# Transverse linear stability of one-dimensional solitary gravity water waves

Frédéric Rousset  Université Paris-Saclay, CNRS, Laboratoire de Mathématiques
d’Orsay (UMR 8628), 91405 Orsay Cedex, France  frederic.rousset@universite-
paris-saclay.fr  and  Changzhen Sun  Université Bourgogne Franche-Comté,
Laboratoire de Mathématiques de Besançon, UMR CNRS 6623  changzhen.sun@univ-
fcomte.fr

###### Abstract.

In this paper, we establish the transverse linear asymptotic stability of one-
dimensional small-amplitude solitary waves of the gravity water-waves system.
More precisely, we show that the semigroup of the linearized operator about
the solitary wave decays exponentially within a spectral subspace
supplementary to the space generated by the spectral projection on continuous
resonant modes. The key element of the proof is to establish suitable uniform
resolvent estimates. To achieve this, we use different arguments depending on
the size of the transverse frequencies, for high transverse frequencies, we
use reductions based on pseudodifferential calculus, for intermediate ones, we
use an energy-based approach relying on the design of various appropriate
energy functionals for different regimes of longitudinal frequencies and for
low frequencies, we use the KP-II approximation. As a corollary of our main
result, we also get the spectral stability in the unweighted energy space.

Keywords: solitary waves; asymptotic stability; water waves system; resolvent
estimates.

2010 MSC: 35B40, 35C08, 35Q31, 76B15, 76B25.

###### Contents

  1. 1 Introduction
    1. 1.1 The line solitary waves
    2. 1.2 Linearization of (1.7) about line solitary waves
    3. 1.3 Outline of the proof and main difficulties
      1. 1.3.1 Resonant modes
      2. 1.3.2 Uniform resolvent estimate
  2. 2 Spectral study in the regime of low transverse frequency
    1. 2.1 KP-II approximation
    2. 2.2 Existence of resonant modes–proof of Theorem 1.2
  3. 3  Spectral stability in the uniform high and intermediate transverse frequency regions.
    1. 3.1 Proof of Proposition 3.2
    2. 3.2 Proof of Proposition 3.3
      1. 3.2.1 Proof of (3.21)
      2. 3.2.2 Proof of (3.22)
      3. 3.2.3 Proof of (3.23)
  4. 4 Spectral stability for low transverse frequencies
    1. 4.1 Proof of the estimate (4.21)
    2. 4.2 Proof of the estimate (4.22)
  5. 5 Spectral stability in the unweighted space
  6. A A technical lemma using pseudodifferential calculus
  7. B Some algebraic properties for the spectrum of the linearized operator
  8. C Study of the transformed Dirichlet-Neumann operator
  9. D Generalized kernel of $L(0)$
  10. E The error between the resonant modes of water waves and KP-II
    1. E.1 Resonant modes for the linearized operator of KP-II equation
    2. E.2 Expansions of the basis and dual basis for water waves

##  1\. Introduction

In this paper, we study the transverse linear asymptotic stability of line
solitons for the three-dimensional gravity water-waves system in a channel
with finite depth. The equations of motions are given by

$\left\\{\begin{array}[]{l}\partial_{t}U+U\cdot\nabla U+\nabla
P+ge_{3}=0,\\\\[3.0pt] \,\mathrm{div}\,U=0,\,\mathop{\rm curl}\nolimits
U=0.\end{array}\right.(t,X,z)\in\Omega_{t}\,.$ (1.1)

Here the fluid domain $\Omega_{t}$ which is also unknown is defined as

$\Omega_{t}=\\{-h\leq z\leq\zeta(t,X)\\},\quad X=(x,y)\in\mathbb{R}^{2}\,,$

where $h$ is a fixed finite constant and $\zeta$ is the free surface, which
satisfies the kinematic condition

$\partial_{t}\zeta(t,X)+U_{X}(t,X,\zeta(X))\cdot\nabla_{X}\zeta(t,X)-U_{3}(t,X,\zeta(X))=0\,.$
(1.2)

We also impose that the pressure $P$ is constant, for example zero on the free
boundary. At the bottom of the fluid, we add the boundary condition:

$U_{3}|_{z=-h}=0\,.$ (1.3)

Given the mathematical challenges posed by its fully nonlinear nature, the
water wave problem has been the subject of numerous studies over the past four
decades. After the breakthrough works of S. Wu, many results have emerged
concerning the local well-posedness and the long-term behavior of the water
wave system, with or without surface tension, [2, 3, 4, 7, 16, 32, 19, 28, 33,
25, 1, 27, 60, 42, 64, 65, 66, 67, 68, 71, 72]. In this list, all the global
in time results can be viewed as proofs of the asymptotic stability of the
constant solution to the water waves problem, that is to say the stationary
flat surface. Note that a byproduct of these results is the non-existence of
small localized non-decaying in time patterns either in infinite depth or in
finite depth in dimension two (in the sense that the free surface is two-
dimensional).

Nevertheless, there are many physically interesting nontrivial patterns such
as solitary waves which are spatially localized traveling waves, we refer for
example to [36, 8, 29, 17, 21, 14] for rigorous mathematical existence results
or periodic waves [58, 59, 37, 39, 62] (also referred to as Stokes waves). To
justify their observation in nature, the study of the dynamics around these
non-trivial patterns is also of much importance. For example, recently, there
has been significant progress about the spectral and modulational nonlinear
instability of the Stokes waves [10, 13, 46, 9, 26, 24].

