$
$\displaystyle-943{\mathtt{c}}_{\mathtt{h}}^{14}-835{\mathtt{c}}_{\mathtt{h}}^{12}+927{\mathtt{c}}_{\mathtt{h}}^{10}+855{\mathtt{c}}_{\mathtt{h}}^{8}-81{\mathtt{c}}_{\mathtt{h}}^{6}+27{\mathtt{c}}_{\mathtt{h}}^{4}-81{\mathtt{c}}_{\mathtt{h}}^{2}-81\big{)}\,,$
$\displaystyle V_{4}^{[2]}:$
$\displaystyle=\frac{1}{384{\mathtt{c}}_{\mathtt{h}}^{19}({\mathtt{c}}_{\mathtt{h}}^{2}+1)}\big{(}12{\mathtt{c}}_{\mathtt{h}}^{26}+36{\mathtt{c}}_{\mathtt{h}}^{24}-9{\mathtt{c}}_{\mathtt{h}}^{22}-45{\mathtt{c}}_{\mathtt{h}}^{20}+357{\mathtt{c}}_{\mathtt{h}}^{18}+285{\mathtt{c}}_{\mathtt{h}}^{16}$
$\displaystyle-1060{\mathtt{c}}_{\mathtt{h}}^{14}-988{\mathtt{c}}_{\mathtt{h}}^{12}+1584{\mathtt{c}}_{\mathtt{h}}^{10}+1584{\mathtt{c}}_{\mathtt{h}}^{8}-243{\mathtt{c}}_{\mathtt{h}}^{6}-135{\mathtt{c}}_{\mathtt{h}}^{4}-81{\mathtt{c}}_{\mathtt{h}}^{2}-81\big{)}\,,$
$\displaystyle B_{4}^{[4]}:$
$\displaystyle=\frac{6{\mathtt{c}}_{\mathtt{h}}^{20}-47{\mathtt{c}}_{\mathtt{h}}^{16}-100{\mathtt{c}}_{\mathtt{h}}^{12}+522{\mathtt{c}}_{\mathtt{h}}^{8}-594{\mathtt{c}}_{\mathtt{h}}^{4}+405}{96{\mathtt{c}}_{\mathtt{h}}^{17}({\mathtt{c}}_{\mathtt{h}}^{4}+5)}\,,$
$\displaystyle V_{4}^{[4]}:$
$\displaystyle=\frac{9{\mathtt{c}}_{\mathtt{h}}^{24}-96{\mathtt{c}}_{\mathtt{h}}^{20}-377{\mathtt{c}}_{\mathtt{h}}^{16}+1484{\mathtt{c}}_{\mathtt{h}}^{12}-1413{\mathtt{c}}_{\mathtt{h}}^{8}+756{\mathtt{c}}_{\mathtt{h}}^{4}+405}{384{\mathtt{c}}_{\mathtt{h}}^{19}({\mathtt{c}}_{\mathtt{h}}^{4}+5)}\,.$

###### Proof.

The first jets of (A.53) in (A.54) were computed in [10, (B.12-B.13)].
On the other hand, in view of (A.40) and (1.9), (A.1), the third order terms
are

$\displaystyle B_{3}(x)$
$\displaystyle=-{\mathtt{c}}_{\mathtt{h}}\eta_{3}^{\prime}(x)+{\mathtt{c}}_{\mathtt{h}}(\eta_{1}^{\prime}(x))^{3}+\psi_{1}^{\prime}(x)\eta_{2}^{\prime}(x)+(\psi_{2}^{\prime}(x)-c_{2})\eta_{1}^{\prime}(x)$
(A.57a)
$\displaystyle=\big{(}{\mathtt{c}}_{\mathtt{h}}\eta_{3}^{[1]}-\frac{3}{4}{\mathtt{c}}_{\mathtt{h}}-\frac{1}{{\mathtt{c}}_{\mathtt{h}}}\eta_{2}^{[2]}+\psi_{2}^{[2]}+c_{2}\big{)}\sin(x)+\big{(}3{\mathtt{c}}_{\mathtt{h}}\eta_{3}^{[3]}+\frac{1}{4}{\mathtt{c}}_{\mathtt{h}}-\frac{1}{{\mathtt{c}}_{\mathtt{h}}}\eta_{2}^{[2]}-\psi_{2}^{[2]}\big{)}\sin(3x)$
and $\displaystyle V_{3}(x)$
$\displaystyle=\psi_{3}^{\prime}(x)-B_{1}(x)\eta_{2}^{\prime}(x)-B_{2}(x)\eta_{1}^{\prime}(x)$
(A.57b)
$\displaystyle=\big{(}\psi_{3}^{[1]}+{\mathtt{c}}_{\mathtt{h}}\eta_{2}^{[2]}+\frac{1}{2}B_{2}^{[2]}\big{)}\cos(x)+\big{(}3\psi_{3}^{[3]}-{\mathtt{c}}_{\mathtt{h}}\eta_{2}^{[2]}-\frac{1}{2}B_{2}^{[2]}\big{)}\cos(3x)\,.$
The fourth order terms are given by $\displaystyle B_{4}(x)$
$\displaystyle=\psi_{3}^{\prime}(x)\eta_{1}^{\prime}(x)+\big{(}\psi_{2}^{\prime}(x)-c_{2}+3{\mathtt{c}}_{\mathtt{h}}(\eta_{1}^{\prime}(x))^{2}\big{)}\eta_{2}^{\prime}(x)+\psi_{1}^{\prime}(x)\eta_{3}^{\prime}(x)-\psi_{1}^{\prime}(x)(\eta_{1}^{\prime}(x))^{3}-{\mathtt{c}}_{\mathtt{h}}\eta_{4}^{\prime}(x)\,,$
$\displaystyle=\big{(}\frac{3}{2}\psi_{3}^{[3]}-\frac{1}{2}\psi_{3}^{[1]}+2c_{2}\eta_{2}^{[2]}-3{\mathtt{c}}_{\mathtt{h}}\eta_{2}^{[2]}-\frac{1}{2{\mathtt{c}}_{\mathtt{h}}}\eta_{3}^{[1]}-\frac{3}{2{\mathtt{c}}_{\mathtt{h}}}\eta_{3}^{[3]}+\frac{1}{4{\mathtt{c}}_{\mathtt{h}}}+2{\mathtt{c}}_{\mathtt{h}}\eta_{4}^{[2]}\big{)}\sin(2x)$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\,\,\begin{aligned}
&\quad\
\big{(}-\frac{3}{2}\psi_{3}^{[3]}-2\eta_{2}^{[2]}\psi_{2}^{[2]}+\frac{3}{2}{\mathtt{c}}_{\mathtt{h}}\eta_{2}^{[2]}-\frac{3}{2{\mathtt{c}}_{\mathtt{h}}}\eta_{3}^{[3]}-\frac{1}{8{\mathtt{c}}_{\mathtt{h}}}+4{\mathtt{c}}_{\mathtt{h}}\eta_{4}^{[4]}\big{)}\sin(4x)\,,\\\
V_{4}(x)&=\psi_{4}^{\prime}(x)-B_{1}(x)\eta_{3}^{\prime}(x)-B_{2}(x)\eta_{2}^{\prime}(x)-B_{3}(x)\eta_{1}^{\prime}(x)\end{aligned}$
(A.57c)
$\displaystyle=\frac{1}{2}{\mathtt{c}}_{\mathtt{h}}\eta_{3}^{[1]}+B_{2}^{[2]}\eta_{2}^{[2]}+\frac{1}{2}B_{3}^{[1]}+\big{(}4\psi_{4}^{[4]}-\frac{3}{2}{\mathtt{c}}_{\mathtt{h}}\eta_{3}^{[3]}-B_{2}^{[2]}\eta_{2}^{[2]}-\frac{1}{2}B_{3}^{[3]}\big{)}\cos(4x)$
$\displaystyle\quad+\big{(}2\psi_{4}^{[2]}-\frac{1}{2}{\mathtt{c}}_{\mathtt{h}}\eta_{3}^{[1]}+\frac{3}{2}{\mathtt{c}}_{\mathtt{h}}\eta_{3}^{[3]}-\frac{1}{2}B_{3}^{[1]}+\frac{1}{2}B_{3}^{[3]}\big{)}\cos(2x)\,.$

From (A.57) we obtain (A.55) with the coefficients in (A.56). ∎

We now provide the fourth order expansion of the functions $p_{\epsilon}(x)$
and $a_{\epsilon}(x)$ in (1.12).

######  Proposition A.7.

The functions $p_{\epsilon}(x)$ and $a_{\epsilon}(x)$ in (1.12) have a Taylor
expansion

$\displaystyle p_{\epsilon}(x)$ $\displaystyle=\epsilon
p_{1}(x)+\epsilon^{2}p_{2}(x)+\epsilon^{3}p_{3}(x)+\epsilon^{4}p_{4}(x)+\mathcal{O}(\epsilon^{5})\,,$
(A.58) $\displaystyle a_{\epsilon}(x)$ $\displaystyle=\epsilon
a_{1}(x)+\epsilon^{2}a_{2}(x)+\epsilon^{3}a_{3}(x)+\epsilon^{4}a_{4}(x)+\mathcal{O}(\epsilon^{5})\,,$

with

$\displaystyle p_{1}(x)=p_{1}^{[1]}\cos(x)\,,\quad
p_{1}^{[1]}:=-2{\mathtt{c}}_{\mathtt{h}}^{-1}\,,$ (A.59a) $\displaystyle
p_{2}(x)=p_{2}^{[0]}+p_{2}^{[2]}\cos(2x)\,,\quad
p_{2}^{[0]}:=\frac{9+12{\mathtt{c}}_{\mathtt{h}}^{4}+5{\mathtt{c}}_{\mathtt{h}}^{8}-2{\mathtt{c}}_{\mathtt{h}}^{12}}{16{\mathtt{c}}_{\mathtt{h}}^{7}}\,,\quad
p_{2}^{[2]}:=-\frac{3+{\mathtt{c}}_{\mathtt{h}}^{4}}{2{\mathtt{c}}_{\mathtt{h}}^{7}}\,,$
$\displaystyle p_{3}(x)=p_{3}^{[1]}\cos(x)+p_{3}^{[3]}\cos(3x)\,,\quad
p_{3}^{[3]}:=-\frac{{\mathtt{c}}_{\mathtt{h}}^{12}+17{\mathtt{c}}_{\mathtt{h}}^{8}+51{\mathtt{c}}_{\mathtt{h}}^{4}+27}{32{\mathtt{c}}_{\mathtt{h}}^{13}}\,,$
(A.59b) $\displaystyle
p_{3}^{[1]}:=\frac{-2{\mathtt{c}}_{\mathtt{h}}^{14}+14{\mathtt{c}}_{\mathtt{h}}^{10}+11{\mathtt{c}}_{\mathtt{h}}^{8}-10{\mathtt{c}}_{\mathtt{h}}^{6}-10{\mathtt{c}}_{\mathtt{h}}^{4}+24{\mathtt{c}}_{\mathtt{h}}^{2}+21}{8{\mathtt{c}}_{\mathtt{h}}^{9}({\mathtt{c}}_{\mathtt{h}}^{2}+1)}\,,$
$\displaystyle
p_{4}(x)=p_{4}^{[0]}+p_{4}^{[2]}\cos(2x)+p_{4}^{[4]}\cos(4x)\,,$ (A.59c)
$\displaystyle
p_{4}^{[0]}:=\frac{1}{1024{\mathtt{c}}_{\mathtt{h}}^{19}({\mathtt{c}}_{\mathtt{h}}^{2}+1)}\big{(}56{\mathtt{c}}_{\mathtt{h}}^{30}+88{\mathtt{c}}_{\mathtt{h}}^{28}-208{\mathtt{c}}_{\mathtt{h}}^{26}-336{\mathtt{c}}_{\mathtt{h}}^{24}+441{\mathtt{c}}_{\mathtt{h}}^{22}+369{\mathtt{c}}_{\mathtt{h}}^{20}-995{\mathtt{c}}_{\mathtt{h}}^{18}$
$\displaystyle\qquad-899{\mathtt{c}}_{\mathtt{h}}^{16}-630{\mathtt{c}}_{\mathtt{h}}^{14}-294{\mathtt{c}}_{\mathtt{h}}^{12}+1026{\mathtt{c}}_{\mathtt{h}}^{10}+1314{\mathtt{c}}_{\mathtt{h}}^{8}-27{\mathtt{c}}_{\mathtt{h}}^{6}+189{\mathtt{c}}_{\mathtt{h}}^{4}+81{\mathtt{c}}_{\mathtt{h}}^{2}+81\big{)}\,,$
$\displaystyle
p_{4}^{[2]}:=\frac{1}{64{\mathtt{c}}_{\mathtt{h}}^{19}({\mathtt{c}}_{\mathtt{h}}^{2}+1)}\big{(}-12{\mathtt{c}}_{\mathtt{h}}^{22}-4{\mathtt{c}}_{\mathtt{h}}^{20}-19{\mathtt{c}}_{\mathtt{h}}^{18}-7{\mathtt{c}}_{\mathtt{h}}^{16}+350{\mathtt{c}}_{\mathtt{h}}^{14}$
$\displaystyle\qquad+314{\mathtt{c}}_{\mathtt{h}}^{12}-256{\mathtt{c}}_{\mathtt{h}}^{10}-268{\mathtt{c}}_{\mathtt{h}}^{8}+198{\mathtt{c}}_{\mathtt{h}}^{6}+162{\mathtt{c}}_{\mathtt{h}}^{4}+27{\mathtt{c}}_{\mathtt{h}}^{2}+27\big{)}\,,$
$\displaystyle
p_{4}^{[4]}:=\frac{-{\mathtt{c}}_{\mathtt{h}}^{20}-39{\mathtt{c}}_{\mathtt{h}}^{16}-366{\mathtt{c}}_{\mathtt{h}}^{12}-850{\mathtt{c}}_{\mathtt{h}}^{8}-657{\mathtt{c}}_{\mathtt{h}}^{4}-135}{64{\mathtt{c}}_{\mathtt{h}}^{19}({\mathtt{c}}_{\mathtt{h}}^{4}+5)}\,,$

