$\gamma\mapsto\vec{\Omega}(\gamma)\cdot\overline{\ell}$ is identically zero,
contradicting Lemma 4.2-1, since $\overline{\ell}\neq 0$.
Case (b). $(j_{m})_{m\in{\mathbb{N}}}$ is bounded and
$|j_{m}^{\prime}|\to\infty$ (or viceversa): this case is excluded by the
momentum condition $\vec{\jmath}\cdot\ell_{m}+j_{m}-j_{m}^{\prime}=0$ in
(4.22) and since $(\ell_{m})$ is bounded.
Case (c). Both $(j_{m})_{m\in{\mathbb{N}}}$,
$(j_{m}^{\prime})_{m\in{\mathbb{N}}}$ are bounded: we have definitively that
$j_{m}=\overline{\jmath}$ and $j_{m}^{\prime}=\overline{\jmath}^{\prime}$,
with $\overline{\jmath},\overline{\jmath}^{\prime}\in{\mathbb{S}}_{0}^{c}$
and, since $j_{m}\neq j_{m}^{\prime}$,

$\overline{\jmath}\neq\overline{\jmath}^{\prime}\,.$ (4.25)

Therefore (4.22) becomes, in the limit $m\rightarrow\infty$,

$\partial_{\gamma}^{n}\big{(}\vec{\Omega}(\gamma)\cdot\overline{\ell}+\Omega_{\overline{\jmath}}(\gamma)-\Omega_{\overline{\jmath}^{\prime}}(\gamma)\big{)}_{|\gamma=\overline{\gamma}}=0\,,\
\forall\,n\in{\mathbb{N}}_{0}\,,\quad\vec{\jmath}\cdot\overline{\ell}+\overline{\jmath}-\overline{\jmath}^{\prime}=0\,.$

By analyticity, we obtain that

$\vec{\Omega}(\gamma)\cdot\overline{\ell}+\Omega_{\overline{\jmath}}(\gamma)-\Omega_{\overline{\jmath}^{\prime}}(\gamma)=0\quad\forall\,\gamma\in\Gamma\,,\quad\vec{\jmath}\cdot\overline{\ell}+\overline{\jmath}-\overline{\jmath}^{\prime}=0\,.$
(4.26)

We distinguish several cases:

  * •

Let $\overline{\jmath},\overline{\jmath}^{\prime}\notin-{\mathbb{S}}$ and
$|\overline{\jmath}|\neq|\overline{\jmath}^{\prime}|$. By (4.26) the vector
$(\vec{\Omega}(\gamma),\Omega_{\overline{\jmath}}(\gamma),\Omega_{\overline{\jmath}^{\prime}}(\gamma))$
is degenerate with $c:=(\overline{\ell},1,-1)\neq 0$, contradicting Lemma
4.2-4.

  * •

Let $\overline{\jmath},\overline{\jmath}^{\prime}\notin-{\mathbb{S}}$ and
$\overline{\jmath}^{\prime}=-\overline{\jmath}$. In view of (4.1), the first
equation in (4.26) becomes

$\vec{\omega}(\gamma)\cdot\overline{\ell}+\frac{\gamma}{2}\Big{(}\sum_{a=1}^{\nu}\overline{\ell}_{a}\frac{G_{\overline{\jmath}_{a}}(0)}{\overline{\jmath}_{a}}+2\frac{G_{\overline{\jmath}}(0)}{\overline{\jmath}}\Big{)}=0\quad\forall\gamma\in\Gamma\,.$

By Lemma 4.3-1 the vector $(\vec{\omega}(\gamma),\gamma)$ is non-degenerate,
thus $\overline{\ell}=0$ and
$2\frac{G_{\overline{\jmath}}(0)}{\overline{\jmath}}=0$, which is a
contradiction.

  * •

Let $\overline{\jmath}^{\prime}\notin-{\mathbb{S}}$ and
$\overline{\jmath}\in-{\mathbb{S}}$. With no loss of generality suppose
$\overline{\jmath}=-\overline{\jmath}_{1}$. In view of (4.1), the first
equation in (4.26) implies that, for any $\gamma\in\Gamma$,

$(\overline{\ell}_{1}+1)\omega_{\overline{\jmath}_{1}}(\gamma)+\sum_{a=2}^{\nu}\overline{\ell}_{a}\omega_{\overline{\jmath}_{a}}(\gamma)-\omega_{\overline{\jmath}^{\prime}}(\gamma)+\frac{\gamma}{2}\Big{(}(\overline{\ell}_{1}-1)\frac{G_{\overline{\jmath}_{1}}(0)}{\overline{\jmath}_{1}}+\sum_{a=2}^{\nu}\overline{\ell}_{a}\frac{G_{\overline{\jmath}_{a}}(0)}{\overline{\jmath}_{a}}-\frac{G_{\overline{\jmath}^{\prime}}(0)}{\overline{\jmath}^{\prime}}\Big{)}=0\,.$

By Lemma 4.3-2 the vector
$\big{(}\vec{\omega}(\gamma),\omega_{\overline{\jmath}^{\prime}}(\gamma),\gamma\big{)}$
is non-degenerate, which is a contradiction.

  * •

Last, let $\overline{\jmath},\overline{\jmath}^{\prime}\in-{\mathbb{S}}$ and
$\overline{\jmath}\neq\overline{\jmath}^{\prime}$, by (4.25). With no loss of
generality suppose $\overline{\jmath}=-\overline{\jmath}_{1}$ and
$\overline{\jmath}^{\prime}=-\overline{\jmath}_{2}$. Then the first equation
in (4.26) reads, for any $\gamma\in\Gamma$,

$\displaystyle(\overline{\ell}_{1}+1)\omega_{\overline{\jmath}_{1}}(\gamma)+\left(\overline{\ell}_{2}-1\right)\omega_{\overline{\jmath}_{2}}+\sum_{a=3}^{\nu}\overline{\ell}_{a}\omega_{\overline{\jmath}_{a}}(\gamma)$
$\displaystyle\ \ \ \
+\frac{\gamma}{2}\Big{(}(\overline{\ell}_{1}-1)\frac{G_{\overline{\jmath}_{1}}(0)}{\overline{\jmath}_{1}}+(\overline{\ell}_{2}+1)\frac{G_{\overline{\jmath}_{2}}(0)}{\overline{\jmath}_{2}}+\sum_{a=3}^{\nu}\overline{\ell}_{a}\frac{G_{\overline{\jmath}_{a}}(0)}{\overline{\jmath}_{a}}\Big{)}=0\,.$

Since the vector $(\vec{\omega}(\gamma),\gamma)$ is non-degenerate by Lemma
4.3-1, it implies $\overline{\ell}_{1}=-1$, $\overline{\ell}_{2}=1$,
$\overline{\ell}_{3}=\ldots=\overline{\ell}_{\nu}=0$. Inserting these values
in the momentum condition in (4.26) we obtain
$-2\overline{\jmath}_{1}+2\overline{\jmath}_{2}=0$. This contradicts
$\overline{\jmath}\neq\overline{\jmath}^{\prime}$.

Step 2. We finally consider the case when $(\ell_{m})_{m\in{\mathbb{N}}}$ is
unbounded. Up to subsequences $\ell_{m}\rightarrow\infty$ as
$m\rightarrow\infty$ and
$\lim_{m\to\infty}\ell_{m}/\braket{\ell_{m}}=:\overline{c}\neq 0$. By (4.1),
Lemma 4.4, (4.23), we have, for any $n\geq 1$,

$\displaystyle\partial_{\gamma}^{n}\frac{1}{\braket{\ell_{m}}}\Big{(}\Omega_{j_{m}}(\gamma)-\Omega_{j_{m}^{\prime}}(\gamma)\Big{)}_{|\gamma=\gamma_{m}}$
$\displaystyle=\partial_{\gamma}^{n}\Big{(}\frac{1}{\braket{\ell_{m}}\sqrt{g}}\Big{(}\frac{c_{j_{m}}(\gamma)}{|j_{m}|^{\frac{1}{2}}}-\frac{c_{j_{m}^{\prime}}(\gamma)}{|j_{m}^{\prime}|^{\frac{1}{2}}}\Big{)}$
$\displaystyle\qquad+\frac{\gamma}{2\braket{\ell_{m}}}\Big{(}\frac{G_{j_{m}}(0)}{j_{m}}-\frac{G_{j_{m}^{\prime}}(0)}{j_{m}^{\prime}}\Big{)}_{|\gamma=\gamma_{m}}\Big{)}\to
0$

as $m\to\infty$. Therefore, for any $n\geq 1$, taking $m\rightarrow\infty$ in
(4.22) we get
$\partial_{\gamma}^{n}\big{(}\vec{\Omega}(\gamma)\cdot\overline{c}\big{)}_{|\gamma=\overline{\gamma}}=0$.
By analyticity this implies
$\vec{\Omega}(\gamma)\cdot\overline{c}=\overline{d}$, for all
$\gamma\in\Gamma$, contradicting Lemma 4.2-2, since $\overline{c}\neq 0$.
Proof of (4.18). It follows as (4.17) and we omit it. ∎

######  Remark 4.6.

For the irrotational gravity water waves equations (1.3) with $\gamma=0$,
quasi-periodic traveling waves solutions exist for most values of the _depth_
${\mathtt{h}}\in[{\mathtt{h}}_{1},{\mathtt{h}}_{2}]$. In detail, the non-
degeneracy of the linear frequencies with respect to the parameter
${\mathtt{h}}$ as in Lemma 4.2 is proved precisely in Lemma 3.2 in [2],
whereas the transversality properties hold by restricting the bounds in Lemma
3.4 in [2] to the Fourier sites satisfying the momentum conditions. We are not
able to use ${\mathtt{h}}$ as a parameter for any value of $\gamma\neq 0$ (in
this case we do not know if the non-degeneracy properties of Lemma 4.2 hold
with respect to ${\mathtt{h}}$).

##  5 Proof of Theorem 1.2

Under the rescaling $(\eta,\zeta)\mapsto(\varepsilon\eta,\varepsilon\zeta)$,
the Hamiltonian system (2.10) transforms into the Hamiltonian system generated
by

${\mathcal{H}}_{\varepsilon}(\eta,\zeta):=\varepsilon^{-2}{\mathcal{H}}(\varepsilon\eta,\varepsilon\zeta)={\mathcal{H}}_{L}(\eta,\zeta)+\varepsilon
P_{\varepsilon}(\eta,\zeta)\,,$ (5.1)

where ${\mathcal{H}}$ is the water waves Hamiltonian (2.9) expressed in the
Wahlén coordinates (2.7), ${\mathcal{H}}_{L}$ is defined in (2.15) and,
denoting ${\mathcal{H}}_{\geq 3}:={\mathcal{H}}-{\mathcal{H}}_{L}$ the cubic
part of the Hamiltonian,

$P_{\varepsilon}(\eta,\zeta):=\varepsilon^{-3}{\mathcal{H}}_{\geq
3}(\varepsilon\eta,\varepsilon\zeta)\,.$

We now study the Hamiltonian system generated by the Hamiltonian
${\mathcal{H}}_{\varepsilon}(\eta,\zeta)$, in the action-angle and normal
coordinates $(\theta,I,w)$ defined in Section 2.2. Thus we consider the
Hamiltonian $H_{\varepsilon}(\theta,I,w)$ defined by

$H_{\varepsilon}:={\mathcal{H}}_{\varepsilon}\circ
A=\varepsilon^{-2}{\mathcal{H}}\circ\varepsilon A$ (5.2)

where $A$ is the map defined in (2.2). The associated symplectic form is given
in (2.43).

By (2.47) (see also (2.30), (2.38)), in the variables $(\theta,I,w)$ the
quadratic Hamiltonian ${\mathcal{H}}_{L}$ defined in (2.15) simply reads, up
to a constant,

${\mathcal{N}}:={\mathcal{H}}_{L}\circ A=\vec{\Omega}(\gamma)\cdot
I+\tfrac{1}{2}\left({\bf{\Omega}}_{W}w,w\right)_{L^{2}}$

where $\vec{\Omega}(\gamma)\in{\mathbb{R}}^{\nu}$ is defined in (1.20) and
${\bf{\Omega}}_{W}$ in (2.14). Thus the Hamiltonian $H_{\varepsilon}$ in (5.2)
is

$H_{\varepsilon}={\mathcal{N}}+\varepsilon P\qquad{\rm with}\qquad
P:=P_{\varepsilon}\circ A\,.$ (5.3)

We look for an embedded invariant torus

$i:{\mathbb{T}}^{\nu}\rightarrow{\mathbb{R}}^{\nu}\times{\mathbb{R}}^{\nu}\times\mathfrak{H}_{{\mathbb{S}}^{+},\Sigma}^{\angle}\,,\quad{\varphi}\mapsto
i({\varphi}):=(\theta({\varphi}),I({\varphi}),w({\varphi}))\,,$

of the Hamiltonian vector field
$X_{H_{\varepsilon}}:=(\partial_{I}H_{\varepsilon},-\partial_{\theta}H_{\varepsilon},\Pi_{{\mathbb{S}}^{+},\Sigma}^{\angle}J\nabla_{w}H_{\varepsilon})$
filled by quasi-periodic solutions with frequency vector
$\omega\in{\mathbb{R}}^{\nu}$.

###  5.1 Nash-Moser theorem of hypothetical conjugation

Instead of looking directly for quasi-periodic solutions of
$X_{H_{\varepsilon}}$ we look for quasi-periodic solutions of the family of
modified Hamiltonians, where $\alpha\in{\mathbb{R}}^{\nu}$ are additional
parameters,

$H_{\alpha}:={\mathcal{N}}_{\alpha}+\varepsilon
P\,,\quad{\mathcal{N}}_{\alpha}:=\alpha\cdot
I+\tfrac{1}{2}\left(w,{\bf{\Omega}}_{W}w\right)_{L^{2}}\,.$ (5.4)

We consider the nonlinear operator

$\displaystyle{\mathcal{F}}(i,\alpha)$
$\displaystyle:={\mathcal{F}}(\omega,\gamma,\varepsilon;i,\alpha):=\omega\cdot\partial_{\varphi}i({\varphi})-X_{H_{\alpha}}(i({\varphi}))$
$\displaystyle=\begin{pmatrix}\omega\cdot\partial_{\varphi}\theta({\varphi})&-\alpha-\varepsilon\partial_{I}P(i({\varphi}))\\\
\omega\cdot\partial_{\varphi}I({\varphi})&+\varepsilon\partial_{\theta}P(i({\varphi}))\\\
\omega\cdot\partial_{\varphi}w({\varphi})&-\,\Pi_{{\mathbb{S}}^{+},\Sigma}^{\angle}J({\bf{\Omega}}_{W}w({\varphi})+\varepsilon\nabla_{w}P(i({\varphi})))\end{pmatrix}\,.$
(5.5)

If ${\mathcal{F}}(i,\alpha)=0$, then the embedding ${\varphi}\mapsto
i({\varphi})$ is an invariant torus for the Hamiltonian vector field
$X_{H_{\alpha}}$, filled with quasi-periodic solutions with frequency
$\omega$.

Each Hamiltonian $H_{\alpha}$ in (5.4) is invariant under the involution
$\vec{\mathcal{S}}$ and the translations $\vec{\tau}_{\varsigma}$,
$\varsigma\in{\mathbb{R}}$, defined respectively in (2.40) and in (2.41):

$H_{\alpha}\circ\vec{\mathcal{S}}=H_{\alpha}\,,\qquad
H_{\alpha}\circ\vec{\tau}_{\varsigma}=H_{\alpha}\,,\quad\forall\,\varsigma\in{\mathbb{R}}\,.$
(5.6)

We look for a reversible traveling torus embedding $i(\varphi)=$
$(\theta({\varphi}),I({\varphi}),w({\varphi}))$, namely satisfying

$\vec{\mathcal{S}}i({\varphi})=i(-{\varphi})\,,\qquad\vec{\tau}_{\varsigma}i({\varphi})=i({\varphi}-\vec{\jmath}\varsigma)\,,\quad\forall\,\varsigma\in{\mathbb{R}}\,.$
(5.7)

Note that, by (5.1) and (5.6), the operator ${\mathcal{F}}(\cdot,\alpha)$ maps
a reversible, respectively traveling, wave into an anti-reversible,
respectively traveling, wave variation, according to Definition 3.26.

The norm of the periodic components of the embedded torus

${\mathfrak{I}}({\varphi}):=i({\varphi})-({\varphi},0,0):=\left(\Theta({\varphi}),I({\varphi}),w({\varphi})\right)\,,\quad\Theta({\varphi}):=\theta({\varphi})-{\varphi}\,,$
(5.8)

is
$\left\|{\mathfrak{I}}\right\|_{s}^{k_{0},\upsilon}:=\left\|\Theta\right\|_{H_{\varphi}^{s}}^{k_{0},\upsilon}+\left\|I\right\|_{H_{\varphi}^{s}}^{k_{0},\upsilon}+\left\|w\right\|_{s}^{k_{0},\upsilon}$,
where

$k_{0}:=m_{0}+2$ (5.9)

and $m_{0}\in{\mathbb{N}}$ is the index of non-degeneracy provided by
Proposition 4.5, which only depends on the linear unperturbed frequencies. We
will often omit to write the dependence of the various constants with respect
to $k_{0}$, which is considered as an absolute constant. We look for quasi-
periodic solutions of frequency $\omega$ belonging to a $\delta$-neighbourhood
(independent of $\varepsilon$)

${\mathtt{\Omega}}:=\big{\\{}\omega\in{\mathbb{R}}^{\nu}\ :\
\operatorname{dist}\big{(}\omega,\vec{\Omega}[\gamma_{1},\gamma_{2}]\big{)}<\delta\big{\\}}\,,\quad\delta>0\,,$

of the curve $\vec{\Omega}[\gamma_{1},\gamma_{2}]$ defined by (1.20).

The next theorem, whose proof is based on an implicit function iterative
scheme of Nash-Moser type, provides, for $\varepsilon$ small enough, a
solution $(i_{\infty},\alpha_{\infty})(\omega,\gamma;\varepsilon)$ of the
nonlinear operator ${\cal F}(\varepsilon,\omega,\gamma;i,\alpha)=0$ for all
the values of $(\omega,\gamma)$ in the Cantor like set ${\cal
C}_{\infty}^{\upsilon}$ below.

######  Theorem 5.1.

(Theorem of hypothetical conjugation) There exist positive constants ${\rm
a_{0}},\varepsilon_{0},C$ depending on ${\mathbb{S}}$, $k_{0}$ and $\tau\geq
1$ such that, for all $\upsilon=\varepsilon^{\rm a}$, ${\rm a}\in(0,{\rm
a}_{0})$ and for all $\varepsilon\in(0,\varepsilon_{0})$, there exist

  1. 1.

a $k_{0}$-times differentiable function of the form
$\alpha_{\infty}:\,{\mathtt{\Omega}}\times[\gamma_{1},\gamma_{2}]\mapsto{\mathbb{R}}^{\nu}$,

$\displaystyle\alpha_{\infty}(\omega,\gamma):=\omega+r_{\varepsilon}(\omega,\gamma)\quad\text{
with }\quad|r_{\varepsilon}|^{k_{0},\upsilon}\leq
C\varepsilon\upsilon^{-1}\,;$ (5.10)

  2. 2.

a family of embedded reversible traveling tori $i_{\infty}({\varphi})$ (cfr.
(5.7)), defined for all
$(\omega,\gamma)\in{\mathtt{\Omega}}\times[\gamma_{1},\gamma_{2}]$, satisfying

$\|i_{\infty}({\varphi})-({\varphi},0,0)\|_{s_{0}}^{k_{0},\upsilon}\leq
C\varepsilon\upsilon^{-1}\,;$ (5.11)

  3. 3.

a sequence of $k_{0}$-times differentiable functions
$\mu_{j}^{\infty}:{\mathbb{R}}^{\nu}\times[\gamma_{1},\gamma_{2}]\rightarrow{\mathbb{R}}$,
$j\in{\mathbb{S}}_{0}^{c}={\mathbb{Z}}\,\setminus\,({\mathbb{S}}\cup\\{0\\})$,
of the form

$\mu_{j}^{\infty}(\omega,\gamma)={\mathtt{m}}_{1}^{\infty}(\omega,\gamma)j+{\mathtt{m}}_{\frac{1}{2}}^{\infty}(\omega,\gamma)\Omega_{j}(\gamma)-{\mathtt{m}}_{0}^{\infty}(\omega,\gamma){\rm
sgn}(j)+{\mathfrak{r}}_{j}^{\infty}(\omega,\gamma)\,,$ (5.12)

with $\Omega_{j}(\gamma)$ defined in (1.13), satisfying

$|{\mathtt{m}}_{1}^{\infty}|^{k_{0},\upsilon}\leq C\varepsilon\,,\
|{\mathtt{m}}_{\frac{1}{2}}^{\infty}-1|^{k_{0},\upsilon}+|{\mathtt{m}}_{0}^{\infty}|^{k_{0},\upsilon}\leq
C\varepsilon\upsilon^{-1}\,,\quad\sup_{j\in{\mathbb{S}}_{0}^{c}}|j|^{\frac{1}{2}}|{\mathfrak{r}}_{j}^{\infty}|^{k_{0},\upsilon}\leq
C\varepsilon\upsilon^{-3}\,,$ (5.13)

such that, for all $(\omega,\gamma)$ in the Cantor-like set

$\displaystyle{\mathcal{C}}_{\infty}^{\upsilon}:=$
$\displaystyle\Big{\\{}(\omega,\gamma)\in{\mathtt{\Omega}}\times[\gamma_{1},\gamma_{2}]\
:\ |\omega\cdot\ell|\geq\ 8\upsilon\langle\ell\rangle^{-\tau}\,,\ \
\forall\,\ell\in{\mathbb{Z}}^{\nu}\setminus\\{0\\}\,,$ (5.14) $\displaystyle\
\left|\omega\cdot\ell-{\mathtt{m}}_{1}^{\infty}(\omega,\gamma)j\right|\geq
8\upsilon\braket{\ell}^{-\tau}\,,\
\forall\,\ell\in{\mathbb{Z}}^{\nu},\,j\in{\mathbb{S}}_{0}^{c}\text{ with
}\vec{\jmath}\cdot\ell+j=0;$ (5.15) $\displaystyle\
\left|\omega\cdot\ell+\mu_{j}^{\infty}(\omega,\gamma)\right|\geq
4\upsilon\left|j\right|^{\frac{1}{2}}\braket{\ell}^{-\tau}\,,\forall\,\ell\in{\mathbb{Z}}^{\nu},\,j\in{\mathbb{S}}_{0}^{c}\text{
with }\vec{\jmath}\cdot\ell+j=0\,;$ (5.16) $\displaystyle\
\left|\omega\cdot\ell+\mu_{j}^{\infty}(\omega,\gamma)-\mu_{j^{\prime}}^{\infty}(\omega,\gamma)\right|\geq
4\upsilon\,\braket{\ell}^{-\tau}\,,$ (5.17) $\displaystyle\
\quad\quad\forall\ell\in{\mathbb{Z}}^{\nu},\,j,j^{\prime}\in{\mathbb{S}}_{0}^{c},\,(\ell,j,j^{\prime})\neq(0,j,j)\text{
with }\vec{\jmath}\cdot\ell+j-j^{\prime}=0\,,$ $\displaystyle\
\left|\omega\cdot\ell+\mu_{j}^{\infty}(\omega,\gamma)+\mu_{j^{\prime}}^{\infty}(\omega,\gamma)\right|\geq
4\upsilon\,\big{(}\left|j\right|^{\frac{1}{2}}+|j^{\prime}|^{\frac{1}{2}}\big{)}\braket{\ell}^{-\tau}\,,$
(5.18) $\displaystyle\
\quad\quad\forall\,\ell\in{\mathbb{Z}}^{\nu},\,j,j^{\prime}\in{\mathbb{S}}_{0}^{c}\,,\text{
with }\vec{\jmath}\cdot\ell+j+j^{\prime}=0\,\Big{\\}}\,,$

the function
$i_{\infty}({\varphi}):=i_{\infty}(\omega,\gamma,\varepsilon;{\varphi})$ is a
solution of
${\mathcal{F}}(\omega,\gamma,\varepsilon;(i_{\infty},\alpha_{\infty})(\omega,\gamma))=0$.
As a consequence, the embedded torus ${\varphi}\mapsto i_{\infty}({\varphi})$
is invariant for the Hamiltonian vector field
$X_{H_{\alpha_{\infty}(\omega,\gamma)}}$ as it is filled by quasi-periodic
reversible traveling wave solutions with frequency $\omega$.

Note that the Cantor-like set ${\cal C}_{\infty}^{\upsilon}$ in (5.14)-(5.18)
is defined in terms of the functions
${\mathtt{m}}_{1}^{\infty}(\omega,\gamma)$ and the “final" perturbed normal
frequencies $\mu_{j}^{\infty}(\omega,\gamma)$, $j\in{\mathbb{S}}_{0}^{c}$,
which are defined for all the values of the parameters $(\omega,\gamma)$. This
formulation completely decouples the Nash-Moser implicit function theorem
construction of $(\alpha_{\infty},i_{\infty})(\omega,\gamma)$ (in Sections
6-9) from the discussion about the measure of the parameters where all the
required “non-resonance" conditions are verified (Section 5.2). This approach
simplifies considerably the presentation because the measure estimates
required to build $(i_{\infty},\alpha_{\infty})(\omega,\gamma)$ are not
verified at each step along the Nash-Moser iteration (the set ${\cal
C}_{\infty}^{\upsilon}$ in (5.14)-(5.18) could be empty, in such a case the
functions $(\alpha_{\infty},i_{\infty})(\omega,\gamma)$ constructed in Theorem
5.1 are obtained by just finitely many sums). In order to define the extended
functions $(i_{\infty},\alpha_{\infty})$ for all the values of
$(\omega,\gamma)$, preserving the weighted norm $\|\ \|^{k_{0},\upsilon}$, we
use the Whitney extension theory reported in Section 3.

