# Rigorous derivation of the Whitham equations from the water waves equations
in the shallow water regime

Louis Emerald

###### Abstract

We derive the Whitham equations from the water waves equations in the shallow
water regime using two different methods, thus obtaining a direct and rigorous
link between these two models. The first one is based on the construction of
approximate Riemann invariants for a Whitham-Boussinesq system and is adapted
to unidirectional waves. The second one is based on a generalisation of
Birkhoff’s normal form algorithm for almost smooth Hamiltonians and is adapted
to bidirectional propagation. In both cases we clarify the improved accuracy
on the fully dispersive Whitham model with respect to the long wave Korteweg-
de Vries approximation.

##  1 Introduction

###  1.1 Motivations

In this paper we work on a unidirectionnal model for surface waves in coastal
oceanography, the Whitham equation, introduced by Whitham in [20] and [21].
This equation, having the same dispersion relation as the general water waves
model, is a full dispersion modification of the Korteweg-de Vries equation
(KdV). As such one expects an improved accuracy and range of validity as a
model for surface waves compared to the long wave KdV approximation. More
precisely, we anticipate that in the presence of weak nonlinearities, the
Whitham equation stays close to the water waves equations, even if the
dispersion effects are not small, as obtained in [10] for the bidirectional
full dispersion systems.

The Whitham model has weaker dispersive effects compared to the KdV model and
thus allows for wavebreaking and Stokes waves of maximal amplitude to occur,
as one would expect of a model for surface waves in coastal oceanography. This
was the historical reason for which Whitham introduced it. A rigorous proof of
the existence of the wavebreaking phenomenon has been established by Hur [11]
and Saut & Wang [17]. Ehrnström & Wahlén [8] and Truong et al. [19] proved the
existence of Stokes waves of maximal amplitude.

Besides [11, 17, 8, 19] we emphasize some other works on the Whitham equation.
Klein et al. [12] compared rigorously the solutions of the Whitham equation
with those of the KdV equation and exhibited three different regimes of
behaviour: scattering for small initial data, finite time blow-up and a KdV
long wave regime. Ehrnström & Wang [9] proved an enhanced existence time for
solutions of the Whitham equation associated with small initial data.
Ehrnström & Kalisch [7] proved the existence of periodic traveling-waves
solutions. Sanford et al. [16] established that large-amplitude periodic
traveling-wave solutions are unstable, while those of small-amplitude are
stable if their wavelength is long enough. Denoting by $\mu$ the shallow water
parameter and by $\varepsilon$ the nonlinearity parameter, Klein et al. [12]
proved the validity of the Whitham equation in the KdV regime
$\\{\mu=\varepsilon\\}$ by obtaining that the Whitham equation approximate the
KdV equation at a precision of order $\varepsilon^{2}$. Moldabayev et al. [15]
derived the Whitham equation from the Hamiltonian of the water waves equations
in the unusual regime $\\{\varepsilon=e^{-\frac{\kappa}{\mu^{\nu/2}}}\\}$,
where $\kappa$ and $\nu$ are parameters numerically inferred from the solitary
waves of the Whitham model. They also assumed the initial data of the water
waves equations prepared to generate unidirectional waves. If we transpose
their derivation in the general shallow water regime
$\\{0\leq\mu\leq\mu_{\max},0\leq\varepsilon\leq 1\\}$, where $\mu_{\max}>0$ is
fixed, we obtain a precision of order $\mu\varepsilon+\varepsilon^{2}$. To our
point of view, the two previous results are not satisfying as the authors did
not characterize the range of validity of the Whitham equation as a model for
water waves with enough accuracy. In this work we propose to improve these
results using two different methods.

The first method is based on an adaptation of the one used to derive the
inviscid Burgers equations from the Nonlinear Shallow Water system using the
Riemann invariants of the latter and requires, as in [15], that initial data
are prepared to generate unidirectional waves. We show from it that the
Whitham equation can be derived from the water waves equations at the order of
precision $\mu\varepsilon$, which is the same order as for the Whitham-
Boussinesq equations when considering general initial data (see [10] for a
rigorous derivation of the latter equations in the shallow water regime). From
this derivation, we prove that solutions of the water waves equations,
associated with well-prepared initial data, can be approximated in the shallow
water regime, to within order $\mu\varepsilon t$ in a time scale of order
$\varepsilon^{-1}$, by a right or left propagating wave solving a Whitham-type
equation.

The second method does not need well-prepared initial data. It is based on a
generalization of Birkhoff’s normal form algorithm for almost smooth functions
(see Definition 3.14) developed by Bambusi in [1] to show that the solutions
of the water waves equations can be approximated in the KdV regime, to within
order $\varepsilon$ in a time scale of order $\varepsilon^{-1}$, by two
counter-propagating waves solving KdV-type equations, thus improving both the
results obtained by Schneider and Wayne in [18], and those obtained by Bona,
Colin and Lannes in [2]. If we transpose Bambusi’s result in the shallow water
regime, we get a precision of order
$\varepsilon^{2}+(\mu^{2}+\mu\varepsilon+\varepsilon^{2})t$ in a time scale of
order $(\max{(\mu,\varepsilon)})^{-1}$, which tells us that the two KdV-type
equations are not appropriate to approximate the water waves equations when
$\mu$ is not small. Bambusi’s method uses the local structure of the KdV
equation. We adapt it to the nonlocal Whitham equation and prove that the
solutions of the water waves equations can be approximated in the shallow
water regime, to within order
$\varepsilon^{2}+(\mu\varepsilon+\varepsilon^{2})t$ in a time scale of order
$(\max{(\mu,\varepsilon)})^{-1}$, by two counter-propagating waves solving
Whitham-type equations. In addition, we express explicitly the loss of
regularity needed to derive the two uncoupled Whitham-type equations solved by
the counter-propagating waves.

We now compare the results from the first method and Bambusi’s method for the
KdV and the Whitham equations. In the presence of strong nonlinearities
($\varepsilon\geq\mu$), we get from the first method a precision order $\mu$
in a time scale of order $\varepsilon^{-1}$, whereas Bambusi’s method for the
KdV equations and for the Whitham equations gives a precision of order
$\mu+\varepsilon$ in the same time scale. The $\varepsilon$ in the latter
order of precision comes from the coupling effects between the two counter-
propagating waves. Hence, if $\mu$ is small, we have a good approximation from
the Whitham equation, and not from the KdV equation, of the solutions of the
water waves equations associated with well-prepared initial data, even if
$\varepsilon$ is not small. In the presence of weak nonlinearities
($\varepsilon\leq\mu$), we get from the first method and Bambusi’s method for
the Whitham equations a precision of order $\varepsilon$ in a time scale of
order $\mu^{-1}$, whereas Bambusi’s method for the KdV equations gives a
precision of order $\mu+\varepsilon$ in the same time scale. Hence, if
$\varepsilon$ is small, we have a good approximation from the Whitham
equation, and not from the KdV equation, of the solutions of the water waves
equations, even if $\mu$ is not small.

###  1.2 Main results

The starting point of our study is the water waves equations:

$\displaystyle\begin{cases}\partial_{t}\zeta-\frac{1}{\mu}\mathcal{G}^{\mu}[\varepsilon\zeta]\psi=0,\\\
\partial_{t}\psi+\zeta+\frac{\varepsilon}{2}(\partial_{x}\psi)^{2}-\frac{\mu\varepsilon}{2}\frac{(\frac{1}{\mu}\mathcal{G}^{\mu}[\varepsilon\zeta]\psi+\varepsilon\partial_{x}\zeta\cdot\partial_{x}\psi)^{2}}{1+\varepsilon^{2}\mu(\partial_{x}\zeta)^{2}}=0.\end{cases}$
(1.1)

Here

  * •

The free surface elevation is the graph of $\zeta$, which is a function of
time $t$ and horizontal space $x\in\mathbb{R}$.

  * •

$\psi(t,x)$ is the trace at the surface of the velocity potential.

  * •

$\mathcal{G}^{\mu}$ is the Dirichlet-Neumann operator defined later in
Definition 1.4.

Every variable and function in (1.2) is compared with physical characteristic
parameters of the same dimension. Among those are the characteristic water
depth $H_{0}$, the characteristic wave amplitude $a_{\operatorname{surf}}$ and
the characteristic wavelength $L$. From these comparisons appear two
adimensional parameters of main importance:

  * •

$\mu\colonequals\frac{H_{0}^{2}}{L^{2}}$: the shallow water parameter,

  * •

$\varepsilon\colonequals\frac{a_{\operatorname{surf}}}{H_{0}}$: the
nonlinearity parameter.

We refer to [13] for details on the derivation of these equations.

In [22], Zakharov proved that the water waves system (1.1) enjoys an
Hamiltonian formulation. Let $H_{\operatorname{WW}}$ be defined by

$\displaystyle
H_{\operatorname{WW}}\colonequals\frac{1}{2}\int_{\mathbb{R}}\zeta^{2}\operatorname{dx}+\frac{1}{2}\int_{\mathbb{R}}\psi\frac{1}{\mu}\mathcal{G}^{\mu}[\varepsilon\zeta]\psi\operatorname{dx}.$
(1.2)

Then (1.1) is equivalent to the Hamilton equations

$\displaystyle\begin{cases}\partial_{t}\zeta=\delta_{\psi}H_{\operatorname{WW}},\\\
\partial_{t}\psi=-\delta_{\zeta}H_{\operatorname{WW}},\end{cases}$

where $\delta_{\zeta}$ and $\delta_{\psi}$ are functional derivatives.

Before giving the main definitions of this section, here is one assumption
maintained throughout this paper.

######  Hypothesis 1.1.

A fundamental hypothesis is the lower boundedness by a positive constant of
the water depth (non-cavitation assumption):

$\displaystyle\exists h_{\min}>0,\forall x\in\mathbb{R},\ \ h\colonequals
1+\varepsilon\zeta(t,x)\geq h_{\min}.$ (1.3)

Moreover, we will always work in the shallow water regime which we define
here.

######  Definition 1.2.

Let $\mu_{\max}>0$, we define the shallow water regime
$\mathcal{A}\colonequals\\{(\mu,\varepsilon),\ \ 0\leq\mu\leq\mu_{\max},\ \
0\leq\varepsilon\leq 1\\}$.

In what follows we need some notations on the functional framework of this
paper.

######  Notations 1.3.

  * •

For any $\alpha\geq 0$ we will respectively denote
$H^{\alpha}(\mathbb{R}),W^{\alpha,1}(\mathbb{R})$ and
$W^{\alpha,\infty}(\mathbb{R})$ the Sobolev spaces of order $\alpha$ in
respectively $L^{2}(\mathbb{R}),L^{1}(\mathbb{R})$ and
$L^{\infty}(\mathbb{R})$. We will denote their associated norms by
$|\cdot|_{H^{\alpha}},|\cdot|_{W^{\alpha,1}}$ and
$|\cdot|_{W^{\alpha,\infty}}$.

  * •

For any $\alpha\geq 0$ we will denote
$\dot{H}^{\alpha+1}(\mathbb{R})\colonequals\\{f\in
L_{\operatorname{loc}}^{2}(\mathbb{R}),\ \ \partial_{x}f\in
H^{\alpha}(\mathbb{R})\\}$ and
$\dot{W}^{\alpha+1,1}(\mathbb{R})\colonequals\\{f\in
L_{\operatorname{loc}}^{1}(\mathbb{R}),\ \ \partial_{x}f\in
W^{\alpha,1}(\mathbb{R})\\}$ the Beppo-Levi spaces of order $\alpha$. Their
associated seminorms are respectively
$|\cdot|_{\dot{H}^{\alpha+1}}\colonequals|\partial_{x}(\cdot)|_{H^{\alpha}}$
and
$|\cdot|_{\dot{W}^{\alpha+1,1}}\colonequals|\partial_{x}(\cdot)|_{W^{\alpha,1}}$.

Now we define the Dirichlet-Neumann operator.

######  Definition 1.4.

For all $\zeta$ and $\psi$ sufficiently smooth, let $\phi$ be the velocity
potential solving the elliptic problem

$\displaystyle\begin{cases}(\mu\partial_{x}^{2}+\partial_{z}^{2})\phi=0,\\\
\phi|_{z=\varepsilon\zeta}=\psi,\ \ \partial_{z}\phi|_{z=-1}=0.\end{cases}$

where $z\in(-1,\varepsilon\zeta)$ is the vertical space variable.

We define the Dirichlet-Neumann operator by the formula

$\displaystyle\mathcal{G}^{\mu}[\varepsilon\zeta]\psi=\sqrt{1+\varepsilon^{2}(\partial_{x}\zeta)^{2}}\partial_{\textbf{n}^{\mu}}\phi|_{\varepsilon\zeta},$

where $\textbf{n}^{\mu}$ is the outward normal vector of the free surface
$\varepsilon\zeta$. It depends on time and space.

$\mathcal{G}^{\mu}$ is linear in $\psi$ and nonlinear in $\zeta$. See [13] for
a thorough study of this operator.

We also recall the definition of a Fourier multiplier.

######  Definition 1.5.

Let $u:\mathbb{R}\to\mathbb{R}$ be a tempered distribution, let $\widehat{u}$
be its Fourier transform. Let $F\in\mathcal{C}^{\infty}(\mathbb{R})$ be a
smooth function such that there exists $m\in\mathbb{Z}$ for which
$\forall\beta\geq 0$,
$|\partial_{\xi}^{\beta}F(\xi)|\leq\widetilde{C}_{\beta}(1+|\xi|)^{m-|\beta|}$,
where $\widetilde{C}_{\beta}>0$ is a constant depending on $\beta$. Then the
Fourier multiplier associated with $F$ is denoted $\mathrm{F}(D)$ (denoted
$\mathrm{F}$ when no confusion is possible) and defined by the formula:

$\displaystyle\widehat{\mathrm{F}(D)u}(\xi)=F(\xi)\widehat{u}(\xi).$

The first method to rigorously justify the Whitham equations from the water
waves equations gives the two following results.

######  Proposition 1.6.

Let $\mu_{\max}>0$. Let also
$\mathrm{F}_{\mu}=\sqrt{\frac{\tanh{(\sqrt{\mu}|D|)}}{\sqrt{\mu}|D|}}$ be a
Fourier multiplier. There exists $n\in\mathbb{N}^{*}$ such that for any
$(\mu,\varepsilon)\in\mathcal{A}$ and $\zeta\in
L^{\infty}_{t}H^{\alpha+n}_{x}(\mathbb{R})$ and $\psi\in
L^{\infty}_{t}\dot{H}^{\alpha+n}_{x}(\mathbb{R})$ solutions of the water waves
equations (1.1) satisfying (1.3), there exists $R_{1},R_{2}\in
H^{\alpha}(\mathbb{R})$ uniformly bounded in $(\mu,\varepsilon)$ such that the
quantities

$\displaystyle
u^{+}=\frac{\sqrt{h}-1}{\varepsilon}+\frac{1}{2}\mathrm{F}_{\mu}^{-1}[v],\ \
u^{-}=-\frac{\sqrt{h}-1}{\varepsilon}+\frac{1}{2}\mathrm{F}_{\mu}^{-1}[v],$
(1.4)

where $v=\mathrm{F}_{\mu}^{2}[\partial_{x}\psi]$, satisfy the equations

$\displaystyle\begin{cases}\partial_{t}u^{+}+(\varepsilon\frac{3u^{+}+u^{-}}{2}+1)\mathrm{F}_{\mu}\partial_{x}u^{+}=\mu\varepsilon
R_{1},\\\
\partial_{t}u^{-}+(\varepsilon\frac{u^{+}+3u^{-}}{2}-1)\mathrm{F}_{\mu}\partial_{x}u^{-}=\mu\varepsilon
R_{2}.\end{cases}$

######  Remark 1.7.

The quantities $u^{+}$ and $u^{-}$ can almost be seen as Riemann invariants of
the Whitham-Boussinesq equations

$\displaystyle\begin{cases}\partial_{t}\zeta+\partial_{x}^{2}\psi+\varepsilon\partial_{x}(\zeta\partial_{x}\psi)+\varepsilon\zeta\partial_{x}^{2}\psi=0,\\\
\partial_{t}\mathrm{F}_{\mu}^{2}[\partial_{x}\psi]+\mathrm{F}_{\mu}^{2}[\partial_{x}\zeta]+\varepsilon\partial_{x}\psi\partial_{x}^{2}\psi=0,\end{cases}$
(1.5)

for which we know the rigorous derivation from the water waves equations (see
[10]). This is the main idea to prove this result.

Using Theorem 4.16 and Theorem 4.18 of [13] (the Rayleigh-Taylor condition is
always satisfied when the bottom is flat, see Subsection 4.3.5 in [13]) we get
the following corollary.

######  Corollary 1.8.

For any $\alpha\geq 0$, there exists $n\in\mathbb{N}^{*}$ such that for any
$(\mu,\varepsilon)\in\mathcal{A}$, we have what follows. Consider the Cauchy
problem for the water wave problem (1.1) with initial conditions
$(\zeta_{0},\psi_{0})\in
H^{\alpha+n}(\mathbb{R})\times\dot{H}^{\alpha+n+1}(\mathbb{R})$ satisfying the
non-cavitation assumption (1.3), and denote by $(\zeta,\psi)$ the
corresponding solution. Let $(u^{+}_{e,0},u^{-}_{e,0})$ be defined by the
formulas (1.4) applied to $(\zeta_{0},\psi_{0})$. There exists a unique
solution $(u^{+}_{e},u^{-}_{e})$ of the exact diagonalized system

$\displaystyle\begin{cases}\partial_{t}u^{+}_{e}+(\varepsilon\frac{3u^{+}_{e}+u^{-}_{e}}{2}+1)\mathrm{F}_{\mu}\partial_{x}u^{+}_{e}=0,\\\
\partial_{t}u^{-}_{e}+(\varepsilon\frac{u^{+}_{e}+3u^{-}_{e}}{2}-1)\mathrm{F}_{\mu}\partial_{x}u^{-}_{e}=0,\end{cases}$
(1.6)

with initial conditions $(u^{+}_{e,0},u^{-}_{e,0})$, which satisfy the
following property: denote by $(\zeta_{\rm{c}},\psi_{\rm{c}})$ the following
quantities

$\displaystyle\zeta_{\rm{c}}=\frac{1}{\varepsilon}\Big{[}(\frac{\varepsilon}{2}(u^{+}_{e}-u^{-}_{e})+1)^{2}-1\Big{]}\
\ \operatorname{and}\ \
\psi_{\rm{c}}=\int_{0}^{x}\mathrm{F}_{\mu}^{-1}[u^{+}_{e}+u^{-}_{e}]\operatorname{dx},$
(1.7)

then for all times $t\in[0,\frac{T}{\varepsilon}]$, one has

$\displaystyle|(\zeta-\zeta_{\rm{c}},\psi-\psi_{\rm{c}})|_{H^{\alpha}\times\dot{H}^{\alpha+1}}\leq\mu\varepsilon
Ct,$

where
$C,T^{-1}=C(\frac{1}{h_{\min}},\mu_{\max},|\zeta_{0}|_{H^{\alpha+n}},|\psi_{0}|_{\dot{H}^{\alpha+n+1}})$.

######  Remark 1.9.

The existence and uniqueness of the solution $(u_{e}^{+},u_{e}^{-})$ of the
exact diagonalized system (1.6) with initial conditions
$(u^{+}_{e,0},u^{-}_{e,0})$ on the required time scale is given by Proposition
2.9.

To write the next result we need another notation.

######  Notation 1.10.

Let $k\in\mathbb{N}$ and $l\in\mathbb{N}$. A function $R$ is said to be of
order $O(\mu^{k}\varepsilon^{l})$, denoted $R=O(\mu^{k}\varepsilon^{l}),$ if
divided by $\mu^{k}\varepsilon^{l}$ this function is uniformly bounded with
respect to $(\mu,\varepsilon)\in\mathcal{A}$ in the Sobolev norms
$|\cdot|_{H^{\alpha}}$, $\alpha\geq 0$.

######  Proposition 1.11.

With the same hypotheses and notations as in Proposition 1.6, if
$u^{-}(0)=O(\mu)$, then there exists $T_{1}>0$ such that for all times
$t\in[0,\frac{T_{1}}{\epsilon}]$, $u^{-}(t)=O(\mu)$. Moreover for these times,
$u^{+}$ satisfies the Whitham equation up to a remainder term of order
$O(\mu\varepsilon)$, i.e.

$\displaystyle\partial_{t}u^{+}+\mathrm{F}_{\mu}[\partial_{x}u^{+}]+\frac{3\varepsilon}{2}u^{+}\partial_{x}u^{+}=O(\mu\epsilon).$

If instead $u^{+}(0)=O(\mu)$, then there exists $T_{2}>0$ such that for all
times $t\in[0,\frac{T_{2}}{\epsilon}]$, $u^{+}(t)=O(\mu)$. Moreover for these
times, $u^{-}$ satisfies the counter propagating Whitham equation up to
remainder term of order $O(\mu\varepsilon)$, i.e.

$\displaystyle\partial_{t}u^{-}-\mathrm{F}_{\mu}[\partial_{x}u^{-}]+\frac{3\varepsilon}{2}u^{-}\partial_{x}u^{-}=O(\mu\varepsilon).$
(1.8)

######  Proposition 1.12.

In Corollary 1.8, if $u^{-}_{e}(0)=O(\mu)$, then one can replace
$\zeta_{\rm{c}}$ and $\psi_{\rm{c}}$ by

$\displaystyle\zeta_{\operatorname{Wh},+}=\frac{1}{\varepsilon}\Big{[}(\frac{\varepsilon}{2}u^{+}+1)^{2}-1\Big{]}\
\ \operatorname{and}\ \
\psi_{\operatorname{Wh},+}=\int_{0}^{x}\mathrm{F}_{\mu}^{-1}[u^{+}]\operatorname{dx},$

where $u^{+}\in\mathcal{C}([0,\frac{T}{\varepsilon}],H^{\alpha}(\mathbb{R}))$
solves the exact Whitham equation

$\displaystyle\partial_{t}u^{+}+\mathrm{F}_{\mu}[\partial_{x}u^{+}]+\frac{3\varepsilon}{2}u^{+}\partial_{x}u^{+}=0,$

and get for all times $t\in[0,\frac{T}{\varepsilon}]$,

$\displaystyle|(\zeta-\zeta_{\operatorname{Wh},+},\psi-\psi_{\operatorname{Wh},+})|_{H^{\alpha}\times\dot{H}^{\alpha+1}}\leq
C(|u_{e}^{-}(0)|_{H^{\alpha+1}}+\mu\varepsilon t),$

where
$C,T^{-1}=C(\frac{1}{h_{\min}},\mu_{\max},|\zeta_{0}|_{H^{\alpha+n}},|\psi_{0}|_{\dot{H}^{\alpha+n+1}})$.

