# Full description of Benjamin-Feir instability
for generalized Korteweg-de Vries equations

Alberto Maspero, Antonio Milosh Radakovic111 International School for Advanced
Studies (SISSA), Via Bonomea 265, 34136, Trieste, Italy. Emails:
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In this paper we consider a family of generalized Korteweg-de Vries equations
and study the linear modulational instability of small amplitude traveling
waves solutions. Under explicit non-degeneracy conditions on the dispersion
relation, we completely describe the spectrum near the origin of the
linearized operator at such solutions and prove that the unstable spectrum
(when present) is composed by branches depicting always a closed figure “8”.
We apply our abstract theorem to several equations such as the Whitham, the
gravity-capillary Whitham and the Kawahara equations, confirming that the
unstable spectrum of the corresponding linearized operators exhibits a figure
“8” instability, as it was observed before only numerically. Our method of
proof uses a symplectic version of Kato’s theory of similarity transformation
to reduce the problem to determine the eigenvalues of a $3\times 3$ complex
Hamiltonian and reversible matrix. Then, via a block-diagonalization
procedure, we conjugate such matrix into a block-diagonal one composed by a
$2\times 2$ Hamiltonian and reversible matrix, describing the unstable
spectrum, and a single purely imaginary element describing the stable
eigenvalue.

##  1 Introduction

The generalized Korteweg-de Vries (gKdV) equations are a family of dispersive
partial differential equations of the form

$\partial_{t}u+{\mathcal{M}}(D)u_{x}+(u^{2})_{x}=0,\quad x\in{\mathbb{R}}\ ,$
(1.1)

where $u(t,x)$ is a real valued function and ${\mathcal{M}}(D)$ a Fourier
multiplier with a real valued symbol ${\mathfrak{m}}(\xi)$, defined by

$\widehat{[{\mathcal{M}}(D)u]}(\xi):={\mathfrak{m}}(\xi)\hat{u}(\xi)\ .$ (1.2)

Here $\hat{u}(\xi):=\int_{\mathbb{R}}u(x)e^{-{\rm i}\xi x}$. The family of
equations (1.1) includes many well-known dispersive PDEs such as the Korteweg-
de-Vries, the Benjamin-Ono, the Whitham, the Kawahara equations. In Table 1
below we report the values of the corresponding symbols.

Under very general assumptions on the symbol ${\mathfrak{m}}(\xi)$, equation
(1.1) admits global in time solutions in the form of space-periodic waves
traveling at constant speed (they are steady in a moving frame). A classical
problem is to determine their stability/instability when subjected to long
wave perturbations. Such problem, usually called modulational instability,
goes back to the pioneering works by Benjamin-Feir [1], Whitham [25],
Lighthill [21], Zakharov [26] in the context of water waves, recently fully
and rigorously justified by [6, 22, 2, 3, 4, 5, 18, 12].

The first, fundamental step to understand modulational instability is to study
the spectrum of the linear operator obtained linearizing (1.1) at a traveling
wave solution: in case it intersects the positive real plane, the traveling
wave is called linearly modulationally unstable. Then, at least for equation
(1.1), the general theory developed by Jin-Liao-Lin in [19] can be used to
upgrade the linear instability to a nonlinear one.

Linear modulational instability for equation (1.1) has been studied both for
small amplitudes waves [20, 15, 16, 23, 17] and even large amplitudes ones in
case of KdV [14, 10, 8] (corresponding to ${\mathfrak{m}}(\xi)=\xi^{2}$). All
these works prove the existence of unstable spectrum near the origin.

The goal of the present paper, following the approach recently developed by
Berti-Maspero-Ventura [2, 3, 4], is to completely characterize the spectrum
near the origin of the complex plane of the operator obtained linearizing
(1.1) at small amplitude traveling waves. We prove that, provided the symbol
${\mathfrak{m}}(\xi)$ fulfills certain explicit and mild non-degeneracy
conditions (see (1.10)–(1.11)), such spectrum is either purely imaginary or it
possesses unstable spectral branches outside the imaginary axis depicting
always a complete figure “8”. This dichotomy is determined by the sign of a
certain coefficient, see (1.16), depending only on few values of
${\mathfrak{m}}(\xi)$ and its derivatives.

We apply our abstract criterion to several equations in the family of
generalized KdV, among which the finite depth Whitham equation and its
gravity-capillary and vorticity-modified variants, and the Kawahara equation.
In all these cases we prove that the unstable spectrum near the origin depicts
a complete figure “8”, as was observed before only numerically [15, 16, 11].

We now describe precisely our result. The first set of assumptions that we
make on the symbol ${\mathfrak{m}}(\xi)$ in (1.2) are rather mild and used to
guarantee the existence of small amplitude traveling wave solutions of (1.1),
see Theorem 1.1 below:

Assumption A: The symbol ${\mathfrak{m}}(\xi)$ of the Fourier multiplier
${\mathcal{M}}(D)$ in (1.2) fulfills:

  * (A1)

${\mathfrak{m}}(\xi)$ is real valued and even,
${\mathfrak{m}}(\xi)={\mathfrak{m}}(-\xi)$ for any $\xi\in{\mathbb{R}}$;

  * (A2)

${\mathfrak{m}}(\xi)$ is a function in
${\mathcal{C}}^{\mathtt{r}}({\mathbb{R}})$, $\mathtt{r}\geq 3$, fulfilling for
some $m\geq-1$ and $\xi$ sufficiently large: there exist $C_{1},C_{2}>0$ such
that

$C_{1}\langle\xi\rangle^{m}\leq{\mathfrak{m}}(\xi)\leq
C_{2}\langle\xi\rangle^{m}\ ,\ \ \forall|\xi|\gg 1\ ;$ (1.3)

here we put $\langle\xi\rangle:=(1+\xi^{2})^{1/2}$;

  * (A3)

there exists $c_{0}>0$ such that

$\inf_{n\in{\mathbb{N}}\setminus\\{1\\}}|{\mathfrak{m}}(n)-{\mathfrak{m}}(1)|\geq
c_{0}\ .$ (1.4)

We list now some symbols ${\mathfrak{m}}(\xi)$ fulfilling Assumption A, and
the corresponding equations (1.1) generated:

Table 1: The correspondence between the symbol and the equation of form (1.1) obtained. Symbol ${\mathfrak{m}}(\xi)$ | Equation
---|---
$\xi^{2}$ | KdV
$|\xi|^{\alpha},\ \ \alpha\geq 3$ | fractional KdV
$|\xi|$ 222Notice that, due to the symmetry of the spectrum $\sigma(A_{-\mu})=\overline{\sigma(A_{\mu})}$, it is equivalent to ask that $m(\xi)$ is ${\mathcal{C}}^{3}$ at any $\xi>0$, has right derivatives up to third order at $\xi=0$, and every derivative is continuous at $\xi=0$. With this extension, Assumption A is satisfied also in the case of the Benjamin-Ono equation. In this case, when in the proof below will appear $\frac{{\rm d}^{n}}{{\rm d}\mu^{n}}{\mathfrak{m}}(0)$, we mean the $n$-th right derivative of ${\mathfrak{m}}$ at $0$, and so $\ddot{\mathfrak{m}}(0)=0$. , | Benjamin-Ono
$\sqrt{\dfrac{\tanh({\mathtt{h}}\xi)}{\xi}}$, ${\mathtt{h}}>0$, | Whitham equation
$\sqrt{(1+\kappa\xi^{2})\frac{\tanh({\mathtt{h}}\xi)}{\xi}}$, ${\mathtt{h}},\kappa>0$, | Whitham with surface tension
$\xi\coth({\mathtt{h}}\xi)-{\mathtt{h}}^{-1}$ | intermediate long-wave
$\xi^{2}+{\mathtt{b}}\xi^{4}$ | Kawahara

Under Assumption A, the gKdV equation in (1.1) admits families of small
amplitude traveling waves solutions, which are periodic in space, i.e. of the
form $u(t,x)=\breve{u}(x-ct)$ for some real $c$ – the speed – and $2\pi$
periodic function $\breve{u}(x)$. The following theorem is essentially known
in the literature, see e.g. [13, 15].

######  Theorem 1.1 ([13, 15]).

Assume Assumption A. There exist ${\epsilon}_{0},{\mathtt{b}}_{0}>0$ and a
locally unique non-zero family of smooth solutions
$(u_{{\epsilon},{\mathtt{b}}}(x),c_{{\epsilon},{\mathtt{b}}})$, analytically
parametrized by $|{\epsilon}|\leq{\epsilon}_{0}$,
$|{\mathtt{b}}|<{\mathtt{b}}_{0}$, of the stationary equation

$c\,u-{\mathcal{M}}(D)u-u^{2}+(1-c^{2}){\mathtt{b}}=0$ (1.5)

so that $u_{{\epsilon},{\mathtt{b}}}(x)$ is $2\pi$-periodic, even, real valued
and having at ${\mathtt{b}}=0$ the asymptotic expansion

$\displaystyle
u_{\epsilon}(x):=u_{{\epsilon},{\mathtt{b}}=0}(x)={\epsilon}\cos(x)+{\epsilon}^{2}\left(u_{2}^{[0]}+u_{2}^{[2]}\cos(2x)\right)+{\mathcal{O}}({\epsilon}^{3}),$
(1.6) $\displaystyle
c_{\epsilon}:=c_{{\epsilon},{\mathtt{b}}=0}={\mathfrak{m}}(1)+{\epsilon}^{2}c_{2}+{\mathcal{O}}({\epsilon}^{3}),$
(1.7)

where

$\displaystyle
u_{2}^{[0]}:=\frac{1}{2}\frac{1}{{\mathfrak{m}}(1)-{\mathfrak{m}}(0)},\qquad
u_{2}^{[2]}:=\frac{1}{2}\frac{1}{{\mathfrak{m}}(1)-{\mathfrak{m}}(2)},$ (1.8)
$\displaystyle
c_{2}:=\frac{1}{{\mathfrak{m}}(1)-{\mathfrak{m}}(0)}+\frac{1}{2}\frac{1}{{\mathfrak{m}}(1)-{\mathfrak{m}}(2)}\
.$ (1.9)

Moreover $u_{{\epsilon},{\mathtt{b}}}$, $c_{{\epsilon},{\mathtt{b}}}$ are
given by

$u_{{\epsilon},{\mathtt{b}}}(x)=u_{\epsilon}(x)+v({\epsilon},{\mathtt{b}})\,,\qquad
c_{{\epsilon},{\mathtt{b}}}=c_{\epsilon}-2v({\epsilon},{\mathtt{b}})\,,$

for a certain real valued smooth function $v=v({\epsilon},{\mathtt{b}})$
defined for $|{\epsilon}|\leq{\epsilon}_{0}$,
$|{\mathtt{b}}|<{\mathtt{b}}_{0}$.

We shall need also a second set of assumptions on the symbol
${\mathfrak{m}}(\xi)$, involving non-degeneracy conditions:

Assumption B: Denote by $\dot{\mathfrak{m}}(\xi):=\frac{{\rm d}}{{\rm
d}\xi}{\mathfrak{m}}(\xi)$ and $\ddot{\mathfrak{m}}(\xi):=\frac{{\rm
d}^{2}}{{\rm d}\xi^{2}}{\mathfrak{m}}(\xi)$. The numbers

$\displaystyle{\mathtt{e}}_{12}:=\dot{\mathfrak{m}}(1),\qquad{\mathtt{e}}_{d}:=\dot{\mathfrak{m}}(1)+{\mathfrak{m}}(1)-{\mathfrak{m}}(0)\
,\qquad{\mathtt{e}}_{b}:=-\dot{\mathfrak{m}}(1)-\frac{1}{2}\ddot{\mathfrak{m}}(1)$
(1.10)

and

${\mathtt{e}}_{w}:=\frac{\dot{\mathfrak{m}}(1)+3{\mathfrak{m}}(1)-2{\mathfrak{m}}(2)-{\mathfrak{m}}(0)}{\big{(}{\mathfrak{m}}(1)-{\mathfrak{m}}(2)\big{)}\,\big{(}\dot{\mathfrak{m}}(1)+{\mathfrak{m}}(1)-{\mathfrak{m}}(0)\big{)}}$
(1.11)

are all well defined and non-zero.

In order to determine the stability/instability of the traveling wave solution
given by Theorem 1.1, we linearize the equations of motion (1.1) at
$u_{\epsilon}$ and put ourself in a moving frame at speed $c_{\epsilon}$,
getting the linear system

$\partial_{t}h={\mathcal{L}}_{\epsilon}h$ (1.12)

where

${\mathcal{L}}_{\epsilon}:=\partial_{x}\circ(c_{\epsilon}-{\mathcal{M}}(D)-2u_{\epsilon})\
.$ (1.13)

Since the operator ${\mathcal{L}}_{\epsilon}$ has $2\pi$-periodic
coefficients, its spectrum on $L^{2}({\mathbb{R}},{\mathbb{C}})$ is mostly
conveniently described by Bloch-Floquet theory, which guarantees that

$\sigma_{L^{2}({\mathbb{R}})}({\mathcal{L}}_{\epsilon})=\bigcup_{\mu\in[-\frac{1}{2},\frac{1}{2})}\sigma_{L^{2}({\mathbb{T}})}({\mathcal{L}}_{\mu,{\epsilon}})$
(1.14)

where the Floquet operator ${\mathcal{L}}_{\mu,{\epsilon}}$ is given by

${\mathcal{L}}_{\mu,{\epsilon}}:=e^{-{\rm i}\mu
x}\,{\mathcal{L}}_{\epsilon}\,e^{{\rm i}\mu x}\ ,$ (1.15)

and acts as an (unbounded) operator on $L^{2}({\mathbb{T}},{\mathbb{C}})$.
Recall that if $\lambda$ is an eigenvalue of ${\mathcal{L}}_{\mu,{\epsilon}}$
on $L^{2}({\mathbb{T}},{\mathbb{C}})$ with eigenvector $v(x)\in
L^{2}({\mathbb{T}},{\mathbb{C}})$, then $h(t,x)=e^{\lambda t}e^{{\rm i}\mu
x}v(x)$ solves the linear problem (1.12).

  * •

Notation: We denote by
$r(\mu^{m_{1}}{\epsilon}^{n_{1}},\dots,\mu^{m_{p}}{\epsilon}^{n_{p}})$ real
valued functions of class at least ${\mathcal{C}}^{1}$ fulfilling, for small
values of $(\mu,{\epsilon})$, the bound
$|r(\mu^{m_{1}}{\epsilon}^{n_{1}},\dots,\mu^{m_{p}}{\epsilon}^{n_{p}})|\leq
C\sum_{j=1}^{p}|\mu|^{m_{j}}|{\epsilon}|^{n_{j}}$.

Our main result describes the spectrum near 0 of the operator
${\mathcal{L}}_{\mu,{\epsilon}}$ for any $(\mu,{\epsilon})$ sufficiently close
to $(0,0)$, value at which ${\mathcal{L}}_{0,0}\equiv{\mathcal{L}}_{0}$ has a
triple eigenvalue 0.

######  Theorem 1.2.

Assume Assumption A and B and denote

${\mathtt{e}}_{\scaleto{\mathrm{WB}}{3pt}}:={\mathtt{e}}_{w}\,{\mathtt{e}}_{b}=\frac{\big{(}\dot{\mathfrak{m}}(1)+\frac{1}{2}\ddot{\mathfrak{m}}(1)\big{)}\
\big{(}\dot{\mathfrak{m}}(1)+3{\mathfrak{m}}(1)-2{\mathfrak{m}}(2)-{\mathfrak{m}}(0)\big{)}}{\big{(}{\mathfrak{m}}(2)-{\mathfrak{m}}(1)\big{)}\,\big{(}\dot{\mathfrak{m}}(1)+{\mathfrak{m}}(1)-{\mathfrak{m}}(0)\big{)}}$
(1.16)

the Whitham-Benjamin coefficient for the generalized KdV (1.1). There exist
${\epsilon}_{1},\mu_{0}>0$ such that the following holds true:

  * (i)

If ${\mathtt{e}}_{\scaleto{\mathrm{WB}}{3pt}}>0$, there exists a function
$\underline{\mu}:[0,{\epsilon}_{1})\to[0,\mu_{0})$ of class
${\mathcal{C}}^{\mathtt{r}-2}$ of the type

$\underline{\mu}({\epsilon})=\sqrt{\frac{{\mathtt{e}}_{\scaleto{\mathrm{WB}}{3pt}}}{{\mathtt{e}}_{b}^{2}}}\,({\epsilon}+r({\epsilon}^{2}))$

such that, for any ${\epsilon}\in[0,{\epsilon}_{1})$, the spectrum of the
operator ${\mathcal{L}}_{\mu,{\epsilon}}$ near zero consists in the purely
imaginary eigenvalue

$\lambda_{0}(\mu,{\epsilon})={\rm
i}\big{(}{\mathfrak{m}}(1)-{\mathfrak{m}}(0)\big{)}\mu+{\rm i}\mu
r({\epsilon}^{2},\mu^{2})$ (1.17)

and two eigenvalues $\lambda_{1}^{\pm}(\mu,{\epsilon})$ of the form

$\lambda_{1}^{\pm}(\mu,{\epsilon})=\begin{cases}-{\rm
i}\mu\big{(}{\mathtt{e}}_{12}+r({\epsilon}^{2},\mu{\epsilon},\mu^{2})\big{)}\pm\mu\sqrt{\Delta_{\scaleto{\mathrm{BF}}{3pt}}}\,,&\mu\in[0,\underline{\mu}({\epsilon}))\\\
-{\rm
i}\mu\big{(}{\mathtt{e}}_{12}+r({\epsilon}^{3})\big{)}\,,&\mu=\underline{\mu}({\epsilon})\\\
-{\rm
i}\mu\big{(}{\mathtt{e}}_{12}+r({\epsilon}^{2},\mu{\epsilon},\mu^{2})\big{)}\pm{\rm
i}\mu\,\sqrt{|\Delta_{\scaleto{\mathrm{BF}}{3pt}}|}\,,&\mu\in[\underline{\mu}({\epsilon}),\mu_{0})\end{cases}$
(1.18)

with

$\Delta_{\scaleto{\mathrm{BF}}{3pt}}(\mu,{\epsilon}):={\mathtt{e}}_{\scaleto{\mathrm{WB}}{3pt}}\,{\epsilon}^{2}(1+\hat{r}_{1}({\epsilon},\mu))-{\mathtt{e}}_{b}^{2}\,\mu^{2}(1+\hat{r}_{2}({\epsilon},\mu))$
(1.19)

and some functions $r,\hat{r}_{1},\hat{r}_{2}$ of class
${\mathcal{C}}^{\mathtt{r}-2}$. The function $\underline{\mu}({\epsilon})$ is
implicitly defined by
$\Delta_{\scaleto{\mathrm{BF}}{3pt}}(\underline{\mu}({\epsilon}),{\epsilon})=0$.

  * (ii)

If ${\mathtt{e}}_{\scaleto{\mathrm{WB}}{3pt}}<0$, then for any
$|\mu|\leq\mu_{0}$ and $|{\epsilon}|\leq{\epsilon}_{1}$, the operator
${\mathcal{L}}_{\mu,{\epsilon}}$ has near the origin three purely imaginary
eigenvalues.

Let us make some comments.

1\. Universality of Figure “8” spectrum for small amplitude traveling waves:
Consider the linearized operator ${\mathcal{L}}_{\epsilon}$ at a traveling
wave of small amplitude ${\epsilon}>0$ and assume the Whitham-Benjamin
coefficient (1.16) is strictly positive. In view of (1.14), the unstable
spectrum of ${\mathcal{L}}_{\epsilon}$ near the origin is completely described
by the two curves $\mu\mapsto\lambda^{\pm}_{1}(\mu,{\epsilon})$. As $\mu$
ranges in the interval $[0,\underline{\mu}({\epsilon}))$, these two curves
depict a closed figure with an “8” shape, which is well approximated by the
curve

$\mu\in[0,\underline{\mu}({\epsilon}))\longrightarrow\left(\pm\mu\sqrt{{\mathtt{e}}_{\scaleto{\mathrm{WB}}{3pt}}{\epsilon}^{2}-{\mathtt{e}}_{b}^{2}\mu^{2}},\,-{\mathtt{e}}_{12}\mu\right)\in{\mathbb{R}}^{2}\
,$ (1.20)

plotted in figure 1 below. In case the coefficient ${\mathtt{e}}_{12}<0$
(recall (1.10)), equation (1.18) parametrizes the upper part of the eight, the
lower part being parametrized by $\mu<0$, and viceversa in case
${\mathtt{e}}_{12}>0$. Nevertheless, the operator
${\mathcal{L}}_{{\epsilon}}$, defined on $L^{2}({\mathbb{R}})$, is real, and
so the spectrum is invariant also by conjugation with respect to the real
axis. So one obtain a complete figure “8”.

We find interesting that, whatever the operator ${\mathcal{M}}(D)$ (provided
Assumptions A and B are met), the spectrum near the origin of
${\mathcal{L}}_{\epsilon}$ is either purely imaginary or it contains an
unstable component having always a figure “8” shape.

Figure 1: The figure represents the plot of the curve parametrized in equation
(1.20).

2\. Generality of choosing a $2\pi$-periodic traveling wave: In Theorem 1.2 we
focused on the modulational instability for a space $2\pi$-periodic traveling
wave. The instability of $\frac{2\pi}{\kappa}$-traveling waves,
$\kappa\in{\mathbb{N}}$, can be easily obtained by rescaling, as we now show.
The linear rescaling

$u(x)=(\Psi^{\kappa}w)(x)=\kappa^{-1}w(\kappa x)$

transforms equation (1.1) in

$\partial_{t}w+{\mathcal{M}}_{\kappa}w_{x}+(w^{2})_{x}=0\
,\quad{\mathcal{M}}_{\kappa}(D):=\kappa{\mathcal{M}}(\kappa D)\ .$ (1.21)

Then a $\frac{2\pi}{\kappa}$-periodic traveling wave solution of (1.1) with
speed $c$, is transformed in a $2\pi$-periodic traveling wave solution of
(1.21) with speed $c/\kappa$. To verify its stability/instability it is then
enough to replace ${\mathfrak{m}}(\xi)$ with
${\mathfrak{m}}_{\kappa}(\xi):={\mathfrak{m}}(\kappa\xi)$ and recompute the
coefficients in (1.10), (1.11) for ${\mathfrak{m}}_{\kappa}(\xi)$. Thus, our
abstract result Theorem 1.2 is also suitable to establish the modulational
instability also for periodic traveling waves of general period
$\frac{2\pi}{\kappa}$, provided that the symbol ${\mathfrak{m}}_{\kappa}$
satisfies Assumptions A and B.

3\. Comparison with Berti-Maspero-Ventura [2, 3, 4]: Although the strategy of
the proof follows the same lines of [2, 3, 4], there are important differences
that we now describe. The first one is that the gKdV equation in (1.1) is
Hamiltonian with a symplectic tensor $\partial_{x}$ which is unbounded,
contrary to the case of water waves in [2, 3] where the symplectic tensor is
the standard matrix $\begin{pmatrix}0&-{\rm Id}\\\ {\rm Id}&0\end{pmatrix}$. A
consequence is that, in our case, the symplectic tensor is conjugated by the
Bloch-Floquet transform to the unbounded operator
${\mathcal{J}}_{\mu}=\partial_{x}+{\rm i}\mu$ which is $\mu$-dependent and not
invertible at $\mu=0$. Then we need to develop a Kato similarity
transformation theory adapted to this situation. In particular a different
notion of symplectic basis is needed (see Definition 3.2) which distinguishes
the cases $\mu=0$ (when ${\mathcal{J}}_{0}$ is not invertible) and $\mu\neq
0$.

A second difference is that, as we shall explain below, the bifurcation
problem is reduced to study the spectrum of a $3\times 3$ complex Hamiltonian
and reversible matrix, and not of a $4\times 4$ one as in [2, 3]. In view of
this, the block-diagonalization procedure performed in Section 5 conjugates
the original $3\times 3$ matrix to a new block-diagonal one in with a $2\times
2$ block and a single diagonal element which is always purely imaginary.

A final difference is that we require only a minimum amount of
${\mathcal{C}}^{3}$ regularity in $\mu$ (contrary to [2, 3, 4] where the
dependence on $\mu$ is analytic). This is needed to deal with the cases of
${\mathfrak{m}}(\xi)$ having only finite regularity, such as for the
fractional KdV. A consequence is that we need extra care in dealing with the
size of the remainders in the expansions of the matrix elements in Proposition
4.3.