We also remind that the conditions on the indexes in (5.14)-(5.18) (where
$\vec{\jmath}\in{\mathbb{Z}}^{\nu}$ is the vector in (2.42)) are due to the
fact that we look for traveling wave solutions. These restrictions are
essential to prove the measure estimates of the next section.

######  Remark 5.2.

The Diophantine condition (5.14) could be weakened requiring only
$|\omega\cdot\ell|\geq\ \upsilon\langle\ell\rangle^{-\tau}$ for any
$\ell\cdot\vec{\jmath}=0$. In such a case the vector $\omega$ could admit one
non-trivial resonance, i.e.
$\overline{\ell}\in{\mathbb{Z}}^{\nu}\setminus\\{0\\}$ such that
$\omega\cdot\overline{\ell}=0$, thus the orbit $\\{\omega
t\\}_{t\in{\mathbb{R}}}$ would densely fill a ($\nu-1$)-dimensional torus,
orthogonal to $\overline{\ell}$. In any case
$\vec{\jmath}\cdot\overline{\ell}\neq 0$ (otherwise
$|\omega\cdot\overline{\ell}|\geq\upsilon\langle\overline{\ell}\rangle^{-\tau}>0$,
contradicting that $\omega\cdot\overline{\ell}=0$) and then the closure of the
set $\\{\omega t-\vec{\jmath}x\\}_{t\in{\mathbb{R}},x\in{\mathbb{R}}}$ is
dense in ${\mathbb{T}}^{\nu}$. This is the natural minimal requirement to look
for traveling quasi-periodic solutions $U(\omega t-\vec{\jmath}x)$ (Definition
3.1).

The next goal is to deduce Theorem 1.2 from Theorem 5.1.

###  5.2 Measure estimates: proof of Theorem 1.2

We now want to prove the existence of quasi-periodic solutions of the original
Hamiltonian system $H_{\varepsilon}$ in (5.3), which is equivalent after a
rescaling to (2.10), and not of just of the Hamiltonian system generated by
the modified Hamiltonian $H_{\alpha_{\infty}}$. We proceed as follows. By
(5.10), the function $\alpha_{\infty}(\,\cdot\,,\gamma)$ from
${\mathtt{\Omega}}$ into its image $\alpha_{\infty}({\mathtt{\Omega}},\gamma)$
is invertible and

$\displaystyle\beta=\alpha_{\infty}(\omega,\gamma)=\omega+r_{\varepsilon}(\omega,\gamma)\
\Leftrightarrow$ (5.19)
$\displaystyle\omega=\alpha_{\infty}^{-1}(\beta,\gamma)=\beta+\breve{r}_{\varepsilon}(\beta,\gamma)\,,\quad\left|\breve{r}_{\varepsilon}\right|^{k_{0},\upsilon}\leq
C\varepsilon\upsilon^{-1}\,.$

Then, for any $\beta\in\alpha_{\infty}({\mathcal{C}}_{\infty}^{\upsilon})$,
Theorem 5.1 proves the existence of an embedded invariant torus filled by
quasi-periodic solutions with Diophantine frequency
$\omega=\alpha_{\infty}^{-1}(\beta,\gamma)$ for the Hamiltonian

$H_{\beta}=\beta\cdot I+\tfrac{1}{2}(w,{\bf{\Omega}}_{W}w)_{L^{2}}+\varepsilon
P\,.$

Consider the curve of the unperturbed tangential frequency vector
$\vec{\Omega}(\gamma)$ in (1.20). In Theorem 5.3 below we prove that for
"most" values of $\gamma\in[\gamma_{1},\gamma_{2}]$ the vector
$(\alpha_{\infty}^{-1}(\vec{\Omega}(\gamma),\gamma),\gamma)$ is in
${\mathcal{C}}_{\infty}^{\upsilon}$, obtaining an embedded torus for the
Hamiltonian $H_{\varepsilon}$ in (5.2), filled by quasi-periodic solutions
with Diophantine frequency vector
$\omega=\alpha_{\infty}^{-1}(\vec{\Omega}(\gamma),\gamma)$, denoted
${\widetilde{\Omega}}$ in Theorem 1.2. Thus $\varepsilon
A(i_{\infty}({\widetilde{\Omega}}t))$, where $A$ is defined in (2.2), is a
quasi-periodic traveling wave solution of the water waves equations (2.10)
written in the Wahlén variables. Finally, going back to the original Zakharov
variables via (2.7) we obtain solutions of (1.3). This proves Theorem 1.2
together with the following measure estimates.

######  Theorem 5.3.

(Measure estimates) Let

$\upsilon=\varepsilon^{\rm a}\,,\quad 0<{\rm a}<\min\\{{\rm
a}_{0},1/(4m_{0}^{2})\\}\,,\quad\tau>m_{0}(2m_{0}\nu+\nu+2)\,,$ (5.20)

where $m_{0}$ is the index of non-degeneracy given in Proposition 4.5 and
$k_{0}:=m_{0}+2$. Then, for $\varepsilon\in(0,\varepsilon_{0})$ small enough,
the measure of the set

${\mathcal{G}}_{\varepsilon}:=\big{\\{}\gamma\in[\gamma_{1},\gamma_{2}]\ :\
\big{(}\alpha_{\infty}^{-1}(\vec{\Omega}(\gamma),\gamma),\gamma\big{)}\in{\mathcal{C}}_{\infty}^{\upsilon}\big{\\}}$
(5.21)

satisfies $|{\mathcal{G}}_{\varepsilon}|\rightarrow\gamma_{2}-\gamma_{1}$ as
$\varepsilon\rightarrow 0$.

The rest of this section is devoted to prove Theorem 5.3. By (5.19) we have

$\vec{\Omega}_{\varepsilon}(\gamma):=\alpha_{\infty}^{-1}(\vec{\Omega}(\gamma),\gamma)=\vec{\Omega}(\gamma)+\vec{r}_{\varepsilon}\,,$
(5.22)

where
$\vec{r}_{\varepsilon}(\gamma):=\breve{r}_{\varepsilon}(\vec{\Omega}(\gamma),\gamma)$
satisfies

$|\partial_{\gamma}^{k}{\vec{r}}_{\varepsilon}(\gamma)|\leq
C\varepsilon\upsilon^{-(1+k)}\,,\quad\forall\,\left|k\right|\leq k_{0}\,,\
\text{uniformly on }[\gamma_{1},\gamma_{2}]\,.$ (5.23)

We also denote, with a small abuse of notation, for all
$j\in{\mathbb{S}}_{0}^{c}$,

$\mu_{j}^{\infty}(\gamma):=\mu_{j}^{\infty}\big{(}\vec{\Omega}_{\varepsilon}(\gamma),\gamma\big{)}:={\mathtt{m}}_{1}^{\infty}(\gamma)j+{\mathtt{m}}_{\frac{1}{2}}^{\infty}(\gamma)\Omega_{j}(\gamma)-{\mathtt{m}}_{0}^{\infty}(\gamma){\rm
sgn}(j)+{\mathfrak{r}}_{j}^{\infty}(\gamma)\,,$ (5.24)

where
${\mathtt{m}}_{1}^{\infty}(\gamma):={\mathtt{m}}_{1}^{\infty}(\vec{\Omega}_{\varepsilon}(\gamma),\gamma)$,
${\mathtt{m}}_{\frac{1}{2}}^{\infty}(\gamma):={\mathtt{m}}_{\frac{1}{2}}^{\infty}(\vec{\Omega}_{\varepsilon}(\gamma),\gamma)$,
${\mathtt{m}}_{0}^{\infty}(\gamma):={\mathtt{m}}_{0}^{\infty}(\vec{\Omega}_{\varepsilon}(\gamma),\gamma)$
and
${\mathfrak{r}}_{j}^{\infty}(\gamma):={\mathfrak{r}}_{j}^{\infty}(\vec{\Omega}_{\varepsilon}(\gamma),\gamma)$.
By (5.13) and (5.23) we get the estimates

$\displaystyle|\partial_{\gamma}^{k}{\mathtt{m}}_{1}^{\infty}(\gamma)|\leq
C\varepsilon\upsilon^{-k}\,,\,\big{|}\partial_{\gamma}^{k}\big{(}{\mathtt{m}}_{\frac{1}{2}}^{\infty}(\gamma)-1\big{)}\big{|}+|\partial_{\gamma}^{k}{\mathtt{m}}_{0}^{\infty}(\gamma)|\leq
C\varepsilon\upsilon^{-k-1},$ (5.25)
$\displaystyle\sup_{j\in{\mathbb{S}}_{0}^{c}}|j|^{\frac{1}{2}}\left|\partial_{\gamma}^{k}{\mathfrak{r}}_{j}^{\infty}(\gamma)\right|\leq
C\varepsilon\upsilon^{-3-k}\,,\quad\forall\,0\leq k\leq k_{0}\,.$ (5.26)

Recalling (5.14)-(5.18), the Cantor set in (5.21) becomes

$\displaystyle{\mathcal{G}}_{\varepsilon}:=$
$\displaystyle\Big{\\{}\gamma\in[\gamma_{1},\gamma_{2}]\ :\
|\vec{\Omega}_{\varepsilon}(\gamma)\cdot\ell|\geq
8\upsilon\braket{\ell}^{-\tau}\,,\
\forall\,\ell\in{\mathbb{Z}}^{\nu}\setminus\\{0\\}\,;$ $\displaystyle\ \
|(\vec{\Omega}_{\varepsilon}(\gamma)-{\mathtt{m}}_{1}^{\infty}(\gamma)\vec{\jmath})\cdot\ell|\geq
8\upsilon\braket{\ell}^{-\tau}\,,\
\forall\,\ell\in{\mathbb{Z}}^{\nu}\setminus\\{0\\}\,;$ $\displaystyle\ \
|\vec{\Omega}_{\varepsilon}(\gamma)\cdot\ell+\mu_{j}^{\infty}(\gamma)|\geq
4\upsilon|j|^{\frac{1}{2}}\braket{\ell}^{-\tau}\,,\
\forall\,\ell\in{\mathbb{Z}}^{\nu}\,,\,j\in{\mathbb{S}}_{0}^{c}\,,\text{ with
}\vec{\jmath}\cdot\ell+j=0\,;$ $\displaystyle\ \
|\vec{\Omega}_{\varepsilon}(\gamma)\cdot\ell+\mu_{j}^{\infty}(\gamma)-\mu_{j^{\prime}}^{\infty}(\gamma)|\geq
4\upsilon\,\braket{\ell}^{-\tau}\,,$ $\displaystyle\ \
\forall\ell\in{\mathbb{Z}}^{\nu},\,j,j^{\prime}\in{\mathbb{S}}_{0}^{c},\,(\ell,j,j^{\prime})\neq(0,j,j)\text{
with }\vec{\jmath}\cdot\ell+j-j^{\prime}=0\,;$ $\displaystyle\ \
|\vec{\Omega}_{\varepsilon}(\gamma)\cdot\ell+\mu_{j}^{\infty}(\gamma)+\mu_{j^{\prime}}^{\infty}(\gamma)|\geq
4\upsilon\,\big{(}|j|^{\frac{1}{2}}+|j^{\prime}|^{\frac{1}{2}}\big{)}\braket{\ell}^{-\tau}\,,$
$\displaystyle\ \
\forall\,\ell\in{\mathbb{Z}}^{\nu},\,j,j^{\prime}\in{\mathbb{S}}_{0}^{c}\text{
with }\vec{\jmath}\cdot\ell+j+j^{\prime}=0\Big{\\}}\,.$

We estimate the measure of the complementary set

$\displaystyle{\mathcal{G}}_{\varepsilon}^{c}$
$\displaystyle:=[\gamma_{1},\gamma_{2}]\setminus{\mathcal{G}}_{\varepsilon}$
(5.27) $\displaystyle=\left(\bigcup_{\ell\neq 0}R_{\ell}^{(0)}\cup
R_{\ell}^{(T)}\right)\cup\left(\bigcup_{\ell\in{\mathbb{Z}}^{\nu},\,j\in{\mathbb{S}}_{0}^{c}\atop\vec{\jmath}\cdot\ell+j=0}R_{\ell,j}^{(I)}\right)\cup\left(\bigcup_{(\ell,j,j^{\prime})\neq(0,j,j),j\neq
j^{\prime}\atop\vec{\jmath}\cdot\ell+j-j^{\prime}=0}R_{\ell,j,j^{\prime}}^{(II)}\right)\cup\left(\bigcup_{\ell\in{\mathbb{Z}}^{\nu},j,j^{\prime}\in{\mathbb{S}}_{0}^{c}\,,\atop\vec{\jmath}\cdot\ell+j+j^{\prime}=0}Q_{\ell,j,j^{\prime}}^{(II)}\right)\,,$

where the “nearly-resonant sets" are, recalling the notation
$\Gamma=[\gamma_{1},\gamma_{2}]$,

$\displaystyle R_{\ell}^{(0)}:=R_{\ell}^{(0)}(\upsilon,\tau):=$
$\displaystyle\big{\\{}\gamma\in\Gamma\,:\,|\vec{\Omega}_{\varepsilon}(\gamma)\cdot\ell|<8\upsilon\braket{\ell}^{-\tau}\big{\\}}\,,$
(5.28) $\displaystyle R_{\ell}^{(T)}:=R_{\ell}^{(T)}(\upsilon,\tau):=$
$\displaystyle\big{\\{}\gamma\in\Gamma\,:\,|(\vec{\Omega}_{\varepsilon}(\gamma)-{\mathtt{m}}_{1}^{\infty}(\gamma)\vec{\jmath})\cdot\ell|<8\upsilon\braket{\ell}^{-\tau}\big{\\}}\,,$
(5.29) $\displaystyle R_{\ell,j}^{(I)}:=R_{\ell,j}^{(I)}(\upsilon,\tau):=$
$\displaystyle\big{\\{}\gamma\in\Gamma\,:\,|\vec{\Omega}_{\varepsilon}(\gamma)\cdot\ell+\mu_{j}^{\infty}(\gamma)|<4\upsilon|j|^{\frac{1}{2}}\braket{\ell}^{-\tau}\big{\\}}\,,$
(5.30) $\displaystyle
R_{\ell,j,j^{\prime}}^{(II)}:=R_{\ell,j,j^{\prime}}^{(II)}(\upsilon,\tau):=$
$\displaystyle\big{\\{}\gamma\in\Gamma\,:\,|\vec{\Omega}_{\varepsilon}(\gamma)\cdot\ell+\mu_{j}^{\infty}(\gamma)-\mu_{j^{\prime}}^{\infty}(\gamma)|<4\upsilon\,\braket{\ell}^{-\tau}\big{\\}}\,,$
(5.31) $\displaystyle
Q_{\ell,j,j^{\prime}}^{(II)}:=Q_{\ell,j,j^{\prime}}^{(II)}(\upsilon,\tau):=$
$\displaystyle\Big{\\{}\gamma\in\Gamma\,:\,|\vec{\Omega}_{\varepsilon}(\gamma)\cdot\ell+\mu_{j}^{\infty}(\gamma)+\mu_{j^{\prime}}^{\infty}(\gamma)|<\frac{4\upsilon\big{(}|j|^{\frac{1}{2}}+|j^{\prime}|^{\frac{1}{2}}\big{)}}{\braket{\ell}^{\tau}}\Big{\\}}\,.$
(5.32)

Note that in the third union in (5.27) we may require $j\neq j^{\prime}$
because $R_{\ell,j,j}^{(II)}\subset R_{\ell}^{(0)}$. In the sequel we shall
always suppose the momentum conditions on the indexes $\ell,j,j^{\prime}$
written in (5.27). Some of the above sets are empty.

######  Lemma 5.4.

For $\varepsilon\in(0,\varepsilon_{0})$ small enough, if
$Q_{\ell,j,j^{\prime}}^{(II)}\neq\emptyset$ then
$|j|^{\frac{1}{2}}+|j^{\prime}|^{\frac{1}{2}}\leq C\braket{\ell}$.

###### Proof.

If $Q_{\ell,j,j^{\prime}}^{(II)}\neq\emptyset$ then there exists
$\gamma\in[\gamma_{1},\gamma_{2}]$ such that

$\left|\mu_{j}^{\infty}(\gamma)+\mu_{j^{\prime}}^{\infty}(\gamma)\right|<\frac{4\upsilon\big{(}|j|^{\frac{1}{2}}+|j^{\prime}|^{\frac{1}{2}}\big{)}}{\braket{\ell}^{\tau}}+C|\ell|\,.$
(5.33)

By (5.24) we have

$\mu_{j}^{\infty}(\gamma)+\mu_{j^{\prime}}^{\infty}(\gamma)={\mathtt{m}}_{1}^{\infty}(\gamma)(j+j^{\prime})+{\mathtt{m}}_{\frac{1}{2}}^{\infty}(\gamma)(\Omega_{j}(\gamma)+\Omega_{j^{\prime}}(\gamma))-{\mathtt{m}}_{0}^{\infty}(\gamma)({\rm
sgn}(j)+{\rm
sgn}(j^{\prime}))+{\mathfrak{r}}_{j}^{\infty}(\gamma)+{\mathfrak{r}}_{j^{\prime}}^{\infty}(\gamma)\,.$

Then, by (5.25)-(5.26) with $k=0$, Lemma 4.4 and the momentum condition
$j+j^{\prime}=-\vec{\jmath}\cdot\ell$, we deduce, for $\varepsilon$ small
enough,

$\displaystyle|\mu_{j}^{\infty}(\gamma)+\mu_{j^{\prime}}^{\infty}(\gamma)|$
$\displaystyle\geq-C\varepsilon|\ell|+\tfrac{\sqrt{g}}{2}\,\big{|}|j|^{\frac{1}{2}}+|j^{\prime}|^{\frac{1}{2}}\big{|}-C^{\prime}-C\varepsilon\upsilon^{-3}\,.$
(5.34)

Combining (5.33) and (5.34), we deduce
$||j|^{\frac{1}{2}}+|j^{\prime}|^{\frac{1}{2}}|\leq C\braket{\ell}$, for
$\varepsilon$ small enough. ∎

In order to estimate the measure of the sets (5.28)-(5.32), the key point is
to prove that the perturbed frequencies satisfy transversality properties
similar to the ones (4.15)-(4.18) satisfied by the unperturbed frequencies. By
Proposition 4.5, (5.22), and the estimates (5.23), (5.25)-(5.26) we deduce the
following lemma (cfr. Lemma 5.5 in [7]).

######  Lemma 5.5.

(Perturbed transversality) For $\varepsilon\in(0,\varepsilon_{0})$ small
enough and for all $\gamma\in[\gamma_{1},\gamma_{2}]$,

$\displaystyle\max_{0\leq n\leq
m_{0}}|\partial_{\gamma}^{n}\vec{\Omega}_{\varepsilon}(\gamma)\cdot\ell|\geq\frac{\rho_{0}}{2}\braket{\ell}\,,\quad\forall\,\ell\in{\mathbb{Z}}^{\nu}\setminus\\{0\\}\,;$
(5.35) $\displaystyle\max_{0\leq n\leq
m_{0}}|\partial_{\gamma}^{n}(\vec{\Omega}_{\varepsilon}(\gamma)-{\mathtt{m}}_{1}^{\infty}(\gamma)\vec{\jmath})\cdot\ell|\geq\frac{\rho_{0}}{2}\braket{\ell}\,,\quad\forall\ell\in{\mathbb{Z}}^{\nu}\setminus\\{0\\}$
(5.36) $\displaystyle\begin{cases}\max_{0\leq n\leq
m_{0}}|\partial_{\gamma}^{n}(\vec{\Omega}_{\varepsilon}(\gamma)\cdot\ell+\mu_{j}^{\infty}(\gamma))|\geq\frac{\rho_{0}}{2}\braket{\ell}\,,\\\
\vec{\jmath}\cdot\ell+j=0\,,\quad\ell\in{\mathbb{Z}}^{\nu}\,,\
j\in{\mathbb{S}}_{0}^{c}\,;\end{cases}$ (5.37)
$\displaystyle\begin{cases}\max_{0\leq n\leq
m_{0}}|\partial_{\gamma}^{n}(\vec{\Omega}_{\varepsilon}(\gamma)\cdot\ell+\mu_{j}^{\infty}(\gamma)-\mu_{j^{\prime}}^{\infty}(\gamma))|\geq\frac{\rho_{0}}{2}\braket{\ell}\\\
\vec{\jmath}\cdot\ell+j-j^{\prime}=0\,,\quad\ell\in{\mathbb{Z}}^{\nu}\,,\
j,j^{\prime}\in{\mathbb{S}}_{0}^{c}\,,\
(\ell,j,j^{\prime})\neq(0,j,j)\,;\end{cases}$ (5.38)
$\displaystyle\begin{cases}\max_{0\leq n\leq
m_{0}}|\partial_{\gamma}^{n}(\vec{\Omega}_{\varepsilon}(\gamma)\cdot\ell+\mu_{j}^{\infty}(\gamma)+\mu_{j^{\prime}}^{\infty}(\gamma))|\geq\frac{\rho_{0}}{2}\braket{\ell}\\\
\vec{\jmath}\cdot\ell+j+j^{\prime}=0\,,\quad\ell\in{\mathbb{Z}}^{\nu}\,,\
j,j^{\prime}\in{\mathbb{S}}_{0}^{c}\,.\end{cases}$ (5.39)

The transversality estimates (5.35)-(5.39) and an application of Rüssmann
Theorem 17.1 in [32], directly imply the following bounds for the sets in
(5.27) (cfr. Lemma 5.6 in [7]).

######  Lemma 5.6.

(Estimates of the resonant sets) The measure of the sets (5.27)- (5.32)
satisfy

$\displaystyle|R_{\ell}^{(0)}|,|R_{\ell}^{(T)}|\lesssim(\upsilon\braket{\ell}^{-(\tau+1)})^{\frac{1}{m_{0}}}\,,$
$\displaystyle\quad|R_{\ell,j}^{(I)}|\lesssim\big{(}\upsilon|j|^{\frac{1}{2}}\braket{\ell}^{-(\tau+1)}\big{)}^{\frac{1}{m_{0}}}\,,$
$\displaystyle|R_{\ell,j,j^{\prime}}^{(II)}|\lesssim\big{(}\upsilon\braket{\ell}^{-(\tau+1)}\big{)}^{\frac{1}{m_{0}}}\,,$
$\displaystyle\quad|Q_{\ell,j,j^{\prime}}^{(II)}|\lesssim\big{(}\upsilon\,\big{(}|j|^{\frac{1}{2}}+|j^{\prime}|^{\frac{1}{2}}\big{)}\braket{\ell}^{-(\tau+1)}\big{)}^{\frac{1}{m_{0}}}\,.$

We now estimate the measure of all the sets in (5.27). By Lemma 5.6, and the
choice of $\tau$ in (5.20), we have

$\displaystyle\Big{|}\bigcup_{\ell\neq 0}R^{(0)}_{\ell}\cup
R^{(T)}_{\ell}\Big{|}\leq\sum_{\ell\neq
0}|R^{(0)}_{\ell}|+|R^{(T)}_{\ell}|\lesssim\sum_{\ell\neq
0}\Big{(}\frac{\upsilon}{\braket{\ell}^{\tau+1}}\Big{)}^{\frac{1}{m_{0}}}\lesssim\upsilon^{\frac{1}{m_{0}}}\,,$
(5.40) $\displaystyle\left|\bigcup_{\ell\neq
0,j=-\vec{\jmath}\cdot\ell}R_{\ell,j}^{(I)}\right|\leq\sum_{\ell\neq
0}|R_{\ell,-\vec{\jmath}\cdot\ell}^{(I)}|\lesssim\sum_{\ell}\Big{(}\frac{\upsilon}{\braket{\ell}^{\tau+\frac{1}{2}}}\Big{)}^{\frac{1}{m_{0}}}\lesssim\upsilon^{\frac{1}{m_{0}}}\,,$
(5.41)

and using also Lemma 5.4,

$\displaystyle\left|\bigcup_{\ell,\,j,j^{\prime}\in{\mathbb{S}}_{0}^{c}\atop\vec{\jmath}\cdot\ell+j+j^{\prime}=0}Q_{\ell,j,j^{\prime}}^{(II)}\right|\leq\sum_{\ell,\left|j\right|\leq
C\braket{\ell}^{2},\atop
j^{\prime}=-\vec{\jmath}\cdot\ell-j}|Q_{\ell,j,j^{\prime}}^{(II)}|\lesssim\sum_{\ell,\left|j\right|\leq
C\braket{\ell}^{2}}\left(\frac{\upsilon}{\braket{\ell}^{\tau}}\right)^{\frac{1}{m_{0}}}\lesssim\upsilon^{\frac{1}{m_{0}}}\,.$
(5.42)

We are left with estimating the measure of

$\displaystyle\bigcup_{(\ell,j,j^{\prime})\neq(0,j,j),j\neq
j^{\prime}\atop\vec{\jmath}\cdot\ell+j-j^{\prime}=0}\\!\\!\\!\\!\\!\\!\\!\\!R_{\ell,j,j^{\prime}}^{(II)}$
$\displaystyle=\left(\bigcup_{j\neq j^{\prime}\,,\ j\cdot
j^{\prime}<0\atop\vec{\jmath}\cdot\ell+j-j^{\prime}=0}R_{\ell,j,j^{\prime}}^{(II)}\right)\cup\left(\bigcup_{j\neq
j^{\prime}\,,\ j\cdot
j^{\prime}>0\atop\vec{\jmath}\cdot\ell+j-j^{\prime}=0}R_{\ell,j,j^{\prime}}^{(II)}\right):={\mathtt{I}}_{1}+{\mathtt{I}}_{2}\,.$
(5.43)

We first estimate the measure of ${\mathtt{I}}_{1}$. For $j\cdot
j^{\prime}<0$, the momentum condition reads $j-j^{\prime}={\rm
sgn}(j)(|j|+|j^{\prime}|)=-\vec{\jmath}\cdot\ell$, thus $|j|,|j^{\prime}|\leq
C\left\langle\ell\right\rangle$. Hence, by Lemma 5.6 and the choice of $\tau$
in (5.20), we have

$\displaystyle|{\mathtt{I}}_{1}|\leq\sum_{\ell,|j|\leq
C\left\langle\ell\right\rangle,j^{\prime}=j+\vec{\jmath}\cdot\ell}|R_{\ell,j,j^{\prime}}^{(II)}|\lesssim\sum_{\ell,\left|j\right|\leq
C\braket{\ell}}\left(\frac{\upsilon}{\braket{\ell}^{\tau+1}}\right)^{\frac{1}{m_{0}}}\lesssim\upsilon^{\frac{1}{m_{0}}}\,.$
(5.44)

Then we estimate the measure of ${\mathtt{I}}_{2}$ in (5.43). The key step is
given in the next lemma. Remind the definition of the sets
$R_{\ell,j,j^{\prime}}^{(II)}$ and $R_{\ell}^{(T)}$ in (5.30) and (5.31).