If instead $u^{+}_{e}(0)=O(\mu)$, then one can replace $\zeta_{\rm{c}}$ and
$\psi_{\rm{c}}$ by

$\displaystyle\zeta_{\operatorname{Wh},-}=\frac{1}{\varepsilon}\Big{[}(\frac{\varepsilon}{2}u^{-}-1)^{2}-1\Big{]}\
\ \operatorname{and}\ \
\psi_{\operatorname{Wh},-}=\int_{0}^{x}\mathrm{F}_{\mu}^{-1}[u^{-}]\operatorname{dx},$

where $u^{-}\in\mathcal{C}([0,\frac{T}{\varepsilon}],H^{\alpha}(\mathbb{R}))$
solves the exact counter-propagating Whitham equation

$\displaystyle\partial_{t}u^{-}-\mathrm{F}_{\mu}[\partial_{x}u^{-}]+\frac{3\varepsilon}{2}u^{-}\partial_{x}u^{-}=0,$

and get for all times $t\in[0,\frac{T}{\varepsilon}]$,

$\displaystyle|(\zeta-\zeta_{\operatorname{Wh},-},\psi-\psi_{\operatorname{Wh},-})|_{H^{\alpha}\times\dot{H}^{\alpha+1}}\leq
C(|u_{e}^{+}(0)|_{H^{\alpha+1}}+\mu\varepsilon t),$

where
$C,T^{-1}=C(\frac{1}{h_{\min}},\mu_{\max},|\zeta_{0}|_{H^{\alpha+n}},|\psi_{0}|_{\dot{H}^{\alpha+n+1}})$.

The second method is based on a generalisation of Birkhoff’s normal form
algorithm developed by Bambusi in [1]. Its application to our case leads us to
define some operator and transformations which we need to write our main
results.

######  Definition 1.13.

  * •

Let $\alpha\geq 0$. We define
$\mathcal{T}_{\rm{I}}:H^{\alpha}(\mathbb{R})\times H^{\alpha}(\mathbb{R})\to
H^{\alpha}(\mathbb{R})\times\dot{H}^{\alpha+1}(\mathbb{R})$ by the formula

$\displaystyle\mathcal{T}_{\rm{I}}(\zeta,v)=\begin{pmatrix}\zeta\\\
\int_{0}^{x}v(y)\operatorname{dy}\end{pmatrix}.$

  * •

Let $\alpha\geq 0$. We define
$\mathcal{T}_{\rm{D}}:H^{\alpha+1}(\mathbb{R})\times
H^{\alpha+1}(\mathbb{R})\to H^{\alpha}(\mathbb{R})\times
H^{\alpha}(\mathbb{R})$ by the formula

$\displaystyle\mathcal{T}_{\rm{D}}(r,s)=\begin{pmatrix}r+s\\\
\mathrm{F}_{\mu}^{-1}[r-s]\end{pmatrix}.$

  * •

We define $\partial^{-1}:L^{1}(\mathbb{R})\to L^{\infty}(\mathbb{R})$ by the
formula

$\displaystyle\partial^{-1}u(y)=\frac{1}{2}\int_{\mathbb{R}}\operatorname{sgn}(y-y_{1})u(y_{1})\operatorname{dy}_{1}.$

  * •

Let $\alpha\geq 0$ be a positive integer. We define
$\mathcal{T}_{\rm{B}}:W^{\alpha+1,1}(\mathbb{R})\times
W^{\alpha+1,1}(\mathbb{R})\to W^{\alpha,1}(\mathbb{R})\times
W^{\alpha,1}(\mathbb{R})$ by the formula

$\displaystyle\mathcal{T}_{\rm{B}}(r,s)=\begin{pmatrix}r+\frac{\varepsilon}{4}\partial_{x}(r)\partial^{-1}(s)+\frac{\varepsilon}{4}rs+\frac{\varepsilon}{8}s^{2}\\\
s+\frac{\varepsilon}{4}\partial_{x}(s)\partial^{-1}(r)+\frac{\varepsilon}{4}rs+\frac{\varepsilon}{8}r^{2}\end{pmatrix}.$

######  Theorem 1.14.

Let $\alpha\geq 0$ be a positive integer and
$(\mu,\varepsilon)\in\mathcal{A}$. Let $r,s$
$\in\mathcal{C}^{1}([0,\frac{T}{\varepsilon}],W^{\alpha+7,1}(\mathbb{R}))$ be
solutions of the Whitham equations

$\displaystyle\begin{cases}\partial_{t}r+\mathrm{F}_{\mu}[\partial_{x}r]+\frac{3\varepsilon}{2}r\partial_{x}r=0,\\\
\partial_{t}s-\mathrm{F}_{\mu}[\partial_{x}s]-\frac{3\varepsilon}{2}s\partial_{x}s=0.\end{cases}$
(1.9)

Let also $H_{\operatorname{WW}}$ be the Hamiltonian of the water waves
equations (1.2) and $J=\begin{pmatrix}0&1\\\ -1&0\end{pmatrix}$ the Poisson
tensor associated with. Then the quantities

$\displaystyle\begin{pmatrix}\zeta_{\operatorname{Wh}}\\\
\psi_{\operatorname{Wh}}\end{pmatrix}\colonequals\mathcal{T}_{\rm{I}}(\mathcal{T}_{\rm{D}}(\mathcal{T}_{\rm{B}}(r,s)),$
(1.10)

satisfy

$\displaystyle\partial_{t}\begin{pmatrix}\zeta_{\operatorname{Wh}}\\\
\psi_{\operatorname{Wh}}\end{pmatrix}=J\nabla(H_{\operatorname{WW}})(\zeta_{\operatorname{Wh}},\psi_{\operatorname{Wh}})+(\mu\varepsilon+\varepsilon^{2})R,\
\ \forall t\in[0,\frac{T}{\varepsilon}],$

where $|R|_{H^{\alpha}\times\dot{H}^{\alpha+1}}\leq
C(\mu_{\max},|r|_{W^{\alpha+7,1}},|s|_{W^{\alpha+7,1}})$.

######  Remark 1.15.

  * •

The existence of
$r,s\in\mathcal{C}^{1}([0,\frac{T}{\varepsilon}],W^{\alpha+6,1}(\mathbb{R}))$
solutions of system (1.9) is given by Lemma 3.23.

  * •

The remainder $R$ is uniformly bounded in $(\mu,\varepsilon)$ for times
$t\in[0,\frac{T}{\max{(\mu,\varepsilon)}}]$ in
$H^{\alpha}(\mathbb{R})\times\dot{H}^{\alpha+1}(\mathbb{R})$. The particular
time interval $[0,\frac{T}{\max{(\mu,\varepsilon)}}]$ comes from the estimates
of $r$ and $s$ in $W^{\alpha+7,1}(\mathbb{R})$ given by Lemma 3.23. Indeed,
due to the dispersive effects one has a loss of decrease of order $O((1+\mu
t)^{\theta})$ for any $\theta>1/2$, in the Sobolev spaces
$W^{\beta,1}(\mathbb{R})$ where $\beta\geq 0$.

  * •

The second equation of (1.9) is different from (1.8). One has to replace the
definition of $u^{-}$ in Proposition 1.6 by
$u^{-}=\frac{\sqrt{h}-1}{\varepsilon}-\frac{1}{2}\mathrm{F}_{\mu}^{-1}[v]$ to
get

$\displaystyle\partial_{t}u^{-}-\mathrm{F}_{\mu}[\partial_{x}u^{-}]-\frac{3\varepsilon}{2}u^{-}\partial_{x}u^{-}=\mu\varepsilon
R,$

when setting $u^{+}(0)=O(\mu)$.

  * •

In Theorem 1.14 we express the loss of regularity needed to pass rigorously
from the Whitham equations (1.9) to the water waves equations (1.1).

Using again Theorem 4.16 and Theorem 4.18 of [13], we get the following
corollary.

######  Corollary 1.16.

There exists $n\in\mathbb{N}^{*}$ such that for any $\alpha\geq 0$ and any
$(\mu,\varepsilon)\in\mathcal{A}$, we have what follows. Consider the Cauchy
problem for the water wave problem (1.1) with initial conditions
$(\zeta_{0},\psi_{0})\in
W^{\alpha+n,1}(\mathbb{R})\times\dot{W}^{\alpha+n+1,1}(\mathbb{R})$ satisfying
the non-cavitation assumption (1.3), and denote by $(\zeta,\psi)$ the
corresponding solution. There exists a solution $(r,s)$ of the Hamiltonian
system (1.9) which satisfies the following property: denote by
$(\zeta_{\operatorname{Wh}},\psi_{\operatorname{Wh}})$ the solution defined by
(1.10), then for all times $t\in[0,\frac{T}{\max{(\mu,\varepsilon)}}]$, one
has

$\displaystyle|(\zeta-\zeta_{\operatorname{Wh}},\psi-\psi_{\operatorname{Wh}})|_{H^{\alpha}\times\dot{H}^{\alpha+1}}\leq
C(\varepsilon^{2}+(\mu\varepsilon+\varepsilon^{2})t),$

where
$C,T^{-1}=C(\frac{1}{h_{\min}},\mu_{\max},|\zeta_{0}|_{W^{\alpha+n}},|\psi_{0}|_{\dot{W}^{\alpha+n+1}})$.
See (3.23) for the initial conditions $(r_{0},s_{0})$ associated with $(r,s)$.

###  1.3 Outline

In section 2 we prove Proposition 1.6 and Proposition 1.11. In subsection 2.1
we focus on the proof of Proposition 1.6 using symbolic calculus to
diagonalize system (1.5) (see remark 1.7). In subsection 2.2 we prove
Proposition 1.11 using a local well-posedness result (see Proposition 2.9) and
a stability result (see Proposition 2.13) on the diagonalized system (1.6).
Then we prove Proposition 1.12.

The section 3 is dedicated to the proof of Theorem 1.14. In subsection 3.8 we
do a first approximation of the water waves’ Hamiltonian by the one of a
specific Whitham-Boussinesq similar to (1.5) (see Proposition 3.2). Then we do
a simple change of unknowns separating the waves into a right and left front
(see Proposition 3.5). In subsection 3.2 we apply the generalised Birkhoff’s
algorithm for almost smooth functions to the Hamiltonian obtained in the
previous subsection. It gives an explicit transformation (3.13) allowing to
pass from the latter Hamiltonian to the one of a system composed of two
decoupled Whitham equations (see Property 3.20 and remark 3.22) at the desired
order of precision. In subsection 3.3 we prove the existence of solutions of
the two decoupled Whitham equations in the suitable Sobolev spaces
$W^{\alpha,1}(\mathbb{R})$ for every integer $\alpha\geq 0$ (see lemma 3.23).
Then we prove an intermediate theorem (see Theorem 3.24) which describes the
action of the transformation resulting from the generalised Birkhoff’s
algorithm on the two decoupled Whitham equations. At the end, we prove Theorem
1.14 and corollary 1.16.

##  2 Derivation of the Whitham equations from the Riemann invariants of a
Whitham-Boussinesq system

The goal of this section is to prove the Propositions 1.6 and 1.11.

We consider the following Whitham-Boussinesq equations.

$\displaystyle\begin{cases}\partial_{t}\zeta+\partial_{x}v+\varepsilon\partial_{x}(\zeta)v+\varepsilon\zeta\partial_{x}v=0,\\\
\partial_{t}v+\mathrm{F}_{\mu}^{2}[\partial_{x}\zeta]+\varepsilon
v\partial_{x}v=0,\end{cases}$ (2.1)

where
$v=\mathrm{F}_{\mu}^{2}[\partial_{x}\psi]=\frac{\tanh{(\sqrt{\mu}|D|)}}{\sqrt{\mu}|D|}[\partial_{x}\psi]$.
Using Proposition 1.15 of [10] we easily get the following proposition.

######  Proposition 2.1.

Let $\mu_{\max}>0$. There exists $n\in\mathbb{N}^{*}$ and $T>0$ such that for
all $\alpha\geq 0$ and $p=(\mu,\varepsilon)\in\mathcal{A}$ (see Definition
1.2), and for every solution $(\zeta,\psi)\in
C([0,\frac{T}{\varepsilon}];H^{\alpha+n}(\mathbb{R})\times\dot{H}^{\alpha+n+1}(\mathbb{R}))$
to the water waves equations (1.1) one has

$\displaystyle\begin{cases}\partial_{t}\zeta+\partial_{x}v+\varepsilon\partial_{x}(\zeta
v)=\mu\varepsilon R_{1},\\\
\partial_{t}v+\mathrm{F}_{\mu}^{2}[\partial_{x}\zeta]+\varepsilon
v\partial_{x}v=\mu\varepsilon R_{2},\end{cases}$ (2.2)

with $|R_{1}|_{H^{\alpha}},|R_{2}|_{H^{\alpha}}\leq
C(\frac{1}{h_{\min}},\mu_{\max},|\zeta|_{H^{\alpha+n}},|\psi|_{\dot{H}^{\alpha+n+1}})$.

We say that the water waves equations are consistent with the Whitham-
Boussinesq equations (2.1) at precision order $O(\mu\varepsilon)$.

######  Remark 2.2.

We only chose a Whitham-Boussinesq system fitted for the computations. There
exists a whole class of these systems. For example, one can add a regularizing
Fourier multiplier such as $\mathrm{F}_{\mu}^{2}$ on the nonlinear terms of
system (2.1) and still be precise at order $O(\mu\varepsilon)$. The system
thus obtained is

$\displaystyle\begin{cases}\partial_{t}\zeta+\partial_{x}v+\varepsilon\mathrm{F}_{\mu}^{2}\partial_{x}[\zeta
v]=0,\\\
\partial_{t}v+\mathrm{F}_{\mu}^{2}[\partial_{x}\zeta]+\frac{\varepsilon}{2}\mathrm{F}_{\mu}^{2}[v\partial_{x}v]=0.\end{cases}$
(2.3)

In [5] the authors proved the local well-posedness of (2.3). The reasoning of
this section is the same for any Whitham-Boussinesq system such as the latter.

The system (2.1) is the starting point of our reasoning. We will adapt the
method used to derive the inviscid Burgers equations from the Nonlinear
Shallow Water equations which use the Riemann invariants of the latter.

Throughout this section we will use Notation 1.10.

###  2.1 Diagonalization of the Whitham-Boussinesq equations

In this subsection we formally diagonalize the system (2.1) to get Proposition
1.6. Because of the consistency of the water waves equations with the system
(2.1), we can take $\zeta$ and $v$ solutions of the latter instead of taking
solutions of the water waves equations. For our purpose, we just need to prove
the following proposition.

######  Proposition 2.3.

Let $\mu_{\max}>0$, there exists $n\in\mathbb{N}$ such that for any
$\alpha\geq 0$ and
$(\zeta,v)\in\mathcal{C}([0,\frac{T}{\varepsilon}];H^{\alpha+n}(\mathbb{R})\times
H^{\alpha+n}(\mathbb{R}))$ solutions of system (2.1) satisfying the non-
cavitation hypothesis (1.3), the quantities

$\displaystyle
u^{+}=2\frac{\sqrt{h}-1}{\varepsilon}+\mathrm{F}_{\mu}^{-1}[v],\ \
u^{-}=-2\frac{\sqrt{h}-1}{\varepsilon}+\mathrm{F}_{\mu}^{-1}[v],$ (2.4)

where $h=1+\varepsilon\zeta$, satisfy

$\displaystyle\begin{cases}\partial_{t}u^{+}+(\varepsilon\frac{3u^{+}+u^{-}}{4}+1)\mathrm{F}_{\mu}\partial_{x}u^{+}=O(\mu\varepsilon),\\\
\partial_{t}u^{-}+(\varepsilon\frac{u^{+}+3u^{-}}{4}-1)\mathrm{F}_{\mu}\partial_{x}u^{-}=O(\mu\varepsilon).\end{cases}$
(2.5)

where $\mathrm{F}_{\mu}=\sqrt{\frac{\tanh{(\sqrt{\mu}|D|)}}{\sqrt{\mu}|D|}}$.

Before proving this result, we recall some definition and classical results we
need.

######  Definition 2.4.

Let $f(x)\in\mathcal{C}^{n}(\mathbb{R})$ with $n\in\mathbb{N^{*}}$ be such
that for any $0\leq\beta\leq n$, $|\partial_{x}^{\beta}f(x)|_{L^{\infty}}\leq
C_{\beta}$, where $C_{\beta}>0$ is a constant depending on $\beta$. Let also
$u(\xi)\in\mathcal{C}^{\infty}(\mathbb{R})$ be a smooth function such that
there exists $m\in\mathbb{Z}$ for which $\forall\beta\geq 0$,
$|\partial_{\xi}^{\beta}u(\xi)|\leq\widetilde{C}_{\beta}(1+|\xi|)^{m-|\beta|}$,
where $\widetilde{C}_{\beta}>0$ is a constant depending on $\beta$.

Then we define the operator $Op(fu)$ as follow: for any sufficiently regular
functions $v$ we have

$\displaystyle Op(fu)v(x)\colonequals f(x)\mathrm{u}(D)[v](x),$

where $\rm{u}(D)$ is the Fourier multiplier associated with the function $u$.

######  Remark 2.5.

The function
$F_{\mu}(\xi)=\sqrt{\frac{\tanh{(\sqrt{\mu}|\xi|)}}{\sqrt{\mu}|\xi|}}$
defining the Fourier multiplier $\mathrm{F}_{\mu}$ satisfies the condition in
the previous Definition 2.4. Idem for its inverse or its square.

######  Lemma 2.6.

Let $\mu_{\max}>0$. Let
$(\varepsilon,\mu)\in\mathcal{A}=\\{(\varepsilon,\mu),\ \ 0\leq\varepsilon\leq
1,\ \ 0\leq\mu\leq\mu_{\max}\\}$. Let $\mathrm{G}_{\mu}$ be a Fourier
multiplier for which there exists $n\in\mathbb{N}^{*}$ such that for any
$\beta\geq 0$ and $u\in H^{\beta+n}(\mathbb{R})$,
$|(\mathrm{G}_{\mu}-1)[u]|_{H^{\beta}}\lesssim\mu|u|_{H^{\beta+n}}$.

  * •

Let $\alpha\geq 0$. Let also $f$ be a function in $H^{\alpha+n}(\mathbb{R})$.
Then for any $u\in H^{\alpha+n}(\mathbb{R})$, the commutator
$[\mathrm{G}_{\mu},f]u\colonequals\mathrm{G}_{\mu}[fu]-f\mathrm{G}_{\mu}[u]$
satisfies

$\displaystyle|[\mathrm{G}_{\mu},f]u|_{H^{\alpha}}=|[\mathrm{G}_{\mu}-1,f]u|_{H^{\alpha}}\lesssim\mu|f|_{H^{\alpha+n}}|u|_{H^{\alpha+n}}.$

  * •

Let $\alpha\geq 0$. Let also $g$ be a function such that
$\frac{g-1}{\varepsilon}$ is uniformly bounded in $\varepsilon$ in
$H^{\alpha+n}(\mathbb{R})$. Then for any function $u\in
H^{\alpha+n}(\mathbb{R})$,

$\displaystyle|[\mathrm{G}_{\mu},g]u|_{H^{\alpha}}=|[\mathrm{G}_{\mu}-1,g-1]u|_{H^{\alpha}}\lesssim\mu\varepsilon|\frac{g-1}{\varepsilon}|_{H^{\alpha+n}}|u|_{H^{\alpha+n}}.$

###### Proof.

Use product estimates A.1. ∎

######  Remark 2.7.

The Fourier multiplier $\mathrm{F}_{\mu}$ satisfies the assumption of Lemma
2.6. Idem for its inverse or its square. The functions $h$, $\sqrt{h}$ and its
inverse satisfy the assumption on $g$.

###### Proof.

The point on the Fourier multipliers $\mathrm{F}_{\mu}$, its inverse and its
square is obvious. Idem for the function $h$. To prove the point on
$\sqrt{h}$, we just need to use composition estimates A.2 with
$G(x)=\sqrt{1+x}-1$.

We now deal with $\frac{1}{\sqrt{h}}$. We remark that
$\frac{1}{\sqrt{h}}-1=\frac{1-\sqrt{h}}{1+\sqrt{h}-1}$. Then we use quotient
estimates A.3 ($\sqrt{h}\geq\sqrt{h_{\min}}$ because $\zeta$ satisfies the
non-cavitation hypothesis (1.3)). We get

$\displaystyle|\frac{1}{\sqrt{h}}-1|_{H^{\alpha}}\leq
C(\frac{1}{\sqrt{h_{\min}}},|\sqrt{h}-1|_{H^{\max{(t_{0},\alpha)}}})|\sqrt{h}-1|_{H^{\alpha}}.$

∎

We now prove Proposition 2.3.

###### Proof.

We can write system (2.1) under the form

$\displaystyle\partial_{t}U+A(U)\partial_{x}U=0,$

where $U\colonequals\begin{pmatrix}\zeta\\\ v\end{pmatrix}$ and
$A(U)\colonequals\begin{pmatrix}\varepsilon v&h\\\
\mathrm{F}_{\mu}^{2}[\circ]&\varepsilon v\end{pmatrix}$, with $h\colonequals
1+\varepsilon\zeta$.

The symbol of operator $A(U)$ is $A(U,\xi)=\begin{pmatrix}\varepsilon v&h\\\
F_{\mu}^{2}(\xi)&\varepsilon v\end{pmatrix}$, $\xi\in\mathbb{R}$ being the
frequency variable. So that for any smooth functions $W=(w_{1},w_{2})^{T}$,
for all $x\in\mathbb{R}$ we have

$\displaystyle
A(U)W(t,x)=\operatorname{Op}(A(U,\xi))W(t,x)=\begin{pmatrix}\varepsilon
v(t,x)w_{1}(t,x)+h(t,x)w_{2}(t,x)\\\
\mathrm{F}_{\mu}^{2}[w_{1}](t,x)+\varepsilon v(t,x)w_{2}(t,x)\end{pmatrix}.$

Now, we diagonalize $A(U,\xi)$.

$\displaystyle\det(A(U,\xi)-\lambda I_{2})=(\varepsilon
v-\lambda)^{2}-F_{\mu}^{2}(\xi)h,$

so that

$\displaystyle A(U,\xi)=PDP^{-1},$

with

$\displaystyle
P=\begin{pmatrix}\frac{\sqrt{h}}{F_{\mu}(\xi)}&-\frac{\sqrt{h}}{F_{\mu}(\xi)}\\\
1&1\end{pmatrix},\ \
P^{-1}=\begin{pmatrix}\frac{F_{\mu}(\xi)}{2\sqrt{h}}&\frac{1}{2}\\\
-\frac{F_{\mu}(\xi)}{2\sqrt{h}}&\frac{1}{2}\end{pmatrix}\ \
\operatorname{and}\ \ D=\begin{pmatrix}\varepsilon v+F_{\mu}(\xi)\sqrt{h}&0\\\
0&\varepsilon v-F_{\mu}(\xi)\sqrt{h}\end{pmatrix}.$

Based on the symbolic calculus we apply
$\mathrm{F}_{\mu}^{-1}\operatorname{Op}(P^{-1})$ to the system:

$\mathrm{F}_{\mu}^{-1}\operatorname{Op}(P^{-1})\partial_{t}U+\mathrm{F}_{\mu}^{-1}\operatorname{Op}(P^{-1})\operatorname{Op}(A(U,\xi))\operatorname{Op}(PP^{-1})\partial_{x}U=0,\\\
\iff\mathrm{F}_{\mu}^{-1}\operatorname{Op}(P^{-1})\partial_{t}U+\mathrm{F}_{\mu}^{-1}\operatorname{Op}(D)\mathrm{F}_{\mu}\mathrm{F}_{\mu}^{-1}\operatorname{Op}(P^{-1})\partial_{x}U=R_{1},$
(2.6)

where

$\displaystyle R_{1}=$
$\displaystyle\mathrm{F}_{\mu}^{-1}(\operatorname{Op}(P^{-1}A(U,\xi))-\operatorname{Op}(P^{-1})\operatorname{Op}(A(U,\xi)))\partial_{x}U$
$\displaystyle+$
$\displaystyle\mathrm{F}_{\mu}^{-1}\operatorname{Op}(P^{-1}A(U,\xi))(\operatorname{Op}(P)\operatorname{Op}(P^{-1})-\operatorname{Op}(PP^{-1}))\partial_{x}U$
$\displaystyle+$
$\displaystyle\mathrm{F}_{\mu}^{-1}(\operatorname{Op}(D)-\operatorname{Op}(P^{-1}A(U,\xi))\operatorname{Op}(P))\operatorname{Op}(P^{-1})\partial_{x}U.$

We remark that for each operators $\mathrm{F}_{\mu}^{-1}$, $\partial_{x}$,
$\operatorname{Op}(P^{-1}A(U,\xi))$ and $\operatorname{Op}(P^{-1})$ there
exists $n\in\mathbb{N}$ such that for any $\beta\geq 0$, the operator is
bounded from $H^{\beta+n}(\mathbb{R})$ to $H^{\beta}(\mathbb{R})$.