The study of the stability of solitary water waves has a long history in the
mathematical literature. Let us first review some results for one-dimensional
water waves (we use again this terminology for the case that the free surface
is one-dimensional) submitted to one-dimensional perturbations. For capillary-
gravity water waves, the spectral stability of one-dimensional solitary waves
subject to finite wave-length perturbation was established in [23]. The
nonlinear orbital stability has been also obtained conditional to the global
existence of the solution by Mielke [40] and Buffoni [11] in the case of
strong surface tension, while Groves and Wahlén proved it for weak surface
tension [22]. These proofs rely either on the general framework of Grillakis-
Shatah-Strauss [20], or on the variational approach of [12] and hence on the
definiteness up to a a finite dimensional subspace of the energy-momentum
functional. For the purely gravity water waves, such a variational approach
does not apply due to the strong undefiniteness of the energy-momentum
functional so that a direct study of the linearized equation and of the
spectrum of the linear operator must be performed. Lin proved in [38] the
linear instability of large amplitude solitary waves. For small amplitude
gravity solitary waves, Pego and Sun [48], in the spirit of the results
previously obtained on the KdV equation and related models in [50], [49],
obtained the linear asymptotic stability by relying on the KdV approximation
in the long wave region in the framework of exponentially weighted spaces.
They also deduced spectral stability in the unweighted energy space. The
question of nonlinear stability for dispersive Hamiltonian PDE for which the
local Cauchy problem involves the control of norms of higher regularity than
the energy space remains a widely open problem.

Here, we shall study the stability of one-dimensional gravity solitary waves
submitted to two-dimensional localized perturbations, this is usually referred
as the study of transverse stability. The study of the transverse stability of
one-dimensional solitary waves also has a long history in dispersive PDE. In
the early 1970’s, by using the theory of integrable systems Zakharov [69]
obtained the transverse instability of the soliton of the KdV equation
considered as a one-dimensional solution of the (two-dimensional) KP-I
equation. A general criterion allowing to get linear transverse instability of
solitary waves has now been obtained in [55]. This criterion applies to a
large class of systems like the KP-I equation and the water-waves system with
strong surface tension (Bond number bigger than $1/3$). One can then obtain
nonlinear instability, see [53], [54] and in particular [56] for the
transverse nonlinear instability of solitary water-waves with strong surface
tension. When the algebraic criterion of [55] is not matched, there are
various model situations. For the hyperbolic Schrödinger equation (where the
usual Laplacian is replaced by $\partial_{x}^{2}-\Delta_{y}$), as established
in [70], there is still transverse linear instability of the one-dimensional
solitary wave whereas for the KP-II equation, there is stability [6]. One can
then conjecture that for small amplitude solitary waves, when the weakly
transverse long-wave regime is given by the KP-II equation, which is the case
of the gravity water waves system (see [35] for the justification for example)
one should get transverse stability. This heuristics was recently justified by
Mizumachi [44] for the Benney-Luke equation which is another asymptotic model
that can be obtained from the water-waves system while keeping the KP-II
equation as a long-wave weakly transverse model. Note that for these
semilinear equations, which are globally well-posed in the energy space, one
can obtain nonlinear asymptotic stability [43], [45]. This remains a very
challenging question for the gravity water-waves system. Here, we thus address
the question of the linear transverse stability of small amplitude solitary
waves for the pure gravity water waves system. Since the solitary wave does
not depend on the transverse variable, one can Fourier transform in the
transverse variable and study a family of linear problems indexed by the
transverse frequency parameter $\eta$. Due to the strongly nonlinear and
nonlocal features of the water waves system, the dependence with respect to
$\eta$ is stronger than in semilinear models like the Benney-Luke equation
studied in [44] and cannot be handled in a perturbative way especially in a
high frequency regime so that new ideas need to be introduced. This will be
explained in more details after the statement of the main results.

###  1.1. The line solitary waves

As the velocity $U$ is divergence and curl free, we can write $U=\nabla\Phi,$
where the velocity potential $\Phi$ solves the elliptic equation:

$\left\\{\begin{array}[]{l}\Delta\Phi=0,\quad(X,z)\in\mathbb{R}^{2}\times\\{-h<z<\zeta(t,X)\\},\\\\[3.0pt]
\Phi|_{z=\zeta}=\varphi(t,X),\quad\partial_{z}\Phi|_{z=-h}=0.\end{array}\right.$
(1.4)

We then define the Dirichlet-Neumann operator

$G[\zeta]\varphi=(\partial_{z}\Phi-\nabla_{X}\zeta\cdot\nabla_{X}\Phi)|_{z=\zeta(t,X)}=\sqrt{1+|\nabla_{X}\zeta|^{2}}\,\nabla\Phi(X,\zeta(t,X))\cdot
n(t,X)$

where

$n(t,X)=\frac{1}{\sqrt{1+|\nabla_{X}\zeta|^{2}}}\big{(}-\nabla_{X}\zeta,1\big{)}^{t}$

is the unit outward normal vector to the free surface. The system (1.1)-(1.3)
is thus equivalent to the following system on $(\zeta,\varphi)^{t}$ evaluated
on the surface:

$\left\\{\begin{array}[]{l}\partial_{t}\zeta=G[\zeta]\varphi,\\\
\partial_{t}\varphi=-\frac{1}{2}|\nabla_{X}\varphi|^{2}+\frac{1}{2}\frac{\big{(}G[\zeta]\varphi+\nabla_{X}\varphi\cdot\nabla_{X}\zeta\big{)}^{2}}{\sqrt{1+|\nabla_{X}\zeta|^{2}}}-g\zeta.\end{array}\right.$
(1.5)