######  Lemma 5.7.

Let $\upsilon_{0}\geq\upsilon$ and $\tau\geq\tau_{0}\geq 1$. There is a
constant $C_{1}>0$ such that, for $\varepsilon$ small enough, for any
$\vec{\jmath}\cdot\ell+j-j^{\prime}=0$, $j\cdot j^{\prime}>0$,

$\min\\{|j|,|j^{\prime}|\\}\geq
C_{1}\upsilon_{0}^{-2}\braket{\ell}^{2(\tau_{0}+1)}\quad\Longrightarrow\quad
R_{\ell,j,j^{\prime}}^{(II)}(\upsilon,\tau)\subset\bigcup_{\ell\neq
0}R_{\ell}^{(T)}(\upsilon_{0},\tau_{0})\,.$ (5.45)

###### Proof.

If $\gamma\in[\gamma_{1},\gamma_{2}]\setminus\bigcup_{\ell\neq
0}R_{\ell}^{(T)}(\upsilon_{0},\tau_{0})$, then

$|(\vec{\Omega}_{\varepsilon}(\gamma)-{\mathtt{m}}_{1}^{\infty}(\gamma)\vec{\jmath})\cdot\ell|\geq
8\upsilon_{0}\braket{\ell}^{-\tau_{0}}\,,\quad\forall\ell\in{\mathbb{Z}}^{\nu}\setminus\\{0\\}\,.$
(5.46)

Then, by (5.24), the momentum condition $j-j^{\prime}=-\vec{\jmath}\cdot\ell$,
(5.25), (5.26), Lemma 4.4, the condition $j\cdot j^{\prime}>0$, (4.23), and
(5.46), we deduce that

$\displaystyle|\vec{\Omega}_{\varepsilon}(\gamma)\cdot\ell+\mu_{j}^{\infty}(\gamma)-\mu_{j^{\prime}}^{\infty}(\gamma)|\geq|\vec{\Omega}_{\varepsilon}(\gamma)\cdot\ell+{\mathtt{m}}_{1}^{\infty}(j-j^{\prime})|-|{\mathtt{m}}_{\frac{1}{2}}^{\infty}||\Omega_{j}(\gamma)-\Omega_{j^{\prime}}(\gamma)|-|{\mathfrak{r}}_{j}^{\infty}(\gamma)-{\mathfrak{r}}_{j^{\prime}}^{\infty}(\gamma)|$
$\displaystyle\geq|(\vec{\Omega}_{\varepsilon}(\gamma)-{\mathtt{m}}_{1}^{\infty}\vec{\jmath})\cdot\ell|-(1-C\varepsilon\upsilon^{-1})\big{|}|j|^{\frac{1}{2}}-|j^{\prime}|^{\frac{1}{2}}\big{|}-C\Big{(}\frac{1}{|j|^{\frac{1}{2}}}+\frac{1}{|j^{\prime}|^{\frac{1}{2}}}\Big{)}-C\frac{\varepsilon}{\upsilon^{3}}\Big{(}\frac{1}{|j|^{\frac{1}{2}}}+\frac{1}{|j^{\prime}|^{\frac{1}{2}}}\Big{)}$
$\displaystyle\geq\frac{8\upsilon_{0}}{\braket{\ell}^{\tau_{0}}}-\frac{1}{2}\frac{|j-j^{\prime}|}{|j|^{\frac{1}{2}}+|j^{\prime}|^{\frac{1}{2}}}-C\Big{(}\frac{1}{|j|^{\frac{1}{2}}}+\frac{1}{|j^{\prime}|^{\frac{1}{2}}}\Big{)}\geq\frac{8\upsilon_{0}}{\braket{\ell}^{\tau_{0}}}-C\Big{(}\frac{\braket{\ell}}{|j|^{\frac{1}{2}}}+\frac{\braket{\ell}}{|j^{\prime}|^{\frac{1}{2}}}\Big{)}$
$\displaystyle\geq\frac{4\upsilon_{0}}{\braket{\ell}^{\tau_{0}}}$

for any
$|j|,|j^{\prime}|>C_{1}\upsilon_{0}^{-2}\braket{\ell}^{2(\tau_{0}+1)}$, for
$C_{1}>C^{2}/64$. Since $\upsilon_{0}\geq\upsilon$ and $\tau\geq\tau_{0}$ we
deduce that

$|\vec{\Omega}_{\varepsilon}(\gamma)\cdot\ell+\mu_{j}^{\infty}(\gamma)-\mu_{j^{\prime}}^{\infty}(\gamma)|\geq
4\upsilon\braket{\ell}^{-\tau}\,,$

namely that $\gamma\not\in R_{\ell,j,j^{\prime}}^{(II)}(\upsilon,\tau)$. ∎

Note that the set of indexes $(\ell,j,j^{\prime})$ such that
$\vec{\jmath}\cdot\ell+j-j^{\prime}=0$ and
$\min\\{|j|,|j^{\prime}|\\}<C_{1}\upsilon_{0}^{-2}\braket{\ell}^{2(\tau_{0}+1)}$
is included, for $\upsilon_{0}$ small enough, into the set

${\cal I}_{\ell}:=\Big{\\{}(\ell,j,j^{\prime})\
:\,\vec{\jmath}\cdot\ell+j-j^{\prime}=0\,,\
|j|,|j^{\prime}|\leq\upsilon_{0}^{-3}\langle\ell\rangle^{2(\tau_{0}+1)}\Big{\\}}$
(5.47)

because
$\max\\{|j|,|j^{\prime}|\\}\leq\min\\{|j|,|j^{\prime}|\\}+|j-j^{\prime}|<C_{1}\upsilon_{0}^{-2}\langle\ell\rangle^{2(\tau_{0}+1)}+C\langle\ell\rangle\leq\upsilon_{0}^{-3}\langle\ell\rangle^{2(\tau_{0}+1)}$.
As a consequence, by Lemma 5.7 we deduce that

$\displaystyle{\mathtt{I}}_{2}=\bigcup_{j\neq j^{\prime}\,,\ j\cdot
j^{\prime}>0\atop\vec{\jmath}\cdot\ell+j-j^{\prime}=0}R_{\ell,j,j^{\prime}}^{(II)}(\upsilon,\tau)\subset\Big{(}\bigcup_{\ell\neq
0}R_{\ell}^{(T)}(\upsilon_{0},\tau_{0})\Big{)}\bigcup\Big{(}\bigcup_{(\ell,j,j^{\prime})\in{\cal
I}_{\ell}}R_{\ell,j,j^{\prime}}^{(II)}(\upsilon,\tau)\Big{)}\,.$ (5.48)

######  Lemma 5.8.

Let $\tau_{0}:=m_{0}\nu$ and $\upsilon_{0}=\upsilon^{\frac{1}{4m_{0}}}$. Then

$|{\mathtt{I}}_{2}|\leq C\upsilon^{\frac{1}{4m_{0}^{2}}}\,.$ (5.49)

###### Proof.

By (5.40) (applied with $\upsilon_{0},\tau_{0}$ instead of $\upsilon,\tau$),
and $\tau_{0}=m_{0}\nu$, the measure of

$\Big{|}\bigcup_{\ell\neq
0}R^{(T)}_{\ell}(\upsilon_{0},\tau_{0})\Big{|}\lesssim\upsilon_{0}^{\frac{1}{m_{0}}}\lesssim\upsilon^{\frac{1}{4m_{0}^{2}}}\,.$
(5.50)

Moreover, recalling (5.47),

$\Big{|}\bigcup_{(\ell,j,j^{\prime})\in{\cal
I}_{\ell}}R_{\ell,j,j^{\prime}}^{(II)}(\upsilon,\tau)\Big{|}\lesssim\sum_{\ell\in{\mathbb{Z}}^{\nu}\atop|j|\leq
C_{1}\upsilon_{0}^{-3}\braket{\ell}^{2(\tau_{0}+1)}}\left(\frac{\upsilon}{\braket{\ell}^{\tau+1}}\right)^{\frac{1}{m_{0}}}\lesssim\sum_{\ell\in{\mathbb{Z}}^{\nu}}\frac{\upsilon^{\frac{1}{m_{0}}}\upsilon_{0}^{-3}}{\braket{\ell}^{\frac{\tau+1}{m_{0}}-2(\tau_{0}+1)}}\leq
C\upsilon^{\frac{1}{4m_{0}}}\,,$ (5.51)

by the choice of $\tau$ in (5.20) and $\upsilon_{0}$. The bound (5.49) follows
by (5.50) and (5.51). ∎

###### Proof of Theorem 5.3 completed..

By (5.27), (5.40), (5.41), (5.42), (5.43), (5.44) and (5.49) we deduce that

$\left|{\mathcal{G}}_{\varepsilon}^{c}\right|\leq
C\upsilon^{\frac{1}{4m_{0}^{2}}}\,.$

For $\upsilon=\varepsilon^{\mathtt{a}}$ as in (5.20), we get
$|{\mathcal{G}}_{\varepsilon}|\geq\gamma_{2}-\gamma_{1}-C\varepsilon^{{\mathtt{a}}/4m_{0}^{2}}$.
The proof of Theorem 5.3 is concluded. ∎

######  Remark 5.9.

We have actually imposed in Lemma 5.8 the stronger non-resonance condition
(5.15) with $\upsilon_{0}=\upsilon^{\frac{1}{4m_{0}}}>\upsilon$. Since it has
no significant importance for Lemma 7.7 we keep $\upsilon$.

##  6 Approximate inverse

In order to implement a convergent Nash-Moser scheme that leads to a solution
of ${\mathcal{F}}(i,\alpha)=0$, where ${\mathcal{F}}(i,\alpha)$ is the
nonlinear operator defined in (5.1), we construct an _almost approximate right
inverse_ of the linearized operator

${\rm
d}_{i,\alpha}{\mathcal{F}}(i_{0},\alpha_{0})[\widehat{\imath},{\widehat{\alpha}}]=\omega\cdot\partial_{\varphi}\widehat{\imath}-{\rm
d}_{i}X_{H_{\alpha}}\left(i_{0}({\varphi})\right)[\widehat{\imath}]-\left({\widehat{\alpha}},0,0\right)\,.$

Note that ${\rm d}_{i,\alpha}{\mathcal{F}}(i_{0},\alpha_{0})={\rm
d}_{i,\alpha}{\mathcal{F}}(i_{0})$ is independent of $\alpha_{0}$. We assume
that the torus
$i_{0}({\varphi})=(\theta_{0}({\varphi}),I_{0}({\varphi}),w_{0}({\varphi}))$
is reversible and traveling, according to (5.7).

In the sequel we shall assume the smallness condition, for some
${\mathtt{k}}:={\mathtt{k}}(\tau,\nu)>0$,
$\varepsilon\upsilon^{-{\mathtt{k}}}\ll 1$.

We closely follow the strategy presented in [6] and implemented for the water
waves equations in [9, 2, 7]. As shown in [7] this construction preserves the
momentum preserving properties needed for the search of traveling waves and
the estimates are very similar. Thus we are short.

First of all, we state tame estimates for the composition operator induced by
the Hamiltonian vector field
$X_{P}=(\partial_{I}P,-\partial_{\theta}P,\Pi_{{\mathbb{S}}^{+},\Sigma}^{\angle}J\nabla_{w}P)$
in (5.1) (see Lemma 6.1 of [7]).

######  Lemma 6.1.

(Estimates of the perturbation $P$) Let ${\mathfrak{I}}({\varphi})$ in (5.8)
satisfy $\left\|{\mathfrak{I}}\right\|_{3s_{0}+2k_{0}+5}^{k_{0},\upsilon}\leq
1$. Then, for any $s\geq s_{0}$,
$\left\|X_{P}(i)\right\|_{s}^{k_{0},\upsilon}\lesssim_{s}1+\left\|{\mathfrak{I}}\right\|_{s+2s_{0}+2k_{0}+3}^{k_{0},\upsilon}$,
and, for all
$\widehat{\imath}:=({\widehat{\theta}},{\widehat{I}},{\widehat{w}})$,

$\displaystyle\left\|{\rm
d}_{i}X_{P}(i)[\widehat{\imath}]\right\|_{s}^{k_{0},\upsilon}$
$\displaystyle\lesssim_{s}\left\|\widehat{\imath}\right\|_{s+1}^{k_{0},\upsilon}+\left\|{\mathfrak{I}}\right\|_{s+2s_{0}+2k_{0}+4}^{k_{0},\upsilon}\left\|\widehat{\imath}\right\|_{s_{0}+1}^{k_{0},\upsilon}\,,$
$\displaystyle\left\|{\rm
d}_{i}^{2}X_{P}(i)[\widehat{\imath},\widehat{\imath}]\right\|_{s}^{k_{0},\upsilon}$
$\displaystyle\lesssim_{s}\left\|\widehat{\imath}\right\|_{s+1}^{k_{0},\upsilon}\left\|\widehat{\imath}\right\|_{s_{0}+1}^{k_{0},\upsilon}+\left\|{\mathfrak{I}}\right\|_{s+2s_{0}+2k_{0}+5}^{k_{0},\upsilon}(\left\|\widehat{\imath}\right\|_{s_{0}+1}^{k_{0},\upsilon})^{2}\,.$

Along this section, we assume the following hypothesis, which is verified by
the approximate solutions obtained at each step of the Nash-Moser Theorem 9.1.

  * •

ANSATZ. The map
$(\omega,\gamma)\mapsto{\mathfrak{I}}_{0}(\omega,\gamma)=i_{0}({\varphi};\omega,\gamma)-({\varphi},0,0)$
is $k_{0}$-times differentiable with respect to the parameters
$(\omega,\gamma)\in{\mathbb{R}}^{\nu}\times[\gamma_{1},\gamma_{2}]$ and, for
some $\mu:=\mu(\tau,\nu)>0$, $\upsilon\in(0,1)$,

$\left\|{\mathfrak{I}}_{0}\right\|_{s_{0}+\mu}^{k_{0},\upsilon}+\left|\alpha_{0}-\omega\right|^{k_{0},\upsilon}\leq
C\varepsilon\upsilon^{-1}\,.$ (6.1)

We first modify the approximate torus $i_{0}({\varphi})$ to obtain a nearby
isotropic torus $i_{\delta}({\varphi})$, namely such that the pull-back 1-form
$i_{\delta}^{*}\Lambda$ is closed, where $\Lambda$ is the Liouville 1-form
defined in (2.44). Consider the pull-back $1$-form

$\displaystyle i_{0}^{*}\Lambda$
$\displaystyle=\sum_{k=1}^{\nu}a_{k}({\varphi}){\rm d}{\varphi}_{k}\,,\quad
a_{k}({\varphi}):=-\big{(}[\partial_{\varphi}\theta_{0}({\varphi})]^{\top}I_{0}({\varphi})\big{)}_{k}+\tfrac{1}{2}\big{(}J_{\angle}^{-1}w_{0}({\varphi}),\partial_{{\varphi}_{k}}w_{0}({\varphi})\big{)}_{L^{2}}\,,$
(6.2)

and define
$A_{kj}({\varphi}):=\partial_{{\varphi}_{k}}a_{j}({\varphi})-\partial_{{\varphi}_{j}}a_{k}({\varphi})$.
The next Lemma follows as in Lemma 5.3 in [2] and Lemma 6.2 in [7]. Let
$Z({\varphi}):={\mathcal{F}}(i_{0},\alpha_{0})({\varphi})=\omega\cdot\partial_{\varphi}i_{0}({\varphi})-X_{H_{\alpha_{0}}}(i_{0}({\varphi}))$.

######  Lemma 6.2.

(Isotropic torus) The torus
$i_{\delta}({\varphi}):=(\theta_{0}({\varphi}),I_{\delta}({\varphi}),w_{0}({\varphi}))$,
defined by

$I_{\delta}({\varphi}):=I_{0}({\varphi})+[\partial_{\varphi}\theta_{0}({\varphi})]^{-\top}\rho({\varphi})\,,\quad\rho=(\rho_{j})_{j=1,\ldots,\nu}\,,\quad\rho_{j}({\varphi}):=\Delta_{\varphi}^{-1}\sum_{k=1}^{\nu}\partial_{{\varphi}_{k}}A_{kj}({\varphi})\,,$

is isotropic. Moreover, there is $\sigma:=\sigma(\nu,\tau)$ such that, for all
$s\geq s_{0}$,

$\displaystyle\left\|I_{\delta}-I_{0}\right\|_{s}^{k_{0},\upsilon}$
$\displaystyle\lesssim_{s}\left\|{\mathfrak{I}}_{0}\right\|_{s+1}^{k_{0},\upsilon}\,,\quad\left\|I_{\delta}-I_{0}\right\|_{s}^{k_{0},\upsilon}\lesssim_{s}\upsilon^{-1}\big{(}\left\|Z\right\|_{s+\sigma}^{k_{0},\upsilon}+\left\|Z\right\|_{s_{0}+\sigma}^{k_{0},\upsilon}\left\|{\mathfrak{I}}_{0}\right\|_{s+\sigma}^{k_{0},\upsilon}\big{)}$
(6.3)
$\displaystyle\left\|{\mathcal{F}}(i_{\delta},\alpha_{0})\right\|_{s}^{k_{0},\upsilon}$
$\displaystyle\lesssim_{s}\left\|Z\right\|_{s+\sigma}^{k_{0},\upsilon}+\left\|Z\right\|_{s_{0}+\sigma}^{k_{0},\upsilon}\left\|{\mathfrak{I}}_{0}\right\|_{s+\sigma}^{k_{0},\upsilon}\,,\quad\left\|{\rm
d}_{i}(i_{\delta})[\widehat{\imath}]\right\|_{s_{1}}\lesssim_{s_{1}}\left\|\widehat{\imath}\right\|_{s_{1}+1}\,,$
(6.4)

for $s_{1}\leq s_{0}+\mu$ (cfr. (6.1)). Furthermore $i_{\delta}({\varphi})$ is
a reversible and traveling torus, cfr. (5.7).

We first find an approximate inverse of the linearized operator ${\rm
d}_{i,\alpha}{\mathcal{F}}(i_{\delta})$. We introduce the symplectic
diffeomorphism $G_{\delta}:(\phi,y,{\mathtt{w}})\rightarrow(\theta,I,w)$ of
the phase space
${\mathbb{T}}^{\nu}\times{\mathbb{R}}^{\nu}\times\mathfrak{H}_{{\mathbb{S}}^{+},\Sigma}^{\angle}$,

$\begin{pmatrix}\theta\\\ I\\\
w\end{pmatrix}:=G_{\delta}\begin{pmatrix}\phi\\\ y\\\
{\mathtt{w}}\end{pmatrix}:=\begin{pmatrix}\theta_{0}(\phi)\\\
I_{\delta}(\phi)+\left[\partial_{\phi}\theta_{0}(\phi)\right]^{-\top}y+\left[(\partial_{\theta}{\widetilde{w}}_{0})(\theta_{0}(\phi))\right]^{\top}J_{\angle}^{-1}{\mathtt{w}}\\\
w_{0}(\phi)+{\mathtt{w}}\end{pmatrix}\,,$ (6.5)

where ${\widetilde{w}}_{0}(\theta):=w_{0}(\theta_{0}^{-1}(\theta))$. It is
proved in Lemma 2 of [6] that $G_{\delta}$ is symplectic, because the torus
$i_{\delta}$ is isotropic (Lemma 6.2). In the new coordinates, $i_{\delta}$ is
the trivial embedded torus $(\phi,y,{\mathtt{w}})=(\phi,0,0)$. Moreover the
diffeomorphism $G_{\delta}$ in (6.5) is reversibility and momentum preserving,
in the sense that (Lemma 6.3 in [7]) $\vec{\mathcal{S}}\circ
G_{\delta}=G_{\delta}\circ\vec{\mathcal{S}}$, $\vec{\tau}_{\varsigma}\circ
G_{\delta}=G_{\delta}\circ\vec{\tau}_{\varsigma}$,
$\forall\,\varsigma\in{\mathbb{R}}$, where $\vec{\mathcal{S}}$ and
$\vec{\tau}_{\varsigma}$ are defined respectively in (2.40), (2.41).

Under the symplectic diffeomorphism $G_{\delta}$, the Hamiltonian vector field
$X_{H_{\alpha}}$ changes into

$X_{K_{\alpha}}=\left(DG_{\delta}\right)^{-1}X_{H_{\alpha}}\circ
G_{\delta}\qquad{\rm where}\qquad K_{\alpha}:=H_{\alpha}\circ G_{\delta}$

is reversible and momentum preserving, in the sense that
$K_{\alpha}\circ\vec{\mathcal{S}}=K_{\alpha}$,
$K_{\alpha}\circ\vec{\tau}_{\varsigma}=K_{\alpha}$,
$\forall\,\varsigma\in{\mathbb{R}}$.

The Taylor expansion of $K_{\alpha}$ at the trivial torus $(\phi,0,0)$ is

$\displaystyle K_{\alpha}(\phi,y,{\mathtt{w}})=$ $\displaystyle\
K_{00}(\phi,\alpha)+K_{10}(\phi,\alpha)\cdot
y+(K_{01}(\phi,\alpha),{\mathtt{w}})_{L^{2}}+\tfrac{1}{2}K_{20}(\phi)y\cdot y$
(6.6)
$\displaystyle+(K_{11}(\phi)y,{\mathtt{w}})_{L^{2}}+\tfrac{1}{2}(K_{02}(\phi){\mathtt{w}},{\mathtt{w}})_{L^{2}}+K_{\geq
3}(\phi,y,{\mathtt{w}})\,,$

where $K_{\geq 3}$ collects all terms at least cubic in the variables
$(y,{\mathtt{w}})$. Here $K_{00}\in{\mathbb{R}}$,
$K_{10}\in{\mathbb{R}}^{\nu}$,
$K_{01}\in\mathfrak{H}_{{\mathbb{S}}^{+},\Sigma}^{\angle}$, whereas $K_{20}$
is a $\nu\times\nu$ symmetric matrix,
$K_{11}\in{\mathcal{L}}({\mathbb{R}}^{\nu},\mathfrak{H}_{{\mathbb{S}}^{+},\Sigma}^{\angle})$
and $K_{02}$ is a self-adjoint operator acting on
$\mathfrak{H}_{{\mathbb{S}}^{+},\Sigma}^{\angle}$.

The Hamilton equations associated to (6.6) are

$\begin{cases}\dot{\phi}=K_{10}(\phi,\alpha)+K_{20}(\phi)y+[K_{11}(\phi)]^{\top}{\mathtt{w}}+\partial_{y}K_{\geq
3}(\phi,y,{\mathtt{w}})\\\
\dot{y}=-\partial_{\phi}K_{00}(\phi,\alpha)-[\partial_{\phi}K_{10}(\phi,\alpha)]^{\top}y-[\partial_{\phi}K_{01}(\phi,\alpha)]^{\top}{\mathtt{w}}\\\
\ \ \ \ \ -\partial_{\phi}\left(\tfrac{1}{2}K_{20}(\phi)y\cdot
y+\left(K_{11}(\phi)y,{\mathtt{w}}\right)_{L^{2}}+\tfrac{1}{2}\left(K_{02}(\phi){\mathtt{w}},{\mathtt{w}}\right)_{L^{2}}+K_{\geq
3}(\phi,y,{\mathtt{w}})\right)\\\
\dot{\mathtt{w}}=J_{\angle}\,\left(K_{01}(\phi,\alpha)+K_{11}(\phi)y+K_{02}(\phi){\mathtt{w}}+\nabla_{{\mathtt{w}}}K_{\geq
3}(\phi,y,{\mathtt{w}})\right)\,,\end{cases}$ (6.7)

where $\partial_{\phi}K_{10}^{\top}$ is the $\nu\times\nu$ transposed matrix
and
$\partial_{\phi}K_{01}^{\top},K_{11}^{\top}:\mathfrak{H}_{{\mathbb{S}}^{+},\Sigma}^{\angle}\rightarrow{\mathbb{R}}^{\nu}$
are defined by the duality relation
$(\partial_{\phi}K_{01}[{\widehat{\phi}}],{\mathtt{w}})_{L^{2}}={\widehat{\phi}}\cdot[\partial_{\phi}K_{01}]^{\top}{\mathtt{w}}$
for any ${\widehat{\phi}}\in{\mathbb{R}}^{\nu}$,
${\mathtt{w}}\in\mathfrak{H}_{{\mathbb{S}}^{+},\Sigma}^{\angle}$.