######  Lemma 2.8.

$R_{1}$ is of order $O(\mu\varepsilon)$.

###### Proof.

There are three terms in $R_{1}$. We deal with each of those separately.

  * •

First term: for any $W=(w_{1},w_{2})\in H^{\beta}(\mathbb{R})\times
H^{\beta}(\mathbb{R})$ with $\beta\geq 0$,

$\displaystyle(\operatorname{Op}(P^{-1}A(U,\xi))-\operatorname{Op}(P^{-1})\operatorname{Op}(A(U,\xi)))W$
$\displaystyle=\frac{1}{2\sqrt{h}}\begin{pmatrix}-[\mathrm{F}_{\mu},\varepsilon
v]w_{1}-[\mathrm{F}_{\mu},h]w_{2}\\\ [\mathrm{F}_{\mu},\varepsilon
v]w_{1}+[\mathrm{F}_{\mu},h]w_{2}\end{pmatrix}.$

Using Lemma 2.6 and $h\geq h_{\min}$, we have

$\displaystyle\frac{1}{2\sqrt{h}}[\mathrm{F}_{\mu},\varepsilon
v]w_{1}=O(\mu\varepsilon)\ \ \operatorname{and}\ \
\frac{1}{2\sqrt{h}}[\mathrm{F}_{\mu},h]w_{2}=O(\mu\varepsilon).$

  * •

Second term: for any $W=(w_{1},w_{2})\in H^{\beta}(\mathbb{R})\times
H^{\beta}(\mathbb{R})$ with $\beta\geq 0$,

$\displaystyle(\operatorname{Op}(P)\operatorname{Op}(P^{-1})-I_{d})W=\begin{pmatrix}\sqrt{h}[\mathrm{F}_{\mu}^{-1},\frac{1}{\sqrt{h}}]\mathrm{F}_{\mu}[w_{1}]\\\
w_{2}\end{pmatrix}.$

Using Lemma 2.6 and the boundedness of $\mathrm{F}_{\mu}$ and $\sqrt{h}$ we
have

$\displaystyle\sqrt{h}[\mathrm{F}_{\mu}^{-1},\frac{1}{\sqrt{h}}]\mathrm{F}_{\mu}[w_{1}]=O(\mu\varepsilon).$

  * •

Third term: for any $W=(w_{1},w_{2})\in H^{\beta}(\mathbb{R})\times
H^{\beta}(\mathbb{R})$ with $\beta\geq 0$,

$(\operatorname{Op}(D)-\operatorname{Op}(P^{-1}A(U,\xi))\operatorname{Op}(P))W\\\
=\begin{pmatrix}-\frac{\varepsilon
v}{2\sqrt{h}}[\mathrm{F}_{\mu}-1,\sqrt{h}]\mathrm{F}_{\mu}^{-1}[w_{1}-w_{2}]-\frac{1}{2}[\mathrm{F}_{\mu}^{2},\sqrt{h}]\mathrm{F}_{\mu}^{-1}[w_{1}-w_{2}]\\\
\frac{\varepsilon
v}{2\sqrt{h}}[\mathrm{F}_{\mu}-1,\sqrt{h}]\mathrm{F}_{\mu}^{-1}[w_{1}-w_{2}]-\frac{1}{2}[\mathrm{F}_{\mu}^{2},\sqrt{h}]\mathrm{F}_{\mu}^{-1}[w_{1}-w_{2}]\end{pmatrix}$

Using Lemma 2.6, $h\geq h_{\min}$ and the boundedness of
$\mathrm{F}_{\mu}^{-1}$, we have

$\displaystyle\begin{cases}\frac{\varepsilon
v}{2\sqrt{h}}[\mathrm{F}_{\mu}-1,\sqrt{h}]\mathrm{F}_{\mu}^{-1}[w_{1}-w_{2}]=O(\mu\varepsilon),\\\
\frac{1}{2}[\mathrm{F}_{\mu}^{2},\sqrt{h}]\mathrm{F}_{\mu}^{-1}[w_{1}-w_{2}]=O(\mu\varepsilon).\end{cases}$

∎

We continue the proof of Proposition 2.3. We compute
$\mathrm{F}_{\mu}^{-1}\operatorname{Op}(P^{-1})\partial_{t}U$:

$\displaystyle\mathrm{F}_{\mu}^{-1}\operatorname{Op}(P^{-1})\partial_{t}U=\mathrm{F}_{\mu}^{-1}\begin{pmatrix}\frac{1}{2\sqrt{h}}\mathrm{F}_{\mu}[\partial_{t}\zeta]+\frac{\partial_{t}v}{2}\\\
-\frac{1}{2\sqrt{h}}\mathrm{F}_{\mu}[\partial_{t}\zeta]+\frac{\partial_{t}v}{2}\end{pmatrix}=\begin{pmatrix}\partial_{t}(\frac{1}{\varepsilon}\sqrt{h}+\frac{1}{2}\mathrm{F}_{\mu}^{-1}[v])\\\
\partial_{t}(-\frac{1}{\varepsilon}\sqrt{h}+\frac{1}{2}\mathrm{F}_{\mu}^{-1}[v])\end{pmatrix}+R_{2},$

where
$R_{2}=\mathrm{F}_{\mu}^{-1}\begin{pmatrix}-[\mathrm{F}_{\mu},\frac{1}{2\sqrt{h}}]\partial_{t}\zeta\\\
[\mathrm{F}_{\mu},\frac{1}{2\sqrt{h}}]\partial_{t}\zeta\end{pmatrix}$. Here,
to prove that $R_{2}$ is of order $O(\mu\varepsilon)$, in addition of Lemma
2.6 we need a control of $|\partial_{t}\zeta|_{H^{\beta}}$ for any $\beta\geq
0$. We get it using the first equation of (2.1) and product estimates A.1.

The same computations for
$\mathrm{F}_{\mu}^{-1}\operatorname{Op}(P^{-1})\partial_{x}U$ gives

$\displaystyle\mathrm{F}_{\mu}^{-1}\operatorname{Op}(P^{-1})\partial_{x}U=\begin{pmatrix}\partial_{x}(\frac{1}{\varepsilon}\sqrt{h}+\frac{1}{2}\mathrm{F}_{\mu}^{-1}[v])\\\
\partial_{x}(-\frac{1}{\varepsilon}\sqrt{h}+\frac{1}{2}\mathrm{F}_{\mu}^{-1}[v])\end{pmatrix}+R_{3},$

where
$R_{3}=\mathrm{F}_{\mu}^{-1}\begin{pmatrix}-[\mathrm{F}_{\mu},\frac{1}{2\sqrt{h}}]\partial_{x}\zeta\\\
[\mathrm{F}_{\mu},\frac{1}{2\sqrt{h}}]\partial_{x}\zeta\end{pmatrix}$ is of
order $O(\mu\varepsilon)$.

So that

$\mathrm{F}_{\mu}^{-1}\operatorname{Op}(D)\mathrm{F}_{\mu}\mathrm{F}_{\mu}^{-1}\operatorname{Op}(P^{-1})\partial_{x}U\\\
=\begin{pmatrix}\varepsilon\mathrm{F}_{\mu}^{-1}[v\mathrm{F}_{\mu}[\circ]]+\mathrm{F}_{\mu}^{-1}[\sqrt{h}\mathrm{F}_{\mu}^{2}[\circ]]&0\\\
0&\varepsilon\mathrm{F}_{\mu}^{-1}[v\mathrm{F}_{\mu}[\circ]]-\mathrm{F}_{\mu}^{-1}[\sqrt{h}\mathrm{F}_{\mu}^{2}[\circ]]\end{pmatrix}\\\
\times\begin{pmatrix}\partial_{x}(\frac{1}{\varepsilon}\sqrt{h}+\frac{1}{2}\mathrm{F}_{\mu}^{-1}[v])\\\
\partial_{x}(-\frac{1}{\varepsilon}\sqrt{h}+\frac{1}{2}\mathrm{F}_{\mu}^{-1}[v])\end{pmatrix}+R_{4},$
(2.7)

where
$R_{4}=\mathrm{F}_{\mu}^{-1}\operatorname{Op}(D)\mathrm{F}_{\mu}[R_{3}]=O(\mu\varepsilon)$
because of the boundedness of the operators
$\mathrm{F}_{\mu}^{-1},\operatorname{Op}(D)$ and $\mathrm{F}_{\mu}$.

Hence, we get

$\displaystyle\begin{cases}\partial_{t}(\frac{1}{\varepsilon}\sqrt{h}+\frac{1}{2}\mathrm{F}_{\mu}^{-1}[v])+\big{(}\varepsilon\mathrm{F}_{\mu}^{-1}[v\circ]+\mathrm{F}_{\mu}^{-1}[\sqrt{h}\mathrm{F}_{\mu}[\circ]]\big{)}\mathrm{F}_{\mu}\partial_{x}(\frac{1}{\varepsilon}\sqrt{h}+\frac{1}{2}\mathrm{F}_{\mu}^{-1}[v])=R_{5},\\\
\partial_{t}(-\frac{1}{\varepsilon}\sqrt{h}+\frac{1}{2}\mathrm{F}_{\mu}^{-1}[v])+\big{(}\varepsilon\mathrm{F}_{\mu}^{-1}[v\circ]-\mathrm{F}_{\mu}^{-1}[\sqrt{h}\mathrm{F}_{\mu}[\circ]]\big{)}\mathrm{F}_{\mu}\partial_{x}(-\frac{1}{\varepsilon}\sqrt{h}+\frac{1}{2}\mathrm{F}_{\mu}^{-1}[v])=R_{6},\end{cases}$

where $R_{5}$ and $R_{6}$ are combinations of $R_{1}$, $R_{2}$, $R_{3}$,
$R_{4}$. They are of order $O(\mu\varepsilon)$. See also that
$\frac{1}{\epsilon}\partial_{x}\sqrt{h}$ and
$\frac{1}{\epsilon}\partial_{x}\sqrt{h}$ are of order $O(1)$

But by Lemma 2.6, we know that for any $w\in H^{\beta}(\mathbb{R})$ with
$\beta\geq 0$, we have

$\displaystyle[\mathrm{F}_{\mu},\sqrt{h}]w=O(\mu\varepsilon)\ \
\operatorname{and}\ \ [\mathrm{F}_{\mu}^{-1},v]w=O(\mu).$

So

$\displaystyle\begin{cases}\partial_{t}(\frac{\sqrt{h}-1}{\varepsilon}+\frac{1}{2}\mathrm{F}_{\mu}^{-1}[v])+\big{(}\varepsilon\mathrm{F}_{\mu}^{-1}[v]+\sqrt{h}\big{)}\mathrm{F}_{\mu}\partial_{x}(\frac{\sqrt{h}-1}{\varepsilon}+\frac{1}{2}\mathrm{F}_{\mu}^{-1}[v])=R_{7},\\\
\partial_{t}(-\frac{\sqrt{h}-1}{\varepsilon}+\frac{1}{2}\mathrm{F}_{\mu}^{-1}[v])+\big{(}\varepsilon\mathrm{F}_{\mu}^{-1}[v]-\sqrt{h}\big{)}\mathrm{F}_{\mu}\partial_{x}(-\frac{\sqrt{h}-1}{\varepsilon}+\frac{1}{2}\mathrm{F}_{\mu}^{-1}[v])=R_{8},\end{cases}$

where $R_{7}$ and $R_{8}$ are of order $O(\mu\varepsilon)$.

We now denote

$\displaystyle
u^{+}=\frac{\sqrt{h}-1}{\varepsilon}+\frac{1}{2}\mathrm{F}_{\mu}^{-1}[v],\ \
u^{-}=-\frac{\sqrt{h}-1}{\varepsilon}+\frac{1}{2}\mathrm{F}_{\mu}^{-1}[v].$

We get from (2.6) and the above:

$\displaystyle\begin{cases}\partial_{t}u^{+}+(\varepsilon\frac{3u^{+}+u^{-}}{2}+1)\mathrm{F}_{\mu}\partial_{x}u^{+}=O(\mu\varepsilon),\\\
\partial_{t}u^{-}+(\varepsilon\frac{u^{+}+3u^{-}}{2}-1)\mathrm{F}_{\mu}\partial_{x}u^{-}=O(\mu\varepsilon).\end{cases}$

∎

###  2.2 From the diagonalized Whitham-Boussinesq equations to the Whitham
equations

In this subsection we prove Proposition 1.11.

To prove the first part of the proposition, i.e. there exists a time $T>0$
such that for all times $t\in[0,\frac{T}{\epsilon}]$, $u^{-}(t,\cdot)$ and
$u^{+}(t,\cdot)$ are of order $O(\mu)$, we need local well-posedness and
stability results on the exact diagonalized system

$\displaystyle\begin{cases}\partial_{t}u^{+}_{e}+(\varepsilon\frac{3u^{+}_{e}+u^{-}_{e}}{2}+1)\mathrm{F}_{\mu}\partial_{x}u^{+}_{e}=0,\\\
\partial_{t}u^{-}_{e}+(\varepsilon\frac{u^{+}_{e}+3u^{-}_{e}}{2}-1)\mathrm{F}_{\mu}\partial_{x}u^{-}_{e}=0.\end{cases}$
(2.8)

We begin by proving the local well-posedness of system (2.8).

######  Proposition 2.9.

(local existence) Let $(\mu,\epsilon)\in\mathcal{A}$ (see Definition 1.2). Let
$1/2<t_{0}\leq 1$, $\alpha\geq t_{0}+1$ and $u^{+}_{0,e},u^{-}_{0,e}\in
H^{\alpha}(\mathbb{R})$. Then there exists a time $T>0$ such that system (2.8)
admits a unique solution
$(u^{+}_{e},u^{-}_{e})\in\mathcal{C}([0,\frac{T}{\varepsilon}];H^{\alpha}(\mathbb{R})^{2})$
with initial conditions $(u^{+}_{0,e},u^{-}_{0,e})$.

Moreover $\frac{1}{T}$ and $\sup_{0\leq
t\leq\frac{T}{\varepsilon}}|u^{+}_{e}|_{H^{\alpha}}+|u^{-}_{e}|_{H^{\alpha}}$
can be estimated by $|u^{+}_{0,e}|_{H^{\alpha}}$,
$|u^{-}_{0,e}|_{H^{\alpha}}$, $t_{0}$, $\mu_{\max}$ (and are independent of
the parameters $(\mu,\varepsilon)\in\mathcal{A}$).

###### Proof.

System (2.8) is very similar to symmetric quasilinear hyperbolic systems. The
only difference is the presence of the non-local operator $\mathrm{F}_{\mu}$.
The well-posedness of such systems relies on the energy estimates (a priori
estimates). We will establish them here. For the rest of the proof, one can
follow the one in [14] where the author uses a method of regularisation
suitable for our system .

First, remark that if we define
$U_{e}(t)\colonequals\begin{pmatrix}u^{+}_{e}(t)\\\ u^{-}_{e}(t)\end{pmatrix}$
and

$\displaystyle
A(U_{e})\colonequals\begin{pmatrix}(\varepsilon\frac{3u^{+}_{e}+u^{-}_{e}}{2}+1)&0\\\
0&(\varepsilon\frac{u^{+}_{e}+3u^{-}_{e}}{2}-1)\end{pmatrix}.$ (2.9)

Then system (2.8) can be written under the form

$\displaystyle\partial_{t}U_{e}+A(U_{e})\mathrm{F}_{\mu}\partial_{x}U_{e}=0.$
(2.10)

It justifies the following framework for energy estimates. Let $\beta\geq 0$,
$R\in L^{\infty}([0,\frac{T}{\varepsilon}];H^{\beta}(\mathbb{R})^{2})$ and
$\underline{U}\in
L^{\infty}([0,\frac{T}{\varepsilon}];H^{\max{(\beta,t_{0}+1)}}(\mathbb{R})^{2})$.
Let
$U\in\mathcal{C}^{0}([0,\frac{T}{\varepsilon}];H^{\beta}(\mathbb{R})^{2})\cap\mathcal{C}^{1}([O,\frac{T}{\varepsilon}];H^{\beta+1}(\mathbb{R})^{2})$
solves the equation

$\displaystyle\partial_{t}U+A(\underline{U})\mathrm{F}_{\mu}\partial_{x}U=\varepsilon
R.$ (2.11)

######  Lemma 2.10.

(Energy estimates of order 0) For all times $t\in[0,\frac{T}{\varepsilon}]$,

$\displaystyle|U(t)|_{2}\leq
e^{\varepsilon\delta_{0}t}|U(0)|_{2}+\varepsilon\gamma_{0}\int_{0}^{t}|R(t^{\prime})|_{2}\operatorname{dt}^{\prime},$

where
$\delta_{0},\gamma_{0}=C(T,|\underline{U}|_{L^{\infty}([0,\frac{T}{\varepsilon}];H^{t_{0}+1})})$

###### Proof.

In what follows we will write $(\cdot,\cdot)_{2}$ the scalar product in
$L^{2}(\mathbb{R})^{2}$. Using the fact that $U$ solves equation (2.11) we get

$\displaystyle\frac{1}{2}\frac{\mathrm{d}}{\operatorname{dt}}|U|_{2}^{2}=(\partial_{t}U,U)_{2}$
$\displaystyle=-(A(\underline{U})\mathrm{F}_{\mu}\partial_{x}U,U)_{2}+\varepsilon(R,U)_{2}.$

But $A(\underline{U})$ is symmetric, so

$\displaystyle(A(\underline{U})\mathrm{F}_{\mu}\partial_{x}U,U)_{2}$
$\displaystyle=(\partial_{x}\mathrm{F}_{\mu}U,A(\underline{U})U)_{2}=-(\mathrm{F}_{\mu}U,(\partial_{x}A(\underline{U}))U)_{2}-(U,\mathrm{F}_{\mu}[A(\underline{U})\partial_{x}U])_{2}$
$\displaystyle=(\mathrm{F}_{\mu}U,(\partial_{x}A(\underline{U}))U)_{2}-(A(\underline{U})\mathrm{F}_{\mu}\partial_{x}U,U)_{2}-([\mathrm{F}_{\mu},A(\underline{U})]\partial_{x}U,U)_{2},$

where
$\partial_{x}A(\underline{U})=\begin{pmatrix}\varepsilon\frac{3\partial_{x}\underline{u_{1}}+\partial_{x}\underline{u_{2}}}{2}&0\\\
0&\varepsilon\frac{\partial_{x}\underline{u_{1}}+3\partial_{x}\underline{u_{2}}}{2}\end{pmatrix}$
and

$\displaystyle[\mathrm{F}_{\mu},A(\underline{U})]=\begin{pmatrix}[\mathrm{F}_{\mu},\varepsilon\frac{3\underline{u}_{1}+\underline{u}_{2}}{2}+1]&0\\\
0&[\mathrm{F}_{\mu},\varepsilon\frac{\underline{u}_{1}+3\underline{u}_{2}}{2}-1]\end{pmatrix}=\varepsilon\begin{pmatrix}[\mathrm{F}_{\mu},\frac{3\underline{u}_{1}+\underline{u}_{2}}{2}]&0\\\
0&[\mathrm{F}_{\mu},\frac{\underline{u}_{1}+3\underline{u}_{2}}{2}]\end{pmatrix}.$

Hence

$\displaystyle\frac{1}{2}\frac{\mathrm{d}}{\operatorname{dt}}|U|_{2}^{2}=\frac{1}{2}(\mathrm{F}_{\mu}U,(\partial_{x}A(\underline{U}))U)_{2}+\frac{1}{2}([\mathrm{F}_{\mu},A(\underline{U})]\partial_{x}U,U)_{2}+\varepsilon(R,U)_{2}.$
(2.12)

Then using Cauchy-Schwarz inequality we get

$\displaystyle|U|_{2}\frac{\mathrm{d}}{\operatorname{dt}}|U|_{2}\leq\frac{1}{2}|\partial_{x}(A(\underline{U}))U|_{2}|U|_{2}+\frac{1}{2}|[\mathrm{F}_{\mu},A(\underline{U})]\partial_{x}U|_{2}|U|_{2}+\varepsilon|R|_{2}|U|_{2}$
$\displaystyle\iff$
$\displaystyle\frac{\mathrm{d}}{\operatorname{dt}}|U|_{2}\leq\frac{1}{2}|\partial_{x}(A(\underline{U}))U|_{2}+\frac{1}{2}|[\mathrm{F}_{\mu},A(\underline{U})]\partial_{x}U|_{2}+\varepsilon|R|_{2}.$

But by product estimates A.1 we have

$\displaystyle|\partial_{x}(A(\underline{U}))U|_{2}\leq\varepsilon
C(|\underline{U}|_{L^{\infty}_{t}H^{t_{0}+1}_{x}})|U|_{2}.$ (2.13)

Moreover by commutator estimates A.5 we also have

$\displaystyle|[\mathrm{F}_{\mu},A(\underline{U})]\partial_{x}U|_{2}\leq\varepsilon
C(|\underline{U}|_{L^{\infty}_{t}H^{t_{0}+1}_{x}})|U|_{2}.$ (2.14)

So that

$\displaystyle\frac{\mathrm{d}}{\operatorname{dt}}|U|_{2}$
$\displaystyle\leq\varepsilon
C(|\underline{U}|_{L^{\infty}_{t}H^{t_{0}+1}_{x}})|U|_{2}+\varepsilon|R|_{2}\leq\varepsilon\delta_{0}|U|_{2}+\varepsilon|R|_{2},$

where $\delta_{0}=C(|\underline{U}|_{L^{\infty}_{t}H^{t_{0}+1}_{x}})$.

We integrate this inequality in time

$\displaystyle|U|_{2}$ $\displaystyle\leq
e^{\varepsilon\delta_{0}t}|U(0)|_{2}+\varepsilon\int_{0}^{t}e^{\varepsilon\delta_{0}(t-t^{\prime})}|R|_{2}\operatorname{dt}^{\prime}$
$\displaystyle\leq e^{\varepsilon\delta_{0}t}|U(0)|_{2}+\varepsilon
e^{\delta_{0}T}\int_{0}^{t}|R|_{2}\operatorname{dt}^{\prime}.$

It is the thesis with $\gamma_{0}=e^{\delta_{0}T}$.

∎

Using this lemma, we get the energy estimates of order $\beta$.

