# Benjamin-Feir instability of
Stokes waves in finite depth

Massimiliano Berti, Alberto Maspero, Paolo Ventura111 International School for
Advanced Studies (SISSA), Via Bonomea 265, 34136, Trieste, Italy. Emails:
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Whitham and Benjamin predicted in 1967 that small-amplitude periodic traveling
Stokes waves of the 2d-gravity water waves equations are linearly unstable
with respect to long-wave perturbations, if the depth ${\mathtt{h}}$ is larger
than a critical threshold $\mathtt{h}_{\scriptscriptstyle{\textsc{WB}}}\approx
1.363$. In this paper we completely describe, for any value of $\mathtt{h}>0$,
the four eigenvalues close to zero of the linearized equations at the Stokes
wave, as the Floquet exponent $\mu$ is turned on. We prove in particular the
existence of a unique depth $\mathtt{h}_{\scriptscriptstyle{\textsc{WB}}}$,
which coincides with the one predicted by Whitham and Benjamin, such that, for
any $0<\mathtt{h}<\mathtt{h}_{\scriptscriptstyle{\textsc{WB}}}$, the
eigenvalues close to zero remain purely imaginary and, for any
$\mathtt{h}>\mathtt{h}_{\scriptscriptstyle{\textsc{WB}}}$, a pair of non-
purely imaginary eigenvalues depicts a closed figure “8”, parameterized by the
Floquet exponent. As
${\mathtt{h}}\to\mathtt{h}_{\scriptscriptstyle{\textsc{WB}}}^{\,+}$ this
figure “8” collapses to the origin of the complex plane. The proof combines a
symplectic version of Kato’s perturbative theory to compute the eigenvalues of
a $4\times 4$ Hamiltonian and reversible matrix, and KAM inspired
transformations to block-diagonalize it. The four eigenvalues have all the
same size $\mathcal{O}(\mu)$ –unlike the infinitely deep water case in [6]–
and the correct Benjamin-Feir phenomenon appears only after one non-
perturbative block-diagonalization step. In addition one has to track, along
the whole proof, the explicit dependence of the entries of the $4\times 4$
reduced matrix with respect to the depth $\mathtt{h}$.

###### Contents

  1. 1 Introduction to main results
  2. 2 The complete Benjamin-Feir spectrum in finite depth
  3. 3 Perturbative approach to the separated eigenvalues
  4. 4 Matrix representation of $\mathscr{L}_{\mu,\epsilon}$ on $\mathcal{V}_{\mu,\epsilon}$
  5. 5 Block-decoupling and emergence of the Whitham-Benjamin function
    1. 5.1 Non-perturbative step of block-decoupling
    2. 5.2 Complete block-decoupling and proof of the main results
  6. A Expansion of the Kato basis
  7. B Expansion of the Stokes waves in finite depth

##  1 Introduction to main results

A classical problem in fluid dynamics, pioneered by the famous work of Stokes
[38] in 1847, concerns the spectral stability/instability of periodic
traveling waves –called Stokes waves– of the gravity water waves equations in
any depth.

Benjamin and Feir [3], Lighthill [32] and Zakharov [43, 45] discovered in the
sixties, through experiments and formal arguments, that Stokes waves in deep
water are unstable, proposing an heuristic mechanism which leads to the
disintegration of wave trains. More precisely, these works predicted unstable
eigenvalues of the linearized equations at the Stokes wave, near the origin of
the complex plane, corresponding to small Floquet exponents $\mu$ or,
equivalently, to long-wave perturbations. The same phenomenon was later
predicted by Whitham [41] and Benjamin [2] for Stokes waves of wavelength
$2\pi\kappa$, in finite depth $\mathtt{h}$, provided that
$\kappa\mathtt{h}>1.363$ approximately. This phenomenon is nowadays called
“Benjamin-Feir” –or modulational– instability, and it is supported by an
enormous amount of physical observations and numerical simulations, see e.g.
[15, 33]. We refer to [46] for an historical survey.

A serious difficulty for a rigorous mathematical proof of the Benjamin-Feir
instability is that the perturbed eigenvalues bifurcate from the eigenvalue
zero, which is defective, with multiplicity four. The first rigorous proof of
a local branch of unstable eigenvalues close to zero for $\kappa\mathtt{h}$
larger than the Whitham-Benjamin threshold $1.363\ldots$ was obtained by
Bridges-Mielke [9] in finite depth (see also the preprint by Hur-Yang [23]).
Their method, based on a spatial dynamics and a center manifold reduction,
breaks down in deep water. For dealing with this case Nguyen-Strauss [35] have
recently developed a new approach, based on a Lyapunov-Schmidt decomposition.
The novel spectral approach developed in Berti-Maspero-Ventura [6] allowed to
fully describe, in deep water, the splitting of the four eigenvalues close to
zero, as the Floquet exponent is turned on, proving in particular the
conjecture that a pair of non-purely imaginary eigenvalues depicts a closed
figure “8”, parameterized by the Floquet exponent.

The goal of this paper is to describe the full Benjamin-Feir instability
phenomenon at any finite value of the depth $\mathtt{h}>0$. This analysis has
fundamental physical importance since real-life experiments are performed in
water tanks (for example the original Benjamin and Feir experiments, in
Feltham’s National Physical Laboratory, had Stokes waves of wavelength 2.2 m
and bottom’s depth of 7.62 m, see [2]). We also remark that the Benjamin-Feir
instability mechanism is a possible responsible of the emergence of rogue
waves in the ocean, we refer to [28, 29] and references therein for a vast
physical literature. A first mathematically rigorous treatment of large waves
is given in [18], via a probabilistic analysis, in the case of NLS.

Along this paper, with no loss of generality, we consider $2\pi$-periodic
Stokes waves, i.e. with wave number $\kappa=1$. In Theorems 2.5 and 1.1 we
prove the existence of a unique depth
$\mathtt{h}_{\scriptscriptstyle{\textsc{WB}}}$, in perfect agreement with the
Benjamin-Feir critical value 1.363…, such that:

  * •

Shallow water case: for any
$0<\mathtt{h}<\mathtt{h}_{\scriptscriptstyle{\textsc{WB}}}$ the eigenvalues
close to zero remain purely imaginary for Stokes waves of sufficiently small
amplitude, see Figure 2(a)-left;

  * •

Sufficiently deep water case: for any
$\mathtt{h}_{\scriptscriptstyle{\textsc{WB}}}<\mathtt{h}<\infty$, there exists
a pair of non-purely imaginary eigenvalues which traces a complete closed
figure “8” (as shown in Figure 2(a)-right) parameterized by the Floquet
exponent $\mu$. By further increasing $\mu$, the eigenvalues recollide far
from the origin on the imaginary axis where then they keep moving. As
${\mathtt{h}}\to\mathtt{h}_{\scriptscriptstyle{\textsc{WB}}}^{\,+}$ the set of
unstable Floquet exponents shrinks to zero and the Benjamin-Feir unstable
eigenvalues collapse to the origin, see Figure 3. This figure ‘8” was first
numerically discovered by Deconink-Oliveras in [15].

We remark that our approach fully describes all the eigenvalues close to $0$,
providing a necessary and sufficient condition for the existence of unstable
eigenvalues, i.e. the positivity of the Benjamin-Feir discriminant function
$\Delta_{\scriptscriptstyle{\textsc{BF}}}(\mathtt{h};\mu,\epsilon)$ defined in
(1.6).

The results of Theorems 2.5 and 1.1 are complementary to those of [6]. In
following the natural spectral approach developed in [6], we encounter a major
difference in the proof, that we now anticipate. In the infinitely deep water
ideal case it turns out that the “reduced” $4\times 4$ matrix obtained by the
Kato reduction procedure is a small perturbation of a block-diagonal matrix
which possesses yet the correct Benjamin-Feir unstable eigenvalues. This is
not the case in finite depth. The correct eigenvalues of the “reduced”
$4\times 4$ matrix emerge only after one non-perturbative step of block
diagonalization. We shall explain in detail this point after the statement of
Theorem 2.5. This is related with the fact that, in infinite deep water, among
the four eigenvalues close to zero of the linearized operator at the Stokes
wave, two are $\mathcal{O}(\mu)$, whereas the other two have much larger size
$\mathcal{O}(\sqrt{\mu})$, whereas in finite depth all four eigenvalues have
size $\mathcal{O}(\mu)$. In addition, along the whole proof, one needs to
carefully track the explicit dependence with respect to $\mathtt{h}$ of the
entries of the reduced $4\times 4$ matrix.

Let us now present rigorously our results.

### Benjamin-Feir instability in finite depth

We consider the pure gravity water waves equations for a bidimensional fluid
occupying a region with finite depth $\mathtt{h}$. With no loss of generality
we set the gravity $g=1$, see Remark 2.4. We consider a $2\pi$-periodic Stokes
wave with amplitude $0<\epsilon\ll 1$ and speed

$c_{\epsilon}={\mathtt{c}}_{\mathtt{h}}+\mathcal{O}(\epsilon^{2})\,,\quad{\mathtt{c}}_{\mathtt{h}}:=\sqrt{\tanh(\mathtt{h})}\,.$

The linearized water waves equations at the Stokes wave are, in the inertial
reference frame moving with speed $c_{\epsilon}$, a linear time independent
system of the form $h_{t}=\mathcal{L}_{\epsilon}h$ where
$\mathcal{L}_{\epsilon}:=\mathcal{L}_{\epsilon}({\mathtt{h}})$ is a linear
operator with $2\pi$-periodic coefficients, see (2.17) (the operator
$\mathcal{L}_{\epsilon}$ in (2.17) is actually obtained conjugating the
linearized water waves equations in the Zakharov formulation at the Stokes
wave via the “good unknown of Alinhac” (2.11) and the Levi-Civita (2.16)
invertible transformations). The operator $\mathcal{L}_{\epsilon}$ possesses
the eigenvalue $0$, which is defective, with multiplicity four, due to
symmetries of the water waves equations. The problem is to prove that the
linear system $h_{t}=\mathcal{L}_{\epsilon}h$ has solutions of the form
$h(t,x)=\text{Re}\left(e^{\lambda t}e^{\mathrm{i}\,\mu x}v(x)\right)$ where
$v(x)$ is a $2\pi$-periodic function, $\mu$ in $\mathbb{R}$ is the Floquet
exponent and $\lambda$ has positive real part, thus $h(t,x)$ grows
exponentially in time. By Bloch-Floquet theory, such $\lambda$ is an
eigenvalue of the operator $\mathcal{L}_{\mu,\epsilon}:=e^{-\mathrm{i}\,\mu
x}\,\mathcal{L}_{\epsilon}\,e^{\mathrm{i}\,\mu x}$ acting on $2\pi$-periodic
functions.

The main result of this paper proves, for any finite value of the depth
$\mathtt{h}$, the full splitting of the four eigenvalues close to zero of the
operator $\mathcal{L}_{\mu,\epsilon}:=\mathcal{L}_{\mu,\epsilon}(\mathtt{h})$
when $\epsilon$ and $\mu$ are small enough, see Theorem 2.5. We first present
Theorem 1.1 which focuses on the figure $``8"$ formed by the Benjamin-Feir
unstable eigenvalues.

We first need to introduce the “Whitham-Benjamin” function

$\mathtt{e}_{\scriptscriptstyle{\textsc{WB}}}:=\mathtt{e}_{\scriptscriptstyle{\textsc{WB}}}(\mathtt{h}):=\frac{1}{{\mathtt{c}}_{\mathtt{h}}}\Big{[}\frac{9{\mathtt{c}}_{\mathtt{h}}^{8}-10{\mathtt{c}}_{\mathtt{h}}^{4}+9}{8{\mathtt{c}}_{\mathtt{h}}^{6}}-\frac{1}{\mathtt{h}-\frac{1}{4}\mathtt{e}_{12}^{2}}\Big{(}1+\frac{1-{\mathtt{c}}_{\mathtt{h}}^{4}}{2}+\frac{3}{4}\frac{(1-{\mathtt{c}}_{\mathtt{h}}^{4})^{2}}{{\mathtt{c}}_{\mathtt{h}}^{2}}\mathtt{h}\Big{)}\Big{]}\,,$
(1.1)

where ${\mathtt{c}}_{\mathtt{h}}=\sqrt{\tanh(\mathtt{h})}$ is the speed of the
linear Stokes wave, and

$\mathtt{e}_{12}:=\mathtt{e}_{12}(\mathtt{h}):={\mathtt{c}}_{\mathtt{h}}+{\mathtt{c}}_{\mathtt{h}}^{-1}(1-{\mathtt{c}}_{\mathtt{h}}^{4})\mathtt{h}>0\,,\quad\forall\mathtt{h}>0\,.$
(1.2)

The function $\mathtt{e}_{\scriptscriptstyle{\textsc{WB}}}(\mathtt{h})$ is
well defined for any $\mathtt{h}>0$ because the denominator
$\mathtt{h}-\tfrac{1}{4}\mathtt{e}_{12}^{2}>0$ in (1.1) is positive for any
$\mathtt{h}>0$, see Lemma 5.7. The function (1.1) coincides, up to a non zero
factor, with the celebrated function obtained by Whitham [41], Benjamin [2]
and Bridges-Mielke [9] which determines the “shallow/sufficiently deep”
threshold regime. In particular the Whitham-Benjamin function
$\mathtt{e}_{\scriptscriptstyle{\textsc{WB}}}(\mathtt{h})$ vanishes at
$\mathtt{h}_{\scriptscriptstyle{\textsc{WB}}}=1.363...$, it is negative for
$0<\mathtt{h}<\mathtt{h}_{\scriptscriptstyle{\textsc{WB}}}$, positive for
$\mathtt{h}>\mathtt{h}_{\scriptscriptstyle{\textsc{WB}}}$ and tends to $1$ as
$\mathtt{h}\to+\infty$, see Figure 1. We also introduce the positive
coefficient

$\mathtt{e}_{22}:=\mathtt{e}_{22}(\mathtt{h}):=\dfrac{(1-{\mathtt{c}}_{\mathtt{h}}^{4})(1+3{\mathtt{c}}_{\mathtt{h}}^{4})\mathtt{h}^{2}+2{\mathtt{c}}_{\mathtt{h}}^{2}({\mathtt{c}}_{\mathtt{h}}^{4}-1)\mathtt{h}+{\mathtt{c}}_{\mathtt{h}}^{4}}{{\mathtt{c}}_{\mathtt{h}}^{3}}>0\,,\quad\forall\mathtt{h}>0\,.$
(1.3)

We remark that the functions
$\mathtt{e}_{12}(\mathtt{h})>{\mathtt{c}}_{\mathtt{h}}$ and
$\mathtt{e}_{22}(\mathtt{h})>0$ are positive for any $\mathtt{h}>0$, tend to
$0$ as $\mathtt{h}\to 0^{+}$ and to $1$ as $\mathtt{h}\to+\infty$, see Lemma
4.8.

Figure 1: Plot of the Whitham-Benjamin function
$\mathtt{e}_{\scriptscriptstyle{\textsc{WB}}}(\mathtt{h})$. The red dot shows
its unique root $\mathtt{h}_{\scriptscriptstyle{\textsc{WB}}}=1.363\dots$.
which is the celebrated “shallow/sufficiently deep” water threshold predicted
independently by Whitham (cfr.[41] p.49) and Benjamin (cfr.[2] p.68), and
recovered in the rigorous proof of Bridges-Mielke [9, p. 183].

Along the paper we denote by
$r(\epsilon^{m_{1}}\mu^{n_{1}},\ldots,\epsilon^{m_{p}}\mu^{n_{p}})$ a real
analytic function fulfilling for some $C>0$ and $\epsilon,\mu$ sufficiently
small, the estimate
$|r(\epsilon^{m_{1}}\mu^{n_{1}},\ldots,\epsilon^{m_{p}}\mu^{n_{p}})|\leq
C\sum_{j=1}^{p}|\epsilon|^{m_{j}}|\mu|^{n_{j}}$, where the constant
$C:=C(\mathtt{h})$ is uniform for $\mathtt{h}$ in any compact set of
$(0,+\infty)$.

######  Theorem 1.1.

(Benjamin-Feir unstable eigenvalues) For any
$\mathtt{h}>\mathtt{h}_{\scriptscriptstyle{\textsc{WB}}}$, there exist
$\epsilon_{1},\mu_{0}>0$ and an analytic function
$\underline{\mu}:[0,\epsilon_{1})\to[0,\mu_{0})$, of the form

$\underline{\mu}(\epsilon)=\mathtt{e}_{\mathtt{h}}\epsilon(1+r(\epsilon))\,,\quad\mathtt{e}_{\mathtt{h}}:=\sqrt{\frac{8\mathtt{e}_{\scriptscriptstyle{\textsc{WB}}}(\mathtt{h})}{\mathtt{e}_{22}(\mathtt{h})}}\,,$
(1.4)

such that, for any $\epsilon\in[0,\epsilon_{1})$, the operator
$\mathcal{L}_{\mu,\epsilon}$ has two eigenvalues
$\lambda^{\pm}_{1}(\mu,\epsilon)$ of the form

$\begin{cases}\mathrm{i}\,\frac{1}{2}\breve{\mathtt{c}}_{\mathtt{h}}\mu+\mathrm{i}\,r_{2}(\mu\epsilon^{2},\mu^{2}\epsilon,\mu^{3})\pm\tfrac{1}{8}\mu\sqrt{\mathtt{e}_{22}(\mathtt{h})}(1+r(\epsilon,\mu))\sqrt{\Delta_{\scriptscriptstyle{\textsc{BF}}}(\mathtt{h};\mu,\epsilon)},&\forall\mu\in[0,\underline{\mu}(\epsilon))\\!\\!\\!\\\\[4.2679pt]
\mathrm{i}\,\frac{1}{2}\breve{\mathtt{c}}_{\mathtt{h}}\underline{\mu}(\epsilon)+\mathrm{i}\,r(\epsilon^{3}),&\mu=\underline{\mu}(\epsilon)\\!\\!\\!\\\\[4.2679pt]
\mathrm{i}\,\frac{1}{2}\breve{\mathtt{c}}_{\mathtt{h}}\mu+\mathrm{i}\,r_{2}(\mu\epsilon^{2},\mu^{2}\epsilon,\mu^{3})\pm\mathrm{i}\,\tfrac{1}{8}\mu\sqrt{\mathtt{e}_{22}(\mathtt{h})}(1+r(\epsilon,\mu))\sqrt{|\Delta_{\scriptscriptstyle{\textsc{BF}}}(\mathtt{h};\mu,\epsilon)|},&\forall\mu\in(\underline{\mu}(\epsilon),\mu_{0})\\!\\!\\!\end{cases}\\!\\!\\!$
(1.5)

where
$\breve{\mathtt{c}}_{\mathtt{h}}:=2{\mathtt{c}}_{\mathtt{h}}-\mathtt{e}_{12}(\mathtt{h})>0$
and $\Delta_{\scriptscriptstyle{\textsc{BF}}}(\mathtt{h};\mu,\epsilon)$ is the
“Benjamin-Feir discriminant” function

$\Delta_{\scriptscriptstyle{\textsc{BF}}}(\mathtt{h};\mu,\epsilon):=8\mathtt{e}_{\scriptscriptstyle{\textsc{WB}}}(\mathtt{h})\epsilon^{2}+r_{1}(\epsilon^{3},\mu\epsilon^{2})-\mathtt{e}_{22}(\mathtt{h})\mu^{2}\big{(}1+r_{1}^{\prime\prime}(\epsilon,\mu)\big{)}\,.$
(1.6)

Note that, for any $0<\epsilon<\epsilon_{1}$ (depending on $\mathtt{h}$) the
function $\Delta_{\scriptscriptstyle{\textsc{BF}}}(\mathtt{h};\mu,\epsilon)>0$
is positive, respectively $<0$, provided $0<\mu<\underline{\mu}(\epsilon)$,
respectively $\mu>\underline{\mu}(\epsilon)$.

Let us make some comments.
1\. Benjamin-Feir unstable eigenvalues. For
$\mathtt{h}>\mathtt{h}_{\scriptscriptstyle{\textsc{WB}}}$, according to (1.5),
for values of the Floquet parameter $0<\mu<\underline{\mu}(\epsilon)$, the
eigenvalues $\lambda^{\pm}_{1}(\mu,\epsilon)$ have opposite non-zero real
part. As $\mu$ tends to $\underline{\mu}(\epsilon)$, the two eigenvalues
$\lambda^{\pm}_{1}(\mu,\epsilon)$ collide on the imaginary axis far from $0$
(in the upper semiplane $\text{Im}(\lambda)>0$), along which they keep moving
for $\mu>\underline{\mu}(\epsilon)$, see Figure 2(a). For $\mu<0$ the operator
${\mathcal{L}}_{\mu,\epsilon}$ possesses the symmetric eigenvalues
$\overline{\lambda_{1}^{\pm}(-\mu,\epsilon)}$ in the semiplane
$\text{Im}(\lambda)<0$. For $\mu\in[0,\underline{\mu}(\epsilon)]$ we obtain
the upper part of the figure “8”, which is well approximated by the curves

$\mu\mapsto\Big{(}\pm\frac{\mu}{8}\sqrt{\mathtt{e}_{22}}\sqrt{8\mathtt{e}_{\scriptscriptstyle{\textsc{WB}}}\epsilon^{2}-\mathtt{e}_{22}\mu^{2}},\
\tfrac{1}{2}\breve{\mathtt{c}}_{\mathtt{h}}\mu\Big{)}\,,$ (1.7)

in accordance with the numerical simulations by Deconinck-Oliveras [15]. Note
that for $\mu>0$ the imaginary part in (1.7) is positive because
$\breve{\mathtt{c}}_{\mathtt{h}}={\mathtt{c}}_{\mathtt{h}}^{-1}(\tanh(\mathtt{h})-(1-\tanh^{2}(\mathtt{h}))\mathtt{h})>0$
for any $\mathtt{h}>0$. The higher order “side-band” corrections of the
eigenvalues $\lambda_{1}^{\pm}(\mu,\epsilon)$ in (1.5), provided by the
analytic functions $r,r_{1},r_{1}^{\prime\prime},r_{2}$, are explicitly
computable. We finally remark that the eigenvalues (1.5) are not analytic in
$(\mu,\epsilon)$ close to the value $(\underline{\mu}(\epsilon),\epsilon)$
where $\lambda^{\pm}_{1}(\mu,\epsilon)$ collide at the top of the figure
$``8"$ far from $0$ (clearly they are continuous).

(a) * (b) * Figure 2: The picture on the left shows, in the “shallow” water
regime $\mathtt{h}<\mathtt{h}_{\scriptscriptstyle{\textsc{WB}}}$, the
eigenvalues $\lambda^{\pm}_{1}(\mu,\epsilon)$ and
$\lambda^{\pm}_{0}(\mu,\epsilon)$ which are purely imaginary. The picture on
the right shows, in the “sufficiently deep” water regime
$\mathtt{h}>\mathtt{h}_{\scriptscriptstyle{\textsc{WB}}}$, the eigenvalues
$\lambda^{\pm}_{1}(\mu,\epsilon)$ in the complex $\lambda$-plane at fixed
$|\epsilon|\ll 1$ as $\mu$ varies. This figure “8 ” depends on $\mathtt{h}$
and shrinks to $0$ as
$\mathtt{h}\to\mathtt{h}_{\scriptscriptstyle{\textsc{WB}}}^{+}$, see Figure 3.
As $\mathtt{h}\to+\infty$ the spectrum resembles the one in deep water found
in [6].

[.4] [.4]

2\. Behaviour near the Whitham-Benjamin depth
$\mathtt{h}_{\scriptscriptstyle{\textsc{WB}}}$.  As
${\mathtt{h}}\to\mathtt{h}_{\scriptscriptstyle{\textsc{WB}}}^{+}$ the constant
$\epsilon_{1}:=\epsilon_{1}(\mathtt{h})>0$ in Theorem 1.1 tends to zero, the
set of unstable Floquet exponents $(0,\underline{\mu}(\epsilon))$ with
$\underline{\mu}(\epsilon)=\mathtt{e}_{\mathtt{h}}\epsilon(1+r(\epsilon))$
given in (1.4) shrinks to zero and the figure “8” of Benjamin-Feir unstable
eigenvalues collapse to zero, see Figure 3. In particular

$\max_{\mu\in[0,\underline{\mu}(\epsilon)]}\text{Re}\,\lambda_{1}^{+}(\mu,\epsilon)=\text{Re}\,\lambda_{1}^{+}(\mu_{\max},\epsilon)=\frac{1}{2}{\mathtt{e}_{\scriptscriptstyle{\textsc{WB}}}}(\mathtt{h})\epsilon^{2}+r(\epsilon^{3})\
\text{ and }$ (1.8)

tends to zero as
$\mathtt{h}\to\mathtt{h}_{\scriptscriptstyle{\textsc{WB}}}^{+}$, since
$0<\epsilon<\epsilon_{1}(\mathtt{h})$ and $\epsilon_{1}(\mathtt{h})\to 0^{+}$.