and

$\displaystyle a_{1}(x)=a_{1}^{[1]}\cos(x)\,,\quad
a_{1}^{[1]}:=-({\mathtt{c}}_{\mathtt{h}}^{2}+{\mathtt{c}}_{\mathtt{h}}^{-2})\,,$
(A.60a) $\displaystyle a_{2}(x)=a_{2}^{[0]}+a_{2}^{[2]}\cos(2x)\,,\quad
a_{2}^{[0]}:=\frac{3}{2}+\frac{1}{2{\mathtt{c}}_{\mathtt{h}}^{4}}\,,\quad
a_{2}^{[2]}:=\frac{9{\mathtt{c}}_{\mathtt{h}}^{8}-14{\mathtt{c}}_{\mathtt{h}}^{4}-3}{4{\mathtt{c}}_{\mathtt{h}}^{8}}\,,$
$\displaystyle\begin{aligned}
&a_{3}(x)=a_{3}^{[1]}\cos(x)+a_{3}^{[3]}\cos(3x)\,,\quad
a_{3}^{[3]}:=\frac{-{\mathtt{c}}_{\mathtt{h}}^{16}-98{\mathtt{c}}_{\mathtt{h}}^{12}+252{\mathtt{c}}_{\mathtt{h}}^{8}-318{\mathtt{c}}_{\mathtt{h}}^{4}-27}{64{\mathtt{c}}_{\mathtt{h}}^{14}}\,,\\\
&a_{3}^{[1]}:=\frac{4{\mathtt{c}}_{\mathtt{h}}^{18}+6{\mathtt{c}}_{\mathtt{h}}^{16}-11{\mathtt{c}}_{\mathtt{h}}^{14}-12{\mathtt{c}}_{\mathtt{h}}^{12}-45{\mathtt{c}}_{\mathtt{h}}^{10}-48{\mathtt{c}}_{\mathtt{h}}^{8}+93{\mathtt{c}}_{\mathtt{h}}^{6}+90{\mathtt{c}}_{\mathtt{h}}^{4}+27{\mathtt{c}}_{\mathtt{h}}^{2}+24}{16{\mathtt{c}}_{\mathtt{h}}^{10}({\mathtt{c}}_{\mathtt{h}}^{2}+1)}\,,\end{aligned}$
(A.60b) $\displaystyle
a_{4}(x)=a_{4}^{[0]}+a_{4}^{[2]}\cos(2x)+a_{4}^{[4]}\cos(4x)\,,$ (A.60c)
$\displaystyle\begin{aligned}
&a_{4}^{[0]}:=\frac{-12{\mathtt{c}}_{\mathtt{h}}^{20}-31{\mathtt{c}}_{\mathtt{h}}^{18}-17{\mathtt{c}}_{\mathtt{h}}^{16}+40{\mathtt{c}}_{\mathtt{h}}^{14}+46{\mathtt{c}}_{\mathtt{h}}^{12}-150{\mathtt{c}}_{\mathtt{h}}^{10}-132{\mathtt{c}}_{\mathtt{h}}^{8}+84{\mathtt{c}}_{\mathtt{h}}^{6}+90{\mathtt{c}}_{\mathtt{h}}^{4}+9{\mathtt{c}}_{\mathtt{h}}^{2}+9}{32{\mathtt{c}}_{\mathtt{h}}^{16}({\mathtt{c}}_{\mathtt{h}}^{2}+1)}\,,\\\
&a_{4}^{[2]}:=\frac{1}{128{\mathtt{c}}_{\mathtt{h}}^{20}({\mathtt{c}}_{\mathtt{h}}^{2}+1)}\big{(}-72{\mathtt{c}}_{\mathtt{h}}^{24}-431{\mathtt{c}}_{\mathtt{h}}^{22}-211{\mathtt{c}}_{\mathtt{h}}^{20}+1767{\mathtt{c}}_{\mathtt{h}}^{18}\\\
&\qquad+1623{\mathtt{c}}_{\mathtt{h}}^{16}-2142{\mathtt{c}}_{\mathtt{h}}^{14}-2070{\mathtt{c}}_{\mathtt{h}}^{12}+1022{\mathtt{c}}_{\mathtt{h}}^{10}+854{\mathtt{c}}_{\mathtt{h}}^{8}+333{\mathtt{c}}_{\mathtt{h}}^{6}+297{\mathtt{c}}_{\mathtt{h}}^{4}+27{\mathtt{c}}_{\mathtt{h}}^{2}+27\big{)}\,,\\\
&a_{4}^{[4]}:=\frac{9{\mathtt{c}}_{\mathtt{h}}^{24}+238{\mathtt{c}}_{\mathtt{h}}^{20}-233{\mathtt{c}}_{\mathtt{h}}^{16}-1676{\mathtt{c}}_{\mathtt{h}}^{12}+743{\mathtt{c}}_{\mathtt{h}}^{8}-3042{\mathtt{c}}_{\mathtt{h}}^{4}-135}{128{\mathtt{c}}_{\mathtt{h}}^{20}({\mathtt{c}}_{\mathtt{h}}^{4}+5)}\,.\end{aligned}$

###### Proof.

The first two jets $p_{1}(x),p_{2}(x)$ in (A.59a) and $a_{1}(x),a_{2}(x)$ in
(LABEL:aino2fd) of the expansion (A.58) have been computed in [10, Lemma 2.2].
Let us compute the third order terms. In view of (A.37), (A.53) and (A.41), we
have

$\displaystyle p_{3}(x)$
$\displaystyle={\mathtt{c}}_{\mathtt{h}}\big{(}-\mathfrak{p}_{3}^{\prime}(x)+2\mathfrak{p}^{\prime}_{1}(x)\mathfrak{p}^{\prime}_{2}(x)-(\mathfrak{p}_{1}^{\prime}(x))^{3}\big{)}+V_{1}(x)\big{(}\mathfrak{p}_{2}^{\prime}(x)-(\mathfrak{p}_{1}^{\prime}(x))^{2}\big{)}$
(A.61a)
$\displaystyle\qquad-\big{(}c_{2}-V_{2}(x)-V_{1}^{\prime}(x)\mathfrak{p}_{1}(x)\big{)}\mathfrak{p}_{1}^{\prime}(x)$
$\displaystyle\qquad-
V_{3}(x)-V_{2}^{\prime}(x)\mathfrak{p}_{1}(x)-V_{1}^{\prime}(x)\mathfrak{p}_{2}(x)-\frac{1}{2}V_{1}^{\prime\prime}(x)\mathfrak{p}_{1}^{{2}}(x)$
$\displaystyle=\big{(}-{\mathtt{c}}_{\mathtt{h}}\mathfrak{p}_{3}^{[1]}+\frac{7}{2{\mathtt{c}}_{\mathtt{h}}}\mathfrak{p}_{2}^{[2]}-\frac{13}{8}{\mathtt{c}}_{\mathtt{h}}^{-5}-\frac{c_{2}}{{\mathtt{c}}_{\mathtt{h}}^{2}}+\frac{1}{2{\mathtt{c}}_{\mathtt{h}}}+\frac{3V_{2}^{[2]}}{2{\mathtt{c}}_{\mathtt{h}}^{2}}-V_{3}^{[1]}\big{)}\cos(x)$
$\displaystyle\qquad+\big{(}-3{\mathtt{c}}_{\mathtt{h}}\mathfrak{p}_{3}^{[3]}+\frac{5}{2{\mathtt{c}}_{\mathtt{h}}}\mathfrak{p}_{2}^{[2]}-\frac{3}{8{\mathtt{c}}_{\mathtt{h}}^{5}}-\frac{V_{2}^{[2]}}{2{\mathtt{c}}_{\mathtt{h}}^{2}}-V_{3}^{[3]}\big{)}\cos(3x)\,,$
$\displaystyle a_{3}(x)$
$\displaystyle=-\mathfrak{p}_{3}^{\prime}(x)+2\mathfrak{p}_{1}^{\prime}(x)\mathfrak{p}_{2}^{\prime}(x)-(\mathfrak{p}_{1}^{\prime}(x))^{3}$
(A.61b)
$\displaystyle\qquad-{\mathtt{c}}_{\mathtt{h}}\big{(}B_{3}^{\prime}(x)+B_{2}^{\prime\prime}(x)\mathfrak{p}_{1}(x)+B_{1}^{\prime\prime}(x)\mathfrak{p}_{2}(x)+\frac{1}{2}B_{1}^{\prime\prime\prime}(x)\mathfrak{p}_{1}^{2}(x)\big{)}$
$\displaystyle\qquad-
p_{1}(x)\big{(}B_{2}^{\prime}(x)+B_{1}^{\prime\prime}(x)\mathfrak{p}_{1}(x)\big{)}-p_{2}(x)B_{1}^{\prime}(x)\,,$
$\displaystyle=\big{(}-\mathfrak{p}_{3}^{[1]}+2{\mathtt{c}}_{\mathtt{h}}^{-2}\mathfrak{p}_{2}^{[2]}-\frac{3}{4}{\mathtt{c}}_{\mathtt{h}}^{-6}-{\mathtt{c}}_{\mathtt{h}}B_{3}^{[1]}+2{\mathtt{c}}_{\mathtt{h}}^{-1}B_{2}^{[2]}+\frac{1}{2}{\mathtt{c}}_{\mathtt{h}}^{2}\mathfrak{p}_{2}^{[2]}$
$\displaystyle\qquad+\frac{1}{8}{\mathtt{c}}_{\mathtt{h}}^{-2}-p_{1}^{[1]}B_{2}^{[2]}+\frac{1}{4}{\mathtt{c}}_{\mathtt{h}}^{-1}p_{1}^{[1]}-{\mathtt{c}}_{\mathtt{h}}p_{2}^{[0]}-\frac{1}{2}{\mathtt{c}}_{\mathtt{h}}p_{2}^{[2]}\big{)}\cos(x)$
$\displaystyle\qquad+\big{(}-3\mathfrak{p}_{3}^{[3]}+2{\mathtt{c}}_{\mathtt{h}}^{-2}\mathfrak{p}_{2}^{[2]}-\frac{1}{4}{\mathtt{c}}_{\mathtt{h}}^{-6}-3{\mathtt{c}}_{\mathtt{h}}B_{3}^{[3]}-2{\mathtt{c}}_{\mathtt{h}}^{-1}B_{2}^{[2]}$
$\displaystyle\qquad-\frac{1}{2}{\mathtt{c}}_{\mathtt{h}}^{2}\mathfrak{p}_{2}^{[2]}-\frac{1}{8}{\mathtt{c}}_{\mathtt{h}}^{-2}-p_{1}^{[1]}B_{2}^{[2]}-\frac{1}{4}{\mathtt{c}}_{\mathtt{h}}^{-1}p_{1}^{[1]}-\frac{1}{2}{\mathtt{c}}_{\mathtt{h}}p_{2}^{[2]}\big{)}\cos(3x)\,,$