######  Lemma 2.11.

(Energy estimates of order $\beta$) For all times
$t\in[0,\frac{T}{\varepsilon}]$,

$\displaystyle|U(t)|_{H^{\beta}}\leq
e^{\varepsilon\delta_{\beta}t}|U(0)|_{H^{\beta}}+\varepsilon\gamma_{\beta}\int_{0}^{t}|R(t^{\prime})|_{H^{\beta}}\operatorname{dt}^{\prime},$

where
$\delta_{\beta},\gamma_{\beta}=C(T,|\underline{U}|_{L^{\infty}([0,\frac{T}{\varepsilon}];H^{\max{(\beta,t_{0}+1)}})})$.

###### Proof.

Let $\Lambda^{\beta}\colonequals(1-\partial_{x}^{2})^{\beta/2}$. Then
$|\Lambda^{\beta}U|_{2}$ is equivalent to $|U|_{H^{\beta}}$. Applying
$\Lambda^{\beta}$ to (2.11) we have

$\displaystyle\Lambda^{\beta}\partial_{t}U+\Lambda^{\beta}[A(\underline{U})\mathrm{F}_{\mu}\partial_{x}U]=\varepsilon\Lambda^{\beta}R$
$\displaystyle\iff$
$\displaystyle\partial_{t}\Lambda^{\beta}U+A(\underline{U})\mathrm{F}_{\mu}\partial_{x}\Lambda^{\beta}U=\varepsilon\Lambda^{\beta}R+\varepsilon
B_{\beta}(\underline{U})\mathrm{F}_{\mu}\partial_{x}U,$

where

$\displaystyle
B_{\beta}(\underline{U})\colonequals\frac{1}{\varepsilon}\begin{pmatrix}[\Lambda^{\beta},\varepsilon\frac{3\underline{u}_{1}+\underline{u}_{2}}{2}+1]&0\\\
0&[\Lambda^{\beta},\varepsilon\frac{\underline{u}_{1}+3\underline{u}_{2}}{2}-1]\end{pmatrix}=\begin{pmatrix}[\Lambda^{\beta},\frac{3\underline{u}_{1}+\underline{u}_{2}}{2}]&0\\\
0&[\Lambda^{\beta},\frac{\underline{u}_{1}+3\underline{u}_{2}}{2}]\end{pmatrix}.$

Using commutator estimates A.5 and the boundedness of $\mathrm{F}_{\mu}$ we
get

$\displaystyle|B_{\beta}(\underline{U})\mathrm{F}_{\mu}\partial_{x}U|_{2}\lesssim|\underline{U}|_{L^{\infty}_{t}H^{\max{(t_{0}+1,\beta)}}_{x}}|U|_{H^{\beta}}.$

Then we use the energy estimates of order 0 (see Lemma 2.10) to get

$\displaystyle|U|_{H^{\beta}}$ $\displaystyle\leq
e^{\varepsilon\delta_{0}t}|U(0)|_{H^{\beta}}+\varepsilon\gamma_{0}\int_{0}^{t}|R|_{H^{\beta}}+\varepsilon\gamma_{0}\int_{0}^{t}|B_{\beta}(\underline{U})\partial_{x}U|_{2}\operatorname{dt}^{\prime}$
$\displaystyle\lesssim
e^{\varepsilon\delta_{0}t}|U(0)|_{H^{\beta}}+\varepsilon
C(T,|\underline{U}|_{L^{\infty}_{t}H^{\max{(t_{0}+1,\beta)}}_{x}})\int_{0}^{t}|U|_{H^{\beta}}\operatorname{dt}^{\prime}+\varepsilon\gamma_{0}\int_{0}^{t}|R|_{H^{\beta}}\operatorname{dt}^{\prime}.$

It only remains to use Grönwall’s lemma to get the result. ∎

This concludes the proof of Proposition 2.9. ∎

######  Corollary 2.12.

Let $t_{0}>1/2$ and $\alpha\geq t_{0}+1$. Let
$U_{e}=\begin{pmatrix}u_{e}^{+}\\\
u_{e}^{-}\end{pmatrix}\in\mathcal{C}([0,\frac{T}{\varepsilon}],H^{\alpha}(\mathbb{R})^{2})$
solves the exact diagonalized system (2.8) with $u^{-}_{e}(0)=O(\mu)$, then
for all times $[0,\frac{T}{\varepsilon}]$, $u^{-}_{e}=O(\mu)$. If instead we
take $u^{+}_{e}(0)=O(\mu)$ then for all times $[0,\frac{T}{\varepsilon}]$,
$u^{+}_{e}=O(\mu)$.

###### Proof.

Using energy estimates of Lemma 2.11 on the equation on $u^{-}_{e}$ of system
(2.8), we get

$\displaystyle|u_{e}^{-}|_{H^{\alpha}}\leq
C(T,|U_{e}|_{L^{\infty}_{t}H^{\alpha}_{x}})|u_{e}^{-}(0)|_{H^{\alpha}}.$

It gives the result on $u_{e}^{-}$. We do the same for $u_{e}^{+}$ with the
other equation of (2.8). ∎

######  Proposition 2.13.

(Stability) Let $t_{0}>1/2$ and $\alpha\geq t_{0}+1$. Let
$U_{e}=\begin{pmatrix}u_{e}^{+}\\\
u_{e}^{-}\end{pmatrix}\in\mathcal{C}([0,\frac{T}{\varepsilon}],H^{\alpha}(\mathbb{R})^{2})$
solve the exact diagonalized system (2.8) and $U=\begin{pmatrix}u^{+}\\\
u^{-}\end{pmatrix}\in\mathcal{C}([0,\frac{T}{\varepsilon}],H^{\alpha}(\mathbb{R})^{2})$,
defined from sufficiently regular solutions of the water waves equations
satisfying the non-cavitation hypothesis (1.3) through the formulas (2.4). By
Proposition 2.3, we have

$\displaystyle\begin{cases}\partial_{t}U_{e}+A(U_{e})\mathrm{F}_{\mu}\partial_{x}U_{e}=0,\\\
\partial_{t}U+A(U)\mathrm{F}_{\mu}\partial_{x}U=\mu\varepsilon R,\end{cases}$
(2.15)

where $A$ is defined by (2.9) and $R\in
L^{\infty}([0,\frac{T}{\varepsilon}],H^{\alpha}(\mathbb{R})^{2})$ is uniformly
bounded in $(\mu,\varepsilon)\in\mathcal{A}$.

Then for all times $t\in[0,\frac{T}{\varepsilon}]$, we have

$\displaystyle|U_{e}-U|_{H^{\alpha}}\leq
c_{\alpha}(|U_{e}(0)-U(0)|_{H^{\alpha}}+\mu\varepsilon
t|R|_{L^{\infty}_{t}H^{\alpha}_{x}}),$ (2.16)

where
$c_{\alpha}=C(T,|U_{e}|_{L^{\infty}_{t}H^{\alpha}_{x}},|U|_{L^{\infty}_{t}H^{\alpha+1}_{x}})$.

In particular, if we take $U_{e}(0)=U(0)$ and $u^{-}(0)=O(\mu)$, then for all
times $t\in[0,\frac{T}{\epsilon}]$,

$\displaystyle u^{-}=O(\mu).$

If instead, we take $u^{+}(0)=O(\mu)$, then for all times
$t\in[0,\frac{T}{\epsilon}]$,

$\displaystyle u^{+}=O(\mu).$

###### Proof.

First we subtract the two equations of (2.15), we get

$\displaystyle\partial_{t}(U_{e}-U)+A(U_{e})\mathrm{F}_{\mu}\partial_{x}(U_{e}-U)=-\varepsilon(\mu
R+\frac{1}{\epsilon}(A(U_{e})-A(U))\mathrm{F}_{\mu}\partial_{x}U).$

We use the estimates of order $\alpha$ from Lemma 2.11

$\displaystyle|U_{e}-U|_{H^{\alpha}}\leq$ $\displaystyle
e^{\varepsilon\delta_{\alpha}t}|U_{e}(0)-U(0)|_{H^{\alpha}}+\varepsilon\gamma_{\alpha}\int_{0}^{t}|\mu
R(t^{\prime})|_{H^{\alpha}}\operatorname{dt}^{\prime}$
$\displaystyle+\varepsilon\gamma_{\alpha}\int_{0}^{t}\frac{1}{\epsilon}|(A(U_{e})-A(U))\mathrm{F}_{\mu}\partial_{x}U|_{H^{\alpha}}\operatorname{dt}^{\prime}.$

But using product estimates A.1, the algebra properties of
$H^{\alpha}(\mathbb{R})$ and the boundedness of $\mathrm{F}_{\mu}$, we have

$\displaystyle|(A(U_{e})-A(U))\mathrm{F}_{\mu}\partial_{x}U|_{H^{\alpha}}\lesssim\epsilon|U_{e}-U|_{H^{\alpha}}|U|_{H^{\alpha+1}}.$

So that using Grönwall’s lemma we get

$\displaystyle|U_{e}-U|_{H^{\alpha}}$
$\displaystyle\leq\Big{[}e^{\varepsilon\delta_{\alpha}t}|U_{e}(0)-U(0)|_{H^{\alpha}}+\varepsilon\gamma_{\alpha}\int_{0}^{t}|\mu
R(t^{\prime})|_{H^{\alpha}}\operatorname{dt}^{\prime}\Big{]}e^{\varepsilon\gamma_{\alpha}\int_{0}^{t}|U|_{H^{\alpha+1}}\operatorname{dt}^{\prime}}$
$\displaystyle\leq
C(T,|U_{e}|_{L^{\infty}_{t}H^{\alpha}_{x}})(|U_{e}(0)-U(0)|_{H^{\alpha}}+\mu\varepsilon
t|R|_{L^{\infty}_{t}H^{\alpha}_{x}})e^{\varepsilon\gamma_{\alpha}t|U|_{L^{\infty}_{t}H^{\alpha+1}_{x}}},$

which is the first part of the result.

Now if we suppose $U_{e}(0)=U(0)$ and $u^{-}(0)=O(\mu)$, then for all times
$t\in[0,\frac{T}{\epsilon}]$, we have

$\displaystyle|u^{-}|_{H^{\alpha}}\leq|u_{e}^{-}-u^{-}|_{H^{\alpha}}+|u_{e}^{-}|_{H^{\alpha}}\leq\mu
c_{\alpha}T|R|_{L^{\infty}_{t}H^{\alpha}_{x}}+|u_{e}^{-}|_{H^{\alpha}}.$

Using Corollary 2.12 we get the result. We do the same if $u^{+}(0)=O(\mu)$. ∎

We now have all the elements to prove Proposition 1.11.

###### Proof.

Supposing $u^{-}(0)=O(\mu)$, we have for all times
$t\in[0,\frac{T}{\varepsilon}]$, $\varepsilon u^{-}(t)=O(\mu\varepsilon)$ (see
Proposition 2.13). But $u^{-}$ and $u^{+}$ solve the first equation of (2.5).
So

$\displaystyle\partial_{t}u^{+}+\mathrm{F}_{\mu}\partial_{x}u^{+}+\frac{3\varepsilon}{2}u^{+}\partial_{x}u^{+}=O(\mu\varepsilon).$
(2.17)

The last equivalence coming from product estimates A.1 and
$|(\mathrm{F}_{\mu}-1)[u]|_{H^{\beta}}\lesssim\mu|u|_{H^{\beta}}$ for any
$\beta\geq 0$ and $u$ in $H^{\beta+2}(\mathbb{R})$.

Supposing instead $u^{+}(0)=O(\mu)$, we have for all times
$t\in[0,\frac{T}{\varepsilon}]$, $\varepsilon u^{+}(t)=O(\mu\varepsilon)$.
Then, the second equation of (2.5) gives

$\displaystyle\partial_{t}u^{-}-\mathrm{F}_{\mu}\partial_{x}u^{-}+\frac{3\varepsilon}{2}u^{-}\partial_{x}u^{-}=O(\mu\varepsilon).$

∎

It only remains to prove Proposition 1.12.

###### Proof.

Let $U_{e}=\begin{pmatrix}u_{e}^{+}\\\
u_{e}^{-}\end{pmatrix}\in\mathcal{C}([0,\frac{T}{\varepsilon}],H^{\alpha}(\mathbb{R})^{2})$
solve the system (2.8), with the initial data $u^{-}_{e,0}=O(\mu)$ and
$u^{+}_{e,0}$, defined by $(\zeta_{0},\psi_{0})$ through the formulas (1.4).
By the energy estimates of Lemma 2.11 applied on the second equation of system
(2.8), we get for all times $t\in[0,\frac{T}{\varepsilon}]$,

$\displaystyle|u^{-}_{e}|_{H^{\alpha}}\leq C|u_{e}^{-}(0)|_{H^{\alpha}},$

where
$C,T^{-1}=C(\frac{1}{h_{\min}},\mu_{\max},|\zeta_{0}|_{H^{\alpha+n}},|\psi_{0}|_{\dot{H}^{\alpha+n+1}})$.
So that
$u^{+}\in\mathcal{C}([0,\frac{T}{\varepsilon}],H^{\alpha}(\mathbb{R}))$,
solution of the Whitham equation (which is well-posed in the Sobolev space
$H^{\alpha}(\mathbb{R})$ [6])

$\displaystyle\partial_{t}u^{+}+\mathrm{F}_{\mu}\partial_{x}u^{+}+\frac{3\varepsilon}{2}u^{+}\partial_{x}u^{+}=0,$

satisfies (see (2.17))

$\displaystyle\partial_{t}u^{+}+(\varepsilon\frac{3u^{+}+u^{-}_{e}}{2}+1)\mathrm{F}_{\mu}\partial_{x}u^{+}=\mu\varepsilon
R,$

where $R$ is uniformly bounded in $(\mu,\varepsilon)$ in
$H^{\alpha}(\mathbb{R})$. Using the stability estimates (2.16), with
$U=\begin{pmatrix}u^{+}\\\ u^{-}_{e}\end{pmatrix}$ and $U_{e}(0)=U(0)$, we
have for all times $t\in[0,\frac{T}{\varepsilon}]$,

$\displaystyle|u^{+}_{e}-u^{+}|_{H^{\alpha}}\leq C\mu\varepsilon t,$

where
$C,T^{-1}=C(\frac{1}{h_{\min}},\mu_{\max},|\zeta_{0}|_{H^{\alpha+n}},|\psi_{0}|_{\dot{H}^{\alpha+n+1}})$.

Thus by Corollary 1.8 we have for all times $t\in[0,\frac{T}{\varepsilon}]$,

$\displaystyle|(\zeta-\zeta_{\operatorname{Wh},+},\psi-\psi_{\operatorname{Wh},+})|_{H^{\alpha}\times\dot{H}^{\alpha+1}}\leq$
$\displaystyle|(\zeta-\zeta_{\rm{c}},\psi-\psi_{\rm{c}})|_{H^{\alpha}\times\dot{H}^{\alpha+1}}$
$\displaystyle+$
$\displaystyle|(\zeta_{\rm{c}}-\zeta_{\operatorname{Wh},+},\psi_{\rm{c}}-\psi_{\operatorname{Wh},+})|_{H^{\alpha}\times\dot{H}^{\alpha+1}}$
$\displaystyle\leq$ $\displaystyle C(|u_{e}^{-}|_{H^{\alpha+1}}+\mu\varepsilon
t)$ $\displaystyle\leq$ $\displaystyle
C(|u_{e,0}^{-}|_{H^{\alpha+1}}+\mu\varepsilon t),$

where
$C,T^{-1}=C(\frac{1}{h_{\min}},\mu_{\max},|\zeta_{0}|_{H^{\alpha+n}},|\psi_{0}|_{\dot{H}^{\alpha+n+1}})$.

The reasoning is the same if instead $u_{e}^{+}(0)=O(\mu)$. ∎

##  3 Decoupling the water waves equations into two Whitham equations

The goal of this section is to prove Theorem 1.14. In the previous section, we
proved the consistency (see Proposition 2.1 for the definition) of the water
waves equations with the Whitham equations at a precision order
$O(\mu\varepsilon)$ in the shallow water regime
$\mathcal{A}\colonequals\\{0\leq\mu\leq\mu_{\max},0\leq\varepsilon\leq 1\\}$
(see Proposition 2.3). For that purpose we made a restrictive hypothesis on
the initial conditions. We supposed either $u^{-}(0)=O(\mu)$ or
$u^{+}(0)=O(\mu)$ (see Notation 1.10). In this section we use a generalisation
of Birkhoff’s normal form algorithm for almost smooth Hamiltonians, introduced
by Bambusi in [1], on the water waves Hamiltonian. It will allow us to
decouple the water waves equations into two Whitham equations satisfied by the
two front waves, at a precision order $O(\mu\varepsilon+\varepsilon^{2})$ in
the shallow water regime, without any assumption of smallness on the initial
conditions.

We will only recall the elements of Bambusi’s theory which are useful to
understand our reasoning. For more details see [1].

In this section we always suppose the parameters $(\mu,\varepsilon)$ in
$\mathcal{A}$. Moreover every $\alpha$ will be a positive integer.

###  3.1 From the water waves equations to a Whitham-Boussinesq system

The starting point is the Hamiltonian of the water waves system

$\displaystyle
H_{\operatorname{WW}}=\frac{1}{2}\int_{\mathbb{R}}\zeta^{2}\operatorname{dx}+\frac{1}{2}\int_{\mathbb{R}}\psi\frac{1}{\mu}\mathcal{G}^{\mu}[\varepsilon\zeta]\psi\operatorname{dx}.$
(3.1)

We use the first shallow water estimates of Proposition 1.8 in [10], which we
recall here for the sake of clarity.

######  Proposition 3.1.

Let $\alpha\geq 0$, and $\zeta\in H^{\alpha+4}(\mathbb{R})$ be such that (1.3)
is satisfied. Let $\psi\in\dot{H}^{\alpha+3}(\mathbb{R})$, then

$\displaystyle|\frac{1}{\mu}\mathcal{G}^{\mu}[\varepsilon\zeta]\psi+\partial_{x}(h\mathrm{F}_{\mu}^{2}[\nabla\psi])|_{H^{\alpha}}\leq\mu\varepsilon
M(\alpha+4)|\psi|_{\dot{H}^{\alpha+4}},$

where $h=1+\varepsilon\zeta$ and for any $\beta\geq 2$, $M(\beta)\colonequals
C(\frac{1}{h_{\min}},\mu_{\max},|\zeta|_{H^{\beta}})$.

######  Proposition 3.2.

Let $H_{\operatorname{WW}}$ be the Hamiltonian of the water waves equations
(3.1). Then for any $\alpha\geq 0$ and any $(\zeta,\psi)\in
H^{\alpha+4}(\mathbb{R})\times\dot{H}^{\alpha+4}(\mathbb{R})$ we have

$\displaystyle|J\nabla H_{\operatorname{WW}}-J\nabla(H_{0}+\varepsilon
H_{1})|_{H^{\alpha}\times\dot{H}^{\alpha+1}}<\mu\varepsilon
M(\alpha+4)|\psi|_{\dot{H}^{\alpha+4}},$ (3.2)

where $J=\begin{pmatrix}0&1\\\ -1&0\end{pmatrix}$,
$H_{0}\colonequals\frac{1}{2}\int_{\mathbb{R}}\zeta^{2}\operatorname{dx}+\frac{1}{2}\int_{\mathbb{R}}(\mathrm{F}_{\mu}[\partial_{x}\psi])^{2}\operatorname{dx}$
and
$H_{1}\colonequals\frac{1}{2}\int_{\mathbb{R}}\zeta(\mathrm{F}_{\mu}[\partial_{x}\psi])^{2}\operatorname{dx}$.

###### Proof.

We defined $H_{\operatorname{WW}}$ as the Hamiltonian of the water waves
equations, so

$\displaystyle J\nabla
H_{\operatorname{WW}}=\begin{pmatrix}\frac{1}{\mu}\mathcal{G}^{\mu}[\varepsilon\zeta]\psi\\\
-\zeta-\frac{\varepsilon}{2}(\partial_{x}\psi)^{2}+\frac{\mu\varepsilon}{2}\frac{(\frac{1}{\mu}\mathcal{G}^{\mu}[\varepsilon\zeta]\psi+\varepsilon\partial_{x}\zeta\partial_{x}\psi)^{2}}{1+\varepsilon^{2}\mu(\partial_{x}\zeta)^{2}}\end{pmatrix}.$
(3.3)

We can easily compute $J\nabla(H_{0}+\varepsilon H_{1})$, we get

$\displaystyle J\nabla(H_{0}+\varepsilon
H_{1})=\begin{pmatrix}-\mathrm{F}_{\mu}^{2}[\partial_{x}^{2}\psi]-\mathrm{F}_{\mu}[\partial_{x}(\varepsilon\zeta\mathrm{F}_{\mu}[\partial_{x}\psi])]\\\
-\zeta-\frac{\varepsilon}{2}(\mathrm{F}_{\mu}[\partial_{x}\psi])^{2}\end{pmatrix}.$
(3.4)

We compare the first term of (3.3) and (3.4).