Figure 3:  The Benjamin-Feir eigenvalue $\lambda^{+}_{1}(\mu_{\max},\epsilon)$
in (1.8) with maximal real part, as well as the whole figure “$8$” shrinks to
zero as $\mathtt{h}\to\mathtt{h}_{\scriptscriptstyle{\textsc{WB}}}^{+}$.

3\. Relation with Bridges-Mielke [9]. Bridges and Mielke describe the unstable
eigenvalues very close to the origin, namely the cross amid the ‘8”. In order
to make a precise comparison with our result let us spell out the relation of
the functions $\mathtt{e}_{\scriptscriptstyle{\textsc{WB}}}$,
$\mathtt{e}_{12}$ and $\mathtt{e}_{22}$ with the coefficients obtained in [9].
The Whitham-Benjamin function $\mathtt{e}_{\scriptscriptstyle{\textsc{WB}}}$
in (4.13) is
$\mathtt{e}_{\scriptscriptstyle{\textsc{WB}}}=({\mathtt{c}}_{\mathtt{h}}\mathtt{h})^{-1}\nu(F)$,
where $\nu(F)$ is defined in [9, formula (6.17)] and
$F={\mathtt{c}}_{\mathtt{h}}\mathtt{h}^{-\frac{1}{2}}$ is the Froude number,
cfr. [9, formula (3.4)]. Moreover the term $\mathtt{e}_{12}$ in (1.2) is
$\mathtt{e}_{12}=2c_{g}$, where
$c_{g}=\frac{1}{2}{\mathtt{c}}_{\mathtt{h}}\big{(}1+F^{-2}\text{sech}^{2}(\mathtt{h})\big{)}$
is the group velocity defined in Bridges-Mielke [9, formula (3.8)]. Finally
$\mathtt{e}_{22}(\mathtt{h})\propto\dot{c}_{g}$ where $\dot{c}_{g}$ is the
derivative of the group velocity defined in [9, formula (6.15)], which for
gravity waves is negative in any depth.
4\. Complete spectrum near $0$. In Theorem 1.1 we have described just the two
unstable eigenvalues of $\mathcal{L}_{\mu,\epsilon}$ close to zero for
$\mathtt{h}>\mathtt{h}_{\scriptscriptstyle{\textsc{WB}}}$. There are also two
larger purely imaginary eigenvalues of order $\mathcal{O}(\mu)$, see Theorem
2.5. We remark that our approach describes all the eigenvalues of
${\mathcal{L}}_{\mu,\epsilon}$ close to $0$ (which are $4$).
5\. Shallow water regime. In the shallow water regime
$0<\mathtt{h}<\mathtt{h}_{\scriptscriptstyle{\textsc{WB}}}$, we prove in
Theorem 2.5 that all the four eigenvalues of ${\mathcal{L}}_{\mu,\epsilon}$
close to zero remain purely imaginary for $\epsilon$ sufficiently small. The
eigenvalue expansions of Theorem 2.5 become singular as $\mathtt{h}\to 0^{+}$.
6\. Behavior at the Whitham-Benjamin threshold
$\mathtt{h}_{\scriptscriptstyle{\textsc{WB}}}$.  The analysis of Theorem 1.1
is not conclusive at the critical depth
$\mathtt{h}=\mathtt{h}_{\scriptscriptstyle{\textsc{WB}}}$. The reason is that
$\mathtt{e}_{\scriptscriptstyle{\textsc{WB}}}(\mathtt{h}_{\scriptscriptstyle{\textsc{WB}}})=0$
and the Benjamin-Feir discriminant function (1.6) reduces to

$\Delta_{\scriptscriptstyle{\textsc{BF}}}(\mathtt{h}_{\scriptscriptstyle{\textsc{WB}}};\mu,\epsilon)=r(\epsilon^{3})+r(\mu\epsilon^{2})-\mathtt{e}_{22}(\mathtt{h}_{\scriptscriptstyle{\textsc{WB}}})\mu^{2}(1+r_{1}^{\prime\prime}(\epsilon,\mu))\,.$
(1.9)

Thus its quadratic expansion is not sufficient anymore to determine the sign
of
$\Delta_{\scriptscriptstyle{\textsc{BF}}}(\mathtt{h}_{\scriptscriptstyle{\textsc{WB}}};\mu,\epsilon)$.
Note that (1.9) could be positive due to the cubic term
$r(\epsilon^{3})=\alpha\epsilon^{3}+\dots$ for $\epsilon$ and $\mu$ small
enough. The coefficient $\alpha$ could be explicitly computed taking into
account the third order expansion of the Stokes waves.
7\. Unstable Floquet exponents and amplitudes $(\mu,\epsilon)$.  In Theorem
2.5 we actually prove that the expansion (1.5) of the eigenvalues of
$\mathcal{L}_{\mu,\epsilon}$ holds for any value of $(\mu,\epsilon)$ in a
larger rectangle $[0,\mu_{0})\times[0,\epsilon_{0})$, and there exist
Benjamin-Feir unstable eigenvalues if and only if the analytic function
$\Delta_{\scriptscriptstyle{\textsc{BF}}}(\mathtt{h};\mu,\epsilon)$ in (1.6)
is positive. The zero set of
$\Delta_{\scriptscriptstyle{\textsc{BF}}}(\mathtt{h};\mu,\epsilon)$ is an
analytic variety which, for
$\mathtt{h}>\mathtt{h}_{\scriptscriptstyle{\textsc{WB}}}$, is, restricted to
the rectangle $[0,\mu_{0})\times[0,\epsilon_{1})$, the graph of the analytic
function
$\underline{\mu}(\epsilon)=\mathtt{e}_{\mathtt{h}}\epsilon(1+r(\epsilon))$ in
(1.4). This function is tangent at $\epsilon=0$ to the straight line
$\mu=\mathtt{e}_{\mathtt{h}}\epsilon$, and divides
$[0,\mu_{0})\times[0,\epsilon_{1})$ in the region where
$\Delta_{\scriptscriptstyle{\textsc{BF}}}(\mathtt{h};\mu,\epsilon)>0$ –and
thus the eigenvalues of ${\mathcal{L}}_{\mu,\epsilon}$ have non-trivial real
part–, from the “stable” one where all the eigenvalues of
${\mathcal{L}}_{\mu,\epsilon}$ are purely imaginary, see Figure 4. In the
region $[0,\mu_{0})\times[\epsilon_{1},\epsilon_{0})$ the higher order
polynomial approximations of
$\Delta_{\scriptscriptstyle{\textsc{BF}}}(\mathtt{h};\mu,\epsilon)$ (which are
computable) will determine the sign of
$\Delta_{\scriptscriptstyle{\textsc{BF}}}(\mathtt{h};\mu,\epsilon)$.

Figure 4: The solid curve portrays the graph of the real analytic function
$\underline{\mu}(\epsilon)$ in (1.4) as
$\mathtt{h}>\mathtt{h}_{\scriptscriptstyle{\textsc{WB}}}$. For values of $\mu$
below this curve, the two eigenvalues $\lambda^{\pm}_{1}(\mu,\epsilon)$ have
non zero real part. For $\mu$ above the curve,
$\lambda^{\pm}_{1}(\mu,\epsilon)$ are purely imaginary. In the region
$[\epsilon_{1},\epsilon_{0})\times[0,\mu_{0})$ the eigenvalues are real/purely
imaginary depending on the higher order corrections given by Theorem 2.5,
which determine the sign of
$\Delta_{\scriptscriptstyle{\textsc{BF}}}(\mathtt{h};\mu,\epsilon)$.

8\. Deep water limit. Theorems 1.1 and 2.5 do not pass to the limit as
$\mathtt{h}\to+\infty$ since the remainders in the expansions of the
eigenvalues are uniform only on any compact set of $\mathtt{h}\in(0,+\infty)$.
From a mathematical point of view, the difference is evident in the asymptotic
behavior of $\tanh(\mathtt{h}\mu)$ (and similar quantities) which, in the
idealized deep water case $\mathtt{h}=+\infty$, is identically equal to $1$
for any arbitrarily small Floquet exponent $\mu$, whereas
$\tanh(\mathtt{h}\mu)=O(\mu\mathtt{h})$ for any $\mathtt{h}$ finite. However
additional intermediate scaling regimes $\mathtt{h}\mu\sim 1$,
$\mathtt{h}\mu\ll 1$, $\mathtt{h}\mu\gg 1$ are possible. It is well-known
(e.g. see [14]) that intermediate long-wave regimes of the water-waves
equations formally lead to different physically-relevant limit equations as
Boussinesq, KdV, NLS, Benjamin-Ono, etc…

We shall describe in detail the ideas of proof and the differences with the
deep water case below the statement of Theorem 2.5.
Further literature. Modulational instability has been studied also for a
variety of approximate water waves models, such as KdV, gKdV, NLS and the
Whitham equation by, for instance, Whitham [42], Segur, Henderson, Carter and
Hammack [37], Gallay and Haragus [17], Haragus and Kapitula [19], Bronski and
Johnson [11], Johnson [25], Hur and Johnson [21], Bronski, Hur and Johnson
[10], Hur and Pandey [22], Leisman, Bronski, Johnson and Marangell [30]. Also
for these approximate models, numerical simulations predict a figure “8”
similar to that in Figure 2(a) for the bifurcation of the unstable eigenvalues
close to zero. We expect the present approach can be adapted to describe the
full bifurcation of the eigenvalues also for these models.

Finally we mention the nonlinear modulational instability result of Jin, Liao,
and Lin [24] for several fluid model equations and the preprint by Chen-Su
[12] for Stokes waves in deep water. Nonlinear transversal instability results
of traveling solitary water waves in finite depth decaying at infinity on
$\mathbb{R}$ have been proved in [36] (in deep water no solitary wave exists
[20, 27]).
Acknowledgments. Research supported by PRIN 2020 (2020XB3EFL001) “Hamiltonian
and dispersive PDEs”.

##  2 The complete Benjamin-Feir spectrum in finite depth

In this section we present in detail the complete spectral Theorem 2.5. We
first introduce the pure gravity water waves equations and the Stokes waves
solutions.
The water waves equations. We consider the Euler equations for a 2-dimensional
incompressible, irrotational fluid under the action of gravity. The fluid
fills the region

${\mathcal{D}}_{\eta}:=\left\\{(x,y)\in\mathbb{T}\times\mathbb{R}\;:\;-\mathtt{h}\leq
y<\eta(t,x)\right\\}\,,\quad\mathbb{T}:=\mathbb{R}/2\pi\mathbb{Z}\,,$

with finite depth and space periodic boundary conditions. The irrotational
velocity field is the gradient of a harmonic scalar potential
$\Phi=\Phi(t,x,y)$ determined by its trace $\psi(t,x)=\Phi(t,x,\eta(t,x))$ at
the free surface $y=\eta(t,x)$. Actually $\Phi$ is the unique solution of the
elliptic equation $\Delta\Phi=0$ in ${\mathcal{D}}_{\eta}$ with Dirichlet
datum $\Phi(t,x,\eta(t,x))=\psi(t,x)$ and $\Phi_{y}(t,x,y)=0$ at
$y=-\mathtt{h}$.

The time evolution of the fluid is determined by two boundary conditions at
the free surface. The first is that the fluid particles remain, along the
evolution, on the free surface (kinematic boundary condition), and the second
one is that the pressure of the fluid is equal, at the free surface, to the
constant atmospheric pressure (dynamic boundary condition). Then, as shown by
Zakharov [44] and Craig-Sulem [13], the time evolution of the fluid is
determined by the following equations for the unknowns
$(\eta(t,x),\psi(t,x))$,

$\eta_{t}=G(\eta)\psi\,,\quad\psi_{t}=-g\eta-\dfrac{\psi_{x}^{2}}{2}+\dfrac{1}{2(1+\eta_{x}^{2})}\big{(}G(\eta)\psi+\eta_{x}\psi_{x}\big{)}^{2}\,,$
(2.1)

where $g>0$ is the gravity constant and $G(\eta):=G(\eta,\mathtt{h})$ denotes
the Dirichlet-Neumann operator
$[G(\eta)\psi](x):=\Phi_{y}(x,\eta(x))-\Phi_{x}(x,\eta(x))\eta_{x}(x)$. In the
sequel, with no loss of generality, we set the gravity constant $g=1$, see
Remark 2.4.

The equations (2.1) are the Hamiltonian system

$\partial_{t}\begin{bmatrix}\eta\\\
\psi\end{bmatrix}=\mathcal{J}\begin{bmatrix}\nabla_{\eta}\mathcal{H}\\\
\nabla_{\psi}\mathcal{H}\end{bmatrix},\quad\quad\mathcal{J}:=\begin{bmatrix}0&\mathrm{Id}\\\
-\mathrm{Id}&0\end{bmatrix},$ (2.2)

where $\nabla$ denote the $L^{2}$-gradient, and the Hamiltonian
$\mathcal{H}(\eta,\psi):=\frac{1}{2}\int_{\mathbb{T}}\left(\psi\,G(\eta)\psi+\eta^{2}\right)\mathrm{d}x$
is the sum of the kinetic and potential energy of the fluid. In addition of
being Hamiltonian, the water waves system (2.1) possesses other important
symmetries. First of all it is time reversible with respect to the involution

$\rho\begin{bmatrix}\eta(x)\\\
\psi(x)\end{bmatrix}:=\begin{bmatrix}\eta(-x)\\\
-\psi(-x)\end{bmatrix},\quad\text{i.e. }\mathcal{H}\circ\rho=\mathcal{H}\,.$
(2.3)

Moreover, the equation (2.1) is space invariant, since, being the bottom flat,

$\tau_{\theta}G(\eta)\psi=G(\tau_{\theta}\eta)[\tau_{\theta}\psi]\,,\quad\forall\theta\in\mathbb{R}\,,\quad\text{where}\quad\tau_{\theta}u(x):=u(x+\theta)\,.$

In addition, the Dirichlet-Neumann operator satisfies
$G(\eta+m,\mathtt{h})=G(\eta,\mathtt{h}+m)$, for any $m\in\mathbb{R}$.
Stokes waves. The Stokes waves are traveling solutions of (2.1) of the form
$\eta(t,x)=\breve{\eta}(x-ct)$ and $\psi(t,x)=\breve{\psi}(x-ct)$ for some
real $c$ and $2\pi$-periodic functions $(\breve{\eta}(x),\breve{\psi}(x))$. In
a reference frame in translational motion with constant speed $c$, the water
waves equations (2.1) become

$\eta_{t}=c\eta_{x}+G(\eta)\psi\,,\quad\psi_{t}=c\psi_{x}-\eta-\dfrac{\psi_{x}^{2}}{2}+\dfrac{1}{2(1+\eta_{x}^{2})}\big{(}G(\eta)\psi+\eta_{x}\psi_{x}\big{)}^{2}$
(2.4)

and the Stokes waves $(\breve{\eta},\breve{\psi})$ are equilibrium steady
solutions of (2.4).

The bifurcation result of small amplitude of Stokes waves is due to Struik
[39] in finite depth, and Levi-Civita [31], and Nekrasov [34] in infinite
depth. We denote by $B(r):=\\{x\in\mathbb{R}\colon\ |x|<r\\}$ the real ball
with center 0 and radius $r$.

######  Theorem 2.1.

(Stokes waves) For any $\mathtt{h}>0$ there exist
$\epsilon_{*}:=\epsilon_{*}(\mathtt{h})>0$ and a unique family of real
analytic solutions $(\eta_{\epsilon}(x),\psi_{\epsilon}(x),c_{\epsilon})$,
parameterized by the amplitude $|\epsilon|\leq\epsilon_{*}$, of

$c\,\eta_{x}+G(\eta)\psi=0\,,\quad
c\,\psi_{x}-\eta-\dfrac{\psi_{x}^{2}}{2}+\dfrac{1}{2(1+\eta_{x}^{2})}\big{(}G(\eta)\psi+\eta_{x}\psi_{x}\big{)}^{2}=0\,,$
(2.5)

such that $\eta_{\epsilon}(x),\psi_{\epsilon}(x)$ are $2\pi$-periodic;
$\eta_{\epsilon}(x)$ is even and $\psi_{\epsilon}(x)$ is odd, of the form

$\displaystyle\eta_{\epsilon}(x)=\epsilon\cos(x)+\epsilon^{2}(\eta_{2}^{[0]}+\eta_{2}^{[2]}\cos(2x))+\mathcal{O}(\epsilon^{3}),$
(2.6)
$\displaystyle\psi_{\epsilon}(x)=\epsilon{\mathtt{c}}_{\mathtt{h}}^{-1}\sin(x)+\epsilon^{2}\psi_{2}^{[2]}\sin(2x)+\mathcal{O}(\epsilon^{3})\,,$
$\displaystyle
c_{\epsilon}={\mathtt{c}}_{\mathtt{h}}+\epsilon^{2}c_{2}+\mathcal{O}(\epsilon^{3})\quad\text{where}\quad{\mathtt{c}}_{\mathtt{h}}=\sqrt{\tanh(\mathtt{h})}\,,$

and

$\displaystyle\eta_{2}^{[0]}:=\frac{{\mathtt{c}}_{\mathtt{h}}^{4}-1}{4{\mathtt{c}}_{\mathtt{h}}^{2}}\,,\qquad\eta_{2}^{[2]}:=\frac{3-{\mathtt{c}}_{\mathtt{h}}^{4}}{4{\mathtt{c}}_{\mathtt{h}}^{6}}\,,\qquad\psi_{2}^{[2]}:=\frac{3+{\mathtt{c}}_{\mathtt{h}}^{8}}{8{\mathtt{c}}_{\mathtt{h}}^{7}}\,,\qquad$
(2.7) $\displaystyle
c_{2}:=\frac{9-10{\mathtt{c}}_{\mathtt{h}}^{4}+9{\mathtt{c}}_{\mathtt{h}}^{8}}{16{\mathtt{c}}_{\mathtt{h}}^{7}}+{\frac{(1-{\mathtt{c}}_{\mathtt{h}}^{4})}{2{\mathtt{c}}_{\mathtt{h}}}}\eta_{2}^{[0]}=\frac{-2{\mathtt{c}}_{\mathtt{h}}^{12}+13{\mathtt{c}}_{\mathtt{h}}^{8}-12{\mathtt{c}}_{\mathtt{h}}^{4}+9}{16{\mathtt{c}}_{\mathtt{h}}^{7}}\,.$
(2.8)

More precisely for any $\sigma\geq 0$ and $s>\frac{5}{2}$, there exists
$\epsilon_{*}>0$ such that the map
$\epsilon\mapsto(\eta_{\epsilon},\psi_{\epsilon},c_{\epsilon})$ is analytic
from $B(\epsilon_{*})\to H^{\sigma,s}_{\mathtt{ev}}(\mathbb{T})\times
H^{\sigma,s}_{\mathtt{odd}}(\mathbb{T})\times\mathbb{R}$, where
$H^{\sigma,s}_{\mathtt{ev}}(\mathbb{T})$, respectively
$H^{\sigma,s}_{\mathtt{odd}}(\mathbb{T})$, denote the space of even,
respectively odd, real valued $2\pi$-periodic analytic functions
$u(x)=\sum_{k\in\mathbb{Z}}u_{k}e^{\mathrm{i}\,kx}$ such that
$\|u\|_{\sigma,s}^{2}:=\sum_{k\in\mathbb{Z}}|u_{k}|^{2}\langle
k\rangle^{2s}e^{2\sigma|k|}<+\infty$.

The expansions (2.6)-(2.8) are derived in the Appendix B for completeness,
although present in the literature (they coincide with [42, section 13,
chapter 13] and [2, section 2]). Note that in the shallow water regime
$\mathtt{h}\to 0^{+}$ the expansions (2.6)-(2.8) become singular. For the
analiticity properties of the maps stated in Theorem 2.1 we refer to [8].

We also mention that more general time quasi-periodic traveling Stokes waves –
which are nonlinear superpositions of multiple Stokes waves traveling with
rationally independent speeds – have been recently proved for (2.1) in [5] in
finite depth, in [16] in infinite depth, and in [4] for capillary-gravity
water waves in any depth.
Linearization at the Stokes waves. In order to determine the
stability/instability of the Stokes waves given by Theorem 2.1, we linearize
the water waves equations (2.4) with $c=c_{\epsilon}$ at
$(\eta_{\epsilon}(x),\psi_{\epsilon}(x))$. In the sequel we closely follow [6]
pointing out the differences of the finite depth case.

By using the shape derivative formula for the differential
$\mathrm{d}_{\eta}G(\eta)[\hat{\eta}]$ of the Dirichlet-Neumann operator one
obtains the autonomous real linear system

$\begin{bmatrix}\hat{\eta}_{t}\\\
\hat{\psi}_{t}\end{bmatrix}=\begin{bmatrix}-G(\eta_{\epsilon})B-\partial_{x}\circ(V-c_{\epsilon})&G(\eta_{\epsilon})\\\
-1+B(V-c_{\epsilon})\partial_{x}-B\partial_{x}\circ(V-c_{\epsilon})-BG(\eta_{\epsilon})\circ
B&-(V-c_{\epsilon})\partial_{x}+BG(\eta_{\epsilon})\end{bmatrix}\begin{bmatrix}\hat{\eta}\\\
\hat{\psi}\end{bmatrix}$ (2.9)

where

$V:=V(x):=-B(\eta_{\epsilon})_{x}+(\psi_{\epsilon})_{x}\,,\ \
B:=B(x):=\frac{G(\eta_{\epsilon})\psi_{\epsilon}+(\psi_{\epsilon})_{x}(\eta_{\epsilon})_{x}}{1+(\eta_{\epsilon})_{x}^{2}}=\frac{(\psi_{\epsilon})_{x}-c_{\epsilon}}{1+(\eta_{\epsilon})_{x}^{2}}(\eta_{\epsilon})_{x}\,.$
(2.10)

The functions $(V,B)$ are the horizontal and vertical components of the
velocity field $(\Phi_{x},\Phi_{y})$ at the free surface. Moreover
$\epsilon\mapsto(V,B)$ is analytic as a map $B(\epsilon_{0})\to
H^{\sigma,s-1}(\mathbb{T})\times H^{\sigma,s-1}(\mathbb{T})$.

The real system (2.9) is Hamiltonian, i.e. of the form
$\mathcal{J}\mathcal{A}$ for a symmetric operator
$\mathcal{A}=\mathcal{A}^{\top}$, where $\mathcal{A}^{\top}$ is the transposed
operator with respect the standard real scalar product of
$L^{2}(\mathbb{T},\mathbb{R})\times L^{2}(\mathbb{T},\mathbb{R})$.

Moreover, since $\eta_{\epsilon}$ is even in $x$ and $\psi_{\epsilon}$ is odd
in $x$, then the functions $(V,B)$ are respectively even and odd in $x$, and
the linear operator in (2.9) is reversible, i.e. it anti-commutes with the
involution $\rho$ in (2.3).

Under the time-independent “good unknown of Alinhac” linear transformation

$\begin{bmatrix}\hat{\eta}\\\ \hat{\psi}\end{bmatrix}:=Z\begin{bmatrix}u\\\
v\end{bmatrix}\,,\qquad Z=\begin{bmatrix}1&0\\\ B&1\end{bmatrix},\quad
Z^{-1}=\begin{bmatrix}1&0\\\ -B&1\end{bmatrix},$ (2.11)

the system (2.9) assumes the simpler form

$\begin{bmatrix}u_{t}\\\
v_{t}\end{bmatrix}=\widetilde{\mathcal{L}}_{\epsilon}\begin{bmatrix}u\\\
v\end{bmatrix},\qquad\widetilde{\mathcal{L}}_{\epsilon}:=\begin{bmatrix}-\partial_{x}\circ(V-c_{\epsilon})&G(\eta_{\epsilon})\\\
-1-(V-c_{\epsilon})B_{x}&-(V-c_{\epsilon})\partial_{x}\end{bmatrix}\,.$ (2.12)

Note that, since the transformation $Z$ is symplectic, i.e.
$Z^{\top}\mathcal{J}Z=\mathcal{J}$, and reversibility preserving, i.e.
$Z\circ\rho=\rho\circ Z$, the linear system (2.12) is Hamiltonian and
reversible as (2.9).