and

$\displaystyle p_{4}(x)$
$\displaystyle=c_{4}-V_{4}(x)-V_{3}^{\prime}(x)\mathfrak{p}_{1}(x)-V_{2}^{\prime}(x)\mathfrak{p}_{2}(x)-V_{1}^{\prime}(x)\mathfrak{p}_{3}(x)$
(A.62a)
$\displaystyle\qquad-\frac{1}{2}V_{2}^{\prime\prime}(x)\mathfrak{p}^{2}_{1}(x)-V_{1}^{\prime\prime}(x)\mathfrak{p}_{2}(x)\mathfrak{p}_{1}(x)-\frac{1}{6}V_{1}^{\prime\prime\prime}(x)\mathfrak{p}_{1}^{3}(x)$
$\displaystyle\qquad+\mathfrak{p}_{1}^{\prime}(x)\big{(}V_{3}(x)+V_{2}^{\prime}(x)\mathfrak{p}_{1}(x)+V_{1}^{\prime}(x)\mathfrak{p}_{2}(x)+\frac{1}{2}V_{1}^{\prime\prime}(x)\mathfrak{p}_{1}^{2}(x)\big{)}$
$\displaystyle\qquad+\big{(}(\mathfrak{p}_{1}^{\prime}(x))^{2}-\mathfrak{p}_{2}^{\prime}(x)\big{)}\big{(}c_{2}-V_{2}(x)-V_{1}^{\prime}(x)\mathfrak{p}_{1}(x)\big{)}$
$\displaystyle\qquad+V_{1}(x)\big{(}\mathfrak{p}_{3}^{\prime}(x)-2\mathfrak{p}_{2}^{\prime}(x)\mathfrak{p}_{1}^{\prime}(x)+(\mathfrak{p}_{1}^{\prime}(x))^{3}\big{)}$
$\displaystyle\qquad+{\mathtt{c}}_{\mathtt{h}}\big{(}-\mathfrak{p}_{4}^{\prime}(x)+2\mathfrak{p}_{3}^{\prime}(x)\mathfrak{p}_{1}^{\prime}(x)+(\mathfrak{p}_{2}^{\prime}(x))^{2}-3\mathfrak{p}_{2}^{\prime}(x)(\mathfrak{p}_{1}^{\prime}(x))^{2}+(\mathfrak{p}_{1}^{\prime}(x))^{4}\big{)}\,,$
$\displaystyle=\frac{3}{4}{\mathtt{c}}_{\mathtt{h}}^{-7}+\frac{1}{2}c_{2}{\mathtt{c}}_{\mathtt{h}}^{-4}-\frac{5}{4}{\mathtt{c}}_{\mathtt{h}}^{-4}V_{2}^{[2]}-2{\mathtt{c}}_{\mathtt{h}}^{-3}\mathfrak{p}_{2}^{[2]}-\frac{1}{4}{\mathtt{c}}_{\mathtt{h}}^{-3}+{\mathtt{c}}_{\mathtt{h}}^{-2}V_{3}^{[1]}+2{\mathtt{c}}_{\mathtt{h}}(\mathfrak{p}_{2}^{[2]})^{2}$
$\displaystyle\
+2{\mathtt{c}}_{\mathtt{h}}^{-1}\mathfrak{p}_{3}^{[1]}+2\mathfrak{p}_{2}^{[2]}V_{2}^{[2]}+c_{4}-V_{4}^{[0]}+\big{(}\frac{13}{12}{\mathtt{c}}_{\mathtt{h}}^{-7}+\frac{1}{2}{\mathtt{c}}_{\mathtt{h}}^{-4}c_{2}+\frac{1}{2}{\mathtt{c}}_{\mathtt{h}}^{-4}V_{2}^{[2]}-6{\mathtt{c}}_{\mathtt{h}}^{-3}\mathfrak{p}_{2}^{[2]}$
$\displaystyle\
-\frac{1}{4}{\mathtt{c}}_{\mathtt{h}}^{-3}+2{\mathtt{c}}_{\mathtt{h}}^{-2}V_{3}^{[3]}+{\mathtt{c}}_{\mathtt{h}}\mathfrak{p}_{2}^{[2]}+{\mathtt{c}}_{\mathtt{h}}^{-1}\mathfrak{p}_{3}^{[1]}+5{\mathtt{c}}_{\mathtt{h}}^{-1}\mathfrak{p}_{3}^{[3]}-2{\mathtt{c}}_{\mathtt{h}}\mathfrak{p}_{4}^{[2]}-2\mathfrak{p}_{2}^{[2]}c_{2}-V_{4}^{[2]}\big{)}\cos(2x)$
$\displaystyle\
+\big{(}\frac{1}{6}{\mathtt{c}}_{\mathtt{h}}^{-7}-\frac{1}{4}{\mathtt{c}}_{\mathtt{h}}^{-4}V_{2}^{[2]}-2{\mathtt{c}}_{\mathtt{h}}^{-3}\mathfrak{p}_{2}^{[2]}-{\mathtt{c}}_{\mathtt{h}}^{-2}V_{3}^{[3]}+2{\mathtt{c}}_{\mathtt{h}}(\mathfrak{p}_{2}^{[2]})^{2}+4{\mathtt{c}}_{\mathtt{h}}^{-1}\mathfrak{p}_{3}^{[3]}-4{\mathtt{c}}_{\mathtt{h}}\mathfrak{p}_{4}^{[4]}-V_{4}^{[4]}\big{)}\cos(4x)$
$\displaystyle a_{4}(x)$
$\displaystyle=-\mathfrak{p}_{4}^{\prime}(x)+(\mathfrak{p}_{2}^{\prime}(x))^{2}+2\mathfrak{p}_{1}^{\prime}(x)\mathfrak{p}_{3}^{\prime}(x)-3(\mathfrak{p}_{1}^{\prime}(x))^{2}\mathfrak{p}_{2}^{\prime}(x)+(\mathfrak{p}_{1}^{\prime}(x))^{4}$
(A.62b)
$\displaystyle\qquad-{\mathtt{c}}_{\mathtt{h}}\big{(}B_{4}^{\prime}(x)+B_{3}^{\prime\prime}(x)\mathfrak{p}_{1}(x)+B_{2}^{\prime\prime}(x)\mathfrak{p}_{2}(x)+B_{1}^{\prime\prime}(x)\mathfrak{p}_{3}(x)$
$\displaystyle\qquad+\frac{1}{2}B_{2}^{\prime\prime\prime}(x)\mathfrak{p}_{1}^{2}(x)+B_{1}^{\prime\prime\prime}(x)\mathfrak{p}_{1}(x)\mathfrak{p}_{2}(x)+\frac{1}{6}B_{1}^{\scriptscriptstyle{\mathit{IV}}}(x)\mathfrak{p}_{1}^{3}(x)\big{)}$
$\displaystyle\qquad-
p_{1}(x)\big{(}B_{3}^{\prime}(x)+B_{2}^{\prime\prime}(x)\mathfrak{p}_{1}(x)+B_{1}^{\prime\prime}(x)\mathfrak{p}_{2}(x)+\frac{1}{2}B_{1}^{\prime\prime\prime}(x)\mathfrak{p}_{1}^{2}(x)\big{)}$
$\displaystyle\qquad-
p_{2}(x)\big{(}B_{2}^{\prime}(x)+B_{1}^{\prime\prime}(x)\mathfrak{p}_{1}(x)\big{)}-p_{3}(x)B_{1}^{\prime}(x)$
$\displaystyle=\frac{3}{8}{\mathtt{c}}_{\mathtt{h}}^{-8}-\frac{1}{16}{\mathtt{c}}_{\mathtt{h}}^{-4}-\frac{3}{2}{\mathtt{c}}_{\mathtt{h}}^{-4}\mathfrak{p}_{2}^{[2]}+\frac{1}{16}{\mathtt{c}}_{\mathtt{h}}^{-3}p_{1}^{[1]}-{\mathtt{c}}_{\mathtt{h}}^{-3}B_{2}^{[2]}+{\mathtt{c}}_{\mathtt{h}}^{-2}B_{2}^{[2]}p_{1}^{[1]}+{\mathtt{c}}_{\mathtt{h}}^{-2}\mathfrak{p}_{3}^{[1]}$
$\displaystyle\
+\frac{1}{2}{\mathtt{c}}_{\mathtt{h}}^{2}\mathfrak{p}_{3}^{[1]}+2{\mathtt{c}}_{\mathtt{h}}\mathfrak{p}_{2}^{[2]}B_{2}^{[2]}+\frac{1}{4}{\mathtt{c}}_{\mathtt{h}}\mathfrak{p}_{2}^{[2]}p_{1}^{[1]}-\frac{1}{2}{\mathtt{c}}_{\mathtt{h}}p_{3}^{[1]}+\frac{1}{2}{\mathtt{c}}_{\mathtt{h}}^{-1}B_{3}^{[1]}+\frac{1}{2}{\mathtt{c}}_{\mathtt{h}}^{-1}p_{2}^{[0]}-\frac{1}{4}{\mathtt{c}}_{\mathtt{h}}^{-1}p_{2}^{[2]}$
$\displaystyle\
+2(\mathfrak{p}_{2}^{[2]})^{2}+\frac{1}{4}\mathfrak{p}_{2}^{[2]}-B_{2}^{[2]}p_{2}^{[2]}-\frac{1}{2}B_{3}^{[1]}p_{1}^{[1]}+\big{(}\frac{1}{2}{\mathtt{c}}_{\mathtt{h}}^{-8}-3{\mathtt{c}}_{\mathtt{h}}^{-4}\mathfrak{p}_{2}^{[2]}+\frac{1}{12}{\mathtt{c}}_{\mathtt{h}}^{-4}+2{\mathtt{c}}_{\mathtt{h}}^{-3}B_{2}^{[2]}$
$\displaystyle\
-\frac{1}{2}{\mathtt{c}}_{\mathtt{h}}^{2}\mathfrak{p}_{3}^{[1]}+{\mathtt{c}}_{\mathtt{h}}^{-2}\mathfrak{p}_{3}^{[1]}+\frac{1}{2}{\mathtt{c}}_{\mathtt{h}}^{2}\mathfrak{p}_{3}^{[3]}+3{\mathtt{c}}_{\mathtt{h}}^{-2}\mathfrak{p}_{3}^{[3]}-\frac{1}{2}{\mathtt{c}}_{\mathtt{h}}^{-1}B_{3}^{[1]}+\frac{9}{2}{\mathtt{c}}_{\mathtt{h}}^{-1}B_{3}^{[3]}-2{\mathtt{c}}_{\mathtt{h}}B_{4}^{[2]}-\frac{1}{2}{\mathtt{c}}_{\mathtt{h}}^{-1}p_{2}^{[0]}$
$\displaystyle\
+\frac{1}{2}{\mathtt{c}}_{\mathtt{h}}^{-1}p_{2}^{[2]}-\frac{1}{2}{\mathtt{c}}_{\mathtt{h}}p_{3}^{[1]}-\frac{1}{2}{\mathtt{c}}_{\mathtt{h}}p_{3}^{[3]}-2B_{2}^{[2]}p_{2}^{[0]}-\frac{1}{2}B_{3}^{[1]}p_{1}^{[1]}-\frac{3}{2}B_{3}^{[3]}p_{1}^{[1]}-2\mathfrak{p}_{4}^{[2]}\big{)}\cos(2x)$
$\displaystyle+\big{(}\frac{1}{8}{\mathtt{c}}_{\mathtt{h}}^{-8}-\frac{3}{2}{\mathtt{c}}_{\mathtt{h}}^{-4}\mathfrak{p}_{2}^{[2]}-\frac{1}{48}{\mathtt{c}}_{\mathtt{h}}^{-4}-{\mathtt{c}}_{\mathtt{h}}^{-3}B_{2}^{[2]}-\frac{1}{16}{\mathtt{c}}_{\mathtt{h}}^{-3}p_{1}^{[1]}-{\mathtt{c}}_{\mathtt{h}}^{-2}B_{2}^{[2]}p_{1}^{[1]}-\frac{1}{2}{\mathtt{c}}_{\mathtt{h}}^{2}\mathfrak{p}_{3}^{[3]}$
$\displaystyle\
+3{\mathtt{c}}_{\mathtt{h}}^{-2}\mathfrak{p}_{3}^{[3]}-2{\mathtt{c}}_{\mathtt{h}}\mathfrak{p}_{2}^{[2]}B_{2}^{[2]}-\frac{1}{4}{\mathtt{c}}_{\mathtt{h}}\mathfrak{p}_{2}^{[2]}p_{1}^{[1]}-\frac{9}{2}{\mathtt{c}}_{\mathtt{h}}^{-1}B_{3}^{[3]}-4{\mathtt{c}}_{\mathtt{h}}B_{4}^{[4]}-\frac{1}{4}{\mathtt{c}}_{\mathtt{h}}^{-1}p_{2}^{[2]}$
$\displaystyle\
-\frac{1}{2}{\mathtt{c}}_{\mathtt{h}}p_{3}^{[3]}+2(\mathfrak{p}_{2}^{[2]})^{2}-\frac{1}{4}\mathfrak{p}_{2}^{[2]}-B_{2}^{[2]}p_{2}^{[2]}-\frac{3}{2}B_{3}^{[3]}p_{1}^{[1]}-4\mathfrak{p}_{4}^{[4]}\big{)}\cos(4x)\,.$

The expansions of $p_{3}(x)$, $a_{3}(x)$, $p_{4}(x)$ and $a_{4}(x)$ in
(A.59a)-(LABEL:aino2fd) descend from (A.61a), (A.61b), (A.62a) and (A.62b)
respectively, in view of (A.46), (A.44) (A.42), (A.54), (A.1b), (A.57b),
(A.57a) and (A.57c)-(A.56b). ∎

######  Remark A.8.

The functions $V$ and $B$ in (A.40) coincide respectively with the horizontal
and vertical derivative of the velocity potential $\Phi$ in [23, formula (12)]
after the rescaling procedure outlined in Remark A.4.

##  Appendix B Expansion of the projector $P_{\mu,\epsilon}$

In this Appendix we prove Lemmata 3.4 and 3.5.

We denote, for any $k\in\mathbb{N}$,

$\displaystyle
f_{k}^{+}:=\begin{bmatrix}{\mathtt{c}}_{\mathtt{h}}^{1/2}\cos(kx)\\\
{\mathtt{c}}_{\mathtt{h}}^{-1/2}\sin(kx)\end{bmatrix}\,,\quad
f_{k}^{-}:=\begin{bmatrix}-{\mathtt{c}}_{\mathtt{h}}^{1/2}\sin(kx)\\\
{\mathtt{c}}_{\mathtt{h}}^{-1/2}\cos(kx)\end{bmatrix}\,,$ (B.1) $\displaystyle
f_{-k}^{+}:=\begin{bmatrix}{\mathtt{c}}_{\mathtt{h}}^{1/2}\cos(kx)\\\
-{\mathtt{c}}_{\mathtt{h}}^{-1/2}\sin(kx)\end{bmatrix}\,,\quad
f_{-k}^{-}:=\begin{bmatrix}{\mathtt{c}}_{\mathtt{h}}^{1/2}\sin(kx)\\\
{\mathtt{c}}_{\mathtt{h}}^{-1/2}\cos(kx)\end{bmatrix}\,,$

and we define for any $k\in\mathbb{Z}$ the spaces

$\mathcal{W}_{k}:=\text{span}\left\\{f_{k}^{+},\ f_{k}^{-},f_{-k}^{+},\
f_{-k}^{-}\right\\}\,,\quad\mathcal{W}_{k}^{\sigma}:=\text{span}_{\mathbb{R}}\\{f_{k}^{\sigma},\
f_{-k}^{\sigma}\\}\,,\quad\sigma=\pm\,.$ (B.2)

We have the following

######  Lemma B.1.

The jets of the operator ${\cal B}_{\mu,\epsilon}$ in (3.8) act on the spaces
in (B.2) as follows333the sum is direct if $j\leq k$, otherwise some spaces
may overlap.

${\cal
B}_{\ell,j}\mathcal{W}_{k}^{\sigma}=\underbrace{\mathrm{i}\,^{\ell}\mathcal{W}_{k-j}^{(-1)^{\ell}\sigma}+_{\mathbb{R}}\mathrm{i}\,^{\ell}\mathcal{W}_{k-j+2}^{(-1)^{\ell}\sigma}+_{\mathbb{R}}\dots+_{\mathbb{R}}\mathrm{i}\,^{\ell}\mathcal{W}_{k+j}^{(-1)^{\ell}\sigma}}_{j+1\
terms}\,,\quad f_{0}^{-}\notin{\cal B}_{0,j}\mathcal{W}_{k}^{-}\,,$ (B.3)

with $\ell,j=0,\dots,4$, while the operator $\mathcal{J}$ in (1.5) acts as
$\mathcal{J}\mathcal{W}_{k}^{\pm}=\mathcal{W}_{k}^{\mp}$.

###### Proof.

The first formula in (B.3) follows by (3.8)-(3.9). Let us prove the second
statement by contradiction supposing that there exists
$g\in\mathcal{W}_{k}^{-}$ such that ${\cal B}_{0,j}g=f_{0}^{-}$. Then
$1=\big{(}f_{0}^{-},f_{0}^{-}\big{)}=\big{(}{\cal
B}_{0,j}g,f_{0}^{-}\big{)}=\big{(}g,{\cal B}_{0,j}f_{0}^{-}\big{)}=0$, by
(3.8), which is a contradiction. ∎

We now include an extended version of [10, Lemma A.2].