The terms $K_{00},K_{01}$, $K_{10}-\omega$ in the Taylor expansion (6.6)
vanish at an exact solution: indeed, arguing as in Lemma 5.4 in [2], there is
$\sigma:=\sigma(\nu,\tau)>0$, such that, for all $s\geq s_{0}$,

$\left\|\partial_{\phi}K_{00}(\cdot,\alpha_{0})\right\|_{s}^{k_{0},\upsilon}+\left\|K_{10}(\cdot,\alpha_{0})-\omega\right\|_{s}^{k_{0},\upsilon}+\left\|K_{01}(\cdot,\alpha_{0})\right\|_{s}^{k_{0},\upsilon}\lesssim_{s}\left\|Z\right\|_{s+\sigma}^{k_{0},\upsilon}+\left\|Z\right\|_{s_{0}+\sigma}^{k_{0},\upsilon}\left\|{\mathfrak{I}}_{0}\right\|_{s+\sigma}^{k_{0},\upsilon}\,.$
(6.8)

Under the linear change of variables

$DG_{\delta}({\varphi},0,0)\begin{pmatrix}{\widehat{\phi}}\\\ {\widehat{y}}\\\
{\widehat{{\mathtt{w}}}}\end{pmatrix}:=\begin{pmatrix}\partial_{\phi}\theta_{0}({\varphi})&0&0\\\
\partial_{\phi}I_{\delta}({\varphi})&[\partial_{\phi}\theta_{0}({\varphi})]^{-\top}&[(\partial_{\theta}{\widetilde{w}}_{0})(\theta_{0}({\varphi}))]^{\top}J_{\angle}^{-1}\\\
\partial_{\phi}w_{0}({\varphi})&0&{\rm
Id}\end{pmatrix}\begin{pmatrix}{\widehat{\phi}}\\\ {\widehat{y}}\\\
{\widehat{{\mathtt{w}}}}\end{pmatrix}\,,$

the linearized operator ${\rm d}_{i,\alpha}{\mathcal{F}}(i_{\delta})$ is
approximately transformed into the one obtained when one linearizes the
Hamiltonian system (6.7) at $(\phi,y,{\mathtt{w}})=({\varphi},0,0)$,
differentiating also in $\alpha$ at $\alpha_{0}$ and changing
$\partial_{t}\rightsquigarrow\omega\cdot\partial_{\varphi}$, namely

$\begin{pmatrix}\widehat{\phi}\\\ \widehat{y}\\\ \widehat{\mathtt{w}}\\\
\widehat{\alpha}\end{pmatrix}\mapsto\begin{pmatrix}\omega\cdot\partial_{\varphi}{\widehat{\phi}}-\partial_{\phi}K_{10}({\varphi})[{\widehat{\phi}}]-\partial_{\alpha}K_{10}({\varphi})[{\widehat{\alpha}}]-K_{20}({\varphi}){\widehat{y}}-[K_{11}({\varphi})]^{\top}{\widehat{{\mathtt{w}}}}\\\
\omega\cdot\partial_{\varphi}{\widehat{y}}+\partial_{\phi\phi}K_{00}({\varphi})[{\widehat{\phi}}]+\partial_{\alpha}\partial_{\phi}K_{00}({\varphi})[{\widehat{\alpha}}]+[\partial_{\phi}K_{10}({\varphi})]^{\top}{\widehat{y}}+[\partial_{\phi}K_{01}({\varphi})]^{\top}{\widehat{{\mathtt{w}}}}\\\
\omega\cdot\partial_{\varphi}{\widehat{{\mathtt{w}}}}-J_{\angle}\,\big{(}\partial_{\phi}K_{01}({\varphi})[{\widehat{\phi}}]+\partial_{\alpha}K_{01}({\varphi})[{\widehat{\alpha}}]+K_{11}({\varphi}){\widehat{y}}+K_{02}({\varphi}){\widehat{{\mathtt{w}}}}\big{)}\end{pmatrix}.$
(6.9)

In order to construct an “almost approximate" inverse of (6.9), we need that

${\mathcal{L}}_{\omega}:=\Pi_{{\mathbb{S}}^{+},\Sigma}^{\angle}\left(\omega\cdot\partial_{\varphi}-JK_{02}({\varphi})\right)|_{\mathfrak{H}_{{\mathbb{S}}^{+},\Sigma}^{\angle}}$
(6.10)

is "almost invertible" (on traveling waves) up to remainders of size
$O(N_{{\mathtt{n}}-1}^{-{{\mathtt{a}}}})$, where, for
${\mathtt{n}}\in{\mathbb{N}}_{0}$

$N_{\mathtt{n}}:=K_{\mathtt{n}}^{p}\,,\quad
K_{\mathtt{n}}:=K_{0}^{\chi^{\mathtt{n}}}\,,\quad\chi=3/2\,.$ (6.11)

The $(K_{\mathtt{n}})_{{\mathtt{n}}\geq 0}$ is the scale used in the nonlinear
Nash-Moser iteration of Section 9 and $(N_{\mathtt{n}})_{{\mathtt{n}}\geq 0}$
is the one in Lemma 7.7 and Theorem 8.2. Let
$H_{\angle}^{s}({\mathbb{T}}^{\nu+1}):=H^{s}({\mathbb{T}}^{\nu+1})\cap\mathfrak{H}_{{\mathbb{S}}^{+},\Sigma}^{\angle}$.

  * (AI)

Almost invertibility of ${\mathcal{L}}_{\omega}$: There exist positive real
numbers $\sigma$, $\mu({\mathtt{b}})$, ${\mathtt{a}}$, $p$, $K_{0}$ and a
subset
${\mathtt{\Lambda}}_{o}\subset{\mathtt{D}}{\mathtt{C}}(\upsilon,\tau)\times[\gamma_{1},\gamma_{2}]$
such that, for all $(\omega,\gamma)\in{\mathtt{\Lambda}}_{o}$, the operator
${\mathcal{L}}_{\omega}$ may be decomposed as

${\mathcal{L}}_{\omega}={\mathcal{L}}_{\omega}^{<}+{\mathcal{R}}_{\omega}+{\mathcal{R}}_{\omega}^{\perp}\,,$
(6.12)

where, for any traveling wave function $g\in
H_{\angle}^{s+\sigma}({\mathbb{T}}^{\nu+1},{\mathbb{R}}^{2})$ and for any
$(\omega,\gamma)\in{\mathtt{\Lambda}}_{o}$, there is a traveling wave solution
$h\in H_{\angle}^{s}({\mathbb{T}}^{\nu+1},{\mathbb{R}}^{2})$ of
${\mathcal{L}}_{\omega}^{<}h=g$ satisfying, for all $s_{0}\leq s\leq
S-\mu({\mathtt{b}})-\sigma$,

$\left\|({\mathcal{L}}_{\omega}^{<})^{-1}g\right\|_{s}^{k_{0},\upsilon}\lesssim_{S}\upsilon^{-1}\big{(}\left\|g\right\|_{s+\sigma}^{k_{0},\upsilon}+\left\|g\right\|_{s_{0}+\sigma}^{k_{0},\upsilon}\left\|{\mathfrak{I}}_{0}\right\|_{s+\mu({{\mathtt{b}}})+\sigma}^{k_{0},\upsilon}\big{)}\,.$
(6.13)

In addition, if $g$ is anti-reversible, then $h$ is reversible. Moreover, for
any $s_{0}\leq s\leq S-\mu({\mathtt{b}})-\sigma$, for any traveling wave
$h\in\mathfrak{H}_{{\mathbb{S}}^{+},\Sigma}^{\angle}$, the operators
${\mathcal{R}}_{\omega},{\mathcal{R}}_{\omega}^{\perp}$ satisfy the estimates

$\displaystyle\left\|{\mathcal{R}}_{\omega}h\right\|_{s}^{k_{0},\upsilon}$
$\displaystyle\lesssim_{S}\varepsilon\upsilon^{-3}N_{{\mathtt{n}}-1}^{-{\mathtt{a}}}\big{(}\left\|h\right\|_{s+\sigma}^{k_{0},\upsilon}+\left\|h\right\|_{s_{0}+\sigma}^{k_{0},\upsilon}\left\|{\mathfrak{I}}_{0}\right\|_{s+\mu({\mathtt{b}})+\sigma}^{k_{0},\upsilon}\big{)}\,,$
(6.14)
$\displaystyle\left\|{\mathcal{R}}_{\omega}^{\perp}h\right\|_{s_{0}}^{k_{0},\upsilon}$
$\displaystyle\lesssim_{S}K_{\mathtt{n}}^{-{\rm
b}}\big{(}\left\|h\right\|_{s_{0}+{\rm
b}+\sigma}^{k_{0},\upsilon}+\left\|h\right\|_{s_{0}+\sigma}^{k_{0},\upsilon}\left\|{\mathfrak{I}}_{0}\right\|_{s_{0}+\mu({\mathtt{b}})+\sigma+{\rm
b}}^{k_{0},\upsilon}\big{)}\,,\ \forall\,{\rm b}>0\,,$ (6.15)
$\displaystyle\left\|{\mathcal{R}}_{\omega}^{\perp}h\right\|_{s}^{k_{0},\upsilon}$
$\displaystyle\lesssim_{S}\left\|h\right\|_{s+\sigma}^{k_{0},\upsilon}+\left\|h\right\|_{s_{0}+\sigma}^{k_{0},\upsilon}\left\|{\mathfrak{I}}_{0}\right\|_{s+\mu({\mathtt{b}})+\sigma}^{k_{0},\upsilon}\,.$
(6.16)

This assumption shall be verified by Theorem 8.9 at each step of the Nash-
Moser iteration.

In order to find an almost approximate inverse of the linear operator in (6.9)
(and so of ${\rm d}_{i,\alpha}{\mathcal{F}}(i_{\delta})$), it is sufficient to
invert the operator

${\mathbb{D}}\big{[}{\widehat{\phi}},{\widehat{y}},{\widehat{{\mathtt{w}}}},{\widehat{\alpha}}\big{]}:=\begin{pmatrix}\omega\cdot\partial_{\varphi}{\widehat{\phi}}-\partial_{\alpha}K_{10}({\varphi})[{\widehat{\alpha}}]-K_{20}({\varphi}){\widehat{y}}-K_{11}^{\top}({\varphi}){\widehat{{\mathtt{w}}}}\\\
\omega\cdot\partial_{\varphi}{\widehat{y}}+\partial_{\alpha}\partial_{\phi}K_{00}({\varphi})[{\widehat{\alpha}}]\\\
{\mathcal{L}}_{\omega}^{<}{\widehat{{\mathtt{w}}}}-J_{\angle}\left(\partial_{\alpha}K_{01}({\varphi})[{\widehat{\alpha}}]+K_{11}({\varphi}){\widehat{y}}\right)\end{pmatrix}$
(6.17)

obtained neglecting in (6.9) the terms $\partial_{\phi}K_{10}$,
$\partial_{\phi\phi}K_{00}$, $\partial_{\phi}K_{00}$, $\partial_{\phi}K_{01}$
(they vanish at an exact solution by (6.8)) and the small remainders
${\mathcal{R}}_{\omega}$, ${\mathcal{R}}_{\omega}^{\perp}$ appearing in
(6.12).

As in section 6 of [7] we have the following result, where we denote
$\|(\phi,y,{\mathtt{w}},\alpha)\|_{s}^{k_{0},\upsilon}:=\max\big{\\{}\|(\phi,y,{\mathtt{w}})\|_{s}^{k_{0},\upsilon},\left|\alpha\right|^{k_{0},\upsilon}\big{\\}}$
(see [7, Proposition 6.5]):

######  Proposition 6.3.

Assume (6.1) (with $\mu=\mu({{\mathtt{b}}})+\sigma$) and (AI). Then, for all
$(\omega,\gamma)\in{\mathtt{\Lambda}}_{o}$, for any anti-reversible traveling
wave variation $g=(g_{1},g_{2},g_{3})$, there exists a unique solution
${\mathbb{D}}^{-1}g:=({\widehat{\phi}},{\widehat{y}},{\widehat{{\mathtt{w}}}},{\widehat{\alpha}})$
of
${\mathbb{D}}({\widehat{\phi}},{\widehat{y}},{\widehat{{\mathtt{w}}}},{\widehat{\alpha}})=g$
where $({\widehat{\phi}},{\widehat{y}},{\widehat{{\mathtt{w}}}})$ is a
reversible traveling wave variation. Moreover, for any $s_{0}\leq s\leq
S-\mu({\mathtt{b}})-\sigma$,
$\|{\mathbb{D}}^{-1}g\|_{s}^{k_{0},\upsilon}\lesssim_{S}\upsilon^{-1}\big{(}\|g\|_{s+\sigma}^{k_{0},\upsilon}+\|{\mathfrak{I}}_{0}\|_{s+\mu({{\mathtt{b}}})+\sigma}^{k_{0},\upsilon}\|g\|_{s_{0}+\sigma}^{k_{0},\upsilon}\big{)}$.

Finally we conclude that the operator

${\bf T}_{0}:={\bf
T}_{0}(i_{0}):=(D{\widetilde{G}}_{\delta})({\varphi},0,0)\circ{\mathbb{D}}^{-1}\circ(DG_{\delta})({\varphi},0,0)^{-1}$
(6.18)

is an almost approximate right inverse for ${\rm
d}_{i,\alpha}{\mathcal{F}}(i_{0})$, where
${\widetilde{G}}_{\delta}(\phi,y,{\mathtt{w}},\alpha):=\left(G_{\delta}(\phi,y,{\mathtt{w}}),\alpha\right)$
is the identity on the $\alpha$-component. Arguing exactly as in Theorem 6.6
in [7] we deduce the following.

######  Theorem 6.4.

(Almost approximate inverse) Assume (AI). Then there is
$\overline{\sigma}:=\overline{\sigma}(\tau,\nu,k_{0})>0$ such that, if (6.1)
holds with $\mu=\mu({\mathtt{b}})+\overline{\sigma}$, then, for all
$(\omega,\gamma)\in{\mathtt{\Lambda}}_{o}$ and for any anti-reversible
traveling wave variation $g:=(g_{1},g_{2},g_{3})$, the operator ${\bf T}_{0}$
defined in (6.18) satisfies, for all $s_{0}\leq s\leq
S-\mu({\mathtt{b}})-\overline{\sigma}$,

$\|{\bf
T}_{0}g\|_{s}^{k_{0},\upsilon}\lesssim_{S}\upsilon^{-1}\big{(}\|g\|_{s+\overline{\sigma}}^{k_{0},\upsilon}+\|{\mathfrak{I}}_{0}\|_{s+\mu({\mathtt{b}})+\overline{\sigma}}^{k_{0},\upsilon}\|g\|_{s_{0}+\overline{\sigma}}^{k_{0},\upsilon}\big{)}\,.$
(6.19)

Moreover, the first three components of ${\bf T}_{0}g$ form a reversible
traveling wave variation. Finally, ${\bf T}_{0}$ is an almost approximate
right inverse of ${\rm d}_{i,\alpha}{\mathcal{F}}(i_{0})$, namely

${\rm d}_{i,\alpha}{\mathcal{F}}(i_{0})\circ{\bf T}_{0}-{\rm
Id}={\mathcal{P}}(i_{0})+{\mathcal{P}}_{\omega}(i_{0})+{\mathcal{P}}_{\omega}^{\perp}(i_{0})\,,$

where, for any traveling wave variation $g$, for all $s_{0}\leq s\leq
S-\mu({\mathtt{b}})-\overline{\sigma}$,

$\displaystyle\|{\mathcal{P}}g\|_{s}^{k_{0},\upsilon}$
$\displaystyle\lesssim_{S}\upsilon^{-1}\Big{(}\|{\mathcal{F}}(i_{0},\alpha_{0})\|_{s_{0}+\overline{\sigma}}^{k_{0},\upsilon}\|g\|_{s+\overline{\sigma}}^{k_{0},\upsilon}$
(6.20)
$\displaystyle\qquad+\,\big{(}\|{\mathcal{F}}(i_{0},\alpha_{0})\|_{s+\overline{\sigma}}^{k_{0},\upsilon}+\|{\mathcal{F}}(i_{0},\alpha_{0})\|_{s_{0}+\overline{\sigma}}^{k_{0},\upsilon}\|{\mathfrak{I}}_{0}\|_{s+\mu({\mathtt{b}})+\overline{\sigma}}^{k_{0},\upsilon}\big{)}\|g\|_{s_{0}+\overline{\sigma}}^{k_{0},\upsilon}\Big{)}\,,$
(6.21) $\displaystyle\|{\mathcal{P}}_{\omega}g\|_{s}^{k_{0},\upsilon}$
$\displaystyle\lesssim_{S}\varepsilon\upsilon^{-4}N_{{\mathtt{n}}-1}^{-{\mathtt{a}}}\big{(}\|g\|_{s+\overline{\sigma}}^{k_{0},\upsilon}+\|{\mathfrak{I}}_{0}\|_{s+\mu({\mathtt{b}})+\overline{\sigma}}^{k_{0},\upsilon}\|g\|_{s_{0}+\overline{\sigma}}^{k_{0},\upsilon}\big{)}\,,$
(6.22)
$\displaystyle\|{\mathcal{P}}_{\omega}^{\perp}g\|_{s_{0}}^{k_{0},\upsilon}$
$\displaystyle\lesssim_{S,b}\upsilon^{-1}K_{\mathtt{n}}^{-b}\left(\|g\|_{s_{0}+\overline{\sigma}+b}^{k_{0},\upsilon}+\|{\mathfrak{I}}_{0}\|_{s_{0}+\mu({\mathtt{b}})+b+\overline{\sigma}}^{k_{0},\upsilon}\|g\|_{s_{0}+\overline{\sigma}}^{k_{0},\upsilon}\right)\,,\quad\forall\,b>0\,,$
(6.23) $\displaystyle\|{\mathcal{P}}_{\omega}^{\perp}g\|_{s}^{k_{0},\upsilon}$
$\displaystyle\lesssim_{S}\upsilon^{-1}\big{(}\|g\|_{s+\overline{\sigma}}^{k_{0},\upsilon}+\|{\mathfrak{I}}_{0}\|_{s+\mu({\mathtt{b}})+\overline{\sigma}}^{k_{0},\upsilon}\|g\|_{s_{0}+\overline{\sigma}}^{k_{0},\upsilon}\big{)}\,.$
(6.24)

##  7 The linearized operator in the normal subspace

We now write an explicit expression of the linear operator
${\mathcal{L}}_{\omega}$ defined in (6.10). As in Lemma 7.1 in [7], since the
diffeomorphism $G_{\delta}$ in (6.5) is just a translation along the infinite
dimensional normal variable $w$, we have the following structural result.

######  Lemma 7.1.

The Hamiltonian operator ${\mathcal{L}}_{\omega}$ defined in (6.10), acting on
the normal subspace $\mathfrak{H}_{{\mathbb{S}}^{+},\Sigma}^{\angle}$, has the
form

${\mathcal{L}}_{\omega}=\Pi_{{\mathbb{S}}^{+},\Sigma}^{\angle}({\mathcal{L}}-\varepsilon
JR)|_{\mathfrak{H}_{{\mathbb{S}}^{+},\Sigma}^{\angle}}\,,$ (7.1)

where:
1\. ${\mathcal{L}}$ is the Hamiltonian operator

${\mathcal{L}}:=\omega\cdot\partial_{\varphi}-J\partial_{u}\nabla_{u}{\mathcal{H}}(T_{\delta}(\varphi))\,,$
(7.2)

where ${\mathcal{H}}$ is the water waves Hamiltonian in the Wahlén variables
defined in (2.9), evaluated at the reversible traveling wave

$T_{\delta}(\phi):=\varepsilon A(i_{\delta}(\phi))=\varepsilon
A\left(\theta_{0}(\phi),I_{\delta}(\phi),w_{0}(\phi)\right)=\varepsilon
v^{\intercal}\left(\theta_{0}(\phi),I_{\delta}(\phi)\right)+\varepsilon
w_{0}(\phi)\,,$ (7.3)

the torus
$i_{\delta}({\varphi}):=(\theta_{0}({\varphi}),I_{\delta}({\varphi}),w_{0}({\varphi}))$
is defined in Lemma 6.2 and $A(\theta,I,w)$, $v^{\intercal}(\theta,I)$ in
(2.2); 2\. $R(\phi)$ has the ‘finite rank" form

$R(\phi)[h]=\sum_{j=1}^{\nu}\left(h,g_{j}\right)_{L^{2}}\chi_{j}\,,\quad\forall\,h\in\mathfrak{H}_{{\mathbb{S}}^{+},\Sigma}^{\angle}\,,$
(7.4)

for functions
$g_{j},\chi_{j}\in\mathfrak{H}_{{\mathbb{S}}^{+},\Sigma}^{\angle}$ which
satisfy, for some $\sigma:=\sigma(\tau,\nu,k_{0})>0$, for all
$j=1,\ldots,\nu$, for all $s\geq s_{0}$,

$\displaystyle\left\|g_{j}\right\|_{s}^{k_{0},\upsilon}+\left\|\chi_{j}\right\|_{s}^{k_{0},\upsilon}$
$\displaystyle\lesssim_{s}1+\left\|{\mathfrak{I}}_{\delta}\right\|_{s+\sigma}^{k_{0},\upsilon}\,,$
(7.5) $\displaystyle\left\|{\rm
d}_{i}g_{j}[\widehat{\imath}]\right\|_{s}+\left\|{\rm
d}_{i}\chi_{j}[\widehat{\imath}]\right\|_{s}$
$\displaystyle\lesssim_{s}\left\|\widehat{\imath}\right\|_{s+\sigma}+\left\|\widehat{\imath}\right\|_{s_{0}+\sigma}\left\|{\mathfrak{I}}_{\delta}\right\|_{s+\sigma}\,.$

The operator ${\mathcal{L}}_{\omega}$ is reversible and momentum preserving.

In order to compute $dX$ we use the "shape derivative" formula, see e.g. [25],

$G^{\prime}(\eta)[{\widehat{\eta}}]\psi:=\lim_{\epsilon\rightarrow
0}\tfrac{1}{\epsilon}\big{(}G(\eta+\epsilon{\widehat{\eta}})\psi-G(\eta)\psi\big{)}=-G(\eta)(B{\widehat{\eta}})-\partial_{x}(V{\widehat{\eta}})\,,$
(7.6)

where

$B(\eta,\psi):=\frac{G(\eta)\psi+\eta_{x}\psi_{x}}{1+\eta_{x}^{2}}\,,\quad
V(\eta,\psi):=\psi_{x}-B(\eta,\psi)\eta_{x}\,.$ (7.7)

Then, recalling (2.9), (2.7), (1.6) and (7.6) the operator ${\mathcal{L}}$ in
(7.2) is given by

$\displaystyle{\mathcal{L}}=\omega\cdot\partial_{\varphi}$
$\displaystyle+\begin{pmatrix}\partial_{x}{\widetilde{V}}+G(\eta)B&-G(\eta)\\\
g+B{\widetilde{V}}_{x}+BG(\eta)B&{\widetilde{V}}\partial_{x}-BG(\eta)\end{pmatrix}$
(7.8)
$\displaystyle+\frac{\gamma}{2}\begin{pmatrix}-G(\eta)\partial_{x}^{-1}&0\\\
\partial_{x}^{-1}G(\eta)B-BG(\eta)\partial_{x}^{-1}-\frac{\gamma}{2}\partial_{x}^{-1}G(\eta)\partial_{x}^{-1}&-\partial_{x}^{-1}G(\eta)\end{pmatrix}\,,$

where

${\widetilde{V}}:=V-\gamma\eta\,,$ (7.9)

and the functions $B:=B(\eta,\psi)$, $V:=V(\eta,\psi)$ in (7.8)-(7.9) are
evaluated at the reversible traveling wave
$(\eta,\psi):=WT_{\delta}({\varphi})$ where $T_{\delta}({\varphi})$ is defined
in (7.3).
Notation. In (7.8) and hereafter the function $B$ is identified with the
corresponding multiplication operators $h\mapsto Bh$, and, where there is no
parenthesis, composition of operators is understood. For example $BG(\eta)B$
means $B\circ G(\eta)\circ B$.

######  Remark 7.2.

We consider the operator ${\mathcal{L}}$ in (7.8) acting on (a dense subspace
of) the whole $L^{2}({\mathbb{T}})\times L^{2}({\mathbb{T}})$. In particular
we extend the operator $\partial_{x}^{-1}$ to act on the whole
$L^{2}({\mathbb{T}})$ as in (3.22).

The following algebraic properties are a direct consequence of the reversible
and space-invariance properties of the water waves equations explained in
Section 2 and the fact that the approximate solution
$(\eta,\zeta)=T_{\delta}({\varphi})$ is a reversible traveling wave (cfr.
Lemma 7.3 in [7]).

######  Lemma 7.3.

The functions $(\eta,\zeta)=T_{\delta}({\varphi})$ and $B,{\widetilde{V}}$
defined in (7.7), (7.9) are quasi-periodic traveling waves. The functions
$(\eta,\zeta)=T_{\delta}({\varphi})$ are $({\rm even}({\varphi},x),{\rm
odd}({\varphi},x))$, $B$ is ${\rm odd}({\varphi},x)$ and ${\widetilde{V}}$ is
${\rm even}({\varphi},x)$. The Hamiltonian operator ${\mathcal{L}}$ is
reversible and momentum preserving.