$\displaystyle|\frac{1}{\mu}\mathcal{G}^{\mu}[\varepsilon\zeta]\psi+\mathrm{F}_{\mu}^{2}[\partial_{x}^{2}\psi]+\mathrm{F}_{\mu}[\partial_{x}(\varepsilon\zeta\mathrm{F}_{\mu}[\partial_{x}\psi])]|_{H^{\alpha}}$
$\displaystyle\leq$
$\displaystyle|\frac{1}{\mu}\mathcal{G}^{\mu}[\varepsilon\zeta]\psi+\partial_{x}(h\mathrm{F}_{\mu}^{2}[\partial_{x}\psi])|_{H^{\alpha}}+|(\mathrm{F}_{\mu}-1)[\partial_{x}(\varepsilon\zeta\mathrm{F}_{\mu}[\partial_{x}\psi])]|_{H^{\alpha}}+|\partial_{x}(\varepsilon\zeta(\mathrm{F}_{\mu}-1)\mathrm{F}_{\mu}[\partial_{x}\psi])|_{H^{\alpha}}.$

Hence, using Proposition 3.1, the fact that for any $u\in
H^{\alpha+2}(\mathbb{R})$,
$|(\mathrm{F}_{\mu}-1)[u]|_{H^{\alpha}}\lesssim\mu|u|_{H^{\alpha+2}}$ and
product estimates A.1, we get

$\displaystyle|\frac{1}{\mu}\mathcal{G}^{\mu}[\varepsilon\zeta]\psi+\mathrm{F}_{\mu}^{2}[\partial_{x}^{2}\psi]+\mathrm{F}_{\mu}[\partial_{x}(\varepsilon\zeta\mathrm{F}_{\mu}[\partial_{x}\psi])]|_{H^{\alpha}}\lesssim\mu\varepsilon
M(\alpha+4)|\psi|_{\dot{H}^{\alpha+4}}.$

Then we compare the second term of (3.3) and (3.4). We first remark that there
is a part of (3.3) which is clearly of order $O(\mu\varepsilon)$, we start by
estimate it using quotient estimates A.3, product estimates A.1 and
Proposition A.6

$\displaystyle|\frac{(\frac{1}{\mu}\mathcal{G}^{\mu}[\varepsilon\zeta]\psi+\varepsilon\partial_{x}\zeta\partial_{x}\psi)^{2}}{1+\varepsilon^{2}\mu(\partial_{x}\zeta)^{2}}|_{\dot{H}^{\alpha+1}}\leq$
$\displaystyle|\frac{(\frac{1}{\mu}\mathcal{G}^{\mu}[\varepsilon\zeta]\psi+\varepsilon\partial_{x}\zeta\partial_{x}\psi)^{2}}{1+\varepsilon^{2}\mu(\partial_{x}\zeta)^{2}}|_{H^{\alpha+1}}$
$\displaystyle\leq$ $\displaystyle
C(\mu_{\max},|(\partial_{x}\zeta)^{2}|_{H^{\alpha+1}})|(\frac{1}{\mu}\mathcal{G}^{\mu}[\varepsilon\zeta]\psi+\varepsilon\partial_{x}\zeta\partial_{x}\psi)^{2}|_{H^{\alpha+1}}$
$\displaystyle\leq$ $\displaystyle
C(\mu_{\max},|\zeta|_{H^{\alpha+2}})|\frac{1}{\mu}\mathcal{G}^{\mu}[\varepsilon\zeta]\psi+\varepsilon\partial_{x}\zeta\partial_{x}\psi|_{H^{\alpha+1}}^{2}$
$\displaystyle\leq$ $\displaystyle M(\alpha+3)|\psi|_{\dot{H}^{\alpha+3}}.$

Then, it only remains to see that using product estimates A.1 and
$|(\mathrm{F}_{\mu}-1)[u]|_{H^{\alpha}}\lesssim\mu|u|_{H^{\alpha+2}}$ for any
$u\in H^{\alpha+2}(\mathbb{R})$, we have

$\displaystyle|(\partial_{x}\psi)^{2}-(\mathrm{F}_{\mu}[\partial_{x}\psi])^{2}|_{\dot{H}^{\alpha+1}}\leq$
$\displaystyle|(\partial_{x}\psi)^{2}-(\mathrm{F}_{\mu}[\partial_{x}\psi])^{2}|_{H^{\alpha+1}}$
$\displaystyle\leq$
$\displaystyle|\partial_{x}\psi-\mathrm{F}_{\mu}[\partial_{x}\psi]|_{H^{\alpha+1}}|\partial_{x}\psi+\mathrm{F}_{\mu}[\partial_{x}\psi]|_{H^{\alpha+1}}\lesssim\mu|\psi|_{\dot{H}^{\alpha+4}}|\psi|_{\dot{H}^{\alpha+2}}.$

This ends the proof. ∎

######  Remark 3.3.

  * •

When deriving the Hamilton’s equations associated with $H_{0}+\varepsilon
H_{1}$ we get the following Whitham-Boussinesq system

$\displaystyle\begin{cases}\partial_{t}\zeta+\mathrm{F}_{\mu}^{2}[\partial_{x}^{2}\psi]+\varepsilon\mathrm{F}_{\mu}\partial_{x}[\zeta\mathrm{F}_{\mu}[\partial_{x}\psi]]=0,\\\
\partial_{t}\psi+\zeta+\frac{\varepsilon}{2}(\mathrm{F}_{\mu}[\partial_{x}\psi])^{2}=0.\end{cases}$

It is equivalent to the system (2.1) at a precision order $O(\mu\varepsilon)$.

  * •

We could approximate the Hamiltonian $H_{\operatorname{WW}}$ with any other
perturbation of $H_{0}+\varepsilon H_{1}$ of order $O(\mu\varepsilon)$. It
wouldn’t change the reasoning. Our choice is made to simplify the
computations.

######  Notations 3.4.

We will denote by $\widetilde{H}_{0}+\varepsilon\widetilde{H}_{1}$ the
Hamiltonian $H_{0}+\varepsilon H_{1}$ rewritten in the variables $(\zeta,v)$
where $v=\partial_{x}\psi$. It is associated with the Poisson tensor
$\widetilde{J}=\begin{pmatrix}0&-\partial_{x}\\\
-\partial_{x}&0\end{pmatrix}$. We recall that a change of unknowns
$\mathcal{T}(\eta,\omega)$ turns the initial Poisson tensor, denoted $J$, into
$\Big{(}\frac{\partial F}{\partial(\eta,\omega)}\Big{)}J\Big{(}\frac{\partial
F}{\partial(\eta,\omega)}\Big{)}^{*}$, where $*$ is the adjoint in
$L^{2}(\mathbb{R})^{2}$.

Before applying Birkhoff’s algorithm, we change the unknowns of the
approximated Hamiltonian $H_{0}+\varepsilon H_{1}$ with those usually used to
diagonalize the linear part of the water waves equations.

######  Proposition 3.5.

Let $r$ and $s$ be defined as

$\displaystyle
r\colonequals\frac{\zeta+\mathrm{F}_{\mu}[\partial_{x}\psi]}{2},\ \
s\colonequals\frac{\zeta-\mathrm{F}_{\mu}[\partial_{x}\psi]}{2}.$ (3.5)

We have

$\displaystyle(\widetilde{H}_{0}+\varepsilon\widetilde{H}_{1})(\zeta,v)=(H_{0}+\varepsilon
H_{1})(\zeta,\psi)=\int_{\mathbb{R}}(r^{2}+s^{2})\operatorname{dx}+\frac{\varepsilon}{2}\int(r^{3}+s^{3})\operatorname{dx}-\frac{\varepsilon}{2}\int_{\mathbb{R}}(r^{2}s+rs^{2})\operatorname{dx}.$
(3.6)

We will denote the later Hamiltonian $H_{\operatorname{BW}}(r,s)$. The Poisson
tensor associated with is
$J_{\mu}=\begin{pmatrix}-\frac{\mathrm{F}_{\mu}\partial_{x}}{2}&0\\\
0&\frac{\mathrm{F}_{\mu}\partial_{x}}{2}\end{pmatrix}$.

###### Proof.

Easy computations. ∎

For later purposes we also define the inverse of the transformation (3.5).

######  Property 3.6.

Let $\alpha\geq 0$ and $\mathcal{T}_{\rm{D}}:H^{\alpha+1}(\mathbb{R})\times
H^{\alpha+1}(\mathbb{R})\to H^{\alpha}(\mathbb{R})\times
H^{\alpha}(\mathbb{R})$ be defined by

$\displaystyle\mathcal{T}_{\rm{D}}(r,s)\colonequals\begin{pmatrix}r+s\\\
\mathrm{F}_{\mu}^{-1}[r-s]\end{pmatrix}.$ (3.7)

Then
$(\widetilde{H}_{0}+\varepsilon\widetilde{H}_{1})(\mathcal{T}_{\rm{D}}(r,s))=H_{\operatorname{BW}}(r,s)$.

###### Proof.

For any $\alpha\geq 0$ and any $u\in H^{\alpha+1}(\mathbb{R})$, we have
$|\mathrm{F}_{\mu}^{-1}[u]|_{H^{\alpha}}\lesssim|u|_{H^{\alpha+1}}$. ∎

To separate each important part of the Hamiltonian $H_{\operatorname{BW}}$ we
set some notations.

######  Notations 3.7.

Let

$\displaystyle\begin{cases}L\colonequals\int_{\mathbb{R}}(r^{2}+s^{2})\operatorname{dx},\\\
Z\colonequals\frac{1}{2}\int(r^{3}+s^{3})\operatorname{dx},\\\
W\colonequals-\frac{1}{2}\int_{\mathbb{R}}(r^{2}s+rs^{2})\operatorname{dx}.\end{cases}$

We have

$\displaystyle H_{\operatorname{BW}}=L+\varepsilon Z+\varepsilon W.$

######  Remark 3.8.

Our goal is to decouple the equations associated with the Hamiltonian
$H_{\operatorname{BW}}$. We remark that only $W$ gives coupled terms in the
Hamilton’s equations.

###  3.2 Application of Birkhoff’s algorithm

We suppose first that all our objects of study are smooth. If $G(r,s)$ is a
smooth function then it’s corresponding Hamilton’s equations are

$\displaystyle\partial_{t}\begin{pmatrix}r\\\ s\end{pmatrix}=J_{\mu}\nabla G.$

We will denote by $\Phi_{G}^{t}$ the corresponding flow.

If $F(r,s)$ is another smooth function, we denote the Lie derivative of $F$
with respect to $G$ by

$\displaystyle\\{F,G\\}_{\mu}\colonequals(\nabla F,J_{\mu}\nabla G),$

where $(\cdot,\cdot)$ denotes the scalar product in $L^{2}(\mathbb{R})^{2}$.

######  Property 3.9.

Let $G(r,s)$ be a smooth function and $\Phi_{G}^{t}$ its flow associated with
the Poisson tensor $J_{\mu}$. Let also $F(r,s)$ be another smooth function.
Then

$\displaystyle
F\circ\Phi_{G}^{\varepsilon}=F+\varepsilon\\{F,G\\}_{\mu}+O(\varepsilon^{2}).$

###### Proof.

See Subsection 4.1 in [1]. ∎

Suppose for now that in our case the Hamiltonian $H_{\operatorname{BW}}$ is
smooth. Then for any smooth function $G(r,s)$ we get

$\displaystyle
H_{\operatorname{BW}}\circ\Phi_{G}^{\varepsilon}=L+\varepsilon\\{L,G\\}_{\mu}+\varepsilon
Z+\varepsilon W+O(\varepsilon^{2}).$

######  Remark 3.10.

Here it seems that to get a normal form at the order of precision
$O(\varepsilon^{2})$, we need $G$ to solve the homological equation

$\displaystyle\\{L,G\\}_{\mu}+W=0.$ (3.8)

In [1], the author gave the solution of such equations. If

$\displaystyle\lim_{\tau\to+\infty}W(\Phi_{L}^{\tau})+W(\Phi_{L}^{-\tau})=0,$
(3.9)

(in our case, we can prove this assumption using a Littlewood-Paley
decomposition) then the solution of (3.8) is

$\displaystyle
G(r,s)=-\frac{1}{2}\int_{\mathbb{R}}\operatorname{sgn}(\tau)W(\Phi_{L}^{\tau}(r,s))d\tau,$
(3.10)

at the additional condition that the above function is well-defined.

However the latter condition is not easy to verify in general. We expose here
a naive attempt to do so in our context.

The flow $\Phi_{L}^{\tau}$ satisfy

$\displaystyle\frac{d}{d\tau}\Phi_{L}^{\tau}=J_{\mu}\nabla
L(\Phi_{L}^{\tau}).$

So that

$\displaystyle\Phi_{L}^{\tau}(r,s)=\begin{pmatrix}e^{-iD\mathrm{F}_{\mu}\tau}r\\\
e^{iD\mathrm{F}_{\mu}\tau}s\end{pmatrix},$

and

$\displaystyle
W(\Phi_{L}^{\tau}(r,s))=-\frac{1}{2}\int_{\mathbb{R}}(e^{-iD\mathrm{F}_{\mu}\tau}r)^{2}e^{iD\mathrm{F}_{\mu}\tau}s\operatorname{dx}-\frac{1}{2}\int_{\mathbb{R}}(e^{iD\mathrm{F}_{\mu}\tau}s)^{2}e^{-iD\mathrm{F}_{\mu}\tau}r\operatorname{dx}.$

So

$\displaystyle|W(\Phi_{L}^{\tau}(r,s))|$
$\displaystyle\leq\int_{\mathbb{R}}|e^{-iD\mathrm{F}_{\mu}\tau}r|^{2}|e^{iD\mathrm{F}_{\mu}\tau}s|\operatorname{dx}+\int_{\mathbb{R}}|e^{iD\mathrm{F}_{\mu}\tau}s|^{2}|e^{-iD\mathrm{F}_{\mu}\tau}r|\operatorname{dx}$
$\displaystyle\leq|e^{iD\mathrm{F}_{\mu}\tau}s|_{\infty}|e^{-iD\mathrm{F}_{\mu}\tau}r|_{2}^{2}+|e^{-iD\mathrm{F}_{\mu}\tau}r|_{\infty}|e^{iD\mathrm{F}_{\mu}\tau}s|_{2}^{2}.$

But $|e^{-iD\mathrm{F}_{\mu}\tau}r|_{2}=|r|_{2}$, and the dispersive estimates
only give a decrease in time of $|e^{iD\mathrm{F}_{\mu}\tau}s|_{\infty}$ and
$|e^{-iD\mathrm{F}_{\mu}\tau}r|_{\infty}$ of order $1/t^{-1/2}$ [4].

The idea to overcome this issue is to only solve an approximation of the
homological equation at order $O(\mu\varepsilon)$.

We define another Lie derivative, associated with the Poisson tensor
$J_{\operatorname{simp}}\colonequals\begin{pmatrix}-\partial_{x}/2&0\\\
0&\partial_{x}/2\end{pmatrix}$:

$\displaystyle\\{L,G\\}_{\operatorname{simp}}\colonequals(\nabla
L,J_{\operatorname{simp}}\nabla G)$

######  Property 3.11.

We have

$\displaystyle\\{L,G\\}_{\mu}=\\{L,G\\}_{\operatorname{simp}}+O(\mu)$

###### Proof.

By definition $\\{L,G\\}_{\mu}=(\nabla L,J_{\mu}\nabla G)$, with
$J_{\mu}=\begin{pmatrix}-\frac{\mathrm{F}_{\mu}\partial_{x}}{2}&0\\\
0&\frac{\mathrm{F}_{\mu}\partial_{x}}{2}\end{pmatrix}$. So

$\displaystyle\\{L,G\\}_{\mu}-\\{L,G\\}_{\operatorname{simp}}=((\nabla
L,(J_{\mu}-J_{\operatorname{simp}})\nabla G),$

with
$J_{\mu}-J_{\operatorname{simp}}=\begin{pmatrix}-\frac{\mathrm{F}_{\mu}-1}{2}&0\\\
0&\frac{\mathrm{F}_{\mu}-1}{2}\end{pmatrix}$. But for any $\alpha\geq 0$ and
any $u\in H^{\alpha+2}(\mathbb{R})$, we have
$|(\mathrm{F}_{\mu}-1)[u]|_{H^{\alpha}}\lesssim\mu|u|_{H^{\alpha+2}}$. ∎

The Lie derivative associated with the Poisson tensor
$J_{\operatorname{simp}}$ is the same one as in Bambusi’s article. To write an
explicit expression of the solution of the simplified homological equation
associated with

$\displaystyle\\{L,G\\}_{\operatorname{simp}}+W=0,$ (3.11)

we need the definition of a classical primitive operator and some properties
on it.

######  Definition/Property 3.12.

Let $\alpha\geq 0$. We define the operator
$\partial^{-1}:W^{\alpha,1}(\mathbb{R})\to W^{\alpha,\infty}(\mathbb{R})$ by
the formula

$\displaystyle\partial^{-1}u(y)=\frac{1}{2}\int_{\mathbb{R}}\operatorname{sgn}(y-y_{1})u(y_{1})\operatorname{dy}_{1}.$
(3.12)

Here are some properties on it:

  * •

It is continuous.

  * •

It is skew-adjoint for the scalar product in $L^{2}(\mathbb{R})$.

  * •

If $\lim_{x\to+\infty}u(x)+u(-x)=0$ (it is the case for any function $u\in
W^{1,1}(\mathbb{R})$, see Corollary 8.9 in [3]), then
$\partial^{-1}(\partial_{x}u)=u$.

  * •

$\partial_{x}(\partial^{-1}u)=u$.

We can now write explicitly the solution of the simplified homological
equation.

######  Proposition 3.13.

Let $W(r,s)=-\frac{1}{2}\int_{\mathbb{R}}(r^{2}s+rs^{2})\operatorname{dx}$,
and $L(r,s)=\int_{\mathbb{R}}(r^{2}+s^{2})\operatorname{dx}$ for any $(r,s)\in
W^{1,1}(\mathbb{R})$. Then the solution of the simplified homological equation
(3.11) is

$\displaystyle
G(r,s)=\frac{1}{4}\int_{\mathbb{R}}[\partial^{-1}(r^{2})s+\partial^{-1}(r)s^{2}]\operatorname{dx}.$

###### Proof.

See Lemma 5.2 in [1]. ∎

Here $G$ is clearly well-defined for $r$ and $s$ in $W^{\alpha,1}(\mathbb{R})$
with $\alpha\geq 1$ large enough. However there is another problem, $G$ does
not generate a flow. For that reason we use the generalization of Birkhoff’s
algorithm in the case of "almost smooth" functions depending on small
parameters (see [1]). We recall here what is an almost smooth function.

######  Definition 3.14.

Let $\\{\mathcal{B}^{\alpha}\\}_{\alpha\geq 0}$ be a Banach scale. A map
$F(r,s,\mu,\varepsilon)$ is said to be almost smooth if
$\forall\beta,\gamma\geq 0$, there exists $\delta$ and an open neighbourhood
of the origin
$\mathcal{U}\subset\mathcal{B}^{\delta}\times\mathbb{R}\times\mathbb{R}$
(where $\mathbb{R}\times\mathbb{R}$ is the domain of $(\mu,\varepsilon)$),
such that

$\displaystyle
F\in\mathcal{C}^{\beta}(\mathcal{U},\mathcal{B}^{\gamma}\times\mathbb{R}\times\mathbb{R}).$

Due to the definition of $\partial^{-1}$, for the rest of the article, we will
work with the Banach scale $\\{W^{\alpha,1}(\mathbb{R})\times
W^{\alpha,1}(\mathbb{R})\\}_{\alpha\geq 0}$. If

  * •

for any $\alpha\geq 0$, there exists $\alpha^{\prime}\geq 0$ such that the
Poisson tensor $J_{\mu}$ is bounded from
$W^{\alpha^{\prime},1}(\mathbb{R})\times W^{\alpha^{\prime},1}(\mathbb{R})$ to
$W^{\alpha,1}(\mathbb{R})\times W^{\alpha,1}(\mathbb{R})$,

  * •

the function $G(r,s)$ defined in Proposition 3.10 has an almost smooth vector
field, i.e. $J_{\mu}\nabla G$ is almost smooth, same for
$\\{L,G\\}_{\operatorname{simp}}$,

then the theory developed in [1] gives the existence of a transformation
$\mathcal{T}_{\rm{B}}$ such that

$\displaystyle H_{\operatorname{BW}}\circ\mathcal{T}_{\rm{B}}$
$\displaystyle=L+\varepsilon\\{L,G\\}_{\mu}+\varepsilon Z+\varepsilon
W+O(\varepsilon^{2})$
$\displaystyle=L+\varepsilon\\{L,G\\}_{\operatorname{simp}}+\varepsilon
Z+\varepsilon W+O(\mu\varepsilon+\varepsilon^{2})$
$\displaystyle=L+\varepsilon Z+O(\mu\varepsilon+\varepsilon^{2}),$

where $A=B+O(\mu^{k}\varepsilon^{l})$ now means
$\frac{A-B}{\mu^{k}\varepsilon^{l}}$ has an almost smooth vector field.

We verify the latter two conditions.

######  Property 3.15.

For any $\alpha\geq 0$ the Fourier multiplier $\mathrm{F}_{\mu}$ is bounded
from $W^{\alpha+1,1}(\mathbb{R})$ to $W^{\alpha,1}(\mathbb{R})$. In
particular, the Poisson tensor $J_{\mu}$ is bounded from
$W^{\alpha+2,1}(\mathbb{R})\times W^{\alpha+2,1}(\mathbb{R})$ to
$W^{\alpha,1}(\mathbb{R})\times W^{\alpha,1}(\mathbb{R})$.

The inverse Fourier multiplier of $\mathrm{F}_{\mu}$, denoted
$\mathrm{F}_{\mu}^{-1}\colonequals\sqrt{\frac{\sqrt{\mu}|D|}{\tanh{(\sqrt{\mu}|D|)}}}$,
is bounded from $W^{\alpha+2,1}(\mathbb{R})$ to $W^{\alpha,1}(\mathbb{R})$.

###### Proof.

First, we remark that we only need to prove the boundedness of
$\mathrm{F}_{\mu}$ in $W^{1,1}(\mathbb{R})\to L^{1}(\mathbb{R})$. To do so, we
consider an even cutoff function $\chi\in\mathcal{C}^{\infty}((-2,2))$ such
that $\chi(\xi)=1$ for $|\xi|\leq 1$. Let $f$ be in $W^{1,1}(\mathbb{R})$, we
can decompose $\mathrm{F}_{\mu}[f]$ into

$\displaystyle\mathrm{F}_{\mu}[f](x)=$
$\displaystyle\int_{\mathbb{R}}\int_{\mathbb{R}}e^{-i(y-x)\xi}F_{\mu}(\xi)\chi(\xi)f(y)\operatorname{dy}d\xi+\int_{\mathbb{R}}\int_{\mathbb{R}}e^{-i(y-x)\xi}F_{\mu}(\xi)(1-\chi(\xi))f(y)\operatorname{dy}d\xi$
$\displaystyle\colonequals$ $\displaystyle I_{1}+I_{2}.$

We deal with each term separately.

  * •

We begin with $I_{1}$. We can write it using the inverse Fourier transform,
denoted here $\mathcal{F}^{-1}$:

$\displaystyle
I_{1}=\int_{\mathbb{R}}\int_{\mathbb{R}}e^{-i(y-x)\xi}F_{\mu}(\xi)\chi(\xi)f(y)\operatorname{dy}d\xi=\mathcal{F}^{-1}[F_{\mu}(\xi)\chi(\xi)]*f.$

Using Young’s inequality, we get

$\displaystyle|\mathcal{F}^{-1}[F_{\mu}(\xi)\chi(\xi)]*f|_{L^{1}}\leq|\mathcal{F}^{-1}[F_{\mu}(\xi)\chi(\xi)]|_{L^{1}}|f|_{L^{1}}.$

But

$\displaystyle|\mathcal{F}^{-1}[F_{\mu}(\xi)\chi(\xi)]|_{L^{1}}\lesssim|F_{\mu}(\xi)\chi(\xi))|_{H^{1}}.$

And $\forall\xi\in\mathbb{R}$ $|F_{\mu}(\xi)|,|F_{\mu}^{\prime}(\xi)|\leq 1$,
so $\mathcal{F}^{-1}[F_{\mu}(\xi)\chi(\xi)]$ is in $L^{1}(\mathbb{R})$. It
finishes the case of $I_{1}$.

  * •

Now we deal with $I_{2}$. Doing an integration by parts, we obtain

$\displaystyle\int_{\mathbb{R}}\int_{\mathbb{R}}e^{-i(y-x)\xi}F_{\mu}(\xi)(1-\chi(\xi))f(y)\operatorname{dy}d\xi$
$\displaystyle=\int_{\mathbb{R}}\int_{\mathbb{R}}e^{-i(y-x)\xi}\frac{F_{\mu}(\xi)}{i\xi}(1-\chi(\xi))\partial_{y}f(y)\operatorname{dy}d\xi$
$\displaystyle=\mathcal{F}^{-1}[\frac{F_{\mu}(\xi)}{i\xi}(1-\chi(\xi))]*\partial_{y}f$

Using again Young’s inequality, we get

$\displaystyle|\mathcal{F}^{-1}[\frac{F_{\mu}(\xi)}{i\xi}(1-\chi(\xi))]*\partial_{y}f|_{L^{1}}\leq|\mathcal{F}^{-1}[\frac{F_{\mu}(\xi)}{i\xi}(1-\chi(\xi))]|_{L^{1}}|\partial_{y}f|_{L^{1}}.$

But

$\displaystyle|\mathcal{F}^{-1}[\frac{F_{\mu}(\xi)}{i\xi}(1-\chi(\xi))]|_{L^{1}}\lesssim|\frac{F_{\mu}(\xi)}{i\xi}(1-\chi(\xi))|_{H^{1}}.$

So $\mathcal{F}^{-1}[\frac{F_{\mu}(\xi)}{i\xi}(1-\chi(\xi))]\in
L^{1}(\mathbb{R})$. It finishes the case of $I_{2}$.