Since we shall study the solitary waves with speed $c>0,$ we first make change
of variable $(x,y)\rightarrow(x-ct,y)$ to change (1.5) as:

$\left\\{\begin{array}[]{l}\partial_{t}\zeta=c\partial_{x}\zeta+G[\zeta]\varphi,\\\
\partial_{t}\varphi=c\partial_{x}\varphi-\frac{1}{2}|\nabla_{X}\varphi|^{2}+\frac{1}{2}\frac{\big{(}G[\zeta]\varphi+\nabla_{X}\varphi\cdot\nabla_{X}\zeta\big{)}^{2}}{\sqrt{1+|\nabla_{X}\zeta|^{2}}}-g\zeta.\end{array}\right.$

As is classical in the study of existence of solitary waves, we perform the
following change of variable to make the equation non-dimensional:

$\hat{t}={ct}/{h},\quad\hat{X}={X}/{h},\quad\tilde{\zeta}=\zeta/h,\quad\tilde{\varphi}=\varphi/ch.$
(1.6)

Then the equation becomes (we omit tilde for the sake of notational
convenience)

$\left\\{\begin{array}[]{l}\partial_{t}\zeta=\partial_{x}\zeta+G[\zeta]\varphi\,,\\\
\partial_{t}\varphi=\partial_{x}\varphi-\frac{1}{2}|\nabla_{X}\varphi|^{2}+\frac{1}{2}\frac{(G[\zeta]\varphi+\nabla_{X}\varphi\cdot\nabla_{X}\zeta)^{2}}{\sqrt{1+|\nabla_{X}\zeta|^{2}}}-\gamma\zeta\,,\end{array}\right.$
(1.7)

where $\gamma=\frac{gh}{c^{2}}.$ Note that the above Zakharov-Craig-Sulem
formulation of the water waves system has the following Hamiltonian structure:

$\partial_{t}\left(\begin{array}[]{l}\zeta\\\
\varphi\end{array}\right)=J\,\delta{\mathcal{H}}[\zeta,\varphi],$

with

$\displaystyle J\,$ $\displaystyle:=\,\begin{pmatrix}0&-1\\\
1&0\end{pmatrix}\,,$ $\displaystyle\mathcal{H}[\zeta,\varphi]$
$\displaystyle=:\frac{1}{2}\int
G[\eta]\varphi\cdot\varphi+\gamma\zeta^{2}-2\,\zeta\,\partial_{x}\varphi\,.$

The existence of one dimensional solitary waves, which are stationary
solutions to (1.7) depending only on the one-dimensional variable $x$, is
established in [8, 31]:

######  Theorem 1.1 (Restatement of Beale’s result [8]).

Assuming $\gamma=1-\epsilon^{2},$ there exists $\epsilon_{0}$ such that for
every $\epsilon\in(0,\epsilon_{0})$ there is a stationary solution
$\zeta_{c}(x),\varphi_{c}(x)$ (independent of $y$), given by

$\zeta_{c}(x)=\epsilon^{2}\Theta(\epsilon
x,\epsilon),\qquad\varphi_{c}(x)=\epsilon\Phi(\epsilon x,\epsilon),$ (1.8)

where $\Theta$ and $\Phi$ satisfy the following:
(1) There exists $d>0$, for any integers $k\geq 0,\,\ell\geq 1,$ there exists
$C_{k},\,C_{\ell}$ such that:

$|\partial_{\hat{x}}^{k}\Theta(\hat{x},\epsilon)|\leq
C_{k}\,e^{-d\,|\hat{x}|},\qquad|\partial_{\hat{x}}^{\ell}\Phi(\hat{x},\epsilon)|\leq
C_{\ell}\,e^{-d\,|\hat{x}|}.$

(2) It holds that when $\epsilon=0,$

$\Theta(\hat{x},0)=\partial_{\hat{x}}\Phi(\hat{x},0)=\Psi_{\textnormal{\scalebox{0.8}{KdV}}}(\hat{x})$
(1.9)

where
$\Psi_{\textnormal{\scalebox{0.8}{KdV}}}(\hat{x})=\mathrm{sech}^{2}\bigg{(}\frac{\sqrt{3}}{2}\hat{x}\bigg{)}$.

Note that $\Psi_{\textnormal{\scalebox{0.8}{KdV}}}$ is the steady solution to
the KdV equation in the moving frame at unit speed:

$-\partial_{x}\Psi+3\Psi\partial_{x}\Psi+\frac{1}{3}\partial_{x}^{3}\Psi=0.$

Note that the existence is proven in [8] by using the Levi-Civita formulation
but the result can be easily transformed into the Zakharov-Craig-Sulem
formulation stated above.

As mentioned previously, the linear stability analysis of small-amplitude
solitary waves of the one dimensional water wave system has been established
in [48] by Pego and Sun. The core of the analysis is performed within the
framework of an exponentially weighted energy space. In this paper, we shall
prove the linear stability of the line solitary wave within the context of the
two-dimensional system described by (1.7).