Next we perform a conformal change of variables to flatten the water surface.
Here the finite depth case induces a modification with respect to the deep
water case. By [1, Appendix A], there exists a diffeomorphism of $\mathbb{T}$,
$x\mapsto x+\mathfrak{p}(x)$, with a small $2\pi$-periodic function
$\mathfrak{p}(x)$, and a small constant $\mathtt{f}_{\epsilon}$, such that, by
defining the associated composition operator
$(\mathfrak{P}u)(x):=u(x+\mathfrak{p}(x))$, the Dirichlet-Neumann operator
writes as [1, Lemma A.5]

$G(\eta_{\epsilon})=\partial_{x}\circ\mathfrak{P}^{-1}\circ{\mathcal{H}}\circ\tanh\big{(}(\mathtt{h}+\mathtt{f}_{\epsilon})|D|\big{)}\circ\mathfrak{P}\,,$
(2.13)

where ${\mathcal{H}}$ is the Hilbert transform, i.e. the Fourier multiplier
operator

$\mathcal{H}(e^{\mathrm{i}\,jx}):=-\mathrm{i}\,\textup{sign}(j)e^{\mathrm{i}\,jx}\,,\quad\forall
j\in\mathbb{Z}\setminus\\{0\\}\,,\quad\mathcal{H}(1):=0\,.$

The function $\mathfrak{p}(x)$ and the constant $\mathtt{f}_{\epsilon}$ are
determined as a fixed point of (see [1, formula (A.15)])

$\mathfrak{p}=\frac{\mathcal{H}}{\tanh\big{(}(\mathtt{h}+\mathtt{f}_{\epsilon})|D|\big{)}}[\eta_{\epsilon}(x+\mathfrak{p}(x))]\,,\qquad\mathtt{f}_{\epsilon}:=\frac{1}{2\pi}\int_{\mathbb{T}}\eta_{\epsilon}(x+\mathfrak{p}(x))\mathrm{d}x\,.$
(2.14)

By the analyticity of the map $\epsilon\to\eta_{\epsilon}\in H^{\sigma,s}$,
$\sigma>0$, $s>1/2$, the analytic implicit function theorem implies the
existence of a solution
$\epsilon\mapsto\mathfrak{p}(x):=\mathfrak{p}_{\epsilon}(x)$,
$\epsilon\mapsto\mathtt{f}_{\epsilon}$, analytic as a map $B(\epsilon_{0})\to
H^{s}(\mathbb{T})\times\mathbb{R}$. Moreover, since $\eta_{\epsilon}$ is even,
the function $\mathfrak{p}(x)$ is odd. In Appendix B we prove the expansion

$\mathfrak{p}(x)=\epsilon{\mathtt{c}}_{\mathtt{h}}^{-2}\sin(x)+\epsilon^{2}\frac{(1+{\mathtt{c}}_{\mathtt{h}}^{4})(3+{\mathtt{c}}_{\mathtt{h}}^{4})}{8{\mathtt{c}}_{\mathtt{h}}^{8}}\sin(2x)+\mathcal{O}(\epsilon^{3})\,,\quad\mathtt{f}_{\epsilon}=\epsilon^{2}\frac{{\mathtt{c}}_{\mathtt{h}}^{4}-3}{4{\mathtt{c}}_{\mathtt{h}}^{2}}+\mathcal{O}(\epsilon^{3})\,.$
(2.15)

Under the symplectic and reversibility-preserving map

$\mathcal{P}:=\begin{bmatrix}(1+\mathfrak{p}_{x})\mathfrak{P}&0\\\
0&\mathfrak{P}\end{bmatrix}\,,$ (2.16)

the system (2.12) transforms, by (2.13), into the linear system
$h_{t}=\mathcal{L}_{\epsilon}h$ where $\mathcal{L}_{\epsilon}$ is the
Hamiltonian and reversible real operator

$\displaystyle\mathcal{L}_{\epsilon}:=\mathcal{P}\,\widetilde{\mathcal{L}}_{\epsilon}\,\mathcal{P}^{-1}$
$\displaystyle=\begin{bmatrix}\partial_{x}\circ({\mathtt{c}}_{\mathtt{h}}+p_{\epsilon}(x))&|D|\tanh((\mathtt{h}+\mathtt{f}_{\epsilon})|D|)\\\
-(1+a_{\epsilon}(x))&({\mathtt{c}}_{\mathtt{h}}+p_{\epsilon}(x))\partial_{x}\end{bmatrix}$
(2.17)
$\displaystyle=\mathcal{J}\begin{bmatrix}1+a_{\epsilon}(x)&-({\mathtt{c}}_{\mathtt{h}}+p_{\epsilon}(x))\partial_{x}\\\
\partial_{x}\circ({\mathtt{c}}_{\mathtt{h}}+p_{\epsilon}(x))&|D|\tanh((\mathtt{h}+\mathtt{f}_{\epsilon})|D|)\end{bmatrix}$

where

${\mathtt{c}}_{\mathtt{h}}+p_{\epsilon}(x):=\displaystyle{\frac{c_{\epsilon}-V(x+\mathfrak{p}(x))}{1+\mathfrak{p}_{x}(x)}}\,,\quad
1+a_{\epsilon}(x):=\displaystyle{\frac{1+(V(x+\mathfrak{p}(x))-c_{\epsilon})B_{x}(x+\mathfrak{p}(x))}{1+\mathfrak{p}_{x}(x)}}\,.$
(2.18)

By the analiticity results of the functions $V,B,\mathfrak{p}(x)$ given above,
the functions $p_{\epsilon}$ and $a_{\epsilon}$ are analytic in $\epsilon$ as
maps $B(\epsilon_{0})\to H^{s}(\mathbb{T})$. In the Appendix B we prove the
following expansions.

######  Lemma 2.2.

The analytic functions $p_{\epsilon}(x)$ and $a_{\epsilon}(x)$ in (2.18) are
even in $x$, and

$p_{\epsilon}(x)=\epsilon
p_{1}(x)+\epsilon^{2}p_{2}(x)+\mathcal{O}(\epsilon^{3})\,,\qquad
a_{\epsilon}(x)=\epsilon
a_{1}(x)+\epsilon^{2}a_{2}(x)+\mathcal{O}(\epsilon^{3})\,,$ (2.19)

where

$\displaystyle p_{1}(x)$ $\displaystyle=p_{1}^{[1]}\cos(x)\,,\qquad\quad\quad
p_{1}^{[1]}:=-2{\mathtt{c}}_{\mathtt{h}}^{-1}\,,$ (2.20) $\displaystyle
p_{2}(x)$ $\displaystyle=p_{2}^{[0]}+p_{2}^{[2]}\cos(2x)\,,\quad
p_{2}^{[0]}:=\frac{9+12{\mathtt{c}}_{\mathtt{h}}^{4}+5{\mathtt{c}}_{\mathtt{h}}^{8}-2{\mathtt{c}}_{\mathtt{h}}^{12}}{16{\mathtt{c}}_{\mathtt{h}}^{7}}\,,\quad
p_{2}^{[2]}:=-\frac{3+{\mathtt{c}}_{\mathtt{h}}^{4}}{2{\mathtt{c}}_{\mathtt{h}}^{7}}\,,$
(2.21)

and

$\displaystyle a_{1}(x)$ $\displaystyle=a_{1}^{[1]}\cos(x)\,,\qquad\qquad
a_{1}^{[1]}:=-({\mathtt{c}}_{\mathtt{h}}^{2}+{\mathtt{c}}_{\mathtt{h}}^{-2})\,,$
(2.22) $\displaystyle a_{2}(x)$
$\displaystyle=a_{2}^{[0]}+a_{2}^{[2]}\cos(2x)\,,\quad
a_{2}^{[0]}:=\frac{3}{2}+\frac{1}{2{\mathtt{c}}_{\mathtt{h}}^{4}}\,,\quad
a_{2}^{[2]}:=\frac{-14{\mathtt{c}}_{\mathtt{h}}^{4}+9{\mathtt{c}}_{\mathtt{h}}^{8}-3}{4{\mathtt{c}}_{\mathtt{h}}^{8}}\,.$
(2.23)

Bloch-Floquet expansion. Since the operator $\mathcal{L}_{\epsilon}$ in (2.17)
has $2\pi$-periodic coefficients, Bloch-Floquet theory guarantees that

$\sigma_{L^{2}(\mathbb{R})}(\mathcal{L}_{\epsilon})=\bigcup_{\mu\in[-\frac{1}{2},\frac{1}{2})}\sigma_{L^{2}(\mathbb{T})}(\mathcal{L}_{\mu,\epsilon})\qquad\text{where}\quad\qquad\mathcal{L}_{\mu,\epsilon}:=e^{-\mathrm{i}\,\mu
x}\,\mathcal{L}_{\epsilon}\,e^{\mathrm{i}\,\mu x}\,.$

The domain $[-\frac{1}{2},\frac{1}{2})$ is called, in solid state physics, the
“first zone of Brillouin”. In particular, if $\lambda$ is an eigenvalue of
$\mathcal{L}_{\mu,\epsilon}$ on $L^{2}(\mathbb{T},\mathbb{C}^{2})$ with
eigenvector $v(x)$, then $h(t,x)=e^{\lambda t}e^{\mathrm{i}\,\mu x}v(x)$
solves $h_{t}=\mathcal{L}_{\epsilon}h$. We remark that:
1\. If $A=\mathrm{Op}(a)$ is a pseudo-differential operator with symbol
$a(x,\xi)$, which is $2\pi$ periodic in the $x$-variable, then
$A_{\mu}:=e^{-\mathrm{i}\,\mu x}Ae^{\mathrm{i}\,\mu
x}=\mathrm{Op}(a(x,\xi+\mu))$.
2\. If $A$ is a real operator then $\overline{A_{\mu}}=A_{-\mu}$. As a
consequence the spectrum $\sigma(A_{-\mu})=\overline{\sigma(A_{\mu})}$ and we
can study $\sigma(A_{\mu})$ just for $\mu>0$. Furthermore $\sigma(A_{\mu})$ is
a 1-periodic set with respect to $\mu$, so one can restrict to
$\mu\in[0,\frac{1}{2})$.

By the previous remarks the Floquet operator associated with the real operator
$\mathcal{L}_{\epsilon}$ in (2.17) is the complex _Hamiltonian_ and
_reversible_ operator

$\displaystyle\mathcal{L}_{\mu,\epsilon}:$
$\displaystyle=\begin{bmatrix}(\partial_{x}+\mathrm{i}\,\mu)\circ({\mathtt{c}}_{\mathtt{h}}+p_{\epsilon}(x))&|D+\mu|\tanh\big{(}(\mathtt{h}+\mathtt{f}_{\epsilon})|D+\mu|\big{)}\\\
-(1+a_{\epsilon}(x))&({\mathtt{c}}_{\mathtt{h}}+p_{\epsilon}(x))(\partial_{x}+\mathrm{i}\,\mu)\end{bmatrix}$
(2.24) $\displaystyle=\underbrace{\begin{bmatrix}0&\mathrm{Id}\\\
-\mathrm{Id}&0\end{bmatrix}}_{\displaystyle{=\mathcal{J}}}\underbrace{\begin{bmatrix}1+a_{\epsilon}(x)&-({\mathtt{c}}_{\mathtt{h}}+p_{\epsilon}(x))(\partial_{x}+\mathrm{i}\,\mu)\\\
(\partial_{x}+\mathrm{i}\,\mu)\circ({\mathtt{c}}_{\mathtt{h}}+p_{\epsilon}(x))&|D+\mu|\tanh\big{(}(\mathtt{h}+\mathtt{f}_{\epsilon})|D+\mu|\big{)}\end{bmatrix}}_{\displaystyle{=:\mathcal{B}_{\mu,\epsilon}}}\,.$

We regard $\mathcal{L}_{\mu,\epsilon}$ as an operator with domain
$H^{1}(\mathbb{T}):=H^{1}(\mathbb{T},\mathbb{C}^{2})$ and range
$L^{2}(\mathbb{T}):=L^{2}(\mathbb{T},\mathbb{C}^{2})$, equipped with the
complex scalar product

$(f,g):=\frac{1}{2\pi}\int_{0}^{2\pi}\left(f_{1}\overline{g_{1}}+f_{2}\overline{g_{2}}\right)\,\text{d}x\,,\quad\forall
f=\begin{bmatrix}f_{1}\\\ f_{2}\end{bmatrix},\ \ g=\begin{bmatrix}g_{1}\\\
g_{2}\end{bmatrix}\in L^{2}(\mathbb{T},\mathbb{C}^{2})\,.$ (2.25)

We also denote $\|f\|^{2}=(f,f)$.

The complex operator $\mathcal{L}_{\mu,\epsilon}$ in (2.24) is _Hamiltonian_
and _Reversible_ , according to the following definition.

######  Definition 2.3.

(Complex Hamiltonian/Reversible operator) A complex operator
$\mathcal{L}:H^{1}(\mathbb{T},\mathbb{C}^{2})\to
L^{2}(\mathbb{T},\mathbb{C}^{2})$ is
($i$) _Hamiltonian_ , if $\mathcal{L}=\mathcal{J}\mathcal{B}$ where
$\mathcal{B}$ is a self-adjoint operator, namely
$\mathcal{B}=\mathcal{B}^{*}$, where $\mathcal{B}^{*}$ (with domain
$H^{1}(\mathbb{T})$) is the adjoint with respect to the complex scalar product
(2.25) of $L^{2}(\mathbb{T})$.
($ii$) _Reversible_ , if

$\mathcal{L}\circ\overline{\rho}=-\overline{\rho}\circ\mathcal{L}\,,$ (2.26)

where $\overline{\rho}$ is the complex involution (cfr. (2.3))

$\overline{\rho}\begin{bmatrix}\eta(x)\\\
\psi(x)\end{bmatrix}:=\begin{bmatrix}\overline{\eta}(-x)\\\
-\overline{\psi}(-x)\end{bmatrix}\,.$ (2.27)

The property (2.26) for $\mathcal{L}_{\mu,\epsilon}$ follows because
$\mathcal{L}_{\epsilon}$ is a real operator which is reversible with respect
to the involution $\rho$ in (2.3). Equivalently, since
$\mathcal{J}\circ\overline{\rho}=-\overline{\rho}\circ\mathcal{J}$, the self-
adjoint operator $\mathcal{B}_{\mu,\epsilon}$ is _reversibility-preserving_ ,
i.e.

$\mathcal{B}_{\mu,\epsilon}\circ\overline{\rho}=\overline{\rho}\circ\mathcal{B}_{\mu,\epsilon}\,.$
(2.28)

In addition
$(\mu,\epsilon)\to\mathcal{L}_{\mu,\epsilon}\in\mathcal{L}(H^{1}(\mathbb{T}),L^{2}(\mathbb{T}))$
is analytic, since the functions $\epsilon\mapsto a_{\epsilon}$,
$p_{\epsilon}$ defined in (2.19) are analytic as maps $B(\epsilon_{0})\to
H^{1}(\mathbb{T})$ and ${\mathcal{L}}_{\mu,\epsilon}$ is analytic with respect
to $\mu$, since, for any $\mu\in[-\frac{1}{2},\frac{1}{2})$,

$|D+\mu|\tanh\big{(}(\mathtt{h}+\mathtt{f}_{\epsilon})|D+\mu|\big{)}=(D+\mu)\tanh\big{(}(\mathtt{h}+\mathtt{f}_{\epsilon})(D+\mu)\big{)}\,.$
(2.29)

We also note that (see [35, Section 5.1])

$|D+\mu|=|D|+\mu(\operatorname*{sgn}(D)+\Pi_{0})\,,\quad\forall\mu>0\,,$
(2.30)

where $\operatorname*{sgn}(D)$ is the Fourier multiplier operator, acting on
$2\pi$-periodic functions, with symbol

$\operatorname*{sgn}(k):=1\ \forall
k>0\,,\quad\operatorname*{sgn}(0):=0\,,\quad\operatorname*{sgn}(k):=-1\
\forall k<0\,,$ (2.31)

and $\Pi_{0}$ is the projector operator on the zero mode,
$\Pi_{0}f(x):=\frac{1}{2\pi}\int_{\mathbb{T}}f(x)\mathrm{d}x.$

######  Remark 2.4.

If $(\eta(x),\psi(x),c)$ solve the traveling wave equations (2.5) then the
rescaled functions
$(\widetilde{\eta}(x),\widetilde{\psi}(x),\widetilde{c}):=(\eta(x),\sqrt{g}\psi(x),\sqrt{g}c)$
solve the same equations with gravity constant $g$ instead of $1$. The
eigenvalues of the corresponding linearized operators (2.9) and (2.24) for a
general gravity $g$ are those of the $g=1$ case multiplied by $\sqrt{g}$.

Our aim is to prove the existence of eigenvalues of
$\mathcal{L}_{\mu,\epsilon}$ in (2.24) with non zero real part. We remark that
the Hamiltonian structure of $\mathcal{L}_{\mu,\epsilon}$ implies that
eigenvalues with non zero real part may arise only from multiple eigenvalues
of $\mathcal{L}_{\mu,0}$ (“Krein criterion”), because if $\lambda$ is an
eigenvalue of $\mathcal{L}_{\mu,\epsilon}$ then also $-\overline{\lambda}$ is,
and the total algebraic multiplicity of the eigenvalues is conserved under
small perturbation. We now describe the spectrum of $\mathcal{L}_{\mu,0}$.
The spectrum of $\mathcal{L}_{\mu,0}$. The spectrum of the Fourier multiplier
matrix operator

$\mathcal{L}_{\mu,0}=\begin{bmatrix}{\mathtt{c}}_{\mathtt{h}}(\partial_{x}+\mathrm{i}\,\mu)&|D+\mu|\,\tanh\big{(}\mathtt{h}|D+\mu|\big{)}\\\
-1&{\mathtt{c}}_{\mathtt{h}}(\partial_{x}+\mathrm{i}\,\mu)\end{bmatrix}$
(2.32)

consists of the purely imaginary eigenvalues
$\\{\lambda_{k}^{\pm}(\mu)\;,\;k\in\mathbb{Z}\\}$, where

$\lambda_{k}^{\pm}(\mu):=\mathrm{i}\,\big{(}{\mathtt{c}}_{\mathtt{h}}(\pm
k+\mu)\mp\sqrt{|k\pm\mu|\tanh(\mathtt{h}|k\pm\mu|)}\big{)}\,.$ (2.33)

For $\mu=0$ the real operator $\mathcal{L}_{0,0}$ possesses the eigenvalue $0$
with algebraic multiplicity $4$,

$\lambda_{0}^{+}(0)=\lambda_{0}^{-}(0)=\lambda_{1}^{+}(0)=\lambda_{1}^{-}(0)=0\,,$

and geometric multiplicity $3$. A real basis of the Kernel of
$\mathcal{L}_{0,0}$ is

$\displaystyle
f_{1}^{+}:=\begin{bmatrix}{\mathtt{c}}_{\mathtt{h}}^{1/2}\cos(x)\\\
{\mathtt{c}}_{\mathtt{h}}^{-1/2}\sin(x)\end{bmatrix},\quad
f_{1}^{-}:=\begin{bmatrix}-{\mathtt{c}}_{\mathtt{h}}^{1/2}\sin(x)\\\
{\mathtt{c}}_{\mathtt{h}}^{-1/2}\cos(x)\end{bmatrix},\qquad
f_{0}^{-}:=\begin{bmatrix}0\\\ 1\end{bmatrix}\,,$ (2.34)

together with the generalized eigenvector

$\displaystyle f_{0}^{+}:=\begin{bmatrix}1\\\
0\end{bmatrix},\qquad\mathcal{L}_{0,0}f_{0}^{+}=-f_{0}^{-}\,.$ (2.35)

Furthermore $0$ is an isolated eigenvalue for $\mathcal{L}_{0,0}$, namely the
spectrum $\sigma\left(\mathcal{L}_{0,0}\right)$ decomposes in two separated
parts

$\sigma\left(\mathcal{L}_{0,0}\right)=\sigma^{\prime}\left(\mathcal{L}_{0,0}\right)\cup\sigma^{\prime\prime}\left(\mathcal{L}_{0,0}\right)\quad\text{where}\quad\sigma^{\prime}(\mathcal{L}_{0,0}):=\\{0\\}$
(2.36)

and

$\sigma^{\prime\prime}(\mathcal{L}_{0,0}):=\big{\\{}\lambda_{k}^{\sigma}(0),\
k=0,1\,,\sigma=\pm\big{\\}}\,.$

We shall also use that, as proved in Theorem 4.1 in [35], the operator
${\mathcal{L}}_{0,\epsilon}$ possesses, for any sufficiently small
$\epsilon\neq 0$, the eigenvalue $0$ with a four dimensional generalized
Kernel, spanned by $\epsilon$-dependent vectors
$U_{1},\tilde{U}_{2},U_{3},U_{4}$ satisfying, for some real constant
$\alpha_{\epsilon},\beta_{\epsilon}$,

${\mathcal{L}}_{0,\epsilon}U_{1}=0\,,\ \
{\mathcal{L}}_{0,\epsilon}\tilde{U}_{2}=0\,,\ \
{\mathcal{L}}_{0,\epsilon}U_{3}=\alpha_{\epsilon}\,\tilde{U}_{2}\,,\ \
{\mathcal{L}}_{0,\epsilon}U_{4}=-U_{1}-\beta_{\epsilon}\tilde{U}_{2}\,,\quad
U_{1}:=\begin{bmatrix}0\\\ 1\end{bmatrix}\,.$ (2.37)

By Kato’s perturbation theory (see Lemma 3.1 below) for any $\mu,\epsilon\neq
0$ sufficiently small, the perturbed spectrum
$\sigma\left(\mathcal{L}_{\mu,\epsilon}\right)$ admits a disjoint
decomposition as

$\sigma\left(\mathcal{L}_{\mu,\epsilon}\right)=\sigma^{\prime}\left(\mathcal{L}_{\mu,\epsilon}\right)\cup\sigma^{\prime\prime}\left(\mathcal{L}_{\mu,\epsilon}\right)\,,$
(2.38)

where $\sigma^{\prime}\left(\mathcal{L}_{\mu,\epsilon}\right)$ consists of 4
eigenvalues close to 0. We denote by $\mathcal{V}_{\mu,\epsilon}$ the spectral
subspace associated with
$\sigma^{\prime}\left(\mathcal{L}_{\mu,\epsilon}\right)$, which has dimension
4 and it is invariant by $\mathcal{L}_{\mu,\epsilon}$. Our goal is to prove
that, for $\epsilon$ small, for values of the Floquet exponent $\mu$ in an
interval of order $\epsilon$, the $4\times 4$ matrix which represents the
operator
$\mathcal{L}_{\mu,\epsilon}:\mathcal{V}_{\mu,\epsilon}\to\mathcal{V}_{\mu,\epsilon}$
possesses a pair of eigenvalues close to zero with opposite non zero real
parts.

Before stating our main result, let us introduce a notation we shall use
through all the paper:

  * $\bullet$

Notation: we denote by
$\mathcal{O}(\mu^{m_{1}}\epsilon^{n_{1}},\dots,\mu^{m_{p}}\epsilon^{n_{p}})$,
$m_{j},n_{j}\in\mathbb{N}$ (for us $\mathbb{N}:=\\{1,2,\dots\\}$), analytic
functions of $(\mu,\epsilon)$ with values in a Banach space $X$ which satisfy,
for some $C>0$ uniform for $\mathtt{h}$ in any compact set of $(0,+\infty)$,
the bound $\|\mathcal{O}(\mu^{m_{j}}\epsilon^{n_{j}})\|_{X}\leq
C\sum_{j=1}^{p}|\mu|^{m_{j}}|\epsilon|^{n_{j}}$ for small values of
$(\mu,\epsilon)$. Similarly we denote
$r_{k}(\mu^{m_{1}}\epsilon^{n_{1}},\dots,\mu^{m_{p}}\epsilon^{n_{p}})$ scalar
functions
$\mathcal{O}(\mu^{m_{1}}\epsilon^{n_{1}},\dots,\mu^{m_{p}}\epsilon^{n_{p}})$
which are also real analytic.