For the sequel we will always assume the following ansatz (satisfied by the
approximate solutions obtained along the nonlinear Nash-Moser iteration of
Section 9): for some constants $\mu_{0}:=\mu_{0}(\tau,\nu)>0$,
$\upsilon\in(0,1)$, (cfr. Lemma 6.2)

$\left\|{\mathfrak{I}}_{0}\right\|_{s_{0}+\mu_{0}}^{k_{0},\upsilon}\,,\
\left\|{\mathfrak{I}}_{\delta}\right\|_{s_{0}+\mu_{0}}^{k_{0},\upsilon}\leq
1\,.$ (7.10)

In order to estimate the variation of the eigenvalues with respect to the
approximate invariant torus, we need also to estimate the variation with
respect to the torus $i({\varphi})$ in another low norm $\left\|\
\right\|_{s_{1}}$ for all Sobolev indexes $s_{1}$ such that

$s_{1}+\sigma_{0}\leq s_{0}+\mu_{0}\,,\quad\text{ for some }\
\sigma_{0}:=\sigma_{0}(\tau,\nu)>0\,.$ (7.11)

Thus, by (7.10), we have
$\left\|{\mathfrak{I}}_{0}\right\|_{s_{1}+\sigma_{0}}^{k_{0},\upsilon}$,
$\left\|{\mathfrak{I}}_{\delta}\right\|_{s_{1}+\sigma_{0}}^{k_{0},\upsilon}\leq
1$. The constants $\mu_{0}$ and $\sigma_{0}$ represent the _loss of
derivatives_ accumulated along the reduction procedure of the next sections.
What is important is that they are independent of the Sobolev index $s$. In
the following sections we shall denote by $\sigma:=\sigma(\tau,\nu,k_{0})>0$,
$\sigma_{N}({\mathtt{q}}_{0}):=\sigma_{N}({\mathtt{q}}_{0},\tau,\nu,k_{0})$,
$\sigma_{M}:=\sigma_{M}(k_{0},\tau,\nu)>0$, $\aleph_{M}(\alpha)$ constants
(which possibly increase from lemma to lemma) representing losses of
derivatives along the finitely many steps of the reduction procedure.

######  Remark 7.4.

In the next sections $\mu_{0}:=\mu_{0}(\tau,\nu,M,\alpha)>0$ will depend also
on indexes $M,\alpha$, whose maximal values will be fixed depending only on
$\tau$ and $\nu$ (and $k_{0}$ which is however considered an absolute constant
along the paper). In particular $M$ is fixed in (8.5), whereas the maximal
value of $\alpha$ depends on $M$, as explained in Remark 7.16.

######  Remark 7.5.

Starting from Section 7.2, we introduce in the estimates upper bounds on the
regularity $s\geq s_{0}$. We shall control the terms in Sobolev spaces $H^{s}$
with $s_{0}\leq s\leq S-\sigma$, where $\sigma$ denotes a loss of derivatives
of the finitely many steps of the reduction (possibly increasing along the
steps), whereas $S>s_{0}+k_{0}$ is any finite Sobolev index. The index $S$ has
to be taken finite in view of Lemma 7.7 (see also Appendix A). The largest
regularity index $S$ will be fixed in (9.3). In particular, it is compatible
with the condition (7.11), namely $s_{1}+\sigma_{0}\leq s_{0}+\mu_{0}<S$.

As a consequence of Moser composition Lemma 3.2 and (6.3), the Sobolev norm of
the function $u=T_{\delta}({\varphi})$ defined in (7.3) satisfies for all
$s\geq s_{0}$

$\left\|u\right\|_{s}^{k_{0},\upsilon}=\left\|\eta\right\|_{s}^{k_{0},\upsilon}+\left\|\zeta\right\|_{s}^{k_{0},\upsilon}\leq\varepsilon
C(s)\big{(}1+\left\|{\mathfrak{I}}_{0}\right\|_{s}^{k_{0},\upsilon}\big{)}$
(7.12)

(the map $A$ defined in (2.2) is smooth). Similarly, using (6.4),

$\left\|\Delta_{12}u\right\|_{s_{1}}\lesssim_{s_{1}}\varepsilon\left\|i_{2}-i_{1}\right\|_{s_{1}}\,,\quad\text{
where }\quad\Delta_{12}u:=u(i_{2})-u(i_{1})\,.$

We finally recall that ${\mathfrak{I}}_{0}={\mathfrak{I}}_{0}(\omega,\gamma)$
is defined for all
$(\omega,\gamma)\in{\mathbb{R}}^{\nu}\times[\gamma_{1},\gamma_{2}]$ and that
the functions $B,{\widetilde{V}}$ and $c$ appearing in ${\mathcal{L}}$ in
(7.8) are ${\mathcal{C}}^{\infty}$ in $({\varphi},x)$, as
$u=(\eta,\zeta)=T_{\delta}({\varphi})$ is.

In Sections 7.1-7.6 we are going to make several transformations, whose aim is
to conjugate the operator ${\cal L}$ in (7.8) to a constant coefficients
Fourier multiplier, up to a pseudo-differential operator of order $-1/2$ plus
a remainder that satisfies tame estimates, see ${\cal L}_{8}$ in (7.138).
Finally, in Section 7.7 we shall conjugate the restricted operator ${\cal
L}_{\omega}$ in (7.1).

###  7.1 Linearized good unknown of Alinhac

The first step is to conjugate the linear operator ${\mathcal{L}}$ in (7.8) by
the symplectic (Definition 3.18) multiplication matrix operator

${\mathcal{Z}}:=\left(\begin{matrix}{\rm Id}&0\\\ B&{\rm
Id}\end{matrix}\right)\ ,\qquad{\mathcal{Z}}^{-1}=\left(\begin{matrix}{\rm
Id}&0\\\ -B&{\rm Id}\end{matrix}\right)\,,$

obtaining

$\displaystyle{\mathcal{L}}_{1}$
$\displaystyle:={\mathcal{Z}}^{-1}{\mathcal{L}}{\mathcal{Z}}=\omega\cdot\partial_{\varphi}+\begin{pmatrix}\partial_{x}{\widetilde{V}}&-G(\eta)\\\
a&{\widetilde{V}}\partial_{x}\end{pmatrix}-\frac{\gamma}{2}\begin{pmatrix}G(\eta)\partial_{x}^{-1}&0\\\
\frac{\gamma}{2}\partial_{x}^{-1}G(\eta)\partial_{x}^{-1}&\partial_{x}^{-1}G(\eta)\end{pmatrix}\,,$
(7.13)

where $a$ is the function

$a:=g+{\widetilde{V}}B_{x}+\omega\cdot\partial_{\varphi}B\,.$ (7.14)

The matrix ${\mathcal{Z}}$ amounts to introduce, as in [25] and [9, 2], a
linearized version of the “good unknown of Alinhac".

######  Lemma 7.6.

The maps ${\mathcal{Z}}^{\pm 1}-{\rm Id}$ are ${\mathcal{D}}^{k_{0}}$-tame
with tame constants satisfying, for some $\sigma:=\sigma(\tau,\nu,k_{0})>0$,
for all $s\geq s_{0}$,

${\mathfrak{M}}_{{\mathcal{Z}}^{\pm 1}-{\rm Id}}(s)\,,\
{\mathfrak{M}}_{({\mathcal{Z}}^{\pm 1}-{\rm
Id})^{*}}(s)\lesssim_{s}\varepsilon\big{(}1+\left\|{\mathfrak{I}}_{0}\right\|_{s+\sigma}^{k_{0},\upsilon}\big{)}\,.$

The function $a$ in (7.14) is a quasi-periodic traveling wave ${\rm
even}({\varphi},x)$. There is $\sigma:=\sigma(\tau,\nu,k_{0})>0$ such that,
for all $s\geq s_{0}$,

$\left\|a-g\right\|_{s}^{k_{0},\upsilon}+\|{\widetilde{V}}\|_{s}^{k_{0},\upsilon}+\left\|B\right\|_{s}^{k_{0},\upsilon}\lesssim_{s}\varepsilon\big{(}1+\|{\mathfrak{I}}_{0}\|_{s+\sigma}^{k_{0},\upsilon}\big{)}\,.$
(7.15)

Moreover, for any $s_{1}$ as in (7.11),

$\displaystyle\left\|\Delta_{12}a\right\|_{s_{1}}+\|\Delta_{12}{\widetilde{V}}\|_{s_{1}}+\left\|\Delta_{12}B\right\|_{s_{1}}\lesssim_{s_{1}}\varepsilon\left\|i_{1}-i_{2}\right\|_{s_{1}+\sigma}\,,$
(7.16) $\displaystyle\|\Delta_{12}({\mathcal{Z}}^{\pm
1})h\|_{s_{1}},\|\Delta_{12}({\mathcal{Z}}^{\pm
1})^{*}h\|_{s_{1}}\lesssim_{s_{1}}\varepsilon\left\|i_{1}-i_{2}\right\|_{s_{1}+\sigma}\left\|h\right\|_{s_{1}}\,.$
(7.17)

The operator ${\mathcal{L}}_{1}$ is Hamiltonian, reversible and momentum
preserving.

###### Proof.

By the expressions of $B,\widetilde{V},a$ in (7.7), (7.9), (7.14) the
composition estimates of Lemma 3.2, (3.7) and the bounds for the Dirichlet-
Neumann operator in Lemma 3.10. Since $B$ is an ${\rm odd}(\varphi,x)$ quasi-
periodic traveling wave, the matrix operator ${\mathcal{Z}}$ is reversibility
and momentum preserving (Definitions 3.19 and 3.22). ∎

###  7.2 Almost-straightening of the first order transport operator

We now write the operator ${\mathcal{L}}_{1}$ in (7.13) as

${\mathcal{L}}_{1}=\omega\cdot\partial_{\varphi}+\begin{pmatrix}\partial_{x}{\widetilde{V}}&0\\\
0&{\widetilde{V}}\partial_{x}\end{pmatrix}+\begin{pmatrix}-\frac{\gamma}{2}G(0)\partial_{x}^{-1}&-G(0)\\\
a-\left(\frac{\gamma}{2}\right)^{2}\partial_{x}^{-1}G(0)\partial_{x}^{-1}&-\frac{\gamma}{2}\partial_{x}^{-1}G(0)\end{pmatrix}+{\bf
R}_{1}\,,$ (7.18)

where, using the decomposition (3.32) of the Dirichlet-Neumann operator,

${\bf
R}_{1}:=-\begin{pmatrix}\frac{\gamma}{2}{\mathcal{R}}_{G}(\eta)\partial_{x}^{-1}&{\mathcal{R}}_{G}(\eta)\\\
\left(\frac{\gamma}{2}\right)^{2}\partial_{x}^{-1}{\mathcal{R}}_{G}(\eta)\partial_{x}^{-1}&\frac{\gamma}{2}\partial_{x}^{-1}{\mathcal{R}}_{G}(\eta)\end{pmatrix}$
(7.19)

is a small remainder in ${\rm OP}S^{-\infty}$. The aim of this section is to
conjugate the variable coefficients quasi-periodic transport operator

${\mathcal{L}}_{\rm
TR}:=\omega\cdot\partial_{\varphi}+\begin{pmatrix}\partial_{x}{\widetilde{V}}&0\\\
0&{\widetilde{V}}\partial_{x}\end{pmatrix}$ (7.20)

to a constant coefficients transport operator
$\omega\cdot\partial_{\varphi}+{\mathtt{m}}_{1,\overline{\mathtt{n}}}\,\partial_{y}$,
up to an exponentially small remainder, see (7.28)-(7.29), where
${\mathtt{n}}\in{\mathbb{N}}_{0}$ and the scale
$(N_{{\mathtt{n}}})_{{\mathtt{n}}\in{\mathbb{N}}_{0}}$ is defined, for
$N_{0}>1$, by

$N_{{\mathtt{n}}}:=N_{0}^{\chi^{{\mathtt{n}}}}\,,\quad\chi=3/2\,,\quad
N_{-1}:=1\,.$ (7.21)

Such small remainder is left because we assume only finitely many non-
resonance conditions, see (7.26). This enables to deduce Lemma 7.9, and then
to formulate the non-resonance condition (5.15), stated in terms of the
“final" function ${\mathtt{m}}_{1}^{\infty}(\omega,\gamma)$, which implies
(7.26) at any step of the nonlinear Nash-Moser iteration of Section 9.

In the next lemma we conjugate ${\mathcal{L}}_{\rm TR}$ by a _symplectic_
(Definition 3.18) transformation

${\mathcal{E}}:=\begin{pmatrix}(1+\beta_{x}({\varphi},x))\circ{\mathcal{B}}&0\\\
0&{\mathcal{B}}\end{pmatrix}\,,\quad{\mathcal{E}}^{-1}:=\begin{pmatrix}{\mathcal{B}}^{-1}\circ(1+\beta_{x}({\varphi},x))^{-1}&0\\\
0&{\mathcal{B}}^{-1}\end{pmatrix}$ (7.22)

where the composition operator

$({\mathcal{B}}u)({\varphi},x):=u\left({\varphi},x+\beta({\varphi},x)\right)$
(7.23)

is induced by a ${\varphi}$-dependent diffeomorphism $y=x+\beta({\varphi},x)$
of the torus ${\mathbb{T}}_{x}$, for some small quasi-periodic traveling wave
$\beta:{\mathbb{T}}_{\varphi}^{\nu}\times{\mathbb{T}}_{x}\to{\mathbb{R}}$,
${\rm odd}({\varphi},x)$. Let

${\mathtt{b}}:=[{\mathtt{a}}]+2\in{\mathbb{N}}\,,\quad{\mathtt{a}}:=3(\tau_{1}+1)\geq
1\,,\quad\tau_{1}:=k_{0}+(k_{0}+1)\tau\,.$ (7.24)

######  Lemma 7.7.

(Almost-Straightening of the transport operator) There exists
$\tau_{2}(\tau,\nu)>\tau_{1}(\tau,\nu)+1+{\mathtt{a}}$ such that, for all
$S>s_{0}+k_{0}$, there are $N_{0}:=N_{0}(S,{\mathtt{b}})\in{\mathbb{N}}$ and
$\updelta:=\updelta(S,{\mathtt{b}})\in(0,1)$ such that, if
$N_{0}^{\tau_{2}}\varepsilon\upsilon^{-1}<\updelta$ the following holds true.
For any $\overline{\mathtt{n}}\in{\mathbb{N}}_{0}$:
1\. There exist a constant
${\mathtt{m}}_{1,\overline{\mathtt{n}}}:={\mathtt{m}}_{1,\overline{\mathtt{n}}}(\omega,\gamma)\in{\mathbb{R}}$,
where ${\mathtt{m}}_{1,0}=0$, defined for any
$(\omega,\gamma)\in{\mathbb{R}}^{\nu}\times[\gamma_{1},\gamma_{2}]$, and a
quasi-periodic traveling wave
$\beta({\varphi},x):=\beta_{\overline{\mathtt{n}}}({\varphi},x)$, ${\rm
odd}({\varphi},x)$, satisfying, for some $\sigma=\sigma(\tau,\nu,k_{0})>0$,
the estimates

$|{\mathtt{m}}_{1,\overline{\mathtt{n}}}|^{k_{0},\upsilon}\lesssim\varepsilon\,,\quad\|\beta\|_{s}^{k_{0},\upsilon}\lesssim_{S}\varepsilon\upsilon^{-1}(1+\|{\mathfrak{I}}_{0}\|_{s+\sigma+{\mathtt{b}}}^{k_{0},\upsilon})\,,\quad\forall\,s_{0}\leq
s\leq S\,,$ (7.25)

independently of $\overline{\mathtt{n}}$;
2\. For any $(\omega,\gamma)$ in

$\displaystyle{\mathtt{T}}{\mathtt{C}}_{\overline{\mathtt{n}}+1}(2\upsilon,\tau)$
$\displaystyle:={\mathtt{T}}{\mathtt{C}}_{\overline{\mathtt{n}}+1}({\mathtt{m}}_{1,\overline{\mathtt{n}}},2\upsilon,\tau)$
(7.26)
$\displaystyle:=\Big{\\{}(\omega,\gamma)\in{\mathbb{R}}^{\nu}\times[\gamma_{1},\gamma_{2}]\,:\,|(\omega-{\mathtt{m}}_{1,\overline{\mathtt{n}}}\vec{\jmath})\cdot\ell|\geq
2\upsilon\braket{\ell}^{-\tau}\,\ \forall\,0<|\ell|\leq
N_{\overline{\mathtt{n}}}\Big{\\}}$

the operator ${\mathcal{L}}_{\rm TR}$ in (7.20) is conjugated to

${\mathcal{E}}^{-1}{\mathcal{L}}_{\rm
TR}{\mathcal{E}}=\omega\cdot\partial_{\varphi}+{\mathtt{m}}_{1,\overline{\mathtt{n}}}\,\partial_{y}+{\bf
P}_{2}^{\perp}\,,$ (7.27)

where

${\bf
P}_{2}^{\perp}:=\begin{pmatrix}\partial_{y}p_{\overline{\mathtt{n}}}&0\\\
0&p_{\overline{\mathtt{n}}}\partial_{y}\end{pmatrix}\,,$ (7.28)

and the real, quasi-periodic traveling wave function
$p_{\overline{\mathtt{n}}}({\varphi},y)$, ${\rm even}({\varphi},y)$,
satisfies, for some $\sigma=\sigma(\tau,\nu,k_{0})>0$ and for any $s_{0}\leq
s\leq S$,

$\|p_{\overline{\mathtt{n}}}\|_{s}^{k_{0},\upsilon}\lesssim_{s,{\mathtt{b}}}\varepsilon\,N_{\overline{\mathtt{n}}-1}^{-{\mathtt{a}}}(1+\|{\mathfrak{I}}_{0}\|_{s+\sigma+{\mathtt{b}}}^{k_{0},\upsilon})\,;$
(7.29)

3\. The operators ${\mathcal{E}}^{\pm}$ are
${\mathcal{D}}^{k_{0}}$-$(k_{0}+1)$-tame, the operators ${\mathcal{E}}^{\pm
1}-{\rm Id}$, $({\mathcal{E}}^{\pm 1}-{\rm Id})^{*}$ are
${\mathcal{D}}^{k_{0}}$-$(k_{0}+2)$-tame with tame constants satisfying, for
some $\sigma:=\sigma(\tau,\nu,k_{0})>0$ and for all $s_{0}\leq s\leq
S-\sigma$,

$\displaystyle{\mathfrak{M}}_{{\mathcal{E}}^{\pm
1}}(s)\lesssim_{S}1+\|{\mathfrak{I}}_{0}\|_{s+\sigma}^{k_{0},\upsilon}\,,\
{\mathfrak{M}}_{{\mathcal{E}}^{\pm 1}-{\rm
Id}}(s)+{\mathfrak{M}}_{\left({\mathcal{E}}^{\pm 1}-{\rm
Id}\right)^{*}}(s)\lesssim_{S}\varepsilon\upsilon^{-1}(1+\|{\mathfrak{I}}_{0}\|_{s+\sigma+{\mathtt{b}}}^{k_{0},\upsilon})\,.$
(7.30)

4\. Furthermore, for any $s_{1}$ as in (7.11),

$\displaystyle|\Delta_{12}{\mathtt{m}}_{1,\overline{\mathtt{n}}}|\lesssim\varepsilon\left\|i_{1}-i_{2}\right\|_{s_{1}+\sigma}\,,\quad\|\Delta_{12}\beta\|_{s_{1}}\lesssim_{s_{1}}\varepsilon\upsilon^{-1}\|i_{1}-i_{2}\|_{s_{1}+\sigma+{\mathtt{b}}}\,,$
(7.31)
$\displaystyle\|\Delta_{12}({\mathcal{A}})h\|_{s_{1}}\lesssim_{s_{1}}\varepsilon\upsilon^{-1}\left\|i_{1}-i_{2}\right\|_{s_{1}+\sigma+{\mathtt{b}}}\left\|h\right\|_{s_{1}+\sigma+{\mathtt{b}}}\,,\quad{\mathcal{A}}\in\\{{\mathcal{E}}^{\pm
1},({\mathcal{E}}^{\pm 1})^{*}\\}\,.$ (7.32)

###### Proof.

We apply Theorem A.2 and Corollary A.4 to the transport operator
$X_{0}=\omega\cdot\partial_{\varphi}+\widetilde{V}\partial_{x}$, which has the
form (A.1) with $p_{0}=\widetilde{V}$. By (7.15) and (7.10), the smallness
conditions (A.3) and (A.10) hold for
$N_{0}^{\tau_{2}}\varepsilon\upsilon^{-1}$ sufficiently small. Therefore there
exist a constant ${\mathtt{m}}_{1,\overline{\mathtt{n}}}\in{\mathbb{R}}$ and a
quasi-periodic traveling wave
$\beta({\varphi},x):=\beta_{\overline{\mathtt{n}}}({\varphi},x)$, ${\rm
odd}({\varphi},x)$, such that, for any $(\omega,\gamma)$ in
${\mathtt{T}}{\mathtt{C}}_{\overline{\mathtt{n}}+1}(2\upsilon,\tau)\subseteq{\mathtt{\Lambda}}_{\overline{\mathtt{n}}+1}^{\upsilon,\rm
T}\subseteq{\mathtt{\Lambda}}_{\overline{\mathtt{n}}}^{\upsilon,\rm T}$ (see
Corollary A.3) we have

${\mathcal{B}}_{\overline{\mathtt{n}}}^{-1}(\omega\cdot\partial_{\varphi}+\widetilde{V}\partial_{x}){\mathcal{B}}_{\overline{\mathtt{n}}}=\omega\cdot\partial_{\varphi}+({\mathtt{m}}_{1,\overline{\mathtt{n}}}+p_{\overline{\mathtt{n}}}({\varphi},y))\partial_{y}$

where the function $p_{\overline{\mathtt{n}}}$ satisfies (7.29) by (A.5) and
(7.15). The estimates (A.6), (A.15), (7.15) imply (7.25) and (7.30). The
conjugated operator of ${\mathcal{L}}_{\rm TR}$ in (7.20) is

${\mathcal{E}}^{-1}{\mathcal{L}}_{\rm
TR}{\mathcal{E}}=\omega\cdot\partial_{\varphi}+\begin{pmatrix}A_{1}&0\\\
0&({\mathtt{m}}_{1,\overline{\mathtt{n}}}+p_{\overline{\mathtt{n}}})\partial_{y}\end{pmatrix}$

where
$\omega\cdot\partial_{\varphi}+A_{1}={\mathcal{B}}^{-1}(1+\beta_{x})^{-1}\big{(}\omega\cdot\partial_{\varphi}+\partial_{x}{\widetilde{V}}\big{)}(1+\beta_{x}){\mathcal{B}}$.
Since ${\mathcal{L}}_{\rm TR}$ is Hamiltonian (Definition 3.18), and the map
${\mathcal{E}}$ is symplectic, we have that
${\mathcal{E}}^{-1}{\mathcal{L}}_{\rm TR}{\mathcal{E}}$ is Hamiltonian as
well. In particular
$A_{1}=-(({\mathtt{m}}_{1,\overline{\mathtt{n}}}+p_{\overline{\mathtt{n}}})\partial_{y})^{*}={\mathtt{m}}_{1,\overline{\mathtt{n}}}\partial_{y}+\partial_{y}p_{\overline{\mathtt{n}}}$.
This proves (7.27)-(7.28). The estimates (7.31)-(7.32) follow by
(A.11)-(A.12), the bound for
$\|\Delta_{12}\beta_{\overline{\mathtt{n}}}\|_{s_{1}}$ in Corollary A.4 and
(7.16)-(7.17). ∎

######  Remark 7.8.

Actually, for any
$(\omega,\gamma)\in{\mathtt{T}}{\mathtt{C}}_{\overline{\mathtt{n}}+1}(2\upsilon,\tau)$
in (7.26), Theorem A.2 and Corollary A.3 would imply also the conjugation of
${\mathcal{L}}_{\rm TR}$ to the operator
$\omega\cdot\partial_{\varphi}+{\mathtt{m}}_{1,\overline{\mathtt{n}}+1}\partial_{y}$
for some ${\mathtt{m}}_{1,\overline{\mathtt{n}}+1}\in{\mathbb{R}}$, up to a
remainder ${\bf P}_{2}^{\perp}=O(\varepsilon
N_{\overline{\mathtt{n}}}^{-{\mathtt{a}}})$. For simplicity we stated only the
conjugation in (7.27). We shall use the non-resonance condition in (7.26) also
later in Sections 7.5, 7.6 .

The next lemma is needed in order to prove the inclusion of the Cantor sets
associated to two nearby approximate solutions.

######  Lemma 7.9.

Let $i_{1},i_{2}$ be close enough and $0<2\upsilon-\rho<2\upsilon<1$. Then

$\varepsilon
C(s_{1})N_{\overline{\mathtt{n}}}^{\tau+1}\|i_{1}-i_{2}\|_{s_{1}+\sigma}\leq\rho\quad\Rightarrow\quad{\mathtt{T}}{\mathtt{C}}_{\overline{\mathtt{n}}+1}(2\upsilon,\tau)(i_{1})\subseteq{\mathtt{T}}{\mathtt{C}}_{\overline{\mathtt{n}}+1}(2\upsilon-\rho,\tau)(i_{2})\,.$

###### Proof.

For any
$(\omega,\gamma)\in{\mathtt{T}}{\mathtt{C}}_{\overline{\mathtt{n}}+1}(2\upsilon,\tau)(i_{1})$,
using also (7.31), we have, for any
$\ell\in{\mathbb{Z}}^{\nu}\setminus\\{0\\}$, $|\ell|\leq
N_{\overline{\mathtt{n}}}$,

$\displaystyle|(\omega-{\mathtt{m}}_{1,\overline{\mathtt{n}}}(i_{2})\vec{\jmath})\cdot\ell|$
$\displaystyle\geq|(\omega-{\mathtt{m}}_{1,\overline{\mathtt{n}}}(i_{1})\vec{\jmath})\cdot\ell|-C|\Delta_{12}{\mathtt{m}}_{1,\overline{\mathtt{n}}}||\ell|$
$\displaystyle\geq\frac{2\upsilon}{\braket{\ell}^{\tau}}-C(s_{1})\varepsilon
N_{\overline{\mathtt{n}}}\|i_{1}-i_{2}\|_{s_{1}+\sigma}\geq\frac{2\upsilon-\rho}{\braket{\ell}^{\tau}}\,.$

We conclude that
$(\omega,\gamma)\in{\mathtt{T}}{\mathtt{C}}_{\overline{\mathtt{n}}+1}(2\upsilon-\rho,\tau)(i_{2})$.
∎

We now conjugate the whole operator ${\mathcal{L}}_{1}$ in (7.18)-(7.19) by
the operator ${\mathcal{E}}$ in (7.22).