At the end, we have
$|\mathrm{F}_{\mu}[f]|_{L^{1}}\leq|I_{1}|_{L^{1}}+|I_{2}|_{L^{1}}\lesssim|f|_{W^{1,1}}$.

With the same process we obtain the result on $\mathrm{F}_{\mu}^{-1}$ (we need
a second integration by parts in the high frequencies to get
$\mathcal{F}^{-1}[\frac{F_{\mu}^{-1}(\xi)}{(i\xi)^{2}}(1-\chi(\xi))]$ in
$L^{1}$). ∎

######  Property 3.16.

The function $G(r,s)$ defined in Proposition 3.13 and
$\\{L,G\\}_{\operatorname{simp}}$ have an almost smooth vector field.

###### Proof.

First we have
$2\partial_{r}G=-r\partial^{-1}(s)-\frac{1}{2}\partial^{-1}(s^{2})$, where we
used the skew-adjoint property of $\partial^{-1}$. So
$-2\mathrm{F}_{\mu}\partial_{x}\partial_{r}G=\mathrm{F}_{\mu}[\partial_{x}(r)\partial^{-1}(s)]+\mathrm{F}_{\mu}[rs]+\frac{1}{2}\mathrm{F}_{\mu}[s^{2}]$.
But for any $\alpha\geq 0$, $\mathrm{F}_{\mu}$ is bounded from
$W^{\alpha+1,1}(\mathbb{R})$ to $W^{\alpha,1}(\mathbb{R})$, and
$W^{\alpha+1,1}(\mathbb{R})$ is an algebra. We also know that $\partial^{-1}$
is a linear operator bounded from $W^{\alpha+1,1}(\mathbb{R})$ to
$W^{\alpha+1,\infty}(\mathbb{R})$ , and the product of a
$W^{\alpha+1,\infty}(\mathbb{R})$ with a $W^{\alpha+1,1}(\mathbb{R})$ function
is still in $W^{\alpha+1,1}(\mathbb{R})$. Combining all these elements, we get
the result for $G$.

The same arguments give the result for $\\{L,G\\}_{\operatorname{simp}}$. ∎

We have an explicit formulation of the transformation $\mathcal{T}_{\rm{B}}$
given by (4.25) in [1], taking $X=J_{\operatorname{simp}}\nabla G$, on which
we can do precise estimates.

######  Proposition 3.17.

Let $\alpha\geq 0$ and $\mathcal{T}_{\rm{B}}:W^{\alpha+1,1}(\mathbb{R})\times
W^{\alpha+1,1}(\mathbb{R})\to W^{\alpha,1}(\mathbb{R})\times
W^{\alpha,1}(\mathbb{R})$ be defined by

$\displaystyle\mathcal{T}_{\rm{B}}(r,s)=\begin{pmatrix}r\\\
s\end{pmatrix}+\varepsilon J_{\operatorname{simp}}\nabla
G(r,s)=\begin{pmatrix}r+\frac{\varepsilon}{4}\partial_{x}(r)\partial^{-1}(s)+\frac{\varepsilon}{4}rs+\frac{\varepsilon}{8}s^{2}\\\
s+\frac{\varepsilon}{4}\partial_{x}(s)\partial^{-1}(r)+\frac{\varepsilon}{4}rs+\frac{\varepsilon}{8}r^{2}\end{pmatrix}.$
(3.13)

Then

$\displaystyle H_{\operatorname{BW}}\circ\mathcal{T}_{\rm{B}}=L+\varepsilon
Z+O(\mu\varepsilon+\varepsilon^{2})$

###### Proof.

Let $r,s\in W^{\alpha+1,1}(\mathbb{R})$. The latter Sobolev space in an
algebra, so the only terms we need to look at are
$\partial_{x}r\partial^{-1}s$ and $\partial_{x}s\partial^{-1}r$. We have

$\displaystyle|\partial_{x}r\partial^{-1}s|_{W^{\alpha,1}}\leq|\partial^{-1}s|_{W^{\alpha,\infty}}|\partial_{x}r|_{W^{\alpha,1}}.$

Here using the continuity of $\partial^{-1}$ from $L^{1}(\mathbb{R})$ to
$L^{\infty}(\mathbb{R})$ for any $\beta\geq 0$ and the third property of
Definition/Property 3.12, we get

$\displaystyle\begin{cases}|\partial^{-1}s|_{W^{\alpha,\infty}}\lesssim|s|_{L^{1}}\
\ \operatorname{if}\ \ \alpha=0,\\\
|\partial^{-1}s|_{W^{\alpha,\infty}}\lesssim|s|_{L^{1}}+\sum\limits_{\beta\leq\alpha}|\partial_{x}^{\beta}\partial^{-1}s|_{L^{\infty}}=|s|_{L^{1}}+\sum\limits_{\beta\leq\alpha}|\partial^{-1}\partial_{x}^{\beta}s|_{L^{\infty}}\lesssim|s|_{W^{\alpha,1}}\
\ \operatorname{if}\ \ \alpha\geq 1.\end{cases}$ (3.14)

Idem for the other term. Thus
$\mathcal{T}_{\rm{B}}:W^{\alpha+1,1}(\mathbb{R})\times
W^{\alpha+1,1}(\mathbb{R})\to W^{\alpha,1}(\mathbb{R})\times
W^{\alpha,1}(\mathbb{R})$.

From the definition of $H_{\operatorname{BW}}$ (see (3.6)) and (3.13) we have

$\displaystyle H_{\operatorname{BW}}(\mathcal{T}_{\rm{B}}(r,s))=$
$\displaystyle\int_{\mathbb{R}}\Big{(}r^{2}+\frac{\varepsilon}{2}r\partial_{x}r\partial^{-1}s+\frac{\varepsilon}{2}r^{2}s+\frac{\varepsilon}{4}rs^{2}\Big{)}\operatorname{dx}$
$\displaystyle+$
$\displaystyle\int_{\mathbb{R}}\Big{(}s^{2}+\frac{\varepsilon}{2}s\partial_{x}s\partial^{-1}r+\frac{\varepsilon}{2}rs^{2}+\frac{\varepsilon}{4}sr^{2}\Big{)}\operatorname{dx}$
$\displaystyle-$
$\displaystyle\frac{\varepsilon}{2}\int_{\mathbb{R}}(r^{2}s+rs^{2})\operatorname{dx}+\frac{\varepsilon}{2}\int_{\mathbb{R}}(r^{3}+s^{3})\operatorname{dx}+O(\varepsilon^{2}).$

But using the skew-adjointness of $\partial^{-1}$ we get

$\displaystyle\begin{cases}\int_{\mathbb{R}}r\partial_{x}r\partial^{-1}s\operatorname{dx}=\frac{1}{2}\int_{\mathbb{R}}\partial_{x}(r^{2})\partial^{-1}s\operatorname{dx}=-\frac{1}{2}\int_{\mathbb{R}}\partial^{-1}(\partial_{x}r^{2})s\operatorname{dx}=-\frac{1}{2}\int_{\mathbb{R}}r^{2}s\operatorname{dx},\\\
\int_{\mathbb{R}}s\partial_{x}s\partial^{-1}r\operatorname{dx}=\frac{1}{2}\int_{\mathbb{R}}\partial_{x}(s^{2})\partial^{-1}r\operatorname{dx}=-\frac{1}{2}\int_{\mathbb{R}}\partial^{-1}(\partial_{x}s^{2})r\operatorname{dx}=-\frac{1}{2}\int_{\mathbb{R}}s^{2}r\operatorname{dx}.\end{cases}$

Thus

$\displaystyle H_{\operatorname{BW}}(\mathcal{T}_{\rm{B}}(r,s))$
$\displaystyle=\int_{\mathbb{R}}(r^{2}+s^{2})\operatorname{dx}+\frac{\varepsilon}{2}\int_{\mathbb{R}}(r^{3}+s^{3})\operatorname{dx}+O(\varepsilon^{2})$
$\displaystyle=L(r,s)+\varepsilon Z(r,s)+O(\varepsilon^{2}).$

∎

######  Remark 3.18.

The theory developped in [1] tells us that the transformation
$\mathcal{T}_{\rm{B}}$ should be
$\mathcal{T}_{\rm{B}}(r,s)=\begin{pmatrix}r\\\ s\end{pmatrix}+\varepsilon
J_{\mu}\nabla G(r,s)$. But the precision we want is
$O(\mu\varepsilon+\varepsilon^{2})$, and $\varepsilon J_{\mu}=\varepsilon
J_{\operatorname{simp}}+O(\mu\varepsilon)$.

The transformation (3.13) preserves the Hamiltonian structure associated with
the Poisson tensor $J_{\mu}$ at the order of precision
$O(\mu\varepsilon+\varepsilon^{2})$, see the following proposition.

######  Proposition 3.19.

Let $\alpha\geq 0$. Let $r,s\in W^{\alpha+4,1}(\mathbb{R})$ Let
$\frac{\partial\mathcal{T}_{\rm{B}}}{\partial(r,s)}$ be the Jacobian matrix of
the transformation $T$. We have

$\displaystyle\Big{(}\frac{\partial\mathcal{T}_{\rm{B}}}{\partial(r,s)}\Big{)}J_{\mu}\Big{(}\frac{\partial\mathcal{T}_{\rm{B}}}{\partial(r,s)}\Big{)}^{*}=J_{\mu}+(\mu\varepsilon+\varepsilon^{2})R,$

where $*$ is the adjoint in $L^{2}(\mathbb{R})^{2}$ and $R$ is linear operator
such that for any $U\in H^{\alpha+4}(\mathbb{R})\times
H^{\alpha+4}(\mathbb{R})$, $|RU|_{H^{\alpha}\times H^{\alpha}}\leq
C(\mu_{\max},|r|_{W^{\alpha+4,1}},|s|_{W^{\alpha+4,1}})|U|_{H^{\alpha+4}\times
H^{\alpha+4}}$.

###### Proof.

We easily compute the Jacobian matrix of $\mathcal{T}_{\rm{B}}$:

$\displaystyle\frac{\partial\mathcal{T}_{\rm{B}}}{\partial(r,s)}=\begin{pmatrix}1+\frac{\varepsilon}{4}s+\frac{\varepsilon}{4}\partial^{-1}(s)\partial_{x}(\circ)&\frac{\varepsilon}{4}\partial_{x}(r)\partial^{-1}(\circ)+\frac{\varepsilon}{4}(r+s)\\\
\frac{\varepsilon}{4}\partial_{x}(s)\partial^{-1}(\circ)+\frac{\varepsilon}{4}(r+s)&1+\frac{\varepsilon}{4}r+\frac{\varepsilon}{4}\partial^{-1}(r)\partial_{x}(\circ)\end{pmatrix}.$

Its $L^{2}$ adjoint is

$\displaystyle\Big{(}\frac{\partial\mathcal{T}_{\rm{B}}}{\partial(r,s)}\Big{)}^{*}=\begin{pmatrix}1+\frac{\varepsilon}{4}s-\frac{\varepsilon}{4}\partial_{x}(\partial^{-1}(s)\circ)&\frac{\varepsilon}{4}(r+s)-\frac{\varepsilon}{4}\partial^{-1}(\partial_{x}(s)\circ)\\\
\frac{\varepsilon}{4}(r+s)-\frac{\varepsilon}{4}\partial^{-1}(\partial_{x}(r)\circ)&1+\frac{\varepsilon}{4}r-\frac{\varepsilon}{4}\partial_{x}(\partial^{-1}(r)\circ).\end{pmatrix}$

Using the fact that both $\frac{\partial\mathcal{T}_{\rm{B}}}{\partial(r,s)}$
and $\Big{(}\frac{\partial\mathcal{T}_{\rm{B}}}{\partial(r,s)}\Big{)}^{*}$ can
be written under the form $I_{d}+O(\varepsilon)$, we write

$\displaystyle\Big{(}\frac{\partial\mathcal{T}_{\rm{B}}}{\partial(r,s)}\Big{)}J_{\mu}\Big{(}\frac{\partial\mathcal{T}_{\rm{B}}}{\partial(r,s)}\Big{)}^{*}$
$\displaystyle=\Big{[}\Big{(}\frac{\partial\mathcal{T}_{\rm{B}}}{\partial(r,s)}-I_{d}\Big{)}+I_{d}\Big{]}J_{\mu}\Big{[}\Big{(}\Big{(}\frac{\partial\mathcal{T}_{\rm{B}}}{\partial(r,s)}\Big{)}^{*}-I_{d}\Big{)}+I_{d}\Big{]}$
$\displaystyle=\Big{(}\frac{\partial\mathcal{T}_{\rm{B}}}{\partial(r,s)}-I_{d}\Big{)}J_{\mu}\Big{(}\Big{(}\frac{\partial\mathcal{T}_{\rm{B}}}{\partial(r,s)}\Big{)}^{*}-I_{d}\Big{)}$
$\displaystyle+\Big{(}\frac{\partial\mathcal{T}_{\rm{B}}}{\partial(r,s)}-I_{d}\Big{)}J_{\mu}+J_{\mu}\Big{(}\Big{(}\frac{\partial\mathcal{T}_{\rm{B}}}{\partial(r,s)}\Big{)}^{*}-I_{d}\Big{)}+J_{\mu}.$

The first term is of order $O(\varepsilon^{2})$, we will estimate it later.
The second and third terms are of order $O(\varepsilon)$. We prove now that
their addition is of order $O(\mu\varepsilon)$. We easily compute both terms:

$\displaystyle\Big{(}\frac{\partial\mathcal{T}_{\rm{B}}}{\partial(r,s)}-I_{d}\Big{)}J_{\mu}=\frac{\varepsilon}{8}\begin{pmatrix}-s\mathrm{F}_{\mu}\partial_{x}[\circ]-\partial^{-1}(s)\mathrm{F}_{\mu}\partial_{x}^{2}[\circ]&(r+s)\mathrm{F}_{\mu}\partial_{x}[\circ]+\partial_{x}(r)\mathrm{F}_{\mu}[\circ]\\\
-(r+s)\mathrm{F}_{\mu}\partial_{x}[\circ]-\partial_{x}(s)\mathrm{F}_{\mu}[\circ]&r\mathrm{F}_{\mu}\partial_{x}[\circ]+\partial^{-1}(r)\mathrm{F}_{\mu}\partial_{x}^{2}[\circ]\end{pmatrix},$

and

$\displaystyle
J_{\mu}\Big{(}\Big{(}\frac{\partial\mathcal{T}_{\rm{B}}}{\partial(r,s)}\Big{)}^{*}-I_{d}\Big{)}=\frac{\varepsilon}{8}\begin{pmatrix}\mathrm{F}_{\mu}\partial_{x}[\partial^{-1}(s)\partial_{x}(\circ)]&-\mathrm{F}_{\mu}\partial_{x}[(r+s)\circ]+\mathrm{F}_{\mu}[\partial_{x}(s)\circ]\\\
\mathrm{F}_{\mu}\partial_{x}[(r+s)\circ]-\mathrm{F}_{\mu}[\partial_{x}(r)\circ]&-\mathrm{F}_{\mu}\partial_{x}[\partial^{-1}(r)\partial_{x}(\circ)]\end{pmatrix}.$

For now, we look at the first term of both matrices. For any $u\in
H^{\alpha+4}(\mathbb{R})$

$\displaystyle|\mathrm{F}_{\mu}\partial_{x}[\partial^{-1}(s)\partial_{x}u]-s\mathrm{F}_{\mu}\partial_{x}[u]-\partial^{-1}(s)\mathrm{F}_{\mu}\partial_{x}^{2}[u]|_{H^{\alpha}}$
$\displaystyle\leq$
$\displaystyle|[\mathrm{F}_{\mu},s]\partial_{x}u|_{H^{\alpha}}+|[\mathrm{F}_{\mu},\partial^{-1}s]\partial_{x}^{2}u|_{H^{\alpha}}$
$\displaystyle=$
$\displaystyle|[\mathrm{F}_{\mu}-1;s]\partial_{x}u|_{H^{\alpha}}+|[\mathrm{F}_{\mu}-1,\partial^{-1}s]\partial_{x}^{2}u|_{H^{\alpha}}.$

But using Remark 2.7 and (3.14) we have

$\displaystyle|[\mathrm{F}_{\mu}-1,\partial^{-1}s]\partial_{x}^{2}u|_{H^{\alpha}}$
$\displaystyle\lesssim\mu|\partial^{-1}(s)\partial_{x}^{2}u|_{H^{\alpha+2}}+|\partial^{-1}s|_{W^{\alpha+2,\infty}}|(\mathrm{F}_{\mu}-1)[\partial_{x}^{2}u|_{H^{\alpha}}$
$\displaystyle\lesssim\mu|\partial^{-1}s|_{W^{\alpha+2,\infty}}|\partial_{x}^{2}u|_{H^{\alpha+2}}$
$\displaystyle\lesssim\mu|s|_{W^{\alpha+2,1}}|u|_{H^{\alpha+4}}.$

Using algebra properties of $H^{\alpha+2}(\mathbb{R})$ and the Sobolev
embedding $W^{\alpha+3,1}(\mathbb{R})\subset H^{\alpha+2}(\mathbb{R})$, we
also have

$\displaystyle|[\mathrm{F}_{\mu}-1,s]\partial_{x}u|_{H^{\alpha}}\lesssim\mu|s|_{H^{\alpha+2}}|u|_{H^{\alpha+3}}\lesssim\mu|s|_{W^{\alpha+3,1}}|u|_{H^{\alpha+3}}.$

So that

$\displaystyle|\mathrm{F}_{\mu}\partial_{x}[\partial^{-1}(s)\partial_{x}u]-s\mathrm{F}_{\mu}\partial_{x}[u]-\partial^{-1}(s)\mathrm{F}_{\mu}\partial_{x}^{2}[u]|_{H^{\alpha}}\lesssim\mu|s|_{W^{\alpha+3,1}}|u|_{H^{\alpha+4}}.$

The same can be done for the addition of the fourth term of both matrices.

Now we look at the addition of the second term of both matrices. For any $u\in
H^{\alpha+3}(\mathbb{R})$ we have

$\displaystyle(r+s)\mathrm{F}_{\mu}[\partial_{x}u]+\partial_{x}(r)\mathrm{F}_{\mu}[u]$
$\displaystyle=\partial_{x}((r+s)\mathrm{F}_{\mu}[u])-\partial_{x}(r+s)\mathrm{F}_{\mu}[u]$
$\displaystyle=\partial_{x}((r+s)\mathrm{F}_{\mu}[u])-\partial_{x}(s)\mathrm{F}_{\mu}[u].$

Using Lemma 2.6 and the Sobolev embedding $W^{\alpha+4,1}(\mathbb{R})\subset
H^{\alpha+3}(\mathbb{R})$, we get

$\displaystyle|\partial_{x}((r+s)\mathrm{F}_{\mu}[u])-\partial_{x}(s)\mathrm{F}_{\mu}[u]-\mathrm{F}_{\mu}[\partial_{x}((r+s)u)]+\mathrm{F}_{\mu}[\partial_{x}(s)u]|_{H^{\alpha}}$
$\displaystyle\leq$
$\displaystyle|\partial_{x}([\mathrm{F}_{\mu},r+s]u)|_{H^{\alpha}}+|[\mathrm{F}_{\mu},\partial_{x}s]u|_{H^{\alpha}}$
$\displaystyle\leq$
$\displaystyle|[\mathrm{F}_{\mu}-1,r+s]u|_{H^{\alpha+1}}+|[\mathrm{F}_{\mu}-1,\partial_{x}s]u|_{H^{\alpha}}$
$\displaystyle\lesssim$
$\displaystyle\mu(|r|_{H^{\alpha+3}}+|s|_{H^{\alpha+3}})|u|_{H^{\alpha+3}}$
$\displaystyle\lesssim$
$\displaystyle\mu(|r|_{W^{\alpha+4,1}}+|s|_{W^{\alpha+4}})|u|_{H^{\alpha+3}}.$

The same can be done for the addition of the third term of both matrices. Thus
we get for any $U\in H^{\alpha+4}(\mathbb{R})\times H^{\alpha+4}(\mathbb{R})$

$\displaystyle|\Big{(}\frac{\partial\mathcal{T}_{\rm{B}}}{\partial(r,s)}-I_{d}\Big{)}J_{\mu}U+J_{\mu}\Big{(}\Big{(}\frac{\partial\mathcal{T}_{\rm{B}}}{\partial(r,s)}\Big{)}^{*}-I_{d}\Big{)}U|_{H^{\alpha}}\lesssim\mu\varepsilon(|r|_{W^{\alpha+4,1}}+|s|_{W^{\alpha+4,1}})|U|_{H^{\alpha+4}}$

It remains to estimate
$\Big{(}\frac{\partial\mathcal{T}_{\rm{B}}}{\partial(r,s)}-I_{d}\Big{)}J_{\mu}\Big{(}\Big{(}\frac{\partial\mathcal{T}_{\rm{B}}}{\partial(r,s)}\Big{)}^{*}-I_{d}\Big{)}$.
Using the boundedness of $\mathrm{F}_{\mu}$ and the same tools as before, we
get for any $U\in H^{\alpha+3}(\mathbb{R})\times H^{\alpha+3}(\mathbb{R})$

$\displaystyle|\Big{(}\frac{\partial\mathcal{T}_{\rm{B}}}{\partial(r,s)}-I_{d}\Big{)}J_{\mu}\Big{(}\Big{(}\frac{\partial\mathcal{T}_{\rm{B}}}{\partial(r,s)}\Big{)}^{*}-I_{d}\Big{)}U|_{H^{\alpha}}\lesssim\varepsilon^{2}C(|r|_{W^{\alpha+4,1}},|r|_{W^{\alpha+4,1}})|U|_{H^{\alpha+3}}.$

∎

###  3.3 From the Hamiltonian of the Whitham-Boussinesq system under normal
form to two decoupled Whitham equations

In the previous subsections, we proved the existence of a transformation
$\mathcal{T}_{\rm{B}}$ such that

$\displaystyle H_{\operatorname{WW}}\circ\mathcal{T}_{\rm{B}}$
$\displaystyle=L+\varepsilon Z+O(\mu\varepsilon+\varepsilon^{2})$
$\displaystyle=\int_{\mathbb{R}}(r^{2}+s^{2})\operatorname{dx}+\varepsilon\frac{1}{2}\int(r^{3}+s^{3})\operatorname{dx}+O(\mu\varepsilon+\varepsilon^{2}).$

The equations associated with the Hamiltonian $L+\varepsilon Z$ and the
Poisson tensor $J_{\mu}$ are easily computed.

######  Property 3.20.

The Hamilton’s equations associated with the normal form $L+\varepsilon Z$ are

$\displaystyle\begin{cases}\partial_{t}r+\mathrm{F}_{\mu}[\partial_{x}r]+\frac{3\varepsilon}{2}\mathrm{F}_{\mu}[r\partial_{x}r]=0,\\\
\partial_{t}s-\mathrm{F}_{\mu}[\partial_{x}s]-\frac{3\varepsilon}{2}\mathrm{F}_{\mu}[s\partial_{x}s]=0.\end{cases}$
(3.15)

###### Proof.

The Hamilton’s equations are

$\displaystyle\partial_{t}\begin{pmatrix}r\\\
s\end{pmatrix}=J_{\mu}\nabla(L+\varepsilon Z).$

∎

######  Notation 3.21.

We write $H_{\operatorname{Wh}}(r,s)\colonequals L(r,s)+\varepsilon Z(r,s)$.