Ideas and scheme of the paper. In order to prove Theorem 1.2 we need to
compute the spectrum near the origin of the operator
${\mathcal{L}}_{\mu,{\epsilon}}$ in (1.15) for every value of ${\epsilon}>0$
and $\mu>0$ sufficiently small. We treat this as a bifurcation problem from
the zero eigenvalue of ${\mathcal{L}}_{0,0}$ which has algebraic multiplicity
three and geometric multiplicity two. In Section 3 we introduce a modified
version of the symplectic Kato’s similarity theory developed in [2] necessary
to handle the fact that the symplectic tensor is not invertible. In Section 4
we exploit this theory to compute the $3\times 3$ complex Hamiltonian and
reversible matrix representing the action of ${\mathcal{L}}_{\mu,{\epsilon}}$
on the invariant subspace ${\mathcal{V}}_{\mu,{\epsilon}}$ associated to the
spectrum of ${\mathcal{L}}_{\mu,{\epsilon}}$ near the origin. Then, in Section
5, we perform a non-perturbative block-diagonalization of the matrix,
conjugating it to a new matrix with two decoupled blocks: a purely imaginary
number, and a $2\times 2$ block, whose eigenvalues can be computed explicitly.
As the parameter $\mu$ varies, these two latter eigenvalues parametrize the
desired figure “8”.
Finally, in Section 6, we apply our abstract Theorem 1.2 to Whitham’s like
equations, Kawahara, the intermediate long-wave and fractional KdV equations
(recovering, for the latter two equations, results already known in the
literature).

Acknowledgments. This work is supported by PRIN 2020 (2020XB3EFL001)
“Hamiltonian and dispersive PDEs”, PRIN 2022 (2022HSSYPN) ”TESEO - Turbulent
Effects vs Stability in Equations from Oceanography”.

##  2 The complete Benjamin-Feir spectrum of generalized KdV

In this section we present the complete spectral Theorem 2.4. We first
introduce the Hamiltonian and reversible structure of the generalized KdV
equation (1.1), which will play a crucial role in our analysis.
We start with the Hamiltonian formulation: equation (1.1) has the Hamiltonian
form

$\partial_{t}u={\mathcal{J}}\nabla H(u)$

where ${\mathcal{J}}$ is the skew-adjoint operator

${\mathcal{J}}:=\partial_{x},$ (2.1)

$H(u)$ is the real valued Hamiltonian function

$H(u):=-\frac{1}{2}\int_{\mathbb{R}}u({\mathcal{M}}(D)u){\rm
d}x-\frac{1}{3}\int_{\mathbb{R}}u^{3}{\rm d}x,$

well defined on $H^{m_{*}}({\mathbb{R}})$, $m_{*}:=\max(1,m)$, and the
gradient is taken with respect to the real $L^{2}$ scalar product

$\langle u,v\rangle_{r}:=\int_{\mathbb{R}}u(x)\,v(x)\,{\rm d}x\ .$ (2.2)

Equation (1.1) enjoys two more symmetries. First it is invariant by
translation, namely its vector field
$X(u):=-\partial_{x}({\mathcal{M}}u+u^{2})$ fulfills

$X\circ\tau_{\varsigma}=\tau_{\varsigma}\circ X\ ,$ (2.3)

where $\tau_{\varsigma}$ is the translation operator

$[\tau_{\varsigma}u](x):=u(x+\varsigma)\ ,\quad\varsigma\in{\mathbb{R}}\ .$
(2.4)

Secondly, the equation is time reversible, namely the vector field $X(u)$
fulfills

$X\circ\rho=-\rho\circ X$ (2.5)

where $\rho$ is the involution

$[\rho u](x):=u(-x)\ .$ (2.6)

The linearized operator ${\mathcal{L}}_{\epsilon}$ in (1.13) inherits such
structures: first it is linearly Hamiltonian, i.e. of the form

${\mathcal{L}}_{\epsilon}={\mathcal{J}}{\mathcal{B}}_{\epsilon}$ (2.7)

where ${\mathcal{B}}_{\epsilon}$ is the symmetric (with respect to the scalar
product (2.2)) operator

${\mathcal{B}}_{\epsilon}:=c_{\epsilon}-{\mathcal{M}}(D)-2u_{\epsilon}\ .$
(2.8)

Moreover it is also reversible, namely

${\mathcal{L}}_{\epsilon}\circ\rho=-\rho\circ{\mathcal{L}}_{\epsilon}\ .$
(2.9)

Bloch-Floquet expansion. Consider the Floquet operator
${\mathcal{L}}_{\mu,{\epsilon}}$ in (1.15). We remark that (see e.g. [2]):

  * (i)

if $A=\mathrm{Op}(a)$ is a pseudo-differential operator with symbol
$a(x,\xi)$, which is $2\pi$ periodic in $x$, then $A_{\mu}:=e^{-{\rm i}\mu
x}Ae^{{\rm i}\mu x}=\mathrm{Op}(a(x,\xi+\mu))$.

  * (ii)

If $A$ is a real operator then $\overline{A_{\mu}}=A_{-\mu}$. We recall that
$\overline{A}u:=\overline{A\overline{u}}$. As a consequence the spectrum
$\sigma(A_{-\mu})=\overline{\sigma(A_{\mu})}$ and we can restrict to study
$\sigma(A_{\mu})$ only for $\mu>0$.

  * (iii)

$\sigma(A_{\mu})$ is a 1-periodic set with respect to $\mu$, so we can
restrict to $\mu\in[0,\frac{1}{2})$.

In view of these properties, the operator ${\mathcal{L}}_{\mu,{\epsilon}}$
takes the form of the complex Hamiltonian and reversible operator

${\mathcal{L}}_{\mu,{\epsilon}}={\mathcal{J}}_{\mu}\,{\mathcal{B}}_{\mu,{\epsilon}}\
,$ (2.10)

where the skew-adjoint operator ${\mathcal{J}}_{\mu}$ and the self-adjoint
operator ${\mathcal{B}}_{\mu,{\epsilon}}$ are defined by

${\mathcal{J}}_{\mu}:=\partial_{x}+{\rm
i}\mu,\quad{\mathcal{B}}_{\mu,{\epsilon}}:=c_{\epsilon}-{\mathcal{M}}(D+\mu)-2u_{\epsilon}$
(2.11)

and ${\mathcal{M}}(D+\mu)$ is the Fourier multiplier

$[{\mathcal{M}}(D+\mu)u]^{\wedge}(\xi):={\mathfrak{m}}(\xi+\mu)\hat{u}(\xi)\
.$ (2.12)

We regard ${\mathcal{L}}_{\mu,{\epsilon}}$ as a closed, unbounded operator
with domain $H^{m_{\star}}({\mathbb{T}},{\mathbb{C}})$, with
$m_{\star}:=\max(1,1+m)$ and $m$ the order of the Fourier multiplier
${\mathfrak{m}}(\xi)$, and range $L^{2}({\mathbb{T}},{\mathbb{C}})$, equipped
with the complex scalar product

$(f,g):=\frac{1}{\pi}\int_{0}^{2\pi}f\overline{g}\,{\rm d}x,\qquad\forall
f,g\in L^{2}({\mathbb{T}},{\mathbb{C}})\ .$ (2.13)

We recall that an operator ${\mathcal{L}}$ is called

  * 1.

complex Hamiltonian, if ${\mathcal{L}}={\mathcal{J}}{\mathcal{B}}$ with
${\mathcal{J}}$ skew-adjoint and ${\mathcal{B}}$ selfadjoint (with respect to
the complex scalar product (2.13)):

${\mathcal{J}}^{*}=-{\mathcal{J}},\qquad{\mathcal{B}}^{*}={\mathcal{B}}\ ;$

  * 2.

complex reversible, if

${\mathcal{L}}\circ\overline{\rho}=-\overline{\rho}\circ{\mathcal{L}}$

where $\overline{\rho}$ is the complex involution

$[\overline{\rho}u](x):=\overline{u}(-x)\ .$ (2.14)

Note that the operator ${\mathcal{B}}_{\mu,{\epsilon}}$ is reversibility
preserving, i.e.

${\mathcal{B}}_{\mu,{\epsilon}}\circ\overline{\rho}=\overline{\rho}\circ{\mathcal{B}}_{\mu,{\epsilon}}\
.$ (2.15)

Due to Assumption A, we have that
$(\mu,{\epsilon})\mapsto{\mathcal{L}}_{\mu,{\epsilon}}$ and
$(\mu,{\epsilon})\mapsto{\mathcal{B}}_{\mu,{\epsilon}}$ are functions of class
${\mathcal{C}}^{\mathtt{r}}$ from
$\\{|\mu|<\mu_{0}\\}\times\\{|{\epsilon}|<{\epsilon}_{0}\\}$ to
${\mathcal{L}}(H^{m_{\star}+1}({\mathbb{T}}),L^{2}({\mathbb{T}})))$.

######  Lemma 2.1.

The operators
$\dot{\mathcal{L}}_{0,{\epsilon}}:=\partial_{\mu}{\mathcal{L}}_{\mu,{\epsilon}}\big{|}_{\mu=0}$
and
$\dot{\mathcal{B}}_{0,{\epsilon}}:=\partial_{\mu}{\mathcal{B}}_{\mu,{\epsilon}}\big{|}_{\mu=0}$
are purely imaginary333An operator $A$ is purely imaginary if
$\overline{A}=-A$, where $\overline{A}f:=\overline{A\overline{f}}$. A purely
imaginary operator maps a real vector into a purely imaginary one. and
respectively reversible and reversibility preserving.

###### Proof.

Consider first $\dot{\mathcal{B}}_{0,{\epsilon}}$. By (2.11), it equals the
Fourier multiplier $-\dot{\mathcal{M}}(D)$, having for symbol the odd function
$-\frac{{\rm d}{\mathfrak{m}}(\xi)}{{\rm d}\xi}$. Regarding
$\dot{\mathcal{L}}_{0,{\epsilon}}$, by (2.10), (2.11) one has
$\dot{\mathcal{L}}_{0,{\epsilon}}={\mathcal{J}}_{0}\dot{\mathcal{B}}_{0,{\epsilon}}+{\rm
i}{\mathcal{B}}_{0,{\epsilon}}$. As ${\mathcal{J}}_{0}$ and
${\mathcal{B}}_{0,{\epsilon}}$ are real operators, whereas
$\dot{\mathcal{B}}_{0,{\epsilon}}$ is purely imaginary,
$\dot{\mathcal{L}}_{0,{\epsilon}}$ is purely imaginary as well. The
reversibility properties follow from those of ${\mathcal{B}}_{\mu,{\epsilon}}$
and ${\mathcal{L}}_{\mu,{\epsilon}}$. ∎

Recall that the spectrum of a complex Hamiltonian and reversible operator is
symmetric with respect to reflections across the imaginary axis, and so the
eigenvalues can exit the imaginary axis only after collisions of two or more
eigenvalues on it. Therefore, we shall now study multiple eigenvalues of the
operators ${\mathcal{L}}_{\mu,0}$ and ${\mathcal{L}}_{0,{\epsilon}}$.

Spectrum of ${\mathcal{L}}_{\mu,0}$. As $c_{0}={\mathfrak{m}}(1)$ and
$u_{0}=0$ by Theorem 1.1, the operator
${\mathcal{L}}_{\mu,0}={\mathcal{J}}_{\mu}{\mathcal{B}}_{\mu,0}$ equals the
Fourier multiplier $(\partial_{x}+{\rm
i}\mu)({\mathfrak{m}}(1)-{\mathcal{M}}(D+\mu))$. Its
$L^{2}({\mathbb{T}})$-spectrum is given by
$\sigma_{L^{2}({\mathbb{T}})}(\mathcal{L}_{\mu,0})=\\{{\rm
i}\,\omega_{j,\mu},\ \ j\in\mathbb{Z}\\}\ $ with

$\omega_{j,\mu}:=(j+\mu)\big{(}{\mathfrak{m}}(1)-{\mathfrak{m}}(j+\mu)\big{)},\qquad
j\in{\mathbb{Z}}\ ,\quad{\mathcal{L}}_{\mu,0}\,e^{{\rm i}jx}={\rm
i}\,\omega_{j,\mu}e^{{\rm i}jx}\,.$ (2.16)

We can notice that at $\mu=0$, in view of Assumption A3, one has

$\omega_{-1,0}=\omega_{0,0}=\omega_{1,0}=0,\quad|\omega_{j,0}|\geq|j|c_{0}>0\
\ \mbox{ for }j\neq 0,\pm 1\ ,$ (2.17)

namely $\ker{\mathcal{L}}_{0,0}$ has dimension three. We can then decompose
the spectrum of ${\mathcal{L}}_{0,0}$ in two disjoint subsets as

$\sigma_{L^{2}({\mathbb{T}})}(\mathcal{L}_{0,0})=\sigma^{\prime}_{L^{2}({\mathbb{T}})}(\mathcal{L}_{0,0})\cup\sigma_{L^{2}({\mathbb{T}})}^{\prime\prime}(\mathcal{L}_{0,0})$
(2.18)

with $\sigma^{\prime}_{L^{2}({\mathbb{T}})}(\mathcal{L}_{0,0})=\\{0\\},$ and,
in view of (1.4),
$\sigma^{\prime\prime}_{L^{2}({\mathbb{T}})}(\mathcal{L}_{0,0})\subseteq\\{{\rm
i}\lambda\text{, s.t. }\lambda\in{\mathbb{R}}\text{, }|\lambda|\geq c_{0}\\}$.
From Kato’s perturbation theory (see Lemma 3.1 below) for any $\mu,{\epsilon}$
sufficiently small, the perturbed spectrum admits a disjoint decomposition as

$\sigma_{L^{2}({\mathbb{T}})}(\mathcal{L}_{\mu,0})=\sigma^{\prime}_{L^{2}({\mathbb{T}})}(\mathcal{L}_{\mu,0})\cup\sigma_{L^{2}({\mathbb{T}})}^{\prime\prime}(\mathcal{L}_{\mu,0})$
(2.19)

with $\sigma^{\prime}_{L^{2}({\mathbb{T}})}(\mathcal{L}_{\mu,0})$ consisting
in three distinguished purely imaginary eigenvalues, which are
${\mathcal{O}}(\mu)$ close to zero.

Spectrum of ${\mathcal{L}}_{0,{\epsilon}}$. The operator
${\mathcal{L}}_{0,{\epsilon}}\equiv{\mathcal{L}}_{{\epsilon}}$ has $0$ as a
defective eigenvalue, as we show in the next lemma.

######  Lemma 2.2.

For $0<|{\epsilon}|\leq{\epsilon}_{0}$, the linear operator
${\mathcal{L}}_{0,{\epsilon}}$ has $0$ as an eigenvalue of algebraic
multiplicity 3 and geometric multiplicity 2. In particular
$g_{\epsilon}^{-}(x):=(u_{\epsilon})_{x}$ and
$g_{\epsilon}^{+}(x):=(\partial_{\epsilon}u_{\epsilon})-\frac{1}{2}\partial_{\epsilon}c_{\epsilon}$
are eigenvectors, whereas $g(x):=\frac{1}{\sqrt{2}}$ is a generalized
eigenvector fulfilling
${\mathcal{L}}_{0,{\epsilon}}g=-\sqrt{2}g_{\epsilon}^{-}$.

###### Proof.

By (1.5), for every ${\epsilon}$ sufficiently small, $u_{\epsilon}$ solves
$F({\epsilon},u_{\epsilon}):=\partial_{x}(c_{\epsilon}u_{\epsilon}-{\mathcal{M}}u_{\epsilon}-u_{\epsilon}^{2})=0$.
Derivating such identity with respect to $x$ yields
${\mathcal{L}}_{0,{\epsilon}}g_{\epsilon}^{-}=0$. Next from (1.13)–(2.8) we
get ${\mathcal{L}}_{0,{\epsilon}}[g]=-\sqrt{2}g_{\epsilon}^{-}$. Finally
derivating $F({\epsilon},u_{\epsilon})=0$ with respect to ${\epsilon}$ yields
${\mathcal{L}}_{0,{\epsilon}}[\partial_{\epsilon}u_{\epsilon}]=-(\partial_{\epsilon}c_{\epsilon})g_{\epsilon}^{-}$,
hence ${\mathcal{L}}_{0,{\epsilon}}[g_{\epsilon}^{+}]=0$. ∎

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

$\sigma\left({\mathcal{L}}_{\mu,{\epsilon}}\right)=\sigma^{\prime}\left({\mathcal{L}}_{\mu,{\epsilon}}\right)\cup\sigma^{\prime\prime}\left({\mathcal{L}}_{\mu,{\epsilon}}\right)\,,$
(2.20)

where $\sigma^{\prime}\left({\mathcal{L}}_{\mu,{\epsilon}}\right)$ consists of
3 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 3 and it is invariant by ${\mathcal{L}}_{\mu,{\epsilon}}$.

######  Remark 2.3.

Lemma 2.2 tells that, for any $0<|{\epsilon}|\leq{\epsilon}_{0}$,
${\mathcal{V}}_{0,{\epsilon}}=\textup{span}\\{g_{\epsilon}^{-},g_{\epsilon}^{+},\frac{1}{\sqrt{2}}\\}$.

Our main result gives the full description of the spectrum of the operator
${\mathcal{L}}_{\mu,{\epsilon}}|_{\mathcal{V}_{\mu,{\epsilon}}}\colon\mathcal{V}_{\mu,{\epsilon}}\to\mathcal{V}_{\mu,{\epsilon}}$
whenever $\mu,{\epsilon}$ are sufficiently small, and characterizes for which
values of $(\mu,{\epsilon})$ eigenvalues with nonzero real parts appear.
Before stating our main result, let us introduce some notations we shall use
through the paper:

  * $\bullet$

Notation: We denote by
${\mathcal{O}}^{\mathtt{r}}(\mu^{m_{1}}{\epsilon}^{n_{1}},\dots,\mu^{m_{p}}{\epsilon}^{n_{p}})$,
$m_{j},n_{j}\in{\mathbb{N}}$ (for us ${\mathbb{N}}:=\\{1,2,\dots\\}$),
functions of $(\mu,{\epsilon})$ of class ${\mathcal{C}}^{\mathtt{r}}$ with
values on a Banach space $X$ which satisfy, for some $C>0$, the bound
$\|{\mathcal{O}}^{\mathtt{r}}(\mu^{m_{j}}{\epsilon}^{n_{j}})\|_{X}\leq
C\sum_{j=1}^{p}|\mu|^{m_{j}}|{\epsilon}|^{n_{j}}$ 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
${\mathcal{O}}^{1}(\mu^{m_{1}}{\epsilon}^{n_{1}},\dots,\mu^{m_{p}}{\epsilon}^{n_{p}})$
which are also real valued.

Our complete spectral result is the following:

######  Theorem 2.4.

Assume Assumption A and B at page 1 and 1. There exist
${\epsilon}_{0},\mu_{0}>0$ such that, for any
$|{\epsilon}|\leq{\epsilon}_{0}$, $0<\mu<\mu_{0}$, the operator
${\mathcal{L}}_{\mu,{\epsilon}}:{\mathcal{V}}_{\mu,{\epsilon}}\to{\mathcal{V}}_{\mu,{\epsilon}}$
can be represented by the $3\times 3$ block-diagonal matrix

$\begin{pmatrix}\mathtt{U}&\vline&\mathtt{0}\\\
\hline\cr\mathtt{0}^{\dagger}&\vline&{\rm i}\,\mathtt{g}\end{pmatrix}$ (2.21)

where $\mathtt{U}$ is the $2\times 2$ matrix

$\mathtt{U}:=-{\rm
i}\mu\big{(}{\mathtt{e}}_{12}+r_{0}({\epsilon}^{2},\mu{\epsilon},\mu^{2})\big{)}+\mu\begin{pmatrix}{\rm
i}\big{(}{\epsilon}^{2}r_{1}(1)+\mu^{2}r_{2}(1)\big{)}&{\mathtt{e}}_{b}+r_{3}(\mu,{\epsilon})\\\
{\mathtt{e}}_{w}\,{\epsilon}^{2}(1+r_{4}(\mu,{\epsilon}))-{\mathtt{e}}_{b}\,\mu^{2}(1+r_{5}(\mu,{\epsilon}))&{\rm
i}\mu^{2}r_{6}(1)\end{pmatrix}$ (2.22)

with ${\mathtt{e}}_{12},{\mathtt{e}}_{w},{\mathtt{e}}_{b}$ in (1.11), (1.10),
$\mathtt{g}$ is the real number

$\mathtt{g}:=\big{(}{\mathfrak{m}}(1)-{\mathfrak{m}}(0)\big{)}\mu+\mu
r({\epsilon}^{2},\mu^{2})\ ,$ (2.23)

and $\mathtt{0}=(0,0)^{\dagger}$. The functions $r_{k}$ are of class
${\mathcal{C}}^{\mathtt{r}-2}$.

The eigenvalues of $\mathtt{U}$ have the form

$\lambda_{1}^{\pm}(\mu,{\epsilon})=-{\rm
i}\mu\big{(}{\mathtt{e}}_{12}+r({\epsilon}^{2},\mu{\epsilon},\mu^{2})\big{)}\pm\mu\,\sqrt{\underbrace{{\mathtt{e}}_{\scaleto{\mathrm{WB}}{3pt}}\,{\epsilon}^{2}(1+\hat{r}_{1}({\epsilon},\mu))-{\mathtt{e}}_{b}^{2}\,\mu^{2}(1+\hat{r}_{2}({\epsilon},\mu))}_{=:\Delta_{\scaleto{\mathrm{BF}}{3pt}}(\mu,{\epsilon})}}$

with $r,\hat{r}_{1},\hat{r}_{2}$ in ${\mathcal{C}}^{\mathtt{r}-2}$. In
particular, they have non trivial real part whenever $\mu\neq 0$ and the
Benjamin-Feir discriminant
$\Delta_{\scaleto{\mathrm{BF}}{3pt}}(\mu,{\epsilon})>0$.

Theorem 1.2 is a direct consequence of Theorem 2.4, as we now show.

###### Proof of Theorem 1.2 assuming Theorem 2.4.

The curve $\underline{\mu}({\epsilon})$ in Theorem 1.2 is implicitly defined
by

$\mu\sqrt{1+\hat{r}_{2}({\epsilon},\mu)}={\epsilon}\sqrt{\dfrac{{\mathtt{e}}_{\scaleto{\mathrm{WB}}{3pt}}}{{\mathtt{e}}_{b}^{2}}}\,\,\sqrt{1+\hat{r}_{1}(\mu,{\epsilon})}\
,$

which has a unique solution of class ${\mathcal{C}}^{\mathtt{r}-2}$ provided
$\mathtt{r}\geq 3$. ∎

##  3 Perturbative approach to the separated eigenvalues

Recall that the operator ${\mathcal{L}}_{\mu,{\epsilon}}:Y\subset X\to X$ has
domain $Y:=H^{m_{\star}}(\mathbb{T},{\mathbb{C}})$, with
$m_{\star}=\max\\{1,1+m\\}$ and $m$ as in Equation (1.3), and range
$X:=L^{2}(\mathbb{T}):=L^{2}(\mathbb{T},{\mathbb{C}})$.
We now state the following Lemma, that is a variant of Lemma 3.1 in [2]. We
denote by $B(r)$, $r>0$, the ball of center 0 and radius $r$ in
${\mathbb{R}}$.