We first compute the conjugation of the matrix

$\displaystyle{\mathcal{E}}^{-1}$
$\displaystyle\begin{pmatrix}-\frac{\gamma}{2}G(0)\partial_{x}^{-1}&-G(0)\\\
a-\left(\frac{\gamma}{2}\right)^{2}\partial_{x}^{-1}G(0)\partial_{x}^{-1}&-\frac{\gamma}{2}\partial_{x}^{-1}G(0)\end{pmatrix}{\mathcal{E}}$
$\displaystyle=\begin{pmatrix}-\frac{\gamma}{2}{\mathcal{B}}^{-1}(1+\beta_{x})^{-1}G(0)\partial_{x}^{-1}(1+\beta_{x}){\mathcal{B}}&-{\mathcal{B}}^{-1}(1+\beta_{x})^{-1}G(0){\mathcal{B}}\\\
{\mathcal{B}}^{-1}\big{(}a-\left(\frac{\gamma}{2}\right)^{2}\partial_{x}^{-1}G(0)\partial_{x}^{-1}\big{)}(1+\beta_{x}){\mathcal{B}}&-\frac{\gamma}{2}{\mathcal{B}}^{-1}\partial_{x}^{-1}G(0){\mathcal{B}}\end{pmatrix}\,.$

The multiplication operator for $a({\varphi},x)$ is transformed into the
multiplication operator for the function

${\mathcal{B}}^{-1}a(1+\beta_{x}){\mathcal{B}}={\mathcal{B}}^{-1}\big{(}a(1+\beta_{x})\big{)}\,.$
(7.33)

We write the Dirichlet-Neumann operator $G(0)$ in (1.9) as

$G(0)=G(0,{\mathtt{h}})=\partial_{x}{\mathcal{H}}T({\mathtt{h}})\,,$ (7.34)

where ${\mathcal{H}}$ is the Hilbert transform defined in (3.21) and

$T({\mathtt{h}}):=\begin{cases}\tanh({\mathtt{h}}|D|)={\rm Id}+{\rm
Op}(r_{\mathtt{h}})&\text{ if }{\mathtt{h}}<+\infty\,,\qquad
r_{{\mathtt{h}}}(\xi):=-\frac{2}{1+e^{2{\mathtt{h}}|\xi|\chi(\xi)}}\in
S^{-\infty}\,,\\\ {\rm Id}&\text{ if }{\mathtt{h}}=\infty\,.\end{cases}$
(7.35)

We have the conjugation formula (see formula (7.42) in [2])

${\mathcal{B}}^{-1}G(0){\mathcal{B}}=\left\\{{\mathcal{B}}^{-1}(1+\beta_{x})\right\\}G(0)+{\mathcal{R}}_{1}\,,$
(7.36)

where

${\mathcal{R}}_{1}:=\left\\{{\mathcal{B}}^{-1}(1+\beta_{x})\right\\}\partial_{y}\big{(}\l{\mathcal{H}}\left({\mathcal{B}}^{-1}{\rm
Op}(r_{\mathtt{h}}){\mathcal{B}}-{\rm
Op}(r_{\mathtt{h}})\right)+\left({\mathcal{B}}^{-1}{\mathcal{H}}{\mathcal{B}}-{\mathcal{H}}\right)({\mathcal{B}}^{-1}T({\mathtt{h}}){\mathcal{B}})\big{)}\,.$

The operator ${\mathcal{R}}_{1}$ is in ${\rm OP}S^{-\infty}$ because both
${\mathcal{B}}^{-1}{\rm Op}(r_{\mathtt{h}}){\mathcal{B}}-{\rm
Op}(r_{\mathtt{h}})$ and
${\mathcal{B}}^{-1}{\mathcal{H}}{\mathcal{B}}-{\mathcal{H}}$ are in ${\rm
OP}S^{-\infty}$ and there is ${\sigma}>0$ such that, for any
$m\in{\mathbb{N}}$, $\alpha\in{\mathbb{N}}_{0}$ and $s\geq s_{0}$,

$\displaystyle\|{\mathcal{B}}^{-1}{\mathcal{H}}{\mathcal{B}}-{\mathcal{H}}\|_{-m,s,\alpha}^{k_{0},\upsilon}\lesssim_{m,s,\alpha,k_{0}}\|\beta\|_{s+m+\alpha+\sigma}^{k_{0},\upsilon}\,,$
(7.37) $\displaystyle\|{\mathcal{B}}^{-1}{\rm
Op}(r_{\mathtt{h}}){\mathcal{B}}-{\rm
Op}(r_{\mathtt{h}})\|_{-m,s,\alpha}^{k_{0},\upsilon}\lesssim_{m,s,\alpha,k_{0}}\|\beta\|_{s+m+\alpha+\sigma}^{k_{0},\upsilon}\,.$

The first estimate is given in Lemmata 2.36 and 2.32 in [9], whereas the
second one follows by that fact that $r_{\mathtt{h}}\in S^{-\infty}$ (see
(7.35)), Lemma 2.18 in [2] and Lemmata 2.34 and 2.32 in [9]. Therefore by
(7.36) we obtain

${\mathcal{B}}^{-1}(1+\beta_{x})^{-1}G(0){\mathcal{B}}=\\{{\mathcal{B}}^{-1}(1+\beta_{x})^{-1}\\}{\mathcal{B}}^{-1}G(0){\mathcal{B}}=G(0)+{\mathcal{R}}_{B}\,,$
(7.38)

where

${\mathcal{R}}_{B}:=\\{{\mathcal{B}}^{-1}(1+\beta_{x})^{-1}\\}\,{\mathcal{R}}_{1}\,.$
(7.39)

Next we transform $G(0)\partial_{x}^{-1}$. By (7.34) and using the identities
${\mathcal{H}}\partial_{x}\partial_{x}^{-1}={\mathcal{H}}$ and
${\mathcal{H}}T({\mathtt{h}})=\partial_{y}^{-1}G(0)$ on the periodic
functions, we have that

$\displaystyle{\mathcal{B}}^{-1}(1+\beta_{x})^{-1}G(0)\partial_{x}^{-1}(1+\beta_{x}){\mathcal{B}}=G(0)\partial_{y}^{-1}+{\mathcal{R}}_{A}$
(7.40)
$\displaystyle{\mathcal{B}}^{-1}\partial_{x}^{-1}G(0){\mathcal{B}}=\partial_{y}^{-1}G(0)+{\mathcal{R}}_{D}\,,$

where

$\displaystyle{\mathcal{R}}_{D}$
$\displaystyle=({\mathcal{B}}^{-1}{\mathcal{H}}{\mathcal{B}}-{\mathcal{H}})({\mathcal{B}}^{-1}T({\mathtt{h}}){\mathcal{B}})+{\mathcal{H}}\big{(}{\mathcal{B}}^{-1}{\rm
Op}(r_{\mathtt{h}}){\mathcal{B}}-{\rm Op}(r_{\mathtt{h}})\big{)}\,,$ (7.41)
$\displaystyle{\mathcal{R}}_{A}$
$\displaystyle=\\{{\mathcal{B}}^{-1}(1+\beta_{x})^{-1}\\}\big{[}{\mathcal{H}}T({\mathtt{h}}),\\{{\mathcal{B}}^{-1}(1+\beta_{x})\\}-1\big{]}$
$\displaystyle\ \
+\\{{\mathcal{B}}^{-1}(1+\beta_{x})^{-1}\\}{\mathcal{R}}_{D}\\{{\mathcal{B}}^{-1}(1+\beta_{x})\\}\,.$

The operator ${\mathcal{R}}_{D}$ is in ${\rm OP}S^{-\infty}$ by (7.37),
(7.35). Also ${\mathcal{R}}_{A}$ is in ${\rm OP}S^{-\infty}$ using that, by
Lemma 2.35 of [9] and (7.35), there is ${\sigma}>0$ such that, for any
$m\in{\mathbb{N}}$, $s\geq s_{0}$, and $\alpha\in{\mathbb{N}}_{0}$,

$\|[{\mathcal{H}}T({\mathtt{h}}),{\widetilde{a}}]\|_{-m,s,\alpha}^{k_{0},\upsilon}\lesssim_{m,s,\alpha,k_{0}}\|{\widetilde{a}}\|_{s+m+\alpha+\sigma}^{k_{0},\upsilon}\,.$
(7.42)

Finally we conjugate $\partial_{x}^{-1}G(0)\partial_{x}^{-1}$. By the Egorov
Proposition 3.9 applied to $\partial_{x}^{-1}$, we have that, for any
$N\in{\mathbb{N}}$,

${\mathcal{B}}^{-1}\partial_{x}^{-1}(1+\beta_{x}){\mathcal{B}}={\mathcal{B}}^{-1}\partial_{x}^{-1}{\mathcal{B}}\,\\{{\mathcal{B}}^{-1}(1+\beta_{x})\\}=\partial_{y}^{-1}+P^{(1)}_{-2,N}(\varphi,x,D)+{\mathtt{R}}_{N}\,,$
(7.43)

where $P^{(1)}_{-2,N}(\varphi,x,D)\in{\rm OP}S^{-2}$ is given by

$P^{(1)}_{-2,N}(\varphi,x,D):=\big{[}\\{{\mathcal{B}}^{-1}(1+\beta_{x})^{-1}\\},\partial_{y}^{-1}\big{]}\\{{\mathcal{B}}^{-1}(1+\beta_{x})\\}+\sum_{j=1}^{N}p_{-1-j}\partial_{y}^{-1-j}\\{{\mathcal{B}}^{-1}(1+\beta_{x})\\}$

with functions $p_{-1-j}(\lambda;\varphi,y)$, $j=0,\ldots,N$, satisfying
(3.30) and ${\mathtt{R}}_{N}$ is a regularizing operator satisfying the
estimate (3.31). So, using (7.40) and (7.43), we obtain

$\displaystyle{\mathcal{B}}^{-1}\partial_{x}^{-1}G(0)\partial_{x}^{-1}(1+\beta_{x}){\mathcal{B}}$
$\displaystyle=({\mathcal{B}}^{-1}\partial_{x}^{-1}G(0){\mathcal{B}})({\mathcal{B}}^{-1}\partial_{x}^{-1}(1+\beta_{x}){\mathcal{B}})$
(7.44)
$\displaystyle=\partial_{y}^{-1}G(0)\partial_{y}^{-1}+P_{-2,N}^{(2)}+{\mathtt{R}}_{2,N}$

where

$\displaystyle P_{-2,N}^{(2)}$
$\displaystyle:=\partial_{y}^{-1}G(0)P^{(1)}_{-2,N}(\varphi,x,D)\in{\rm
OP}S^{-2}$ (7.45)

and ${\mathtt{R}}_{2,N}$ is the regularizing operator

${\mathtt{R}}_{2,N}:={\mathcal{R}}_{D}({\mathcal{B}}^{-1}\partial_{x}^{-1}(1+\beta_{x}){\mathcal{B}})+G(0)\partial_{y}^{-1}{\mathtt{R}}_{N}\,.$
(7.46)

In conclusion, by Lemma 7.7, (7.33), (7.38), (7.40) and (7.44) we obtain the
following lemma, which summarizes the main result of this section.

######  Lemma 7.10.

Let $N\in{\mathbb{N}}$. For any $\overline{\mathtt{n}}\in{\mathbb{N}}_{0}$ and
for all
$(\omega,\gamma)\in{\mathtt{T}}{\mathtt{C}}_{\overline{\mathtt{n}}+1}(2\upsilon,\tau)$,
the operator ${\mathcal{L}}_{1}$ in (7.18) is conjugated to the real,
Hamiltonian, reversible and momentum preserving operator

$\displaystyle{\mathcal{L}}_{2}:={\mathcal{E}}^{-1}{\mathcal{L}}_{1}{\mathcal{E}}=\omega\cdot\partial_{\varphi}+{\mathtt{m}}_{1,\overline{\mathtt{n}}}\partial_{y}\,+$
$\displaystyle\begin{pmatrix}-\frac{\gamma}{2}G(0)\partial_{y}^{-1}&-G(0)\\\
a_{1}-\left(\frac{\gamma}{2}\right)^{2}\partial_{y}^{-1}G(0)\partial_{y}^{-1}&-\frac{\gamma}{2}\partial_{y}^{-1}G(0)\end{pmatrix}$
(7.47) $\displaystyle+\begin{pmatrix}0&0\\\
-\left(\frac{\gamma}{2}\right)^{2}P_{-2,N}^{(2)}&0\end{pmatrix}+{\bf
R}_{2}^{\Psi}+{\bf T}_{2,N}+{\bf P}_{2}^{\perp}\,,$

defined for any
$(\omega,\gamma)\in{\mathbb{R}}^{\nu}\times[\gamma_{1},\gamma_{2}]$, where:
1\. The constant
${\mathtt{m}}_{1,\overline{\mathtt{n}}}={\mathtt{m}}_{1,\overline{\mathtt{n}}}(\omega,\gamma)\in{\mathbb{R}}$
satisfies
$|{\mathtt{m}}_{1,\overline{\mathtt{n}}}|^{k_{0},\upsilon}\lesssim\varepsilon$,
independently on $\overline{\mathtt{n}}$;
2\. The real quasi-periodic traveling wave
$a_{1}:={\mathcal{B}}^{-1}\big{(}a(1+\beta_{x})\big{)}$, ${\rm
even}({\varphi},x)$, satisfies, for some $\sigma:=\sigma(k_{0},\tau,\nu)>0$
and for all $s_{0}\leq s\leq S-\sigma$,

$\|a_{1}-g\|_{s}^{k_{0},\upsilon}\lesssim_{s}\varepsilon\upsilon^{-1}(1+\|{\mathfrak{I}}_{0}\|_{s+\sigma}^{k_{0},\upsilon})\,;$
(7.48)

3\. The operator $P_{-2,N}^{(2)}$ is a pseudodifferential operator in ${\rm
OP}S^{-2}$, reversibility and momentum preserving, and satisfies, for some
$\sigma_{N}:=\sigma_{N}(\tau,\nu,N)>0$, for finitely many
$0\leq\alpha\leq\alpha(M)$ (fixed in Remark 7.16) and for all $s_{0}\leq s\leq
S-\sigma_{N}-\alpha$,

$\|P_{-2,N}^{(2)}\|_{-2,s,\alpha}^{k_{0},\upsilon}\lesssim_{s,N,\alpha}\varepsilon\upsilon^{-1}(1+\|{\mathfrak{I}}_{0}\|_{s+\sigma_{N}+\alpha}^{k_{0},\upsilon})\,;$
(7.49)

4\. For any ${\mathtt{q}}\in{\mathbb{N}}^{\nu}_{0}$ with
$|{\mathtt{q}}|\leq{\mathtt{q}}_{0}$, $n_{1},n_{2}\in{\mathbb{N}}_{0}$ with
$n_{1}+n_{2}\leq N-(k_{0}+{\mathtt{q}}_{0})+2$, the operator $\langle
D\rangle^{n_{1}}\partial_{\varphi}^{\mathtt{q}}({\bf
R}_{2}^{\Psi}(\varphi)+{\bf T}_{2,N}({\varphi}))\langle D\rangle^{n_{2}}$ is
${\mathcal{D}}^{k_{0}}$-tame with a tame constant satisfying, for some
$\sigma_{N}({\mathtt{q}}_{0}):=\sigma_{N}({\mathtt{q}}_{0},k_{0},\tau,\nu)>0$
and for any $s_{0}\leq s\leq S-\sigma_{N}({\mathtt{q}}_{0})$,

${\mathfrak{M}}_{\langle D\rangle^{n_{1}}\partial_{\varphi}^{\mathtt{q}}({\bf
R}_{2}^{\Psi}(\varphi)+{\bf T}_{2,N}({\varphi}))\langle
D\rangle^{n_{2}}}(s)\lesssim_{S,N,{\mathtt{q}}_{0}}\varepsilon\upsilon^{-1}\big{(}1+\|{\mathfrak{I}}_{0}\|_{s+\sigma_{N}({\mathtt{q}}_{0})}^{k_{0},\upsilon}\big{)}\,;$
(7.50)

5\. The operator ${\bf P}_{2}^{\perp}$ is defined in (7.28) and the function
$p_{\overline{\mathtt{n}}}$ satisfies (7.29);
6\. Furthermore, for any $s_{1}$ as in (7.11), finitely many
$0\leq\alpha\leq\alpha(M)$, ${\mathtt{q}}\in{\mathbb{N}}_{0}^{\nu}$, with
$\left|{\mathtt{q}}\right|\leq{\mathtt{q}}_{0}$, and
$n_{1},n_{2}\in{\mathbb{N}}_{0}$, with $n_{1}+n_{2}\leq N-{\mathtt{q}}_{0}+1$,

$\displaystyle|\Delta_{12}{\mathtt{m}}_{1,\overline{\mathtt{n}}}|\lesssim_{s_{1}}\varepsilon\left\|i_{1}-i_{2}\right\|_{s_{1}+\sigma}\,,\
\|\Delta_{12}a_{1}\|_{s_{1}}\lesssim\varepsilon\upsilon^{-1}\left\|i_{1}-i_{2}\right\|_{s_{1}+\sigma}\,,$
(7.51)
$\displaystyle\|\Delta_{12}P_{-2,N}^{(2)}\|_{-2,s_{1},\alpha}\lesssim_{s_{1},N,\alpha}\varepsilon\upsilon^{-1}\left\|i_{1}-i_{2}\right\|_{s_{1}+\sigma_{N}+\alpha}\,,$
(7.52)
$\displaystyle\left\|\braket{D}^{n_{1}}\partial_{\varphi}^{\mathtt{q}}\Delta_{12}({\bf
R}_{2}^{\Psi}(\varphi)+{\bf
T}_{2,N}({\varphi}))\braket{D}^{n_{2}}\right\|_{{\mathcal{L}}(H^{s_{1}})}\lesssim_{s_{1},N,{\mathtt{q}}_{0}}\varepsilon\upsilon^{-1}\left\|i_{1}-i_{2}\right\|_{s_{1}+\sigma_{N}({\mathtt{q}}_{0})}\,.$
(7.53)

###### Proof.

Item 1 follows by Lemma 7.7. The function $a_{1}$ satisfies (7.48) by (7.14),
(3.7), (7.15), (7.30), (7.25). The estimate (7.49) follows by (7.45),
Proposition 3.9 and Lemmata 3.5, 3.6, 3.8, 7.7. The operators ${\bf
R}_{2}^{\Psi}$ and ${\bf T}_{2,N}$ in (7.47) are

${\bf
R}_{2}^{\Psi}:=-\begin{pmatrix}\frac{\gamma}{2}{\mathcal{R}}_{A}&{\mathcal{R}}_{B}\\\
0&\frac{\gamma}{2}{\mathcal{R}}_{D}\end{pmatrix}+{\mathcal{E}}^{-1}{\bf
R}_{1}{\mathcal{E}}\,,\qquad{\bf
T}_{2,N}:=-\left(\frac{\gamma}{2}\right)^{2}\begin{pmatrix}0&0\\\
{\mathtt{R}}_{2,N}&0\end{pmatrix}\,,$

where ${\mathcal{R}}_{B}$, ${\mathcal{R}}_{A}$, ${\mathcal{R}}_{D}$, are
defined in (7.39), (7.41), and ${\bf R}_{1}$, ${\mathtt{R}}_{2,N}$ in (7.19),
(7.46). Thus the estimate (7.50) holds by Lemmata 3.12, 3.13, 7.7, (7.37),
(7.42), Proposition 3.9, Lemma 3.10, (7.25) and Lemmata 2.34, 2.32 in [9]. The
estimates (7.51)-(7.53) are proved similarly. ∎

###  7.3 Symmetrization of the order $1/2$

The goal of this section is to symmetrize the order $1/2$ of the quasi-
periodic Hamiltonian operator ${\mathcal{L}}_{2}$ in (7.47). From now on, we
neglect the contribution of the operator ${\bf P}_{2}^{\perp}$, which will be
conjugated in Section 7.7. For simplicity of notation we denote such operator
${\mathcal{L}}_{2}$ as well.
Step 1: We first conjugate the operator ${\mathcal{L}}_{2}$ in (7.47), where
we relabel the space variable $y\rightsquigarrow x$, by the real, symplectic,
reversibility preserving and momentum preserving transformation

${\widetilde{{\mathcal{M}}}}:=\begin{pmatrix}\Lambda&0\\\
0&\Lambda^{-1}\end{pmatrix}\,,\quad{\widetilde{{\mathcal{M}}}}^{-1}:=\begin{pmatrix}\Lambda^{-1}&0\\\
0&\Lambda\end{pmatrix}\,,$ (7.54)

where $\Lambda\in{\rm OP}S^{\frac{1}{4}}$ is the Fourier multiplier

$\Lambda:=\tfrac{1}{\sqrt{g}}\pi_{0}+M(D)\,,\quad\text{with
inverse}\quad\Lambda^{-1}:=\sqrt{g}\pi_{0}+M(D)^{-1}\in{\rm
OP}S^{-\frac{1}{4}}\,,$ (7.55)

with $\pi_{0}$ defined in (3.23) and (cfr. (2.16))

$M(D):=G(0)^{\frac{1}{4}}\big{(}g-(\tfrac{\gamma}{2})^{2}\partial_{x}^{-1}G(0)\partial_{x}^{-1}\big{)}^{-\frac{1}{4}}\in{\rm
OP}S^{\frac{1}{4}}\,.$ (7.56)

We have the identities $\Lambda^{-1}G(0)\Lambda^{-1}=\omega(\gamma,D)$ and

$\Lambda\big{(}g-\big{(}\tfrac{\gamma}{2}\big{)}^{2}\partial_{x}^{-1}G(0)\partial_{x}^{-1}\big{)}\Lambda=\Lambda^{-1}G(0)\Lambda^{-1}+\pi_{0}=\omega(\gamma,D)+\pi_{0}\,,$
(7.57)

where $\omega(\gamma,D)\in{\rm OP}S^{\frac{1}{2}}$ is defined in (2.18).