######  Remark 3.22.

The two equations (3.15) are almost two Whitham equations. One just need to
use that for any $\alpha\geq 0$ and any $u\in H^{\alpha+2}(\mathbb{R})$ one
has $|(\mathrm{F}_{\mu}-1)[u]|_{H^{\alpha}}\lesssim\mu|u|_{H^{\alpha+2}}$ to
get the existence of a remainder $(R_{1},R_{2})$ controllable in Sobolev
spaces $H^{\alpha}$ such that

$\displaystyle\begin{cases}\partial_{t}r+\mathrm{F}_{\mu}[\partial_{x}r]+\frac{3\varepsilon}{2}r\partial_{x}r=\mu\varepsilon
R_{1},\\\
\partial_{t}s-\mathrm{F}_{\mu}[\partial_{x}s]-\frac{3\varepsilon}{2}s\partial_{x}s=\mu\varepsilon
R_{2}.\end{cases}$

######  Lemma 3.23.

Let $\theta>1/2$ and $\alpha\geq 0$. Then for any $r_{0},s_{0}\in
W^{\alpha+5,1}(\mathbb{R})$ there exists a time $T>0$ such that both equations
(3.15) admit a unique solution
$r,s\in\mathcal{C}([0,\frac{T}{\varepsilon}],H^{\alpha+4}(\mathbb{R}))$, with
initial conditions $r_{0}$ and $s_{0}$, which satisfy for any times
$t\in[0,\frac{T}{\epsilon}]$

$\displaystyle\begin{cases}|r(t)|_{W^{\alpha,1}}\lesssim(1+\mu
t)^{\theta}C(\mu_{\max},T,|r_{0}|_{W^{\alpha+5,1}}),\\\
|s(t)|_{W^{\alpha,1}}\lesssim(1+\mu
t)^{\theta}C(\mu_{\max},T,|s_{0}|_{W^{\alpha+5,1}}).\end{cases}$

In particular
$r,s\in\mathcal{C}([0,\frac{T}{\varepsilon}],W^{\alpha,1}(\mathbb{R}))$ are
uniformly bounded in $(\mu,\varepsilon)$ for times
$t\in[0,\frac{T}{\max{(\mu,\varepsilon)}}]$ in $W^{\alpha,1}(\mathbb{R})$.

Moreover, for any $r_{0},s_{0}\in W^{\alpha+7,1}(\mathbb{R})$, both equations
(3.15) admit a unique solution
$r,s\in\mathcal{C}^{1}([0,\frac{T}{\max{(\mu,\varepsilon)}}],W^{\alpha,1}(\mathbb{R}))\cap\mathcal{C}([0,\frac{T}{\max{(\mu,\varepsilon)}}],W^{\alpha+2,1}(\mathbb{R}))$
with initial conditions $r_{0}$ and $s_{0}$.

###### Proof.

We have the Sobolev embeddings $W^{\alpha+5,1}(\mathbb{R})\subset
H^{\alpha+4}(\mathbb{R})$ and the equations (3.15) are well-posed in these
Sobolev spaces (see [6]). So there exists $T>0$ such that the latter equations
admit a unique solution in
$\mathcal{C}([0,\frac{T}{\varepsilon}],H^{\alpha+2}(\mathbb{R}))$.

We do the rest of the reasoning for $r$. We use the Duhamel formula to say
that

$\displaystyle
r(t)=e^{-itD\mathrm{F}_{\mu}}r_{0}+\frac{3\varepsilon}{2}\int_{0}^{t}e^{-i(t-t^{\prime})D\mathrm{F}_{\mu}}\mathrm{F}_{\mu}[r\partial_{x}r]\operatorname{dt}^{\prime}.$

We prove first that
$|e^{-itD\mathrm{F}_{\mu}}r_{0}|_{W^{\alpha,1}}\lesssim(1+\mu
t)^{\theta}|r_{0}|_{W^{\alpha+2,1}}$.

We begin by doing the same reasoning as in the proof of Property 3.15. We
remark that we only need to prove the previous estimate in
$L^{1}(\mathbb{R})$. Let $\chi\in\mathcal{C}^{\infty}((-2,2))$ be a cutoff
function such that $\chi(\xi)=1$ for $|\xi|\leq 1$. We decompose
$e^{-itD\mathrm{F}_{\mu}}r_{0}=e^{-itD(\mathrm{F}_{\mu}-1)}e^{-itD}r_{0}$ into

$\displaystyle e^{-itD(\mathrm{F}_{\mu}-1)}e^{-itD}r_{0}(x)$
$\displaystyle=\mathcal{F}^{-1}[e^{-it\xi(F_{\mu}(\xi)-1)}\chi(\xi)]*(e^{-itD}r_{0})$
$\displaystyle+\mathcal{F}^{-1}[e^{{}_{i}t\xi(F_{\mu}(\xi)-1)}\frac{1-\chi(\xi)}{(i\xi)^{2}}]*\partial_{y}^{2}e^{-itD}r_{0}\colonequals
I_{1}+I_{2}.$

  * •

We prove here that $|I_{1}|_{L^{1}}\lesssim(1+\mu
t)^{\frac{1}{2}+}|r_{0}|_{L^{1}}$. Using Young’s inequality, we get

$\displaystyle|I_{1}|_{L^{1}}\leq|\mathcal{F}^{-1}[e^{-it\xi(F_{\mu}(\xi)-1)}\chi(\xi)]|_{L^{1}}|e^{-itD}r_{0}|_{L^{1}}\leq|\mathcal{F}^{-1}[e^{-it\xi(F_{\mu}(\xi)-1)}\chi(\xi)]|_{L^{1}}|r_{0}|_{L^{1}}.$

Now we pose $G(x)=\mathcal{F}^{-1}[e^{-it\xi(F_{\mu}(\xi)-1)}\chi(\xi)]$. Let
$\beta\in\mathcal{C}^{\infty}((-2,2))$ be a cutoff function such that
$\beta(x)=1$ for $|x|\leq 1$. We have for any $\theta>1/2$

$\displaystyle|G|_{L^{1}}$ $\displaystyle\leq|\beta
G|_{L^{1}}+|(1-\beta)G|_{L^{1}}$ $\displaystyle\lesssim
1+|\frac{1-\beta(x)}{x^{\theta}}|_{L^{2}}|x^{\theta}G|_{L^{2}}.$

We will prove the inequality $|xG|_{L^{2}}\lesssim 1+\mu t$ and interpolate it
with the obvious one $|G|_{L^{2}}\lesssim 1$.

We remark that

$\displaystyle|xG|_{L^{2}}=|\partial_{\xi}\widehat{G}|_{L^{2}}=|\partial_{\xi}(e^{-it\xi(F_{\mu}(\xi)-1)}\chi(\xi))|_{L^{2}}.$

But we have the two following estimates on $F_{\mu}$: for any
$\xi\in\mathbb{R}$

$\displaystyle|F_{\mu}(\xi)-1|\lesssim\mu\xi^{2},\ \ \ \
|F_{\mu}^{\prime}(\xi)|\lesssim\mu\xi^{2}.$ (3.16)

So

$\displaystyle|\partial_{\xi}(e^{-it\xi(F_{\mu}(\xi)-1)}\chi(\xi))|_{L^{2}}$
$\displaystyle\leq t|(F_{\mu}(\xi)-1)\chi(\xi)|_{L^{2}}+t|\xi
F_{\mu}^{\prime}(\xi)\chi(\xi)|_{L^{2}}+|\chi^{\prime}(\xi)|_{L^{2}}\lesssim
1+\mu t.$

Then Hölder’s inequality gives us

$\displaystyle|x^{\theta}G|_{L^{2}}=|(xG)^{\theta}G^{1-\theta}|_{L^{2}}\leq|(xG)^{\theta}|_{L^{p}}|G^{1-\theta}|_{L^{q}},$

with $\frac{1}{2}=\frac{1}{p}+\frac{1}{q}$. It remains to take
$p=\frac{2}{\theta}$ and $q=\frac{2}{1-\theta}$ to get

$\displaystyle|x^{\theta}G|_{L^{2}}\leq|xG|_{L^{2}}^{\theta}|G|_{L^{2}}^{1-\theta}\lesssim(1+\mu
t)^{\theta}.$

  * •

We prove here that $|I_{2}|_{L^{1}}\lesssim(1+\mu
t)^{\theta}|r_{0}|_{W^{2,1}}$. Again we use Young’s inequality to get

$\displaystyle|I_{2}|_{L^{1}}\leq|\mathcal{F}^{-1}[e^{{}_{i}t\xi(F_{\mu}(\xi)-1)}\frac{1-\chi(\xi)}{(i\xi)^{2}}]|_{L^{1}}|r_{0}|_{W^{2,1}}.$

Then the rest of the proof is the same, with a little exception: here, using
the estimates (3.16), we have

$\displaystyle|\partial_{\xi}(e^{-it\xi(F_{\mu}(\xi)-1)}\frac{1-\chi(\xi)}{\xi^{2}})|_{L^{2}}$
$\displaystyle\leq$
$\displaystyle|\frac{t(F_{\mu}(\xi)-1)(1-\chi(\xi))\xi^{2}+t\xi
F_{\mu}^{\prime}(\xi)(1-\chi(\xi))\xi^{2}-\chi^{\prime}(\xi)\xi^{2}-2(1-\chi(\xi))\xi}{\xi^{4}}|_{L^{2}}$
$\displaystyle\lesssim$ $\displaystyle 1+\mu t.$

Now we prove
$|\int_{0}^{t}e^{-i(t-t^{\prime})D\mathrm{F}_{\mu}}\mathrm{F}_{\mu}[r\partial_{x}r]\operatorname{dt}^{\prime}|_{W^{\alpha,1}}\lesssim(1+\mu
t)^{\theta}\int_{0}^{t}|r|^{2}_{H^{\alpha+4}}\operatorname{dt}^{\prime}$, so
that for all times $t\in[0,\frac{T}{\varepsilon}]$ one has

$\displaystyle|\frac{3\varepsilon}{2}\int_{0}^{t}e^{-i(t-t^{\prime})D\mathrm{F}_{\mu}}\mathrm{F}_{\mu}[r\partial_{x}r]\operatorname{dt}^{\prime}|_{W^{\alpha,1}}$
$\displaystyle\leq(1+\mu t)^{\theta}C(\mu_{\max},T,|r_{0}|_{H^{\alpha+4}})$
$\displaystyle\leq(1+\mu t)^{\theta}C(\mu_{\max},T,|r_{0}|_{W^{\alpha+5,1}}).$

Using what we did before, we have

$\displaystyle|\int_{0}^{t}e^{-i(t-t^{\prime})D\mathrm{F}_{\mu}}\mathrm{F}_{\mu}[r\partial_{x}r]\operatorname{dt}^{\prime}|_{W^{\alpha,1}}$
$\displaystyle\leq\int_{0}^{t}|e^{-i(t-t^{\prime})D\mathrm{F}_{\mu}}\mathrm{F}_{\mu}[r\partial_{x}r]|_{W^{\alpha,1}}\operatorname{dt}^{\prime}$
$\displaystyle\lesssim\int_{0}^{t}(1+\mu(t-t^{\prime}))^{\theta}|\mathrm{F}_{\mu}[r\partial_{x}r]|_{W^{\alpha+2,1}}\operatorname{dt}^{\prime}.$

Using the continuity of $\mathrm{F}_{\mu}$ from $W^{\alpha+3,1}(\mathbb{R})$
to $W^{\alpha+2,1}(\mathbb{R})$ (see Property 3.15), we get

$\displaystyle|\int_{0}^{t}e^{-i(t-t^{\prime})D\mathrm{F}_{\mu}}\mathrm{F}_{\mu}[r\partial_{x}r]\operatorname{dt}^{\prime}|_{W^{\alpha,1}}$
$\displaystyle\lesssim(1+\mu
t)^{\theta}\int_{0}^{t}|r\partial_{x}r|_{W^{\alpha+3,1}}\operatorname{dt}^{\prime}$
$\displaystyle\lesssim(1+\mu
t)^{\theta}\int_{0}^{t}|r|_{H^{\alpha+4}}^{2}\operatorname{dt}^{\prime}.$

It remains to prove that taking $r_{0}$ and $s_{0}$ in
$W^{\alpha+7}(\mathbb{R})$, $\partial_{t}r$ and $\partial_{t}s$ are in
$W^{\alpha,1}(\mathbb{R})$. We do the reasoning for $r$. From what we proved
above, $r$ is in $W^{\alpha+2,1}(\mathbb{R})$. It is also solution of the
equation

$\displaystyle\partial_{t}r+\mathrm{F}_{\mu}[\partial_{x}r]+\frac{3\varepsilon}{2}\mathrm{F}_{\mu}[r\partial_{x}r]=0.$

So that

$\displaystyle|\partial_{t}r|_{W^{\alpha,1}}\lesssim$
$\displaystyle|\mathrm{F}_{\mu}[\partial_{x}r]|_{W^{\alpha,1}}+|\mathrm{F}_{\mu}[r\partial_{x}r]|_{W^{\alpha,1}}$
$\displaystyle\leq$
$\displaystyle|r|_{W^{\alpha+2,1}}+|r\partial_{x}r|_{W^{\alpha+1,1}}\lesssim|r|^{2}_{W^{\alpha+2,1}}.$

∎

######  Theorem 3.24.

Let $\alpha\geq 0$. Let
$r,s\in\mathcal{C}^{1}([0,\frac{T}{\varepsilon}],W^{\alpha+6,1}(\mathbb{R}))$
be solutions of the equations (3.15). Let also $\mathcal{T}_{\rm{B}}$ be the
transformation (3.13), and define:

$\displaystyle\begin{pmatrix}r_{\rm{c}}\\\
s_{\rm{c}}\end{pmatrix}\colonequals\mathcal{T}_{\rm{B}}(r,s).$ (3.17)

Then, there exists
$R\in\mathcal{C}^{1}([0,\frac{T}{\varepsilon}],H^{\alpha}(\mathbb{R})\times
H^{\alpha})(\mathbb{R})$, such that one has:

$\displaystyle\partial_{t}\begin{pmatrix}r_{\rm{c}}\\\
s_{\rm{c}}\end{pmatrix}=J_{\mu}\nabla(H_{\operatorname{BW}})(r_{\rm{c}},s_{\rm{c}})+(\mu\varepsilon+\varepsilon^{2})R,\
\ \forall t\in[0,\frac{T}{\varepsilon}],$

where $H_{\operatorname{BW}}$ is the Hamiltonian defined in (3.6) and
$|R|_{H^{\alpha}\times H^{\alpha}}\leq
C(\mu_{\max},|r|_{W^{\alpha+6,1}},|s|_{W^{\alpha+6,1}})$. In particular, $R$
is uniformly bounded in $(\mu,\varepsilon)$ for times
$t\in[0,\frac{T}{\max{(\mu,\varepsilon)}}]$ in $H^{\alpha}(\mathbb{R})\times
H^{\alpha}(\mathbb{R})$.

###### Proof.

The existence of the solutions $(r,s)$ is given by the previous lemma 3.23.

We write $\mathcal{T}_{\rm{B}}(r,s)=\begin{pmatrix}T_{1}(r,s)\\\
T_{2}(r,s)\end{pmatrix}=\begin{pmatrix}r+\frac{\varepsilon}{4}\partial_{x}r\partial^{-1}s+\frac{\varepsilon}{4}rs+\frac{\varepsilon}{8}s^{2}\\\
s+\frac{\varepsilon}{4}\partial_{x}s\partial^{-1}r+\frac{\varepsilon}{4}rs+\frac{\varepsilon}{8}r^{2}\end{pmatrix}$.
By definition of $H_{\operatorname{BW}}$ (see (3.6)) and
$H_{\operatorname{Wh}}$ (see Notation 3.21), and the computations in the proof
of Proposition 3.17 we have

$\displaystyle H_{\operatorname{BW}}(\mathcal{T}_{\rm{B}}(r,s))$
$\displaystyle=\int_{\mathbb{R}}(T_{1}(r,s)^{2}+T_{2}(r,s)^{2})\operatorname{dx}+\frac{\varepsilon}{2}\int(T_{1}(r,s)^{3}+T_{2}(r,s)^{3})\operatorname{dx}$
$\displaystyle-\frac{\varepsilon}{2}\int_{\mathbb{R}}(T_{1}(r,s)^{2}T_{2}(r,s)+T_{1}(r,s)T_{2}(r,s)^{2})\operatorname{dx}$
$\displaystyle=H_{\operatorname{Wh}}(r,s)$
$\displaystyle+\int_{\mathbb{R}}[(\frac{\varepsilon}{4}\partial_{x}(r)\partial^{-1}(s)+\frac{\varepsilon}{4}rs+\frac{\varepsilon}{8}s^{2})^{2}+(\frac{\varepsilon}{4}\partial_{x}(s)\partial^{-1}(r)+\frac{\varepsilon}{4}rs+\frac{\varepsilon}{8}r^{2})^{2}]\operatorname{dx}$
$\displaystyle+\frac{\varepsilon}{2}\int_{\mathbb{R}}(T_{1}(r,s)^{3}-r^{3}+T_{2}(r,s)^{3}-s^{3})\operatorname{dx}$
$\displaystyle-\frac{\varepsilon}{2}\int_{\mathbb{R}}(T_{1}(r,s)^{2}T_{2}(r,s)-r^{2}s+T_{1}(r,s)T_{2}(r,s)^{2}-rs^{2})\operatorname{dx}$

So
$H_{\operatorname{BW}}(\mathcal{T}_{\rm{B}}(r,s))=H_{\operatorname{Wh}}(r,s)+\varepsilon^{2}\int_{\mathbb{R}}p(r,s,\partial_{x}(r)\partial^{-1}(s),\partial_{x}(s)\partial^{-1}(r))\operatorname{dx}$,
where $p:\mathbb{R}^{4}\to\mathbb{R}$ is polynomial.

But $(r_{\rm{c}},s_{\rm{c}})=\mathcal{T}_{\rm{B}}(r,s)$. So

$\displaystyle\partial_{t}\begin{pmatrix}r_{\rm{c}}\\\
s_{\rm{c}}\end{pmatrix}=\frac{\partial\mathcal{T}_{\rm{B}}}{\partial(r,s)}\partial_{t}\begin{pmatrix}r\\\
s\end{pmatrix},$

where $\frac{\partial\mathcal{T}_{\rm{B}}}{\partial(r,s)}$ is the Jacobian
matrix of $\mathcal{T}_{\rm{B}}(r,s)$ computed in the proof of Proposition
3.19. Since $r$ and $s$ are solutions of the Hamilton’s equations associated
with $H_{\operatorname{Wh}}$, we have

$\displaystyle\partial_{t}\begin{pmatrix}r_{\rm{c}}\\\
s_{\rm{c}}\end{pmatrix}$
$\displaystyle=\frac{\partial\mathcal{T}_{\rm{B}}}{\partial(r,s)}J_{\mu}\nabla(H_{\operatorname{Wh}})(r,s)$
$\displaystyle=\frac{\partial\mathcal{T}_{\rm{B}}}{\partial(r,s)}J_{\mu}\nabla(H_{\operatorname{BW}}(\mathcal{T}_{\rm{B}}(r,s))-\varepsilon^{2}\frac{\partial\mathcal{T}_{\rm{B}}}{\partial(r,s)}J_{\mu}\nabla\int_{\mathbb{R}}p(r,s,\partial_{x}r\partial^{-1}s,\partial_{x}s\partial^{-1}r)\operatorname{dx}$
$\displaystyle=\frac{\partial\mathcal{T}_{\rm{B}}}{\partial(r,s)}J_{\mu}\Big{(}\frac{\partial\mathcal{T}_{\rm{B}}}{\partial(r,s)}\Big{)}^{*}\nabla(H_{\operatorname{BW}})(r_{\rm{c}},s_{\rm{c}})-\varepsilon^{2}\frac{\partial\mathcal{T}_{\rm{B}}}{\partial(r,s)}J_{\mu}\nabla\int_{\mathbb{R}}p(r,s,\partial_{x}r\partial^{-1}s,\partial_{x}s\partial^{-1}r)\operatorname{dx}.$

Moreover by Proposition 3.19, we know that
$\frac{\partial\mathcal{T}_{\rm{B}}}{\partial(r,s)}J_{\mu}\Big{(}\frac{\partial\mathcal{T}_{\rm{B}}}{\partial(r,s)}\Big{)}^{*}=J_{\mu}+(\mu\varepsilon+\varepsilon^{2})R$,
with $|RU|_{H^{\alpha}\times H^{\alpha}}\leq
C(\mu_{\max},|r|_{W^{\alpha+4,1}},|s|_{W^{\alpha+4,1}})|U|_{H^{\alpha+4}}$ for
any $U\in H^{\alpha+4}(\mathbb{R})\times H^{\alpha+4}(\mathbb{R})$. So, here,
we need to estimate $\nabla(H_{\operatorname{BW}})(r_{\rm{c}},s_{\rm{c}})$ in
$H^{\alpha+4}(\mathbb{R})\times H^{\alpha+4}(\mathbb{R})$.

$\displaystyle\nabla(H_{\operatorname{BW}})(\mathcal{T}_{\rm{B}}(r,s))=\begin{pmatrix}2T_{1}(r,s)+\frac{3\varepsilon}{2}T_{1}(r,s)^{2}-\varepsilon
T_{1}(r,s)T_{2}(r,s)-\frac{\varepsilon}{2}T_{2}(r,s)^{2}\\\
2T_{2}(r,s)+\frac{3\varepsilon}{2}T_{2}(r,s)^{2}-\varepsilon
T_{1}(r,s)T_{2}(r,s)-\frac{\varepsilon}{2}T_{1}(r,s)^{2}\end{pmatrix}.$

Combining the algebra properties of $H^{\alpha+4}(\mathbb{R})$ with the
estimates

$\displaystyle\begin{cases}|\partial_{x}(r)\partial^{-1}(s)|_{H^{\alpha+4}}\lesssim|\partial^{-1}s|_{W^{\alpha+4,\infty}}|\partial_{x}r|_{H^{\alpha+4}}\lesssim|s|_{W^{\alpha+4,1}}|r|_{H^{\alpha+5}}\lesssim|s|_{W^{\alpha+4,1}}|r|_{W^{\alpha+6,1}},\\\
|\partial_{x}(s)\partial^{-1}(r)|_{H^{\alpha+4}}\lesssim|r|_{W^{\alpha+4,1}}|s|_{W^{\alpha+6,1}}\end{cases}$

(see (3.14)), we have

$\displaystyle|\nabla(H_{\operatorname{BW}})(\mathcal{T}_{\rm{B}}(r,s))|_{H^{\alpha+4}}\leq
C(||r|_{W^{\alpha+6,1}},|s|_{W^{\alpha+6,1}}).$

It remains to prove that
$\frac{\partial\mathcal{T}_{\rm{B}}}{\partial(r,s)}J_{\mu}\nabla\int_{\mathbb{R}}p(r,s,\partial_{x}(r)\partial^{-1}(s),\partial_{x}(s)\partial^{-1}(r))\operatorname{dx}$
is in $\mathcal{C}^{1}([0,\frac{T}{\varepsilon}],H^{\alpha}(\mathbb{R})\times
H^{\alpha}(\mathbb{R}))$.