######  Lemma 3.1.

(Kato theory for separated eigenvalues) Let $\Gamma$ be a closed,
counterclockwise-oriented curve around $0$ in the complex plane separating
$\sigma^{\prime}\left({\mathcal{L}}_{0,0}\right)=\\{0\\}$ and the other part
of the spectrum $\sigma^{\prime\prime}\left({\mathcal{L}}_{0,0}\right)$ in
(2.18). There exist ${\epsilon}_{0},\mu_{0}>0$ such that for any
$(\mu,{\epsilon})\in B(\mu_{0})\times B({\epsilon}_{0})$ (where $B(x,r)$ is
the ball centered at $x$ with radius $r$ inside ${\mathbb{R}}$) the following
statements hold:
1\. The curve $\Gamma$ belongs to the resolvent set of the operator
${\mathcal{L}}_{\mu,{\epsilon}}:Y\subset X\to X$ defined in (1.15).
2\. The operators

$P_{\mu,{\epsilon}}:=-\frac{1}{2\pi{\rm
i}}\oint_{\Gamma}({\mathcal{L}}_{\mu,{\epsilon}}-\lambda)^{-1}{\rm
d}\lambda:X\to Y$ (3.1)

are well defined projectors commuting with ${\mathcal{L}}_{\mu,{\epsilon}}$,
i.e. $P_{\mu,{\epsilon}}^{2}=P_{\mu,{\epsilon}}$ and
$P_{\mu,{\epsilon}}{\mathcal{L}}_{\mu,{\epsilon}}={\mathcal{L}}_{\mu,{\epsilon}}P_{\mu,{\epsilon}}$.
The map $(\mu,\epsilon)\mapsto P_{\mu,\epsilon}$ is of class
${\mathcal{C}}^{\mathtt{r}}$.
3\. The domain $Y$ of the operator ${\mathcal{L}}_{\mu,{\epsilon}}$ decomposes
as the direct sum

$Y=\mathcal{V}_{\mu,{\epsilon}}\oplus\text{Ker}(P_{\mu,{\epsilon}})\,,\quad\mathcal{V}_{\mu,{\epsilon}}:=\text{Rg}(P_{\mu,{\epsilon}})=\text{Ker}({\rm
Id}-P_{\mu,{\epsilon}})\,,$ (3.2)

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

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

4\. The projectors $P_{\mu,{\epsilon}}$ are similar one to each other: the
transformation operators

$U_{\mu,{\epsilon}}:=\big{(}{\rm
Id}-(P_{\mu,{\epsilon}}-P_{0,0})^{2}\big{)}^{-1/2}\big{[}P_{\mu,{\epsilon}}P_{0,0}+({\rm
Id}-P_{\mu,{\epsilon}})({\rm Id}-P_{0,0})\big{]}$ (3.3)

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

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

and

$U_{\mu,{\epsilon}}P_{0,0}U_{\mu,{\epsilon}}^{-1}=P_{\mu,{\epsilon}}\ ,\qquad
U_{\mu,{\epsilon}}^{-1}P_{\mu,{\epsilon}}U_{\mu,{\epsilon}}=P_{0,0}\ .$

The map $(\mu,\epsilon)\mapsto U_{\mu,{\epsilon}}$ is of class
${\mathcal{C}}^{\mathtt{r}}$.
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 $\dim\mathcal{V}_{\mu,{\epsilon}}=\dim\mathcal{V}_{0,0}=3$, for any
$(\mu,{\epsilon})\in B(\mu_{0})\times B({\epsilon}_{0})$.

###### Proof.

The proof follows the same lines as Lemma 3.1 in [2], replacing the
analyticity of the map $(\mu,{\epsilon})\mapsto{\mathcal{L}}_{\mu,{\epsilon}}$
with the regularity ${\mathcal{C}}^{\mathtt{r}}$. ∎

We now prove that Kato’s transformation operator $U_{\mu,{\epsilon}}$ is
symplectic and reversibility preserving. We first introduce, for $\mu\neq 0$
the inverse operator

${\mathcal{E}}_{\mu}:={\mathcal{J}}_{\mu}^{-1}\colon L^{2}({\mathbb{T}})\to
H^{1}({\mathbb{T}}),\qquad\mu>0\ ,$ (3.4)

which acts as a Fourier multiplier of symbol $\dfrac{1}{{\rm i}(\xi+\mu)}$.
When $\mu=0$, the operator ${\mathcal{J}}_{0}=\partial_{x}$ is invertible in
the subspace

$H^{s}_{0}({\mathbb{T}}):=\\{u\in
H^{s}({\mathbb{T}})\colon\int_{\mathbb{T}}u(x){\rm d}x=0\\}\ .$

Hence, we define

${\mathcal{E}}_{0}={\mathcal{J}}_{0}^{-1}:L^{2}_{0}({\mathbb{T}})\to
H^{1}_{0}({\mathbb{T}})\ .$ (3.5)

We now define $\mu$-symplectic and reversible basis.

######  Definition 3.2 ($\mu$-symplectic and reversible basis).

A linearly independent set $\\{f_{1}^{+},f_{1}^{-},f_{0}\\}$ is

  * (i)

$\mu$-symplectic, $\mu\neq 0$, if one has

$\displaystyle({\mathcal{E}}_{\mu}f_{1}^{+},f_{1}^{+})=({\mathcal{E}}_{\mu}f_{1}^{-},f_{1}^{-})={\rm
i}\frac{\mu}{1-\mu^{2}},\qquad({\mathcal{E}}_{\mu}f_{0},f_{0})=-\frac{{\rm
i}}{\mu}\ ,$ (3.6)
$\displaystyle({\mathcal{E}}_{\mu}f_{1}^{+},f_{1}^{-})=\frac{1}{1-\mu^{2}}\
,\qquad({\mathcal{E}}_{\mu}f_{0},f_{1}^{+})=({\mathcal{E}}_{\mu}f_{0},f_{1}^{-})=0\
,$

and $0$-symplectic if both $f_{1}^{+},f_{1}^{-}\in L_{0}^{2}({\mathbb{T}})$,
$f_{0}\in\ker{\mathcal{J}}_{0}=\textup{span}\langle 1\rangle$ and

$({\mathcal{E}}_{0}f_{1}^{+},f_{1}^{-})=1,\hskip
8.5359pt({\mathcal{E}}_{0}f_{1}^{+},f_{1}^{+})=({\mathcal{E}}_{0}f_{1}^{-},f_{1}^{-})=0\
;$ (3.7)

  * (ii)

reversible if $\overline{\rho}f_{1}^{+}=f_{1}^{+},$
$\overline{\rho}f_{1}^{-}=-f_{1}^{-},$ $\overline{\rho}f_{0}=f_{0}$, and
$\overline{\rho}$ is the complex involution in (2.14).

######  Remark 3.3.

The basis $f_{1}^{+}=\cos x$, $f_{1}^{-}=\sin x$, $f_{0}=\frac{1}{\sqrt{2}}$
is $\mu$-symplectic for any $\mu\in[0,\frac{1}{2})$.

In the previous Lemma 3.1, it has been shown that the operator
$U_{\mu,{\epsilon}}$ is an isomorphism between ${\mathcal{V}}_{0,0}$ and
${\mathcal{V}}_{\mu,{\epsilon}}$. In the following Lemma we show that, in
addition, the previous isomorphism preserves the symplectic and reversible
structure of a basis. Here the properties are different with respect to the
case of [2], since here the symplectic tensor ${\mathcal{J}}_{\mu}$ is not
invertible at $\mu=0$.

######  Lemma 3.4.

For any $(\mu,{\epsilon})\in B(\mu_{0})\times B({\epsilon}_{0})$ we have that

  * (i)

$P_{\mu,{\epsilon}}$ is skew-Hamiltonian, i.e.

${\mathcal{E}}_{\mu}P_{\mu,{\epsilon}}=P_{\mu,{\epsilon}}^{*}{\mathcal{E}}_{\mu},\quad\mbox{for
}\mu\neq
0,\qquad{\mathcal{E}}_{0}P_{0,{\epsilon}}=P_{0,{\epsilon}}^{*}{\mathcal{E}}_{0}\quad\mbox{
in }L^{2}_{0}({\mathbb{T}}),$ (3.8)

and reversibility preserving, i.e.

$\overline{\rho}P_{\mu,{\epsilon}}=P_{\mu,{\epsilon}}\overline{\rho}\ .$ (3.9)

Moreover, $P_{0,{\epsilon}}[1]=1$,
$P_{0,{\epsilon}}{\mathcal{J}}_{0}={\mathcal{J}}_{0}P_{0,{\epsilon}}^{*}$ and
thus $P_{0,{\epsilon}}$ leaves $L^{2}_{0}({\mathbb{T}})$ invariant.

  * (ii)

$U_{\mu,{\epsilon}}$ are symplectic, i.e.

$U_{\mu,{\epsilon}}^{*}{\mathcal{E}}_{\mu}U_{\mu,{\epsilon}}={\mathcal{E}}_{\mu}\quad\mbox{for
}\mu\neq 0\ ,\qquad
U_{0,{\epsilon}}^{*}{\mathcal{E}}_{0}U_{0,{\epsilon}}={\mathcal{E}}_{0}\quad\mbox{
in }L^{2}_{0}({\mathbb{T}})\ ,$ (3.10)

and reversibility preserving. Moreover, $U_{0,{\epsilon}}[1]=1$,
$U_{0,{\epsilon}}{\mathcal{J}}_{0}={\mathcal{J}}_{0}U_{0,{\epsilon}}^{-*}$ and
thus $U_{0,{\epsilon}}$ leaves $L^{2}_{0}({\mathbb{T}})$ invariant.

  * (iii)

$P_{0,{\epsilon}}$ and $U_{0,{\epsilon}}$ are real operators, i.e.
$\overline{P_{0,{\epsilon}}}=P_{0,{\epsilon}},\hskip
8.5359pt\overline{U_{0,{\epsilon}}}=U_{0,{\epsilon}}$.

  * (iv)

The operators
$\dot{P}_{0,{\epsilon}}:=\partial_{\mu}P_{\mu,{\epsilon}}\big{|}_{\mu=0}$ and
$\dot{U}_{0,{\epsilon}}:=\partial_{\mu}U_{\mu,{\epsilon}}|_{\mu=0}$ are purely
imaginary.

###### Proof.

$(i)$ First we have that for any $\mu\in[0,\frac{1}{2})$

${\mathcal{L}}_{\mu,{\epsilon}}{\mathcal{J}}_{\mu}={\mathcal{J}}_{\mu}{\mathcal{B}}_{\mu,{\epsilon}}{\mathcal{J}}_{\mu}={\mathcal{J}}_{\mu}({\mathcal{J}}_{\mu}^{*}{\mathcal{B}}_{\mu,{\epsilon}}^{*})^{*}={\mathcal{J}}_{\mu}(-{\mathcal{J}}_{\mu}{\mathcal{B}}_{\mu,{\epsilon}})^{*}=-{\mathcal{J}}_{\mu}{\mathcal{L}}_{\mu,{\epsilon}}^{*}\
,$ (3.11)

which implies that

$({\mathcal{L}}_{\mu,{\epsilon}}-\lambda){\mathcal{J}}_{\mu}=-{\mathcal{J}}_{\mu}({\mathcal{L}}_{\mu,{\epsilon}}^{*}+\lambda),\quad\forall\mu\in\big{[}0,\frac{1}{2}\big{)}\
.$ (3.12)

Consider now $\mu>0$. Then ${\mathcal{J}}_{\mu}$ is invertible and from (3.12)
we deduce that
$\lambda\in\rho({\mathcal{L}}_{\mu,{\epsilon}})\iff-\lambda\in\rho({\mathcal{L}}_{\mu,{\epsilon}}^{*})$;
so inverting (3.12) and integrating along a circuit
$\Gamma\subseteq\rho({\mathcal{L}}_{\mu,{\epsilon}})$ we obtain that

${\mathcal{E}}_{\mu}P_{\mu,{\epsilon}}={\mathcal{E}}_{\mu}\oint_{\Gamma}({\mathcal{L}}_{\mu,{\epsilon}}-\lambda)^{-1}\frac{-{\rm
d}\lambda}{2\pi{\rm
i}}=\oint_{\Gamma}({\mathcal{L}}_{\mu,{\epsilon}}^{*}+\lambda)^{-1}\frac{{\rm
d}\lambda}{2\pi{\rm
i}}\circ{\mathcal{E}}_{\mu}=P_{\mu,{\epsilon}}^{*}\,{\mathcal{E}}_{\mu}\ ,$
(3.13)

where we used that
$P_{\mu,{\epsilon}}^{*}=\oint_{\Gamma}({\mathcal{L}}_{\mu,{\epsilon}}^{*}+\lambda)^{-1}\frac{{\rm
d}\lambda}{2\pi{\rm i}}$. This proves (3.8) in case $\mu>0$.

Take now $\mu=0$. First we have that $P_{0,{\epsilon}}[1]=1$
$\forall{\epsilon}$, since
$1\in{\mathcal{V}}_{0,{\epsilon}}=RgP_{0,{\epsilon}}$ (see Remark 2.3). Next
consider (3.12) at $\mu=0$, apply
$({\mathcal{L}}_{0,{\epsilon}}-\lambda)^{-1}$ to the left,
$({\mathcal{L}}_{0,{\epsilon}}^{*}+\lambda)^{-1}$ to the right and integrate
along the circuit $\Gamma$ to get the identity
$P_{0,{\epsilon}}{\mathcal{J}}_{0}={\mathcal{J}}_{0}P_{0,{\epsilon}}^{*}$.
This last identity implies that $P_{0,{\epsilon}}$ leaves
$L^{2}_{0}({\mathbb{T}})$ invariant; indeed if $u\in L^{2}_{0}({\mathbb{T}})$
then

$(P_{0,{\epsilon}}u,1)=(P_{0,{\epsilon}}{\mathcal{J}}_{0}\,{\mathcal{E}}_{0}u,1)=({\mathcal{J}}_{0}P_{0,{\epsilon}}^{*}\,{\mathcal{E}}_{0}u,1)=-(P_{0,{\epsilon}}^{*}\,{\mathcal{E}}_{0}u,{\mathcal{J}}_{0}[1])=0\
,$ (3.14)

showing that $P_{0,{\epsilon}}\colon L^{2}_{0}({\mathbb{T}})\to
L^{2}_{0}({\mathbb{T}})$. Then, on $L^{2}_{0}({\mathbb{T}})$, apply
${\mathcal{E}}_{0}$ to the left and to the right of the identity
$P_{0,{\epsilon}}{\mathcal{J}}_{0}={\mathcal{J}}_{0}P_{0,{\epsilon}}^{*}$ to
get the second of (3.8). Property (3.9) follows as in [2, Lemma 3.2].

$(ii)$ Consider first the case $\mu>0$. By (2.11),
${\mathcal{L}}_{0,0}=\partial_{x}({\mathfrak{m}}(1)-{\mathcal{M}}(D))$, hence
${\mathcal{L}}_{0,0}{\mathcal{J}}_{\mu}=-{\mathcal{J}}_{\mu}{\mathcal{L}}_{0,0}^{*}$.
Then arguing as above we deduce that
${\mathcal{E}}_{\mu}P_{0,0}=P_{0,0}^{*}{\mathcal{E}}_{\mu}$. Using also (3.8)
we deduce

$\begin{split}{\mathcal{E}}_{\mu}U_{\mu,{\epsilon}}={\mathcal{E}}_{\mu}\big{(}{\rm
Id}-(P_{\mu,{\epsilon}}-P_{0,0})^{2}\big{)}^{-1/2}\big{[}P_{\mu,{\epsilon}}P_{0,0}+({\rm
Id}-P_{\mu,{\epsilon}})({\rm Id}-P_{0,0})\big{]}\\\ =\big{(}{\rm
Id}-(P_{\mu,{\epsilon}}^{*}-P_{0,0}^{*})^{2}\big{)}^{-1/2}\big{[}P_{\mu,{\epsilon}}^{*}P_{0,0}^{*}+({\rm
Id}-P_{\mu,{\epsilon}}^{*})({\rm Id}-P_{0,0}^{*})\big{]}{\mathcal{E}}_{\mu}\\\
=\big{(}{\rm
Id}-(P_{\mu,{\epsilon}}-P_{0,0})^{2}\big{)}^{-*/2}\big{[}P_{0,0}P_{\mu,{\epsilon}}+({\rm
Id}-P_{0,0})({\rm Id}-P_{\mu,{\epsilon}})\big{]}^{*}{\mathcal{E}}_{\mu}\\\
=\left(\big{[}P_{0,0}P_{\mu,{\epsilon}}+({\rm Id}-P_{0,0})({\rm
Id}-P_{\mu,{\epsilon}})\big{]}\big{(}{\rm
Id}-(P_{\mu,{\epsilon}}-P_{0,0})^{2}\big{)}^{-1/2}\right)^{*}{\mathcal{E}}_{\mu}\end{split}$
(3.15)

By Lemma 3.1, point $4$, this is equal to
$U^{-*}_{\mu,{\epsilon}}{\mathcal{E}}_{\mu}$, concluding the proof of (3.10)
in case $\mu\neq 0$.

Consider now $\mu=0$. Using the expression (3.3) of $U_{0,{\epsilon}}$ and the
fact that $P_{0,{\epsilon}}[1]=1$ $\forall\epsilon$, we obtain that
$U_{0,{\epsilon}}[1]=1$. Then, using
$P_{0,{\epsilon}}{\mathcal{J}}_{0}={\mathcal{J}}_{0}P_{0,{\epsilon}}^{*}$ and
arguing as in (3.15) we get
$U_{0,{\epsilon}}{\mathcal{J}}_{0}={\mathcal{J}}_{0}U_{0,{\epsilon}}^{-*}$,
from which we deduce that $U_{0,{\epsilon}}$ leaves $L^{2}_{0}({\mathbb{T}})$
invariant. The second of (3.10) follows easily from
$U_{0,{\epsilon}}{\mathcal{J}}_{0}={\mathcal{J}}_{0}U_{0,{\epsilon}}^{-*}$ on
$L^{2}_{0}({\mathbb{T}})$.

$(iii)$ It follows as in [2, Lemma 3.2].

$(iv)$ By (3.1),

$\dot{P}_{0,{\epsilon}}=\frac{1}{2\pi{\rm
i}}\oint_{\Gamma}({\mathcal{L}}_{0,{\epsilon}}-\lambda)^{-1}\dot{\mathcal{L}}_{0,{\epsilon}}({\mathcal{L}}_{0,{\epsilon}}-\lambda)^{-1}{\rm
d}\lambda$

Since ${\mathcal{L}}_{0,{\epsilon}}$ is real,

$\overline{\dot{P}_{0,{\epsilon}}}=-\frac{1}{2\pi{\rm
i}}\oint_{\overline{\Gamma}}({\mathcal{L}}_{0,{\epsilon}}-\lambda)^{-1}\overline{\dot{\mathcal{L}}_{0,{\epsilon}}}({\mathcal{L}}_{0,{\epsilon}}-\lambda)^{-1}{\rm
d}\lambda\ ,$

where $\overline{\Gamma}$ denotes the curve $\Gamma$ with inverted
orientation. Since $\dot{\mathcal{L}}_{0,{\epsilon}}$ is purely imaginary by
Lemma 2.1, in conclusion
$\overline{\dot{P}_{0,{\epsilon}}}=-\dot{P}_{0,{\epsilon}}$, proving that
$\dot{P}_{0,{\epsilon}}$ is purely imaginary.

Then consider $\dot{U}_{0,{\epsilon}}$. Denote $g(A):=({\rm
Id}-A^{2})^{-\frac{1}{2}}$, defined for $\|A\|<1$. Then
$\overline{g(A)}=g(\overline{A})$, and differentiating one gets
$\overline{{\rm d}g(A)[B]}={\rm d}g(\overline{A})[\overline{B}]$. Thus,

$\dot{U}_{0,{\epsilon}}={\rm
d}g(P_{0,{\epsilon}}-P_{0,0})[\dot{P}_{0,{\epsilon}}]\left(P_{0,{\epsilon}}P_{0,0}+({\rm
Id}-P_{0,{\epsilon}})({\rm
Id}-P_{0,0})\right)+g(P_{0,{\epsilon}}-P_{0,0})\,\dot{P}_{0,{\epsilon}}(2P_{0,0}-{\rm
Id})$

Using that $P_{0,{\epsilon}}$ is real and $\dot{P}_{0,{\epsilon}}$ is purely
imaginary, one gets the thesis. ∎

In the next lemma we show how a general vector in
${\mathcal{V}}_{\mu,{\epsilon}}$ is decomposed on a $\mu$-symplectic basis.

######  Lemma 3.5.

Let $\\{f_{1}^{+},f_{1}^{-},f_{0}\\}$ a $\mu$-symplectic basis of
${\mathcal{V}}_{\mu,{\epsilon}}$ according to Definition 3.2. Then for every
$f\in{\mathcal{V}}_{\mu,{\epsilon}}$, it holds that

  * (i)

if $\mu\neq 0$, then
$f=\alpha_{1}^{+}f_{1}^{+}+\alpha_{1}^{-}f_{1}^{-}+\alpha_{0}f_{0}$ with

$\begin{pmatrix}\alpha_{1}^{+}\\\ \alpha_{1}^{-}\\\
\alpha_{0}\end{pmatrix}=\begin{pmatrix}{\rm i}\mu&1&0\\\ -1&{\rm i}\mu&0\\\
0&0&{\rm
i}\mu\end{pmatrix}\cdot\begin{pmatrix}({\mathcal{E}}_{\mu}f,f_{1}^{+})\\\
({\mathcal{E}}_{\mu}f,f_{1}^{-})\\\ ({\mathcal{E}}_{\mu}f,f_{0})\end{pmatrix}\
;$ (3.16)

  * (ii)

if $\mu=0$, then, denoting $\breve{f}:=f-(f,f_{0})f_{0}\in
L^{2}_{0}({\mathbb{T}})$, one has

$f=({\mathcal{E}}_{0}\breve{f},f_{1}^{-})f_{1}^{+}-({\mathcal{E}}_{0}\breve{f},f_{1}^{+})f_{1}^{-}+(f,f_{0})f_{0}\
.$ (3.17)

###### Proof.

$(i)$ Write
$f=\alpha_{1}^{+}f_{1}^{+}+\alpha_{1}^{-}f_{1}^{-}+\alpha_{0}f_{0}$ with some
coefficients $\alpha_{1}^{\pm}$, $\alpha_{0}$ to be determined. Taking the
inner product of ${\mathcal{E}}_{\mu}f$ with every element of the basis gives
the system

$\begin{pmatrix}({\mathcal{E}}_{\mu}f,f_{1}^{+})\\\
({\mathcal{E}}_{\mu}f,f_{1}^{-})\\\
({\mathcal{E}}_{\mu}f,f_{0})\end{pmatrix}=\begin{pmatrix}({\mathcal{E}}_{\mu}f_{1}^{+},f_{1}^{+})&({\mathcal{E}}_{\mu}f_{1}^{-},f_{1}^{+})&({\mathcal{E}}_{\mu}f_{0},f_{1}^{+})\\\
({\mathcal{E}}_{\mu}f_{1}^{+},f_{1}^{-})&({\mathcal{E}}_{\mu}f_{1}^{-},f_{1}^{-})&({\mathcal{E}}_{\mu}f_{0},f_{1}^{-})\\\
({\mathcal{E}}_{\mu}f_{1}^{+},f_{0})&({\mathcal{E}}_{\mu}f_{1}^{-},f_{0})&({\mathcal{E}}_{\mu}f_{0},f_{0})\end{pmatrix}\cdot\begin{pmatrix}\alpha_{1}^{+}\\\
\alpha_{1}^{-}\\\
\alpha_{0}\end{pmatrix}\stackrel{{\scriptstyle\eqref{Emubasis}}}{{=}}\begin{pmatrix}{\rm
i}\frac{\mu}{1-\mu^{2}}&-\frac{1}{1-\mu^{2}}&0\\\ \frac{1}{1-\mu^{2}}&{\rm
i}\frac{\mu}{1-\mu^{2}}&0\\\ 0&0&-\frac{{\rm
i}}{\mu}\end{pmatrix}\cdot\begin{pmatrix}\alpha_{1}^{+}\\\ \alpha_{1}^{-}\\\
\alpha_{0}\end{pmatrix}$

Inverting the previous relation gives (3.16).

$(ii)$ By definition of $0$-symplectic basis, $f_{1}^{\pm}\in
L^{2}_{0}({\mathbb{T}})\cap Y$, whereas $f_{0}$ is a constant function.
Writing
$\breve{f}=\alpha_{1}^{+}f_{1}^{+}+\alpha_{1}^{-}f_{1}^{-}+\alpha_{0}f_{0}$
and taking the scalar product with $f_{0}$, we deduce that $\alpha_{0}=0$.
Then take the scalar product of ${\mathcal{E}}_{0}\breve{f}$ with $f_{1}^{+}$
and $f_{1}^{-}$ and use (3.7) to get $(ii)$. ∎

Next we outline a property of a reversible basis. 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 $x$.