By (7.47) we compute

$\displaystyle{\mathcal{L}}_{3}:={\widetilde{{\mathcal{M}}}}^{-1}{\mathcal{L}}_{2}{\widetilde{{\mathcal{M}}}}=$
$\displaystyle\
\omega\cdot\partial_{\varphi}+{\mathtt{m}}_{1,\overline{\mathtt{n}}}\partial_{x}+\begin{pmatrix}-\frac{\gamma}{2}G(0)\partial_{x}^{-1}&-\Lambda^{-1}G(0)\Lambda^{-1}\\\
\Lambda\big{(}a_{1}-(\frac{\gamma}{2})^{2}\partial_{x}^{-1}G(0)\partial_{x}^{-1}\big{)}\Lambda&-\frac{\gamma}{2}G(0)\partial_{x}^{-1}\end{pmatrix}$
(7.58) $\displaystyle+\begin{pmatrix}0&0\\\ -(\frac{\gamma}{2})^{2}\Lambda
P_{-2,N}^{(2)}\Lambda&0\end{pmatrix}+{\widetilde{{\mathcal{M}}}}^{-1}{\bf
R}_{2}^{\Psi}{\widetilde{{\mathcal{M}}}}+{\widetilde{{\mathcal{M}}}}^{-1}{\bf
T}_{2,N}{\widetilde{{\mathcal{M}}}}\,.$

By (7.57), (7.55) and (7.56), we get

$\displaystyle\Lambda\big{(}a_{1}-(\tfrac{\gamma}{2})^{2}\partial_{x}^{-1}G(0)\partial_{x}^{-1}\big{)}\Lambda=\Lambda\big{(}g-(\tfrac{\gamma}{2})^{2}\partial_{x}^{-1}G(0)\partial_{x}^{-1}\big{)}\Lambda+\Lambda(a_{1}-g)\Lambda$
(7.59) $\displaystyle\ \ \ \ \ \ \
=\omega(\gamma,D)+(a_{1}-g)\Lambda^{2}+[\Lambda,a_{1}]\Lambda+\pi_{0}$
$\displaystyle\ \ \ \ \ \ \
=\big{(}1+\tfrac{a_{1}-g}{g}\big{)}\omega(\gamma,D)+\tfrac{a_{1}-g}{g}\big{(}g\Lambda^{2}-\omega(\gamma,D)\big{)}+[\Lambda,a_{1}]\Lambda+\pi_{0}$
$\displaystyle\ \ \ \ \ \ \
=a_{2}^{2}\omega(\gamma,D)+\tfrac{a_{1}-g}{g}(\tfrac{\gamma}{2})^{2}M(D)^{2}\partial_{x}^{-1}G(0)\partial_{x}^{-1}+[\Lambda,a_{1}]\Lambda+\pi_{0}+\tfrac{a_{1}-g}{g}\pi_{0}$

where $a_{2}$ is the real quasi-periodic traveling wave function (with $a_{1}$
defined in Lemma 7.10)

$a_{2}:=\sqrt{\tfrac{a_{1}}{g}}=\sqrt{1+\tfrac{a_{1}-g}{g}}\,,\quad{\rm
even}({\varphi},x)\,\,.$ (7.60)

Therefore, by (7.58), (7.57), (7.59) we obtain

$\displaystyle{\mathcal{L}}_{3}$
$\displaystyle=\omega\cdot\partial_{\varphi}+{\mathtt{m}}_{1,\overline{\mathtt{n}}}\partial_{x}+\begin{pmatrix}-\frac{\gamma}{2}G(0)\partial_{x}^{-1}&-\omega(\gamma,D)\\\
a_{2}\omega(\gamma,D)a_{2}&-\frac{\gamma}{2}\partial_{x}^{-1}G(0)\end{pmatrix}+\begin{pmatrix}0&0\\\
\pi_{0}&0\end{pmatrix}$ (7.61) $\displaystyle+\begin{pmatrix}0&0\\\
C_{3}&0\end{pmatrix}+{\bf R}_{3}^{\Psi}+{\bf T}_{3,N}\,,$

where

$C_{3}:=a_{2}[a_{2},\omega(\gamma,D)]+\tfrac{a_{1}-g}{g}(\tfrac{\gamma}{2})^{2}M(D)^{2}\partial_{x}^{-1}G(0)\partial_{x}^{-1}+[\Lambda,a_{1}]\Lambda-(\tfrac{\gamma}{2})^{2}\Lambda
P_{-2,N}^{(2)}\Lambda$ (7.62)

is in ${\rm OP}S^{-\frac{1}{2}}$ and

${\bf R}_{3}^{\Psi}:={\widetilde{{\mathcal{M}}}}^{-1}{\bf
R}_{2}^{\Psi}{\widetilde{{\mathcal{M}}}}+\begin{pmatrix}0&0\\\
(\tfrac{a_{1}}{g}-1)\pi_{0}&0\end{pmatrix}\,,\quad{\bf
T}_{3,N}:={\widetilde{{\mathcal{M}}}}^{-1}{\bf
T}_{2,N}{\widetilde{{\mathcal{M}}}}\,.$ (7.63)

The operator ${\mathcal{L}}_{3}$ in (7.61) is Hamiltonian, reversible and
momentum preserving.
Step 2: We now conjugate the operator ${\mathcal{L}}_{3}$ in (7.61) with the
symplectic matrix of multiplication operators

${\mathcal{Q}}:=\begin{pmatrix}q&0\\\ 0&q^{-1}\end{pmatrix}\
,\qquad{\mathcal{Q}}^{-1}:=\begin{pmatrix}q^{-1}&0\\\ 0&q\end{pmatrix}\,,$

where $q$ is a real function, close to $1$, to be determined, see (7.69). We
have that

$\displaystyle{\mathcal{L}}_{4}:={\mathcal{Q}}^{-1}{\mathcal{L}}_{3}{\mathcal{Q}}=\omega\cdot\partial_{\varphi}+{\mathtt{m}}_{1,\overline{\mathtt{n}}}\partial_{x}+\begin{pmatrix}A&B\\\
C&D\end{pmatrix}+{\mathcal{Q}}^{-1}({\bf R}_{3}^{\Psi}+{\bf
T}_{3,N}){\mathcal{Q}}\,,$ (7.64)

where (actually $D=-A^{*}$, see Definition 3.18)

$\displaystyle
A:=-\tfrac{\gamma}{2}q^{-1}G(0)\partial_{x}^{-1}q+{\mathtt{m}}_{1,\overline{\mathtt{n}}}q^{-1}q_{x}+q^{-1}(\omega\cdot\partial_{\varphi}q)\,,$
(7.65) $\displaystyle B:=-q^{-1}\omega(\gamma,D)q^{-1}\,,$ (7.66)
$\displaystyle C:=qa_{2}\omega(\gamma,D)a_{2}q+q\pi_{0}q+qC_{3}q\,,$ (7.67)
$\displaystyle
D:=-\tfrac{\gamma}{2}q\partial_{x}^{-1}G(0)q^{-1}-{\mathtt{m}}_{1,\overline{\mathtt{n}}}q^{-1}q_{x}-q^{-1}(\omega\cdot\partial_{\varphi}q)\,.$
(7.68)

We choose the function $q$ so that the coefficients of the highest order terms
of the off-diagonal self-adjoint operators $B$ and $C$ satisfy
$q^{-1}=qa_{2}$, namely as the real quasi-periodic traveling wave, ${\rm
even}({\varphi},x)$

$q({\varphi},x):=a_{2}({\varphi},x)^{-\frac{1}{2}}\,.$ (7.69)

Thus ${\mathcal{Q}}$ is reversibility and momentum preserving.

In view of (7.65)-(7.68) and (7.69) the operator ${\mathcal{L}}_{4}$ in (7.64)
becomes

$\displaystyle{\mathcal{L}}_{4}$
$\displaystyle=\omega\cdot\partial_{\varphi}+{\mathtt{m}}_{1,\overline{\mathtt{n}}}\partial_{x}+\begin{pmatrix}-\frac{\gamma}{2}G(0)\partial_{x}^{-1}&-a_{2}^{\frac{1}{2}}\omega(\gamma,D)a_{2}^{\frac{1}{2}}\\\
a_{2}^{\frac{1}{2}}\omega(\gamma,D)a_{2}^{\frac{1}{2}}&-\frac{\gamma}{2}\partial_{x}^{-1}G(0)\end{pmatrix}$
(7.70) $\displaystyle\ \ \ +\begin{pmatrix}0&0\\\
\pi_{0}&0\end{pmatrix}+\begin{pmatrix}a_{3}&0\\\
C_{4}&-a_{3}\end{pmatrix}+{\bf R}_{4}^{\Psi}+{\bf T}_{4,N}\,,$

where $a_{3}$ is the real quasi-periodic traveling wave function, ${\rm
odd}({\varphi},x)$,

$\displaystyle a_{3}$
$\displaystyle:={\mathtt{m}}_{1,\overline{\mathtt{n}}}q^{-1}q_{x}+q^{-1}(\omega\cdot\partial_{\varphi}q)\,,\quad
C_{4}:=qC_{3}q\in{\rm OP}S^{-\frac{1}{2}}\,,$ (7.71)

and ${\bf R}_{4}^{\Psi},{\bf T}_{4,N}$ are the smoothing remainders (recall
that $G(0)\partial_{x}^{-1}={\mathcal{H}}T({\mathtt{h}})$)

$\displaystyle{\bf
R}_{4}^{\Psi}:=\begin{pmatrix}-\frac{\gamma}{2}q^{-1}[{\mathcal{H}}T({\mathtt{h}}),q-1]&0\\\
q\pi_{0}q-\pi_{0}&-\frac{\gamma}{2}[q-1,{\mathcal{H}}T({\mathtt{h}})]q^{-1}\end{pmatrix}+{\mathcal{Q}}^{-1}{\bf
R}_{3}^{\Psi}{\mathcal{Q}}\in{\rm OP}S^{-\infty}\,,$ (7.72) $\displaystyle{\bf
T}_{4,N}:={\mathcal{Q}}^{-1}{\bf T}_{3,N}{\mathcal{Q}}\,.$

The operator ${\mathcal{L}}_{4}$ in (7.70) is Hamiltonian, reversible and
momentum preserving.
Step 3: We finally move in complex coordinates, conjugating the operator
${\mathcal{L}}_{4}$ in (7.70) via the transformation ${\mathcal{C}}$ defined
in (2.19). The main result of this section is the following lemma.

######  Lemma 7.11.

Let $N\in{\mathbb{N}}$, ${\mathtt{q}}_{0}\in{\mathbb{N}}_{0}$. We have that

$\displaystyle{\mathcal{L}}_{5}:=$
$\displaystyle\,({\widetilde{{\mathcal{M}}}}{\mathcal{Q}}{\mathcal{C}})^{-1}{\mathcal{L}}_{2}{\widetilde{{\mathcal{M}}}}{\mathcal{Q}}{\mathcal{C}}$
(7.73) $\displaystyle=$
$\displaystyle\,\omega\cdot\partial_{\varphi}+{\mathtt{m}}_{1,\overline{\mathtt{n}}}\partial_{x}+{\rm
i}\,a_{2}({\varphi},x){\bf{\Omega}}(\gamma,D)+{\rm
i}\,{\bf{\Pi}}_{0}+a_{4}{\mathcal{H}}+{\bf R}_{5}^{(-\frac{1}{2},d)}+{\bf
R}_{5}^{(0,o)}+{\bf T}_{5,N}\,,$

where:
1\. The real quasi-periodic traveling wave $a_{2}({\varphi},x)$ defined in
(7.60), ${\rm even}({\varphi},x)$, satisfies, for some
$\sigma=\sigma(k_{0},\tau,\nu)>$ and for any $s_{0}\leq s\leq S-\sigma$,

$\|a_{2}-1\|_{s}^{k_{0},\upsilon}\lesssim\varepsilon\upsilon^{-1}(1+\|{\mathfrak{I}}_{0}\|_{s+\sigma}^{k_{0},\upsilon})\,;$
(7.74)

2\. ${\bf{\Omega}}(\gamma,D)$ is the matrix of Fourier multipliers (see
(2.20), (2.21))

${\bf{\Omega}}(\gamma,D)=\begin{pmatrix}\Omega(\gamma,D)&0\\\
0&-\overline{\Omega(\gamma,D)}\end{pmatrix},\quad\Omega(\gamma,D)=\omega(\gamma,D)+{\rm
i}\,\frac{\gamma}{2}\partial_{x}^{-1}G(0)\,;$ (7.75)

3\. The operator

${\bf{\Pi}}_{0}:=\frac{1}{2}\begin{pmatrix}\pi_{0}&\pi_{0}\\\
-\pi_{0}&-\pi_{0}\end{pmatrix}\,.$

4\. The real quasi-periodic traveling wave
$a_{4}({\varphi},x):=\tfrac{\gamma}{2}(a_{2}({\varphi},x)-1)$, ${\rm
even}({\varphi},x)$, satisfies, for some $\sigma:=$ $\sigma(k_{0},\tau,\nu)>0$
and for all $s_{0}\leq s\leq S-\sigma$,

$\|a_{4}\|_{s}^{k_{0},\upsilon}\lesssim_{s}\varepsilon\upsilon^{-1}(1+\|{\mathfrak{I}}_{0}\|_{s+\sigma}^{k_{0},\upsilon})\,;$
(7.76)

5\. ${\bf R}_{5}^{(-\frac{1}{2},d)}\in{\rm OP}S^{-\frac{1}{2}}$ and ${\bf
R}_{5}^{(0,o)}\in{\rm OP}S^{0}$ are pseudodifferential operators of the form

$\displaystyle\footnotesize{\bf
R}_{5}^{(-\frac{1}{2},d)}:=\begin{pmatrix}r_{5}^{(d)}({\varphi},x,D)&0\\\
0&\overline{r_{5}^{(d)}({\varphi},x,D)}\end{pmatrix},\quad{\bf
R}_{5}^{(0,o)}:=\begin{pmatrix}0&r_{5}^{(o)}({\varphi},x,D)\\\
\overline{r_{5}^{(o)}({\varphi},x,D)}&0\end{pmatrix}\,,$

reversibility and momentum preserving, satisfying, for some
$\sigma_{N}:=\sigma(\tau,\nu,N)>0$, for finitely many
$0\leq\alpha\leq\alpha(M)$ (fixed in Remark 7.16), and for all $s_{0}\leq
s\leq S-\sigma_{N}-3\alpha$,

$\|{\bf
R}_{5}^{(-\frac{1}{2},d)}\|_{-\frac{1}{2},s,\alpha}^{k_{0},\upsilon}+\|{\bf
R}_{5}^{(0,o)}\|_{0,s,\alpha}^{k_{0},\upsilon}\lesssim_{s,N,\alpha}\varepsilon\upsilon^{-1}(1+\|{\mathfrak{I}}_{0}\|_{s+\sigma_{N}+3\alpha}^{k_{0},\upsilon})\,;$
(7.77)

6\. For any ${\mathtt{q}}\in{\mathbb{N}}^{\nu}_{0}$ with
$|{\mathtt{q}}|\leq{\mathtt{q}}_{0}$, $n_{1},n_{2}\in{\mathbb{N}}_{0}$ with
$n_{1}+n_{2}\leq N-(k_{0}+{\mathtt{q}}_{0})+\frac{3}{2}$, the operator
$\langle D\rangle^{n_{1}}\partial_{\varphi}^{\mathtt{q}}{\bf
T}_{5,N}(\varphi)\langle D\rangle^{n_{2}}$ is ${\mathcal{D}}^{k_{0}}$-tame
with a tame constant satisfying, for some
$\sigma_{N}({\mathtt{q}}_{0}):=\sigma_{N}({\mathtt{q}}_{0},k_{0},\tau,\nu)>0$
and for any $s_{0}\leq s\leq S-\sigma_{N}({\mathtt{q}}_{0})$,

${\mathfrak{M}}_{\langle D\rangle^{n_{1}}\partial_{\varphi}^{\mathtt{q}}{\bf
T}_{5,N}(\varphi)\langle
D\rangle^{n_{2}}}(s)\lesssim_{S,N,{\mathtt{q}}_{0}}\varepsilon\upsilon^{-1}\big{(}1+\|{\mathfrak{I}}_{0}\|_{s+\sigma_{N}({\mathtt{q}}_{0})}^{k_{0},\upsilon}\big{)}\,;$
(7.78)

7\. The operators ${\mathcal{Q}}^{\pm 1}$, ${\mathcal{Q}}^{\pm 1}-{\rm Id}$,
$({\mathcal{Q}}^{\pm 1}-{\rm Id})^{*}$ are ${\mathcal{D}}^{k_{0}}$-tame with
tame constants satisfying, for some $\sigma:=\sigma(\tau,\nu,k_{0})>0$ and for
all $s_{0}\leq s\leq S-\sigma$,

$\displaystyle{\mathfrak{M}}_{{\mathcal{Q}}^{\pm
1}}(s)\lesssim_{S}1+\|{\mathfrak{I}}_{0}\|_{s+\sigma}^{k_{0},\upsilon}\,,\ \
{\mathfrak{M}}_{{\mathcal{Q}}^{\pm 1}-{\rm
Id}}(s)+{\mathfrak{M}}_{\left({\mathcal{Q}}^{\pm 1}-{\rm
Id}\right)^{*}}(s)\lesssim_{S}\varepsilon\upsilon^{-1}(1+\|{\mathfrak{I}}_{0}\|_{s+\sigma}^{k_{0},\upsilon})\,.$
(7.79)

8\. Furthermore, for any $s_{1}$ as in (7.11), finitely many
$0\leq\alpha\leq\alpha(M)$, ${\mathtt{q}}\in{\mathbb{N}}_{0}^{\nu}$, with
$\left|{\mathtt{q}}\right|\leq{\mathtt{q}}_{0}$, and
$n_{1},n_{2}\in{\mathbb{N}}_{0}$, with $n_{1}+n_{2}\leq
N-{\mathtt{q}}_{0}+\frac{1}{2}$,

$\displaystyle\|\Delta_{12}({\mathcal{A}})h\|_{s_{1}}\lesssim_{s_{1}}\varepsilon{\upsilon^{-1}}\left\|i_{1}-i_{2}\right\|_{s_{1}+\sigma}\left\|h\right\|_{s_{1}+\sigma}\,,\quad{\mathcal{A}}\in\\{{\mathcal{Q}}^{\pm
1}=({\mathcal{Q}}^{\pm 1})^{*}\\}\,,$ (7.80)
$\displaystyle\|\Delta_{12}a_{2}\|_{s_{1}}\lesssim_{s_{1}}\varepsilon{\upsilon^{-1}}\left\|i_{1}-i_{2}\right\|_{s_{1}+\sigma}\,,\
\|\Delta_{12}a_{4}\|_{s_{1}}\lesssim\varepsilon{\upsilon^{-1}}\left\|i_{1}-i_{2}\right\|_{s_{1}+\sigma}\,,$
(7.81) $\displaystyle\|\Delta_{12}{\bf
R}_{5}^{(-\frac{1}{2},d)}\|_{-\frac{1}{2},s_{1},\alpha}+\|\Delta_{12}{\bf
R}_{5}^{(0,o)}\|_{0,s_{1},\alpha}\lesssim_{s_{1},N,\alpha}\varepsilon{\upsilon^{-1}}\left\|i_{1}-i_{2}\right\|_{s_{1}+\sigma_{N}+2\alpha}\,,$
(7.82)
$\displaystyle\left\|\braket{D}^{n_{1}}\partial_{\varphi}^{\mathtt{q}}\Delta_{12}{\bf
T}_{5,N}({\varphi})\braket{D}^{n_{2}}\right\|_{{\mathcal{L}}(H^{s_{1}})}\lesssim_{s_{1},N,{\mathtt{q}}_{0}}\varepsilon{\upsilon^{-1}}\left\|i_{1}-i_{2}\right\|_{s_{1}+\sigma_{N}({\mathtt{q}}_{0})}\,.$
(7.83)

The real operator ${\mathcal{L}}_{5}$ is Hamiltonian, reversible and momentum
preserving.

###### Proof.

By the expression of ${\mathcal{L}}_{4}$ in (7.70) and (3.17) we obtain that
${\cal L}_{5}$ has the form (7.73) with

$\displaystyle r_{5}^{(d)}$
$\displaystyle:=\tfrac{\gamma}{2}(a_{2}-1){\mathcal{H}}(T({\mathtt{h}})-1)+{\rm
i}\big{(}\tfrac{1}{2}C_{4}+a_{2}^{\frac{1}{2}}[\omega(\gamma,D),a_{2}^{\frac{1}{2}}]\big{)}\in{\rm
OP}S^{-\frac{1}{2}}\,,$ (7.84) $\displaystyle r_{5}^{(o)}$
$\displaystyle:=a_{3}+\tfrac{{\rm i}}{2}C_{4}\in{\rm OP}S^{0}$

(with $C_{4}$ given in (7.71)) and ${\bf T}_{5,N}:={\mathcal{C}}^{-1}({\bf
R}_{4}^{\Psi}+{\bf T}_{4,N}){\mathcal{C}}$. The function $q$ defined in
(7.69), with $a_{2}$ in (7.60), satisfies, by (7.48) and Lemma 3.2, for all
$s_{0}\leq s\leq S-\sigma$,

$\|q^{\pm
1}-1\|_{s}^{k_{0},\upsilon}\lesssim_{s}\varepsilon{\upsilon^{-1}}(1+\|{\mathfrak{I}}_{0}\|_{s+\sigma}^{k_{0},\upsilon})\,.$
(7.85)

The estimates (7.74) and (7.76) follows by (7.60) and (7.85). The estimate
(7.77) follows by (7.84), (7.74), (7.69), (7.62), (7.60), (7.48), (7.49),
(7.71), (7.55), (2.16), Lemma 7.10. The estimate (7.78) follows by (7.72),
(7.63), (7.42), (7.50), (7.48) Lemmata 3.12, 3.13, (7.85). The estimates
(7.79) follow by Lemmata 3.13 and (7.85). The estimates (7.80)- (7.83) are
proved similarly. ∎

###  7.4 Symmetrization up to smoothing remainders

The goal of this section is to transform the operator ${\mathcal{L}}_{5}$ in
(7.73) into the operator ${\mathcal{L}}_{6}$ in (7.88) which is block diagonal
up to a regularizing remainder. From this step we do not preserve any further
the Hamiltonian structure, but only the reversible and momentum preserving one
(it is sufficient for proving Theorem 5.1).

######  Lemma 7.12.

Fix ${\mathfrak{m}},N\in{\mathbb{N}}$, ${\mathtt{q}}_{0}\in{\mathbb{N}}_{0}$.
There exist real, reversibility and momentum preserving operator matrices
$\\{{\bf X}_{k}\\}_{k=1}^{{\mathfrak{m}}}$ of the form

${\bf X}_{k}:=\begin{pmatrix}0&\chi_{k}({\varphi},x,D)\\\
\overline{\chi_{k}({\varphi},x,D)}&0\end{pmatrix},\qquad\chi_{k}({\varphi},x,\xi)\in
S^{-\frac{k}{2}}\,,$ (7.86)

such that, conjugating the operator ${\mathcal{L}}_{5}$ in (7.73) via the map

${\bf{\Phi}}_{{\mathfrak{m}}}:=e^{{\bf X}_{1}}\circ\cdots\circ e^{{\bf
X}_{{\mathfrak{m}}}}\,,$ (7.87)

we obtain the real, reversible and momentum preserving operator

$\displaystyle{\mathcal{L}}_{6}$
$\displaystyle:={\mathcal{L}}_{6}^{({\mathfrak{m}})}:={\bf{\Phi}}_{{\mathfrak{m}}}^{-1}\,{\mathcal{L}}_{5}\,{\bf{\Phi}}_{{\mathfrak{m}}}$
(7.88)
$\displaystyle=\omega\cdot\partial_{\varphi}+{\mathtt{m}}_{1,\overline{\mathtt{n}}}\partial_{x}+{\rm
i}\,a_{2}{\bf{\Omega}}(\gamma,D)+{\rm i}{\bf{\Pi}}_{0}+a_{4}{\mathcal{H}}+{\bf
R}_{6}^{(-\frac{1}{2},d)}+{\bf R}_{6}^{(-\frac{{\mathfrak{m}}}{2},o)}+{\bf
T}_{6,N}\,,$

where:
1\. ${\bf R}_{6}^{(-\frac{1}{2},d)}$ is a block-diagonal operator

$\displaystyle{\bf R}_{6}^{(-\frac{1}{2},d)}:={\bf
R}_{6,{\mathfrak{m}}}^{(-\frac{1}{2},d)}$
$\displaystyle:=\begin{pmatrix}r_{6}^{(d)}({\varphi},x,D)&0\\\
0&\overline{r_{6}^{(d)}({\varphi},x,D)}\end{pmatrix}\in{\rm
OP}S^{-\frac{1}{2}}\,,$

${\bf R}_{6}^{(-\frac{{\mathtt{m}}}{2},o)}$ is a smoothing off diagonal
remainder

$\displaystyle{\bf R}_{6}^{(-\frac{{\mathtt{m}}}{2},o)}:={\bf
R}_{6,{\mathfrak{m}}}^{(-\frac{{\mathfrak{m}}}{2},o)}$
$\displaystyle:=\begin{pmatrix}0&r_{6}^{(o)}({\varphi},x,D)\\\
\overline{r_{6}^{(o)}({\varphi},x,D)}&0\end{pmatrix}\in{\rm
OP}S^{-\frac{{\mathfrak{m}}}{2}}\,,$ (7.89)

satisfying for finitely many $0\leq\alpha\leq\alpha({\mathfrak{m}})$ (fixed in
Remark 7.16), for some $\sigma_{N}:=\sigma_{N}(k_{0},\tau,\nu,N)>0$,
$\aleph_{{\mathfrak{m}}}(\alpha)>0$ and for all $s_{0}\leq s\leq
S-\sigma_{N}-\aleph_{{\mathfrak{m}}}(\alpha)$,

$\displaystyle\|{\bf
R}_{6}^{(-\frac{1}{2},d)}\|_{-\frac{1}{2},s,\alpha}^{k_{0},\upsilon}+\|{\bf
R}_{6}^{(-\frac{{\mathfrak{m}}}{2},o)}\|_{-\frac{{\mathfrak{m}}}{2},s,\alpha}^{k_{0},\upsilon}\lesssim_{s,{\mathfrak{m}},N,\alpha}\varepsilon{\upsilon^{-1}}\big{(}1+\|{\mathfrak{I}}_{0}\|_{s+\sigma_{N}+\aleph_{{\mathfrak{m}}}(\alpha)}^{k_{0},\upsilon}\big{)}\,.$
(7.90)

Both ${\bf R}_{6}^{(-\frac{1}{2},d)}$ and ${\bf
R}_{6}^{(-\frac{{\mathfrak{m}}}{2},o)}$ are reversible and momentum
preserving;
2\. For any ${\mathtt{q}}\in{\mathbb{N}}^{\nu}_{0}$ with
$|{\mathtt{q}}|\leq{\mathtt{q}}_{0}$, $n_{1},n_{2}\in{\mathbb{N}}_{0}$ with
$n_{1}+n_{2}\leq N-(k_{0}+{\mathtt{q}}_{0})+\frac{3}{2}$, the operator
$\langle D\rangle^{n_{1}}\partial_{\varphi}^{\mathtt{q}}{\bf
T}_{6,N}(\varphi)\langle D\rangle^{n_{2}}$ is ${\mathcal{D}}^{k_{0}}$-tame
with a tame constant satisfying, for some
$\sigma_{N}({\mathtt{q}}_{0}):=\sigma_{N}(k_{0},\tau,\nu,{\mathtt{q}}_{0})$,
for any $s_{0}\leq s\leq
S-\sigma_{N}({\mathtt{q}}_{0})-\aleph_{{\mathfrak{m}}}(0)$,

${\mathfrak{M}}_{\langle D\rangle^{n_{1}}\partial_{\varphi}^{\mathtt{q}}{\bf
T}_{6,N}(\varphi)\langle
D\rangle^{n_{2}}}(s)\lesssim_{S,{\mathfrak{m}},N,{\mathtt{q}}_{0}}\varepsilon{\upsilon^{-1}}(1+\|{\mathfrak{I}}_{0}\|_{s+\sigma_{N}({\mathtt{q}}_{0})+\aleph_{{\mathfrak{m}}}(0)}^{k_{0},\upsilon})\,.$
(7.91)

3\. The conjugation map ${\bf{\Phi}}_{{\mathfrak{m}}}$ in (7.87) satisfies,
for all $s_{0}\leq s\leq S-\sigma_{N}-\aleph_{{\mathfrak{m}}}(0)$,

$\|{\bf{\Phi}}_{{\mathfrak{m}}}^{\pm 1}-{\rm
Id}\|_{0,s,0}^{k_{0},\upsilon}+\|\left({\bf{\Phi}}_{{\mathfrak{m}}}^{\pm
1}-{\rm
Id}\right)^{*}\|_{0,s,0}^{k_{0},\upsilon}\lesssim_{s,{\mathfrak{m}},N}\varepsilon{\upsilon^{-1}}(1+\|{\mathfrak{I}}_{0}\|_{s+\sigma_{N}+\aleph_{{\mathfrak{m}}}(0)}^{k_{0},\upsilon})\,.$
(7.92)