######  Lemma B.2.

The space $H^{1}(\mathbb{T})$ decomposes as
$H^{1}(\mathbb{T})=\mathcal{V}_{0,0}\oplus\mathcal{U}\oplus{\mathcal{W}_{H^{1}}}$,
with $\mathcal{W}_{H^{1}}=\overline{\bigoplus\limits_{k\geq
2}^{{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1},}}\mathcal{W}_{k}}^{{}_{\footnotesize\begin{matrix}H^{1}\end{matrix}}}$
where the subspaces $\mathcal{V}_{0,0},\mathcal{U}$ and $\mathcal{W}_{k}$,
defined below, are invariant under $\mathscr{L}_{0,0}$ and the following
properties hold:

  * (i)

$\mathcal{V}_{0,0}=\text{span}\\{f^{+}_{1},f^{-}_{1},f^{+}_{0},f^{-}_{0}\\}$
is the generalized kernel of $\mathscr{L}_{0,0}$. For any $\lambda\neq 0$ the
operator $\mathscr{L}_{0,0}-\lambda:\mathcal{V}_{0,0}\to\mathcal{V}_{0,0}$ is
invertible and

$\displaystyle(\mathscr{L}_{0,0}-\lambda)^{-1}f_{1}^{+}=-\frac{1}{\lambda}f_{1}^{+}\,,\
(\mathscr{L}_{0,0}-\lambda)^{-1}f_{1}^{-}=-\frac{1}{\lambda}f_{1}^{-},\
(\mathscr{L}_{0,0}-\lambda)^{-1}f_{0}^{-}=-\frac{1}{\lambda}f_{0}^{-}\,,$
(B.4a)
$\displaystyle(\mathscr{L}_{0,0}-\lambda)^{-1}f_{0}^{+}=-\frac{1}{\lambda}f_{0}^{+}+\frac{1}{\lambda^{2}}f_{0}^{-}\,.$
(B.4b)

  * (ii)

$\mathcal{U}:=\text{span}\left\\{f_{-1}^{+},f_{-1}^{-}\right\\}$. For any
$\lambda\neq\pm 2\mathrm{i}\,$ the operator
$\mathscr{L}_{0,0}-\lambda:\mathcal{U}\to\mathcal{U}$ is invertible and

$\displaystyle(\mathscr{L}_{0,0}-\lambda)^{-1}f_{-1}^{+}=\frac{1}{\lambda^{2}+4{\mathtt{c}}_{\mathtt{h}}^{2}}\left(-\lambda
f_{-1}^{+}+2{\mathtt{c}}_{\mathtt{h}}f_{-1}^{-}\right),$ (B.4c)
$\displaystyle(\mathscr{L}_{0,0}-\lambda)^{-1}f_{-1}^{-}=\frac{1}{\lambda^{2}+4{\mathtt{c}}_{\mathtt{h}}^{2}}\left(-2{\mathtt{c}}_{\mathtt{h}}f_{-1}^{+}-\lambda
f_{-1}^{-}\right)\,.$

  * (iii)

Each subspace $\mathcal{W}_{k}$ in (B.2) is invariant under
$\mathscr{L}_{0,0}$. For any $|\lambda|<\delta(\mathtt{h})$ small enough and
any natural $k\geq 2$, the operator
${\mathscr{L}_{0,0}-\lambda:\mathcal{W}_{k}\to\mathcal{W}_{k}}$ is invertible
and for any $f\in\mathcal{W}_{k}$ and any natural number $N$

$\displaystyle(\mathscr{L}_{0,0}-\lambda)^{-1}f=\mathscr{L}_{0,0}^{\,\rotatebox[origin={c}]{90.0}{$\perp$}}f+\lambda\big{(}\mathscr{L}_{0,0}^{\,\rotatebox[origin={c}]{90.0}{$\perp$}}\big{)}^{2}f+\dots+\lambda^{N-1}\big{(}\mathscr{L}_{0,0}^{\,\rotatebox[origin={c}]{90.0}{$\perp$}}\big{)}^{N}f+\lambda^{N}\varphi_{f,N}(\lambda,x)\,,$
(B.4d)

for some analytic function
$\lambda\mapsto\varphi_{f,N}(\lambda,\cdot)\in\mathcal{W}_{k}$, where
$\mathscr{L}_{0,0}^{\,\rotatebox[origin={c}]{90.0}{$\perp$}}:\mathcal{W}_{k}\to\mathcal{W}_{k}$
is

${\footnotesize\mathscr{L}_{0,0}^{\,\rotatebox[origin={c}]{90.0}{$\perp$}}:=\begin{matrix}\big{(}{\mathtt{c}}_{\mathtt{h}}^{2}\partial_{x}^{2}+|D|\tanh(\mathtt{h}|D|)\big{)}^{-1}\end{matrix}\begin{bmatrix}{\mathtt{c}}_{\mathtt{h}}\partial_{x}&-|D|\tanh(\mathtt{h}|D|)\\\
1&{\mathtt{c}}_{\mathtt{h}}\partial_{x}\end{bmatrix},\quad\mathscr{L}_{0,0}^{\,\rotatebox[origin={c}]{90.0}{$\perp$}}\mathcal{W}^{\pm}_{k}=\mathcal{W}^{\mp}_{k}\,.}$
(B.5)

######  Remark B.3.

We will use in the sequel the following decomposition formula, for any,

$\begin{aligned}
&{\footnotesize\begin{matrix}\begin{bmatrix}\mathtt{a}\cos(x)\\\
\mathtt{b}\sin(x)\end{bmatrix}\end{matrix}}=\frac{1}{2}({\mathtt{a}}{\mathtt{c}}_{\mathtt{h}}^{-\frac{1}{2}}+{\mathtt{b}}{\mathtt{c}}_{\mathtt{h}}^{\frac{1}{2}})f_{1}^{+}+\frac{1}{2}({\mathtt{a}}{\mathtt{c}}_{\mathtt{h}}^{-\frac{1}{2}}-{\mathtt{b}}{\mathtt{c}}_{\mathtt{h}}^{\frac{1}{2}})f_{-1}^{+}\,,\\\
&{\footnotesize\begin{matrix}\begin{bmatrix}\mathtt{a}\sin(x)\\\
\mathtt{b}\cos(x)\end{bmatrix}\end{matrix}}=\frac{1}{2}({\mathtt{b}}{\mathtt{c}}_{\mathtt{h}}^{\frac{1}{2}}-{\mathtt{a}}{\mathtt{c}}_{\mathtt{h}}^{-\frac{1}{2}})f_{1}^{-}+\frac{1}{2}({\mathtt{b}}{\mathtt{c}}_{\mathtt{h}}^{\frac{1}{2}}+{\mathtt{a}}{\mathtt{c}}_{\mathtt{h}}^{-\frac{1}{2}})f_{-1}^{-}\,,\end{aligned}\quad\forall{\mathtt{a}}\,,{\mathtt{b}}\in\mathbb{C}\,.$
(B.6)

Notation. We denote by $\mathcal{O}(\lambda)$ an analytic function having a
zero of order $1$ at $\lambda=0$ and $\mathcal{O}_{Z}(\lambda^{m})$ an
analytic function with valued in a subspace $Z$ having a zero of order $m$ at
$\lambda=0$. We denote with $\mathcal{O}(\lambda^{-1}:\lambda)$ any function
having a Laurent series at $\lambda=0$ of the form
$\sum_{\begin{subarray}{c}j\in\mathbb{Z}\setminus\\{0\\}\end{subarray}}a_{j}\lambda^{j}$.
We denote by $f_{{\cal W}_{k}}$ a function in ${\cal W}_{k}$.

If $h(\lambda)=h_{0}+\mathcal{O}(\lambda^{-1}:\lambda)$, $h_{0}\in\mathbb{C}$,
then, by the residue theorem,

$\frac{1}{2\pi\mathrm{i}\,}\oint_{\Gamma}\frac{h(\lambda)}{\lambda}\mathrm{d}\lambda=h_{0}\,.$
(B.7)

We prepend to the proof of Lemma 3.4 a list of results given by straight-
forward computations.

######  Lemma B.4 (Action of $(\mathscr{L}_{0,0}-\lambda)^{-1}\mathcal{J}{\cal
B}_{0,1}$ on $\mathcal{V}_{0,0}$, $\mathcal{U}$ and $\mathcal{W}_{2}$).

One has

$\displaystyle(\mathscr{L}_{0,0}-\lambda)^{-1}\mathcal{J}{\cal
B}_{0,1}f_{1}^{+}=\frac{\zeta_{1}^{+}}{\lambda}f_{0}^{-}+\mathrm{A}_{2}^{+}+\lambda\mathrm{B}_{2}^{-}+\lambda^{2}f_{\mathcal{W}_{2}^{+}}+\lambda^{3}f_{\mathcal{W}_{2}^{-}}+\mathcal{O}_{\mathcal{W}_{2}}(\lambda^{4})\,,$
(B.8) $\displaystyle(\mathscr{L}_{0,0}-\lambda)^{-1}\mathcal{J}{\cal
B}_{0,1}f_{1}^{-}=\mathrm{A}_{2}^{-}+\lambda\mathrm{B}_{2}^{+}+\lambda^{2}f_{\mathcal{W}_{2}^{-}}+\lambda^{3}f_{\mathcal{W}_{2}^{+}}+\mathcal{O}_{\mathcal{W}_{2}}(\lambda^{4})\,,$
$\displaystyle(\mathscr{L}_{0,0}-\lambda)^{-1}\mathcal{J}{\cal
B}_{0,1}f_{0}^{+}=\frac{\zeta_{0}^{+}}{\lambda}f_{1}^{-}+\frac{\alpha_{0}^{+}}{\lambda^{2}+4{\mathtt{c}}_{\mathtt{h}}^{2}}f_{-1}^{+}+\lambda\frac{\beta_{0}^{+}}{\lambda^{2}+4{\mathtt{c}}_{\mathtt{h}}^{2}}f_{-1}^{-}\,,$
$\displaystyle(\mathscr{L}_{0,0}-\lambda)^{-1}\mathcal{J}{\cal
B}_{0,1}f_{0}^{-}=0\,,$
$\displaystyle(\mathscr{L}_{0,0}-\lambda)^{-1}\mathcal{J}{\cal
B}_{0,1}f^{+}_{-1}=\frac{\zeta_{-1}^{+}}{\lambda}f_{0}^{-}+\mathrm{A}_{-2}^{+}+\lambda
f_{\mathcal{W}_{2}^{-}}+\mathcal{O}_{\mathcal{W}_{2}}(\lambda^{2})\,,$
$\displaystyle(\mathscr{L}_{0,0}-\lambda)^{-1}\mathcal{J}{\cal
B}_{0,1}f^{-}_{-1}=f_{\mathcal{W}_{2}^{-}}+\mathcal{O}_{\mathcal{W}_{2}}(\lambda)\,,$

where $\zeta^{+}_{1},\zeta^{+}_{0}$,
$\alpha_{0}^{+},\beta_{0}^{+},\zeta_{-1}^{+}$ are real numbers, and

$\displaystyle\mathrm{A}_{2}^{+}(x):={\footnotesize\begin{bmatrix}\dfrac{-a_{1}^{[1]}{\mathtt{c}}_{\mathtt{h}}+(2+{\mathtt{c}}_{\mathtt{h}}^{4})p_{1}^{[1]}}{2{\mathtt{c}}_{\mathtt{h}}^{9/2}}\cos(2x)\\\\[11.38109pt]
-\dfrac{({\mathtt{c}}_{\mathtt{h}}^{4}+1)(a_{1}^{[1]}{\mathtt{c}}_{\mathtt{h}}-2p_{1}^{[1]})}{4{\mathtt{c}}_{\mathtt{h}}^{11/2}}\sin(2x)\end{bmatrix}}\,,\
\mathrm{B}_{2}^{-}(x):={\footnotesize\begin{bmatrix}\dfrac{({\mathtt{c}}_{\mathtt{h}}^{4}+1)\big{(}({\mathtt{c}}_{\mathtt{h}}^{4}+4)p_{1}^{[1]}-2a_{1}^{[1]}{\mathtt{c}}_{\mathtt{h}}\big{)}}{4{\mathtt{c}}_{\mathtt{h}}^{19/2}}\sin(2x)\\\\[11.38109pt]
\dfrac{({\mathtt{c}}_{\mathtt{h}}^{4}+1)\big{(}a_{1}^{[1]}{\mathtt{c}}_{\mathtt{h}}({\mathtt{c}}_{\mathtt{h}}^{4}+2)-(3{\mathtt{c}}_{\mathtt{h}}^{4}+4)p_{1}^{[1]}\big{)}}{8{\mathtt{c}}_{\mathtt{h}}^{21/2}}\cos(2x)\end{bmatrix}}\,,$
$\displaystyle\mathrm{A}_{2}^{-}(x):={\footnotesize\begin{bmatrix}\dfrac{\big{(}a_{1}^{[1]}{\mathtt{c}}_{\mathtt{h}}-({\mathtt{c}}_{\mathtt{h}}^{4}+2)p_{1}^{[1]}\big{)}}{2{\mathtt{c}}_{\mathtt{h}}^{9/2}}\sin(2x)\\\\[11.38109pt]
-\dfrac{({\mathtt{c}}_{\mathtt{h}}^{4}+1)(a_{1}^{[1]}{\mathtt{c}}_{\mathtt{h}}-2p_{1}^{[1]})}{4{\mathtt{c}}_{\mathtt{h}}^{11/2}}\cos(2x)\end{bmatrix}}\,,\
\
\mathrm{B}_{2}^{+}(x):={\footnotesize\begin{bmatrix}\dfrac{({\mathtt{c}}_{\mathtt{h}}^{4}+1)\big{(}({\mathtt{c}}_{\mathtt{h}}^{4}+4)p_{1}^{[1]}-2a_{1}^{[1]}{\mathtt{c}}_{\mathtt{h}}\big{)}}{4{\mathtt{c}}_{\mathtt{h}}^{19/2}}\cos(2x)\\\\[11.38109pt]
-\dfrac{({\mathtt{c}}_{\mathtt{h}}^{4}+1)\big{(}a_{1}^{[1]}{\mathtt{c}}_{\mathtt{h}}({\mathtt{c}}_{\mathtt{h}}^{4}+2)-(3{\mathtt{c}}_{\mathtt{h}}^{4}+4)p_{1}^{[1]}\big{)}}{8{\mathtt{c}}_{\mathtt{h}}^{21/2}}\sin(2x)\end{bmatrix}}\,,$
$\displaystyle\mathrm{A}_{-2}^{+}(x):={\footnotesize\begin{bmatrix}\dfrac{({\mathtt{c}}_{\mathtt{h}}^{3}p_{1}^{[1]}-a_{1}^{[1]})}{2{\mathtt{c}}_{\mathtt{h}}^{7/2}}\cos(2x)\\\\[11.38109pt]
-\dfrac{a_{1}^{[1]}({\mathtt{c}}_{\mathtt{h}}^{4}+1)}{4{\mathtt{c}}_{\mathtt{h}}^{9/2}}\sin(2x)\end{bmatrix}}\,.$
(B.9)