Our complete spectral result is the following:

######  Theorem 2.5.

(Complete Benjamin-Feir spectrum) There exist $\epsilon_{0},\mu_{0}>0$,
uniformly for the depth $\mathtt{h}$ in any compact set of $(0,+\infty)$, such
that, for any $0\,<\,\mu<\mu_{0}$ and $0\leq\epsilon<\epsilon_{0}$, the
operator
$\mathcal{L}_{\mu,\epsilon}:\mathcal{V}_{\mu,\epsilon}\to\mathcal{V}_{\mu,\epsilon}$
can be represented by a $4\times 4$ matrix of the form

$\begin{pmatrix}\mathtt{U}&\vline&0\\\ \hline\cr
0&\vline&\mathtt{S}\end{pmatrix},$ (2.39)

where $\mathtt{U}$ and $\mathtt{S}$ are $2\times 2$ matrices, with identical
diagonal entries each, of the form

$\displaystyle\mathtt{U}={\begin{pmatrix}\mathrm{i}\,\big{(}({\mathtt{c}}_{\mathtt{h}}-\tfrac{1}{2}\mathtt{e}_{12})\mu+r_{2}(\mu\epsilon^{2},\mu^{2}\epsilon,\mu^{3})\big{)}&-\mathtt{e}_{22}\frac{\mu}{8}(1+r_{5}(\epsilon,\mu))\\\
-\mu\epsilon^{2}\mathtt{e}_{\scriptscriptstyle{\textsc{WB}}}+r_{1}^{\prime}(\mu\epsilon^{3},\mu^{2}\epsilon^{2})+\mathtt{e}_{22}\frac{\mu^{3}}{8}(1+r_{1}^{\prime\prime}(\epsilon,\mu))&\mathrm{i}\,\big{(}({\mathtt{c}}_{\mathtt{h}}-\tfrac{1}{2}\mathtt{e}_{12})\mu+r_{2}(\mu\epsilon^{2},\mu^{2}\epsilon,\mu^{3})\big{)}\end{pmatrix}}\,,$
$\displaystyle\mathtt{S}=\begin{pmatrix}\mathrm{i}\,{\mathtt{c}}_{\mathtt{h}}\mu+\mathrm{i}\,{r_{9}(\mu\epsilon^{2},\mu^{2}\epsilon)}&\tanh(\mathtt{h}\mu)+{r_{10}(\mu\epsilon)}\\\
-\mu+{r_{8}(\mu\epsilon^{2},\mu^{3}\epsilon)}&\mathrm{i}\,{\mathtt{c}}_{\mathtt{h}}\mu+\mathrm{i}\,{r_{9}(\mu\epsilon^{2},\mu^{2}\epsilon)}\end{pmatrix}\,,$
(2.40)

where $\mathtt{e}_{\scriptscriptstyle{\textsc{WB}}}$,
$\mathtt{e}_{12},\mathtt{e}_{22}$ are defined in (1.1), (1.2), (1.3). The
eigenvalues of $\mathtt{U}$ have the form

$\displaystyle\lambda_{1}^{\pm}(\mu,\epsilon)$
$\displaystyle=\mathrm{i}\,\frac{1}{2}\breve{\mathtt{c}}_{\mathtt{h}}\mu+\mathrm{i}\,r_{2}(\mu\epsilon^{2},\mu^{2}\epsilon,\mu^{3})\pm\tfrac{1}{8}\mu\sqrt{\mathtt{e}_{22}(\mathtt{h})(1+r_{5}(\epsilon,\mu))}\sqrt{\Delta_{\scriptscriptstyle{\textsc{BF}}}(\mathtt{h};\mu,\epsilon)}\,,$
(2.41)

where
$\breve{\mathtt{c}}_{\mathtt{h}}:=2{\mathtt{c}}_{\mathtt{h}}-\mathtt{e}_{12}(\mathtt{h})$
and $\Delta_{\scriptscriptstyle{\textsc{BF}}}(\mathtt{h};\mu,\epsilon)$ is the
Benjamin-Feir discriminant function (1.6) (with
$r_{1}(\epsilon^{3},\mu\epsilon^{2}):=-8r_{1}^{\prime}(\epsilon^{3},\mu\epsilon^{2})$).
As $\mathtt{e}_{22}(\mathtt{h})>0$, they have non-zero real part if and only
if $\Delta_{\scriptscriptstyle{\textsc{BF}}}(\mathtt{h};\mu,\epsilon)>0$.

The eigenvalues of the matrix $\mathtt{S}$ are a pair of purely imaginary
eigenvalues of the form

$\lambda_{0}^{\pm}(\mu,\epsilon)=\mathrm{i}\,{\mathtt{c}}_{\mathtt{h}}\mu\big{(}1+{r_{9}(\epsilon^{2},\mu\epsilon)\big{)}}\mp\mathrm{i}\,\sqrt{\mu\tanh(\mathtt{h}\mu)}\big{(}1+{r(\epsilon)}\big{)}\,.$
(2.42)

For $\epsilon=0$ the eigenvalues
$\lambda_{1}^{\pm}(\mu,0),\lambda_{0}^{\pm}(\mu,0)$ coincide with those in
(2.33).

######  Remark 2.6.

At $\epsilon=0$, the eigenvalues in (2.41) have the Taylor expansion

$\lambda^{\pm}_{1}(\mu,0)=\mathrm{i}\,({\mathtt{c}}_{\mathtt{h}}-\frac{1}{2}\mathtt{e}_{12}(\mathtt{h}))\mu\pm\mathrm{i}\,\frac{\mathtt{e}_{22}(\mathtt{h})}{8}\mu^{2}+\mathcal{O}(\mu^{3})\,,$

which coincides with the one of $\lambda^{\pm}_{1}(\mu)$ in (2.33), in view of
the coefficients $\mathtt{e}_{12}(\mathtt{h})$ and
$\mathtt{e}_{22}(\mathtt{h})$ defined in (1.2), (1.3).

We conclude this section describing in detail our approach.
Ideas and scheme of proof. The proof follows the general ideas of the
infinitely deep water case [6], although important differences arise in finite
depth and require a different approach. The first step is to exploit Kato’s
theory to prolong the unperturbed symplectic basis
$\\{f_{1}^{\pm},f_{0}^{\pm}\\}$ of $\mathcal{V}_{0,0}$ in (2.34)-(2.35) into a
symplectic basis of the spectral subspace $\mathcal{V}_{\mu,\epsilon}$
associated with $\sigma^{\prime}\left(\mathcal{L}_{\mu,\epsilon}\right)$ in
(2.38), depending analytically on $\mu,\epsilon$. The transformation operator
$U_{\mu,\epsilon}$ in Lemma 3.1 is symplectic, analytic in $\mu,\epsilon$, and
maps isomorphically $\mathcal{V}_{0,0}$ into $\mathcal{V}_{\mu,\epsilon}$. The
vectors $f^{\sigma}_{k}(\mu,\epsilon):=U_{\mu,\epsilon}f_{k}^{\sigma}$,
$k=0,1$, $\sigma=\pm$, are the required symplectic basis of the symplectic
subspace $\mathcal{V}_{\mu,\epsilon}$. Its expansion in $\mu,\epsilon$ is
provided in Lemma 4.2. This procedure reduces our spectral problem to
determine the eigenvalues of the $4\times 4$ Hamiltonian and reversible matrix
$\mathtt{L}_{\mu,\epsilon}$ (cfr. Lemma 3.4), representing the action of the
operator $\mathcal{L}_{\mu,\epsilon}-\mathrm{i}\,{\mathtt{c}}_{\mathtt{h}}\mu$
on the basis $\\{f_{k}^{\sigma}(\mu,\epsilon)\\}$. In Proposition 4.3 we prove
that

$\mathtt{L}_{\mu,\epsilon}=\mathtt{J}_{4}\begin{pmatrix}E&F\\\
F^{*}&G\end{pmatrix}=\begin{pmatrix}\mathtt{J}_{2}E&\mathtt{J}_{2}F\\\
\mathtt{J}_{2}F^{*}&\mathtt{J}_{2}G\end{pmatrix}\qquad\text{where}\qquad\mathtt{J}_{4}=\begin{pmatrix}\mathtt{J}_{2}&0\\\
0&\mathtt{J}_{2}\end{pmatrix}\,,\ \ \mathtt{J}_{2}=\begin{pmatrix}0&1\\\
-1&0\end{pmatrix}\,,$ (2.43)

and the $2\times 2$ matrices $E,G,F$ have the expansions (4.10)-(4.12). In
finite depth this computation is much more involved than in deep water, as we
need to track the exact dependence of the matrix entries with respect to
$\mathtt{h}$. In particular the matrix $E$ is

$E=\begin{pmatrix}\mathtt{e}_{11}\epsilon^{2}(1+r_{1}^{\prime}(\epsilon,\mu\epsilon))-\mathtt{e}_{22}\frac{\mu^{2}}{8}(1+r_{1}^{\prime\prime}(\epsilon,\mu))&\mathrm{i}\,\big{(}\frac{1}{2}\mathtt{e}_{12}\mu+r_{2}(\mu\epsilon^{2},\mu^{2}\epsilon,\mu^{3})\big{)}\\\
-\mathrm{i}\,\big{(}\frac{1}{2}\mathtt{e}_{12}\mu+r_{2}(\mu\epsilon^{2},\mu^{2}\epsilon,\mu^{3})\big{)}&-\mathtt{e}_{22}\frac{\mu^{2}}{8}(1+r_{5}(\epsilon,\mu))\end{pmatrix}$
(2.44)

where the coefficients $\mathtt{e}_{11}$ and $\mathtt{e}_{22}$, defined in
(4.13) and (1.3), are strictly positive for any value of $\mathtt{h}>0$. Thus
the submatrix $\mathtt{J}_{2}E$ has a pair of eigenvalues with nonzero real
part, for any value of $\mathtt{h}>0$, provided
$0<\mu<\overline{\mu}(\epsilon)\sim\epsilon$. On the other hand, it has to
come out that the complete $4\times 4$ matrix $\mathtt{L}_{\mu,\epsilon}$
possesses unstable eigenvalues if and only if the depth exceeds the celebrated
Whitham-Benjamin threshold $\mathtt{h}_{\scriptscriptstyle{\textsc{WB}}}\sim
1.363\ldots$. Indeed the correct eigenvalues of $\mathtt{L}_{\mu,\epsilon}$
are not a small perturbation of those of
$\footnotesize{\begin{pmatrix}\mathtt{J}_{2}E&0\\\
0&\mathtt{J}_{2}G\end{pmatrix}}$ and will emerge only after one non-
perturbative step of block diagonalization. This was not the case in the
infinitely deep water case [6], where at this stage the corresponding
submatrix $\mathtt{J}_{2}E$ had already the correct Benjamin-Feir eigenvalues,
and we only had to check their stability under perturbation.

######  Remark 2.7.

We also stress that (2.44) is not a simple Taylor expansion in $\mu,\epsilon$:
note that the $(2,2)$-entry in (2.44) does not have any term
$\mathcal{O}(\epsilon^{m})$ nor $\mathcal{O}(\mu\epsilon^{m})$ for any
$m\in\mathbb{N}$. These terms would be dangerous because they could change the
sign of the entry $(2,2)$ which instead, in (2.44), is always negative (recall
that $\mathtt{e}_{22}(\mathtt{h})>0$). We prove the absence of terms
$\epsilon^{m}$, $m\in\mathbb{N}$, fully exploiting (as in [6]) the structural
information (2.37) concerning the four dimensional generalized Kernel of the
operator $\mathcal{L}_{0,\epsilon}$ for any $\epsilon>0$, see Lemma 4.4.
Moreover, in finite depth it turns out that there are no terms of order
$\mu\epsilon^{m}$, $m\in\mathbb{N}$, which instead are present in deep water,
and were eliminated in [6] via a further change of basis. We also note that
the $2\times 2$ matrices $\mathtt{J}_{2}E$ and $\mathtt{J}_{2}G$ in (2.43)
have both eigenvalues of size $\mathcal{O}(\mu)$. As already mentioned in the
introduction, this is a crucial difference with the deep water case, where the
eigenvalues of $\mathtt{J}_{2}G$ have the much larger size
$\mathcal{O}(\sqrt{\mu})$.

In order to determine the correct spectrum of the matrix
$\mathtt{L}_{\mu,\epsilon}$ in (2.43), we perform a block diagonalization of
$\mathtt{L}_{\mu,\epsilon}$ to eliminate the coupling term $\mathtt{J}_{2}F$
(which has size $\epsilon$, see (4.12)). We proceed, in Section 5, in three
steps:
1\. _Symplectic rescaling_. We first perform a symplectic rescaling which is
singular at $\mu=0$, see Lemma 5.1, obtaining the matrix
$\mathtt{L}_{\mu,\epsilon}^{(1)}$. The effects are twofold: (i) the diagonal
elements of

$E^{(1)}=\begin{pmatrix}\mathtt{e}_{11}\mu\epsilon^{2}(1+r_{1}^{\prime}(\epsilon,\mu\epsilon))-\mathtt{e}_{22}\frac{\mu^{3}}{8}(1+r_{1}^{\prime\prime}(\epsilon,\mu))&\mathrm{i}\,\big{(}\frac{1}{2}\mathtt{e}_{12}\mu+r_{2}(\mu\epsilon^{2},\mu^{2}\epsilon,\mu^{3})\big{)}\\\
-\mathrm{i}\,\big{(}\frac{1}{2}\mathtt{e}_{12}\mu+r_{2}(\mu\epsilon^{2},\mu^{2}\epsilon,\mu^{3})\big{)}&-\mathtt{e}_{22}\frac{\mu}{8}(1+r_{5}(\epsilon,\mu))\end{pmatrix}$
(2.45)

have size $\mathcal{O}(\mu)$, as well as those of $G^{(1)}$, and (ii) the
matrix $F^{(1)}$ has the smaller size ${\mathcal{O}(\mu\epsilon)}$.
2\. _Non-perturbative step of block-diagonalization (Section 5.1)_. Inspired
by KAM theory, we perform one step of block decoupling to decrease further the
size of the off-diagonal blocks. This step modifies the matrix
$\mathtt{J}_{2}E^{(1)}$ in a substantial way, by a term
$\mathcal{O}(\mu\epsilon^{2})$. Let us explain better this step. In order to
reduce the size of $\mathtt{J}_{2}F^{(1)}$, we conjugate
$\mathtt{L}_{\mu,\epsilon}^{(1)}$ by the symplectic matrix $\exp(S^{(1)})$,
where $S^{(1)}$ is a Hamiltonian matrix with the same form of
$\mathtt{J}_{2}F^{(1)}$, see (5.9). The transformed matrix
$\mathtt{L}_{\mu,\epsilon}^{(2)}=\exp(S^{(1)})\mathtt{L}_{\mu,\epsilon}^{(1)}\exp(-S^{(1)})$
has the Lie expansion222recall that $\exp(S)L\exp(-S)=\sum_{n\geq
0}\frac{1}{n!}\textup{ad}_{S}^{n}(L)$, where $\textup{ad}_{S}^{0}(L):=L$,
$\textup{ad}_{S}^{n}(L)=[S,\textup{ad}_{S}^{n-1}(L)]$ for $n\geq 1$.

$\displaystyle\mathtt{L}_{\mu,\epsilon}^{(2)}$
$\displaystyle=\begin{pmatrix}\mathtt{J}_{2}E^{(1)}&0\\\
0&\mathtt{J}_{2}G^{(1)}\end{pmatrix}$ (2.46)
$\displaystyle\quad+\begin{pmatrix}0&\mathtt{J}_{2}F^{(1)}\\\
\mathtt{J}_{2}[F^{(1)}]^{*}&0\end{pmatrix}+\left[S^{(1)}\,,\,\begin{pmatrix}\mathtt{J}_{2}E^{(1)}&0\\\
0&\mathtt{J}_{2}G^{(1)}\end{pmatrix}\right]$
$\displaystyle\quad+\frac{1}{2}\Big{[}S^{(1)},\Big{[}S^{(1)},\begin{pmatrix}\mathtt{J}_{2}E^{(1)}&0\\\
0&\mathtt{J}_{2}G^{(1)}\end{pmatrix}\Big{]}\Big{]}+\Big{[}S^{(1)},\begin{pmatrix}0&\mathtt{J}_{2}F^{(1)}\\\
\mathtt{J}_{2}[F^{(1)}]^{*}&0\end{pmatrix}\Big{]}+\mbox{h.o.t.}$

The first line in the right hand side of (2.46) is the previous block-diagonal
matrix, the second line of (2.46) is a purely off-diagonal matrix and the
third line is the sum of two block-diagonal matrices and “h.o.t.” collects
terms of much smaller size. $S^{(1)}$ is determined in such a way that the
second line of (2.46) vanishes, and therefore the remaining off-diagonal
matrices (appearing in the h.o.t. remainder) are smaller in size. Unlike the
infinitely deep water case [6], the block-diagonal corrections in the third
line of (2.46) are not perturbative and they modify substantially the block-
diagonal part. More precisely we obtain that $\mathtt{L}_{\mu,\epsilon}^{(2)}$
has the form (5.10) with

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

Note the appearance of the Whitham-Benjamin function
$\mathtt{e}_{\scriptscriptstyle{\textsc{WB}}}(\mathtt{h})$ in the (1,1)-entry
of $E^{(2)}$, which changes sign at the critical depth
$\mathtt{h}_{\scriptscriptstyle{\textsc{WB}}}$, see Figure 1, unlike the
coefficient $\mathtt{e}_{11}(\mathtt{h})>0$ in (2.45). If
$\mathtt{e}_{\scriptscriptstyle{\textsc{WB}}}(\mathtt{h})>0$ and $\epsilon$
and $\mu$ are sufficiently small, the matrix $\mathtt{J}_{2}E^{(2)}$ has
eigenvalues with non-zero real part (recall that
$\mathtt{e}_{22}(\mathtt{h})>0$ for any $\mathtt{h}$). On the contrary, if
$\mathtt{e}_{\scriptscriptstyle{\textsc{WB}}}(\mathtt{h})<0$, then the
eigenvalues of $\mathtt{J}_{2}E^{(2)}$ lay on the imaginary axis.
3\. _Complete block-diagonalization (Section 5.2)_. In Lemma 5.9 we completely
block-diagonalize $\mathtt{L}^{(2)}_{\mu,\epsilon}$ by means of a standard
implicit function theorem. By this procedure the original matrix
$\mathtt{L}_{\mu,\epsilon}$ is conjugated into the Hamiltonian and reversible
matrix (2.39), proving Theorem 2.5.

##  3 Perturbative approach to the separated eigenvalues

We apply Kato’s similarity transformation theory [26, I-§4-6, II-§4] to study
the splitting of the eigenvalues of $\mathcal{L}_{\mu,\epsilon}$ close to $0$
for small values of $\mu$ and $\epsilon$, following [6]. First of all, it is
convenient to decompose the operator $\mathcal{L}_{\mu,\epsilon}$ in (2.24) as

$\mathcal{L}_{\mu,\epsilon}=\mathrm{i}\,{\mathtt{c}}_{\mathtt{h}}\mu+\mathscr{L}_{\mu,\epsilon}\,,\qquad\mu>0\,,$
(3.1)

where, using also (2.30),

$\mathscr{L}_{\mu,\epsilon}:=\begin{bmatrix}\partial_{x}\circ({\mathtt{c}}_{\mathtt{h}}+p_{\epsilon}(x))+\mathrm{i}\,\mu\,p_{\epsilon}(x)&|D+\mu|\,\tanh\big{(}(\mathtt{h}+\mathtt{f}_{\epsilon})|D+\mu|\big{)}\\\
-(1+a_{\epsilon}(x))&({\mathtt{c}}_{\mathtt{h}}+p_{\epsilon}(x))\partial_{x}+\mathrm{i}\,\mu\,p_{\epsilon}(x)\end{bmatrix}\,.$
(3.2)

The operator $\mathscr{L}_{\mu,\epsilon}$ is still Hamiltonian, having the
form

$\mathscr{L}_{\mu,\epsilon}=\mathcal{J}\,{\mathcal{B}}_{\mu,\epsilon}\,,\quad{\mathcal{B}}_{\mu,\epsilon}:=\begin{bmatrix}1+a_{\epsilon}(x)&-({\mathtt{c}}_{\mathtt{h}}+p_{\epsilon}(x))\partial_{x}-\mathrm{i}\,\mu\,p_{\epsilon}(x)\\\
\partial_{x}\circ({\mathtt{c}}_{\mathtt{h}}+p_{\epsilon}(x))+\mathrm{i}\,\mu\,p_{\epsilon}(x)&|D+\mu|\,\tanh\big{(}(\mathtt{h}+\mathtt{f}_{\epsilon})|D+\mu|\big{)}\end{bmatrix}$
(3.3)

with ${\mathcal{B}}_{\mu,\epsilon}$ selfadjoint, and it is also reversible,
namely it satisfies, by (2.26),

$\mathscr{L}_{\mu,\epsilon}\circ\overline{\rho}=-\overline{\rho}\circ\mathscr{L}_{\mu,\epsilon}\,,\qquad\overline{\rho}\mbox{
defined in }\eqref{reversibilityappears}\,,$ (3.4)

whereas ${\mathcal{B}}_{\mu,\epsilon}$ is reversibility-preserving, i.e.
fulfills (2.28). Note also that ${\mathcal{B}}_{0,\epsilon}$ is a real
operator.

The scalar operator
$\mathrm{i}\,{\mathtt{c}}_{\mathtt{h}}\mu\equiv\mathrm{i}\,{\mathtt{c}}_{\mathtt{h}}\mu\,\text{Id}$
just translates the spectrum of $\mathscr{L}_{\mu,\epsilon}$ along the
imaginary axis of the quantity $\mathrm{i}\,{\mathtt{c}}_{\mathtt{h}}\mu$,
that is, in view of (3.1),
$\sigma({\mathcal{L}}_{\mu,\epsilon})=\mathrm{i}\,{\mathtt{c}}_{\mathtt{h}}\mu+\sigma(\mathscr{L}_{\mu,\epsilon})\,.$
Thus in the sequel we focus on studying the spectrum of
$\mathscr{L}_{\mu,\epsilon}$.

Note also that $\mathscr{L}_{0,\epsilon}=\mathcal{L}_{0,\epsilon}$ for any
$\epsilon\geq 0$. In particular $\mathscr{L}_{0,0}$ has zero as isolated
eigenvalue with algebraic multiplicity 4, geometric multiplicity 3 and
generalized kernel spanned by the vectors
$\\{f^{+}_{1},f^{-}_{1},f^{+}_{0},f^{-}_{0}\\}$ in (2.34), (2.35). Furthermore
its spectrum is separated as in (2.36). For any $\epsilon\neq 0$ small,
$\mathscr{L}_{0,\epsilon}$ has zero as isolated eigenvalue with geometric
multiplicity $2$, and two generalized eigenvectors satisfying (2.37).

We remark that, in view of (2.30), the operator $\mathscr{L}_{\mu,\epsilon}$
is analytic with respect to $\mu$. The operator
$\mathscr{L}_{\mu,\epsilon}:Y\subset X\to X$ has domain
$Y:=H^{1}(\mathbb{T}):=H^{1}(\mathbb{T},\mathbb{C}^{2})$ and range
$X:=L^{2}(\mathbb{T}):=L^{2}(\mathbb{T},\mathbb{C}^{2})$.