4\. Furthermore, for any $s_{1}$ as in (7.11), finitely many
$0\leq\alpha\leq\alpha({\mathfrak{m}})$,
${\mathtt{q}}\in{\mathbb{N}}_{0}^{\nu}$, with
$\left|{\mathtt{q}}\right|\leq{\mathtt{q}}_{0}$, and
$n_{1},n_{2}\in{\mathbb{N}}_{0}$, with $n_{1}+n_{2}\leq
N-{\mathtt{q}}_{0}+\frac{1}{2}$, we have

$\displaystyle\|\Delta_{12}{\bf
R}_{6}^{(-\frac{1}{2},d)}\|_{-\frac{1}{2},s_{1},\alpha}+\|\Delta_{12}{\bf
R}_{6}^{(-\frac{{\mathfrak{m}}}{2},o)}\|_{-\frac{{\mathfrak{m}}}{2},s_{1},\alpha}\lesssim_{s_{1},{\mathfrak{m}},N,\alpha}\varepsilon{\upsilon^{-1}}\left\|i_{1}-i_{2}\right\|_{s_{1}+\sigma_{N}+\aleph_{{\mathfrak{m}}}(\alpha)}\,,$
(7.93)
$\displaystyle\|\braket{D}^{n_{1}}\partial_{\varphi}^{\mathtt{q}}\Delta_{12}{\bf
T}_{6,N}\braket{D}^{n_{2}}\|_{{\mathcal{L}}(H^{s_{1}})}\lesssim_{s_{1},{\mathfrak{m}},N,{\mathtt{q}}_{0}}\varepsilon{\upsilon^{-1}}\left\|i_{1}-i_{2}\right\|_{s_{1}+\sigma_{N}({\mathtt{q}}_{0})+\aleph_{{\mathfrak{m}}}(0)}\,,$
(7.94) $\displaystyle\|\Delta_{12}{\bf{\Phi}}_{{\mathfrak{m}}}^{\pm
1}\|_{0,s_{1},0}+\|\Delta_{12}({\bf{\Phi}}_{{\mathfrak{m}}}^{\pm
1})^{*}\|_{0,s_{1},0}\lesssim_{s_{1},{\mathfrak{m}},N}\varepsilon{\upsilon^{-1}}\left\|i_{1}-i_{2}\right\|_{s_{1}+\sigma_{N}+\aleph_{{\mathfrak{m}}}(0)}\,.$
(7.95)

###### Proof.

The proof is inductive. The operator
${\mathcal{L}}_{6}^{(0)}:={\mathcal{L}}_{5}$ satisfies (7.90)-(7.91) with
$\aleph_{0}(\alpha):=3\alpha$, by (7.77)-(7.78). Suppose we have done already
${\mathfrak{m}}$ steps obtaining an operator
${\mathcal{L}}_{6}^{({\mathfrak{m}})}$ as in (7.88) with ${\bf
R}_{6,{\mathfrak{m}}}^{(-\frac{1}{2},d)}:={\bf R}_{6}^{(-\frac{1}{2},d)}$ and
${\bf R}_{6,{\mathfrak{m}}}^{(-\frac{1}{2},o)}:={\bf
R}_{6}^{(-\frac{{\mathfrak{m}}}{2},o)}$ and the remainder
${\bf\Phi}_{{\mathfrak{m}}}^{-1}{\bf T}_{5,N}{\bf\Phi}_{{\mathfrak{m}}}$,
instead of ${\bf T}_{6,N}$. We now show how to define
${\mathcal{L}}_{6}^{({\mathfrak{m}}+1)}$. Let

$\chi_{{\mathfrak{m}}+1}({\varphi},x,\xi):=-\big{(}2{\rm
i}\,a_{2}({\varphi},x)\omega(\gamma,\xi)\big{)}^{-1}r_{6,{\mathfrak{m}}}^{(o)}({\varphi},x,\xi)\chi(\xi)\in
S^{-\frac{{\mathfrak{m}}}{2}-\frac{1}{2}}\,,$ (7.96)

where $\chi$ is the cut-off function defined in (3.11) and
$\omega(\gamma,\xi)$ is the symbol (cfr. (2.18))

$\omega(\gamma,\xi):=\sqrt{G(0;\xi)\Big{(}g+\frac{\gamma^{2}}{4}\frac{G(0;\xi)}{\xi^{2}}\Big{)}}\in
S^{\frac{1}{2}}\,,\ \
G(0;\xi):=\begin{cases}\chi(\xi)|\xi|\tanh({\mathtt{h}}|\xi|)\,,\
{\mathtt{h}}<+\infty\cr\chi(\xi)|\xi|\,,\qquad\qquad\ \,\
{\mathtt{h}}=+\infty\,.\end{cases}$

Note that $\chi_{{\mathfrak{m}}+1}$ in (7.96) is well defined because
$\omega(\gamma,\xi)$ is positive on the support of $\chi(\xi)$ and
$a_{2}({\varphi},x)$ is close to 1. We conjugate the operator
${\mathcal{L}}_{6}^{({\mathfrak{m}})}$ in (7.88) by the flow generated by
${\bf X}_{{\mathfrak{m}}+1}$ of the form (7.86) with
$\chi_{{\mathfrak{m}}+1}(\varphi,x,\xi)$ defined in (7.96). By (7.90) and
(7.75), for suitable constants
$\aleph_{{\mathfrak{m}}+1}(\alpha)>\aleph_{{\mathfrak{m}}}(\alpha)$, for
finitely many $\alpha\in{\mathbb{N}}_{0}$ and for any $s_{0}\leq s\leq
S-\sigma_{N}-\aleph_{{\mathfrak{m}}+1}(\alpha)$,

$\|{\bf
X}_{{\mathfrak{m}}+1}\|_{-\frac{{\mathfrak{m}}}{2}-\frac{1}{2},s,\alpha}^{k_{0},\upsilon}\lesssim_{s,{\mathfrak{m}},\alpha}\varepsilon{\upsilon^{-1}}\big{(}1+\|{\mathfrak{I}}_{0}\|_{s+\sigma_{N}+\aleph_{{\mathfrak{m}}+1}(\alpha)}^{k_{0},\upsilon}\big{)}\,.$
(7.97)

Therefore, by Lemmata 3.7, 3.5 and the induction assumption (7.92) for
${\bf{\Phi}}_{{\mathfrak{m}}}$, the conjugation map
${\bf{\Phi}}_{{\mathfrak{m}}+1}:={\bf{\Phi}}_{{\mathfrak{m}}}e^{{\bf
X}_{{\mathfrak{m}}+1}}$ is well defined and satisfies estimate (7.92) with
${\mathfrak{m}}+1$. By the Lie expansion (3.18) we have

$\displaystyle{\mathcal{L}}_{6}^{({\mathfrak{m}}+1)}$ $\displaystyle:=e^{-{\bf
X}_{{\mathfrak{m}}+1}}\,{\mathcal{L}}_{6}^{({\mathfrak{m}})}\,e^{{\bf
X}_{{\mathfrak{m}}+1}}$ (7.98)
$\displaystyle=\omega\cdot\partial_{\varphi}+{\mathtt{m}}_{1,\overline{\mathtt{n}}}\partial_{x}+{\rm
i}a_{2}{\bf{\Omega}}(\gamma,D)+{\rm i}{\bf{\Pi}}_{0}+a_{4}{\mathcal{H}}+{\bf
R}_{6,{\mathfrak{m}}}^{(-\frac{1}{2},d)}$ $\displaystyle-\big{[}{\bf
X}_{{\mathfrak{m}}+1},{\mathtt{m}}_{1,\overline{\mathtt{n}}}\partial_{x}+{\rm
i}\,a_{2}{\bf{\Omega}}(\gamma,D)\big{]}+{\bf
R}_{6,{\mathfrak{m}}}^{(-\frac{{\mathfrak{m}}}{2},o)}+{\bf\Phi}_{{\mathfrak{m}}+1}^{-1}{\bf
T}_{5,N}{\bf\Phi}_{{\mathfrak{m}}+1}$ $\displaystyle-\int_{0}^{1}e^{-\tau{\bf
X}_{{\mathfrak{m}}+1}}\big{[}{\bf
X}_{{\mathfrak{m}}+1}\,,\,\omega\cdot\partial_{\varphi}+{\rm
i}{\bf{\Pi}}_{0}+a_{4}{\mathcal{H}}+{\bf
R}_{6,{\mathfrak{m}}}^{(-\frac{1}{2},d)}\big{]}e^{\tau{\bf
X}_{{\mathfrak{m}}+1}}\,{\rm d}{\tau}$ (7.99)
$\displaystyle-\int_{0}^{1}e^{-\tau{\bf X}_{{\mathfrak{m}}+1}}\left[{\bf
X}_{{\mathfrak{m}}+1},{\bf
R}_{6,{\mathfrak{m}}}^{(-\frac{{\mathfrak{m}}}{2},o)}\right]e^{\tau{\bf
X}_{{\mathfrak{m}}+1}}\,{\rm d}{\tau}$ (7.100)
$\displaystyle+\int_{0}^{1}(1-\tau)e^{-\tau{\bf
X}_{{\mathfrak{m}}+1}}\left[{\bf X}_{{\mathfrak{m}}+1},\left[{\bf
X}_{{\mathfrak{m}}+1},{\mathtt{m}}_{1,\overline{\mathtt{n}}}\partial_{x}+{\rm
i}\,a_{2}{\bf{\Omega}}(\gamma,D)\right]\right]e^{\tau{\bf
X}_{{\mathfrak{m}}+1}}\,{\rm d}{\tau}\,.$ (7.101)

In view of (7.86), (7.75) and (7.89), we have that

$-\big{[}{\bf
X}_{{\mathfrak{m}}+1},{\mathtt{m}}_{1,\overline{\mathtt{n}}}\partial_{x}+{\rm
i}\,a_{2}{\bf{\Omega}}(\gamma,D)\big{]}+{\bf
R}_{6,{\mathfrak{m}}}^{(-\frac{{\mathfrak{m}}}{2},o)}=\begin{pmatrix}0&Z_{{\mathfrak{m}}+1}\\\
\overline{Z_{{\mathfrak{m}}+1}}&0\end{pmatrix}=:{\bf Z}_{{\mathfrak{m}}+1}\,,$

where, denoting for brevity
$\chi_{{\mathfrak{m}}+1}:=\chi_{{\mathfrak{m}}+1}({\varphi},x,\xi)$, it
results

$\displaystyle Z_{{\mathfrak{m}}+1}$ $\displaystyle={\rm i}\left({\rm
Op}(\chi_{{\mathfrak{m}}+1})a_{2}\,\omega(\gamma,D)+a_{2}\,\omega(\gamma,D){\rm
Op}(\chi_{{\mathfrak{m}}+1})\right)$ $\displaystyle\quad+\left[{\rm
Op}(\chi_{{\mathfrak{m}}+1}),-{\mathtt{m}}_{1,\overline{\mathtt{n}}}\partial_{x}+a_{2}\,\tfrac{\gamma}{2}\partial_{x}^{-1}G(0)\right]+{\rm
Op}(r_{6,{\mathfrak{m}}}^{(o)})\,.$

By (3.24), (3.26) and since $\chi_{{\mathfrak{m}}+1}\in
S^{-\frac{{\mathfrak{m}}}{2}-\frac{1}{2}}$ by (7.96), we have that

${\rm
Op}(\chi_{{\mathfrak{m}}+1})a_{2}\omega(\gamma,D)+a_{2}\omega(\gamma,D){\rm
Op}(\chi_{{\mathfrak{m}}+1})={\rm
Op}\big{(}2a_{2}\omega(\gamma,\xi)\chi_{{\mathfrak{m}}+1}\big{)}+{\mathtt{r}}_{{\mathfrak{m}}+1}\,,$

where ${\mathtt{r}}_{{\mathfrak{m}}+1}$ is in ${\rm
OP}S^{-\frac{{\mathfrak{m}}}{2}-1}$. By (7.96) and (7.4)

$Z_{{\mathfrak{m}}+1}={\rm i}{\mathtt{r}}_{{\mathfrak{m}}+1}+\left[{\rm
Op}(\chi_{{\mathfrak{m}}+1}),-{\mathtt{m}}_{1,\overline{\mathtt{n}}}\partial_{x}+a_{2}\tfrac{\gamma}{2}\partial_{x}^{-1}G(0)\right]+{\rm
Op}(r_{6,{\mathfrak{m}}}^{(o)}(1-\chi(\xi)))\in{\rm
OP}S^{-\frac{{\mathfrak{m}}}{2}-\frac{1}{2}}\,.$

The remaining pseudodifferential operators in (7.99)-(7.101) have order ${\rm
OP}S^{-\frac{{\mathfrak{m}}+1}{2}}$. Therefore the operator
${\mathcal{L}}_{6}^{({\mathfrak{m}}+1)}$ in (7.98) has the form (7.88) at
${\mathfrak{m}}+1$ with

${\bf R}_{6,{\mathfrak{m}}+1}^{(-\frac{1}{2},d)}+{\bf
R}_{6,{\mathfrak{m}}+1}^{(-\frac{{\mathfrak{m}}+1}{2},o)}:={\bf
R}_{6,{\mathfrak{m}}}^{(-\frac{1}{2},d)}+{\bf
Z}_{{\mathfrak{m}}+1}+\eqref{geno12}+\eqref{geno13}+\eqref{geno14}$ (7.102)

and a smoothing remainder ${\bf\Phi}_{{\mathfrak{m}}+1}^{-1}{\bf
T}_{5,N}{\bf\Phi}_{{\mathfrak{m}}+1}$. By Lemmata 3.5, 3.6, (7.90), (7.97),
(7.76), we conclude that ${\bf R}_{6,{\mathfrak{m}}+1}^{(-\frac{1}{2},d)}$ and
${\bf R}_{6,{\mathfrak{m}}+1}^{(-\frac{{\mathfrak{m}}+1}{2},o)}$ satisfy
(7.90) at order ${\mathfrak{m}}+1$ for suitable constants
$\aleph_{{\mathfrak{m}}+1}(\alpha)>\aleph_{{\mathfrak{m}}}(\alpha)$. Moreover
the operator ${\bf{\Phi}}_{{\mathfrak{m}}+1}^{-1}{\bf
T}_{5,N}{\bf{\Phi}}_{{\mathfrak{m}}+1}$ satisfies (7.91) at order
${\mathfrak{m}}+1$ by Lemmata 3.12, 3.13 and (7.78), (7.92). Estimates
(7.93)-(7.95) follow similarly. By (7.96), Lemmata 3.20, 3.24, and the
induction assumption that ${\bf
R}_{6,{\mathfrak{m}}}^{(-\frac{{\mathfrak{m}}}{2},o)}$ is reversible and
momentum preserving, we get that ${\bf X}_{{\mathfrak{m}}+1}$ is reversibility
and momentum preserving, and so are $e^{\pm{\bf X}_{{\mathfrak{m}}+1}}$. We
deduce that ${\mathcal{L}}_{6}^{({\mathfrak{m}}+1)}$ is reversible and
momentum preserving, in particular ${\bf
R}_{6,{\mathfrak{m}}+1}^{(-\frac{{\mathfrak{m}}+1}{2},o)}$ in (7.102). ∎

######  Remark 7.13.

The number of regularizing iterations ${\mathfrak{m}}\in{\mathbb{N}}$ will be
fixed by the KAM reduction scheme in Section 8, more precisely we take
${\mathfrak{m}}=2M$ with $M$ in (8.5). Note that it is independent of the
Sobolev index $s$.

So far the operator ${\mathcal{L}}_{6}$ of Lemma 7.12 depends on two indexes
${\mathfrak{m}},N$ which provide respectively the order of the regularizing
off-diagonal remainder ${\bf R}_{6}^{(-\frac{{\mathfrak{m}}}{2},o)}$ and of
the smoothing tame operator ${\bf T}_{6,N}$. From now on we fix

${\mathfrak{m}}:=2M\,,\ M\in{\mathbb{N}}\,,\quad N=M\,.$ (7.103)

###  7.5 Reduction of the order 1/2

The goal of this section is to transform the operator ${\mathcal{L}}_{6}$ in
(7.88) with ${\mathfrak{m}}:=2M$, $N=M$ (cfr. (7.103)), into the operator
${\mathcal{L}}_{7}$ in (7.117) whose coefficient in front of
${\bf{\Omega}}(\gamma,D)$ is constant. First we rewrite

${\mathcal{L}}_{6}=\omega\cdot\partial_{\varphi}+\begin{pmatrix}P_{6}&0\\\
0&\overline{P_{6}}\end{pmatrix}+{\rm i}{\bf{\Pi}}_{0}+{\bf
R}_{6}^{(-M,o)}+{\bf T}_{6,M}\,,$

having denoted

$P_{6}:=P_{6}({\varphi},x,D):={\mathtt{m}}_{1,\overline{\mathtt{n}}}\partial_{x}+{\rm
i}a_{2}({\varphi},x)\Omega(\gamma,D)+a_{4}{\mathcal{H}}+r_{6}^{(d)}({\varphi},x,D)\,.$
(7.104)

We conjugate ${\mathcal{L}}_{6}$ through the real operator

${\bf{\Phi}}({\varphi}):=\begin{pmatrix}\Phi({\varphi})&0\\\
0&\overline{\Phi}({\varphi})\end{pmatrix}$ (7.105)

where $\Phi({\varphi}):=\Phi^{\tau}({\varphi})|_{\tau=1}$ is the time $1$-flow
of the PDE

$\begin{cases}\partial_{\tau}\Phi^{\tau}({\varphi})={\rm
i}A({\varphi})\Phi^{\tau}({\varphi})\,,\\\ \Phi^{0}({\varphi})={\rm
Id}\,,\end{cases}\qquad A({\varphi}):=b({\varphi},x)|D|^{\frac{1}{2}}\,,$
(7.106)

and $b({\varphi},x)$ is a real quasi-periodic traveling wave, ${\rm
odd}({\varphi},x)$, chosen later, see (7.114). Thus ${\rm
i}b({\varphi},x)|D|^{\frac{1}{2}}$ is reversibility and momentum preserving as
well as ${\bf{\Phi}}({\varphi})$. Moreover
$\Phi\pi_{0}=\pi_{0}=\Phi^{-1}\pi_{0}$, which implies

${\bf{\Phi}}^{-1}{\bf{\Pi}}_{0}{\bf{\Phi}}={\bf{\Pi}}_{0}{\bf{\Phi}}\,.$
(7.107)

By the Lie expansion (3.18) we have

$\displaystyle\Phi^{-1}P_{6}\Phi$ $\displaystyle=P_{6}-{\rm
i}[A,P_{6}]-\frac{1}{2}[A,[A,P_{6}]]+\sum_{n=3}^{2M+1}\frac{(-{\rm
i})^{n}}{n!}{\rm ad}_{A({\varphi})}^{n}(P_{6})+T_{M}\,,$ (7.108)
$\displaystyle T_{M}$ $\displaystyle:=\frac{(-{\rm
i})^{2M+2}}{(2M+1)!}\int_{0}^{1}(1-\tau)^{2M+1}\Phi^{-\tau}({\varphi})\,{\rm
ad}_{A({\varphi})}^{2M+2}(P_{6})\,\Phi^{\tau}({\varphi}){\rm d}\tau\,,$

and, by (3.19),

$\displaystyle\Phi^{-1}\circ\omega\cdot\partial_{\varphi}\circ\Phi$
$\displaystyle=\omega\cdot\partial_{\varphi}+{\rm
i}(\omega\cdot\partial_{\varphi}A)+\frac{1}{2}[A,\omega\cdot\partial_{\varphi}A]-\sum_{n=3}^{2M+1}\frac{(-{\rm
i})^{n}}{n!}{\rm
ad}_{A({\varphi})}^{n-1}(\omega\cdot\partial_{\varphi}A({\varphi}))+T_{M}^{\prime}\,,$
$\displaystyle T_{M}^{\prime}$ $\displaystyle:=-\frac{(-{\rm
i})^{2M+2}}{(2M+1)!}\int_{0}^{1}(1-\tau)^{2M+1}\Phi^{-\tau}({\varphi})\,{\rm
ad}_{A({\varphi})}^{2M+1}(\omega\cdot\partial_{\varphi}A({\varphi}))\,\Phi^{\tau}({\varphi}){\rm
d}\tau\,.$ (7.109)

Note that ${\rm ad}_{A({\varphi})}^{2M+2}(P_{6})$ and ${\rm
ad}_{A({\varphi})}^{2M+1}(\omega\cdot\partial_{\varphi}A({\varphi}))$ are in
${\rm OP}S^{-M}$. We now determine the pseudo-differential term of order $1/2$
in (7.108)-(7.109). We use the expansion of the linear dispersion operator
$\Omega(\gamma,D)$, defined by (4.1), (1.10), and, since $j\to
c_{j}(\gamma)\in S^{0}$ (see (4.14)),

$\Omega(\gamma,D)=\sqrt{g}|D|^{\frac{1}{2}}+{\rm
i}\,\tfrac{\gamma}{2}{\mathcal{H}}+r_{-\frac{1}{2}}(\gamma,D)\,,\quad
r_{-\frac{1}{2}}(\gamma,D)\in{\rm OP}S^{-\frac{1}{2}}\,,$ (7.110)

where ${\mathcal{H}}$ is the Hilbert transform in (3.21). By (7.104), that
$A=b|D|^{\frac{1}{2}}$, (3.27), (7.110) we get

$\displaystyle[A,P_{6}]$
$\displaystyle=\big{[}b|D|^{\frac{1}{2}},{\mathtt{m}}_{1,{\mathtt{n}}}\partial_{x}+{\rm
i}\,\sqrt{g}a_{2}|D|^{\frac{1}{2}}+(a_{4}-\tfrac{\gamma}{2}a_{2}){\mathcal{H}}+r_{6}^{(d)}(x,D)+{\rm
i}\,a_{2}r_{-\frac{1}{2}}(\gamma,D)\big{]}$
$\displaystyle=-{\mathtt{m}}_{1,\overline{\mathtt{n}}}b_{x}|D|^{\frac{1}{2}}-{\rm
i}\tfrac{\sqrt{g}}{2}(b_{x}a_{2}-(a_{2})_{x}b){\mathcal{H}}+{\rm
Op}(r_{b,-\frac{1}{2}})\,,$ (7.111)

where $r_{b,-\frac{1}{2}}\in S^{-\frac{1}{2}}$ is small with $b$. As a
consequence, the contribution at order $\frac{1}{2}$ of the operator ${\rm
i}\,\omega\cdot\partial_{\varphi}A+P_{6}-{\rm i}[A,P_{6}]$ is ${\rm
i}\big{(}\omega\cdot\partial_{\varphi}b+{\mathtt{m}}_{1,\overline{\mathtt{n}}}b_{x}+\sqrt{g}\,a_{2})|D|^{\frac{1}{2}}$.
We choose $b({\varphi},x)$ as the solution of

$(\omega\cdot\partial_{\varphi}+{\mathtt{m}}_{1,\overline{\mathtt{n}}}\partial_{x})b+\sqrt{g}\,\Pi_{N_{\overline{\mathtt{n}}}}\,a_{2}=\sqrt{g}\,{\mathtt{m}}_{\frac{1}{2}}$
(7.112)

where ${\mathtt{m}}_{\frac{1}{2}}$ is the average (see (3.6))

${\mathtt{m}}_{\frac{1}{2}}:=\braket{a_{2}}_{{\varphi},x}\,.$ (7.113)

We define $b({\varphi},x)$ to be the real, ${\rm odd}({\varphi},x)$, quasi-
periodic traveling wave

$b({\varphi},x):=-\sqrt{g}(\omega\cdot\partial_{\varphi}+{\mathtt{m}}_{1,\overline{\mathtt{n}}}\partial_{x})_{\rm
ext}^{-1}\big{(}\Pi_{N_{\overline{\mathtt{n}}}}a_{2}({\varphi},x)-{\mathtt{m}}_{\frac{1}{2}}\big{)}$
(7.114)

recall (3.10). Note that $b({\varphi},x)$ and ${\mathtt{m}}_{\frac{1}{2}}$ are
defined for any
$(\omega,\gamma)\in{\mathbb{R}}^{\nu}\times[\gamma_{1},\gamma_{2}]$ and that,
for any
$(\omega,\gamma)\in{\mathtt{T}}{\mathtt{C}}_{\overline{\mathtt{n}}+1}(2\upsilon,\tau)$
defined in (7.26), it solves (7.112).

We deduce by (7.108), (7.109), (7.104), (7.5)-(7.114), that, for any
$(\omega,\gamma)\in{\mathtt{T}}{\mathtt{C}}_{\overline{\mathtt{n}}+1}(2\upsilon,\tau)$,

$\displaystyle L_{7}$
$\displaystyle:=\Phi^{-1}({\varphi})\left(\omega\cdot\partial_{\varphi}+P_{6}\right)\Phi({\varphi})$
$\displaystyle=\omega\cdot\partial_{\varphi}+{\mathtt{m}}_{1,\overline{\mathtt{n}}}\partial_{x}+{\rm
i}\,{\mathtt{m}}_{\frac{1}{2}}\Omega(\gamma,D)+a_{5}{\mathcal{H}}+{\rm
Op}(r_{7}^{(d)})+T_{M}+T_{M}^{\prime}+{\rm
i}\sqrt{g}(\Pi_{N_{\overline{\mathtt{n}}}}^{\perp}a_{2})|D|^{\frac{1}{2}}\,,$

where $a_{5}({\varphi},x)$ is the real function (using that
$a_{4}=\frac{\gamma}{2}(a_{2}-1)$)

$\displaystyle a_{5}:=$
$\displaystyle\,\tfrac{\gamma}{2}({\mathtt{m}}_{\frac{1}{2}}-1)-\tfrac{\sqrt{g}}{2}(b_{x}a_{2}-(a_{2})_{x}b)$
(7.115)