###  1.2. Linearization of (1.7) about line solitary waves

We now linearize the system (1.7) about a solitary wave
$(\zeta_{c},\varphi_{c})^{t}$ given by Theorem 1.1. By using the expression of
the derivative of the Dirichlet-Neumann operator with respect to the surface
[Theorem 3.2.1, [34]], we find the following linearized system:

$\displaystyle\left\\{\begin{array}[]{l}\displaystyle\partial_{t}\zeta=\partial_{x}\zeta+G[\zeta_{c}]\varphi-G[\zeta_{c}](Z_{\epsilon}\zeta)-\partial_{x}(v_{c}\zeta),\\\\[5.0pt]
\partial_{t}\varphi=(1-v_{c})\partial_{x}\varphi+Z_{c}G[\zeta_{c}](\varphi-
Z_{c}\zeta)-(\gamma+Z_{c}\partial_{x}v_{c})\eta\,,\end{array}\right.$

with

$Z_{c}=\frac{G[\zeta_{c}]\varphi_{c}+\partial_{x}\zeta_{c}\partial_{x}\varphi_{c}}{1+|\partial_{x}\zeta_{c}|^{2}},\qquad\qquad
v_{c}=\partial_{x}\varphi_{c}-Z_{c}\partial_{x}\zeta_{c}.$ (1.10)

We then set ${V}_{1}=\zeta,{V}_{2}=\varphi-Z_{c}\zeta$, and then obtain that
${V}=({V}_{1},{V}_{2})^{t}$ solves:

$\partial_{t}{V}={L}{V},\qquad\qquad{L}=\left(\begin{array}[]{cc}\partial_{x}(d_{c}\cdot)&G[\zeta_{c}]\\\\[5.0pt]
-w_{c}&d_{c}\partial_{x}\end{array}\right)\,,$ (1.11)

where

$d_{c}=1-v_{c},\quad w_{c}=\gamma-d_{c}\partial_{x}Z_{c}.$ (1.12)

Let us now introduce the adapted functional framework. For any
$a\in\mathbb{R},$ we shall use the weighted $L^{2}$ space
$L_{a}^{2}(\mathbb{R}^{2})=\colon
L^{2}(\mathbb{R}^{2};e^{2ax}\mathrm{d}x\mathrm{d}y\,)\,$ endowed with the
norm:

$\|f\|_{L_{a}^{2}(\mathbb{R}^{2})}=\colon\bigg{(}\int|f(x)|^{2}e^{2ax}\mathrm{d}x\mathrm{d}y\bigg{)}^{\frac{1}{2}}.$

We also define the weighted space $H_{a,\star}^{{1}/{2}}(\mathbb{R}^{2})$ as
the closure of smooth compactly supported functions endowed with the norm

$\displaystyle\|f\|_{H_{a,\star}^{{1}/{2}}(\mathbb{R}^{2})}=\colon\bigg{(}\int\big{|}\frac{\nabla}{\langle
D\rangle^{1/2}}f(x,y)\big{|}^{2}e^{2ax}\mathrm{d}x\mathrm{d}y\bigg{)}^{\frac{1}{2}}.$

Let us point out that $\|\cdot\|_{H_{a,\star}^{{1}/{2}}(\mathbb{R}^{2})}$ is
indeed a norm thanks to the Poincaré inequality

$\|\nabla
f\|_{L_{a}^{2}(\mathbb{R}^{2})}\geq|a|\|f\|_{L_{a}^{2}(\mathbb{R}^{2})}.$

We shall investigate the linear asymptotic behavior of the system (1.11) in
the weighted space:

$X_{a}=\colon L_{a}^{2}(\mathbb{R}^{2})\times
H_{a,\star}^{{1}/{2}}(\mathbb{R}^{2}).$ (1.13)

We consider $L$ as a linear operator on $X_{a}$ with domain

$H_{a}^{1}(\mathbb{R}^{2})\times
H_{a,\star}^{{3}/{2}}(\mathbb{R}^{2})=:\big{\\{}F\,\big{|}\,\langle D\rangle
F\in X_{a}\big{\\}}.$

To state our main results, it is convenient to also use weighted energy spaces
in the one-dimensional variable $x$ only but involving in a quantitative way a
parameter $\eta\in\mathbb{R}$ which stands for the transverse frequency:

$L_{a}^{2}(\mathbb{R})=\colon L^{2}(\mathbb{R};e^{2ax}\mathrm{d}x),$
$\displaystyle
H_{a,\star,\eta}^{{1}/{2}}(\mathbb{R})=\bigg{\\{}f\,\big{|}\,\|f\|_{H^{{1}/{2}}_{a,\star}(\mathbb{R})}=\colon\bigg{(}\int\big{|}\frac{(\partial_{x},\eta)}{\langle(D_{x},\eta)\rangle^{1/2}}f(x)\big{|}^{2}e^{2ax}\mathrm{d}x\bigg{)}^{\frac{1}{2}}<+\infty\bigg{\\}}$
(1.14)

and set

$Y_{a,\eta}=L_{a}^{2}(\mathbb{R})\times
H_{a,\star,\eta}^{{1}/{2}}(\mathbb{R}).$ (1.15)

We then consider the following unbounded operator $L({\eta})$ on $Y_{a,\eta}$
defined for any smooth function $U(x)$ by
$L({\eta})(U(x))=e^{-iy\eta}L(e^{iy\eta}U(x)),$ so that $L({\eta})$ takes the
form

$L({\eta})=\left(\begin{array}[]{cc}\partial_{x}(d_{c}\cdot)&G_{\eta}[\zeta_{c}]\\\\[5.0pt]
-w_{c}&d_{c}\partial_{x}\end{array}\right)$ (1.16)

with $G_{\eta}[\zeta_{c}]=e^{-iy\eta}G[\zeta_{c}]e^{iy\eta}$ being defined as
$G_{\eta}[\zeta_{c}]f(x)=\sqrt{1+|\nabla\zeta_{c}|^{2}}(-\partial_{x}\zeta_{c}\cdot\partial_{x}\Phi_{\eta}+\partial_{z}\Phi_{\eta})$
where $\Phi_{\eta}(x,z)$ solves the elliptic problem in
$\Omega=\mathbb{R}\times[-1,\zeta_{c}]:$

$(\Delta_{x,z}-\eta^{2})\Phi_{\eta}=0,\quad\Phi_{\eta}|_{z=\zeta_{c}}=f,\quad\partial_{z}\phi_{\eta}|_{z=-1}=0.$

An important motivation for working in this weighted framework is that the
essential spectrum of the operator $L({\eta})$ which is the imaginary axis in
unweighted spaces is pushed to the left so that the first step in order to
establish linear stability is to investigate the presence of eigenvalues close
to the imaginary axis.