######  Lemma 3.1.

(Kato theory for separated eigenvalues) Let $\Gamma$ be a closed,
counterclockwise-oriented curve around $0$ in the complex plane separating
$\sigma^{\prime}\left(\mathscr{L}_{0,0}\right)=\\{0\\}$ and the other part of
the spectrum $\sigma^{\prime\prime}\left(\mathscr{L}_{0,0}\right)$ in (2.36).
There exist $\epsilon_{0},\mu_{0}>0$ such that for any $(\mu,\epsilon)\in
B(\mu_{0})\times B(\epsilon_{0})$ the following statements hold:
1\. The curve $\Gamma$ belongs to the resolvent set of the operator
$\mathscr{L}_{\mu,\epsilon}:Y\subset X\to X$ defined in (3.2).
2\. The operators

$P_{\mu,\epsilon}:=-\frac{1}{2\pi\mathrm{i}\,}\oint_{\Gamma}(\mathscr{L}_{\mu,\epsilon}-\lambda)^{-1}\mathrm{d}\lambda:X\to
Y$ (3.5)

are well defined projectors commuting with $\mathscr{L}_{\mu,\epsilon}$, i.e.
$P_{\mu,\epsilon}^{2}=P_{\mu,\epsilon}$ and
$P_{\mu,\epsilon}\mathscr{L}_{\mu,\epsilon}=\mathscr{L}_{\mu,\epsilon}P_{\mu,\epsilon}$.
The map $(\mu,\epsilon)\mapsto P_{\mu,\epsilon}$ is analytic from
$B({\mu_{0}})\times B({\epsilon_{0}})$ to $\mathcal{L}(X,Y)$.
3\. The domain $Y$ of the operator $\mathscr{L}_{\mu,\epsilon}$ decomposes as
the direct sum

$Y=\mathcal{V}_{\mu,\epsilon}\oplus\text{Ker}(P_{\mu,\epsilon})\,,\quad\mathcal{V}_{\mu,\epsilon}:=\text{Rg}(P_{\mu,\epsilon})=\text{Ker}(\mathrm{Id}-P_{\mu,\epsilon})\,,$

of closed invariant subspaces, namely
$\mathscr{L}_{\mu,\epsilon}:\mathcal{V}_{\mu,\epsilon}\to\mathcal{V}_{\mu,\epsilon}$,
$\mathscr{L}_{\mu,\epsilon}:\text{Ker}(P_{\mu,\epsilon})\to\text{Ker}(P_{\mu,\epsilon})$.
Moreover

$\displaystyle\sigma(\mathscr{L}_{\mu,\epsilon})\cap\\{z\in\mathbb{C}\mbox{
inside
}\Gamma\\}=\sigma(\mathscr{L}_{\mu,\epsilon}|_{{\mathcal{V}}_{\mu,\epsilon}})=\sigma^{\prime}(\mathscr{L}_{\mu,\epsilon}),$
$\displaystyle\sigma(\mathscr{L}_{\mu,\epsilon})\cap\\{z\in\mathbb{C}\mbox{
outside
}\Gamma\\}=\sigma(\mathscr{L}_{\mu,\epsilon}|_{Ker(P_{\mu,\epsilon})})=\sigma^{\prime\prime}(\mathscr{L}_{\mu,\epsilon})\
,$

proving the “semicontinuity property” (2.38) of separated parts of the
spectrum.
4\. The projectors $P_{\mu,\epsilon}$ are similar one to each other: the
transformation operators333 The operator $(\mathrm{Id}-R)^{-\frac{1}{2}}$ is
defined, for any operator $R$ satisfying $\|R\|_{{\mathcal{L}}(Y)}<1$, by the
power series
$\displaystyle(\mathrm{Id}-R)^{-\frac{1}{2}}:=\sum_{k=0}^{\infty}{-1/2\choose
k}(-R)^{k}=\mathrm{Id}+\frac{1}{2}R+\frac{3}{8}R^{2}+\mathcal{O}(R^{3})\,.$
(3.6)

$U_{\mu,\epsilon}:=\big{(}\mathrm{Id}-(P_{\mu,\epsilon}-P_{0,0})^{2}\big{)}^{-1/2}\big{[}P_{\mu,\epsilon}P_{0,0}+(\mathrm{Id}-P_{\mu,\epsilon})(\mathrm{Id}-P_{0,0})\big{]}$
(3.7)

are bounded and invertible in $Y$ and in $X$, with inverse

$U_{\mu,\epsilon}^{-1}=\big{[}P_{0,0}P_{\mu,\epsilon}+(\mathrm{Id}-P_{0,0})(\mathrm{Id}-P_{\mu,\epsilon})\big{]}\big{(}\mathrm{Id}-(P_{\mu,\epsilon}-P_{0,0})^{2}\big{)}^{-1/2}\,,$

and $U_{\mu,\epsilon}P_{0,0}U_{\mu,\epsilon}^{-1}=P_{\mu,\epsilon}$ as well as
$U_{\mu,\epsilon}^{-1}P_{\mu,\epsilon}U_{\mu,\epsilon}=P_{0,0}$.

The map $(\mu,\epsilon)\mapsto U_{\mu,\epsilon}$ is analytic from
$B(\mu_{0})\times B(\epsilon_{0})$ to $\mathcal{L}(Y)$.
5\. The subspaces $\mathcal{V}_{\mu,\epsilon}=\text{Rg}(P_{\mu,\epsilon})$ are
isomorphic one to each other:
$\mathcal{V}_{\mu,\epsilon}=U_{\mu,\epsilon}\mathcal{V}_{0,0}.$ In particular
$\dim\mathcal{V}_{\mu,\epsilon}=\dim\mathcal{V}_{0,0}=4$, for any
$(\mu,\epsilon)\in B(\mu_{0})\times B(\epsilon_{0})$.

###### Proof.

For any $\lambda\in\mathbb{C}$ we decompose
$\mathscr{L}_{\mu,\epsilon}-\lambda=\mathscr{L}_{0,0}-\lambda+{\mathcal{R}}_{\mu,\epsilon}$
where
$\footnotesize\mathscr{L}_{0,0}=\begin{bmatrix}{\mathtt{c}}_{\mathtt{h}}\partial_{x}&|D|\tanh(\mathtt{h}|D|)\\\
-1&{\mathtt{c}}_{\mathtt{h}}\partial_{x}\end{bmatrix}$ and

${\mathcal{R}}_{\mu,\epsilon}:=\mathscr{L}_{\mu,\epsilon}-\mathscr{L}_{0,0}=\begin{bmatrix}(\partial_{x}+\mathrm{i}\,\mu)p_{\epsilon}(x)&f_{\mu,\epsilon}(D)\\\
-a_{\epsilon}(x)&p_{\epsilon}(x)(\partial_{x}+\mathrm{i}\,\mu)\end{bmatrix}:Y\to
X\,,$

having used also (2.30) and setting

$f_{\mu,\epsilon}(D):=|D+\mu|\,\tanh\big{(}(\mathtt{h}+\mathtt{f}_{\epsilon})|D+\mu|\big{)}-|D|\tanh(\mathtt{h}|D|)\in\mathcal{L}(Y)\,,\
\ \ {\|f_{\mu,\epsilon}(D)\|}_{\mathcal{L}(Y)}=\mathcal{O}(\mu,\epsilon)\,.$

For any $\lambda\in\Gamma$, the operator $\mathscr{L}_{0,0}-\lambda$ is
invertible and its inverse is the Fourier multiplier matrix operator

$(\mathscr{L}_{0,0}-\lambda)^{-1}=\text{Op}\left(\frac{1}{(\mathrm{i}\,{\mathtt{c}}_{\mathtt{h}}k-\lambda)^{2}+|k|\tanh(\mathtt{h}|k|)}\begin{bmatrix}\mathrm{i}\,{\mathtt{c}}_{\mathtt{h}}k-\lambda&-|k|\tanh(\mathtt{h}|k|)\\\
1&\mathrm{i}\,{\mathtt{c}}_{\mathtt{h}}k-\lambda\end{bmatrix}\right):X\to
Y\,.$

Hence, for $|\epsilon|<\epsilon_{0}$ and $|\mu|<\mu_{0}$ small enough,
uniformly on the compact set $\Gamma$, the operator
$(\mathscr{L}_{0,0}-\lambda)^{-1}{\mathcal{R}}_{\mu,\epsilon}:Y\to Y$ is
bounded, with small operatorial norm. Then
$\mathscr{L}_{\mu,\epsilon}-\lambda$ is invertible by Neumann series and
$\Gamma$ belongs to the resolvent set of $\mathscr{L}_{\mu,\epsilon}$. The
remaining part of the proof follows exactly as in Lemma 3.1 in [6]. ∎

The Hamiltonian and reversible nature of the operator
$\mathscr{L}_{\mu,\epsilon}$, see (3.3) and (3.4), imply additional algebraic
properties for spectral projectors $P_{\mu,\epsilon}$ and the transformation
operators $U_{\mu,\epsilon}$. By Lemma 3.2 in [6] we have that:

######  Lemma 3.2.

For any $(\mu,\epsilon)\in B(\mu_{0})\times B(\epsilon_{0})$, the following
holds true:
($i$) The projectors $P_{\mu,\epsilon}$ defined in (3.5) are skew-Hamiltonian,
namely $\mathcal{J}P_{\mu,\epsilon}=P_{\mu,\epsilon}^{*}\mathcal{J}$, and
reversibility preserving, i.e.
$\overline{\rho}P_{\mu,\epsilon}=P_{\mu,\epsilon}\overline{\rho}$.
(ii) The transformation operators $U_{\mu,\epsilon}$ in (3.7) are symplectic,
namely $U_{\mu,\epsilon}^{*}\mathcal{J}U_{\mu,\epsilon}=\mathcal{J}$, and
reversibility preserving.
(iii) $P_{0,\epsilon}$ and $U_{0,\epsilon}$ are real operators, i.e.
$\overline{P_{0,\epsilon}}=P_{0,\epsilon}$ and
$\overline{U_{0,\epsilon}}=U_{0,\epsilon}$.

By the previous lemma, the linear involution $\overline{\rho}$ commutes with
the spectral projectors $P_{\mu,\epsilon}$ and then $\overline{\rho}$ leaves
invariant the subspace
$\mathcal{V}_{\mu,\epsilon}=\text{Rg}(P_{\mu,\epsilon})$.
Symplectic and reversible basis of $\mathcal{V}_{\mu,\epsilon}$. It is
convenient to represent the Hamiltonian and reversible operator
$\mathscr{L}_{\mu,\epsilon}:\mathcal{V}_{\mu,\epsilon}\to\mathcal{V}_{\mu,\epsilon}$
in a basis which is symplectic and reversible, according to the following
definition.

######  Definition 3.3.

(Symplectic and reversible basis) A basis
$\mathtt{F}:=\\{\mathtt{f}^{+}_{1},\,\mathtt{f}^{-}_{1},\,\mathtt{f}^{+}_{0},\,\mathtt{f}^{-}_{0}\\}$
of $\mathcal{V}_{\mu,\epsilon}$ is

  * •

_symplectic_ if, for any $k,k^{\prime}=0,1$,

$\left(\mathcal{J}\mathtt{f}_{k}^{-}\,,\,\mathtt{f}_{k}^{+}\right)=1\,,\ \
\big{(}\mathcal{J}\mathtt{f}_{k}^{\sigma},\mathtt{f}_{k}^{\sigma}\big{)}=0\,,\
\forall\sigma=\pm\,;\ \ \text{if}\ k\neq k^{\prime}\ \text{then}\
\big{(}\mathcal{J}\mathtt{f}_{k}^{\sigma},\mathtt{f}_{k^{\prime}}^{\sigma^{\prime}}\big{)}=0\,,\
\forall\sigma,\sigma^{\prime}=\pm\,.$ (3.8)

  * •

_reversible_ if

$\overline{\rho}\mathtt{f}^{+}_{1}=\mathtt{f}^{+}_{1},\quad\overline{\rho}\mathtt{f}^{-}_{1}=-\mathtt{f}^{-}_{1},\quad\overline{\rho}\mathtt{f}^{+}_{0}=\mathtt{f}^{+}_{0},\quad\overline{\rho}\mathtt{f}^{-}_{0}=-\mathtt{f}^{-}_{0},\quad\text{i.e.
}\overline{\rho}\mathtt{f}_{k}^{\sigma}=\sigma\mathtt{f}_{k}^{\sigma}\,,\
\forall\sigma=\pm,k=0,1\,.$ (3.9)

We use the following notation along the paper: we denote by $even(x)$ a real
$2\pi$-periodic function which is even in $x$, and by $odd(x)$ a real
$2\pi$-periodic function which is odd in $x$.

By the definition of the involution $\overline{\rho}$ in (2.27), the real and
imaginary parts of a reversible basis $\mathtt{F}=\\{\mathtt{f}^{\pm}_{k}\\}$,
$k=0,1$, enjoy the following parity properties (cfr. Lemma 3.4 in [6])

$\mathtt{f}_{k}^{+}(x)=\begin{bmatrix}even(x)+\mathrm{i}\,odd(x)\\\
odd(x)+\mathrm{i}\,even(x)\end{bmatrix},\quad\mathtt{f}_{k}^{-}(x)=\begin{bmatrix}odd(x)+\mathrm{i}\,even(x)\\\
even(x)+\mathrm{i}\,odd(x)\end{bmatrix}.$ (3.10)

By Lemmata 3.5 and 3.6 in [6] we have the following result.

######  Lemma 3.4.

The $4\times 4$ matrix that represents the Hamiltonian and reversible operator
$\mathscr{L}_{\mu,\epsilon}=\mathcal{J}{\mathcal{B}}_{\mu,\epsilon}:\mathcal{V}_{\mu,\epsilon}\to\mathcal{V}_{\mu,\epsilon}$
with respect to a symplectic and reversible basis
$\mathtt{F}=\\{\mathtt{f}_{1}^{+},\mathtt{f}_{1}^{-},\mathtt{f}_{0}^{+},\mathtt{f}_{0}^{-}\\}$
of $\mathcal{V}_{\mu,\epsilon}$ is

$\displaystyle\mathtt{J}_{4}\mathtt{B}_{\mu,\epsilon}\,,\quad\mathtt{J}_{4}:=\begin{pmatrix}\mathtt{J}_{2}&\vline&0\\\
\hline\cr
0&\vline&\mathtt{J}_{2}\end{pmatrix},\quad{\small\mathtt{J}_{2}:=\begin{pmatrix}0&1\\\
-1&0\end{pmatrix}},\quad\text{where
}\quad\mathtt{B}_{\mu,\epsilon}=\mathtt{B}_{\mu,\epsilon}^{*}$ (3.11)

is the self-adjoint matrix

$\mathtt{B}_{\mu,\epsilon}=\begin{pmatrix}\left({\mathcal{B}}_{\mu,\epsilon}\,\mathtt{f}^{+}_{{1}}\
,\mathtt{f}^{+}_{1}\right)&\left({\mathcal{B}}_{\mu,\epsilon}\,\mathtt{f}^{-}_{{1}}\
,\mathtt{f}^{+}_{1}\right)&\left({\mathcal{B}}_{\mu,\epsilon}\,\mathtt{f}^{+}_{{0}}\
,\mathtt{f}^{+}_{1}\right)&\left({\mathcal{B}}_{\mu,\epsilon}\,\mathtt{f}^{-}_{{0}}\
,\mathtt{f}^{+}_{1}\right)\\\
\left({\mathcal{B}}_{\mu,\epsilon}\,\mathtt{f}^{+}_{{1}}\
,\mathtt{f}^{-}_{1}\right)&\left({\mathcal{B}}_{\mu,\epsilon}\,\mathtt{f}^{-}_{{1}}\
,\mathtt{f}^{-}_{1}\right)&\left({\mathcal{B}}_{\mu,\epsilon}\,\mathtt{f}^{+}_{{0}}\
,\mathtt{f}^{-}_{1}\right)&\left({\mathcal{B}}_{\mu,\epsilon}\,\mathtt{f}^{-}_{{0}}\
,\mathtt{f}^{-}_{1}\right)\\\
\left({\mathcal{B}}_{\mu,\epsilon}\,\mathtt{f}^{+}_{{1}}\
,\mathtt{f}^{+}_{0}\right)&\left({\mathcal{B}}_{\mu,\epsilon}\,\mathtt{f}^{-}_{{1}}\
,\mathtt{f}^{+}_{0}\right)&\left({\mathcal{B}}_{\mu,\epsilon}\,\mathtt{f}^{+}_{{0}}\
,\mathtt{f}^{+}_{0}\right)&\left({\mathcal{B}}_{\mu,\epsilon}\,\mathtt{f}^{-}_{{0}}\
,\mathtt{f}^{+}_{0}\right)\\\
\left({\mathcal{B}}_{\mu,\epsilon}\,\mathtt{f}^{+}_{{1}}\
,\mathtt{f}^{-}_{0}\right)&\left({\mathcal{B}}_{\mu,\epsilon}\,\mathtt{f}^{-}_{{1}}\
,\mathtt{f}^{-}_{0}\right)&\left({\mathcal{B}}_{\mu,\epsilon}\,\mathtt{f}^{+}_{{0}}\
,\mathtt{f}^{-}_{0}\right)&\left({\mathcal{B}}_{\mu,\epsilon}\,\mathtt{f}^{-}_{{0}}\
,\mathtt{f}^{-}_{0}\right)\\\ \end{pmatrix}.$ (3.12)

The entries of the matrix $\mathtt{B}_{\mu,\epsilon}$ are alternatively real
or purely imaginary: for any $\sigma=\pm$, $k=0,1$,

$\left({\mathcal{B}}_{\mu,\epsilon}\,\mathtt{f}^{\sigma}_{k}\,,\,\mathtt{f}^{\sigma}_{k^{\prime}}\right)\text{
is
real},\qquad\left({\mathcal{B}}_{\mu,\epsilon}\,\mathtt{f}^{\sigma}_{k}\,,\,\mathtt{f}^{-\sigma}_{k^{\prime}}\right)\text{
is purely imaginary}\,.$ (3.13)

It is convenient to give a name to the matrices of the form obtained in Lemma
3.4.

######  Definition 3.5.

A $2n\times 2n$, $n=1,2,$ matrix of the form
$\mathtt{L}=\mathtt{J}_{2n}\mathtt{B}$ is
1\. _Hamiltonian_ if $\mathtt{B}$ is a self-adjoint matrix, i.e.
$\mathtt{B}=\mathtt{B}^{*}$;
2\. _Reversible_ if $\mathtt{B}$ is reversibility-preserving, i.e.
$\rho_{2n}\circ\mathtt{B}=\mathtt{B}\circ\rho_{2n}$, where

$\rho_{4}:=\begin{pmatrix}\rho_{2}&0\\\
0&\rho_{2}\end{pmatrix},\qquad\rho_{2}:=\begin{pmatrix}\mathfrak{c}&0\\\
0&-\mathfrak{c}\end{pmatrix},$ (3.14)

and $\mathfrak{c}:z\mapsto\overline{z}$ is the conjugation of the complex
plane. Equivalently, $\rho_{2n}\circ\mathtt{L}=-\mathtt{L}\circ\rho_{2n}$.

In the sequel we shall mainly deal with $4\times 4$ Hamiltonian and reversible
matrices. The transformations preserving the Hamiltonian structure are called
_symplectic_ , and satisfy

$\displaystyle Y^{*}\mathtt{J}_{4}Y=\mathtt{J}_{4}\,.$ (3.15)

If $Y$ is symplectic then $Y^{*}$ and $Y^{-1}$ are symplectic as well. A
Hamiltonian matrix $\mathtt{L}=\mathtt{J}_{4}\mathtt{B}$, with
$\mathtt{B}=\mathtt{B}^{*}$, is conjugated through $Y$ in the new Hamiltonian
matrix

$\mathtt{L}_{1}=Y^{-1}\mathtt{L}Y=Y^{-1}\mathtt{J}_{4}Y^{-*}Y^{*}\mathtt{B}Y=\mathtt{J}_{4}\mathtt{B}_{1}\quad\text{where
}\quad\mathtt{B}_{1}:=Y^{*}\mathtt{B}Y=\mathtt{B}_{1}^{*}\,.$ (3.16)

Note that the matrix $\rho_{4}$ in (3.14) represents the action of the
involution
$\overline{\rho}:{\mathcal{V}}_{\mu,\epsilon}\to{\mathcal{V}}_{\mu,\epsilon}$
defined in (2.27) in a reversible basis (cfr. (3.9)). A $4\times 4$ matrix
$\mathtt{B}=(\mathtt{B}_{ij})_{i,j=1,\dots,4}$ is reversibility-preserving if
and only if its entries are alternatively real and purely imaginary, namely
$\mathtt{B}_{ij}$ is real when $i+j$ is even and purely imaginary otherwise,
as in (3.13). A $4\times 4$ complex matrix
$\mathtt{L}=(\mathtt{L}_{ij})_{i,j=1,\ldots,4}$ is reversible if and only if
$\mathtt{L}_{ij}$ is purely imaginary when $i+j$ is even and real otherwise.

We finally mention that the flow of a Hamiltonian reversibility-preserving
matrix is symplectic and reversibility-preserving (see Lemma 3.8 in [6]).

##  4 Matrix representation of $\mathscr{L}_{\mu,\epsilon}$ on
$\mathcal{V}_{\mu,\epsilon}$

Using the transformation operators $U_{\mu,\epsilon}$ in (3.7), we construct
the basis of $\mathcal{V}_{\mu,\epsilon}$

$\mathcal{F}:=\big{\\{}f_{1}^{+}(\mu,\epsilon),\ f_{1}^{-}(\mu,\epsilon),\
f_{0}^{+}(\mu,\epsilon),\ f_{0}^{-}(\mu,\epsilon)\big{\\}}\,,\quad
f_{k}^{\sigma}(\mu,\epsilon):=U_{\mu,\epsilon}f_{k}^{\sigma}\,,\
\sigma=\pm\,,\,k=0,1\,,$ (4.1)

where

$f_{1}^{+}=\begin{bmatrix}{\mathtt{c}}_{\mathtt{h}}^{1/2}\cos(x)\\\
{\mathtt{c}}_{\mathtt{h}}^{-1/2}\sin(x)\end{bmatrix},\quad
f_{1}^{-}=\begin{bmatrix}-{\mathtt{c}}_{\mathtt{h}}^{1/2}\sin(x)\\\
{\mathtt{c}}_{\mathtt{h}}^{-1/2}\cos(x)\end{bmatrix},\quad
f_{0}^{+}=\begin{bmatrix}1\\\ 0\end{bmatrix},\quad
f_{0}^{-}=\begin{bmatrix}0\\\ 1\end{bmatrix}\,,$ (4.2)

form a basis of $\mathcal{V}_{0,0}=\mathrm{Rg}(P_{0,0})$, cfr. (2.34)-(2.35).
Note that the real valued vectors $\\{f_{1}^{\pm},f_{0}^{\pm}\\}$ form a
symplectic and reversible basis for $\mathcal{V}_{0,0}$, according to
Definition 3.3. Then, by Lemma 3.2 and 3.1 we deduce that (cfr. Lemma 4.1 in
[6]):

######  Lemma 4.1.

The basis $\mathcal{F}$ of $\mathcal{V}_{\mu,\epsilon}$ defined in (4.1), is
symplectic and reversible, i.e. satisfies (3.8) and (3.9). Each map
$(\mu,\epsilon)\mapsto f^{\sigma}_{k}(\mu,\epsilon)$ is analytic as a map
$B(\mu_{0})\times B(\epsilon_{0})\to H^{1}(\mathbb{T})$.