Moreover

$\displaystyle(\mathscr{L}_{0,0}-\lambda)^{-1}\mathcal{J}{\cal
B}_{0,1}\mathrm{A}_{2}^{+}=\frac{\zeta_{2}^{+}}{\lambda}f_{1}^{-}+\frac{\alpha_{2}^{+}}{\lambda^{2}+4{\mathtt{c}}_{\mathtt{h}}^{2}}f_{-1}^{+}+\mathrm{A}_{3}^{+}+\frac{\lambda\beta_{2}^{+}}{\lambda^{2}+4{\mathtt{c}}_{\mathtt{h}}^{2}}f_{-1}^{-}+\lambda
f_{\mathcal{W}_{3}^{-}}+\mathcal{O}_{\mathcal{W}_{3}}(\lambda^{2})\,,$
$\displaystyle(\mathscr{L}_{0,0}-\lambda)^{-1}\mathcal{J}{\cal
B}_{0,1}\mathrm{B}_{2}^{-}=\frac{\zeta_{3}^{+}}{\lambda}f_{1}^{+}+\frac{\alpha_{3}^{+}}{\lambda^{2}+4{\mathtt{c}}_{\mathtt{h}}^{2}}f_{-1}^{-}+f_{\mathcal{W}_{3}^{-}}+\frac{\lambda\beta_{3}^{+}}{\lambda^{2}+4{\mathtt{c}}_{\mathtt{h}}^{2}}f_{-1}^{+}+\mathcal{O}_{\mathcal{W}_{3}}(\lambda)\,,$
$\displaystyle(\mathscr{L}_{0,0}-\lambda)^{-1}\mathcal{J}{\cal
B}_{0,1}\mathrm{A}_{2}^{-}=\frac{\zeta_{2}^{-}}{\lambda}f_{1}^{+}+\frac{\alpha_{2}^{-}}{\lambda^{2}+4{\mathtt{c}}_{\mathtt{h}}^{2}}f_{-1}^{-}+f_{\mathcal{W}_{3}^{-}}+\frac{\lambda\beta_{2}^{-}}{\lambda^{2}+4{\mathtt{c}}_{\mathtt{h}}^{2}}f_{-1}^{+}+\lambda
f_{\mathcal{W}_{3}^{+}}+\mathcal{O}_{\mathcal{W}_{3}}(\lambda^{2})\,,$
$\displaystyle(\mathscr{L}_{0,0}-\lambda)^{-1}\mathcal{J}{\cal
B}_{0,1}\mathrm{B}_{2}^{+}=\frac{\zeta_{3}^{-}}{\lambda}f_{1}^{-}+\frac{\alpha_{3}^{-}}{\lambda^{2}+4{\mathtt{c}}_{\mathtt{h}}^{2}}f_{-1}^{+}+f_{\mathcal{W}_{3}^{+}}+\frac{\lambda\beta_{3}^{-}}{\lambda^{2}+4{\mathtt{c}}_{\mathtt{h}}^{2}}f_{-1}^{-}+\mathcal{O}_{\mathcal{W}_{3}}(\lambda)\,,$
(B.10)

where $\zeta_{2}^{-}$, $\beta_{2}^{\pm}$, $\alpha_{3}^{\pm}$ and
$\beta_{3}^{\pm}$ are real numbers and

$\displaystyle\zeta_{2}^{+}:=-\frac{(a_{1}^{[1]})^{2}{\mathtt{c}}_{\mathtt{h}}^{2}-2a_{1}^{[1]}({\mathtt{c}}_{\mathtt{h}}^{4}+2){\mathtt{c}}_{\mathtt{h}}p_{1}^{[1]}+(3{\mathtt{c}}_{\mathtt{h}}^{4}+4)(p_{1}^{[1]})^{2}}{8{\mathtt{c}}_{\mathtt{h}}^{5}}\,,$
(B.11)
$\displaystyle\alpha_{2}^{+}:=-\alpha_{2}^{-}:=-\frac{(a_{1}^{[1]})^{2}{\mathtt{c}}_{\mathtt{h}}-2a_{1}^{[1]}({\mathtt{c}}_{\mathtt{h}}^{4}+1)p_{1}^{[1]}+{\mathtt{c}}_{\mathtt{h}}^{3}(p_{1}^{[1]})^{2}}{4{\mathtt{c}}_{\mathtt{h}}^{3}}\,,$
$\displaystyle\zeta_{3}^{+}:=\zeta_{3}^{-}:=\frac{({\mathtt{c}}_{\mathtt{h}}^{4}+1)(a_{1}^{[1]}{\mathtt{c}}_{\mathtt{h}}-2p_{1}^{[1]})\big{(}({\mathtt{c}}_{\mathtt{h}}^{4}+2)p_{1}^{[1]}-a_{1}^{[1]}{\mathtt{c}}_{\mathtt{h}}\big{)}}{8{\mathtt{c}}_{\mathtt{h}}^{10}}\,,$
$\displaystyle\mathrm{A}_{3}^{+}(x):=\begin{bmatrix}\dfrac{(a_{1}^{[1]})^{2}({\mathtt{c}}_{\mathtt{h}}^{4}+3){\mathtt{c}}_{\mathtt{h}}^{2}-2{\mathtt{c}}_{\mathtt{h}}p_{1}^{[1]}a_{1}^{[1]}({\mathtt{c}}_{\mathtt{h}}^{8}+9{\mathtt{c}}_{\mathtt{h}}^{4}+6)+(11{\mathtt{c}}_{\mathtt{h}}^{8}+29{\mathtt{c}}_{\mathtt{h}}^{4}+12)(p_{1}^{[1]})^{2}}{32{\mathtt{c}}_{\mathtt{h}}^{19/2}}\cos(3x)\\\\[11.38109pt]
\dfrac{(3{\mathtt{c}}_{\mathtt{h}}^{4}+1)\big{(}(a_{1}^{[1]})^{2}{\mathtt{c}}_{\mathtt{h}}^{2}-2{\mathtt{c}}_{\mathtt{h}}p_{1}^{[1]}a_{1}^{[1]}({\mathtt{c}}_{\mathtt{h}}^{4}+2)+(3{\mathtt{c}}_{\mathtt{h}}^{4}+4)(p_{1}^{[1]})^{2}\big{)}}{32{\mathtt{c}}_{\mathtt{h}}^{21/2}}\sin(3x)\end{bmatrix}\,.$

###### Proof.

Use the operator ${\cal B}_{0,1}$ in (3.8b), Lemma B.2 and that

$\frac{1}{2\tanh(2\mathtt{h})-4{\mathtt{c}}_{\mathtt{h}}^{2}}=-\frac{1+{\mathtt{c}}_{\mathtt{h}}^{4}}{4{\mathtt{c}}_{\mathtt{h}}^{6}}\,,\quad\frac{1}{3\tanh(3\mathtt{h})-9{\mathtt{c}}_{\mathtt{h}}^{2}}=-\frac{1+3{\mathtt{c}}_{\mathtt{h}}^{4}}{24{\mathtt{c}}_{\mathtt{h}}^{6}}\,,$

which comes from the identities in (A.11). ∎

######  Lemma B.5 (Action of $(\mathscr{L}_{0,0}-\lambda)^{-1}\mathcal{J}{\cal
B}_{1,0}$ on $\mathcal{V}_{0,0}$ and $\mathcal{U}$).

One has

$\displaystyle(\mathscr{L}_{0,0}-\lambda)^{-1}\mathcal{J}{\cal
B}_{1,0}f_{1}^{+}=-\mathrm{i}\,\frac{\mu_{\mathtt{h}}}{\lambda}f_{1}^{+}+\mathrm{i}\,\frac{2{\mathtt{c}}_{\mathtt{h}}\mu_{\mathtt{h}}}{\lambda^{2}+4{\mathtt{c}}_{\mathtt{h}}^{2}}f_{-1}^{-}-\mathrm{i}\,\frac{\lambda\mu_{\mathtt{h}}}{\lambda^{2}+4{\mathtt{c}}_{\mathtt{h}}^{2}}f_{-1}^{+}\,,$
(B.12) $\displaystyle(\mathscr{L}_{0,0}-\lambda)^{-1}\mathcal{J}{\cal
B}_{1,0}f_{1}^{-}=-\mathrm{i}\,\frac{\mu_{\mathtt{h}}}{\lambda}f_{1}^{-}+\mathrm{i}\,\frac{2{\mathtt{c}}_{\mathtt{h}}\mu_{\mathtt{h}}}{\lambda^{2}+4{\mathtt{c}}_{\mathtt{h}}^{2}}f_{-1}^{+}+\mathrm{i}\,\lambda\frac{\mu_{\mathtt{h}}}{\lambda^{2}+4{\mathtt{c}}_{\mathtt{h}}^{2}}f_{-1}^{-}\,,$
$\displaystyle(\mathscr{L}_{0,0}-\lambda)^{-1}\mathcal{J}{\cal
B}_{1,0}f_{0}^{\pm}=0\,,$
$\displaystyle(\mathscr{L}_{0,0}-\lambda)^{-1}\mathcal{J}{\cal
B}_{1,0}f_{-1}^{+}=\mathrm{i}\,\frac{\mu_{\mathtt{h}}}{\lambda}f_{1}^{+}-\mathrm{i}\,\frac{2{\mathtt{c}}_{\mathtt{h}}\mu_{\mathtt{h}}}{\lambda^{2}+4{\mathtt{c}}_{\mathtt{h}}^{2}}f_{-1}^{-}+\mathrm{i}\,\frac{\lambda\mu_{\mathtt{h}}}{\lambda^{2}+4{\mathtt{c}}_{\mathtt{h}}^{2}}f_{-1}^{+}\,,$
$\displaystyle(\mathscr{L}_{0,0}-\lambda)^{-1}\mathcal{J}{\cal
B}_{1,0}f_{-1}^{-}=-\mathrm{i}\,\frac{\mu_{\mathtt{h}}}{\lambda}f_{1}^{-}+\mathrm{i}\,\frac{2{\mathtt{c}}_{\mathtt{h}}\mu_{\mathtt{h}}}{\lambda^{2}+4{\mathtt{c}}_{\mathtt{h}}^{2}}f_{-1}^{+}+\mathrm{i}\,\lambda\frac{\mu_{\mathtt{h}}}{\lambda^{2}+4{\mathtt{c}}_{\mathtt{h}}^{2}}f_{-1}^{-}\,,$

with

$\mu_{\mathtt{h}}:=\dfrac{({\mathtt{c}}_{\mathtt{h}}^{4}-1)\mathtt{h}-{\mathtt{c}}_{\mathtt{h}}^{2}}{2{\mathtt{c}}_{\mathtt{h}}}\,.$
(B.13)

###### Proof.

We apply the operator ${\cal B}_{1,0}=\ell_{1,0}(|D|)\Pi_{\mathtt{s}}$ in
(3.8b) to the vectors in $\mathcal{V}_{0,0}$ and use (B.6) and (B.4a)-(B.4c).
∎

######  Lemma B.6 (Action of $(\mathscr{L}_{0,0}-\lambda)^{-1}\mathcal{J}{\cal
B}_{0,2}$ on $\mathcal{V}_{0,0}$).

One has

$\displaystyle(\mathscr{L}_{0,0}-\lambda)^{-1}\mathcal{J}{\cal
B}_{0,2}f_{1}^{+}=\frac{\tau_{1}^{+}}{\lambda}f_{1}^{-}+\frac{\ell_{1}^{+}}{\lambda^{2}+4{\mathtt{c}}_{\mathtt{h}}^{2}}f_{-1}^{+}+\mathrm{L}_{3}^{+}+\frac{\lambda
m_{1}^{+}}{\lambda^{2}+4{\mathtt{c}}_{\mathtt{h}}^{2}}f_{-1}^{-}+O_{\mathcal{W}_{3}}(\lambda)\,,$
$\displaystyle(\mathscr{L}_{0,0}-\lambda)^{-1}\mathcal{J}{\cal
B}_{0,2}f_{1}^{-}=\frac{\tau_{1}^{-}}{\lambda}f_{1}^{+}+\frac{\ell_{1}^{-}}{\lambda^{2}+4{\mathtt{c}}_{\mathtt{h}}^{2}}f_{-1}^{-}+f_{\mathcal{W}_{3}^{-}}+\mathcal{O}_{\mathcal{U}\oplus\mathcal{W}_{3}}(\lambda)\,,$
(B.14) $\displaystyle(\mathscr{L}_{0,0}-\lambda)^{-1}\mathcal{J}{\cal
B}_{0,2}f_{0}^{+}=\frac{\tau_{0}^{+}}{\lambda}f_{0}^{-}+f_{\mathcal{W}_{2}^{+}}+\mathcal{O}_{\mathcal{W}_{2}}(\lambda)\,,\quad(\mathscr{L}_{0,0}-\lambda)^{-1}\mathcal{J}{\cal
B}_{0,2}f_{0}^{-}=0\,,$

where $m_{1}^{+}$, $\tau_{1}^{-}$, $\tau_{0}^{+}$, are real numbers, and

$\displaystyle\tau_{1}^{+}:=\frac{1}{4{\mathtt{c}}_{\mathtt{h}}}\big{(}2a_{2}^{[0]}{\mathtt{c}}_{\mathtt{h}}^{2}+a_{2}^{[2]}{\mathtt{c}}_{\mathtt{h}}^{2}+2\mathtt{f}_{2}(1-{\mathtt{c}}_{\mathtt{h}}^{4})-4{\mathtt{c}}_{\mathtt{h}}p_{2}^{[0]}-2{\mathtt{c}}_{\mathtt{h}}p_{2}^{[2]}\big{)}\,,$
$\displaystyle\ell_{1}^{+}:=\frac{1}{2}\big{(}{\mathtt{c}}_{\mathtt{h}}^{2}(2a_{2}^{[0]}+a_{2}^{[2]})-2\mathtt{f}_{2}(1-{\mathtt{c}}_{\mathtt{h}}^{4})\big{)}\,,\quad\ell_{1}^{-}:=\frac{1}{2}\big{(}{\mathtt{c}}_{\mathtt{h}}^{2}(-2a_{2}^{[0]}+a_{2}^{[2]})+2\mathtt{f}_{2}(1-{\mathtt{c}}_{\mathtt{h}}^{4})\big{)}\,,$
$\displaystyle\mathrm{L}_{3}^{+}(x):=\begin{bmatrix}-\dfrac{\big{(}a_{2}^{[2]}{\mathtt{c}}_{\mathtt{h}}({\mathtt{c}}_{\mathtt{h}}^{4}+3)-2(5{\mathtt{c}}_{\mathtt{h}}^{4}+3)p_{2}^{[2]})}{16{\mathtt{c}}_{\mathtt{h}}^{9/2}}\cos(3x)\\\\[11.38109pt]
-\dfrac{(3{\mathtt{c}}_{\mathtt{h}}^{4}+1)(a_{2}^{[2]}{\mathtt{c}}_{\mathtt{h}}-2p_{2}^{[2]})}{16{\mathtt{c}}_{\mathtt{h}}^{11/2}}\sin(3x)\end{bmatrix}\,.$
(B.15)

###### Proof.

We apply the operators ${\cal B}_{0,2}$ in (3.8c) and $\mathcal{J}$ in (1.5)
to the vectors in $\mathcal{V}_{0,0}$. Then we use (B.6) and Lemma B.2 to
obtain (B.14)-(B.15). ∎

######  Lemma B.7 (Action of $(\mathscr{L}_{0,0}-\lambda)^{-1}\mathcal{J}{\cal
B}_{1,1}$ on $\mathcal{V}_{0,0}$ and $\mathcal{U}$).