We can observe that when $\eta=0,$ $L(0)$ is the linearized operator for 1D
water waves about the line soliton $(\zeta_{c},\,\varphi_{c}-Z_{c}\zeta_{c})$
studied in [48]. By the translational and Galilean invariances, there is an
eigenvalue $\lambda=0$ with algebraic multiplicity two (see appendix).
Consequently, for nonzero and small $\eta,$ we expect that the operator
$L(\eta)$ will have two small eigenvalues $\lambda(\eta)$ and
$\overline{\lambda(\eta)}$ in the space $Y_{a,\eta}$ bifurcating from
$\lambda(0)=0.$ This will be our first result:

######  Theorem 1.2.

Let $0<a=\hat{a}\epsilon<\frac{\sqrt{3}}{4}\epsilon.$ There exist
$\epsilon_{0}>0,\hat{\eta}_{0}=\hat{\eta}_{0}(\hat{a})>0$ such that for any
$\epsilon\in(0,\epsilon_{0}],\eta\in[-\epsilon^{2}\hat{\eta}_{0},\epsilon^{2}\hat{\eta}_{0}],$
the operators $L(\eta)$ has two pairs of eigenmodes
$(\lambda(\pm\eta),\,U(\cdot,\pm\eta))$ in the space $Y_{a,\eta},$ where

$\lambda(\eta)\in
C^{\infty}\big{(}[-\epsilon^{2}\hat{\eta}_{0},\epsilon^{2}\hat{\eta}_{0}]\big{)}\quad
U(\cdot,\eta)\in
C^{\infty}\big{(}[-\epsilon^{2}\hat{\eta}_{0},\epsilon^{2}\hat{\eta}_{0}],Y_{a,\eta}\big{)}\,.$

Moreover, for any
$\eta\in[-\epsilon^{2}\hat{\eta}_{0},\epsilon^{2}\hat{\eta}_{0}],$ it holds
that

$\displaystyle\lambda(\eta)=i\lambda_{1}\eta-\lambda_{2}\,\eta^{2}+{\mathcal{O}}(\eta^{2})\,,$
$\displaystyle
U(\cdot,\eta)=U^{1}+i\lambda_{1}\eta\,U^{2}+{\mathcal{O}}(\eta^{2})\,\,\,\text{
in }Y_{a,\eta}\,,$
$\displaystyle\overline{\lambda(\eta)}=\lambda(-\eta),\quad\overline{U(\cdot,\eta)}=U(\cdot,-\eta),$

where $\lambda_{1},\lambda_{2}$ are two positive constants.

In the same way, the adjoint operator $L^{*}(\eta)$ has eigenmodes under the
form $(\lambda(\mp\eta),U^{*}(\cdot,\pm\eta))$ where
$\overline{U^{*}(\cdot,\eta)}=U^{*}(\cdot,-\eta).$

######  Remark 1.3.

By the KP-II approximation, it is expected that the resonant modes in the long
wave weakly transverse scaling $U(\epsilon^{-1}\cdot,\epsilon^{2}\hat{\eta})$
should be approximated by the resonant modes of the linearized KP-II equation
(see appendix E). Note that, as shown in [43], the latter is exponentially
growing at $-\infty$ once $\hat{\eta}\neq 0$ (see (E.1)) and thus
$U(\cdot,\eta)$ does not belong to the unweighted space $L^{2}(\mathbb{R}).$

As shown in the Section 2, one can then choose two smooth in $\eta$ functions
$(g_{1}(\cdot,\eta),g_{2}(\cdot,\eta))$ such that for $\eta\neq 0$ they form a
basis of the two dimensional space generated in the space $Y_{a,\eta}$ by
$U(\pm\eta)$ and for $\eta=0$ a basis of the generalized kernel of $L(0).$ Let
$g_{1}^{*}(\cdot,\eta),g_{2}^{*}(\cdot,\eta)\in Y_{a,\eta}^{*}$ be the
corresponding dual basis defined in (2.30). We can then define the spectral
projector

$\displaystyle\mathbb{P}(\eta_{0})f=\sum_{k=1,\,2}\int_{-\eta_{0}}^{\eta_{0}}\langle{\mathcal{F}}_{y}(f)(\cdot,\eta),g_{k}^{*}(\cdot,\eta)\rangle_{{}_{Y_{a,\eta}\times
Y_{a,\eta}^{*}}}\,g_{k}(\cdot,\eta)\,e^{iy\eta}\,\mathrm{d}\eta$ (1.17)

which projects the elements in $X_{a}$ onto the spectral subspaces associated
to the continuous family of eigenvalues
$\\{\lambda(\pm\eta)\\}_{|\eta|\leq\eta_{0}}.$ We refer to (2.28) for the
definition of $\langle\cdot\rangle_{Y_{{a,\eta}}\times Y_{a,\eta}^{*}}.$
Define $\mathbb{Q}(\eta_{0})={\rm Id}-\mathbb{P}(\eta_{0}).$ It stems from the
symmetries of basis and dual basis that the projections $\mathbb{P}(\eta_{0})$
and $\mathbb{Q}(\eta_{0})$ indeed send real functions to real functions. Our
second result which is the main technical one is the obtention of uniform
resolvent bounds on the space $\mathbb{Q}(\eta_{0})X_{a}$, which is an
infinite dimensional spectral subspace supplementary to the continuous
resonant eigenspaces. These estimates will allow to get exponential decay for
the semigroup on this space.