In the next lemma we expand the vectors $f_{k}^{\sigma}(\mu,\epsilon)$ in
$(\mu,\epsilon)$. We denote by $even_{0}(x)$ a real, even, $2\pi$-periodic
function with zero space average. In the sequel
$\mathcal{O}(\mu^{m}\epsilon^{n})\footnotesize\begin{bmatrix}even(x)\\\
odd(x)\end{bmatrix}$ denotes an analytic map in $(\mu,\epsilon)$ with values
in $H^{1}(\mathbb{T},\mathbb{C}^{2})$, whose first component is $even(x)$ and
the second one $odd(x)$; similar meaning for
$\mathcal{O}(\mu^{m}\epsilon^{n})\footnotesize\begin{bmatrix}odd(x)\\\
even(x)\end{bmatrix}$, etc…

######  Lemma 4.2.

(Expansion of the basis $\mathcal{F}$) For small values of $(\mu,\epsilon)$
the basis $\mathcal{F}$ in (4.1) has the expansion

$\displaystyle f^{+}_{1}(\mu,\epsilon)$
$\displaystyle=\begin{bmatrix}{\mathtt{c}}_{\mathtt{h}}^{\frac{1}{2}}\cos(x)\\\
{\mathtt{c}}_{\mathtt{h}}^{-\frac{1}{2}}\sin(x)\end{bmatrix}+\mathrm{i}\,\frac{\mu}{4}\gamma_{\mathtt{h}}\begin{bmatrix}{\mathtt{c}}_{\mathtt{h}}^{\frac{1}{2}}\sin(x)\\\
{\mathtt{c}}_{\mathtt{h}}^{-\frac{1}{2}}\cos(x)\end{bmatrix}+\epsilon\begin{bmatrix}\alpha_{\mathtt{h}}\cos(2x)\\\
\beta_{\mathtt{h}}\sin(2x)\end{bmatrix}$ (4.3)
$\displaystyle+\mathcal{O}(\mu^{2})\begin{bmatrix}even_{0}(x)+\mathrm{i}\,odd(x)\\\
odd(x)+\mathrm{i}\,even_{0}(x)\end{bmatrix}+\mathcal{O}(\epsilon^{2})\begin{bmatrix}even_{0}(x)\\\
odd(x)\end{bmatrix}+\mathrm{i}\,\mu\epsilon\begin{bmatrix}odd(x)\\\
even(x)\end{bmatrix}+\mathcal{O}(\mu^{2}\epsilon,\mu\epsilon^{2})\,,$
$\displaystyle f^{-}_{1}(\mu,\epsilon)$
$\displaystyle=\begin{bmatrix}-{\mathtt{c}}_{\mathtt{h}}^{\frac{1}{2}}\sin(x)\\\
{\mathtt{c}}_{\mathtt{h}}^{-\frac{1}{2}}\cos(x)\end{bmatrix}+\mathrm{i}\,\frac{\mu}{4}\gamma_{\mathtt{h}}\begin{bmatrix}{\mathtt{c}}_{\mathtt{h}}^{\frac{1}{2}}\cos(x)\\\
-{\mathtt{c}}_{\mathtt{h}}^{-\frac{1}{2}}\sin(x)\end{bmatrix}+\epsilon\begin{bmatrix}-\alpha_{\mathtt{h}}\sin(2x)\\\
\beta_{\mathtt{h}}\cos(2x)\end{bmatrix}$ (4.4)
$\displaystyle+\mathcal{O}(\mu^{2})\begin{bmatrix}odd(x)+\mathrm{i}\,even_{0}(x)\\\
even_{0}(x)+\mathrm{i}\,odd(x)\end{bmatrix}+\mathcal{O}(\epsilon^{2})\begin{bmatrix}odd(x)\\\
even(x)\end{bmatrix}+\mathrm{i}\,\mu\epsilon\begin{bmatrix}even(x)\\\
odd(x)\end{bmatrix}+\mathcal{O}(\mu^{2}\epsilon,\mu\epsilon^{2})\,,$
$\displaystyle f^{+}_{0}(\mu,\epsilon)$ $\displaystyle=\begin{bmatrix}1\\\
0\end{bmatrix}+\epsilon\delta_{\mathtt{h}}\begin{bmatrix}{\mathtt{c}}_{\mathtt{h}}^{\frac{1}{2}}\cos(x)\\\
-{\mathtt{c}}_{\mathtt{h}}^{-\frac{1}{2}}\sin(x)\end{bmatrix}+\mathcal{O}(\epsilon^{2})\begin{bmatrix}even_{0}(x)\\\
odd(x)\end{bmatrix}+\mathrm{i}\,\mu\epsilon\begin{bmatrix}odd(x)\\\
even_{0}(x)\end{bmatrix}+\mathcal{O}(\mu^{2}\epsilon,\mu\epsilon^{2})\,,$
(4.5) $\displaystyle f^{-}_{0}(\mu,\epsilon)$
$\displaystyle=\begin{bmatrix}0\\\
1\end{bmatrix}+\mathrm{i}\,\mu\epsilon\begin{bmatrix}even_{0}(x)\\\
odd(x)\end{bmatrix}+\mathcal{O}(\mu^{2}\epsilon,\mu\epsilon^{2})\,,$ (4.6)

where the remainders $\mathcal{O}()$ are vectors in $H^{1}(\mathbb{T})$ and

$\alpha_{\mathtt{h}}:=\frac{1}{2}{\mathtt{c}}_{\mathtt{h}}^{-\frac{11}{2}}(3+{\mathtt{c}}_{\mathtt{h}}^{4})\,,\quad\beta_{\mathtt{h}}:=\frac{1}{4}{\mathtt{c}}_{\mathtt{h}}^{-\frac{13}{2}}(1+{\mathtt{c}}_{\mathtt{h}}^{4})(3-{\mathtt{c}}_{\mathtt{h}}^{4})\,,\quad\gamma_{\mathtt{h}}:=1+\frac{\mathtt{h}(1-{\mathtt{c}}_{\mathtt{h}}^{4})}{{\mathtt{c}}_{\mathtt{h}}^{2}}\,,\quad\delta_{\mathtt{h}}:=\frac{3+{\mathtt{c}}_{\mathtt{h}}^{4}}{4{\mathtt{c}}_{\mathtt{h}}^{\frac{5}{2}}}\,.$
(4.7)

For $\mu=0$ the basis $\\{f_{k}^{\pm}(0,\epsilon),k=0,1\\}$ is real and

$f^{+}_{1}(0,\epsilon)=\begin{bmatrix}even_{0}(x)\\\ odd(x)\end{bmatrix},\
f^{-}_{1}(0,\epsilon)=\begin{bmatrix}odd(x)\\\ even(x)\end{bmatrix},\
f^{+}_{0}(0,\epsilon)=\begin{bmatrix}1\\\
0\end{bmatrix}+\begin{bmatrix}even_{0}(x)\\\ odd(x)\end{bmatrix}\,,\
f^{-}_{0}(0,\epsilon)=\begin{bmatrix}0\\\ 1\end{bmatrix}\,.$ (4.8)

###### Proof.

The long calculations are given in Appendix A. ∎

We now state the main result of this section.

######  Proposition 4.3.

The matrix that represents the Hamiltonian and reversible operator
$\mathscr{L}_{\mu,\epsilon}:\mathcal{V}_{\mu,\epsilon}\to\mathcal{V}_{\mu,\epsilon}$
in the symplectic and reversible basis $\mathcal{F}$ of
$\mathcal{V}_{\mu,\epsilon}$ defined in (4.1), is a Hamiltonian matrix
$\mathtt{L}_{\mu,\epsilon}=\mathtt{J}_{4}\mathtt{B}_{\mu,\epsilon}$, where
$\mathtt{B}_{\mu,\epsilon}$ is a self-adjoint and reversibility preserving
(i.e. satisfying (3.13)) $4\times 4$ matrix of the form

$\mathtt{B}_{\mu,\epsilon}=\begin{pmatrix}E&F\\\ F^{*}&G\end{pmatrix},\qquad
E=E^{*}\,,\ \ G=G^{*}\,,$ (4.9)

where $E,F,G$ are the $2\times 2$ matrices

$\displaystyle
E:=\begin{pmatrix}\mathtt{e}_{11}\epsilon^{2}(1+r_{1}^{\prime}(\epsilon,\mu\epsilon))-\mathtt{e}_{22}\frac{\mu^{2}}{8}(1+r_{1}^{\prime\prime}(\epsilon,\mu))&\mathrm{i}\,\big{(}\frac{1}{2}\mathtt{e}_{12}\mu+r_{2}(\mu\epsilon^{2},\mu^{2}\epsilon,\mu^{3})\big{)}\\\
-\mathrm{i}\,\big{(}\frac{1}{2}\mathtt{e}_{12}\mu+r_{2}(\mu\epsilon^{2},\mu^{2}\epsilon,\mu^{3})\big{)}&-\mathtt{e}_{22}\frac{\mu^{2}}{8}(1+r_{5}(\epsilon,\mu))\end{pmatrix}$
(4.10) $\displaystyle
G:=\begin{pmatrix}1+r_{8}(\epsilon^{2},\mu^{2}\epsilon)&-\mathrm{i}\,r_{9}(\mu\epsilon^{2},\mu^{2}\epsilon)\\\
\mathrm{i}\,r_{9}(\mu\epsilon^{2},\mu^{2}\epsilon)&\mu\tanh(\mathtt{h}\mu)+r_{10}(\mu^{2}\epsilon)\end{pmatrix}$
(4.11) $\displaystyle
F:=\begin{pmatrix}\mathtt{f}_{11}\epsilon+r_{3}(\epsilon^{3},\mu\epsilon^{2},\mu^{2}\epsilon)&\mathrm{i}\,\mu\epsilon{\mathtt{c}}_{\mathtt{h}}^{-\frac{1}{2}}+\mathrm{i}\,r_{4}({\mu\epsilon^{2}},\mu^{2}\epsilon)\\\
\mathrm{i}\,r_{6}(\mu\epsilon)&r_{7}(\mu^{2}\epsilon)\end{pmatrix}\,,$ (4.12)

with $\mathtt{e}_{12}$ and $\mathtt{e}_{22}$ given in (1.2) and (1.3)
respectively, and

$\displaystyle\mathtt{e}_{11}$
$\displaystyle:=\dfrac{9{\mathtt{c}}_{\mathtt{h}}^{8}-10{\mathtt{c}}_{\mathtt{h}}^{4}+9}{8{\mathtt{c}}_{\mathtt{h}}^{7}}=\dfrac{9(1-{\mathtt{c}}_{\mathtt{h}}^{4})^{2}+8{\mathtt{c}}_{\mathtt{h}}^{4}}{8{\mathtt{c}}_{\mathtt{h}}^{7}}>0\,,\qquad\mathtt{f}_{11}:=\tfrac{1}{2}{\mathtt{c}}_{\mathtt{h}}^{-\frac{3}{2}}(1-{\mathtt{c}}_{\mathtt{h}}^{4})\,.$
(4.13)

The rest of this section is devoted to the proof of Proposition 4.3.

We decompose ${\mathcal{B}}_{\mu,\epsilon}$ in (3.3) as

${\mathcal{B}}_{\mu,\epsilon}={\mathcal{B}}_{\epsilon}+{\mathcal{B}}^{\flat}+{\mathcal{B}}^{\sharp}\,,$

where ${\mathcal{B}}_{\epsilon}$, ${\mathcal{B}}^{\flat}$,
${\mathcal{B}}^{\sharp}$ are the self-adjoint and reversibility preserving
operators

$\displaystyle{\mathcal{B}}_{\epsilon}:={\mathcal{B}}_{0,\epsilon}:=\left[\begin{array}[]{cc}1+a_{\epsilon}(x)&-({\mathtt{c}}_{\mathtt{h}}+p_{\epsilon}(x))\partial_{x}\\\
\partial_{x}\circ({\mathtt{c}}_{\mathtt{h}}+p_{\epsilon}(x))&|D|\tanh((\mathtt{h}+\mathtt{f}_{\epsilon})|D|)\end{array}\right],$
(4.16) $\displaystyle{\mathcal{B}}^{\flat}:=\begin{bmatrix}0&0\\\
0&|D+\mu|\tanh((\mathtt{h}+\mathtt{f}_{\epsilon})|D+\mu|)-|D|\tanh((\mathtt{h}+\mathtt{f}_{\epsilon})|D|)\end{bmatrix},\,$
(4.17)
$\displaystyle{\mathcal{B}}^{\sharp}:=\mu\begin{bmatrix}0&-\mathrm{i}\,p_{\epsilon}\\\
\mathrm{i}\,p_{\epsilon}&0\end{bmatrix}\,.$ (4.18)

In view of (2.29) the operator ${\mathcal{B}}^{\flat}$ is analytic in $\mu$.

######  Lemma 4.4.

(Expansion of $\mathtt{B}_{\epsilon}$) The self-adjoint and reversibility
preserving matrix $\mathtt{B}_{\epsilon}:=\mathtt{B}_{\epsilon}(\mu)$
associated, as in (3.12), with the self-adjoint and reversibility preserving
operator ${\mathcal{B}}_{\epsilon}$ defined in (4.16), with respect to the
basis $\mathcal{F}$ of ${\mathcal{V}}_{\mu,\epsilon}$ in (4.1), expands as

$\displaystyle\mathtt{B}_{\epsilon}=\begin{pmatrix}\mathtt{e}_{11}\epsilon^{2}+\zeta_{\mathtt{h}}\mu^{2}+r_{1}(\epsilon^{3},\mu\epsilon^{3})&\mathrm{i}\,r_{2}(\mu\epsilon^{2})&\vline&\mathtt{f}_{11}\epsilon+r_{3}(\epsilon^{3},\mu\epsilon^{2})&\mathrm{i}\,r_{4}(\mu\epsilon^{3})\\\
-\mathrm{i}\,r_{2}(\mu\epsilon^{2})&\zeta_{\mathtt{h}}\mu^{2}&\vline&\mathrm{i}\,r_{6}(\mu\epsilon)&0\\\
\hline\cr\mathtt{f}_{11}\epsilon+r_{3}(\epsilon^{3},\mu\epsilon^{2})&-\mathrm{i}\,r_{6}(\mu\epsilon)&\vline&1+r_{8}(\epsilon^{2},\mu\epsilon^{2})&\mathrm{i}\,r_{9}(\mu\epsilon^{2})\\\
-\mathrm{i}\,r_{4}(\mu\epsilon^{3})&0&\vline&-\mathrm{i}\,r_{9}(\mu\epsilon^{2})&0\\\
\end{pmatrix}+\mathcal{O}(\mu^{2}\epsilon,\mu^{3})\,,$ (4.19)

where $\mathtt{e}_{11}$, $\mathtt{f}_{11}$ are defined respectively in (4.13),
and

$\zeta_{\mathtt{h}}:=\tfrac{1}{8}{\mathtt{c}}_{\mathtt{h}}\gamma_{\mathtt{h}}^{2}\,.$
(4.20)

###### Proof.

We expand the matrix $\mathtt{B}_{\epsilon}(\mu)$ as

$\mathtt{B}_{\epsilon}(\mu)=\mathtt{B}_{\epsilon}(0)+\mu(\partial_{\mu}\mathtt{B}_{\epsilon})(0)+\frac{\mu^{2}}{2}(\partial_{\mu}^{2}\mathtt{B}_{0})(0)+\mathcal{O}(\mu^{2}\epsilon,\mu^{3})\,.$
(4.21)

The matrix $\mathtt{B}_{\epsilon}(0)$. The main result of this long paragraph
is to prove that the matrix $\mathtt{B}_{\epsilon}(0)$ has the expansion
(4.25). The matrix $\mathtt{B}_{\epsilon}(0)$ is real, because the operator
${\mathcal{B}}_{\epsilon}$ is real and the basis
$\\{f_{k}^{\pm}(0,\epsilon)\\}_{k=0,1}$ is real. Consequently, by (3.13), its
matrix elements $(\mathtt{B}_{\epsilon}(0))_{i,j}$ are real whenever $i+j$ is
even and vanish for $i+j$ odd. In addition
$f^{-}_{0}(0,\epsilon)=\footnotesize\begin{bmatrix}0\\\ 1\end{bmatrix}$ by
(4.8), and, by (4.16), we get
${\mathcal{B}}_{\epsilon}f^{-}_{0}(0,\epsilon)=0$, for any $\epsilon$. We
deduce that the self-adjoint matrix $\mathtt{B}_{\epsilon}(0)$ has the form

$\mathtt{B}_{\epsilon}(0)=\left({\mathcal{B}}_{\epsilon}\,f^{\sigma}_{k}(0,\epsilon),\,f^{\sigma^{\prime}}_{k^{\prime}}(0,\epsilon)\right)_{k,k^{\prime}=0,1,\sigma,\sigma^{\prime}=\pm}=\begin{pmatrix}E_{11}(0,\epsilon)&0&\vline&F_{11}(0,\epsilon)&0\\\
0&E_{22}(0,\epsilon)&\vline&0&0\\\ \hline\cr
F_{11}(0,\epsilon)&0&\vline&G_{11}(0,\epsilon)&0\\\
0&0&\vline&0&0\end{pmatrix}\,,$ (4.22)

where $E_{11}(0,\epsilon)$, $E_{22}(0,\epsilon)$, $G_{11}(0,\epsilon)$,
$F_{11}(0,\epsilon)$ are real. We claim that $E_{22}(0,\epsilon)=0$ for any
$\epsilon$. As a first step, following [6], we prove that

$\text{ either }\ E_{22}(0,\epsilon)\equiv 0\,,\qquad\text{ or }\
E_{11}(0,\epsilon)\equiv 0\equiv F_{11}(0,\epsilon)\,.$ (4.23)

Indeed, by (2.37), the operator
$\mathscr{L}_{0,\epsilon}\equiv{\mathcal{L}}_{0,\epsilon}$ possesses, for any
sufficiently small $\epsilon\neq 0$, the eigenvalue $0$ with a four
dimensional generalized Kernel
$\mathcal{W}_{\epsilon}:=\text{span}\\{U_{1},\tilde{U}_{2},U_{3},U_{4}\\}$,
spanned by $\epsilon$-dependent vectors $U_{1},\tilde{U}_{2},U_{3},U_{4}$. By
Lemma 3.1 it results that
$\mathcal{W}_{\epsilon}={\mathcal{V}}_{0,\epsilon}=\text{Rg}(P_{0,\epsilon})$
and by (2.37) we have $\mathscr{L}_{0,\epsilon}^{2}=0$ on
$\mathcal{V}_{0,\epsilon}$. Thus the matrix

$\mathtt{L}_{\epsilon}(0):=\mathtt{J}_{4}\mathtt{B}_{\epsilon}(0)=\begin{pmatrix}0&E_{22}(0,\epsilon)&\vline&0&0\\\
-E_{11}(0,\epsilon)&0&\vline&-F_{11}(0,\epsilon)&0\\\ \hline\cr
0&0&\vline&0&0\\\
-F_{11}(0,\epsilon)&0&\vline&-G_{11}(0,\epsilon)&0\end{pmatrix}\,,$ (4.24)

which represents
$\mathscr{L}_{0,\epsilon}:\mathcal{V}_{0,\epsilon}\to\mathcal{V}_{0,\epsilon}$,
satisfies $\mathtt{L}^{2}_{\epsilon}(0)=0$, namely

$\mathtt{L}^{2}_{\epsilon}(0)=-\begin{pmatrix}(E_{11}E_{22})(0,\epsilon)&0&\vline&(F_{11}E_{22})(0,\epsilon)&0\\\
0&(E_{11}E_{22})(0,\epsilon)&\vline&0&0\\\ \hline\cr 0&0&\vline&0&0\\\
0&(F_{11}E_{22})(0,\epsilon)&\vline&0&0\end{pmatrix}=0$

which implies (4.23). We now prove that the matrix $\mathtt{B}_{\epsilon}(0)$
defined in (4.22) expands as

$\mathtt{B}_{\epsilon}(0)=\begin{pmatrix}\mathtt{e}_{11}\epsilon^{2}+{r(\epsilon^{3})}&0&\vline&\mathtt{f}_{11}\epsilon+r(\epsilon^{3})&0\\\
0&0&\vline&0&0\\\
\hline\cr\mathtt{f}_{11}\epsilon+r(\epsilon^{3})&0&\vline&1+r(\epsilon^{2})&0\\\
0&0&\vline&0&0\end{pmatrix}$ (4.25)

where $\mathtt{e}_{11}$ and $\mathtt{f}_{11}$ are in (4.31) and (4.34). We
expand the operator ${\mathcal{B}}_{\epsilon}$ in (4.16) as

$\displaystyle{\mathcal{B}}_{\epsilon}={\mathcal{B}}_{0}+\epsilon{\mathcal{B}}_{1}+\epsilon^{2}{\mathcal{B}}_{2}+\mathcal{O}(\epsilon^{3}),\quad{\mathcal{B}}_{0}:=\begin{bmatrix}1&-{\mathtt{c}}_{\mathtt{h}}\partial_{x}\\\
{\mathtt{c}}_{\mathtt{h}}\partial_{x}&|D|\tanh(\mathtt{h}|D|)\end{bmatrix}\,,$
(4.26)
$\displaystyle{\mathcal{B}}_{1}:=\begin{bmatrix}a_{1}(x)&-p_{1}(x)\partial_{x}\\\
\partial_{x}\circ p_{1}(x)&0\end{bmatrix}\,,\
\;{\mathcal{B}}_{2}:=\begin{bmatrix}a_{2}(x)&-p_{2}(x)\partial_{x}\\\
\partial_{x}\circ
p_{2}(x)&-\mathtt{f}_{2}\partial_{x}^{2}\big{(}1-\tanh^{2}(\mathtt{h}|D|)\big{)}\end{bmatrix}\,,$

where the remainder term $\mathcal{O}(\epsilon^{3})\in\mathcal{L}(Y,X)$, the
functions $a_{1}$, $p_{1}$, $a_{2}$, $p_{2}$ are given in (2.20)-(2.23) and,
in view of (2.15),
$\mathtt{f}_{2}:=\tfrac{1}{4}{\mathtt{c}}_{\mathtt{h}}^{-2}({\mathtt{c}}_{\mathtt{h}}^{4}-3)$.

$\bullet$ Expansion of
$E_{11}(0,\epsilon)=\mathtt{e}_{11}\epsilon^{2}+r(\epsilon^{3})$.  By (4.3) we
split the real function $f_{1}^{+}(0,\epsilon)$ as

$\displaystyle\qquad\qquad f_{1}^{+}(0,\epsilon)=f_{1}^{+}+\epsilon
f_{1_{1}}^{+}+\epsilon^{2}f_{1_{2}}^{+}+\mathcal{O}(\epsilon^{3})\,,$ (4.27)
$\displaystyle
f_{1}^{+}=\begin{bmatrix}{\mathtt{c}}_{\mathtt{h}}^{\frac{1}{2}}\cos(x)\\\
{\mathtt{c}}_{\mathtt{h}}^{-\frac{1}{2}}\sin(x)\end{bmatrix},\
f_{1_{1}}^{+}:=\begin{bmatrix}\alpha_{\mathtt{h}}\cos(2x)\\\
\beta_{\mathtt{h}}\sin(2x)\end{bmatrix}\,,\
f_{1_{2}}^{+}:=\begin{bmatrix}even_{0}(x)\\\ odd(x)\end{bmatrix},$

where both $f_{1_{2}}^{+}$ and $\mathcal{O}(\epsilon^{3})$ are vectors in
$H^{1}(\mathbb{T})$. Since
${\mathcal{B}}_{0}f_{1}^{+}=\mathcal{J}^{-1}\mathscr{L}_{0,0}f_{1}^{+}=0$, and
both ${\mathcal{B}}_{0}$, ${\mathcal{B}}_{1}$ are self-adjoint real operators,
it results

$\displaystyle E_{11}(0,\epsilon)$
$\displaystyle=\left({\mathcal{B}}_{\epsilon}f^{+}_{1}(0,\epsilon)\,,\,f^{+}_{1}(0,\epsilon)\right)$
$\displaystyle=\epsilon\left({\mathcal{B}}_{1}f_{1}^{+}\,,\,f_{1}^{+}\right)+\epsilon^{2}\left[\left({\mathcal{B}}_{2}f_{1}^{+}\,,\,f_{1}^{+}\right)+2\left({\mathcal{B}}_{1}f_{1}^{+}\,,\,f_{1_{1}}^{+}\right)+\left({\mathcal{B}}_{0}f_{1_{1}}^{+}\,,\,f_{1_{1}}^{+}\right)\right]+\mathcal{O}(\epsilon^{3})\,.$
(4.28)

By (4.26) one has

$\displaystyle{\mathcal{B}}_{1}f_{1}^{+}=\begin{bmatrix}A_{1}(1+\cos(2x))\\\
B_{1}\sin(2x)\end{bmatrix},\ \
{\mathcal{B}}_{2}f_{1}^{+}=\begin{bmatrix}A_{2}\cos(x)+A_{3}\cos(3x)\\\
B_{2}\sin(x)+B_{3}\sin(3x)\end{bmatrix},\ \
{\mathcal{B}}_{0}f_{1_{1}}^{+}=\begin{bmatrix}A_{4}\cos(2x)\\\
B_{4}\sin(2x)\end{bmatrix}\,,$ (4.29)

with

$\displaystyle
A_{1}:=\tfrac{1}{2}(a_{1}^{[1]}{\mathtt{c}}_{\mathtt{h}}^{\frac{1}{2}}-p_{1}^{[1]}{\mathtt{c}}_{\mathtt{h}}^{-\frac{1}{2}}),\qquad
B_{1}:=-p_{1}^{[1]}{\mathtt{c}}_{\mathtt{h}}^{\frac{1}{2}}\,,$ (4.30)
$\displaystyle
A_{2}:={\mathtt{c}}_{\mathtt{h}}^{\frac{1}{2}}a_{2}^{[0]}-{\mathtt{c}}_{\mathtt{h}}^{-\frac{1}{2}}p_{2}^{[0]}+\tfrac{1}{2}{\mathtt{c}}_{\mathtt{h}}^{\frac{1}{2}}a_{2}^{[2]}-\tfrac{1}{2}{\mathtt{c}}_{\mathtt{h}}^{-\frac{1}{2}}p_{2}^{[2]}\,,\qquad
A_{4}:=\alpha_{\mathtt{h}}-2\beta_{\mathtt{h}}{\mathtt{c}}_{\mathtt{h}}\,,$
$\displaystyle
B_{2}:=-{\mathtt{c}}_{\mathtt{h}}^{\frac{1}{2}}p_{2}^{[0]}-\tfrac{1}{2}{\mathtt{c}}_{\mathtt{h}}^{\frac{1}{2}}p_{2}^{[2]}+{{\mathtt{c}}_{\mathtt{h}}^{-\frac{1}{2}}}\mathtt{f}_{2}(1-{\mathtt{c}}_{\mathtt{h}}^{4})\,,\qquad
B_{4}:=-2\alpha_{\mathtt{h}}{\mathtt{c}}_{\mathtt{h}}+{\frac{4{\mathtt{c}}_{\mathtt{h}}^{2}}{1+{\mathtt{c}}_{\mathtt{h}}^{4}}}{\beta_{\mathtt{h}}}\,.$