######  Lemma 3.6.

Let $\\{f_{1}^{+},f_{1}^{-},f_{0}\\}$ be a reversible basis according to
Definition 3.2. Then

$f_{1}^{+}(x)=even(x)+{\rm i}\,odd(x)\ ,\quad f_{1}^{-}(x)=odd(x)+{\rm
i}\,even(x)\ ,\quad f_{0}(x)=even(x)+{\rm i}\,odd(x)\ .$ (3.18)

###### Proof.

Using the definition of the involution $\overline{\rho}$ in (2.14) one has
$f_{1}^{+}(x)=a(x)+{\rm i}b(x)=\overline{\rho}f_{1}^{+}(x)=a(-x)-{\rm
i}b(-x)$, implying that $a(x)$ is even and $b(x)$ odd. The other cases are
analogous. ∎

In the next lemma we express the operator
${\mathcal{L}}_{\mu,{\epsilon}}\big{|}_{{\mathcal{V}}_{\mu,{\epsilon}}}$ on
any $\mu$-symplectic and reversible basis according to Definition 3.2.

######  Lemma 3.7 (Matrix representation of ${\mathcal{L}}_{\mu,{\epsilon}}$
on ${\mathcal{V}}_{\mu,{\epsilon}}$).

The $3\times 3$ matrix that represents the Hamiltonian and reversible operator
${\mathcal{L}}_{\mu,{\epsilon}}={\mathcal{J}}_{\mu}{\mathcal{B}}_{\mu,{\epsilon}}$
with respect to any $\mu$-symplectic and reversible basis ${\mathtt{F}}$ of
${\mathcal{V}}_{\mu,{\epsilon}}$, (according to Definition 3.2) is

${\mathtt{L}}_{\mu,{\epsilon}}={\mathtt{J}}_{\mu}{\mathtt{B}}_{\mu,{\epsilon}}$
(3.19)

where

${\mathtt{J}}_{\mu}:=\begin{pmatrix}{\rm i}\mu&1&0\\\ -1&{\rm i}\mu&0\\\
0&0&{\rm
i}\mu\end{pmatrix},\qquad{\mathtt{B}}_{\mu,{\epsilon}}:=\begin{pmatrix}({\mathcal{B}}_{\mu,{\epsilon}}f_{1}^{+},f_{1}^{+})&({\mathcal{B}}_{\mu,{\epsilon}}f_{1}^{-},f_{1}^{+})&({\mathcal{B}}_{\mu,{\epsilon}}f_{0},f_{1}^{+})\\\
({\mathcal{B}}_{\mu,{\epsilon}}f_{1}^{+},f_{1}^{-})&({\mathcal{B}}_{\mu,{\epsilon}}f_{1}^{-},f_{1}^{-})&({\mathcal{B}}_{\mu,{\epsilon}}f_{0},f_{1}^{-})\\\
({\mathcal{B}}_{\mu,{\epsilon}}f_{1}^{+},f_{0})&({\mathcal{B}}_{\mu,{\epsilon}}f_{1}^{-},f_{0})&({\mathcal{B}}_{\mu,{\epsilon}}f_{0},f_{0})\end{pmatrix}\
.$ (3.20)

Moreover, the entries of the matrix ${\mathtt{B}}_{\mu,{\epsilon}}$ are
alternatively real or purely imaginary:

$({\mathcal{B}}_{\mu,{\epsilon}}f_{k}^{\sigma},f_{k^{\prime}}^{\sigma})\in{\mathbb{R}},\qquad({\mathcal{B}}_{\mu,{\epsilon}}f_{k}^{\sigma},f_{k^{\prime}}^{-\sigma})\in{\rm
i}{\mathbb{R}}\ .$ (3.21)

###### Proof.

We first consider $\mu\neq 0$. By Lemma 3.5 $(i)$ and using that
${\mathcal{E}}_{\mu}{\mathcal{L}}_{\mu,{\epsilon}}={\mathcal{B}}_{\mu,{\epsilon}}$
we have
${\mathcal{L}}_{\mu,{\epsilon}}f=\alpha_{1}^{+}f_{1}^{+}+\alpha_{1}^{-}f_{1}^{-}+\alpha_{0}f_{0}$
with

$\begin{pmatrix}\alpha_{1}^{+}\\\ \alpha_{1}^{-}\\\
\alpha_{0}\end{pmatrix}=\begin{pmatrix}{\rm i}\mu&1&0\\\ -1&{\rm i}\mu&0\\\
0&0&{\rm
i}\mu\end{pmatrix}\cdot\begin{pmatrix}({\mathcal{B}}_{\mu,{\epsilon}}f,f_{1}^{+})\\\
({\mathcal{B}}_{\mu,{\epsilon}}f,f_{1}^{-})\\\
({\mathcal{B}}_{\mu,{\epsilon}}f,f_{0})\end{pmatrix}\ ,$ (3.22)

hence the matrix representing the action
${\mathcal{L}}_{\mu,{\epsilon}}\colon{\mathcal{V}}_{\mu,{\epsilon}}\to{\mathcal{V}}_{\mu,{\epsilon}}$
is given by (3.19)–(3.20).

Now let $\mu=0$. Recall that by definition of $0$-symplectic basis, $f_{0}$ is
a constant function. Then
$({\mathcal{L}}_{0,{\epsilon}}f,f_{0})=-({\mathcal{B}}_{0,{\epsilon}}f,{\mathcal{J}}_{0}f_{0})=0$,
hence Lemma 3.5 gives
${\mathcal{L}}_{0,{\epsilon}}f=\alpha_{1}^{+}f_{1}^{+}+\alpha_{1}^{-}f_{1}^{-}$
with

$\begin{pmatrix}\alpha_{1}^{+}\\\
\alpha_{1}^{-}\end{pmatrix}=\begin{pmatrix}0&1\\\
-1&0\end{pmatrix}\cdot\begin{pmatrix}({\mathcal{B}}_{0,{\epsilon}}f,f_{1}^{+})\\\
({\mathcal{B}}_{0,{\epsilon}}f,f_{1}^{-})\end{pmatrix}\ ,$ (3.23)

hence the matrix representing the action
${\mathcal{L}}_{0,{\epsilon}}\colon{\mathcal{V}}_{0,{\epsilon}}\to{\mathcal{V}}_{0,{\epsilon}}$
is given by (3.19)–(3.20) with $\mu=0$.

To prove (3.21) we observe that
$(f,g)=\overline{(\overline{\rho}f,\overline{\rho}g)}$ for any $f,g\in
L^{2}({\mathbb{T}})$. Then by (2.15) and the properties of reversible basis

$({\mathcal{B}}_{\mu,{\epsilon}}f_{k}^{\sigma},f_{k^{\prime}}^{\sigma^{\prime}})=\overline{(\overline{\rho}{\mathcal{B}}_{\mu,{\epsilon}}f_{k}^{\sigma},\overline{\rho}f_{k^{\prime}}^{\sigma^{\prime}})}=\overline{({\mathcal{B}}_{\mu,{\epsilon}}\overline{\rho}f_{k}^{\sigma},\overline{\rho}f_{k^{\prime}}^{\sigma^{\prime}})}=\sigma\sigma^{\prime}\overline{({\mathcal{B}}_{\mu,{\epsilon}}f_{k}^{\sigma},f_{k^{\prime}}^{\sigma^{\prime}})}\
,$ (3.24)

proving (3.21). ∎

It will be important in the following to act with coordinate transformations
that preserve the symplectic and reversible structure of
${\mathtt{L}}_{\mu,{\epsilon}}$. This necessity inspires the following
definition:

######  Definition 3.8 ($\mu$-symplectic and reversible matrix).

A $3\times 3$ matrix ${\mathtt{L}}={\mathtt{J}}{\mathtt{B}}$ is

  * •

complex Hamiltonian, if ${\mathtt{B}}$ is self-adjoint and ${\mathtt{J}}$ is
skew-adjoint with respect to the standard scalar product of
${\mathbb{C}}^{3}$;

  * •

reversible, if
${\mathtt{L}}\circ\overline{\rho}=-\overline{\rho}\circ{\mathtt{L}}$, where

$\overline{\rho}=\begin{pmatrix}\mathfrak{c}&0&0\\\ 0&-\mathfrak{c}&0\\\
0&0&\mathfrak{c}\end{pmatrix}$

and $\mathfrak{c}:z\mapsto\overline{z}$ is the conjugation in the complex
plane.

Moreover, we say that an invertible matrix $Y$ is ${\mathtt{J}}$-symplectic if
$Y{\mathtt{J}}Y^{*}={\mathtt{J}}$.

In our context we will have that ${\mathtt{J}}$ is reversible and
${\mathtt{B}}$ is reversibility preserving, i.e.
${\mathtt{B}}\circ\overline{\rho}=\overline{\rho}\circ{\mathtt{B}}$.

######  Lemma 3.9.

Let $\Sigma$ be a self-adjoint and reversible matrix and ${\mathtt{J}}$ be a
skew-adjoint and reversible matrix. Then $\forall\tau\in{\mathbb{R}}$,
$\exp{(\tau{\mathtt{J}}\Sigma)}$ is ${\mathtt{J}}$-symplectic and
reversibility preserving.

###### Proof.

Let $\varphi(\tau)=\exp{(\tau{\mathtt{J}}\Sigma)}$. Let us prove that
$\psi(\tau):=\varphi(\tau){\mathtt{J}}\varphi(\tau)^{*}-{\mathtt{J}}=0$.
Indeed, $\psi(0)=0$ and

$\frac{{\rm d}}{{\rm
d}\tau}\psi(\tau)=\varphi(\tau)({\mathtt{J}}\Sigma{\mathtt{J}}+{\mathtt{J}}\Sigma^{*}{\mathtt{J}}^{*})\varphi(\tau)=0.$

To show the reversibility preserving property, let us observe that
${\mathtt{J}}\Sigma$ is reversibility preserving since both ${\mathtt{J}}$ and
$\Sigma$ are reversible. Then $\exp{(\tau{\mathtt{J}}\Sigma)}=\sum_{n\geq
0}\frac{\tau^{n}}{n!}({\mathtt{J}}\Sigma)^{n}$ is reversibility preserving as
well. ∎

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

In this section we use the transformation operators $U_{\mu,{\epsilon}}$
obtained in the previous section to construct a symplectic and reversible
basis of ${\mathcal{V}}_{\mu,{\epsilon}}$, and we compute in Proposition 4.3
the $3\times 3$ Hamiltonian and reversible matrix representing the action of
${\mathcal{L}}_{\mu,{\epsilon}}\colon{\mathcal{V}}_{\mu,{\epsilon}}\to{\mathcal{V}}_{\mu,{\epsilon}}$
on such basis. The symplectic and reversible basis of
${\mathcal{V}}_{\mu,{\epsilon}}$ we choose is
${\mathtt{F}}=\\{f_{1}^{+}(\mu,{\epsilon}),f_{1}^{-}(\mu,{\epsilon}),f_{0}(\mu,{\epsilon})\\}$,
with

$f_{k}^{\sigma}(\mu,{\epsilon}):=U_{\mu,{\epsilon}}f_{k}^{\sigma},\qquad
f_{1}^{+}=\cos x,\ \ f_{1}^{-}=\sin x,\ \ f_{0}^{+}=\frac{1}{\sqrt{2}}\ .$
(4.1)

######  Lemma 4.1.

The Kato basis
$\\{f_{1}^{+}(\mu,{\epsilon}),f_{1}^{-}(\mu,{\epsilon}),f_{0}(\mu,{\epsilon})\\}$
in (4.1) is $\mu$-symplectic $\forall(\mu,{\epsilon})\in B(\mu_{0})\times
B({\epsilon}_{0})$ and reversible, according to Definition 3.2. Each map
$(\mu,{\epsilon})\mapsto f_{k}^{\sigma}(\mu,{\epsilon})$ is
${\mathcal{C}}^{\mathtt{r}}$ as a map $B(\mu_{0})\times B({\epsilon}_{0})\to
H^{m_{\star}}({\mathbb{T}})$.

###### Proof.

The basis $\\{f_{1}^{\pm},f_{0}\\}$ is $\mu$-symplectic $\forall\mu$ and
reversible. Then the result follows from Lemma 3.4. ∎

In the next lemma we expand the vectors $f_{k}^{\sigma}(\mu,{\epsilon})$ in
$\mu,{\epsilon}$.

######  Lemma 4.2 (Expansion of the basis ${\mathtt{F}}$).

For small values of $\mu,{\epsilon}$ the vectors
$f_{k}^{\sigma}(\mu,{\epsilon})$ in (4.1) expand as

$\displaystyle f_{1}^{+}(\mu,{\epsilon})=\cos
x+{\epsilon}\,{\mathfrak{a}}\cos(2x)-{\rm
i}\mu{\epsilon}\,{\mathfrak{b}}\sin(2x)+{\epsilon}^{2}even_{0}(x)+{\mathcal{O}}^{\mathtt{r}}(\mu^{2}{\epsilon},\mu{\epsilon}^{2},{\epsilon}^{3})$
(4.2) $\displaystyle f_{1}^{-}(\mu,{\epsilon})=\sin
x+{\epsilon}\,{\mathfrak{a}}\sin(2x)+{\rm
i}\mu{\epsilon}\,{\mathfrak{b}}\cos(2x)+{\epsilon}^{2}odd(x)+{\mathcal{O}}^{\mathtt{r}}(\mu^{2}{\epsilon},\mu{\epsilon}^{2},{\epsilon}^{3})$
$\displaystyle
f_{0}^{+}(\mu,{\epsilon})=\frac{1}{\sqrt{2}}+{\mathcal{O}}^{\mathtt{r}}(\mu^{2}{\epsilon},\mu{\epsilon}^{2})$

with ${\mathfrak{b}}$ a real number and

${\mathfrak{a}}:=\frac{1}{{\mathfrak{m}}(1)-{\mathfrak{m}}(2)}\ ,$ (4.3)

and $even_{0}(x)$ is a real valued, even function with zero average, and the
${\mathcal{O}}(\cdot)$s are vectors in $H^{m_{\star}}({\mathbb{T}})$.
Moreover, for any $|{\epsilon}|\leq{\epsilon}_{0}$,
$f_{0}^{+}(0,{\epsilon})=f_{0}^{+}$.

Finally, one has the expansions

$\dot{f}_{1}^{-}(0,{\epsilon})={\rm
i}{\epsilon}{\mathfrak{b}}\cos(2x)+{\mathcal{O}}({\epsilon}^{2}),\quad\dot{f}_{0}(0,{\epsilon})={\mathcal{O}}({\epsilon}^{2})\
,\quad\ddot{f}_{k}^{\sigma}(0,{\epsilon})={\mathcal{O}}({\epsilon})\ .$ (4.4)

###### Proof.

The long computations are left in Appendix A. ∎

We now state the main result of this section:

######  Proposition 4.3.

The action of the Hamiltonian and reversible operator
${\mathcal{L}}_{\mu,{\epsilon}}$ in the symplectic and reversible basis
$\\{f_{1}^{+}(\mu,{\epsilon}),f_{1}^{-}(\mu,{\epsilon}),f_{0}^{+}(\mu,{\epsilon})\\}$
of ${\mathcal{V}}_{\mu,{\epsilon}}$, defined in (4.1), is represented by the
$3\times 3$ Hamiltonian and reversible matrix

${\mathtt{L}}_{\mu,{\epsilon}}={\mathtt{J}}_{\mu}{\mathtt{B}}_{\mu,{\epsilon}},$
(4.5)

where ${\mathtt{J}}_{\mu}$ is in (3.20) and ${\mathtt{B}}_{\mu,{\epsilon}}$ is
the self-adjoint and reversibility preserving $3\times 3$-matrix

${\mathtt{B}}_{\mu,{\epsilon}}=\begin{pmatrix}E&\vline&{\mathtt{f}}\\\
\hline\cr{\mathtt{f}}^{\dagger}&\vline&g\end{pmatrix}\ .$ (4.6)

Here $E$ is the $2\times 2$ selfadjoint matrix

$E=\begin{pmatrix}-{\mathfrak{a}}{\epsilon}^{2}(1+r_{1}^{\prime}({\epsilon}))+{\mathtt{e}}_{22}\mu^{2}(1+r_{1}^{\prime\prime}({\epsilon},\mu))&{\rm
i}\mu({\mathtt{e}}_{12}+r_{2}({\epsilon}^{2},\mu{\epsilon},\mu^{2}))\\\ -{\rm
i}\mu({\mathtt{e}}_{12}+r_{2}({\epsilon}^{2},\mu{\epsilon},\mu^{2}))&{\mathtt{e}}_{22}\mu^{2}(1+r_{4}({\epsilon},\mu))\end{pmatrix}\
,$ (4.7)

the vector ${\mathtt{f}}$ and the number $g$ are given by

${\mathtt{f}}=\begin{pmatrix}-\sqrt{2}{\epsilon}\left(1+r_{3}({\epsilon}^{2},\mu^{2})\right)\\\
{\rm i}\mu{\epsilon}\,r_{5}({\epsilon},\mu)\\\ \end{pmatrix},\qquad
g={\mathtt{e}}_{33}+\frac{{\mathfrak{a}}}{2}{\epsilon}^{2}+{\mathtt{g}}_{33}\mu^{2}+r_{6}({\epsilon}^{3},\mu^{2}{\epsilon},\mu^{3})\
,$ (4.8)

with ${\mathfrak{a}}$ in (4.3) and

${\mathtt{e}}_{22}:=-\frac{1}{2}\ddot{\mathfrak{m}}(1),\qquad{\mathtt{e}}_{12}:=\dot{\mathfrak{m}}(1),\qquad{\mathtt{e}}_{33}:={\mathfrak{m}}(1)-{\mathfrak{m}}(0),\qquad{\mathtt{g}}_{33}:=-\frac{1}{2}\ddot{\mathfrak{m}}(0)\,.$
(4.9)

The reminders $r_{k}$ are functions of class ${\mathcal{C}}^{\mathtt{r}-2}$.

The remaining part of the section will be devoted to the proof of Proposition
4.3.

By Lemma 4.1, the matrix
${\mathtt{B}}_{\mu,{\epsilon}}:=({\mathcal{B}}_{\mu,{\epsilon}}f_{k}^{\sigma}(\mu,{\epsilon}),f_{k^{\prime}}^{\sigma^{\prime}}(\mu,{\epsilon}))$
is of class ${\mathcal{C}}^{\mathtt{r}}$, $\mathtt{r}\geq 3$. This allows us
to expand the matrix ${\mathtt{B}}_{\mu,{\epsilon}}$ in $\mu$, up to second
order:

${\mathtt{B}}_{\mu,{\epsilon}}={\mathtt{B}}_{0,{\epsilon}}+\mu\dot{\mathtt{B}}_{0,{\epsilon}}+\frac{\mu^{2}}{2}\ddot{\mathtt{B}}_{0,{\epsilon}}+\mu^{2}\varphi(\mu,{\epsilon}),\quad\varphi(\mu,{\epsilon}):=\frac{1}{2}\int_{0}^{1}(1-\tau)^{2}\,\left(\ddot{\mathtt{B}}_{\tau\mu,{\epsilon}}-\ddot{\mathtt{B}}_{0,{\epsilon}}\right){\rm
d}\tau\ \in{\mathcal{C}}^{\mathtt{r}-2}\ .$ (4.10)

We now consider each term separately.

Expansion of ${\mathtt{B}}_{0,{\epsilon}}$. We start with the matrix
${\mathtt{B}}_{0,{\epsilon}}=\\{({\mathcal{B}}_{0,{\epsilon}}f_{k}^{\sigma}(0,{\epsilon}),f_{k^{\prime}}^{\sigma^{\prime}}(0,{\epsilon}))\\}$.

######  Lemma 4.4 (Expansion of the matrix ${\mathtt{B}}_{0,{\epsilon}}$).

The $3\times 3$ selfadjoint, real and reversibility preserving matrix
${\mathtt{B}}_{0,{\epsilon}}$ expands as

${\mathtt{B}}_{0,{\epsilon}}=\begin{pmatrix}-{\mathfrak{a}}{\epsilon}^{2}+r_{1}({\epsilon}^{3})&0&-\sqrt{2}{\epsilon}+r_{3}({\epsilon}^{3})\\\
0&0&0\\\
-\sqrt{2}{\epsilon}+r_{3}({\epsilon}^{3})&0&{\mathfrak{m}}(1)-{\mathfrak{m}}(0)+\frac{1}{2}{\mathfrak{a}}{\epsilon}^{2}+r_{6}({\epsilon}^{3})\end{pmatrix}\
,$ (4.11)

with ${\mathfrak{a}}$ in (4.3) and $r_{k}$ are functions of class
${\mathcal{C}}^{\mathtt{r}}$ independent of $\mu$.

###### Proof.

The matrix ${\mathtt{B}}_{0,{\epsilon}}$ is real since
${\mathcal{B}}_{0,{\epsilon}}$ is a real operator and the vectors
$f_{k}^{\sigma}(0,{\epsilon})$ are real, being $U_{0,{\epsilon}}$ a real
operator by Lemma 3.4 $(iii)$. Moreover, by Lemma 3.7, the entries
$({\mathcal{B}}_{0,{\epsilon}}f_{k}^{\sigma}(0,{\epsilon}),f_{k^{\prime}}^{-\sigma}(0,{\epsilon}))$
are purely imaginary, so we get

${\mathtt{B}}_{0,{\epsilon}}=\begin{pmatrix}E_{11}({\epsilon})&0&E_{13}({\epsilon})\\\
0&E_{22}({\epsilon})&0\\\
E_{13}({\epsilon})&0&E_{33}({\epsilon})\end{pmatrix}$ (4.12)

for some real functions $E_{jk}({\epsilon})$.

Step 1: proof that $E_{22}({\epsilon})\equiv 0$ for any
$|{\epsilon}|<{\epsilon}_{0}$. Recall that, by Lemma 2.2, the operator
${\mathcal{L}}_{0,{\epsilon}}$ has 0 as eigenvalue with algebraic multiplicity
3 and geometric multiplicity 2. The matrix
${\mathtt{J}}_{0}{\mathtt{B}}_{0,{\epsilon}}$, that represents the action of
this operator on the generalized kernel ${\mathcal{V}}_{0,{\epsilon}}$, has
therefore two Jordan blocks: one of dimension $1$ and the other of dimension
$2$. Then, we have that ${\mathtt{J}}_{0}{\mathtt{B}}_{0,{\epsilon}}$ is
nilpotent of index $2$, i.e.