By using the notation $\|\cdot\|_{B(X)}$ for the operator norm for linear
operators on $X$ and $\rho(\cdot\,;X)$ for the resolvent set of an operator on
$X$, we get the following:

######  Theorem 1.4.

Let $0<a=\hat{a}\epsilon<\frac{\sqrt{3}}{4}\epsilon$ and $X_{a}=\colon
L_{a}^{2}\times H_{a,*}^{{1}/{2}}.$ There exist
$\epsilon_{0}>0,\beta_{0}=\beta_{0}(\hat{a})>0,\hat{\eta}_{0}=\hat{\eta}_{0}(\hat{a})$
such that for any $0<\epsilon\leq\epsilon_{0},\,0<\beta\leq\beta_{0},\,$ we
have

$\Omega_{\beta,\,\epsilon}=\colon\big{\\{}\lambda\in\mathbb{C}\,\big{|}\,{\rm
Re}\,\lambda>-\beta\epsilon^{3}\big{\\}}\subset\rho(L;\mathbb{Q}(\epsilon^{2}\hat{\eta}_{0})X_{a}).$

Moreover, there is
$C_{0}=C_{0}(\hat{a},\hat{\eta}_{0},\beta_{0},\epsilon_{0})>0,$ such that for
any $\lambda\in\Omega_{\beta,\epsilon},$ the operator $(\lambda-L)$ is
invertible on the space $\mathbb{Q}(\epsilon^{2}\hat{\eta}_{0})X_{a}$ and

$\|(\lambda-L)^{-1}\|_{B(\mathbb{Q}(\eta_{0})X_{a})}\leq C_{0}.$ (1.18)

In view of the above uniform resolvent estimate (1.18), we have the following
semigroup estimate which is a consequence of the Gearhart-Prüss theorem [18,
52, 15]:

######  Corollary 1.5.

Let $\epsilon_{0},\beta_{0},\hat{\eta}_{0}$ be defined in Theorem 1.4. For any
$\epsilon\leq\epsilon_{0},\,\beta<\beta_{0},$ it holds that:

$\|e^{tL}\mathbb{Q}(\epsilon^{2}\hat{\eta}_{0})\|_{B(X_{a})}\lesssim
e^{-\beta\epsilon^{3}t}.$ (1.19)

As a consequence of Theorem 1.4, and its proof, we can also get the following
spectral stability result in unweighted spaces:

######  Theorem 1.6.

Let $L(\eta)$ be defined in (1.16) and denote
$Y_{0}(\eta)=L^{2}(\mathbb{R})\times H^{1/2}_{0,*,\eta}(\mathbb{R}).$ For
every $\epsilon$, $0<\epsilon\leq\epsilon_{0}$, the spectrum of $L(\eta)$ in
the space $Y_{0}(\eta)$ is contained in the imaginary axis.

Note that one has to be careful with the space $X_{a}=L_{a}^{2}\times
H_{a,*}^{1/2}$ and the definition of the Dirichlet-Neumann operator on it.
Indeed, due to the possible exponential growth, this space is not contained in
the Schwarz space of distributions and hence one cannot use the Fourier
transform. We can nevertheless use the Fourier transform of $e^{ax}f$ but not
the one of $f$ when $f$ is in $L^{2}_{a}$. In order to study $L$ defined in
(1.11) on the weighted space $X_{a}=L_{a}^{2}\times H_{a,*}^{1/2},$ one thus
need to be careful with the definition and the properties of the Dirichlet-
Neumann operator $G[\zeta_{c}]$ on $H_{a,*}^{1/2}.$ It will actually be more
convenient to work on an unweighted space by considering the transformed
operator $G_{a}[\zeta_{c}]=\colon e^{ax}\,G[\zeta_{c}]\,e^{-ax}$ and study it
in the usual Sobolev space:

$\displaystyle
H_{\star}^{{1}/{2}}(\mathbb{R}^{2})=\bigg{\\{}f\,\big{|}\,\|f\|_{H_{\star}^{{1}/{2}}(\mathbb{R}^{2})}=\colon\bigg{(}\int\bigg{(}\big{|}\frac{\nabla}{\langle
D\rangle^{1/2}}f\big{|}^{2}+a^{2}|f|^{2}\bigg{)}(x,y)\,\mathrm{d}x\mathrm{d}y\bigg{)}^{\frac{1}{2}}<+\infty\bigg{\\}}.$
(1.20)

Note that $H_{\star}^{{1}/{2}}(\mathbb{R}^{2})$ still depends on $a$ but we
neglect this dependence in the notation in order to distinguish it with the
weighted space $H_{a,\star}^{{1}/{2}}(\mathbb{R}^{2}).$ An equivalent
definition of $G_{a}[\zeta_{c}]$ is that, for any $f\in H_{*}^{1/2},$