By (4.29) and (4.27), we deduce

$E_{11}(0,\epsilon)=\mathtt{e}_{11}\epsilon^{2}+r(\epsilon^{3})\,,\quad\mathtt{e}_{11}:=\frac{1}{2}\big{(}A_{2}{\mathtt{c}}_{\mathtt{h}}^{\frac{1}{2}}+B_{2}{\mathtt{c}}_{\mathtt{h}}^{-\frac{1}{2}}+2\alpha_{\mathtt{h}}A_{1}+2B_{1}\beta_{\mathtt{h}}+\alpha_{\mathtt{h}}A_{4}+\beta_{\mathtt{h}}B_{4}\big{)}\,.$
(4.31)

By (4.31), (4.30), (4.7), (2.20)-(2.23) we obtain (4.13). Since
$\mathtt{e}_{11}>0$ the second alternative in (4.23) is ruled out, implying
$E_{22}(0,\epsilon)\equiv 0$.
$\bullet$ Expansion of $G_{11}(0,\epsilon)=1+r(\epsilon^{2})$.  By (4.5) we
split the real-valued function $f_{0}^{+}(0,\epsilon)$ as

$f_{0}^{+}(0,\epsilon)=f_{0}^{+}+\epsilon
f_{0_{1}}^{+}+\epsilon^{2}f_{0_{2}}^{+}+\mathcal{O}(\epsilon^{3})\,,\ \
f_{0}^{+}=\begin{bmatrix}1\\\ 0\end{bmatrix}\,,\
f_{0_{1}}^{+}:=\delta_{\mathtt{h}}\begin{bmatrix}{\mathtt{c}}_{\mathtt{h}}^{\frac{1}{2}}\cos(x)\\\
-{\mathtt{c}}_{\mathtt{h}}^{-\frac{1}{2}}\sin(x)\end{bmatrix}\,,\
f_{0_{2}}^{+}:=\begin{bmatrix}even_{0}(x)\\\ odd(x)\end{bmatrix}\,.$ (4.32)

Since, by (2.34) and (4.26), ${\mathcal{B}}_{0}f_{0}^{+}=f_{0}^{+}$, using
that ${\mathcal{B}}_{0}$, ${\mathcal{B}}_{1}$ are self-adjoint real operators,
and $\|f_{0}^{+}\|=1$, $(f_{0}^{+},f_{0_{1}}^{+})$, we have
$G_{11}(0,\epsilon)=\left({\mathcal{B}}_{\epsilon}f^{+}_{0}(0,\epsilon)\,,\,f^{+}_{0}(0,\epsilon)\right)=1+\epsilon\left({\mathcal{B}}_{1}f_{0}^{+}\,,\,f_{0}^{+}\right)+r(\epsilon^{2})$.
By (4.26) and (2.20)-(2.23) one has

${\mathcal{B}}_{1}f_{0}^{+}=\begin{bmatrix}a_{1}^{[1]}\cos(x)\\\
-p_{1}^{[1]}\sin(x)\end{bmatrix}$ (4.33)

and, by (4.32), we deduce $G_{11}(0,\epsilon)=1+r(\epsilon^{2})$.
$\bullet$ Expansion of
$F_{11}(0,\epsilon)=\mathtt{f}_{11}\epsilon+r(\epsilon^{3})$. By (4.26),
(4.27), (4.32), using that ${\mathcal{B}}_{0},{\mathcal{B}}_{1}$ are self-
adjoint and real, and ${\mathcal{B}}_{0}f_{1}^{+}=0$,
${\mathcal{B}}_{0}f_{0}^{+}=f_{0}^{+}$, we obtain

$\displaystyle F_{11}(0,\epsilon)$
$\displaystyle=\epsilon\left[\left({\mathcal{B}}_{1}f_{1}^{+}\,,\,f_{0}^{+}\right)+\left(f_{1_{1}}^{+}\,,\,f_{0}^{+}\right)\right]$
$\displaystyle\quad+\epsilon^{2}\big{[}\left({\mathcal{B}}_{2}f_{1}^{+}\,,\,f_{0}^{+}\right)+\left({\mathcal{B}}_{1}f_{1}^{+}\,,\,f_{0_{1}}^{+}\right)+\left({\mathcal{B}}_{1}f_{0}^{+}\,,\,f_{1_{1}}^{+}\right)+\left(f_{1_{2}}^{+}\,,\,f_{0}^{+}\right)+\left({\mathcal{B}}_{0}f_{1_{1}}^{+}\,,\,f_{0_{1}}^{+}\right)\big{]}+r(\epsilon^{3})\,.$

By (4.27), (4.29), (4.30), (4.32), (4.33), all these scalar products vanish
but the first one, and then

$F_{11}(0,\epsilon)=\mathtt{f}_{11}\epsilon+r(\epsilon^{3})\,,\quad\mathtt{f}_{11}:=A_{1}=\tfrac{1}{2}(a_{1}^{[1]}{\mathtt{c}}_{\mathtt{h}}^{\frac{1}{2}}-p_{1}^{[1]}{\mathtt{c}}_{\mathtt{h}}^{-\frac{1}{2}})\,,$
(4.34)

which, by substituting the expressions of $a_{1}^{[1]}$, $p_{1}^{[1]}$ in
Lemma 2.2, gives the expression in (4.13).

The expansion (4.25) in proved.
Linear terms in $\mu$. We now compute the terms of
$\mathtt{B}_{\epsilon}(\mu)$ that are linear in $\mu$. It results

$\partial_{\mu}\mathtt{B}_{\epsilon}(0)=X+X^{*}\qquad\text{where}\qquad
X:=\big{(}{\mathcal{B}}_{\epsilon}f_{k}^{\sigma}(0,\epsilon),(\partial_{\mu}f^{\sigma^{\prime}}_{k^{\prime}})(0,\epsilon)\big{)}_{k,k^{\prime}=0,1,\sigma,\sigma^{\prime}=\pm}\,.$
(4.35)

We now prove that

$X=\begin{pmatrix}\mathcal{O}(\epsilon^{3})&0&\vline&\mathcal{O}(\epsilon^{2})&0\\\
\mathcal{O}(\epsilon^{2})&0&\vline&\mathcal{O}(\epsilon)&0\\\
\hline\cr\mathcal{O}(\epsilon^{3})&0&\vline&\mathcal{O}(\epsilon^{2})&0\\\
\mathcal{O}(\epsilon^{3})&0&\vline&\mathcal{O}(\epsilon^{2})&0\end{pmatrix}.$
(4.36)

The matrix $\mathtt{L}_{\epsilon}(0)$ in (4.24) where $E_{22}(0,\epsilon)=0$,
represents the action of the operator
$\mathscr{L}_{0,\epsilon}:\mathcal{V}_{0,\epsilon}\to\mathcal{V}_{0,\epsilon}$
in the basis $\\{f^{\sigma}_{k}(0,\epsilon)\\}$ and then we deduce that
$\mathscr{L}_{0,\epsilon}f_{1}^{-}(0,\epsilon)=0$,
$\mathscr{L}_{0,\epsilon}f_{0}^{-}(0,\epsilon)=0$. Thus also
${\mathcal{B}}_{\epsilon}f_{1}^{-}(0,\epsilon)=0$,
${\mathcal{B}}_{\epsilon}f_{0}^{-}(0,\epsilon)=0$, and the second and the
fourth column of the matrix $X$ in (4.36) are zero. To compute the other two
columns we use the expansion of the derivatives. In view of (4.3)-(4.6) and by
denoting with a dot the derivative w.r.t. $\mu$, one has

$\displaystyle\dot{f}^{+}_{1}(0,\epsilon)=\frac{\mathrm{i}\,}{4}\gamma_{\mathtt{h}}\begin{bmatrix}{\mathtt{c}}_{\mathtt{h}}^{\frac{1}{2}}\sin(x)\\\
{\mathtt{c}}_{\mathtt{h}}^{-\frac{1}{2}}\cos(x)\end{bmatrix}+\mathrm{i}\,\epsilon\begin{bmatrix}odd(x)\\\
even(x)\end{bmatrix}+\mathcal{O}(\epsilon^{2})\,,\quad\dot{f}^{+}_{0}(0,\epsilon)=\mathrm{i}\,\epsilon\begin{bmatrix}odd(x)\\\
even_{0}(x)\end{bmatrix}+\mathcal{O}(\epsilon^{2})\,,$ (4.37)
$\displaystyle\dot{f}^{-}_{1}(0,\epsilon)=\frac{\mathrm{i}\,}{4}\gamma_{\mathtt{h}}\begin{bmatrix}{\mathtt{c}}_{\mathtt{h}}^{\frac{1}{2}}\cos(x)\\\
-{\mathtt{c}}_{\mathtt{h}}^{-\frac{1}{2}}\sin(x)\end{bmatrix}+\mathrm{i}\,\epsilon\begin{bmatrix}even(x)\\\
odd(x)\end{bmatrix}+\mathcal{O}(\epsilon^{2})\,,\ \
\dot{f}^{-}_{0}(0,\epsilon)=\mathrm{i}\,\epsilon\begin{bmatrix}even_{0}(x)\\\
odd(x)\end{bmatrix}+\mathcal{O}(\epsilon^{2})\,.$

In view of (2.2), (4.3)-(4.6), (4.24), (4.8), (4.31),(4.34), and since
${\mathcal{B}}_{\epsilon}f_{k}^{\sigma}(0,\epsilon)=-\mathcal{J}\mathscr{L}_{\epsilon}f_{k}^{\sigma}(0,\epsilon)$,
we have

$\displaystyle{\mathcal{B}}_{\epsilon}f_{1}^{+}(0,\epsilon)$
$\displaystyle=E_{11}(0,\epsilon)\,\mathcal{J}f_{1}^{-}(0,\epsilon)+F_{11}(0,\epsilon)\,\mathcal{J}f_{0}^{-}=\epsilon\begin{bmatrix}\mathtt{f}_{11}\\\
0\end{bmatrix}+\epsilon^{2}\mathtt{e}_{11}\begin{bmatrix}{\mathtt{c}}_{\mathtt{h}}^{-\frac{1}{2}}\cos(x)\\\
{\mathtt{c}}_{\mathtt{h}}^{\frac{1}{2}}\sin(x)\end{bmatrix}+\mathcal{O}(\epsilon^{3})\,,$
(4.38) $\displaystyle{\mathcal{B}}_{\epsilon}f_{0}^{+}(0,\epsilon)$
$\displaystyle=F_{11}(0,\epsilon)\,\mathcal{J}f_{1}^{-}(0,\epsilon)+G_{11}(0,\epsilon)\,\mathcal{J}f_{0}^{-}=\begin{bmatrix}1\\\
0\end{bmatrix}+\epsilon\mathtt{f}_{11}\begin{bmatrix}{\mathtt{c}}_{\mathtt{h}}^{-\frac{1}{2}}\cos(2x)\\\
{\mathtt{c}}_{\mathtt{h}}^{\frac{1}{2}}\sin(2x)\end{bmatrix}+\mathcal{O}(\epsilon^{2})\,.$

We deduce (4.36) by (4.37) and (4.38).
Quadratic terms in $\mu$. By denoting with a double dot the double derivative
w.r.t. $\mu$, we have

$\partial_{\mu}^{2}\mathtt{B}_{0}(0)=\left({\mathcal{B}}_{0}f_{k}^{\sigma}\,,\,\ddot{f}_{k^{\prime}}^{\sigma^{\prime}}(0,0)\right)+\left(\ddot{f}_{k}^{\sigma}(0,0)\,,\,{\mathcal{B}}_{0}f_{k}^{\sigma^{\prime}}\right)+2\left({\mathcal{B}}_{0}\dot{f}_{k}^{\sigma}(0,0)\,,\,\dot{f}_{k^{\prime}}^{\sigma^{\prime}}(0,0)\right)=:Y+Y^{*}+2Z\,.$
(4.39)

We claim that $Y=0$. Indeed, its first, second and fourth column are zero,
since ${\mathcal{B}}_{0}f_{k}^{\sigma}=0$ for
$f_{k}^{\sigma}\in\\{f_{1}^{+},f_{1}^{-},f_{0}^{-}\\}$. The third column is
also zero by noting that ${\mathcal{B}}_{0}f_{0}^{+}=f_{0}^{+}$ and

$\ddot{f}_{1}^{+}(0,0)=\begin{bmatrix}even_{0}(x)+\mathrm{i}\,odd(x)\\\
odd(x)+\mathrm{i}\,even_{0}(x)\end{bmatrix},\ \
\ddot{f}_{1}^{-}(0,0)=\begin{bmatrix}odd(x)+\mathrm{i}\,even_{0}(x)\\\
even_{0}(x)+\mathrm{i}\,odd(x)\end{bmatrix},\ \
\ddot{f}_{0}^{+}(0,0)=\ddot{f}_{0}^{-}(0,0)=0\,.$

We claim that

$\displaystyle
Z=\left({\mathcal{B}}_{0}\dot{f}_{k}^{\sigma}(0,0)\,,\,\dot{f}_{k^{\prime}}^{\sigma^{\prime}}(0,0)\right)_{\begin{subarray}{c}k,k^{\prime}=0,1,\\\
\sigma,\sigma^{\prime}=\pm\end{subarray}}=\begin{pmatrix}\zeta_{\mathtt{h}}&0&\vline&0&0\\\
0&\zeta_{\mathtt{h}}&\vline&0&0\\\ \hline\cr 0&0&\vline&0&0\\\
0&0&\vline&0&0\\\ \end{pmatrix}\,,$ (4.40)

with $\zeta_{\mathtt{h}}$ as in (4.20). Indeed, by (4.37), we have
$\dot{f}^{+}_{0}(0,0)=\dot{f}^{-}_{0}(0,0)=0$. Therefore the last two columns
of $Z$, and by self-adjointness the last two rows, are zero. By (4.26), (4.37)
we obtain the matrix (4.40) with

$\zeta_{\mathtt{h}}:=\left({\mathcal{B}}_{0}\dot{f}^{+}_{1}(0,0)\,,\,\dot{f}^{+}_{1}(0,0)\right)=\left({\mathcal{B}}_{0}\dot{f}^{-}_{1}(0,0)\,,\,\dot{f}^{-}_{1}(0,0)\right)=\tfrac{1}{8}{\mathtt{c}}_{\mathtt{h}}\gamma_{\mathtt{h}}^{2}\,.$

In conclusion (4.21), (4.35), (4.36), (4.39), the fact that $Y=0$ and (4.40)
imply (4.19), using also the selfadjointness of $\mathtt{B}_{\epsilon}$ and
(3.13). ∎

We now consider ${\mathcal{B}}^{\flat}$.

######  Lemma 4.5.

(Expansion of $\mathtt{B}^{\flat}$) The self-adjoint and reversibility-
preserving matrix $\mathtt{B}^{\flat}$ associated, as in (3.12), to the self-
adjoint and reversibility-preserving operator ${\mathcal{B}}^{\flat}$, defined
in (4.17), with respect to the basis $\mathcal{F}$ of
${\mathcal{V}}_{\mu,\epsilon}$ in (4.1), admits the expansion

$\mathtt{B}^{\flat}=\begin{pmatrix}-\frac{\mu^{2}}{4}\text{{b}}_{\mathtt{h}}&\mathrm{i}\,(\frac{\mu}{2}\mathtt{e}_{12}+r_{2}(\mu\epsilon^{2}))&\vline&0&0\\\
-\mathrm{i}\,(\frac{\mu}{2}\mathtt{e}_{12}+r_{2}(\mu\epsilon^{2}))&-\frac{\mu^{2}}{4}\text{{b}}_{\mathtt{h}}&\vline&\mathrm{i}\,r_{6}(\mu\epsilon)&0\\\
\hline\cr 0&-\mathrm{i}\,r_{6}(\mu\epsilon)&\vline&0&0\\\
0&0&\vline&0&\mu\tanh(\mathtt{h}\mu)\end{pmatrix}+\mathcal{O}(\mu^{2}\epsilon,\mu^{3})$
(4.41)

where $\mathtt{e}_{12}$ is defined in (1.2) and

$\text{{b}}_{\mathtt{h}}:=\gamma_{\mathtt{h}}{\mathtt{c}}_{\mathtt{h}}+{\mathtt{c}}_{\mathtt{h}}^{-1}\mathtt{h}(1-{\mathtt{c}}_{\mathtt{h}}^{4})(\gamma_{\mathtt{h}}-2(1-{\mathtt{c}}_{\mathtt{h}}^{2}\mathtt{h}))\,\,.$
(4.42)

###### Proof.

We have to compute the expansion of the matrix entries
$({\mathcal{B}}^{\flat}f^{\sigma}_{k}(\mu,\epsilon),f^{\sigma^{\prime}}_{k^{\prime}}(\mu,\epsilon))$.
First, by (4.6), (4.17) and since $\mathtt{f}_{\epsilon}=O(\epsilon^{2})$
(cfr. (2.15)) we have

$\displaystyle{\mathcal{B}}^{\flat}f^{-}_{0}(\mu,\epsilon)=\begin{bmatrix}0\\\
\mu\tanh\big{(}\mathtt{h}\mu\big{)}\end{bmatrix}+\begin{bmatrix}0\\\
\mathcal{O}(\mu^{2}\epsilon)\end{bmatrix}\,.$

Hence, by (4.3)-(4.6), the entries of the last column (and row) of
$\mathtt{B}^{\flat}$ are

$\displaystyle\big{(}{\mathcal{B}}^{\flat}f^{-}_{0}(\mu,\epsilon),f^{+}_{1}(\mu,\epsilon)\big{)}=\mathcal{O}(\mu^{2}\epsilon)\
,\quad\big{(}{\mathcal{B}}^{\flat}f^{-}_{0}(\mu,\epsilon),f^{-}_{1}(\mu,\epsilon)\big{)}=\mu\tanh(\mathtt{h}\mu)\mathcal{O}(\epsilon^{2})+\mathcal{O}(\mu^{2}\epsilon^{2})=\mathcal{O}(\mu^{2}\epsilon^{2})$
$\displaystyle\big{(}{\mathcal{B}}^{\flat}f^{-}_{0}(\mu,\epsilon),f^{+}_{0}(\mu,\epsilon)\big{)}=\mathcal{O}(\mu^{2}\epsilon,\mu^{3})\
,\quad\big{(}{\mathcal{B}}^{\flat}f^{-}_{0}(\mu,\epsilon),f^{-}_{0}(\mu,\epsilon)\big{)}=\mu\tanh(\mathtt{h}\mu)+\mathcal{O}(\mu^{2}\epsilon)\,,$

in agreement with (4.41).

In order to compute the other matrix entries we expand ${\mathcal{B}}^{\flat}$
in (4.17) at $\mu=0$, obtaining

$\displaystyle{\mathcal{B}}^{\flat}=\mu{\mathcal{B}}^{\flat}_{1}{(0)}+\mu{\mathcal{R}}^{\flat}(\epsilon)+\mu^{2}{\mathcal{B}}^{\flat}_{2}+{\mathcal{O}(\mu^{2}\epsilon,\mu^{3})\,,\quad\text{where}}$
(4.43)
$\displaystyle{\mathcal{B}}^{\flat}_{1}(0):=\Big{[}\mathtt{h}D\big{(}1-\tanh^{2}(\mathtt{h}|D|)\big{)}+\operatorname*{sgn}(D)\tanh(\mathtt{h}|D|)\Big{]}\Pi_{{\mathrm{I\\!I}}}\,,\quad\Pi_{{\mathrm{I\\!I}}}:=\begin{bmatrix}0&0\\\
0&\mathrm{Id}\end{bmatrix}\,,$
$\displaystyle{\mathcal{R}}^{\flat}(\epsilon):=\mathcal{O}(\epsilon^{2})\Pi_{{\mathrm{I\\!I}}}\,,\qquad{\mathcal{B}}^{\flat}_{2}:={\Big{[}\mathtt{h}\big{(}1-\tanh^{2}(\mathtt{h}|D|)\big{)}\big{(}1-\mathtt{h}\tanh(\mathtt{h}|D|)|D|\big{)}\Big{]}\Pi_{{\mathrm{I\\!I}}}\,.}$

We note that

$\mu\big{(}{\mathcal{R}}^{\flat}(\epsilon)f^{\sigma}_{k}(\mu,\epsilon),f^{\sigma^{\prime}}_{k^{\prime}}(\mu,\epsilon)\big{)}=\mu\big{(}{\mathcal{R}}^{\flat}f^{\sigma}_{k}(0,\epsilon),f^{\sigma^{\prime}}_{k^{\prime}}(0,\epsilon)\big{)}+\mathcal{O}(\mu^{2}\epsilon^{2})=\begin{cases}\mathcal{O}(\mu^{2}\epsilon^{2})&\mbox{if
}\sigma=\sigma^{\prime}\,,\\\ \mathcal{O}(\mu\epsilon^{2})&\mbox{if
}\sigma\neq\sigma^{\prime}\,.\end{cases}$ (4.44)

Indeed, if $\sigma=\sigma^{\prime}$,
$\big{(}{\mathcal{R}}^{\flat}f^{\sigma}_{k}(0,\epsilon),f^{\sigma^{\prime}}_{k^{\prime}}(0,\epsilon)\big{)}$
is real by (3.13), but purely imaginary444 An operator $\mathcal{A}$ is
_purely imaginary_ if $\overline{\mathcal{A}}=-\mathcal{A}$. A purely
imaginary operator sends real functions into purely imaginary ones. too, since
the operator ${\mathcal{R}}^{\flat}$ is purely imaginary (as
${\mathcal{B}}^{\flat}$ is) and the basis
$\\{f_{k}^{\pm}(0,\epsilon)\\}_{k=0,1}$ is real. The terms (4.44) contribute
to $r_{2}(\mu\epsilon^{2})$ and $r_{6}(\epsilon\mu)$ in (4.41).