$0=({\mathtt{J}}_{0}{\mathtt{B}}_{0,{\epsilon}})^{2}=\begin{pmatrix}-E_{11}({\epsilon})E_{22}({\epsilon})&0&-E_{22}({\epsilon})E_{13}({\epsilon})\\\
0&-E_{11}({\epsilon})E_{22}({\epsilon})&0\\\ 0&0&0\end{pmatrix}\ .$

We show below that $E_{11}({\epsilon})$ and $E_{13}({\epsilon})$ are different
from zero for ${\epsilon}$ sufficiently small, therefore

$E_{22}({\epsilon})\equiv 0\ .$ (4.13)

Step 2: expansion of $E_{11}({\epsilon})$, $E_{33}({\epsilon})$ and
$E_{13}({\epsilon})$. We start by Taylor expanding both the operator
${\mathcal{B}}_{0,{\epsilon}}$ and the elements of the basis ${\mathtt{F}}$ in
orders of ${\epsilon}$:

${\mathcal{B}}_{0,{\epsilon}}={\mathcal{B}}_{0}+{\epsilon}{\mathcal{B}}_{1}+{\epsilon}^{2}{\mathcal{B}}_{2}+{\mathcal{O}}^{\mathtt{r}}({\epsilon}^{3}),\qquad
f_{k}^{\sigma}(0,{\epsilon})=f_{k}^{\sigma}+{\epsilon}f_{k_{1}}^{\sigma}+{\epsilon}^{2}f_{k_{2}}^{\sigma}+{\mathcal{O}}^{\mathtt{r}}({\epsilon}^{3}),$
(4.14)

with

${\mathcal{B}}_{0}:={\mathfrak{m}}(1)-{\mathcal{M}}(D),\qquad{\mathcal{B}}_{1}:=-2\cos
x,\qquad{\mathcal{B}}_{2}:=c_{2}-2u_{2}^{[0]}-2u_{2}^{[2]}\cos(2x)$ (4.15)

with $c_{2},u_{2}^{[0]}$, $u_{2}^{[2]}$ given in Theorem 1.1 and
$f_{k_{1}}^{\sigma}$ in Lemma 4.2. We will see that $f_{k_{2}}^{\sigma}$ does
not need to be computed explicitly, since it won’t bring any relevant
contribution in the expansion. Moreover, a direct computation gives

$\displaystyle{\mathcal{B}}_{0}f_{1}^{\sigma}=0\
,\quad{\mathcal{B}}_{0}f_{0}^{+}=({\mathfrak{m}}(1)-{\mathfrak{m}}(0))f_{0}^{+}\
,\quad{\mathcal{B}}_{0}f_{1_{1}}^{+}=({\mathfrak{m}}(1)-{\mathfrak{m}}(2)){\mathfrak{a}}\cos(2x)\
,$ (4.16)
$\displaystyle{\mathcal{B}}_{1}f_{1}^{+}=-(1+\cos(2x)),\quad{\mathcal{B}}_{1}f_{1_{1}}^{+}=-{\mathfrak{a}}(\cos
x+\cos(3x))\ ,\quad{\mathcal{B}}_{1}f_{0}^{+}=-\sqrt{2}\cos(x)\ ,$
$\displaystyle{\mathcal{B}}_{2}f_{1}^{+}=(c_{2}-2u_{2}^{[0]}-u_{2}^{[2]})\cos
x-u_{2}^{[2]}\cos(3x)\
,\quad{\mathcal{B}}_{2}f_{0}^{+}=(c_{2}-2u_{2}^{[0]}-2u_{2}^{[2]}\cos(2x))f_{0}^{+}\
.$

Using the decomposition (4.14) we get

$\begin{split}({\mathcal{B}}_{0,{\epsilon}}f_{k}^{\sigma}(0,{\epsilon}),&f_{k^{\prime}}^{\sigma^{\prime}}(0,{\epsilon}))=({\mathcal{B}}_{0}f_{k}^{\sigma},f_{k^{\prime}}^{\sigma^{\prime}})+{\epsilon}\left[({\mathcal{B}}_{1}f_{k}^{\sigma},f_{k^{\prime}}^{\sigma^{\prime}})+({\mathcal{B}}_{0}f_{k_{1}}^{\sigma},f_{k^{\prime}}^{\sigma^{\prime}})+({\mathcal{B}}_{0}f_{k}^{\sigma},f_{k_{1}^{\prime}}^{\sigma^{\prime}})\right]+\\\
+{\epsilon}^{2}&\left[({\mathcal{B}}_{2}f_{k}^{\sigma},f_{k}^{\sigma^{\prime}})+({\mathcal{B}}_{1}f_{k_{1}}^{\sigma},f_{k^{\prime}}^{\sigma^{\prime}})+({\mathcal{B}}_{1}f_{k}^{\sigma},f_{k_{1}^{\prime}}^{\sigma^{\prime}})+({\mathcal{B}}_{0}f_{k_{1}}^{\sigma},f_{k_{1}^{\prime}}^{\sigma^{\prime}})+({\mathcal{B}}_{0}f_{k_{2}}^{\sigma},f_{k^{\prime}}^{\sigma^{\prime}})+({\mathcal{B}}_{0}f_{k}^{\sigma},f_{k_{2}^{\prime}}^{\sigma^{\prime}})\right]\end{split}$
(4.17)

We are now ready to compute the expansion of the elements
$E_{11}({\epsilon})$, $E_{33}({\epsilon})$ and $E_{13}({\epsilon})$.
$\bullet$ Expansion of
$E_{1,1}({\epsilon})=({\mathcal{B}}_{0,{\epsilon}}f_{1}^{+}(0,{\epsilon}),f_{1}^{+}(0,{\epsilon}))$.
By (4.16) and the selfadjointness of ${\mathcal{B}}_{0}$, the term of order
zero in (4.17) does not contribute, together with the following others:

$({\mathcal{B}}_{0}f_{1}^{+},f_{1_{1}}^{+})=({\mathcal{B}}_{0}f_{1_{1}}^{+},f_{1}^{+})=({\mathcal{B}}_{0}f_{1}^{+},f_{1_{2}}^{+})=({\mathcal{B}}_{0}f_{1_{2}}^{+},f_{1}^{+})=({\mathcal{B}}_{1}f_{1}^{+},f_{1}^{+})=0$

The nonvanishing terms, which we compute again using (4.16), are

$({\mathcal{B}}_{2}f_{1}^{+},f_{1}^{+})=c_{2}-2u_{2}^{[0]}-u_{2}^{[2]}\,,\qquad({\mathcal{B}}_{1}f_{1_{1}}^{+},f_{1}^{+})=-{\mathfrak{a}}=({\mathcal{B}}_{1}f_{1}^{+},f_{1_{1}}^{+})\,,\qquad({\mathcal{B}}_{0}f_{1_{1}}^{+},f_{1_{1}}^{+})=({\mathfrak{m}}(1)-{\mathfrak{m}}(2)){\mathfrak{a}}^{2}\
.$

Summing up, and using the expressions of $u_{2}^{[0]}$ and $u_{2}^{[2]}$ in
Theorem 1.1 and of ${\mathfrak{a}}$ in (4.3), we get

$E_{1,1}({\epsilon})=(\underbrace{c_{2}-2u_{2}^{[0]}-u_{2}^{[2]}}_{=0}-\underbrace{2{\mathfrak{a}}+({\mathfrak{m}}(1)-{\mathfrak{m}}(2)){\mathfrak{a}}^{2}}_{=-{\mathfrak{a}}}){\epsilon}^{2}+r({\epsilon}^{3})=-{\mathfrak{a}}{\epsilon}^{2}+r({\epsilon}^{3})\
,$ (4.18)

where $r({\epsilon}^{3})$ is of class ${\mathcal{C}}^{\mathtt{r}}$.
$\bullet$ Expansion of
$E_{1,3}({\epsilon})=({\mathcal{B}}_{0,{\epsilon}}f_{0}^{+}(0,{\epsilon}),f_{1}^{+}(0,{\epsilon}))$.
Again the term of order $0$ in (4.17) vanishes. About the terms of order
${\epsilon}$, by (4.16) the only one not vanishing is
$({\mathcal{B}}_{1}f_{1}^{+},f_{0}^{-})=-\sqrt{2}$. Finally, all terms of
order ${\epsilon}^{2}$ vanish, using also that $f_{1_{2}}^{+}$ has zero
average by Lemma 4.2. In conclusion

$E_{1,3}({\epsilon})=-\sqrt{2}{\epsilon}+r({\epsilon}^{3})\ $ (4.19)

with $r({\epsilon}^{3})$ of class ${\mathcal{C}}^{\mathtt{r}}$.
$\bullet$ Expansion of
$E_{3,3}({\epsilon})=({\mathcal{B}}_{0,{\epsilon}}f_{0}^{+}(0,{\epsilon}),f_{0}^{+}(0,{\epsilon}))$:
Recall that $f_{0}^{+}(0,{\epsilon})=f_{0}^{+}$ for any $\epsilon$ by Lemma
4.2. Since
${\mathcal{B}}_{0,{\epsilon}}f_{0}=(c_{\epsilon}-{\mathfrak{m}}(0)-2u_{\epsilon})f_{0}^{+}$,
by Theorem 1.1 we readily get

$E_{3,3}({\epsilon})={\mathfrak{m}}(1)-{\mathfrak{m}}(0)+{\epsilon}^{2}\underbrace{(c_{2}-2u_{2}^{[0]})}_{\frac{1}{2}{\mathfrak{a}}}+r({\epsilon}^{3})$
(4.20)

with $r({\epsilon}^{3})$ of class ${\mathcal{C}}^{\mathtt{r}}$.
Then the expansion (4.11) follows from (4.12), (4.13), (4.18), (4.19) and
(4.20). ∎

Expansion of $\dot{\mathtt{B}}_{0,{\epsilon}}$. Next we compute the expansion
of the matrix $\dot{\mathtt{B}}_{0,{\epsilon}}$ in (4.10). Remark that

$\dot{\mathtt{B}}_{0,{\epsilon}}=((\dot{\mathcal{B}}_{0,{\epsilon}}f_{k}^{\sigma}(0,{\epsilon}),f_{k^{\prime}}^{\sigma^{\prime}}(0,{\epsilon}))+2\textup{Sym}({\mathcal{B}}_{0,{\epsilon}}f_{k}^{\sigma}(0,{\epsilon}),\dot{f}_{k^{\prime}}^{\sigma^{\prime}}(0,{\epsilon})))=\dot{\mathtt{B}}_{0,{\epsilon}}^{[1]}+\dot{\mathtt{B}}_{0,{\epsilon}}^{[2]}+\dot{\mathtt{B}}_{0,{\epsilon}}^{[2]*},$
(4.21)

where $X^{*}$ denotes the transpose conjugate matrix to $X$.

We now prove:

######  Lemma 4.5 (Expansion of the matrix $\dot{\mathtt{B}}_{0,{\epsilon}}$).

The $3\times 3$ selfadjoint, purely imaginary and reversibility preserving
matrix $\dot{\mathtt{B}}_{0,{\epsilon}}$ in (4.21) expands as

$\dot{\mathtt{B}}_{0,{\epsilon}}=\begin{pmatrix}0&{\rm
i}(\dot{\mathfrak{m}}(1)+r_{2}({\epsilon}^{2}))&0\\\ -{\rm
i}(\dot{\mathfrak{m}}(1)+r_{2}({\epsilon}^{2}))&0&{\rm
i}r_{5}({\epsilon}^{2})\\\ 0&-{\rm i}r_{5}({\epsilon}^{2})&0\end{pmatrix}\ $
(4.22)

with $r_{k}$ are functions of class ${\mathcal{C}}^{\mathtt{r}-1}$ independent
of $\mu$.

###### Proof.

We consider separately the matrices $\dot{\mathtt{B}}_{0,{\epsilon}}^{[1]}$
and $\dot{\mathtt{B}}_{0,{\epsilon}}^{[2]}$ in (4.21).

Expansion of $\dot{\mathtt{B}}_{0,{\epsilon}}^{[1]}$. Recall
$\dot{\mathtt{B}}_{0,{\epsilon}}^{[1]}=(\dot{\mathcal{B}}_{0,{\epsilon}}f_{k}^{\sigma}(0,{\epsilon}),f_{k^{\prime}}^{\sigma^{\prime}}(0,{\epsilon}))_{k,k^{\prime}=0,1}^{\sigma,\sigma^{\prime}=\pm}$.
By Lemma 2.1, $\dot{\mathcal{B}}_{0,{\epsilon}}$ is reversibility preserving,
so the entries of the matrix are alternatively real and purely imaginary. But
$\dot{\mathcal{B}}_{0,{\epsilon}}$ is also purely imaginary, so every entry
must also be purely imaginary. As a result, the entries of the type
$(\dot{\mathcal{B}}_{0,{\epsilon}}f_{k}^{\sigma}(0,{\epsilon}),f_{k^{\prime}}^{\sigma}(0,{\epsilon}))$
are zero for any $|{\epsilon}|<{\epsilon}_{0}$:

$\dot{\mathtt{B}}_{0,{\epsilon}}^{[1]}=\begin{pmatrix}0&\dot{E}_{12}^{[1]}({\epsilon})&0\\\
-\dot{E}_{12}^{[1]}({\epsilon})&0&\dot{E}_{23}^{[1]}({\epsilon})\\\
0&-\dot{E}_{23}^{[1]}({\epsilon})&0\end{pmatrix}\ .$ (4.23)

We now compute the expansion of the remaining elements. Recall that
$\dot{\mathcal{B}}_{0,{\epsilon}}=-\dot{\mathcal{M}}(D)$.
$\bullet$ Expansion of
$\dot{E}_{12}^{[1]}({\epsilon})=(\dot{\mathcal{B}}_{0,{\epsilon}}f_{1}^{-}(0,{\epsilon}),f_{1}^{+}(0,{\epsilon}))$:
By (4.2),

$\dot{E}_{12}^{[1]}({\epsilon})=(\dot{\mathcal{B}}_{0,{\epsilon}}f_{1}^{-}(0,{\epsilon}),f_{1}^{+}(0,{\epsilon}))=({\rm
i}\dot{\mathfrak{m}}(1)\cos x+{\rm
i}{\epsilon}\dot{\mathfrak{m}}(2){\mathfrak{a}}\cos(2x)+{\mathcal{O}}^{\mathtt{r}-1}({\epsilon}^{2}),f_{1}^{+}(0,{\epsilon}))={\rm
i}(\dot{\mathfrak{m}}(1)+r({\epsilon}^{2}))\,,$ (4.24)

where $r({\epsilon}^{2})$ is of class ${\mathcal{C}}^{\mathtt{r}-1}$.
$\bullet$ Expansion of
$\dot{E}_{23}^{[1]}({\epsilon})=(\dot{\mathcal{B}}_{0,{\epsilon}}f_{0}^{+}(0,{\epsilon}),f_{1}^{-}(0,{\epsilon}))$:
By Lemma 4.2,
$\dot{\mathcal{B}}_{0,{\epsilon}}f_{0}^{+}(0,{\epsilon})=-\dot{\mathcal{M}}(D)f_{0}^{+}=0$,
thus

$\dot{E}_{23}^{[1]}({\epsilon})=0\ .$ (4.25)

Expansion of $\dot{\mathtt{B}}_{0,{\epsilon}}^{[2]}$. Recall
$\dot{\mathtt{B}}_{0,{\epsilon}}^{[2]}=\big{(}({\mathcal{B}}_{0,{\epsilon}}f_{k}^{\sigma}(0,{\epsilon}),\dot{f}_{k^{\prime}}^{\sigma^{\prime}}(0,{\epsilon}))\big{)}$.
Since ${\mathtt{F}}$ is a reversible basis,
$\dot{f}_{k}^{\sigma}(0,{\epsilon})$ have the same parity properties in
(3.18). But
$\dot{f}_{k}^{\sigma}(0,{\epsilon})=\dot{U}_{0,{\epsilon}}f_{k}^{\sigma}$ is
purely imaginary, as $\dot{U}_{0,{\epsilon}}$ is purely imaginary by Lemma 3.4
and $f_{k}^{\sigma}$ is real. Being instead $f_{k}^{\sigma}(0,{\epsilon})$
real, we conclude that

$f_{k}^{+}(0,{\epsilon})=even(x)\ ,\quad f_{k}^{-}(0,{\epsilon})=odd(x)\
,\quad\dot{f}_{k}^{+}(0,{\epsilon})={\rm i}\,odd(x)\
,\qquad\dot{f}_{k}^{-}(0,{\epsilon})={\rm i}\,even(x)\ .$ (4.26)

Now we use that ${\mathcal{B}}_{0,{\epsilon}}$ is parity preserving, as
${\mathcal{B}}_{0,{\epsilon}}=c_{\epsilon}-{\mathcal{M}}(D)-2u_{\epsilon}$ and
${\mathcal{M}}(D)$ is an even Fourier multiplier (Assumption (A1)) and
$u_{\epsilon}$ of Theorem 1.1 is an even function. So the matrix
$\dot{\mathtt{B}}_{0,{\epsilon}}^{[2]}+\dot{\mathtt{B}}_{0,{\epsilon}}^{[2]*}$
has the form

$\dot{\mathtt{B}}_{0,{\epsilon}}^{[2]}+\dot{\mathtt{B}}_{0,{\epsilon}}^{[2]*}=\begin{pmatrix}0&\dot{E}_{12}^{[2]}({\epsilon})&0\\\
-\dot{E}_{12}^{[2]}({\epsilon})&0&\dot{E}_{23}^{[2]}({\epsilon})\\\
0&-\dot{E}_{23}^{[2]}({\epsilon})&0\end{pmatrix}$ (4.27)

with some coefficients that we now compute.
$\bullet$ Expansion of
$\dot{E}_{12}^{[2]}({\epsilon})=({\mathcal{B}}_{0,{\epsilon}}f_{1}^{-}(0,{\epsilon}),\dot{f}_{1}^{+}(0,{\epsilon}))+({\mathcal{B}}_{0,{\epsilon}}\dot{f}_{1}^{-}(0,{\epsilon}),f_{1}^{+}(0,{\epsilon}))$:
First note that the action of ${\mathcal{L}}_{0,{\epsilon}}$ on the basis
$\\{f_{1}^{\pm}(0,{\epsilon}),f_{0}^{+}(0,{\epsilon})\\}$ is given by the
matrix ${\mathtt{J}}_{0}{\mathtt{B}}_{0,{\epsilon}}$, with
${\mathtt{B}}_{0,{\epsilon}}$ in (4.11). Then one checks that
${\mathcal{L}}_{0,{\epsilon}}f_{1}^{-}(0,{\epsilon})=0$, as it is clear by
Lemma 4.4. Using this identity together with the fact that
$\dot{f}_{1}^{+}(0,{\epsilon})$ is mean free (see (4.26)), and (4.14), (4.15),
and the expression for $\dot{f}_{1}^{-}(0,{\epsilon})$ in (4.4), we get

$\dot{E}_{12}^{[2]}({\epsilon})=\underbrace{({\mathcal{B}}_{0,{\epsilon}}f_{1}^{-}(0,{\epsilon}),\dot{f}_{1}^{+}(0,{\epsilon}))}_{=-({\mathcal{L}}_{0,{\epsilon}}f_{1}^{-}(0,{\epsilon}),{\mathcal{E}}_{0}\dot{f}_{1}^{+}(0,{\epsilon}))=0}+\overline{({\mathcal{B}}_{0,{\epsilon}}f_{1}^{+}(0,{\epsilon}),\dot{f}_{1}^{-}(0,{\epsilon}))}={\rm
i}{\epsilon}^{2}{\mathfrak{b}}\underbrace{\left({\mathfrak{a}}({\mathfrak{m}}(1)-{\mathfrak{m}}(2))-1\right)}_{=0}+{\rm
i}r({\epsilon}^{3})\,$ (4.28)

where $r({\epsilon}^{3})$ is of class ${\mathcal{C}}^{\mathtt{r}-1}$.
$\bullet$ Expansion of
$\dot{E}_{23}^{[2]}({\epsilon})=({\mathcal{B}}_{0,{\epsilon}}f_{0}(0,{\epsilon}),\dot{f}_{1}^{-}(0,{\epsilon}))+({\mathcal{B}}_{0,{\epsilon}}\dot{f}_{0}^{+}(0,{\epsilon}),f_{1}^{-}(0,{\epsilon}))$:
As $f_{0}^{+}(0,{\epsilon})=f_{0}^{+}$ $\forall{\epsilon}$ small enough, we
have by (4.16) that
${\mathcal{B}}_{0,{\epsilon}}f_{0}^{+}(0,{\epsilon})=a+b{\epsilon}f_{1}^{+}+{\mathcal{O}}({\epsilon}^{2})$
for some constants $a,b$, and by Lemma 4.2 that
$\dot{f}_{1}^{-}(0,{\epsilon})={\rm
i}{\epsilon}{\mathfrak{b}}\cos(2x)+{\mathcal{O}}({\epsilon}^{2})$ and
$\dot{f}_{0}(0,{\epsilon})={\mathcal{O}}({\epsilon}^{2})$. So

$\dot{E}_{23}^{[2]}({\epsilon})=({\mathcal{B}}_{0,{\epsilon}}f_{0}(0,{\epsilon}),\dot{f}_{1}^{-}(0,{\epsilon}))+\overline{({\mathcal{B}}_{0,{\epsilon}}f_{1}^{-}(0,{\epsilon}),\dot{f}_{0}(0,{\epsilon}))}={\rm
i}r({\epsilon}^{2})$ (4.29)

where $r({\epsilon}^{2})$ is of class ${\mathcal{C}}^{\mathtt{r}-1}$.

In conclusion the expansion (4.22) follows from (4.23), (4.24), (4.25),
(4.27), (4.28), (4.29). ∎

Expansion of $\ddot{\mathtt{B}}_{0,{\epsilon}}$. Finally, we compute the
expansion of the matrix $\ddot{\mathtt{B}}_{0,{\epsilon}}$ in (4.10). Such
matrix is given by

$\ddot{\mathtt{B}}_{0,{\epsilon}}=\left((\ddot{\mathcal{B}}_{0,{\epsilon}}f_{k}^{\sigma},f_{k^{\prime}}^{\sigma^{\prime}})+({\mathcal{B}}_{0,{\epsilon}}\ddot{f}_{k}^{\sigma},f_{k^{\prime}}^{\sigma^{\prime}})+({\mathcal{B}}_{0,{\epsilon}}f_{k}^{\sigma},\ddot{f}_{k^{\prime}}^{\sigma^{\prime}})+2(\dot{\mathcal{B}}_{0,{\epsilon}}\dot{f}_{k}^{\sigma},f_{k^{\prime}}^{\sigma^{\prime}})+2(\dot{\mathcal{B}}_{0,{\epsilon}}f_{k}^{\sigma},\dot{f}_{k^{\prime}}^{\sigma^{\prime}})+2({\mathcal{B}}_{0,{\epsilon}}\dot{f}_{k}^{\sigma},\dot{f}_{k^{\prime}}^{\sigma^{\prime}})\right)|_{\mu=0}$
(4.30)

######  Lemma 4.6 (Expansion of the matrix
$\ddot{\mathtt{B}}_{0,{\epsilon}}$).

The $3\times 3$ selfadjoint, real and reversibility preserving matrix
$\ddot{\mathtt{B}}_{0,{\epsilon}}$ in (4.30) expands as

$\ddot{\mathtt{B}}_{0,{\epsilon}}=\begin{pmatrix}-\ddot{\mathfrak{m}}(1)&0&0\\\
0&-\ddot{\mathfrak{m}}(1)&0\\\
0&0&-\ddot{\mathfrak{m}}(0)\end{pmatrix}+{\mathcal{O}}^{\mathtt{r}-2}({\epsilon})\
$ (4.31)

with ${\mathcal{O}}^{\mathtt{r}-2}({\epsilon})$ independent of $\mu$.

###### Proof.

By Lemma 4.2,
$\dot{f}_{k}^{\sigma}(0,{\epsilon}),\ddot{f}_{k}^{\sigma}(0,{\epsilon})={\mathcal{O}}({\epsilon})$.
Hence, all the terms in (4.30) are ${\mathcal{O}}^{\mathtt{r}-2}({\epsilon})$,
except
$(\ddot{\mathcal{B}}_{0,{\epsilon}}f_{k}^{\sigma},f_{k^{\prime}}^{\sigma^{\prime}})$.
Since $\ddot{\mathcal{B}}_{0,{\epsilon}}=-\ddot{\mathcal{M}}(D)$, the result
follows. ∎

In the following lemma we improve the order of the remainders of the off-
diagonal terms. This will be necessary in order to block diagonalize the
matrix ${\mathtt{B}}_{\mu,{\epsilon}}$.

######  Lemma 4.7 (off-diagonal errors).

The off diagonal terms of the matrix ${\mathtt{B}}_{\mu,{\epsilon}}$ have the
following form:

$({\mathtt{B}}_{\mu,{\epsilon}})_{1,3}=-\sqrt{2}{\epsilon}\big{(}1+r_{1}({\epsilon}^{2},\mu^{2})\big{)},\quad({\mathtt{B}}_{\mu,{\epsilon}})_{2,3}={\rm
i}\mu{\epsilon}\,r_{2}(\mu,{\epsilon})$ (4.32)

with remainders $r_{k}\in{\mathcal{C}}^{\mathtt{r}-2}$.

###### Proof.

So far, from (4.10) using Lemmata 4.4, 4.5 and 4.6, we have that

$({\mathtt{B}}_{\mu,{\epsilon}})_{1,3}=-\sqrt{2}{\epsilon}+r_{3}({\epsilon}^{3})+\mu^{2}\varphi_{1}(\mu,{\epsilon})\quad\text{and}\quad({\mathtt{B}}_{\mu,{\epsilon}})_{2,3}={\rm
i}\mu\,r_{5}({\epsilon}^{2})+{\rm i}\mu^{2}\varphi_{2}(\mu,{\epsilon})$ (4.33)

with $r_{3}\in{\mathcal{C}}^{\mathtt{r}}$,
$r_{5}\in{\mathcal{C}}^{\mathtt{r}-1}$,
$\varphi_{i}(\mu,{\epsilon})\in{\mathcal{C}}^{\mathtt{r}-2}$, $i=1,2$.