One has

$\displaystyle(\mathscr{L}_{0,0}-\lambda)^{-1}\mathcal{J}{\cal
B}_{1,1}f_{1}^{+}=-\frac{\mathrm{i}\,{\mathtt{c}}_{\mathtt{h}}^{-\frac{1}{2}}}{\lambda^{2}}f_{0}^{-}+\frac{\mathrm{i}\,{\mathtt{c}}_{\mathtt{h}}^{-\frac{1}{2}}}{\lambda}f_{0}^{+}+\mathrm{i}\,f_{\mathcal{W}_{2}^{-}}+\mathcal{O}_{\mathcal{W}}(\lambda)\,,$
(B.16) $\displaystyle(\mathscr{L}_{0,0}-\lambda)^{-1}\mathcal{J}{\cal
B}_{1,1}f_{1}^{-}=\frac{\mathrm{i}\,{\mathtt{c}}_{\mathtt{h}}^{-\frac{3}{2}}}{\lambda}f_{0}^{-}+\mathrm{i}\,\mathrm{Q}_{2}^{+}+\lambda\mathrm{i}\,f_{\mathcal{W}_{2}^{-}}+\mathcal{O}_{\mathcal{W}_{2}}(\lambda^{2})\,,$
$\displaystyle(\mathscr{L}_{0,0}-\lambda)^{-1}\mathcal{J}{\cal
B}_{1,1}f_{0}^{+}=\frac{\mathrm{i}\,{\mathtt{c}}_{\mathtt{h}}^{-\frac{3}{2}}}{\lambda}f_{1}^{+}-\frac{2\mathrm{i}\,{\mathtt{c}}_{\mathtt{h}}^{-\frac{1}{2}}}{\lambda^{2}+4{\mathtt{c}}_{\mathtt{h}}^{2}}f_{-1}^{-}+\lambda\frac{\mathrm{i}\,{\mathtt{c}}_{\mathtt{h}}^{-\frac{3}{2}}}{\lambda^{2}+4{\mathtt{c}}_{\mathtt{h}}^{2}}f_{-1}^{+}\,,$
$\displaystyle(\mathscr{L}_{0,0}-\lambda)^{-1}\mathcal{J}{\cal
B}_{1,1}f_{0}^{-}=\frac{\mathrm{i}\,{\mathtt{c}}_{\mathtt{h}}^{-\frac{1}{2}}}{\lambda}f_{1}^{-}+\frac{2\mathrm{i}\,{\mathtt{c}}_{\mathtt{h}}^{\frac{1}{2}}}{\lambda^{2}+4{\mathtt{c}}_{\mathtt{h}}^{2}}f_{-1}^{+}+\lambda\frac{\mathrm{i}\,{\mathtt{c}}_{\mathtt{h}}^{-\frac{1}{2}}}{\lambda^{2}+4{\mathtt{c}}_{\mathtt{h}}^{2}}f_{-1}^{-}\,,$

where

$\displaystyle\mathrm{Q}_{2}^{+}(x):=\begin{bmatrix}\dfrac{({\mathtt{c}}_{\mathtt{h}}^{4}+3)p_{1}^{[1]}}{4{\mathtt{c}}_{\mathtt{h}}^{9/2}}\cos(2x)\\\\[11.38109pt]
\dfrac{3({\mathtt{c}}_{\mathtt{h}}^{4}+1)p_{1}^{[1]}}{8{\mathtt{c}}_{\mathtt{h}}^{11/2}}\sin(2x)\end{bmatrix}\,.$
(B.17)

###### Proof.

We have ${\cal B}_{1,1}=-\mathrm{i}\,p_{1}(x)\mathcal{J}$ by (3.8c), with
$p_{1}(x)=p_{1}^{[1]}\cos(x)$ in (A.59a) and
$p_{1}^{[1]}=-2{\mathtt{c}}_{\mathtt{h}}^{-1}$. Use also Lemma B.2. ∎

######  Lemma B.8 (Action of $(\mathscr{L}_{0,0}-\lambda)^{-1}\mathcal{J}{\cal
B}_{2,0}$ on $f_{0}^{-}$).

One has

$(\mathscr{L}_{0,0}-\lambda)^{-1}\mathcal{J}{\cal
B}_{2,0}f_{0}^{-}=\frac{\mathtt{h}}{\lambda^{2}}f_{0}^{-}-\frac{\mathtt{h}}{\lambda}f_{0}^{+}\,.$
(B.18)

###### Proof.

We apply ${\cal B}_{2,0}=\ell_{2,0}(|D|)\Pi_{\mathtt{ev}}$ in (3.8c) and
(B.4b). ∎

######  Lemma B.9 (Action of
$((\mathscr{L}_{0,0}-\lambda)^{-1}\mathcal{J}{\cal B}_{0,1})^{2}$ on
$\mathcal{V}_{0,0}$).

One has

$\displaystyle\begin{aligned}
&[(\mathscr{L}_{0,0}-\lambda)^{-1}\mathcal{J}{\cal
B}_{0,1}]^{2}f_{1}^{+}=\frac{\zeta_{2}^{+}}{\lambda}f_{1}^{-}+\frac{\alpha_{2}^{+}}{\lambda^{2}+4{\mathtt{c}}_{\mathtt{h}}^{2}}f_{-1}^{+}+\zeta_{3}^{+}f_{1}^{+}+\mathrm{A}_{2}^{+}\\\
&\qquad\qquad\qquad\qquad\qquad\qquad+\lambda\big{(}f_{\mathcal{W}_{1}^{-}}+f_{\mathcal{W}_{3}^{-}}\big{)}+\lambda^{2}f_{\mathcal{W}_{1}^{+}}+\mathcal{O}_{\mathcal{W}_{3}}(\lambda^{2})+\mathcal{O}(\lambda^{3})\,,\\\
&[(\mathscr{L}_{0,0}-\lambda)^{-1}\mathcal{J}{\cal
B}_{0,1}]^{2}f_{1}^{-}=\frac{\zeta_{2}^{-}}{\lambda}f_{1}^{+}+\frac{\alpha_{2}^{-}}{\lambda^{2}+4{\mathtt{c}}_{\mathtt{h}}^{2}}f_{-1}^{-}+\zeta_{3}^{-}f_{1}^{-}+f_{\mathcal{W}_{3}^{-}}+\lambda\big{(}f_{\mathcal{W}_{1}^{+}}+f_{\mathcal{W}_{3}^{+}}\big{)}\\\
&\qquad\qquad\qquad\qquad\qquad\qquad+\lambda^{2}f_{\mathcal{W}_{1}^{-}}+\mathcal{O}_{\mathcal{W}_{3}}(\lambda^{2})+\mathcal{O}(\lambda^{3})\,,\end{aligned}$
(B.19) $\displaystyle[(\mathscr{L}_{0,0}-\lambda)^{-1}\mathcal{J}{\cal
B}_{0,1}]^{2}f_{0}^{+}=\zeta_{0}^{+}\mathrm{B}_{2}^{+}+\frac{\alpha_{0}^{+}}{4{\mathtt{c}}_{\mathtt{h}}^{2}}\mathrm{A}_{-2}^{+}+\mathcal{O}(\lambda^{-1}:\lambda)\,,\
\ [(\mathscr{L}_{0,0}-\lambda)^{-1}\mathcal{J}{\cal
B}_{0,1}]^{2}f_{0}^{-}=0\,.$

###### Proof.

Apply twice Lemma B.4 and use that

$\displaystyle\lambda^{2}\,(\mathscr{L}_{0,0}-\lambda)^{-1}\mathcal{J}{\cal
B}_{0,1}\,f_{\mathcal{W}_{2}^{\pm}}=\lambda\alpha_{5}^{\pm}f_{1}^{\mp}+\lambda^{2}f_{\mathcal{W}_{1}^{\pm}}+\mathcal{O}_{\mathcal{W}_{3}}(\lambda^{2})+\mathcal{O}(\lambda^{3})\
,$ $\displaystyle\lambda^{3}\,(\mathscr{L}_{0,0}-\lambda)^{-1}\mathcal{J}{\cal
B}_{0,1}\
f_{\mathcal{W}_{2}^{\mp}}=\lambda^{2}\alpha_{6}^{\pm}f_{1}^{\pm}+\mathcal{O}(\lambda^{3})\
,$ $\displaystyle(\mathscr{L}_{0,0}-\lambda)^{-1}\mathcal{J}{\cal
B}_{0,1}\mathcal{O}_{\mathcal{W}_{2}}(\lambda^{4})=\mathcal{O}(\lambda^{3})$

where $\alpha_{5}^{\pm}$, $\alpha_{6}^{\pm}$ are real numbers. ∎

We further list a series of identities to exploit later.

By applying first Lemma B.5 and then Lemma B.4 we get

$\displaystyle(\mathscr{L}_{0,0}-\lambda)^{-1}\mathcal{J}{\cal
B}_{0,1}(\mathscr{L}_{0,0}-\lambda)^{-1}\mathcal{J}{\cal
B}_{1,0}f_{1}^{+}=-\mathrm{i}\,\mu_{\mathtt{h}}\Big{[}\frac{\zeta_{1}^{+}}{\lambda^{2}}f_{0}^{-}+\frac{1}{\lambda}\mathrm{A}_{2}^{+}+f_{\mathcal{W}_{2}^{-}}+\alpha_{7}f_{0}^{-}+\mathcal{O}_{\mathcal{W}_{2}}(\lambda)+\mathcal{O}_{\mathcal{W}_{0}}(\lambda^{2})\Big{]}\,,$
$\displaystyle{(\mathscr{L}_{0,0}-\lambda)^{-1}\mathcal{J}{\cal
B}_{0,1}(\mathscr{L}_{0,0}-\lambda)^{-1}\mathcal{J}{\cal
B}_{1,0}f_{1}^{-}=-\mathrm{i}\,\mu_{\mathtt{h}}\Big{[}\frac{1}{\lambda}\Big{(}\mathrm{A}_{2}^{-}-\frac{2{\mathtt{c}}_{\mathtt{h}}\zeta_{-1}^{+}}{\lambda^{2}+4{\mathtt{c}}_{\mathtt{h}}^{2}}f_{0}^{-}\Big{)}+\mathrm{J}_{2}^{+}+\lambda
f_{\mathcal{W}_{2}^{-}}+\mathcal{O}_{\mathcal{W}_{2}}(\lambda^{2})}\Big{]}\,,$
$\displaystyle(\mathscr{L}_{0,0}-\lambda)^{-1}\mathcal{J}{\cal
B}_{0,1}(\mathscr{L}_{0,0}-\lambda)^{-1}\mathcal{J}{\cal
B}_{1,0}f_{0}^{\pm}=0\,,$ (B.20)

where $\alpha_{7}$ is a real number and

$\mathrm{J}_{2}^{+}(x):=\mathrm{B}_{2}^{+}(x)-\frac{1}{2{\mathtt{c}}_{\mathtt{h}}}\mathrm{A}_{-2}^{+}(x)\stackrel{{\scriptstyle\eqref{risolventeintermedio.1}}}{{=}}\begin{bmatrix}\dfrac{(5{\mathtt{c}}_{\mathtt{h}}^{4}+4)p_{1}^{[1]}-a_{1}^{[1]}{\mathtt{c}}_{\mathtt{h}}({\mathtt{c}}_{\mathtt{h}}^{4}+2)}{4{\mathtt{c}}_{\mathtt{h}}^{19/2}}\cos(2x)\\\\[11.38109pt]
\dfrac{({\mathtt{c}}_{\mathtt{h}}^{4}+1)\big{(}(3{\mathtt{c}}_{\mathtt{h}}^{4}+4)p_{1}^{[1]}-2a_{1}^{[1]}{\mathtt{c}}_{\mathtt{h}}\big{)}}{8{\mathtt{c}}_{\mathtt{h}}^{21/2}}\sin(2x)\end{bmatrix}\,.$
(B.21)

By applying first Lemma B.4 and then Lemma B.5, and since $\mathcal{J}{\cal
B}_{1,0}\mathcal{W}_{k}\subseteq\mathcal{W}_{k}$, we get

$\displaystyle(\mathscr{L}_{0,0}-\lambda)^{-1}\mathcal{J}{\cal
B}_{1,0}(\mathscr{L}_{0,0}-\lambda)^{-1}\mathcal{J}{\cal
B}_{0,1}f_{1}^{+}=\mathrm{i}\,f_{\mathcal{W}_{2}^{-}}+\mathcal{O}_{\mathcal{W}_{2}}(\lambda)\,,$
(B.22) $\displaystyle(\mathscr{L}_{0,0}-\lambda)^{-1}\mathcal{J}{\cal
B}_{1,0}(\mathscr{L}_{0,0}-\lambda)^{-1}\mathcal{J}{\cal
B}_{0,1}f_{1}^{-}=\mathrm{i}\,\mathrm{S}_{2}^{+}+\mathrm{i}\,\lambda
f_{\mathcal{W}_{2}^{-}}+\mathcal{O}_{\mathcal{W}_{2}}(\lambda^{2})\,,$
$\displaystyle(\mathscr{L}_{0,0}-\lambda)^{-1}\mathcal{J}{\cal
B}_{1,0}(\mathscr{L}_{0,0}-\lambda)^{-1}\mathcal{J}{\cal
B}_{0,1}f_{0}^{+}=\mathrm{i}\,f_{\mathcal{W}_{1}^{-}}+\mathcal{O}(\lambda^{-1}:\lambda)\,,$
$\displaystyle(\mathscr{L}_{0,0}-\lambda)^{-1}\mathcal{J}{\cal
B}_{1,0}(\mathscr{L}_{0,0}-\lambda)^{-1}\mathcal{J}{\cal
B}_{0,1}f_{0}^{-}=0\,,$

where, using also (B.5), (3.8b)–(3.10a), (B.9)

$\mathrm{i}\,\mathrm{S}_{2}^{+}(x):=(\mathscr{L}_{0,0})^{\,\rotatebox[origin={c}]{90.0}{$\perp$}}\mathcal{J}{\cal
B}_{1,0}\mathrm{A}_{2}^{-}=\mathrm{i}\,\begin{bmatrix}\dfrac{({\mathtt{c}}_{\mathtt{h}}^{8}\mathtt{h}+{\mathtt{c}}_{\mathtt{h}}^{6}-2{\mathtt{c}}_{\mathtt{h}}^{4}\mathtt{h}+{\mathtt{c}}_{\mathtt{h}}^{2}+\mathtt{h})(a_{1}^{[1]}{\mathtt{c}}_{\mathtt{h}}-2p_{1}^{[1]})}{4{\mathtt{c}}_{\mathtt{h}}^{21/2}}\cos(2x)\\\\[11.38109pt]
\dfrac{({\mathtt{c}}_{\mathtt{h}}^{8}\mathtt{h}+{\mathtt{c}}_{\mathtt{h}}^{6}-2{\mathtt{c}}_{\mathtt{h}}^{4}\mathtt{h}+{\mathtt{c}}_{\mathtt{h}}^{2}+\mathtt{h})(a_{1}^{[1]}{\mathtt{c}}_{\mathtt{h}}-2p_{1}^{[1]})}{8{\mathtt{c}}_{\mathtt{h}}^{23/2}}\sin(2x)\end{bmatrix}\,.$
(B.23)

We are now in position to prove Lemmata 3.4 and 3.5.
Proof of Lemma 3.4. The proof is divided in three parts, one for each group of
formulas in (3.19).