$G_{a}[\zeta_{c}]f(x)=\sqrt{1+|\nabla\zeta_{c}|^{2}}\big{(}\partial_{z}\Phi_{a}-\partial_{x}\zeta_{c}\,(\partial_{x}-a)\Phi_{a}\big{)},$
(1.21)

where $\Phi_{a}(x,z)$ solves the following equation in
$\Omega=\mathbb{R}^{2}\times[-1,\zeta_{c}]:$

$-\big{(}(\partial_{x}-a)^{2}+\Delta_{y,z}\big{)}\Phi_{a}=0,\quad\Phi_{a}|_{z=\zeta_{c}}=f,\quad\partial_{z}\Phi_{a}|_{z=-1}=0.$
(1.22)

When the surface is flat, that is when $\zeta_{c}=0,$ we can find explicitly
the solution of the above problem by using the Fourier transform which is now
legitimate:

$\displaystyle
E[0](f)(\cdot,z)=\frac{\cosh\big{(}{\mu_{a}(D)(z+1)}\big{)}}{\cosh{\mu_{a}(D)}}f\,,\quad\text{
with \, }\mu_{a}(D)=\sqrt{(D_{x}+ia)^{2}+D_{y}^{2}}\,,$ (1.23)

where $D$ stands for $(D_{x},D_{y})$ and for a complex number $c,$ we define
$\sqrt{c}$ as the limit of the square root with positive real part of the
complex numbers $c+i\delta$ when $0<\delta\rightarrow 0^{+}.$ Thanks to
(1.23), we derive that $G_{a}[0]=(\mu_{a}\tanh\mu_{a})(D)\,.$ By using the
notation ${\mathcal{S}}=\mathbb{R}^{2}\times[-1,0]$, note that for any
positive constant $a,$ the solution $E[0]f$ belongs only to the anisotropic
homogeneous space
$\dot{H}_{x,z}^{1}(\mathcal{S})=\big{\\{}f\,|\,\partial_{x}f,\partial_{z}f\in
L^{2}({\mathcal{S}})\big{\\}}.$ However, its derivative in the transverse
variable $y$ cannot be bounded in $L^{2}({\mathcal{S}}).$ Indeed, the symbol
$\mu_{a}^{-1}(\xi,\eta)=\big{(}\sqrt{(\xi+ia)^{2}+\eta^{2}}\big{)}^{-1}$ may
exhibit singularity around nonzero points $(\xi,\eta)=(0,\pm a).$ This
degeneracy is a source of new difficulties in the study of the transverse
stability, in particular, we cannot choose our functional framework as
$L_{a}^{2}\times\sqrt{G_{a}[0]}^{-1}L_{a}^{2}$ as in [48] where the asymptotic
linear stability is studied for one-dimensional water waves.

To get useful properties of the Dirichlet-Neumann operator for a non-zero
surface, as it is classically done, we can use a change of variable to reduce
the problem to the fixed domain ${\mathcal{S}}$ and investigate the
solvability of the difference between the resulting unknown and $E[0]f.$ An
important observation is that, although the corresponding linear operator is
not elliptic, one can take benefits of the Poincaré inequality (due to the
zero upper boundary condition) when studying the perturbation to control its
$L^{2}({\mathcal{S}})$ norm by the $\dot{H}^{1}({\mathcal{S}})$ norm. Another
key point is that the source term does not contain any $\partial_{y}$
derivative of $E[0]f$ which does not belong to $L^{2}({\mathcal{S}}).$ This
provides the solvability and uniqueness of the solution to (1.22) in the new
variables in the space $H_{0,up}^{1}({\mathcal{S}})+E[0]f.$ We refer to Lemma
C.1 for more details.

The rough study of $G_{a}[\zeta_{c}]$ above also motivates us to consider the
transformed linear operator

${L}_{a}=\colon
e^{ax}L\,e^{-ax}=\left(\begin{array}[]{cc}(\partial_{x}-a)(d_{c}\cdot)&G_{a}[\zeta_{c}]\\\\[5.0pt]
-w_{c}&d_{c}(\partial_{x}-a)\end{array}\right).\,\,$ (1.24)

Since the space $X=\colon L^{2}(\mathbb{R}^{2})\times
H_{\star}^{{1}/{2}}(\mathbb{R}^{2})$ can be identified with $e^{ax}X_{a},$ we
see that the study of $L$ on the weighted space $X_{a}$ is equivalent to the
study of $L_{a}$ on the unweighted space $X.$ Consequently, Theorem 1.4 is
equivalent to the following theorem:

######  Theorem 1.7.

Let $X=\colon L^{2}(\mathbb{R}^{2})\times
H_{\star}^{{1}/{2}}(\mathbb{R}^{2}),\,\mathbb{Q}_{a}(\eta_{0})=e^{ax}\,\mathbb{Q}(\eta_{0})\,e^{-ax}.$
There exist
$\epsilon_{0}>0,\beta_{0}=\beta_{0}(\hat{a})>0,\hat{\eta}_{0}=\hat{\eta}_{0}(\hat{a})>0,C_{0}>0$
such that for any $0<\epsilon\leq\epsilon_{0},0<\beta\leq\beta_{0},\,$ and any
$\lambda\in\Omega_{\beta,\,\epsilon},$ the operator $(\lambda-L_{a})$ is
invertible on the space $\mathbb{Q}_{a}(\epsilon^{2}\hat{\eta}_{0})X$ and

$\|(\lambda-L_{a})^{-1}\|_{B(\mathbb{Q}_{a}(\epsilon^{2}\hat{\eta}_{0})X)}\leq
C_{0}\,.$ (1.25)

###  1.3. Outline of the proof and main difficulties

In this subsection, we explain the main issues and ideas for the proof of the
existence of the resonant modes in Theorem 1.2 and the resolvent estimate in
Theorem 1.7.