Consider first
$({\mathtt{B}}_{\mu,{\epsilon}})_{1,3}\in{\mathcal{C}}^{\mathtt{r}}$. By Lemma
B.1 $(ii)$, write
$r_{3}({\epsilon}^{3})={\epsilon}\,\tilde{r}_{3}({\epsilon}^{2})$ for
$\tilde{r}_{3}\in{\mathcal{C}}^{\mathtt{r}-1}$. Next put

$h_{1}(\mu,{\epsilon}):=\mu^{2}\varphi_{1}(\mu,{\epsilon})=({\mathtt{B}}_{\mu,{\epsilon}})_{1,3}+\sqrt{2}{\epsilon}-{\epsilon}\,\tilde{r}_{3}({\epsilon}^{2})\in{\mathcal{C}}^{\mathtt{r}}\
.$ (4.34)

We claim that

$|h_{1}(\mu,{\epsilon})|\leq C|\mu^{2}{\epsilon}|\
,\quad\forall(\mu,{\epsilon})\in B(\mu_{0})\times B({\epsilon}_{0})\ .$ (4.35)

Then, from Lemma B.1 $(ii)$, it follows that
$h_{1}(\mu,{\epsilon})={\epsilon}\,r(\mu^{2})$ for some
$r\in{\mathcal{C}}^{\mathtt{r}-1}$, proving the first of (4.32).
We prove now (4.35). First we show that $h_{1}(\mu,0)\equiv 0$. Indeed,
$f_{k}^{\sigma}(\mu,0)=f_{k}^{\sigma}$ (see Lemma A.2) and therefore
$({\mathcal{B}}_{\mu,0}f_{1}^{\pm}(\mu,0),f_{0}^{+}(\mu,0))=0$ $\,\forall\mu$
(recall that ${\mathcal{B}}_{\mu,0}$ a Fourier multiplier). It follows that

$({\mathtt{B}}_{\mu,0})_{i,3}=0\ \ \ \forall\mu\ ,\quad i=1,2\ .$ (4.36)

From (4.34) we then deduce $h_{1}(\mu,0)\equiv 0$. Then, since from (4.34)
$\mu^{2}\varphi_{1}(\mu,0)=h_{1}(\mu,0)=0$, it follows that also
$\varphi_{1}(\mu,0)\equiv 0$ $\,\forall\mu$. Then by Lemma B.1
$\varphi_{1}(\mu,{\epsilon})={\epsilon}\,\tilde{\varphi}_{1}(\mu,{\epsilon})$
for some $\tilde{\varphi}_{1}\in{\mathcal{C}}^{\mathtt{r}-3}$, and in
conclusion
$h_{1}(\mu,{\epsilon})=\mu^{2}{\epsilon}\,\tilde{\varphi}_{1}(\mu,{\epsilon})$.
The local boundedness of $\tilde{\varphi}_{1}$ proves the claim (4.35).

Next consider $({\mathtt{B}}_{\mu,{\epsilon}})_{2,3}$. By (4.36), (4.33) and
arguing as before, we have again that
$\varphi_{2}(\mu,{\epsilon})={\epsilon}\,\tilde{\varphi}_{2}(\mu,{\epsilon})$
with $\tilde{\varphi}_{2}\in{\mathcal{C}}^{\mathtt{r}-3}$. Thus from (4.33) we
have that

$|\,({\mathtt{B}}_{\mu,{\epsilon}})_{2,3}|\,\leq C|\mu\,{\epsilon}|\
(|{\epsilon}|+|\mu|)\ ,\quad\forall(\mu,{\epsilon})\in B(\mu_{0})\times
B({\epsilon}_{0})\ $

and by Lemma B.1 $(ii)$ it follows
$({\mathtt{B}}_{\mu,{\epsilon}})_{2,3}=\mu{\epsilon}\,r(\mu,{\epsilon})$ for
some $r\in{\mathcal{C}}^{\mathtt{r}-2}$, as claimed. ∎

We are finally ready to prove Proposition 4.3.

###### Proof of Proposition 4.3.

The expansions in (4.7), (4.8) follow from (4.10) and Lemmata 4.4, 4.5, 4.6
and 4.7.
In particular, the entries of the matrix ${\mathtt{B}}_{\mu,{\epsilon}}$ have
the expressions

$\begin{split}({\mathtt{B}}_{\mu,{\epsilon}})_{1,1}&=-{\mathfrak{a}}{\epsilon}^{2}+r^{\mathtt{r}}_{1}({\epsilon}^{3})+\frac{1}{2}\mu^{2}(-\ddot{\mathfrak{m}}(1)+r^{\mathtt{r}-2}_{2}({\epsilon}))+\mu^{2}\varphi_{1,1}(\mu,{\epsilon})\\\
({\mathtt{B}}_{\mu,{\epsilon}})_{1,2}&={\rm i}(\dot{\mathfrak{m}}(1)\mu+\mu
r^{\mathtt{r}-1}_{2}({\epsilon}^{2})+\mu^{2}r^{\mathtt{r}-2}_{2}({\epsilon}))+\mu^{2}\varphi_{1,2}(\mu,{\epsilon})\\\
({\mathtt{B}}_{\mu,{\epsilon}})_{2,2}&=-\frac{1}{2}\ddot{\mathfrak{m}}(1)\mu^{2}+\mu^{2}r^{\mathtt{r}-2}_{4}({\epsilon})+\mu^{2}\varphi_{2,2}(\mu,{\epsilon})\\\
({\mathtt{B}}_{\mu,{\epsilon}})_{3,3}&={\mathfrak{m}}(1)-{\mathfrak{m}}(0)+\frac{1}{2}{\mathfrak{a}}{\epsilon}^{2}-\frac{1}{2}\mu^{2}(\ddot{\mathfrak{m}}(0)+r^{\mathtt{r}-2}_{6}({\epsilon}))+\mu^{2}\varphi_{3,3}(\mu,{\epsilon})\end{split}$

where $r^{\mathtt{r}}({\epsilon}^{a})$ denotes a real valued function in
${\mathcal{O}}^{\mathtt{r}}({\epsilon}^{a})$ and $\varphi(\mu,{\epsilon})$ is
the remainder matrix in equation (4.10). Since the matrix-valued function
$\varphi(\mu,{\epsilon})\in{\mathcal{C}}^{\mathtt{r}-2}$ in (4.10) fulfills
$\varphi(0,0)=0$, by Lemma B.1-$(iii)$
$\varphi_{i,j}(\mu,{\epsilon})=r^{\mathtt{r}-2}(\mu,{\epsilon})$.
Lemma B.1-$(i)$ allows gathering two orders of ${\epsilon}$ from
$r^{\mathtt{r}}_{1}({\epsilon}^{3})$, by lowering its regularity to
${\mathcal{C}}^{\mathtt{r}-2}$, proving equation (4.7). Finally, the
expansions in Lemma 4.7 complete the proof of Proposition 4.3. ∎

##  5 Block-decoupling

The aim of this section is to block diagonalize the matrix
${\mathtt{L}}_{\mu,{\epsilon}}$ in (4.5). The first step is a singular scaling
transformation, which is not symplectic (according to Definition 3.2) and thus
we compute how the Poisson tensor ${\mathtt{J}}_{\mu}$ is transformed.

######  Lemma 5.1.

The conjugation of the Hamiltonian and reversible matrix
${\mathtt{L}}_{\mu,{\epsilon}}$ with the reversibility preserving matrix
444From now on we use the symbol $f^{\dagger}$ for the conjugate transpose of
vector

$Y:=\begin{pmatrix}Q&\vline&{\tt 0}\\\ \hline\cr{\tt
0}^{\dagger}&\vline&\sqrt{\mu}\end{pmatrix},\quad
Q=\begin{pmatrix}\sqrt{\mu}&0\\\
0&\frac{1}{\sqrt{\mu}}\end{pmatrix},\quad\mu>0$ (5.1)

yields the Hamiltonian and reversible matrix

${\mathtt{L}}_{\mu,{\epsilon}}^{(1)}:=Y^{-1}{\mathtt{L}}_{\mu,{\epsilon}}Y=\mu\,{\mathtt{J}}_{\mu}^{(1)}{\mathtt{B}}_{\mu,{\epsilon}}^{(1)}$
(5.2)

where ${\mathtt{J}}_{\mu}^{(1)}$ is the skew-adjoint and reversible matrix

${\mathtt{J}}_{\mu}^{(1)}:=Y^{-1}{\mathtt{J}}_{\mu}Y^{-*}=\begin{pmatrix}{\rm
i}&1&0\\\ -1&{\rm i}\mu^{2}&0\\\ 0&0&{\rm i}\end{pmatrix}\,\ ,$ (5.3)

${\mathtt{B}}_{\mu,{\epsilon}}^{(1)}$ is the selfadjoint and reversibility
preserving matrix

${\mathtt{B}}_{\mu,{\epsilon}}^{(1)}:=Y^{*}{\mathtt{B}}_{\mu,{\epsilon}}Y=\begin{pmatrix}E^{(1)}&\vline&{\mathtt{f}}^{(1)}\\\
\hline\cr{\mathtt{f}}^{(1){\dagger}}&\vline&g^{(1)}\end{pmatrix}$ (5.4)

and the $2\times 2$ symmetric and reversibility preserving matrix $E^{(1)}$,
the vector ${\mathtt{f}}^{(1)}$ and the number $g^{(1)}$ expand as

$\begin{split}&E^{(1)}=\begin{pmatrix}-{\mathfrak{a}}{\epsilon}^{2}(1+r_{1}^{\prime}({\epsilon}))+{\mathtt{e}}_{22}\mu^{2}(1+r_{1}^{\prime\prime}({\epsilon},\mu))&{\rm
i}({\mathtt{e}}_{12}+r_{2}({\epsilon}^{2},\mu{\epsilon},\mu^{2}))\\\ -{\rm
i}({\mathtt{e}}_{12}+r_{2}({\epsilon}^{2},\mu{\epsilon},\mu^{2}))&({\mathtt{e}}_{22}+r_{4}({\epsilon},\mu))\end{pmatrix}\,,\\\
&{\mathtt{f}}^{(1)}={\epsilon}\begin{pmatrix}-\sqrt{2}+r_{3}({\epsilon}^{2},\mu^{2})\\\
{\rm i}r_{5}({\epsilon},\mu)\end{pmatrix}\,,\quad
g^{(1)}={\mathtt{e}}_{33}+\frac{{\mathfrak{a}}}{2}{\epsilon}^{2}+{\mathtt{g}}_{33}\mu^{2}+r_{6}({\epsilon}^{3},\mu^{2}{\epsilon},\mu^{3})\
.\end{split}$ (5.5)

The reminders $r_{k}$ are functions of class ${\mathcal{C}}^{\mathtt{r}-2}$.

###### Proof.

Compute
$Y^{*}{\mathtt{B}}_{\mu,{\epsilon}}Y=\begin{pmatrix}QEQ&\vline&\sqrt{\mu}Q\vec{\mathtt{f}}\\\
\hline\cr(\sqrt{\mu}Q\vec{\mathtt{f}})^{\dagger}&\vline&\mu g\end{pmatrix}$
and collect a common factor $\mu$ from each entry. ∎

######  Remark 5.2.

1\. Before the conjugation, every entry of the matrix
${\mathtt{B}}_{\mu,{\epsilon}}$ is of class ${\mathcal{C}}^{\mathtt{r}}$ in
$(\mu,{\epsilon})$. After the conjugation, the (2,2)-entry is divided by
$\mu$, and thus it’s regularity is decreased to ${\mathcal{C}}^{\mathtt{r}-1}$
at $\mu=0$.
2\. The spectrum of ${\mathtt{L}}_{\mu,{\epsilon}}^{(1)}$ coincides with that
of ${\mathtt{L}}_{\mu,{\epsilon}}$ whenever $\mu\neq 0$.

###  5.1 Non-perturbative step of block decoupling

The next step is to block-diagonalize the matrix
${\mathtt{L}}_{\mu,{\epsilon}}^{(1)}$ in (5.2). The main result of this
section is the following.

######  Lemma 5.3.

There exists a vector ${\mathtt{s}}(\mu,{\epsilon})$, with values in
${\mathbb{C}}^{2}$ and of class ${\mathcal{C}}^{\mathtt{r}-2}$ of the form

$\displaystyle{\mathtt{s}}:={\mathtt{s}}(\mu,{\epsilon})$
$\displaystyle=\frac{\sqrt{2}\,{\epsilon}}{{\mathtt{e}}_{d}^{2}}\begin{pmatrix}-{\rm
i}({\mathtt{e}}_{b}+r_{1}({\epsilon}^{2},\mu{\epsilon},\mu^{2}))\vspace{.3em}\\\
{\mathtt{e}}_{b}+r_{2}({\epsilon},\mu)\end{pmatrix}$ (5.6)

with ${\mathtt{e}}_{b},{\mathtt{e}}_{d}$, defined in (1.10), real and non-zero
by Assumption B, such that the following holds true. Conjugating the
Hamiltonian and reversible matrix ${\mathtt{L}}_{\mu,{\epsilon}}^{(1)}$ in
(5.2) with the symplectic and reversibility-preserving matrix

$\exp{(S)},\qquad
S:={\mathtt{J}}_{\mu}^{(1)}\begin{pmatrix}\mathbf{0}&\vline&{\mathtt{s}}(\mu,{\epsilon})\\\
\hline\cr{\mathtt{s}}(\mu,{\epsilon})^{\dagger}&\vline&0\end{pmatrix}={\mathtt{J}}_{\mu}^{(1)}X\
,$ (5.7)

we obtain the Hamiltonian and reversible matrix

${\mathtt{L}}_{\mu,{\epsilon}}^{(2)}:=\mu\exp{(S)}\,{\mathtt{L}}^{(1)}_{\mu,{\epsilon}}\,\exp{(-S)}=\mu\,{\mathtt{J}}_{\mu}^{(1)}{\mathtt{B}}_{\mu,{\epsilon}}^{(2)}$
(5.8)

where ${\mathtt{J}}_{\mu}^{(1)}$ in (5.3) and

${\mathtt{B}}_{\mu,{\epsilon}}^{(2)}:=\exp{(S)}^{-*}\,{\mathtt{B}}_{\mu,{\epsilon}}^{(1)}\,\exp{(S)}^{-1}=\begin{pmatrix}E^{(2)}&\vline&{\mathtt{f}}^{(2)}\\\
\hline\cr{\mathtt{f}}^{(2){\dagger}}&\vline&g^{(2)}\end{pmatrix}$ (5.9)

with the $2\times 2$ symmetric and reversibility preserving matrix $E^{(2)}$,
the vector ${\mathtt{f}}^{(2)}$ and the number $g^{(2)}$ expanding as

$\begin{split}E^{(2)}=&\begin{pmatrix}-{\mathtt{e}}_{w}{\epsilon}^{2}(1+r_{1}^{\prime}({\epsilon},\mu))+{\mathtt{e}}_{22}\mu^{2}(1+r_{1}^{\prime\prime}({\epsilon},\mu))&{\rm
i}({\mathtt{e}}_{12}+r_{2}({\epsilon}^{2},\mu{\epsilon},\mu^{2}))\\\ -{\rm
i}({\mathtt{e}}_{12}+r_{2}({\epsilon}^{2},\mu{\epsilon},\mu^{2}))&{\mathtt{e}}_{22}+r_{4}({\epsilon},\mu)\end{pmatrix}\,,\\\
&{\mathtt{f}}^{(2)}={\epsilon}^{3}\begin{pmatrix}r_{3}(1)\\\ {\rm
i}r_{5}(1)\end{pmatrix}\,,\qquad
g^{(2)}={\mathtt{e}}_{33}+r_{6}({\epsilon}^{2},\mu^{2})\end{split}$ (5.10)

and

${\mathtt{e}}_{w}:={\mathfrak{a}}+\frac{2}{{\mathtt{e}}_{d}}\
,\quad{\mathfrak{a}}\mbox{ in }\eqref{fa}\ .$ (5.11)

The functions $r_{k}$ are all ${\mathcal{C}}^{\mathtt{r}-2}$ in
$(\mu,{\epsilon})$.

The rest of the section is devoted to prove Lemma 5.3.
We look for ${\mathtt{s}}=\begin{pmatrix}{\rm i}s_{1}(\mu,{\epsilon})\\\
s_{2}(\mu,{\epsilon})\end{pmatrix}$ and $s_{j}(\mu,{\epsilon})$ real valued,
so that the matrix $X$ is symmetric and reversible. By (5.3) and Lemma 3.9,
the matrix $\exp{(S)}$ is ${\mathtt{J}}^{(1)}$-symplectic and reversibility
preserving for any $\mu\neq 0$. Next we compute the Lie series of the matrix
${\mathtt{L}}_{\mu,{\epsilon}}^{(2)}$ in (5.8). We first split
${\mathtt{L}}_{\mu,{\epsilon}}^{(1)}$ in the block-diagonal and off diagonal
parts:

${\mathtt{L}}_{\mu,{\epsilon}}^{(1)}=\mu\left(D^{(1)}+R^{(1)}\right),\qquad
D^{(1)}:=\begin{pmatrix}\hat{{\mathtt{J}}}_{\mu}E^{(1)}&\vline&{\tt 0}\\\
\hline\cr{\tt 0}^{\dagger}&\vline&{\rm i}g^{(1)}\end{pmatrix},\qquad
R^{(1)}:=\begin{pmatrix}{\bf
0}&\vline&\hat{{\mathtt{J}}}_{\mu}{\mathtt{f}}^{(1)}\\\ \hline\cr{\rm
i}{\mathtt{f}}^{(1){\dagger}}&\vline&0\end{pmatrix}$ (5.12)

where we adopted the notation

${\mathtt{J}}_{\mu}^{(1)}=\begin{pmatrix}\hat{{\mathtt{J}}}_{\mu}&\vline&\mathtt{0}\\\
\hline\cr\mathtt{0}^{\dagger}&\vline&{\rm
i}\end{pmatrix},\qquad\hat{{\mathtt{J}}}_{\mu}=\begin{pmatrix}{\rm i}&1\\\
-1&{\rm i}\mu^{2}\end{pmatrix}\ .$ (5.13)

The Lie expansion of ${\mathtt{L}}_{\mu,{\epsilon}}^{(2)}$ is

$\begin{split}&{\mathtt{L}}_{\mu,{\epsilon}}^{(2)}=\mu\Big{(}D^{(1)}+R^{(1)}+[S,D^{(1)}]+[S,R^{(1)}]+\frac{1}{2}[S,[S,D^{(1)}]]+\\\
+\frac{1}{2}\int_{0}^{1}(1-\tau)^{2}&\exp{(\tau
S)}\,\textup{ad}_{S}^{3}(D^{(1)})\,\exp{(-\tau S)}{\rm
d}\tau+\int_{0}^{1}(1-\tau)\exp{(\tau
S)}\,\textup{ad}_{S}^{2}(R^{(1)})\,\exp{(-\tau S)}{\rm
d}\tau\Big{)}\,.\end{split}$ (5.14)

We look for $S$ in order to solve the homological equation
$R^{(1)}+[S,D^{(1)}]=0$, which, by the explicit expressions (5.6), (5.12),
reads

$\begin{pmatrix}{\bf 0}&\vline&({\rm
i}g^{(1)}-\hat{{\mathtt{J}}}_{\mu}E^{(1)})\hat{{\mathtt{J}}}_{\mu}{\mathtt{s}}+\hat{{\mathtt{J}}}_{\mu}{\mathtt{f}}^{(1)}\\\
\hline\cr{\rm i}{\mathtt{s}}^{\dagger}(\hat{{\mathtt{J}}}_{\mu}E^{(1)}-{\rm
i}g^{(1)})+{\rm i}{\mathtt{f}}^{(1){\dagger}}&\vline&0\end{pmatrix}=0\ .$
(5.15)

One readily checks that the two equations are equivalent, so it is enough to
solve

$(\hat{{\mathtt{J}}}_{\mu}E^{(1)}-{\rm
i}g^{(1)})\hat{{\mathtt{J}}}_{\mu}{\mathtt{s}}=\hat{{\mathtt{J}}}_{\mu}{\mathtt{f}}^{(1)}\
.$ (5.16)

By Proposition 5.1 the matrix $\hat{{\mathtt{J}}}_{\mu}E^{(1)}-{\rm i}g^{(1)}$
has the form

$\hat{{\mathtt{J}}}_{\mu}E^{(1)}-{\rm i}g^{(1)}=\begin{pmatrix}-{\rm
i}{\mathtt{e}}_{d}+{\rm
i}r_{1}({\epsilon}^{2},\mu{\epsilon},\mu^{2})&{\mathtt{e}}_{b}+r_{2}({\epsilon},\mu)\\\
{\mathfrak{a}}{\epsilon}^{2}(1+r_{3}^{\prime}({\epsilon}))-{\mathtt{e}}_{b}\mu^{2}(1+r_{3}^{\prime\prime}({\epsilon},\mu))&-{\rm
i}{\mathtt{e}}_{d}+{\rm
i}r_{4}({\epsilon}^{2},\mu{\epsilon},\mu^{2})\end{pmatrix}$ (5.17)

where we used that by (4.9)

${\mathtt{e}}_{12}+{\mathtt{e}}_{33}\equiv{\mathtt{e}}_{d},\quad{\mathtt{e}}_{22}-{\mathtt{e}}_{12}\equiv{\mathtt{e}}_{b}\
$ (5.18)

with ${\mathtt{e}}_{d}$, ${\mathtt{e}}_{b}$ the numbers in (1.10). Its
determinant is

$\det(\hat{{\mathtt{J}}}_{\mu}E^{(1)}-{\rm
i}g^{(1)})=-\left({\mathtt{e}}_{d}^{2}+r({\epsilon}^{2},\mu{\epsilon},\mu^{2})\right)$
(5.19)

which is not zero for sufficiently small $(\mu,{\epsilon})$ since
${\mathtt{e}}_{d}\neq 0$ by Assumption B. A direct computation than gives the
following result:

######  Lemma 5.4.

The vector ${\mathtt{s}}(\mu,{\epsilon})$ in (5.6) solves the homological
equation (5.16) and it is ${\mathcal{C}}^{\mathtt{r}-2}$ .

Since the matrix $S$ solves the homological equation $[S,D^{(1)}]+R^{(1)}=0$,
the Lie expansion of ${\mathtt{L}}^{(2)}_{\mu,{\epsilon}}$ in (5.14) reduces
to

${\mathtt{L}}^{(2)}_{\mu,{\epsilon}}=\mu\Big{(}D^{(1)}+\frac{1}{2}[S,R^{(1)}]+\frac{1}{2}\int_{0}^{1}(1-\tau^{2})\exp{(\tau
S)}\,\textup{ad}_{S}^{2}(R^{(1)})\,\exp{(-\tau S)}{\rm d}\tau\Big{)}\ .$
(5.20)

In particular, the block-diagonal correction $\frac{1}{2}[S,R^{(1)}]$ is the
Hamiltonian and reversible matrix

$\frac{1}{2}{\mathtt{J}}_{\mu}^{(1)}\begin{pmatrix}{\rm
i}({\mathtt{s}}{\mathtt{f}}^{(1){\dagger}}-{\mathtt{f}}^{(1)}{\mathtt{s}}^{\dagger})&\vline&\mathtt{0}\\\
\hline\cr\mathtt{0}&\vline&{\mathtt{s}}^{\dagger}\hat{\mathtt{J}}_{\mu}{\mathtt{f}}^{(1)}-{\mathtt{f}}^{(1){\dagger}}\hat{\mathtt{J}}_{\mu}{\mathtt{s}}\end{pmatrix}=:{\mathtt{J}}_{\mu}^{(1)}\begin{pmatrix}\Delta
E^{(1)}&\vline&0\\\ \hline\cr 0&\vline&\ \Delta g^{(1)}\end{pmatrix}$ (5.21)

where the $2\times 2$ selfadjoint a reversibility preserving matrix $\Delta
E^{(1)}$ and the real number $\Delta g^{(1)}$ expand as

$\Delta
E^{(1)}={\epsilon}^{2}\begin{pmatrix}-2{\mathtt{e}}_{d}^{-1}(1+r_{1}({\epsilon},\mu))&{\rm
i}{\mathtt{e}}_{b}{\mathtt{e}}_{d}^{-2}(1+r_{2}({\epsilon},\mu))\vspace{0,15cm}\\\
-{\rm
i}{\mathtt{e}}_{b}{\mathtt{e}}_{d}^{-2}(1+r_{2}({\epsilon},\mu))&0\end{pmatrix}\
,\quad\Delta g^{(1)}=r_{3}({\epsilon}^{2})\ .$ (5.22)

Notice that, whereas the other corrective terms are perturbative (i.e. they
are of the same order of the remainders in ${\mathtt{B}}_{\mu,{\epsilon}}$),
the $(1,1)$ entrance of the matrix $\Delta E^{(1)}$ is of the same order
${\epsilon}^{2}$ as that of $E^{(1)}$, and will give an essential contribute
to the Benjamin-Feir instability. In particular, the block-diagonal matrix
$D^{(1)}+\frac{1}{2}[S,R^{(1)}]$ is now of the form

${\mathtt{J}}_{\mu}^{(1)}\begin{pmatrix}E^{(1)}+\Delta E^{(1)}&\vline&0\\\
\hline\cr 0&\vline&g^{(1)}+\Delta
g^{(1)}\end{pmatrix}=:{\mathtt{J}}_{\mu}^{(1)}\begin{pmatrix}E^{(2)}&\vline&0\\\
\hline\cr 0&\vline&g^{(2)}\end{pmatrix}$ (5.23)

with $E^{(2)}$ and $g^{(2)}$ given in Lemma 5.3. In particular note the new
coefficient ${\mathtt{e}}_{w}$ in (5.11) at order ${\epsilon}^{2}$ in the
entrance (1,1) of $E^{(2)}$.
Finally, we show that the reminder of the Lie expansion in (5.20) is small.