The polynomial $p$ is a linear combination of terms of the form
$r^{n_{1}}s^{n_{2}}(\partial_{x}(r)\partial^{-1}(s))^{n_{3}}(\partial_{x}(s)\partial^{-1}(r))^{n_{4}}$
where $n_{1},n_{2},n_{3}$ and $n_{4}$ are non-negative integers such that
$n_{1}+n_{2}+n_{3}+n_{4}\geq 2$. Moreover

$\displaystyle
J_{\mu}\nabla\int_{\mathbb{R}}r^{n_{1}}s^{n_{2}}(\partial_{x}(r)\partial^{-1}(s))^{n_{3}}(\partial_{x}(s)\partial^{-1}(r))^{n_{4}}\operatorname{dx}=\begin{pmatrix}M_{1}\\\
M_{2}\end{pmatrix},$

where

$\displaystyle\begin{cases}M_{1}=&-\frac{1}{2}\mathrm{F}_{\mu}\Big{[}\partial_{x}\Big{(}n_{1}r^{n_{1}-1}s^{n_{2}}(\partial_{x}(r)\partial^{-1}(s))^{n_{3}}(\partial_{x}(s)\partial^{-1}(r))^{n_{4}}\Big{)}\Big{]}\\\
&+\frac{1}{2}\mathrm{F}_{\mu}\Big{[}\partial_{x}^{2}\Big{(}r^{n_{1}}s^{n_{2}}n_{3}(\partial_{x}r)^{n_{3}-1}(\partial^{-1}s)^{n_{3}}(\partial_{x}(s)\partial^{-1}(r))^{n_{4}}\Big{)}\Big{]}\\\
&+\frac{1}{2}\mathrm{F}_{\mu}\Big{[}r^{n_{1}}s^{n_{2}}(\partial_{x}(r)\partial^{-1}(s))^{n_{3}}(\partial_{x}s)^{n_{4}}n_{4}(\partial^{-1}r)^{n_{4}-1}\Big{]},\\\
M_{2}=&\frac{1}{2}\mathrm{F}_{\mu}\Big{[}\partial_{x}\Big{(}r^{n_{1}}n_{2}s^{n_{2}-1}(\partial_{x}(r)\partial^{-1}(s))^{n_{3}}(\partial_{x}(s)\partial^{-1}(r))^{n_{4}}\Big{)}\Big{]}\\\
&-\frac{1}{2}\mathrm{F}_{\mu}\Big{[}r^{n_{1}}s^{n_{2}}(\partial_{x}r)^{n_{3}}n_{3}(\partial^{-1}s)^{n_{3}-1}(\partial_{x}(s)\partial^{-1}(r))^{n_{4}}\Big{]}\\\
&-\frac{1}{2}\mathrm{F}_{\mu}\Big{[}\partial_{x}^{2}\Big{(}r^{n_{1}}s^{n_{2}}(\partial_{x}(r)\partial^{-1}(s))^{n_{3}}n_{4}(\partial_{x}s)^{n_{4}-1}(\partial^{-1}r)^{n_{4}}\Big{)}\Big{]},\end{cases}$

where we used $\partial_{x}\partial^{-1}u=u$ when $u\in L^{1}(\mathbb{R})$
(see Definition/Property 3.12). The property $n_{1}+n_{2}+n_{3}+n_{4}\geq 2$
implies that the terms containing $\partial^{-1}s$ or $\partial^{-1}r$ are
always multiplied by a function in a Sobolev space $W^{\beta,1}(\mathbb{R})$
with $\beta\geq 0$.

We recall now the expression of
$\frac{\partial\mathcal{T}_{\rm{B}}}{\partial(r,s)}$:

$\displaystyle\frac{\partial\mathcal{T}_{\rm{B}}}{\partial(r,s)}=\begin{pmatrix}1+\frac{\varepsilon}{4}s+\frac{\varepsilon}{4}\partial^{-1}(s)\partial_{x}(\circ)&\frac{\varepsilon}{4}\partial_{x}(r)\partial^{-1}(\circ)+\frac{\varepsilon}{4}(r+s)\\\
\frac{\varepsilon}{4}\partial_{x}(s)\partial^{-1}(\circ)+\frac{\varepsilon}{4}(r+s)&1+\frac{\varepsilon}{4}r+\frac{\varepsilon}{4}\partial^{-1}(r)\partial_{x}(\circ)\end{pmatrix}.$

From the above computations we get

$\displaystyle\frac{\partial\mathcal{T}_{\rm{B}}}{\partial(r,s)}J_{\mu}\nabla\int_{\mathbb{R}}p(r,s,\partial_{x}(r)\partial^{-1}(s),\partial_{x}(s)\partial^{-1}(r))\operatorname{dx}$
(3.18) $\displaystyle=$
$\displaystyle\begin{pmatrix}M_{1}+\frac{\varepsilon}{4}sM_{1}+\frac{\varepsilon}{4}\partial^{-1}(s)\partial_{x}(M_{1})+\frac{\varepsilon}{4}\partial_{x}(r)\partial^{-1}(M_{2})+\frac{\varepsilon}{4}(r+s)M_{2}\\\
\frac{\varepsilon}{4}\partial_{x}(s)\partial^{-1}(M_{1})+\frac{\varepsilon}{4}(r+s)M_{1}+M_{2}+\frac{\varepsilon}{4}rM_{2}+\frac{\varepsilon}{4}\partial^{-1}(r)\partial_{x}(M_{2})\end{pmatrix},$
(3.19)

We estimate the terms thus obtained. Using product estimates A.1 and (3.14) we
have

$\displaystyle\begin{cases}|sM_{1}|_{H^{\alpha}}\lesssim|s|_{H^{\alpha+1}}|M_{1}|_{H^{\alpha}},\\\
|\partial^{-1}(s)\partial_{x}(M_{1})|_{H^{\alpha}}\lesssim|\partial^{-1}s|_{W^{\alpha,\infty}}|M_{1}|_{H^{\alpha+1}}\lesssim|s|_{W^{\alpha,1}}|M_{1}|_{H^{\alpha+1}},\\\
|\partial_{x}(r)\partial^{-1}(M_{1})|_{H^{\alpha}}\lesssim|r|_{H^{\alpha+1}}|\partial^{-1}M_{1}|_{W^{\alpha,\infty}}\lesssim|r|_{W^{\alpha+2,1}}|M_{1}|_{W^{\alpha,1}}.\end{cases}$

Idem for the terms with $M_{2}$. So that it only remains to estimate $M_{1}$
and $M_{2}$. We do it for $M_{1}$. Using product estimates A.1 and
$n_{1}+n_{2}+n_{3}+n_{4}\geq 2$, we get

$\displaystyle|M_{1}|_{H^{\alpha}}\lesssim|\partial^{-1}r|_{W^{\alpha+2,\infty}}^{N_{1}}|\partial^{-1}s|_{W^{\alpha+2,\infty}}^{N_{2}}|r|_{H^{\alpha+3}}^{N_{3}}|s|_{H^{\alpha+3}}^{N_{4}},$

where $N_{1},N_{2},N_{3}$ and $N_{4}$ are non negative integers such that
$(N_{3},N_{4})\neq(0,0)$. Then, using (3.14) and Sobolev embeddings, we have

$\displaystyle|M_{1}|_{H^{\alpha}}\lesssim|r|_{W^{\alpha+4,1}}^{N_{1}+N_{3}}|s|_{W^{\alpha+4,1}}^{N_{2}+N_{4}}.$

Moreover, using the continuity of $\mathrm{F}_{\mu}$ from
$W^{\alpha+1,1}(\mathbb{R})$ to $W^{\alpha,1}(\mathbb{R})$ and the algbra
properties of $W^{\beta,1}(\mathbb{R})$ for any $\beta\geq 1$, we get

$\displaystyle|M_{1}|_{W^{\alpha,1}}\lesssim|\partial^{-1}r|_{W^{\alpha+3,\infty}}^{N_{1}}|\partial^{-1}s|_{W^{\alpha+3,\infty}}^{N_{2}}|r|_{W^{\alpha+4,1}}^{N_{3}}|s|_{W^{\alpha+4,1}}^{N_{4}}\lesssim|r|_{W^{\alpha+4,1}}^{N_{1}+N_{3}}|s|_{W^{\alpha+4,1}}^{N_{2}+N_{4}}.$

∎

Combining Proposition 3.2 and Theorem 3.24 we get Theorem 1.14 which we recall
here for the sake of clarity.

######  Theorem 3.25.

Let $\alpha\geq 0$. Let
$r,s\in\mathcal{C}^{1}([0,\frac{T}{\varepsilon}],W^{\alpha+7,1}(\mathbb{R}))$
be solutions of the equations (3.15). Let also
$\mathcal{T}_{\rm{I}},\mathcal{T}_{\rm{D}}$ and $\mathcal{T}_{B}$ be the
transformations of Definition 1.13 and define:

$\displaystyle\begin{pmatrix}\zeta_{\operatorname{Wh}}\\\
\psi_{\operatorname{Wh}}\end{pmatrix}\colonequals\mathcal{T}_{\rm{I}}(\mathcal{T}_{\rm{D}}(\mathcal{T}_{\rm{B}}(r,s))).$
(3.20)

Then, there exists
$R\in\mathcal{C}^{1}([0,\frac{T}{\varepsilon}],H^{\alpha}(\mathbb{R})\times
H^{\alpha}(\mathbb{R}))$, such that one has:

$\displaystyle\partial_{t}\begin{pmatrix}\zeta_{\operatorname{Wh}}\\\
\psi_{\operatorname{Wh}}\end{pmatrix}=J\nabla(H_{\operatorname{WW}})(\zeta_{\operatorname{Wh}},\psi_{\operatorname{Wh}})+(\mu\varepsilon+\varepsilon^{2})R,\
\ \forall t\in[0,\frac{T}{\varepsilon}],$

where $H_{\operatorname{WW}}$ is the Hamiltonian defined in (1.2) and
$|R|_{H^{\alpha}\times\dot{H}^{\alpha+1}}\leq
C(\mu_{\max},|r|_{W^{\alpha+7,1}},|s|_{W^{\alpha+7,1}})$. In particular, $R$
is uniformly bounded in $(\mu,\varepsilon)$ for times
$t\in[0,\frac{T}{\max{(\mu,\varepsilon)}}]$ in
$H^{\alpha}(\mathbb{R})\times\dot{H}^{\alpha+1}(\mathbb{R})$.

###### Proof.

We start by defining the quantities

$\displaystyle\begin{pmatrix}\zeta_{\operatorname{Wh}}\\\
v_{\operatorname{Wh}}\end{pmatrix}\colonequals\mathcal{T}_{\rm{D}}(\mathcal{T}_{\rm{B}}(r,s)).$
(3.21)

Differentiating (3.21) in time and using Theorem 3.24 we get

$\displaystyle\partial_{t}\begin{pmatrix}\zeta_{\operatorname{Wh}}\\\
v_{\operatorname{Wh}}\end{pmatrix}=$
$\displaystyle\frac{\partial\mathcal{T}_{\rm{D}}}{\partial(r,s)}(\mathcal{T}_{\rm{B}}(r,s))\frac{\partial\mathcal{T}_{\rm{B}}}{\partial(r,s)}\partial_{t}\begin{pmatrix}r\\\
s\end{pmatrix}$ $\displaystyle=$
$\displaystyle\frac{\partial\mathcal{T}_{\rm{D}}}{\partial(r,s)}(\mathcal{T}_{\rm{B}}(r,s))J_{\mu}\nabla(H_{\operatorname{BW}})(\mathcal{T}_{\rm{B}}(r,s))+(\mu\varepsilon+\varepsilon^{2})\frac{\partial\mathcal{T}_{\rm{D}}}{\partial(r,s)}(\mathcal{T}_{\rm{B}}(r,s))R_{1},$

where
$\frac{\partial\mathcal{T}_{\rm{D}}}{\partial(r,s)}(\mathcal{T}_{\rm{B}}(r,s))=\begin{pmatrix}1&1\\\
\mathrm{F}_{\mu}^{-1}[\circ]&-\mathrm{F}_{\mu}^{-1}[\circ]\end{pmatrix}$ and
$R_{1}$ is the rest in Theorem 3.24. Using now Proposition 3.6 we get

$\displaystyle\partial_{t}\begin{pmatrix}\zeta_{\operatorname{Wh}}\\\
v_{\operatorname{Wh}}\end{pmatrix}$
$\displaystyle=\widetilde{J}\nabla(\widetilde{H}_{0}+\varepsilon\widetilde{H}_{1})(\mathcal{T}_{\rm{D}}(\mathcal{T}_{\rm{B}}(r,s))+(\mu\varepsilon+\varepsilon^{2})R_{2}$
$\displaystyle=\widetilde{J}\nabla(\widetilde{H}_{0}+\varepsilon\widetilde{H}_{1})(\zeta_{\operatorname{Wh}},v_{\operatorname{Wh}})+(\mu\varepsilon+\varepsilon^{2})R_{2},$

where $\widetilde{J}=\begin{pmatrix}0&-\partial_{x}\\\
-\partial_{x}&0\end{pmatrix}$ and $|R_{2}|_{H^{\alpha}\times H^{\alpha}}\leq
C(|r|_{W^{\alpha+7,1}},|s|_{W^{\alpha+7,1}})$ (because for any $\beta\geq 0$
and any $u\in W^{\beta+1,1}(\mathbb{R})$,
$|\mathrm{F}_{\mu}^{-1}[u]|_{H^{\beta}}\lesssim|u|_{H^{\beta+1/2}}\lesssim|u|_{W^{\beta+1}}$).
Changing the unknowns into
$(\zeta_{\operatorname{Wh}},\psi_{\operatorname{Wh}})$ we obtain

$\displaystyle\partial_{t}\begin{pmatrix}\zeta_{\operatorname{Wh}}\\\
\psi_{\operatorname{Wh}}\end{pmatrix}=J\nabla(H_{0}+\varepsilon
H_{1})(\zeta_{\operatorname{Wh}},\psi_{\operatorname{Wh}})+(\mu\varepsilon+\varepsilon^{2})R_{3},$

where $J=\begin{pmatrix}0&1\\\ -1&0\end{pmatrix}$ and
$|R_{3}|_{H^{\alpha}\times\dot{H}^{\alpha+1}}\leq
C(|r|_{W^{\alpha+7,1}},|s|_{W^{\alpha+7,1}})$ So that

$\displaystyle\partial_{t}\begin{pmatrix}\zeta_{\operatorname{Wh}}\\\
\psi_{\operatorname{Wh}}\end{pmatrix}$
$\displaystyle=J\nabla(H_{\operatorname{WW}})(\zeta_{\operatorname{Wh}},\psi_{\operatorname{Wh}})+(J\nabla(H_{0}+\varepsilon
H_{1})(\zeta_{\operatorname{Wh}},\psi_{\operatorname{Wh}})-J\nabla(H_{\operatorname{WW}})(\zeta_{\operatorname{Wh}},\psi_{\operatorname{Wh}}))$
$\displaystyle+(\mu\varepsilon+\varepsilon^{2})R_{2}.$

Using Proposition 3.2 we get the result. The uniform boundedness in
$(\mu,\varepsilon)$ for times $t\in[0,\frac{T}{\max{(\mu,\varepsilon)}}]$ of
the remainder comes from the lemma 3.23. ∎

It only remains to prove Corollary 1.16.

###### Proof.

We construct $(r_{0},s_{0})$ such that

$\displaystyle|\mathcal{T}_{\rm{I}}(\mathcal{T}_{\rm{D}}(\mathcal{T}_{\rm{B}}(r_{0},s_{0})))-(\zeta_{0},\psi_{0})|_{H^{\alpha}\times\dot{H}^{\alpha+1}}\lesssim\varepsilon^{2}.$
(3.22)

We remark first that the transformations $\mathcal{T}_{\rm{I}}$ and
$\mathcal{T}_{\rm{D}}$ are invertible with, for any $\beta\geq 0$,
$\mathcal{T}_{\rm{I}}^{-1}:W^{\beta,1}(\mathbb{R})\times\dot{W}^{\beta+1,1}(\mathbb{R})\to
W^{\beta,1}(\mathbb{R})\times W^{\beta,1}(\mathbb{R})$ and
$\mathcal{T}_{\rm{D}}^{-1}:W^{\beta+1,1}(\mathbb{R})\times
W^{\beta+1,1}(\mathbb{R})\to W^{\beta,1}(\mathbb{R})\times
W^{\beta,1}(\mathbb{R})$ due to the continuity of $\mathrm{F}_{\mu}$ from
$W^{\beta+1,1}(\mathbb{R})$ to $W^{\beta,1}(\mathbb{R})$. Moreover, by
definition (see (3.13)), we can write $\mathcal{T}_{\rm{B}}$ under the form

$\displaystyle\mathcal{T}_{\rm{B}}=\rm{I}d+\varepsilon\widetilde{\mathcal{T}_{\rm{B}}},$

where $\rm{I}d$ is the identity and for any $\beta\geq 0$,
$\widetilde{\mathcal{T}_{\rm{B}}}:W^{\beta+1,1}(\mathbb{R})\times
W^{\beta+1,1}(\mathbb{R})\to W^{\beta,1}(\mathbb{R})\times
W^{\beta,1}(\mathbb{R})$. We define the transformation
$\mathcal{T}_{\rm{B}}^{\operatorname{inv}}:W^{\beta+1,1}(\mathbb{R})\times
W^{\beta+1,1}(\mathbb{R})\to W^{\beta,1}(\mathbb{R})\times
W^{\beta,1}(\mathbb{R})$ by

$\displaystyle\mathcal{T}_{\rm{B}}^{\operatorname{inv}}=\rm{I}d-\varepsilon\widetilde{\mathcal{T}_{\rm{B}}}.$

One has for any $(\eta,w)\in W^{\beta+2,1}(\mathbb{R})\times
W^{\beta+2,1}(\mathbb{R})$,
$\mathcal{T}_{\rm{B}}(\mathcal{T}_{\rm{B}}^{\operatorname{inv}}(\eta,w))=\begin{pmatrix}\eta\\\
w\end{pmatrix}+\varepsilon^{2}R$ where $|R|_{W^{\beta,1}\times
W^{\beta,1}}\leq C(|\eta|_{W^{\beta+2,1}},|w|_{W^{\beta+2,1}})$. So that

$\displaystyle(r_{0},s_{0})=\mathcal{T}_{\rm{B}}^{\operatorname{inv}}(\mathcal{T}_{\rm{D}}^{-1}(\mathcal{T}_{\rm{I}}^{-1}(\zeta_{0},\psi_{0})))$
(3.23)

satisfy (3.22). ∎

##  Appendix A Technical tools

######  Proposition A.1.

(Product estimates)

  1. 1.

Let $t_{0}>1/2$, $s\geq-t_{0}$ and $f\in H^{\alpha}\cap
H^{t_{0}}(\mathbb{R}),g\in H^{\alpha}(\mathbb{R})$. Then $fg\in
H^{\alpha}(\mathbb{R})$ and

$\displaystyle|fg|_{H^{\alpha}}\lesssim|f|_{H^{\max{(t_{0},\alpha)}}}|g|_{H^{\alpha}}$

  2. 2.

Let $\alpha_{1},\alpha_{2}\in\mathbb{R}$ be such that
$\alpha_{1}+\alpha_{2}\geq 0$. Then for all $\alpha\leq\alpha_{j}$ $(j=1,2)$
and $\alpha<\alpha_{1}+\alpha_{2}-1/2$, and all $f\in
H^{\alpha_{1}}(\mathbb{R}),g\in H^{\alpha_{2}}(\mathbb{R})$, one has $fg\in
H^{\alpha}(\mathbb{R})$ and

$\displaystyle|fg|_{H^{\alpha}}\lesssim|f|_{H^{\alpha_{1}}}|g|_{H^{\alpha_{2}}}$

###### Proof.

See Appendix B.1 in [13]. ∎

######  Proposition A.2.

(Composition estimates) Let $G:\mathbb{R}\to\mathbb{R}$ be a smooth function
such that $G(0)=0$. Also, let $t_{0}>1/2$, $\alpha\geq 0$ and $f\in
H^{\max{(t_{0},\alpha)}}(\mathbb{R})$. Then $G(f)\in H^{\alpha}(\mathbb{R})$
and

$\displaystyle|G(f)|_{H^{\alpha}}\leq
C(|f|_{H^{\max{(t_{0},\alpha)}}})|f|_{H^{\alpha}}.$

###### Proof.

See Appendix B.1 in [13]. ∎

######  Proposition A.3.

(Quotient estimates) Let $t_{0}>1/2,\alpha\geq-t_{0}$ and $c_{0}>0$. Also let
$f\in H^{\alpha}(\mathbb{R})$ and $g\in H^{\alpha}\cap H^{t_{0}}(\mathbb{R})$
be such that for all $x\in\mathbb{R}$, one has $1+g(X)\geq c_{0}$. Then
$\frac{f}{1+g}$ belongs to $H^{\alpha}(\mathbb{R})$ and

$\displaystyle|\frac{f}{1+g}|_{H^{\alpha}}\leq
C(\frac{1}{c_{0}},|g|_{H^{\max{(t_{0},\alpha)}}})|f|_{H^{\alpha}}$

###### Proof.

See Appendix B.1 in [13]. ∎

######  Definition A.4.

We say that a Fourier multiplier $\mathrm{F}(D)$ is of order $\alpha$
$(\alpha\in\mathbb{R})$ and write $\mathrm{F}\in\mathcal{S}^{\alpha}$ if
$\xi\in\mathbb{R}\mapsto F(\xi)\in\mathbb{C}$ is smooth and satisfies

$\displaystyle\forall\xi\in\mathbb{R},\forall\beta\in\mathbb{N},\ \
\sup_{\xi\in\mathbb{R}}\langle\xi\rangle^{\beta-\alpha}|\partial^{\beta}F(\xi)|<\infty.$

We also introduce the seminorm

$\displaystyle\mathcal{N}^{\alpha}(F)=\sup_{\beta\in\mathbb{N},\beta\leq
4}\sup_{\xi\in\mathbb{R}}\langle\xi\rangle^{\beta-\alpha}|\partial^{\beta}F(\xi)|.$

######  Proposition A.5.

Let $t_{0}>1/2$, $\alpha\geq 0$ and $F\in\mathcal{S}^{\alpha}$. If $f\in
H^{\alpha}\cap H^{t_{0}+1}$ then, for all $g\in H^{\alpha-1}$,

$\displaystyle|[F(D),f]g|_{2}\leq\mathcal{N}^{\alpha}(F)|f|_{H^{\max{(t_{0}+1,\alpha)}}}|g|_{H^{\alpha-1}}.$

###### Proof.

See Appendix B.2 in [13] for a proof of this proposition. ∎

######  Proposition A.6.

Let $\alpha\geq 2$. Let $\zeta\in H^{\alpha+2}(\mathbb{R})$ be such that (1.3)
is satisfied and $\psi\in\dot{H}^{\alpha+2}(\mathbb{R})$. Then one has

$\displaystyle|\frac{1}{\mu}\mathcal{G}^{\mu}[\varepsilon\zeta]\psi|_{H^{\alpha}}\leq
M(s+2)|\psi|_{\dot{H}^{\alpha+2}}.$

###### Proof.

This is a direct consequence of Theorem 3.15 in [13]. ∎

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