Next we compute the other scalar products. By (4.3), (4.43), and the
identities $\operatorname*{sgn}(D)\sin(kx)=-\mathrm{i}\,\cos(kx)$ and
$\operatorname*{sgn}(D)\cos(kx)=\mathrm{i}\,\sin(kx)$ for any
$k\in\mathbb{N}$, we have

$\mu{\mathcal{B}}^{\flat}_{1}(0)f^{+}_{1}(\mu,\epsilon)=\footnotesize-\mathrm{i}\,\mu\text{\large\bf$\flat$}_{1}\begin{bmatrix}0\\\
\cos(x)\end{bmatrix}-\frac{\mu^{2}}{4}\gamma_{\mathtt{h}}\text{\large\bf$\flat$}_{1}\begin{bmatrix}0\\\
\sin(x)\end{bmatrix}-\mathrm{i}\,\mu\epsilon\text{\large\bf$\flat$}_{2}\begin{bmatrix}0\\\
\cos(2x)\end{bmatrix}+\mathrm{i}\,\mathcal{O}(\mu\epsilon^{2})\begin{bmatrix}0\\\
even_{0}(x)\end{bmatrix}+\mathcal{O}(\mu^{2}\epsilon,\mu^{3})$

where

$\displaystyle\text{\large\bf$\flat$}_{1}:={\mathtt{c}}_{\mathtt{h}}^{-\frac{1}{2}}({\mathtt{c}}_{\mathtt{h}}^{2}+(1-{\mathtt{c}}_{\mathtt{h}}^{4})\mathtt{h})$
(4.45)
$\displaystyle\text{\large\bf$\flat$}_{2}:=\beta_{\mathtt{h}}\Big{(}\tanh(2\mathtt{h})+2\mathtt{h}(1-\tanh^{2}(2\mathtt{h}))\Big{)}=\beta_{\mathtt{h}}\Big{(}\frac{2{\mathtt{c}}_{\mathtt{h}}^{2}}{1+{\mathtt{c}}_{\mathtt{h}}^{4}}+2\mathtt{h}\big{(}1-\frac{4{\mathtt{c}}_{\mathtt{h}}^{4}}{(1+{\mathtt{c}}_{\mathtt{h}}^{4})^{2}}\big{)}\Big{)}\,.$

Similarly
$\mu^{2}{\mathcal{B}}^{\flat}_{2}f^{+}_{1}(\mu,\epsilon)=\mu^{2}\text{\large\bf$\flat$}_{3}\footnotesize{\begin{bmatrix}0\\\
\sin(x)\end{bmatrix}}+\mathcal{O}(\mu^{2}\epsilon,\mu^{3})$, where

$\text{\large\bf$\flat$}_{3}:=\mathtt{h}\big{(}1-\tanh^{2}(\mathtt{h})\big{)}\big{(}1-\tanh(\mathtt{h})\mathtt{h}\big{)}{\mathtt{c}}_{\mathtt{h}}^{-\frac{1}{2}}=\mathtt{h}(1-{\mathtt{c}}_{\mathtt{h}}^{4})(1-{\mathtt{c}}_{\mathtt{h}}^{2}\mathtt{h}){\mathtt{c}}_{\mathtt{h}}^{-\frac{1}{2}}\,.$
(4.46)

Analogously, using (4.4),

$\footnotesize\mu{\mathcal{B}}^{\flat}_{1}(0)f^{-}_{1}(\mu,\epsilon)=\mathrm{i}\,\mu\text{\large\bf$\flat$}_{1}\begin{bmatrix}0\\\
\sin(x)\end{bmatrix}-\frac{\mu^{2}}{4}\gamma_{\mathtt{h}}\text{\large\bf$\flat$}_{1}\begin{bmatrix}0\\\
\cos(x)\end{bmatrix}+\mathrm{i}\,\mu\epsilon\text{\large\bf$\flat$}_{3}\begin{bmatrix}0\\\
\sin(2x)\end{bmatrix}+\mathrm{i}\,\mathcal{O}(\mu\epsilon^{2})\begin{bmatrix}0\\\
odd(x)\end{bmatrix}+\mathcal{O}(\mu^{2}\epsilon,\mu^{3})\,,$

and
$\mu^{2}{\mathcal{B}}^{\flat}_{2}f^{-}_{1}(\mu,\epsilon)=\mu^{2}\text{\large\bf$\flat$}_{3}\footnotesize{\begin{bmatrix}0\\\
\cos(x)\end{bmatrix}}+\mathcal{O}(\mu^{2}\epsilon,\mu^{3})$, with
$\text{\large\bf$\flat$}_{j}$, $j=1,2,3$, defined in (4.45) and (4.46). In
addition, by (4.5)-(4.6), we get that

$\mu{\mathcal{B}}^{\flat}_{1}(0)f^{+}_{0}(\mu,\epsilon)=\mathrm{i}\,\mu\epsilon\delta_{\mathtt{h}}\text{\large\bf$\flat$}_{1}\begin{bmatrix}0\\\
\cos(x)\end{bmatrix}+\mathrm{i}\,\mathcal{O}(\mu\epsilon^{2})\begin{bmatrix}0\\\
even_{0}(x)\end{bmatrix}+\mathcal{O}(\mu^{2}\epsilon)\,,\ \
\mu^{2}{\mathcal{B}}^{\flat}_{2}f^{+}_{0}(\mu,\epsilon)=\begin{bmatrix}0\\\
\mathcal{O}(\mu^{2}\epsilon)\end{bmatrix}\,$

with $\text{\large\bf$\flat$}_{1}$ in (4.45). By taking the scalar products of
the above expansions of ${\mathcal{B}}^{\flat}f^{\sigma}_{k}(\mu,\epsilon)$
with the functions $f^{\sigma^{\prime}}_{k^{\prime}}(\mu,\epsilon)$ expanded
as in (4.3)-(4.6) we obtain that (recall that the scalar product is conjugate-
linear in the second component)

$\displaystyle\big{(}\mu{\mathcal{B}}^{\flat}_{1}(0)f^{+}_{1}(\mu,\epsilon),f^{+}_{1}(\mu,\epsilon)\big{)}\,,\
\big{(}\mu{\mathcal{B}}^{\flat}_{1}(0)f^{-}_{1}(\mu,\epsilon),f^{-}_{1}(\mu,\epsilon)\big{)}={-\frac{\mu^{2}}{4}\gamma_{\mathtt{h}}\text{\large\bf$\flat$}_{1}{\mathtt{c}}_{\mathtt{h}}^{-\frac{1}{2}}}+\mathcal{O}(\mu^{2}\epsilon,\mu^{3})$
$\displaystyle\big{(}\mu^{2}{\mathcal{B}}^{\flat}_{2}f^{+}_{1}(\mu,\epsilon),f^{+}_{1}(\mu,\epsilon)\big{)}\,,\
\big{(}\mu^{2}{\mathcal{B}}^{\flat}_{2}f^{-}_{1}(\mu,\epsilon),f^{-}_{1}(\mu,\epsilon)\big{)}={\frac{\mu^{2}}{2}\text{\large\bf$\flat$}_{3}{\mathtt{c}}_{\mathtt{h}}^{-\frac{1}{2}}}+\mathcal{O}(\mu^{2}\epsilon,\mu^{3})$

and, recalling (4.43), (4.45), (4.46), we deduce the expansion of the entries
$(1,1)$ and $(2,2)$ of the matrix $\mathtt{B}^{\flat}$ in (4.41) with
$\text{{b}}_{\mathtt{h}}={\mathtt{c}}_{\mathtt{h}}^{-\frac{1}{2}}(\gamma_{\mathtt{h}}\text{\large\bf$\flat$}_{1}-2\text{\large\bf$\flat$}_{3})$
in (4.42). Moreover

$\big{(}\mu{\mathcal{B}}^{\flat}_{1}(0)f^{-}_{1}(\mu,\epsilon),f^{+}_{1}(\mu,\epsilon)\big{)}={\mathrm{i}\,\frac{\mu}{2}\mathtt{e}_{12}}+\mathcal{O}(\mu\epsilon^{2},\mu^{2}\epsilon,\mu^{3})\,,\
\
\big{(}\mu^{2}{\mathcal{B}}^{\flat}_{2}f^{-}_{1}(\mu,\epsilon),f^{+}_{1}(\mu,\epsilon)\big{)}=\mathcal{O}(\mu^{3},\mu^{2}\epsilon)\,,$

where
$\mathtt{e}_{12}:=\text{\large\bf$\flat$}_{1}{\mathtt{c}}_{\mathtt{h}}^{-\frac{1}{2}}$
is equal to (1.2). Finally we obtain

$\displaystyle\big{(}\mu({\mathcal{B}}^{\flat}_{1}(0)+\mu{\mathcal{B}}^{\flat}_{2})f^{-}_{1}(\mu,\epsilon),f^{+}_{0}(\mu,\epsilon)\big{)}=\mathcal{O}(\mu\epsilon,\mu^{3})$
$\displaystyle(\mu({\mathcal{B}}^{\flat}_{1}(0)+\mu{\mathcal{B}}^{\flat}_{2})f^{+}_{1}(\mu,\epsilon),f^{+}_{0}(\mu,\epsilon))=\mathcal{O}(\mu^{3},\mu^{2}\epsilon)\,,$
$\displaystyle\big{(}\mu({\mathcal{B}}^{\flat}_{1}(0)+\mu{\mathcal{B}}^{\flat}_{2})f^{+}_{0}(\mu,\epsilon),f^{+}_{0}(\mu,\epsilon)\big{)}=\mathcal{O}(\mu^{2}\epsilon^{2})\,.$

The expansion (4.41) is proved. ∎

Finally we consider ${\mathcal{B}}^{\sharp}$.

######  Lemma 4.6.

(Expansion of $\mathtt{B}^{\sharp}$) The self-adjoint and reversibility-
preserving matrix $\mathtt{B}^{\sharp}$ associated, as in (3.12), to the self-
adjoint and reversibility-preserving operators ${\mathcal{B}}^{\sharp}$,
defined in (4.18), with respect to the basis $\mathcal{F}$ of
${\mathcal{V}}_{\mu,\epsilon}$ in (4.1), admits the expansion

$\mathtt{B}^{\sharp}=\begin{pmatrix}0&\mathrm{i}\,r_{2}(\mu\epsilon^{2})&\vline&0&\mathrm{i}\,\mu\epsilon{\mathtt{c}}_{\mathtt{h}}^{-\frac{1}{2}}+\mathrm{i}\,r_{4}(\mu\epsilon^{2})\\\
-\mathrm{i}\,r_{2}(\mu\epsilon^{2})&0&\vline&-\mathrm{i}\,r_{6}(\mu\epsilon)&0\\\
\hline\cr
0&\mathrm{i}\,r_{6}(\mu\epsilon)&\vline&0&-\mathrm{i}\,r_{9}(\mu\epsilon^{2})\\\
-\mathrm{i}\,\mu\epsilon{\mathtt{c}}_{\mathtt{h}}^{-\frac{1}{2}}-\mathrm{i}\,r_{4}(\mu\epsilon^{2})&0&\vline&\mathrm{i}\,r_{9}(\mu\epsilon^{2})&0\end{pmatrix}+\mathcal{O}(\mu^{2}\epsilon)\,.$
(4.47)

###### Proof.

Since ${\mathcal{B}}^{\sharp}=-\mathrm{i}\,\mu p_{\epsilon}\mathcal{J}$ and
$p_{\epsilon}=\mathcal{O}(\epsilon)$ by (2.19), we have the expansion

$\big{(}{\mathcal{B}}^{\sharp}f_{k}^{\sigma}(\mu,\epsilon),f_{k^{\prime}}^{\sigma^{\prime}}(\mu,\epsilon)\big{)}=\big{(}{\mathcal{B}}^{\sharp}f_{k}^{\sigma}(0,\epsilon),f_{k^{\prime}}^{\sigma^{\prime}}(0,\epsilon)\big{)}+\mathcal{O}(\mu^{2}\epsilon)\,.$
(4.48)

The matrix entries
$({\mathcal{B}}^{\sharp}f^{\sigma}_{k}(0,\epsilon),f^{\sigma}_{k^{\prime}}(0,\epsilon))$,
$k,k^{\prime}=0,1$, $\sigma=\\{\pm\\}$ are zero, because they are
simultaneously real by (3.13), and purely imaginary, being the operator
${\mathcal{B}}^{\sharp}$ purely imaginary and the basis
$\\{f_{k}^{\pm}(0,\epsilon)\\}_{k=0,1}$ real. Hence $\mathtt{B}^{\sharp}$ has
the form

$\mathtt{B}^{\sharp}=\begin{pmatrix}0&\mathrm{i}\,\beta&\vline&0&\mathrm{i}\,\delta\\\
-\mathrm{i}\,\beta&0&\vline&-\mathrm{i}\,\gamma&0\\\ \hline\cr
0&\mathrm{i}\,\gamma&\vline&0&\mathrm{i}\,\eta\\\
-\mathrm{i}\,\delta&0&\vline&-\mathrm{i}\,\eta&0\end{pmatrix}+\mathcal{O}(\mu^{2}\epsilon)\quad\text{where}\quad\left\\{\begin{matrix}\left({\mathcal{B}}^{\sharp}f_{1}^{-}(0,\epsilon)\,,\,f_{1}^{+}(0,\epsilon)\right)=:\mathrm{i}\,\beta\,,\\\
\left({\mathcal{B}}^{\sharp}f_{1}^{-}(0,\epsilon)\,,\,f_{0}^{+}(0,\epsilon)\right)=:\mathrm{i}\,\gamma\,,\\\
\left({\mathcal{B}}^{\sharp}f_{0}^{-}(0,\epsilon)\,,\,f_{1}^{+}(0,\epsilon)\right)=:\mathrm{i}\,\delta\,,\\\
\left({\mathcal{B}}^{\sharp}f_{0}^{-}(0,\epsilon)\,,\,f_{0}^{+}(0,\epsilon)\right)=:\mathrm{i}\,\eta\,,\end{matrix}\right.$
(4.49)

and $\alpha$, $\beta$, $\gamma$, $\delta$ are real numbers. As
${\mathcal{B}}^{\sharp}=\mathcal{O}(\mu\epsilon)$ in $\mathcal{L}(Y)$, we
deduce that $\gamma=r(\mu\epsilon)$. Let us compute the expansion of $\beta$,
$\delta$ and $\eta$. By (2.20) and (2.2) we write the operator
${\mathcal{B}}^{\sharp}$ in (4.18) as

${\mathcal{B}}^{\sharp}=\mathrm{i}\,\mu\epsilon{\mathcal{B}}_{1}^{\sharp}+\mathcal{O}(\mu\epsilon^{2})\,,\quad{\mathcal{B}}_{1}^{\sharp}:=2{\mathtt{c}}_{\mathtt{h}}^{-1}\cos(x)\begin{bmatrix}0&\mathrm{Id}\\\
-\mathrm{Id}&0\end{bmatrix}\,,$ (4.50)

with $\mathcal{O}(\mu\epsilon^{2})\in\mathcal{L}(Y)$. In view of (4.3)-(4.6),
$f_{1}^{\pm}(0,\epsilon)=f_{1}^{\pm}+\mathcal{O}(\epsilon)$,
$f_{0}^{+}(0,\epsilon)=f_{0}^{+}+\mathcal{O}(\epsilon)$,
$f_{0}^{-}(0,\epsilon)=\footnotesize\begin{bmatrix}0\\\ 1\end{bmatrix}$, where
$f_{k}^{\sigma}$ are in (4.2). By (4.50) we have
$\footnotesize{\mathcal{B}}_{1}^{\sharp}f_{1}^{-}=\begin{bmatrix}{\mathtt{c}}_{\mathtt{h}}^{-\frac{3}{2}}(1+\cos(2x))\\\
{\mathtt{c}}_{\mathtt{h}}^{-\frac{1}{2}}\sin(2x)\end{bmatrix}$,
$\footnotesize{\mathcal{B}}_{1}^{\sharp}f_{0}^{-}=\begin{bmatrix}2{\mathtt{c}}_{\mathtt{h}}^{-1}\cos(x)\\\
0\end{bmatrix}$ and then

$\displaystyle\beta=\mu\epsilon\left({\mathcal{B}}_{1}^{\sharp}f_{1}^{-}\,,\,f_{1}^{+}\right)+r(\mu\epsilon^{2})=r(\mu\epsilon^{2})\,,$
$\displaystyle\delta=\mu\epsilon\left({\mathcal{B}}_{1}^{\sharp}f_{0}^{-}\,,\,f_{1}^{+}\right)+r(\mu\epsilon^{2})=\mu\epsilon{\mathtt{c}}_{\mathtt{h}}^{-\frac{1}{2}}+r(\mu\epsilon^{2})\,,$
$\displaystyle\eta=\mu\epsilon\left({\mathcal{B}}_{1}^{\sharp}f_{0}^{-}\,,\,f_{0}^{+}\right)+r(\mu\epsilon^{2})=r(\mu\epsilon^{2})\,.$

This proves (4.47). ∎

Lemmata 4.4, 4.5, 4.6 imply (4.9) where the matrix $E$ has the form (4.10) and

$\mathtt{e}_{22}:=2(\textbf{b}_{\mathtt{h}}-4\zeta_{\mathtt{h}})=2\gamma_{\mathtt{h}}{\mathtt{c}}_{\mathtt{h}}+2{\mathtt{c}}_{\mathtt{h}}^{-1}\mathtt{h}(1-{\mathtt{c}}_{\mathtt{h}}^{4})(\gamma_{\mathtt{h}}-2(1-{\mathtt{c}}_{\mathtt{h}}^{2}\mathtt{h}))-{\mathtt{c}}_{\mathtt{h}}\gamma_{\mathtt{h}}^{2}\,,$

with $\textbf{b}_{\mathtt{h}}$ in (4.42) and $\zeta_{\mathtt{h}}$ in (4.20).
The term $\mathtt{e}_{22}$ has the expansion in (1.3). Moreover

$\displaystyle
G:=G(\mu,\epsilon)=\begin{pmatrix}1+r_{8}(\epsilon^{2},\mu^{2}\epsilon,\mu^{3})&-\mathrm{i}\,r_{9}(\mu\epsilon^{2},\mu^{2}\epsilon,\mu^{3})\\\
\mathrm{i}\,r_{9}(\mu\epsilon^{2},\mu^{2}\epsilon,\mu^{3})&\mu\tanh(\mathtt{h}\mu)+r_{10}(\mu^{2}\epsilon,\mu^{3})\end{pmatrix}$
(4.51) $\displaystyle
F:=F(\mu,\epsilon)=\begin{pmatrix}\mathtt{f}_{11}\epsilon+r_{3}(\epsilon^{3},\mu\epsilon^{2},\mu^{2}\epsilon,\mu^{3})&\mathrm{i}\,\mu\epsilon{\mathtt{c}}_{\mathtt{h}}^{-\frac{1}{2}}+\mathrm{i}\,r_{4}({\mu\epsilon^{2}},\mu^{2}\epsilon,\mu^{3})\\\
\mathrm{i}\,r_{6}(\mu\epsilon,\mu^{3})&r_{7}(\mu^{2}\epsilon,\mu^{3})\end{pmatrix}\,.$
(4.52)

In order to deduce the expansion (4.11)-(4.12) of the matrices $F,G$ we
exploit further information for

$\mathscr{L}_{\mu,0}:=\mathcal{J}{\mathcal{B}}_{\mu,0}\,,\quad{\mathcal{B}}_{\mu,0}:=\begin{bmatrix}1&-{\mathtt{c}}_{\mathtt{h}}\partial_{x}\\\
{\mathtt{c}}_{\mathtt{h}}\partial_{x}&|D+\mu|\,\tanh\big{(}\mathtt{h}|D+\mu|\big{)}\end{bmatrix}\,.$
(4.53)

We have

######  Lemma 4.7.

At $\epsilon=0$ the matrices are $F(\mu,0)=0$ and
$G(\mu,0)=\begin{pmatrix}1&0\\\ 0&\mu\tanh(\mathtt{h}\mu)\end{pmatrix}$.

###### Proof.

By Lemma A.5 and (4.53) we have
${\mathcal{B}}_{\mu,0}f_{0}^{+}(\mu,0)=f_{0}^{+}$ and
${\mathcal{B}}_{\mu,0}f_{0}^{-}(\mu,0)=\mu\tanh(\mathtt{h}\mu)f_{0}^{-}$, for
any $\mu$. Then the lemma follows recalling (3.12) and the fact that
$f_{1}^{+}(\mu,0)$ and $f_{1}^{-}(\mu,0)$ have zero space average by Lemma
A.5. ∎

In view of Lemma 4.7 we deduce that the matrices (4.51) and (4.52) have the
form (4.11) and (4.12). This completes the proof of Proposition 4.3.

We now show that the constant $\mathtt{e}_{22}$ in (1.3) is positive for any
depth $\mathtt{h}>0$.

######  Lemma 4.8.

For any $\mathtt{h}>0$ the term $\mathtt{e}_{22}$ in (1.3) is positive,
$\mathtt{e}_{22}\to 0$ as $\mathtt{h}\to 0^{+}$ and $\mathtt{e}_{22}\to 1$ as
$\mathtt{h}\to+\infty$. As a consequence for any $\mathtt{h}_{0}>0$ the term
$\mathtt{e}_{22}$ is bounded from below uniformly in
$\mathtt{h}>\mathtt{h}_{0}$.

###### Proof.

The quantity $z:={\mathtt{c}}_{\mathtt{h}}^{2}=\tanh(\mathtt{h})$ is in
$(0,1)$ for any $\mathtt{h}>0$. Then the quadratic polynomial
$(0,+\infty)\ni\mathtt{h}\mapsto(1-z^{2})(1+3z^{2})\mathtt{h}^{2}+2z(z^{2}-1)\mathtt{h}+z^{2}$
is positive because its discriminant $-4z^{4}(1-z^{2})$ is negative as
$0<z^{2}<1$. The limits for $\mathtt{h}\to 0^{+}$ and $\mathtt{h}\to+\infty$
follow by inspection. ∎

##  5 Block-decoupling and emergence of the Whitham-Benjamin function

In this section we block-decouple the $4\times 4$ Hamiltonian matrix
$\mathtt{L}_{\mu,\epsilon}=\mathtt{J}_{4}\mathtt{B}_{\mu,\epsilon}$ obtained
in Proposition 4.3.

We first perform a singular symplectic and reversibility-preserving change of
coordinates.

######  Lemma 5.1.

(Singular symplectic rescaling) The conjugation of the Hamiltonian and
reversible matrix
$\mathtt{L}_{\mu,\epsilon}=\mathtt{J}_{4}\mathtt{B}_{\mu,\epsilon}$ obtained
in Proposition 4.3 through the symplectic and reversibility-preserving
$4\times 4$-matrix

$Y:=\begin{pmatrix}Q&0\\\ 0&Q\end{pmatrix}\quad\text{with}\quad
Q:=\begin{pmatrix}\mu^{\frac{1}{2}}&0\\\ 0&\mu^{-\frac{1}{2}}\end{pmatrix}\,,\
\ \mu>0\,,$ (5.1)

yields the Hamiltonian and reversible matrix

$\displaystyle\mathtt{L}_{\mu,\epsilon}^{(1)}:=Y^{-1}\mathtt{L}_{\mu,\epsilon}Y=\mathtt{J}_{4}\mathtt{B}^{(1)}_{\mu,\epsilon}=\begin{pmatrix}\mathtt{J}_{2}E^{(1)}&\mathtt{J}_{2}F^{(1)}\\\
\mathtt{J}_{2}[F^{(1)}]^{*}&\mathtt{J}_{2}G^{(1)}\end{pmatrix}$ (5.2)

where $\mathtt{B}_{\mu,\epsilon}^{(1)}$ is a self-adjoint and reversibility-
preserving $4\times 4$ matrix

$\mathtt{B}_{\mu,\epsilon}^{(1)}=\begin{pmatrix}E^{(1)}&F^{(1)}\\\
[F^{(1)}]^{*}&G^{(1)}\end{pmatrix},\quad E^{(1)}=[E^{(1)}]^{*}\,,\
G^{(1)}=[G^{(1)}]^{*}\,,$ (5.3)

where the $2\times 2$ reversibility-preserving matrices $E^{(1)}$, $G^{(1)}$
and $F^{(1)}$ extend analytically at $\mu=0$ with the following expansion

$\displaystyle
E^{(1)}=\begin{pmatrix}\mathtt{e}_{11}\mu\epsilon^{2}(1+r_{1}^{\prime}(\epsilon,\mu\epsilon))-\mathtt{e}_{22}\frac{\mu^{3}}{8}(1+r_{1}^{\prime\prime}(\epsilon,\mu))&\mathrm{i}\,\big{(}\frac{1}{2}\mathtt{e}_{12}\mu+r_{2}(\mu\epsilon^{2},\mu^{2}\epsilon,\mu^{3})\big{)}\\\
-\mathrm{i}\,\big{(}\frac{1}{2}\mathtt{e}_{12}\mu+r_{2}(\mu\epsilon^{2},\mu^{2}\epsilon,\mu^{3})\big{)}&-\mathtt{e}_{22}\frac{\mu}{8}(1+r_{5}(\epsilon,\mu))\end{pmatrix}\,,$
(5.4) $\displaystyle
G^{(1)}=\begin{pmatrix}\mu+r_{8}(\mu\epsilon^{2},\mu^{3}\epsilon)&-\mathrm{i}\,r_{9}(\mu\epsilon^{2},\mu^{2}\epsilon)\\\
\mathrm{i}\,r_{9}(\mu\epsilon^{2},\mu^{2}\epsilon)&\tanh(\mathtt{h}\mu)+r_{10}(\mu\epsilon)\end{pmatrix}\,,$
(5.5) $\displaystyle
F^{(1)}=\begin{pmatrix}\mathtt{f}_{11}\mu\epsilon+r_{3}(\mu\epsilon^{3},\mu^{2}\epsilon^{2},\mu^{3}\epsilon)&\mathrm{i}\,\mu\epsilon{\mathtt{c}}_{\mathtt{h}}^{-\frac{1}{2}}+\mathrm{i}\,r_{4}(\mu\epsilon^{2},\mu^{2}\epsilon)\\\
\mathrm{i}\,r_{6}(\mu\epsilon)&r_{7}(\mu\epsilon)\end{pmatrix}$ (5.6)

where $\mathtt{e}_{11},\mathtt{e}_{12},\mathtt{e}_{22},\mathtt{f}_{11}$ are
defined in (4.13), (1.2), (1.3).

######  Remark 5.2.

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

###### Proof.

The matrix $Y$ is symplectic, i.e. (3.15) holds, and since $\mu$ is real, it
is reversibility preserving, i.e. satisfies (3.13). By (3.16),

$\mathtt{B}_{\mu,\epsilon}^{(1)}=Y^{*}\mathtt{B}_{\mu,\epsilon}Y=\begin{pmatrix}E^{(1)}&F^{(1)}\\\
[F^{(1)}]^{*}&G^{(1)}\end{pmatrix},$

with, $Q$ being self-adjoint, $E^{(1)}=QEQ=[E^{(1)}]^{*}$,
$G^{(1)}=QGQ=[G^{(1)}]^{*}$ and $F^{(1)}=QFQ$. In view of (4.10)-(4.12), we
obtain (5.4)-(5.6). ∎

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

We first verify that the quantity
$D_{\mathtt{h}}:=\mathtt{h}-\tfrac{1}{4}\mathtt{e}_{12}^{2}$ is nonzero for
any $\mathtt{h}>0$. In view of the comment 3 after Theorem 1.1, we have that