Computation of $P_{0,2}f_{j}^{\sigma}$. Since
$P_{0,\epsilon}f_{0}^{-}=f_{0}^{-}$ (cfr. [10, Lemma A.4]) we have

$P_{0,2}f_{0}^{-}=0\,.$ (B.24)

On the other hand, for $f_{j}^{\sigma}\in\\{f_{1}^{+},f_{1}^{-},f_{0}^{+}\\}$,
in view of (3.13) we have, by (B.4a)-(B.4b) and since ${\cal
B}_{0,1}f_{0}^{-}={\cal B}_{0,2}f_{0}^{-}=0$,

$\displaystyle P_{0,2}f_{j}^{\sigma}$
$\displaystyle=-\frac{1}{2\pi\mathrm{i}\,}\oint_{\Gamma}\frac{(\mathscr{L}_{0,0}-\lambda)^{-1}}{\lambda}\mathcal{J}{\cal
B}_{0,2}f_{j}^{\sigma}\mathrm{d}\lambda$ (B.25)
$\displaystyle+\frac{1}{2\pi\mathrm{i}\,}\oint_{\Gamma}\frac{(\mathscr{L}_{0,0}-\lambda)^{-1}}{\lambda}\mathcal{J}{\cal
B}_{0,1}(\mathscr{L}_{0,0}-\lambda)^{-1}\mathcal{J}{\cal
B}_{0,1}f_{j}^{\sigma}\mathrm{d}\lambda=:\mathrm{I}_{j}^{\sigma}+\mathrm{I\\!I}_{j}^{\sigma}\,.$

In case $f_{j}^{\sigma}=f_{0}^{+}$ one readily sees, in view of (B.14) for
$\mathrm{I}_{0}^{+}$ and (LABEL:2104_1749) for $\mathrm{I\\!I}_{0}^{+}$, that
$P_{0,2}f_{0}^{+}\in\mathcal{W}_{2}^{+}$ which implies the second statement in
(3.19a). We now compute the remaining four terms.

First by Lemma B.6 and the residue theorem

$\mathrm{I}_{1}^{+}=-\frac{\ell_{1}^{+}}{{4{\mathtt{c}}_{\mathtt{h}}^{2}}}f_{-1}^{+}-\mathrm{L}_{3}^{+}\,,\qquad\mathrm{I}_{1}^{-}=-\frac{\ell_{1}^{-}}{{4{\mathtt{c}}_{\mathtt{h}}^{2}}}f_{-1}^{-}+f_{\mathcal{W}_{3}^{-}}\,.$
(B.26)

Then by (LABEL:2104_1749) and the residue theorem

$\mathrm{I\\!I}_{1}^{+}=\frac{\alpha_{2}^{+}}{4{\mathtt{c}}_{\mathtt{h}}^{2}}f_{-1}^{+}+\zeta_{3}^{+}f_{1}^{+}+\mathrm{A}_{3}^{+}\,,\qquad\mathrm{I\\!I}_{1}^{-}=\frac{\alpha_{2}^{-}}{4{\mathtt{c}}_{\mathtt{h}}^{2}}f_{-1}^{-}+\zeta_{3}^{-}f_{1}^{-}+f_{\mathcal{W}_{3}^{-}}\
.$ (B.27)

In conclusion we have formulae (3.19a), (3.19b) with

$P_{0,2}f_{1}^{+}=\underbrace{\frac{\alpha_{2}^{+}-\ell_{1}^{+}}{4{\mathtt{c}}_{\mathtt{h}}^{2}}}_{\mathfrak{u}_{0,2}^{+}}f_{-1}^{+}+\underbrace{\zeta_{3}^{+}}_{\mathfrak{n}_{0,2}}f_{1}^{+}\underbrace{-\mathrm{L}_{3}^{+}+\mathrm{A}_{3}^{+}}_{\scriptsize\begin{bmatrix}\mathfrak{a}_{0,2}\cos(3x)\\\
\mathfrak{b}_{0,2}\sin(3x)\end{bmatrix}},\quad
P_{0,2}f_{1}^{-}=\underbrace{\frac{\alpha_{2}^{-}-\ell_{1}^{-}}{4{\mathtt{c}}_{\mathtt{h}}^{2}}}_{\mathfrak{u}_{0,2}^{-}}f_{-1}^{-}+\underbrace{\zeta_{3}^{-}}_{\mathfrak{n}_{0,2}}f_{1}^{-}+f_{\mathcal{W}_{3}^{-}}\,,$

and we obtain the explicit expression (3.19c) of the coefficients given by

$\displaystyle\mathfrak{n}_{0,2}:=\zeta_{3}^{+}\,,\ \
\mathfrak{u}_{0,2}^{+}:=\frac{\alpha_{2}^{+}-\ell_{1}^{+}}{4{\mathtt{c}}_{\mathtt{h}}^{2}}\,,\
\
\mathfrak{u}_{0,2}^{-}:=\frac{\alpha_{2}^{-}-\ell_{1}^{-}}{4{\mathtt{c}}_{\mathtt{h}}^{2}}\,,\
\ {\footnotesize\begin{bmatrix}\mathfrak{a}_{0,2}\cos(3x)\\\
\mathfrak{b}_{0,2}\sin(3x)\end{bmatrix}}:=\mathrm{A}_{3}^{+}(x)-\mathrm{L}_{3}^{+}(x)\,,$

with $\zeta_{3}^{+}$, $\alpha_{2}^{\pm}$, $\mathrm{A}_{3}^{+}$ in (B.11) and
$\ell_{1}^{\pm}$, $\mathrm{L}_{3}^{+}$ in (B.15).

Computation of $P_{2,0}f_{0}^{-}$. Since $P_{\mu,0}f_{0}^{-}=f_{0}^{-}$ (cfr.
[10, Lemma A.5]) we have

$P_{2,0}f_{0}^{-}=0\,.$ (B.28)

Computation of $P_{1,1}f_{j}^{\sigma}$. In case
$f_{j}^{\sigma}\in\\{f_{1}^{+},f_{1}^{-},f_{0}^{-}\\}$ by (3.13), (3.14) and
(B.4a), we have

$\displaystyle P_{1,1}f_{j}^{\sigma}$
$\displaystyle=-\frac{1}{2\pi\mathrm{i}\,}\oint_{\Gamma}\frac{(\mathscr{L}_{0,0}-\lambda)^{-1}}{\lambda}\mathcal{J}{\cal
B}_{1,1}f_{j}^{\sigma}\mathrm{d}\lambda+\frac{1}{2\pi\mathrm{i}\,}\oint_{\Gamma}\frac{(\mathscr{L}_{0,0}-\lambda)^{-1}}{\lambda}\mathcal{J}{\cal
B}_{0,1}(\mathscr{L}_{0,0}-\lambda)^{-1}\mathcal{J}{\cal
B}_{1,0}f_{j}^{\sigma}\mathrm{d}\lambda$
$\displaystyle+\frac{1}{2\pi\mathrm{i}\,}\oint_{\Gamma}\frac{(\mathscr{L}_{0,0}-\lambda)^{-1}}{\lambda}\mathcal{J}{\cal
B}_{1,0}(\mathscr{L}_{0,0}-\lambda)^{-1}\mathcal{J}{\cal
B}_{0,1}f_{j}^{\sigma}\mathrm{d}\lambda=:\mathrm{I\\!I\\!I}_{j}^{\sigma}+\mathrm{I\\!V}_{j}^{\sigma}+\mathrm{V}_{j}^{\sigma}\,,$
(B.29a) whereas, by (B.4b), (B.12), (B.8) $\displaystyle P_{1,1}f_{0}^{+}$
$\displaystyle=-\frac{1}{2\pi\mathrm{i}\,}\oint_{\Gamma}\frac{(\mathscr{L}_{0,0}-\lambda)^{-1}}{\lambda}\mathcal{J}{\cal
B}_{1,1}f_{0}^{+}\mathrm{d}\lambda+\frac{1}{2\pi\mathrm{i}\,}\oint_{\Gamma}\frac{(\mathscr{L}_{0,0}-\lambda)^{-1}}{\lambda^{2}}\mathcal{J}{\cal
B}_{1,1}f_{0}^{-}\mathrm{d}\lambda$
$\displaystyle+\frac{1}{2\pi\mathrm{i}\,}\oint_{\Gamma}\frac{(\mathscr{L}_{0,0}-\lambda)^{-1}}{\lambda}\mathcal{J}{\cal
B}_{1,0}(\mathscr{L}_{0,0}-\lambda)^{-1}\mathcal{J}{\cal
B}_{0,1}f_{0}^{+}\mathrm{d}\lambda=:\mathrm{I\\!I\\!I}_{0}^{+}+\mathrm{I\\!V}_{0}^{+}+\mathrm{V}_{0}^{+}\,.$
(B.29b)

When $f_{j}^{\sigma}=f_{1}^{+}$ one readily sees, in view of (B.16) for
$\mathrm{I\\!I\\!I}_{1}^{+}$, (B.20) for $\mathrm{I\\!V}_{1}^{+}$ and (B.22)
for $\mathrm{V}_{1}^{+}$, that
$P_{1,1}f_{1}^{+}\in\mathrm{i}\,\mathcal{W}_{0}^{-}\oplus_{\mathbb{R}}\mathrm{i}\,\mathcal{W}_{2}^{-}$
as stated in (3.19e). Similarly, when $f_{j}^{\sigma}=f_{0}^{+}$ one has, in
view of (B.16) for $\mathrm{I\\!I\\!I}_{0}^{+}$ and $\mathrm{I\\!V}_{0}^{+}$
and (B.22) for $\mathrm{V}_{0}^{+}$, that
$P_{1,1}f_{0}^{+}\in\mathrm{i}\,\mathcal{W}_{1}^{-}$ as stated in (3.19e).

We now compute the remaining terms. By Lemma B.7 and the residue theorem

$\mathrm{I\\!I\\!I}_{1}^{-}=-{\mathrm{i}\,\mathrm{Q}_{2}^{+}}\,,\quad\mathrm{I\\!I\\!I}_{0}^{-}=-\frac{\mathrm{i}\,}{2{\mathtt{c}}_{\mathtt{h}}^{3/2}}f_{-1}^{+}\,.$
(B.30)

By (B.20) we have

$\mathrm{I\\!V}_{1}^{-}=-\mathrm{i}\,\mu_{\mathtt{h}}{\mathrm{J}_{2}^{+}},\quad\mathrm{I\\!V}_{0}^{-}=0\,.$
(B.31)

By (B.22) we have

$\mathrm{V}_{1}^{-}={\mathrm{i}\,\mathrm{S}_{2}^{+}}\,,\quad\mathrm{V}_{0}^{-}=0\,.$
(B.32)

In conclusion we have formulae (3.19e) with

$P_{1,1}f_{1}^{-}=\underbrace{-{\mathrm{i}\,\mathrm{Q}_{2}^{+}}+{\mathrm{i}\,\mathrm{S}_{2}^{+}}-\mathrm{i}\,\mu_{\mathtt{h}}{\mathrm{J}_{2}^{+}}}_{\mathrm{i}\,{\scriptsize\begin{bmatrix}\mathfrak{a}_{1,1}\cos(2x)\\\
\mathfrak{b}_{1,1}\sin(2x)\end{bmatrix}}}\,,\quad{P_{1,1}f_{0}^{-}=-\frac{\mathrm{i}\,}{2{\mathtt{c}}_{\mathtt{h}}^{3/2}}f_{-1}^{+}}\,,$
(B.33)

and we obtain the explicit expression (3.20) of the coefficients given by

$\begin{bmatrix}\mathfrak{a}_{1,1}\cos(2x)\\\
\mathfrak{b}_{1,1}\sin(2x)\end{bmatrix}:=-\mathrm{Q}_{2}^{+}+\mathrm{S}_{2}^{+}-\mu_{\mathtt{h}}\mathrm{J}_{2}^{+}$

with $\mathrm{S}_{2}^{+}$ in (B.23), $\mu_{\mathtt{h}}$ in (B.13),
$\mathrm{Q}_{2}^{+}$ in (B.17) and $\mathrm{J}_{2}^{+}$ in (B.21).
This concludes the proof of Lemma 3.4. ∎

Proof of Lemma 3.5.