######  Lemma 5.5.

The $3\times 3$ Hamiltonian and reversible matrix

$\frac{1}{2}\int_{0}^{1}(1-\tau^{2})\exp{(\tau
S)}\,\textup{ad}_{S}^{2}(R^{(1)})\,\exp{(-\tau S)}{\rm
d}\tau={\epsilon}^{3}{\mathtt{J}}_{\mu}^{(1)}\begin{pmatrix}\widehat{E}&\vline&\widehat{\mathtt{f}}^{(2)}\\\
\hline\cr\widehat{\mathtt{f}}^{(2){\dagger}}&\vline&\widehat{g}\end{pmatrix}$

with $\widehat{E}$, $\widehat{g}$ and $\widehat{\mathtt{f}}^{(2)}$ functions
${\mathcal{C}}^{\mathtt{r}-2}$ of $(\mu,{\epsilon})$.

###### Proof.

Each entry of $S$ in (5.7), (5.6) and $R^{(1)}$ in (5.12), (5.5) has the form
${\epsilon}\varphi(\mu,{\epsilon})$ with
$\varphi\in{\mathcal{C}}^{\mathtt{r}-2}$. The entries of $\exp{(\tau S)}$ are
${\mathcal{C}}^{\mathtt{r}-2}$ functions of $(\mu,{\epsilon})$, and so
$\textup{ad}_{S}^{2}(R^{(1)})$ has entries of the form
${\epsilon}^{3}\varphi(\mu,{\epsilon})$ with
$\varphi\in{\mathcal{C}}^{\mathtt{r}-2}$. ∎

This concludes the proof of Lemma 5.3.

###  5.2 Complete block decoupling

We now aim to completely block diagonalize the matrix
${\mathtt{L}}_{\mu,{\epsilon}}$. In order to do that, after the first step of
non-perturbative block decoupling in the previous section, we will perform a
further ${\mathtt{J}}^{(1)}_{\mu}$-symplectic change of basis, that will
completely remove the off-diagonal blocks of
${\mathtt{L}}_{\mu,{\epsilon}}^{(2)}$. This change of basis will be found
solving a nonlinear equation in the entries of the matrix
${\mathtt{B}}_{\mu,{\epsilon}}^{(2)}$. First we split

$\displaystyle{\mathtt{L}}^{(2)}_{\mu,{\epsilon}}$
$\displaystyle=\mu\left(D^{(2)}+R^{(2)}\right)\ ,$ (5.24) $\displaystyle
D^{(2)}:=\begin{pmatrix}\hat{{\mathtt{J}}}_{\mu}E^{(2)}&\vline&{\tt 0}\\\
\hline\cr{\tt 0}^{\dagger}&\vline&{\rm i}g^{(2)}\end{pmatrix},\qquad
R^{(2)}:=\begin{pmatrix}{\bf
0}&\vline&\hat{{\mathtt{J}}}_{\mu}{\mathtt{f}}^{(2)}\\\ \hline\cr{\rm
i}{\mathtt{f}}^{(2){\dagger}}&\vline&0\end{pmatrix}$

with $\hat{{\mathtt{J}}}_{\mu}$ in (5.13), $E^{(2)}$, $g^{(2)}$,
${\mathtt{f}}^{(2)}$ in (5.10).

######  Lemma 5.6.

There exist $\mu_{0}>0,{\epsilon}_{0}>0$ such that for any $|\mu|<\mu_{0}$,
$|{\epsilon}|<{\epsilon}_{0}$ there exist a reversibility preserving,
Hamiltonian matrix $S^{(2)}=S^{(2)}(\mu,{\epsilon})$ and a Hamiltonian,
reversible, block-diagonal matrix $P=P(\mu,{\epsilon})$, both of class
${\mathcal{C}}^{\mathtt{r}-2}$ in $(\mu,{\epsilon})\in B(\mu_{0})\times
B({\epsilon}_{0})$, such that

${\mathtt{L}}_{\mu,{\epsilon}}^{(3)}:=\exp(S^{(2)})\,{\mathtt{L}}_{\mu,{\epsilon}}^{(2)}\,\exp(-S^{(2)})=\mu\big{(}D^{(2)}+{\epsilon}^{6}P\big{)}\
.$ (5.25)

###### Proof.

We write for brevity $S^{(2)}=S$, and we look for a Hamiltonian reversibility
preserving matrix of the form

$S={\epsilon}^{3}{\mathtt{J}}_{\mu}^{(1)}\begin{pmatrix}\mathbf{0}&\vline&{\mathtt{s}}\\\
\hline\cr{\mathtt{s}}^{\dagger}&\vline&0\end{pmatrix}\
,\quad{\mathtt{s}}=\begin{pmatrix}{\rm i}s_{1}(\mu,{\epsilon})\\\
s_{2}(\mu,{\epsilon})\end{pmatrix}\ ,\quad
s_{1},s_{2}\in{\mathcal{C}}^{\mathtt{r}-2}\ .$ (5.26)

Decomposing $\exp(S){\mathtt{L}}_{\mu,{\epsilon}}^{(2)}\exp(-S)$ in the
diagonal and off-diagonal blocks we solve for $S$ and $P$ the system

$\begin{cases}\Pi_{D}(\exp(S){\mathtt{L}}_{\mu,{\epsilon}}^{(2)}\exp(-S))=\mu\big{(}D^{(2)}+{\epsilon}^{6}P\big{)}\\\
\Pi_{\varnothing}(\exp(S){\mathtt{L}}_{\mu,{\epsilon}}^{(2)}\exp(-S))=0\end{cases}\
,$ (5.27)

where we denoted $\Pi_{D},\,\Pi_{\varnothing}$ the projections respectively
onto the diagonal and off diagonal matrices.
Rewriting the second one in Lie expansion, the equation reads

$R^{(2)}+[S,D^{(2)}]+\underbrace{\Pi_{\varnothing}(\int_{0}^{1}(1-\tau)\exp(\tau
S)\textup{ad}_{S}^{2}(D^{(2)}+R^{(2)})\exp(-\tau S){\rm
d}\tau)}_{{\mathcal{R}}({\mathtt{s}},\mu,{\epsilon})}=0\,.$ (5.28)

By (5.10),

$R^{(2)}={\epsilon}^{3}{\widetilde{R}}^{(2)}={\epsilon}^{3}\begin{pmatrix}{\bf
0}&\vline&\hat{{\mathtt{J}}}_{\mu}\widetilde{\mathtt{f}}^{(2)}\\\
\hline\cr{\rm
i}{\widetilde{{\mathtt{f}}}}^{(2){\dagger}}&\vline&0\end{pmatrix}$

with $\widetilde{\mathtt{f}}^{(2)}$ of class ${\mathcal{C}}^{\mathtt{r}-2}$.
Writing also $S={\epsilon}^{3}{\widetilde{S}}$, (5.28) reads

${\widetilde{R}}^{(2)}+[{\widetilde{S}},D^{(2)}]+{\epsilon}^{3}\underbrace{\Pi_{\varnothing}(\int_{0}^{1}(1-\tau)\exp(\tau{\epsilon}^{3}{\widetilde{S}})\textup{ad}_{{\widetilde{S}}}^{2}(D^{(2)}+{\epsilon}^{3}{\widetilde{R}}^{(2)})\exp(-\tau{\epsilon}^{3}{\widetilde{S}}){\rm
d}\tau)}_{{\mathcal{R}}({\mathtt{s}},\mu,{\epsilon})}=0\,,$ (5.29)

where, due to the operator $\Pi_{\varnothing}$,
${\mathcal{R}}({\mathtt{s}},\mu,{\epsilon})$ has the form

${\mathcal{R}}({\mathtt{s}}^{\prime},\mu,{\epsilon})={\mathtt{J}}_{\mu}^{(1)}\begin{pmatrix}0&\vline&\eta({\mathtt{s}},\mu,{\epsilon})\\\
\hline\cr\eta({\mathtt{s}},\mu,{\epsilon})^{\dagger}&\vline&0\end{pmatrix}$

with $\eta({\mathtt{s}},\mu,{\epsilon})$ of class
${\mathcal{C}}^{\mathtt{r}-2}$ in its variables, and at least quadratic in
${\mathtt{s}}$. Hence, equation (5.29) reduces to

$F({\mathtt{s}},\mu,{\epsilon})=({\rm
i}g^{(2)}-\hat{{\mathtt{J}}}_{\mu}E^{(2)})\hat{{\mathtt{J}}}_{\mu}{\mathtt{s}}+\hat{{\mathtt{J}}}_{\mu}{\widetilde{{\mathtt{f}}}}^{(2)}+\hat{{\mathtt{J}}}_{\mu}\eta({\mathtt{s}},\mu,{\epsilon})=0$
(5.30)

where $F\in{\mathcal{C}}^{\mathtt{r}-2}(B_{r}(0)\times B_{\mu_{0}}(0)\times
B_{{\epsilon}_{0}}(0))$. Now

${\rm d}_{{\mathtt{s}}}F(0,0,0)=({\rm
i}g^{(2)}-\hat{{\mathtt{J}}}_{\mu}E^{(2)})\hat{{\mathtt{J}}}_{\mu}\quad\mbox{
and }\quad\det({\rm
i}g^{(2)}-\hat{{\mathtt{J}}}_{\mu}E^{(2)})\bigg{|}_{{\epsilon}=\mu=0}=-{\mathtt{e}}_{d}^{2}\neq
0$

by Assumption B. By the Implicit Function Theorem we find a
${\mathcal{C}}^{\mathtt{r}-2}$ vector ${\mathtt{s}}(\mu,{\epsilon})$ that
solves equation 5.30. Finally, we prove that ${\mathtt{s}}$ has the claimed
form (5.26). Indeed, a direct computation shows that the matrix
$\overline{\rho}\circ S\circ\overline{\rho}$ solves (5.28), and so by
uniqueness of the solution $S=\overline{\rho}\circ S\circ\overline{\rho}$,
proving that $S$ is reversibility preserving as claimed in (5.26).

Consider now the first equation in (5.27). Again Lie expanding we find

$\displaystyle\Pi_{D}$
$\displaystyle(\exp(S){\mathtt{L}}_{\mu,{\epsilon}}^{(2)}\exp(-S))$ (5.31)
$\displaystyle=\mu\Big{(}D^{(2)}+{\epsilon}^{6}\underbrace{\big{(}[{\widetilde{S}},{\widetilde{R}}^{(2)}]+\Pi_{D}\int_{0}^{1}(1-\tau)\exp(\tau
S)\,\textup{ad}_{{\widetilde{S}}}^{2}(D^{(2)}+R^{(2)})\,\exp(-\tau S){\rm
d}\tau\big{)}}_{=:P(\mu,{\epsilon})\in{\mathcal{C}}^{\mathtt{r}-2}}\big{)}\Big{)}$

concluding the proof of the lemma. ∎

We are now ready to prove the main result:

###### Proof of Theorem 2.4.

By Lemma 5.6,
${\mathtt{L}}_{\mu,{\epsilon}}^{(3)}=\mu\,{\mathtt{J}}_{\mu}^{(1)}\begin{pmatrix}E^{(3)}&\vline&0\\\
\hline\cr 0&\vline&g^{(3)}\end{pmatrix}$ with $E^{(3)}$ and $g^{(3)}$
expanding as $E^{(2)}$ and $g^{(2)}$ in (5.10). Then the matrix
${\mathtt{U}}:=\mu\hat{\mathtt{J}}_{\mu}E^{(3)}$ expands as in (2.22) (recall
${\mathtt{e}}_{12}=\dot{\mathfrak{m}}(1)$ by (1.10) and
${\mathtt{e}}_{22}-{\mathtt{e}}_{12}={\mathtt{e}}_{b}$ by (5.18)) and the
number ${\mathtt{g}}:=\mu{\rm i}g^{(3)}$ as in (2.23) (recall
${\mathtt{e}}_{33}={\mathfrak{m}}(1)-{\mathfrak{m}}(0)$ by (4.9)). ∎

##  6 Applications

In this section we apply Theorem 1.2 to several generalized KdV equations,
listed in the table in Section 1

###  6.1 Whitham’s type equations

We first focus on Whitham’s type equations. The Whitham equation is a famous
nonlinear dispersive partial differential equation, modelling unidirectional,
two-dimensional water waves at the surface of an irrotational, inviscid fluid.
It is characterized by the same dispersion relation as the original water
waves equation, and can be adapted to include effects such as surface tension
or a constant vorticity, by modifying the dispersion relation. In particular,
it has the form (1.1), where the operator ${\mathcal{M}}(D)$ has symbol

$\displaystyle{\mathfrak{m}}_{\mathtt{h}}(\xi)=\sqrt{\frac{\tanh({\mathtt{h}}\xi)}{\xi}}\hskip
70.0001pt\mbox{(classical Whitham)}$ (6.1)
$\displaystyle{\mathfrak{m}}_{{\mathtt{h}},\kappa}(\xi)=\sqrt{(1+\kappa\xi^{2})\frac{\tanh({\mathtt{h}}\xi)}{\xi}}\qquad\mbox{(gravity-
capillary Whitham)}$ (6.2)
$\displaystyle{\mathfrak{m}}_{{\mathtt{h}},\gamma}(\xi)=\frac{\gamma}{2}\frac{\tanh({\mathtt{h}}\xi)}{\xi}+\sqrt{\frac{\tanh({\mathtt{h}}\xi)}{\xi}+\frac{\gamma^{2}}{4}\frac{\tanh^{2}({\mathtt{h}}\xi)}{\xi^{2}}}\qquad\mbox{(vorticity-
modified Whitham)}$ (6.3)

where ${\mathtt{h}}>0$ is a parameter representing the depth of the fluid,
$\kappa>0$ the surface tension coefficients and $\gamma\in{\mathbb{R}}$ a
constant vorticity.

Modulational instability for the Whitham equations in (6.1)–(6.3) has been
studied in [15, 16], where it has been proved that, for certain values of the
parameters ${\mathtt{h}},\kappa,\gamma$, the linearized Whitham equations at a
small amplitude traveling wave solutions have unstable eigenvalues near the
origin.

Applying our abstract Theorem 1.2 we now show that the spectrum of the
linearized operator actually depicts a closed ”figure 8”.

#### Whitham equation.

We first consider the classical Witham equation with symbol
${\mathfrak{m}}_{\mathtt{h}}(\xi)$ in (6.1). For any ${\mathtt{h}}>0$, the
function $\xi\mapsto{\mathfrak{m}}_{\mathtt{h}}(\xi)$ is real valued, even,
smooth, it fulfills (1.3) with $m=-\frac{1}{2}$ and it is strictly decreasing
on ${\mathbb{R}}_{>0}$. Hence, condition (1.4) is verified for all
${\mathtt{h}}>0$ and so is Assumption A. Then, by Theorem 1.1, there exists
${\epsilon}_{0}={\epsilon}_{0}({\mathtt{h}})>0$ and for any
${\epsilon}\in(0,{\epsilon}_{0})$ a traveling wave solution $u_{\epsilon}$ of
(1.1). Denote by ${\mathcal{L}}_{\epsilon}$ the linearized operator (1.13) at
$u_{\epsilon}$. We obtain:

######  Theorem 6.1 (Whitham).

Consider equation (1.1) with ${\mathcal{M}}(D)$ having symbol (6.1). There
exists a critical threshold ${\mathtt{h}}_{\scaleto{\mathrm{WB}}{3pt}}\sim
1.146\dots$ such that:

  * •

if ${\mathtt{h}}>{\mathtt{h}}_{\scaleto{\mathrm{WB}}{3pt}}$, there exists
${\epsilon}_{1}({\mathtt{h}})>0$ such that for any
${\epsilon}\in[0,{\epsilon}_{1}({\mathtt{h}}))$, the spectrum of the operator
${\mathcal{L}}_{\epsilon}$ depicts near the origin a complete figure “8” as in
Figure 1.

  * •

If ${\mathtt{h}}<{\mathtt{h}}_{\scaleto{\mathrm{WB}}{3pt}}$, the spectrum of
${\mathcal{L}}_{\epsilon}$ near the origin is purely imaginary.

###### Proof.

We have already verified Assumption A, and now we check Assumption B. The
coefficient ${\mathtt{e}}_{12}=\dot{\mathfrak{m}}_{\mathtt{h}}(1)<0$ since
$\xi\mapsto{\mathfrak{m}}_{\mathtt{h}}(\xi)$ is strictly decreasing for
$\xi\geq 0$. Also,

${\mathtt{e}}_{d}=\dot{\mathfrak{m}}_{\mathtt{h}}(1)+{\mathfrak{m}}_{\mathtt{h}}(1)-{\mathfrak{m}}_{\mathtt{h}}(0)<0,\quad{\mathfrak{m}}(1)-{\mathfrak{m}}(2)>0$
(6.4)

for the same reason. The coefficient ${\mathtt{e}}_{b}$ is

${\mathtt{e}}_{b}=\frac{\big{(}{\mathtt{h}}(1-{\mathtt{c}}_{\mathtt{h}}^{4})-{\mathtt{c}}_{\mathtt{h}}^{2}\big{)}^{2}+4{\mathtt{h}}^{2}{\mathtt{c}}_{\mathtt{h}}^{4}(1-{\mathtt{c}}_{\mathtt{h}}^{4})}{8{\mathtt{c}}_{\mathtt{h}}^{3}}>0\
,\qquad{\mathtt{c}}_{\mathtt{h}}:=\sqrt{\tanh({\mathtt{h}})}\ .$ (6.5)

Finally, the coefficient

${\mathtt{e}}_{w}=\frac{\left({\mathtt{h}}(1-{{\mathtt{c}}_{\mathtt{h}}}^{4})+5{{\mathtt{c}}_{\mathtt{h}}}^{2}-2{{\mathtt{c}}_{\mathtt{h}}}(\sqrt{{\mathtt{h}}}+2{{\mathtt{c}}_{\mathtt{h}}}\sqrt{\frac{1}{1+{{\mathtt{c}}_{\mathtt{h}}}^{4}}})\right)}{2{{\mathtt{c}}_{\mathtt{h}}}({\mathtt{h}}(1-{{\mathtt{c}}_{\mathtt{h}}}^{4})-2\sqrt{{\mathtt{h}}}{{\mathtt{c}}_{\mathtt{h}}}-3{{\mathtt{c}}_{\mathtt{h}}}^{2})(1-\sqrt{\frac{1}{1+{{\mathtt{c}}_{\mathtt{h}}}^{4}}})}\
$

is plotted in Figure 2 as a function of ${\mathtt{h}}$. Numerically one sees
that it is a strictly increasing function of ${\mathtt{h}}$. Moreover,
$\lim_{{\mathtt{h}}\to 0+}{\mathtt{e}}_{w}=-\infty$ and
$\lim_{{\mathtt{h}}\to\infty}{\mathtt{e}}_{w}=\frac{\sqrt{2}}{\sqrt{2}-1}$. It
has only one zero at the critical depth
${\mathtt{h}}_{\scaleto{\mathrm{WB}}{3pt}}\approx 1.146...$

Sign of ${\mathtt{e}}_{\scaleto{\mathrm{WB}}{3pt}}$: The function
${\mathtt{e}}_{\scaleto{\mathrm{WB}}{3pt}}={\mathtt{e}}_{w}{\mathtt{e}}_{b}$
is plotted in Figure 2. Since ${\mathtt{e}}_{b}$ is strictly positive by
(6.5), the function ${\mathtt{e}}_{\scaleto{\mathrm{WB}}{3pt}}$ has a zero at
${\mathtt{h}}_{\scaleto{\mathrm{WB}}{3pt}}\approx 1.146...$. Therefore,
whenever ${\mathtt{h}}>{\mathtt{h}}_{\scaleto{\mathrm{WB}}{3pt}}$,
${\mathtt{e}}_{\scaleto{\mathrm{WB}}{3pt}}>0$ and item $(i)$ of Theorem 1.2
applies. Otherwise, item $(ii)$ applies.

∎

Figure 2:  In the left figure is represented the dependence of
${\mathtt{e}}_{w}$ on ${\mathtt{h}}$. We can see that it is in accordance with
the one in [15]. In the right figure is represented the dependence of
${\mathtt{e}}_{\scaleto{\mathrm{WB}}{3pt}}$ on ${\mathtt{h}}$.

Relation with Hur-Johnson [15]. Hur and Johnson proves the existence of
unstable eigenvalues of the ${\mathcal{L}}_{\mu,{\epsilon}}$ near zero
provided the stability-index function

$\Gamma({\mathtt{h}}):=3\alpha({\mathtt{h}})-2\alpha(2{\mathtt{h}})-1-{\mathtt{h}}\alpha^{\prime}({\mathtt{h}})\,,\qquad\alpha({\mathtt{h}}):=\sqrt{\frac{\tanh({\mathtt{h}})}{{\mathtt{h}}}}$

is negative. A direct computation shows that

$\sqrt{{\mathtt{h}}}\,\Gamma({\mathtt{h}})=\frac{{\mathtt{e}}_{\scaleto{\mathrm{WB}}{3pt}}}{{\mathtt{e}}_{b}}\,({\mathfrak{m}}_{\mathtt{h}}(1)-{\mathfrak{m}}_{\mathtt{h}}(2))\,{\mathtt{e}}_{d}\
,$

and so, in view of (6.4) and (6.5), $\Gamma({\mathtt{h}})<0$ if and only if
${\mathtt{e}}_{\scaleto{\mathrm{WB}}{3pt}}>0$. So our work shows that whenever
the stability-index function is negative, actually the spectrum of the
operator ${\mathcal{L}}_{\epsilon}$ depicts a complete figure 8 around the
origin.