Computation of $P_{0,3}f_{j}^{\sigma}$. Similarly to (B.24) we have
$P_{0,3}f_{0}^{-}=0$ (as stated in (3.21)). Let us now compute
$P_{0,3}f_{1}^{+}$. By (3.13), (3.14) and (B.4a)

$\displaystyle P_{0,3}f_{1}^{+}$
$\displaystyle=-\frac{1}{2\pi\mathrm{i}\,}\oint_{\Gamma}\frac{(\mathscr{L}_{0,0}-\lambda)^{-1}}{\lambda}\mathcal{J}{\cal
B}_{0,3}f_{1}^{+}\mathrm{d}\lambda$
$\displaystyle+\frac{1}{2\pi\mathrm{i}\,}\oint_{\Gamma}\frac{(\mathscr{L}_{0,0}-\lambda)^{-1}}{\lambda}\mathcal{J}{\cal
B}_{0,2}(\mathscr{L}_{0,0}-\lambda)^{-1}\mathcal{J}{\cal
B}_{0,1}f_{1}^{+}\mathrm{d}\lambda$
$\displaystyle+\frac{1}{2\pi\mathrm{i}\,}\oint_{\Gamma}\frac{(\mathscr{L}_{0,0}-\lambda)^{-1}}{\lambda}\mathcal{J}{\cal
B}_{0,1}(\mathscr{L}_{0,0}-\lambda)^{-1}\mathcal{J}{\cal
B}_{0,2}f_{1}^{+}\mathrm{d}\lambda$
$\displaystyle-\frac{1}{2\pi\mathrm{i}\,}\oint_{\Gamma}\frac{(\mathscr{L}_{0,0}-\lambda)^{-1}}{\lambda}\mathcal{J}{\cal
B}_{0,1}(\mathscr{L}_{0,0}-\lambda)^{-1}\mathcal{J}{\cal
B}_{0,1}(\mathscr{L}_{0,0}-\lambda)^{-1}\mathcal{J}{\cal
B}_{0,1}f_{1}^{+}\mathrm{d}\lambda$
$\displaystyle=:\mathrm{V\\!I}+\mathrm{V\\!I\\!I}+\mathrm{V\\!I\\!I\\!I}+\mathrm{I\\!X}\,.$
(B.34)

We now compute these four terms.

$\mathrm{V\\!I}\big{)}$ By (3.8d) we have ${\cal
B}_{0,3}=\begin{bmatrix}a_{3}(x)&-p_{3}(x)\partial_{x}\\\ \partial_{x}\circ
p_{3}(x)&0\end{bmatrix}$ with $p_{3}(x),a_{3}(x)$ in (A.59b)-(LABEL:aino3).
Then ${\cal B}_{0,3}f_{0}^{-}=0$ whereas $\mathcal{J}{\cal
B}_{0,3}f_{1}^{+}=\alpha_{10}f_{0}^{-}+{\mathrm{W}_{2}^{-}}+f_{\mathcal{W}_{4}^{-}}$
where $\alpha_{10}\in\mathbb{R}$ and

${\mathrm{W}_{2}^{-}}(x):=\begin{bmatrix}-{\mathtt{c}}_{\mathtt{h}}^{\frac{1}{2}}(p_{3}^{[1]}+p_{3}^{[3]})\,\sin(2x)\\\\[11.38109pt]
\dfrac{-{\mathtt{c}}_{\mathtt{h}}(a_{3}^{[1]}+a_{3}^{[3]})+(p_{3}^{[1]}+p_{3}^{[3]})}{2{\mathtt{c}}_{\mathtt{h}}^{\frac{1}{2}}}\cos(2x)\end{bmatrix}\
.$ (B.35)

Hence by (B.4d) we get

$(\mathscr{L}_{0,0}-\lambda)^{-1}\mathcal{J}{\cal
B}_{0,3}f_{1}^{+}=\mathscr{L}_{0,0}^{\,\rotatebox[origin={c}]{90.0}{$\perp$}}({\mathrm{W}_{2}^{-}}+f_{\mathcal{W}_{4}^{-}})+\mathcal{O}(\lambda^{-1}:\lambda)\,,$

and, by (B.7), the term $\mathrm{V\\!I}$ in (B.34) is

$\mathrm{V\\!I}=-\mathscr{L}_{0,0}^{\,\rotatebox[origin={c}]{90.0}{$\perp$}}{\mathrm{W}_{2}^{-}}+\widetilde{f}_{\mathcal{W}_{4}^{+}}\,,$
(B.36)

with

$\mathscr{L}_{0,0}^{\,\rotatebox[origin={c}]{90.0}{$\perp$}}{\mathrm{W}_{2}^{-}}(x)=-\begin{bmatrix}\kappa_{1}\cos(2x)\\\\[11.38109pt]
\varsigma_{1}\sin(2x)\end{bmatrix}\,,\quad\begin{aligned}
\kappa_{1}:&=\dfrac{{\mathtt{c}}_{\mathtt{h}}(a_{3}^{[1]}+a_{3}^{[3]})-({\mathtt{c}}_{\mathtt{h}}^{4}+2)(p_{3}^{[1]}+p_{3}^{[3]})}{2{\mathtt{c}}_{\mathtt{h}}^{9/2}}\,,\\\
\varsigma_{1}:&=\dfrac{({\mathtt{c}}_{\mathtt{h}}^{4}+1)({\mathtt{c}}_{\mathtt{h}}(a_{3}^{[1]}+a_{3}^{[3]})-2(p_{3}^{[1]}+p_{3}^{[3]}))}{4{\mathtt{c}}_{\mathtt{h}}^{11/2}}\,.\end{aligned}$
(B.37)

$\mathrm{V\\!I\\!I}\big{)}$ By Lemma B.4 and since ${\cal
B}_{0,2}f_{0}^{-}=0$, we get

$\displaystyle(\mathscr{L}_{0,0}-\lambda)^{-1}\mathcal{J}{\cal
B}_{0,2}(\mathscr{L}_{0,0}-\lambda)^{-1}\mathcal{J}{\cal
B}_{0,1}f_{1}^{+}=(\mathscr{L}_{0,0}-\lambda)^{-1}\mathcal{J}{\cal
B}_{0,2}\mathrm{A}_{2}^{+}$ (B.38)
$\displaystyle\qquad+\lambda(\mathscr{L}_{0,0}-\lambda)^{-1}\mathcal{J}{\cal
B}_{0,2}f_{\mathcal{W}_{2}^{-}}+\lambda^{2}(\mathscr{L}_{0,0}-\lambda)^{-1}\mathcal{J}{\cal
B}_{0,2}f_{\mathcal{W}_{2}^{+}}+\mathcal{O}(\lambda)\,.$

Applying the operators ${\cal B}_{0,2}$ in (3.8c) and $\mathcal{J}$ in (1.5)
to the vector $\mathrm{A}_{2}^{+}$ in (B.9) one obtains

$\mathcal{J}{\cal
B}_{0,2}\mathrm{A}_{2}^{+}={\mathrm{X}_{2}^{-}}+f_{\mathcal{W}_{0}^{-}}+f_{\mathcal{W}_{4}^{-}}\,,$
(B.39a) where
${\mathrm{X}_{2}^{-}}(x):=\begin{bmatrix}-\dfrac{({\mathtt{c}}_{\mathtt{h}}^{4}-1)^{2}\mathtt{f}_{2}(a_{1}^{[1]}{\mathtt{c}}_{\mathtt{h}}-2p_{1}^{[1]})+{\mathtt{c}}_{\mathtt{h}}({\mathtt{c}}_{\mathtt{h}}^{4}+1)p_{2}^{[0]}\big{(}({\mathtt{c}}_{\mathtt{h}}^{4}+2)p_{1}^{[1]}-a_{1}^{[1]}{\mathtt{c}}_{\mathtt{h}}\big{)}}{{\mathtt{c}}_{\mathtt{h}}^{11/2}({\mathtt{c}}_{\mathtt{h}}^{4}+1)}\sin(2x)\\\\[11.38109pt]
-\dfrac{a_{2}^{[0]}{\mathtt{c}}_{\mathtt{h}}\big{(}({\mathtt{c}}_{\mathtt{h}}^{4}+2)p_{1}^{[1]}-a_{1}^{[1]}{\mathtt{c}}_{\mathtt{h}}\big{)}+({\mathtt{c}}_{\mathtt{h}}^{4}+1)p_{2}^{[0]}(a_{1}^{[1]}{\mathtt{c}}_{\mathtt{h}}-2p_{1}^{[1]})}{2{\mathtt{c}}_{\mathtt{h}}^{11/2}}\cos(2x)\end{bmatrix}\
.$ (B.39b) On the other hand, in view of (B.3) and Lemma B.2, one has
$\lambda(\mathscr{L}_{0,0}-\lambda)^{-1}\mathcal{J}{\cal
B}_{0,2}f_{\mathcal{W}_{2}^{-}}=\mathcal{O}(\lambda)\,,\quad\lambda^{2}(\mathscr{L}_{0,0}-\lambda)^{-1}\mathcal{J}{\cal
B}_{0,2}f_{\mathcal{W}_{2}^{+}}=\mathcal{O}(\lambda)\,.$ (B.39c)

Then Lemmata B.6, B.2, (B.38) and (B.39) give

$\displaystyle(\mathscr{L}_{0,0}-\lambda)^{-1}\mathcal{J}{\cal
B}_{0,2}(\mathscr{L}_{0,0}-\lambda)^{-1}\mathcal{J}{\cal
B}_{0,1}f_{1}^{+}=\mathscr{L}_{0,0}^{\,\rotatebox[origin={c}]{90.0}{$\perp$}}({\mathrm{X}_{2}^{-}}+f_{\mathcal{W}_{4}^{-}})+\mathcal{O}(\lambda^{-1}:\lambda)\,,$

and, by (B.7), the term $\mathrm{V\\!I\\!I}$ in (B.34) is

$\displaystyle\mathrm{V\\!I\\!I}$
$\displaystyle=\mathscr{L}_{0,0}^{\,\rotatebox[origin={c}]{90.0}{$\perp$}}{\mathrm{X}_{2}^{-}}+\widetilde{f}_{\mathcal{W}_{4}^{+}}\,,$
(B.40)

with, by (B.5) and (B.39b),

$\mathscr{L}_{0,0}^{\,\rotatebox[origin={c}]{90.0}{$\perp$}}{\mathrm{X}_{2}^{-}}(x)=\begin{bmatrix}\kappa_{2}\cos(2x)\\\
\varsigma_{2}\sin(2x)\end{bmatrix}$ (B.41)

with

$\displaystyle{\footnotesize\kappa_{2}:=}{\scriptsize\begin{matrix}\dfrac{a_{1}^{[1]}{\mathtt{c}}_{\mathtt{h}}\big{(}a_{2}^{[0]}{\mathtt{c}}_{\mathtt{h}}^{2}+({\mathtt{c}}_{\mathtt{h}}^{4}-1)^{2}\mathtt{f}_{2}-2({\mathtt{c}}_{\mathtt{h}}^{4}+1){\mathtt{c}}_{\mathtt{h}}p_{2}^{[0]}\big{)}+p_{1}^{[1]}\big{(}-a_{2}^{[0]}({\mathtt{c}}_{\mathtt{h}}^{4}+2){\mathtt{c}}_{\mathtt{h}}^{2}-2({\mathtt{c}}_{\mathtt{h}}^{4}-1)^{2}\mathtt{f}_{2}+({\mathtt{c}}_{\mathtt{h}}^{8}+5{\mathtt{c}}_{\mathtt{h}}^{4}+4){\mathtt{c}}_{\mathtt{h}}p_{2}^{[0]}\big{)}}{2{\mathtt{c}}_{\mathtt{h}}^{21/2}}\end{matrix}}\,,$
$\displaystyle{\footnotesize\varsigma_{2}}:={\scriptsize\begin{matrix}\dfrac{a_{1}^{[1]}{\mathtt{c}}_{\mathtt{h}}\big{(}a_{2}^{[0]}{\mathtt{c}}_{\mathtt{h}}^{2}({\mathtt{c}}_{\mathtt{h}}^{4}+1)+({\mathtt{c}}_{\mathtt{h}}^{4}-1)^{2}\mathtt{f}_{2}-{\mathtt{c}}_{\mathtt{h}}({\mathtt{c}}_{\mathtt{h}}^{8}+3{\mathtt{c}}_{\mathtt{h}}^{4}+2)p_{2}^{[0]}\big{)}+p_{1}^{[1]}\big{(}-a_{2}^{[0]}({\mathtt{c}}_{\mathtt{h}}^{8}+3{\mathtt{c}}_{\mathtt{h}}^{4}+2){\mathtt{c}}_{\mathtt{h}}^{2}-2({\mathtt{c}}_{\mathtt{h}}^{4}-1)^{2}\mathtt{f}_{2}+(3{\mathtt{c}}_{\mathtt{h}}^{8}+7{\mathtt{c}}_{\mathtt{h}}^{4}+4){\mathtt{c}}_{\mathtt{h}}p_{2}^{[0]}\big{)}}{4{\mathtt{c}}_{\mathtt{h}}^{23/2}}\end{matrix}.}$

$\mathrm{V\\!I\\!I\\!I}\big{)}$ By Lemma B.6, (B.15) and Lemma B.2 we get

$\displaystyle(\mathscr{L}_{0,0}$ $\displaystyle-\lambda)^{-1}\mathcal{J}{\cal
B}_{0,1}(\mathscr{L}_{0,0}-\lambda)^{-1}\mathcal{J}{\cal B}_{0,2}f_{1}^{+}=$
(B.42)
$\displaystyle\frac{\tau_{1}^{+}}{\lambda}(\mathscr{L}_{0,0}-\lambda)^{-1}\mathcal{J}{\cal
B}_{0,1}f_{1}^{-}+\frac{\ell_{1}^{+}}{\lambda^{2}+4\text{${\mathtt{c}}_{\mathtt{h}}^{2}$}}(\mathscr{L}_{0,0}-\lambda)^{-1}\mathcal{J}\text{${\cal
B}_{0,1}$}f_{-1}^{+}+\mathscr{L}_{0,0}^{\,\rotatebox[origin={c}]{90.0}{$\perp$}}\mathcal{J}{\cal
B}_{0,1}{\mathrm{L}_{3}^{+}}$ $\displaystyle+$ $\displaystyle\frac{\lambda
m_{1}^{+}}{\lambda^{2}+4{\mathtt{c}}_{\mathtt{h}}^{2}}(\mathscr{L}_{0,0}-\lambda)^{-1}\mathcal{J}{\cal
B}_{0,1}f_{-1}^{-}+(\mathscr{L}_{0,0}-\lambda)^{-1}\mathcal{J}{\cal
B}_{0,1}\mathcal{O}_{\mathcal{W}_{3}}(\lambda)+\mathcal{O}(\lambda)\,.$

Applying the operators ${\cal B}_{0,1}$ in (3.8b) and $\mathcal{J}$ in (1.5)
to the vector $\mathrm{L}_{3}^{+}$ in (B.15) one obtains

$\mathcal{J}{\cal