#### Gravity-capillary Whitham equation.

We now consider the Whitham equation with gravity and surface tension, namely
(1.1) with symbol ${\mathfrak{m}}_{{\mathtt{h}},\kappa}(\xi)$ in (6.2). Let us
focus first on Assumption A. The symbol
${\mathfrak{m}}_{{\mathtt{h}},\kappa}(\xi)$ is even, smooth and of order
$\frac{1}{2}$. Regarding Assumption (A3), it fails whenever
$({\mathtt{h}},\kappa)$ provoke a resonance, namely an integer
$n\in{\mathbb{N}}_{0}\setminus\\{1\\}$ such that
${\mathfrak{m}}_{{\mathtt{h}},\kappa}(1)={\mathfrak{m}}_{{\mathtt{h}},\kappa}(n)$.
These conditions define on the plane $(\kappa,{\mathtt{h}})$ a family of
curves

$\kappa_{n}({\mathtt{h}})=\begin{cases}\dfrac{{\mathtt{h}}}{\tanh({\mathtt{h}})}-1\,\qquad&\mbox{
if }n=0\\\
\dfrac{n\tanh({\mathtt{h}})-\tanh(n{\mathtt{h}})}{n^{2}\tanh(n{\mathtt{h}})-n\tanh({\mathtt{h}})}\,\qquad&\mbox{
if }n\geq 2\end{cases}$ (6.6)

that are represented in dotted lines in Figure 3 below, and move from top to
bottom as $n$ increases. In particular, curve (c) in Figure 3 is due to the
resonance between the first and second mode, that actively produce a change in
the stability of the periodic traveling wave. Denote by ${\mathcal{R}}$ the
resonant set

${\mathcal{R}}:=\\{({\mathtt{h}},\kappa)\in{\mathbb{R}}_{>0}\times{\mathbb{R}}_{>0}\colon\kappa=\kappa_{n}({\mathtt{h}})\
\ \mbox{ for some }n\in{\mathbb{N}}_{0}\setminus\\{1\\}\\}\ .$ (6.7)

In case $({\mathtt{h}},\kappa)\not\in{\mathcal{R}}$, (1.4) is verified.
Indeed, ${\mathfrak{m}}_{{\mathtt{h}},\kappa}(\xi)\mapsto+\infty$ for
$\xi\mapsto+\infty$, and thus $\exists\ \overline{n}>0$ such that $\forall
n>\overline{n}$,
$|{\mathfrak{m}}_{{\mathtt{h}},\kappa}(n)-{\mathfrak{m}}_{{\mathtt{h}},\kappa}(1)|>1$.
Therefore,
$|{\mathfrak{m}}_{{\mathtt{h}},\kappa}(n)-{\mathfrak{m}}_{{\mathtt{h}},\kappa}(1)|>\min\\{1,\min\\{|{\mathfrak{m}}_{{\mathtt{h}},\kappa}(n)-{\mathfrak{m}}_{{\mathtt{h}},\kappa}(1)|\
:\ n\leq\overline{n}\\}\\}$. Then, by Theorem 1.1, for any
$({\mathtt{h}},\kappa)\not\in{\mathcal{R}}$ there exists
${\epsilon}_{0}={\epsilon}_{0}({\mathtt{h}},\kappa)>0$ and for any
${\epsilon}\in(0,{\epsilon}_{0})$ a traveling wave solution $u_{\epsilon}$ of
(1.1). Denote by ${\mathcal{L}}_{\epsilon}$ the linearized operator (1.13) at
$u_{\epsilon}$. We obtain:

######  Theorem 6.2 (Gravity-capillary Whitham).

Consider equation (1.1) with ${\mathcal{M}}(D)$ having symbol (6.2). There
exist two open regions ${\mathcal{U}}$ and ${\mathcal{S}}$, delimited by
analytic curves in ${\mathbb{R}}_{>0}\times{\mathbb{R}}_{>0}$, such that:

  * •

for any $({\mathtt{h}},\kappa)\in{\mathcal{U}}\setminus{\mathcal{R}}$ there
exists ${\epsilon}_{1}({\mathtt{h}},\kappa)>0$ such that for any
${\epsilon}\in[0,{\epsilon}_{1}({\mathtt{h}},\kappa))$, the spectrum of the
operator ${\mathcal{L}}_{\epsilon}$ depicts near the origin a complete figure
“8” as in Figure 1.

  * •

for any $({\mathtt{h}},\kappa)\in{\mathcal{S}}\setminus{\mathcal{R}}$ the
spectrum of ${\mathcal{L}}_{\epsilon}$ near the origin is purely imaginary.

###### Proof.

Assumption A has already been verified. The coefficient ${\mathtt{e}}_{12}$ is
given by

${\mathtt{e}}_{12}=\sqrt{1+\kappa}\,{{\mathtt{c}}_{\mathtt{h}}}\ .$ (6.8)

Its zero set gives a curve $\gamma_{12}$ plotted in purple in Figure 3. The
function

${\mathtt{e}}_{d}=\frac{\left({\mathtt{h}}(1+\kappa)(1-{{\mathtt{c}}_{\mathtt{h}}}^{4})-(3+\kappa){{\mathtt{c}}_{\mathtt{h}}}^{2}-2\sqrt{{\mathtt{h}}(1+\kappa)}{{\mathtt{c}}_{\mathtt{h}}}\right)}{2{{\mathtt{c}}_{\mathtt{h}}}\sqrt{1+\kappa}}$
(6.9)

has zero set given by curve (d) in Figure 3, the function

$\begin{split}{\mathtt{e}}_{b}=\frac{1-3\kappa(2+\kappa)}{(2\sqrt{1+\kappa})^{3}}{{\mathtt{c}}_{\mathtt{h}}}+\frac{1}{2}{\mathtt{h}}^{2}(1+\kappa)^{\frac{1}{2}}(1-{{\mathtt{c}}_{\mathtt{h}}}^{4}){{\mathtt{c}}_{\mathtt{h}}}^{-3}-\frac{1}{4}{\mathtt{h}}(1+k)^{-\frac{1}{2}}(1+3k)(1-{{\mathtt{c}}_{\mathtt{h}}}^{4})\end{split}$
(6.10)

has zero set given by curve (b) in Figure 3 and finally the function

${\mathtt{e}}_{w}=\frac{1}{{\mathtt{e}}_{d}({\mathfrak{m}}_{{\mathtt{h}},\kappa}(1)-{\mathfrak{m}}_{{\mathtt{h}},\kappa}(2))}\left(\frac{\left({\mathtt{h}}(1+\kappa)(1-{{\mathtt{c}}_{\mathtt{h}}}^{4})+(5+7\kappa){{\mathtt{c}}_{\mathtt{h}}}\right)}{2\sqrt{1+\kappa}{{\mathtt{c}}_{\mathtt{h}}}}-\sqrt{\mathtt{h}}+\sqrt{1+4\kappa}\frac{2{{\mathtt{c}}_{\mathtt{h}}}}{\sqrt{1+{{\mathtt{c}}_{\mathtt{h}}}^{4}}}\right)$
(6.11)

has zero set given by two connected components (a) and (e).

Finally, the function

${\mathtt{e}}_{\scaleto{\mathrm{WB}}{3pt}}={\mathtt{e}}_{w}{\mathtt{e}}_{b}$

Then put
${\mathcal{U}}:=\\{{\mathtt{e}}_{\scaleto{\mathrm{WB}}{3pt}}>0\\}\setminus\gamma_{12}$,
and
${\mathcal{S}}:=\\{{\mathtt{e}}_{\scaleto{\mathrm{WB}}{3pt}}<0\\}\setminus\gamma_{12}$.
∎

Figure 3: In the figure is represented the sign of the function
${\mathtt{e}}_{\scaleto{\mathrm{WB}}{3pt}}$ dependent on the parameters
${\mathtt{h}}$ (on the horizontal axis) and $\kappa$ (on the vertical axis).
The areas denoted by the letter U are the parts of the plane where
${\mathtt{e}}_{\scaleto{\mathrm{WB}}{3pt}}$ is positive, the areas denoted by
the letter S are the ones where it is negative. Moreover, in the figure are
reported the curves where Assumption A fails as dotted lines, namely there
exists $n\in{\mathbb{N}}\setminus\\{1\\}$ such that
${\mathfrak{m}}(1)={\mathfrak{m}}(n)$. In addition, in correspondence of the
curves in blue and the curve in purple, Assumption B fails. In particular, the
curve represented in blue are the ones where the sign of
${\mathtt{e}}_{\scaleto{\mathrm{WB}}{3pt}}$ changes: (a) and (e) are given by
the condition ${\mathtt{e}}_{w}=0$; (b) is given by ${\mathtt{e}}_{b}=0$; (c)
is given by ${\mathfrak{m}}(1)-{\mathfrak{m}}(2)=0$ and it is a resonance
contributing to the sign of ${\mathtt{e}}_{\scaleto{\mathrm{WB}}{3pt}}$, and
finally (d) is given by ${\mathtt{e}}_{d}=0$. The curve in purple is where
${\mathtt{e}}_{12}=0$.

Relation with Hur, Johnson [16]. In [16] the authors prove that whenever the
index function

$\Gamma({\mathtt{h}},\kappa):=\frac{(zm(z))^{\prime\prime}\,\big{(}(zm(z))^{\prime}-m(0)\big{)}}{m(z)-m(2z)}\cdot\big{(}2(m(z)-m(2z))+(zm(z))^{\prime}-m(0)\big{)}\big{|}_{z={\mathtt{h}}}\
,\quad m(z):={\mathfrak{m}}_{{\mathtt{h}},\kappa}(z/{\mathtt{h}})$

is negative, the operator ${\mathcal{L}}_{\mu,{\epsilon}}$ has two unstable
eigenvalues near zero. Again a direct computation using (1.16) shows that

${\mathtt{h}}\Gamma({\mathtt{h}},\kappa)=-2{\mathtt{e}}_{\scaleto{\mathrm{WB}}{3pt}}\,{\mathtt{e}}_{d}^{2}\
,$

so $\Gamma({\mathtt{h}},\kappa)<0$ iff
${\mathtt{e}}_{\scaleto{\mathrm{WB}}{3pt}}>0$. As a consequence, the
stability-instability regions that we find in Figure 3 are in one to one
correspondence with those of [16], see e.g. Figure 2.

#### Vorticity-modified Whitham equation.

We now consider the Whitham equation with constant vorticity $\gamma$, namely
(1.1) with symbol ${\mathfrak{m}}_{{\mathtt{h}},\gamma}(\xi)$ in (6.3). Let us
focus first on Assumption A. Similarly to the original Whitham equation, the
Fourier multiplier ${\mathcal{M}}(D)$ has order $-\frac{1}{2}$, and the symbol
${\mathfrak{m}}_{{\mathtt{h}},\gamma}(\xi)$ is smooth
$\forall({\mathtt{h}},\gamma)\in{\mathbb{R}}_{>0}\times{\mathbb{R}}$.
Moreover, ${\mathfrak{m}}_{{\mathtt{h}},\gamma}(\xi)$ is strictly decreasing
in $\xi>0$, $\forall\
({\mathtt{h}},\gamma)\in{\mathbb{R}}_{>0}\times{\mathbb{R}}$: indeed,
$\xi\in(0,+\infty)\mapsto\frac{\tanh({\mathtt{h}}\xi)}{\xi}$ is strictly
decreasing $\forall{\mathtt{h}}\in{\mathbb{R}}_{>0}$, whereas
$x\in(0,+\infty)\mapsto\frac{\gamma}{2}x+\sqrt{x+\frac{\gamma^{2}}{4}x^{2}}$
is strictly increasing $\forall\gamma\in{\mathbb{R}}$. Thus, no resonance
occur, verifying Assumption A. Then, by Theorem 1.1, there exists
${\epsilon}_{0}={\epsilon}_{0}({\mathtt{h}})>0$ and a traveling wave solution
$u_{\epsilon}$ of (1.1), parametrized by any ${\epsilon}\in(0,{\epsilon}_{0})$
. Denote by ${\mathcal{L}}_{\epsilon}$ the linearized operator (1.13) at
$u_{\epsilon}$. We obtain:

######  Theorem 6.3 (Vorticity-modified Whitham).

Let ${\mathcal{M}}(D)$ having symbol (6.3). There exists an analytic curve in
the semiplane $({\mathtt{h}},\gamma)$, defined by equation

$\dot{\mathfrak{m}}_{{\mathtt{h}},\gamma}(1)+3{\mathfrak{m}}_{{\mathtt{h}},\gamma}(1)-2{\mathfrak{m}}_{{\mathtt{h}},\gamma}(2)-{\mathfrak{m}}_{{\mathtt{h}},\gamma}(0)=0$

separating the regions ${\mathcal{U}}$ and ${\mathcal{S}}$, such that:

  * •

if $({\mathtt{h}},\gamma)\in{\mathcal{U}}$, there exists
${\epsilon}_{1}({\mathtt{h}})>0$ such that, for any
${\epsilon}\in[0,{\epsilon}_{1}({\mathtt{h}}))$, the spectrum of the operator
${\mathcal{L}}_{\epsilon}$ depicts near the origin a complete figure “8” as in
Figure 1.

  * •

If $({\mathtt{h}},\gamma)\in{\mathcal{S}}$, the spectrum of
${\mathcal{L}}_{\epsilon}$ near the origin is purely imaginary.

###### Proof.

We only need to show that, outside the critical curve, Assumption B is
verified. In particular, by the monotonicity of
${\mathfrak{m}}_{\gamma,{\mathtt{h}}}$ it is clear that

$\dot{\mathfrak{m}}_{\gamma,{\mathtt{h}}}(1)<0,\quad{\mathfrak{m}}_{\gamma,{\mathtt{h}}}(1)-{\mathfrak{m}}_{\gamma,{\mathtt{h}}}(2)>0,\quad{\mathtt{e}}_{d}=\dot{\mathfrak{m}}_{\gamma,{\mathtt{h}}}(1)+{\mathfrak{m}}_{\gamma,{\mathtt{h}}}(1)-{\mathfrak{m}}_{\gamma,{\mathtt{h}}}(0)<0.$

Moreover, a direct computation shows that

$\begin{split}{\mathtt{e}}_{b}=\frac{{{\mathtt{c}}_{\mathtt{h}}}}{2}\left(2+8{\mathtt{h}}^{2}\left(\frac{1-{{\mathtt{c}}_{\mathtt{h}}}^{4}}{2{{\mathtt{c}}_{\mathtt{h}}}^{2}}\right)^{2}-4{\mathtt{h}}\frac{1-{{\mathtt{c}}_{\mathtt{h}}}^{4}}{{{\mathtt{c}}_{\mathtt{h}}}}+{\mathtt{h}}^{2}(1-{{\mathtt{c}}_{\mathtt{h}}}^{4})(4+\gamma^{2}{{\mathtt{c}}_{\mathtt{h}}}^{2})(2+\gamma^{2}{{\mathtt{c}}_{\mathtt{h}}}^{2}+\gamma{{\mathtt{c}}_{\mathtt{h}}}\sqrt{4+\gamma^{2}{{\mathtt{c}}_{\mathtt{h}}}^{2}})\right)\\\
/(4+\gamma^{2}{{\mathtt{c}}_{\mathtt{h}}}^{2})^{\frac{3}{2}}\ .\end{split}$

A numerical evaluation show that
${\mathtt{e}}_{b}>0\quad\forall({\mathtt{h}},\gamma)\in{\mathbb{R}}_{>0}\times{\mathbb{R}}$,
and thus Assumption B is verified outside the curve defined by
${\mathtt{e}}_{w}=0$. ∎

Figure 4:  In the figure is represented the sign of the function
${\mathtt{e}}_{\scaleto{\mathrm{WB}}{3pt}}$ dependent on the parameters
${\mathtt{h}}$ (on the horizontal axis) and $\gamma$ (on the vertical axis).
The areas denoted by the letter U are the parts of the plane where
${\mathtt{e}}_{\scaleto{\mathrm{WB}}{3pt}}$ is positive, and thus where the
equation is unstable; the areas denoted by the letter S are the ones where it
is negative, and thus the equation results to be stable. The curve in blue
represent the critical line where
$\dot{\mathfrak{m}}_{{\mathtt{h}},\gamma}(1)+3{\mathfrak{m}}_{{\mathtt{h}},\gamma}(1)-2{\mathfrak{m}}_{{\mathtt{h}},\gamma}(2)-{\mathfrak{m}}_{{\mathtt{h}},\gamma}(0)=0$,
sepearating the two regimes.

###  6.2 Kawahara equation

In this section we consider the Kawahara equation, namely equation 1.1 with
Fourier multiplier ${\mathcal{M}}(D)$ having as symbol

${\mathfrak{m}}_{{\mathtt{a}},{\mathtt{b}}}(\xi)={\mathtt{a}}\xi^{2}+{\mathtt{b}}\xi^{4},\qquad{\mathtt{a}}\in{\mathbb{R}},\
\ {\mathtt{b}}>0\ .$ (6.12)

Kawahara equation is a model for nonlinear dispersive waves, that emerges as a
correction to the well known KdV equation in the regimes where the fifth order
term of the dispersion relation of the water waves equation is relevant.
We first verify Assumption A. Clearly the symbol
${\mathfrak{m}}_{{\mathtt{a}},{\mathtt{b}}}$ is even, smooth, and fulfills
(1.3) with $m=4$. Assumption (A3) is verified provided
$({\mathtt{a}},{\mathtt{b}})$ is outside the resonant set

${\mathcal{R}}=\\{({\mathtt{a}},{\mathtt{b}})\in{\mathbb{R}}\times{\mathbb{R}}_{>0}\colon\
\ {\mathtt{a}}=-(n^{2}+1){\mathtt{b}},\ \forall
n\in{\mathbb{N}}\setminus\\{1\\}\\}\ ,$ (6.13)

composed by infinitely many critical lines. Then for any
$({\mathtt{a}},{\mathtt{b}})\not\in{\mathcal{R}}$, Theorem 1.1 guarantees the
existence of ${\epsilon}_{0}={\epsilon}_{0}({\mathtt{a}},{\mathtt{b}})>0$ and
for any ${\epsilon}\in(0,{\epsilon}_{0})$ a traveling wave solution
$u_{\epsilon}$ of (1.1). Denote by ${\mathcal{L}}_{\epsilon}$ the linearized
operator (1.13) at $u_{\epsilon}$. We obtain:

######  Theorem 6.4 (Kawahara).

Consider equation (1.1) with ${\mathcal{M}}(D)$ having symbol (6.12). Let

$\displaystyle{\mathcal{U}}:=\\{({\mathtt{a}},{\mathtt{b}})\in{\mathbb{R}}\times{\mathbb{R}}_{>0}\colon-\frac{25}{3}{\mathtt{b}}<{\mathtt{a}}<-5{\mathtt{b}}\
\ \vee\ \ -\frac{10}{3}{\mathtt{b}}<{\mathtt{a}}<-\frac{5}{3}{\mathtt{b}}\
,{\mathtt{a}}\neq-2{\mathtt{b}}\\}\setminus{\mathcal{R}}\ ,$ (6.14)
$\displaystyle{\mathcal{S}}:=\\{({\mathtt{a}},{\mathtt{b}})\in{\mathbb{R}}\times{\mathbb{R}}_{>0}\colon{\mathtt{a}}<-\frac{25}{3}{\mathtt{b}}\
\ \vee\ \ -5{\mathtt{b}}<{\mathtt{a}}<-\frac{10}{3}{\mathtt{b}}\ \ \vee\ \
{\mathtt{a}}>-\frac{5}{3}{\mathtt{b}}\
,{\mathtt{a}}\neq-2{\mathtt{b}}\\}\setminus{\mathcal{R}}$

where ${\mathcal{R}}$ is the resonant set in (6.13).

  * •

For any $({\mathtt{a}},{\mathtt{b}})\in{\mathcal{U}}$ there exists
${\epsilon}_{1}={\epsilon}_{1}({\mathtt{a}},{\mathtt{b}})>0$ such that for any
${\epsilon}\in[0,{\epsilon}_{1})$, the spectrum of the operator
${\mathcal{L}}_{\epsilon}$ depicts near the origin a complete figure “8” as in
Figure 1.

  * •

For any $({\mathtt{a}},{\mathtt{b}})\in{\mathcal{S}}$ the spectrum of
${\mathcal{L}}_{\epsilon}$ near the origin is purely imaginary.

###### Proof.

We check Assumption B. Since

$\displaystyle{\mathtt{e}}_{12}=2{\mathtt{a}}+4{\mathtt{b}}\
,\quad{\mathtt{e}}_{d}=3{\mathtt{a}}+5{\mathtt{b}}\
,\quad{\mathtt{e}}_{b}=-3{\mathtt{a}}-10{\mathtt{b}}\
,\quad{\mathtt{e}}_{w}=\frac{3{\mathtt{a}}+25{\mathtt{b}}}{3({\mathtt{a}}+5{\mathtt{b}})(3{\mathtt{a}}+5{\mathtt{b}})}\
.$ (6.15)

hence, the non-degeneracy condition defines the 4 critical lines
${\mathtt{a}}=-c_{j}{\mathtt{b}}$,
$c_{j}=-\frac{5}{3},-2,-\frac{10}{3},-\frac{25}{3}$ where Assumption B is not
met (note that the line ${\mathtt{a}}=-5{\mathtt{b}}$ is already included in
${\mathcal{R}}$). We plot these lines in Figure 5. The positive superlevel of
the function

${\mathtt{e}}_{\scaleto{\mathrm{WB}}{3pt}}=-\frac{(3{\mathtt{a}}+10{\mathtt{b}})(3{\mathtt{a}}+25{\mathtt{b}})}{3({\mathtt{a}}+5{\mathtt{b}})(3{\mathtt{a}}+5{\mathtt{b}})}$
(6.16)

defines the region of parameters $({\mathtt{a}},{\mathtt{b}})$ where
${\mathcal{L}}_{\epsilon}$ has unstable spectrum near the origin, whereas the
negative one defines the region where ${\mathcal{L}}_{\epsilon}$ is stable,
see Figure 5. ∎

Figure 5:  The blue lines are the critical lines
${\mathtt{a}}=-\frac{5}{3}{\mathtt{b}}$,
${\mathtt{a}}=-\frac{10}{3}{\mathtt{b}}$, ${\mathtt{a}}=-5{\mathtt{b}}$,
${\mathtt{a}}=-\frac{25}{3}{\mathtt{b}}$, where ${\mathtt{e}}_{d}$,
${\mathtt{e}}_{b}$, ${\mathtt{e}}_{w}$ are identically zero or not defined,
the green one is ${\mathtt{a}}=-2{\mathtt{b}}$ where ${\mathtt{e}}_{12}\equiv
0$, ${\mathtt{e}}_{d}$, and the dotted line are the resonant lines in
${\mathcal{R}}$ in (6.13). The regions $U$ are where
${\mathtt{e}}_{\scaleto{\mathrm{WB}}{3pt}}>0$ and the $S$ ones where
${\mathtt{e}}_{\scaleto{\mathrm{WB}}{3pt}}<0$.

Relation with Creedon, Deconinck, Trichtchenko [11]. The paper [11] studies
the spectrum of ${\mathcal{L}}_{\epsilon}$ both near and away from zero. They
prove the existence of unstable eigenvalues of
${\mathcal{L}}_{\mu,{\epsilon}}$ near zero when (6.16) is strictly positive.
Moreover, they prove the existence of isola of instabilities also away from
zero.

###  6.3 Intermediate long-wave equation

The intermediate long-wave equation is equation (1.1) with ${\mathcal{M}}(D)$
having symbol

${\mathfrak{m}}_{\mathtt{h}}(\xi)=\frac{\xi}{\tanh({\mathtt{h}}\xi)}-\frac{1}{{\mathtt{h}}}\
,\qquad{\mathtt{h}}>0$ (6.17)

It models unidirectional long waves at the surface of a fluid of finite depth,
and it is particularly famous as an example of integrable Hamiltonian system.
It can be derived from the Whitham equation, as it is described in [24].
To apply our abstract result to the intermediate long-wave equation, we first
verify Assumption A. The symbol is smooth, even, of order 1. Moreover,
$\xi\mapsto{\mathfrak{m}}_{\mathtt{h}}(\xi)$ is strictly increasing for
$\xi\geq 0$, and so Assumption A is satisfied. Theorem 1.1 guarantees the
existence of small amplitude traveling waves. In this case our abstract
Theorem 1.2 yields:

######  Theorem 6.5 (Intermediate long-wave).

Consider equation (1.1) with ${\mathcal{M}}(D)$ having symbol (6.17). For any
${\mathtt{h}}>0$ the spectrum of ${\mathcal{L}}_{\epsilon}$ near the origin is
purely imaginary.

###### Proof.

Since $\xi\mapsto{\mathfrak{m}}_{\mathtt{h}}(\xi)$ is strictly increasing,
${\mathtt{e}}_{12}$ and ${\mathtt{e}}_{d}$ are strictly positive.
${\mathtt{e}}_{b}$ is strictly negative: by direct computations one has

${\mathtt{e}}_{b}=-{{\mathtt{c}}_{\mathtt{h}}}^{-2}+{\mathtt{h}}({{\mathtt{c}}_{\mathtt{h}}}^{-4}-1+(1-{\mathtt{h}}{{\mathtt{c}}_{\mathtt{h}}}^{-2}-{\mathtt{h}}{{\mathtt{c}}_{\mathtt{h}}}^{2})(\frac{1}{4}(1+{{\mathtt{c}}_{\mathtt{h}}}^{4})^{2}{{\mathtt{c}}_{\mathtt{h}}}^{-4}-1))$

and

${\mathtt{e}}_{w}=\frac{2{\mathtt{h}}(-2+{\mathtt{h}}{{\mathtt{c}}_{\mathtt{h}}}^{-2})(1-{{\mathtt{c}}_{\mathtt{h}}}^{4})+2{{\mathtt{c}}_{\mathtt{h}}}^{2}}{-(1+2{\mathtt{h}}^{2})(1-{{\mathtt{c}}_{\mathtt{h}}}^{4})+1+{{\mathtt{c}}_{\mathtt{h}}}^{4}}$

The sign of ${\mathtt{e}}_{b}$ and ${\mathtt{e}}_{\scaleto{\mathrm{WB}}{3pt}}$
is represented in Figure 6 below, proving the Theorem.

∎

Figure 6:  In the figure is represented of the functions ${\mathtt{e}}_{b}$
and ${\mathtt{e}}_{\scaleto{\mathrm{WB}}{3pt}}$ in dependence of the
parameters ${\mathtt{h}}>0$.