####  1.3.1. Resonant modes

The resonant modes

$\big{(}\lambda(\eta),e^{iy\,\eta}U(\cdot,\eta)\big{)}_{-\eta_{0}\leq\eta\leq\eta_{0}}$

are searched by looking for solutions of the eigenproblems

$(\lambda(\eta)-L(\eta))\,U(\cdot,\eta)=0.$

As mentioned previously, $L(0)$ corresponds to the linearized operator of one-
dimensional water waves about the 1-d soliton $(\zeta_{c},\varphi_{c}),$ which
has a nontrivial generalized kernel of dimension two. To find the curves
$\lambda(\eta)$ bifurcating from $\eta=0$ and the eigenfunctions
$U(\cdot,\eta),$ we use the classical Lyapunov-Schmidt reduction approach.
While featuring some algebraic differences, the proof share some similarities
with the the arguments presented in [44] where transverse linear stability of
line solitary waves is established for the Benney-Luke equation.

####  1.3.2. Uniform resolvent estimate

We now give more details about the proof of the uniform resolvent estimate
(1.25), which is the core of this paper. In order to get (1.25), the first
thing we have to detect is the presence of some damping effects once we work
in the weighted space $X_{a}$ due to the sign of the group velocity for the
constant coefficients part of the original linearized operator $L,$ or
equivalently the transformed operator $L_{a}$ on the unweighted space $X.$
This has been used since the pioneer work of Pego and Weinstein [50, 51] on
the asymptotic stability of KdV equation and Boussinesq equations. To be more
precise, consider the constant operator

$L_{a}^{0}=\colon\left(\begin{array}[]{cc}\partial_{x}-a&G_{a}[0]\\\\[5.0pt]
-\gamma&\partial_{x}-a\end{array}\right),$

which is obtained by setting $(\zeta_{c},\varphi_{c})=0$ in the definition of
$L_{a}$ (1.24). The two eigenvalues of its symbol are

$\lambda_{\pm}^{0}=\colon
i(\xi+ia)\pm\sqrt{-\gamma\,\mu_{a}\tanh\mu_{a}(\xi,\eta)}\,,\,\qquad\big{(}\,\mu_{a}(\xi,\eta)=\sqrt{(\xi+ia)^{2}+\eta^{2}}\,\big{)}\,.$

Some algebraic computations show that for any $(\xi,\eta)\in\mathbb{R}^{2},$
there exists a positive constant
$\beta(\hat{\alpha})={\mathcal{O}}(\hat{\alpha}),$ such that

${\rm Re}\,\lambda_{\pm}^{0}\leq-\beta(\hat{\alpha})\,\epsilon^{3}\,.$

Consequently, for any $\lambda$ such that ${\rm
Re}\,\lambda\geq-\beta(\hat{\alpha})\,\epsilon^{3}/2,$ it holds that

$\big{|}(\lambda-\lambda_{\pm}^{0})^{-1}\big{|}\leq\frac{2}{\beta(\hat{\alpha})\epsilon^{3}}\,,$

which enables us to conclude that $(\lambda-L_{a}^{0})$ is invertible and that
the resolvent bounds

$\sup_{{\rm Re}\,\lambda\geq-\beta(\hat{\alpha})\,\epsilon^{3}/2}\|(\lambda-
L_{a}^{0})^{-1}\|_{B(X)}<+\infty\,$

holds. The next step is to obtain the same result (1.25) for $L_{a}$ in which
the terms depending on the solitary waves $(\zeta_{c},\varphi_{c})$ are
nonzero but are small. The main difficulties in order to obtain this result in
the water-waves case are the following.

In the case of the water waves system which is fully nonlinear, the part of
the system involving coefficients depending on the solitary wave cannot be
seen as a lower order small perturbation as in the study of the linearization
of semilinear problems, see for instance [44]. More precisely, here the
appearance of $v_{c}\partial_{x}$ in the diagonal terms of $L_{a}$ results in
operators that are not bounded and the difference between $G_{a}[\zeta_{c}]$
and $G_{a}[0]$ is not a bounded operator from
$H_{*}^{\frac{1}{2}}(\mathbb{R}^{2})$ to $L^{2}(\mathbb{R}^{2}).$ To address
this issue, we cut frequencies and use pseudo-differential calculus and energy
estimates in the high-frequency regions. Further details will be provided
later.

Another difficulty lies in the presence of the Dirichlet-Neumann operator
$G_{a}[\zeta_{c}].$ If $\zeta_{c}$ is $C^{\infty}(\mathbb{R})$ smooth which is
the case in the current study, one can expect to adapt well-known results for
the case $a=0$ (see [5] for example) to get that $G_{a}[\zeta_{c}]$ is a
pseudo-differential operator with symbol

$\lambda_{\zeta_{c}}(x,\xi,\eta)=\lambda_{\zeta_{c}}^{1}(x,\xi,\eta)+\lambda_{\zeta_{c}}^{0}(x,\xi,\eta)+\cdots\,,$

where the principal symbol is

$\lambda^{1}_{\zeta_{c}}(x,\xi,\eta)=\sqrt{(\xi+ia)^{2}+\eta^{2}(1+(\partial_{x}\zeta_{c})^{2})}\,.$

However, there is no exact formula for the symbol $\lambda_{\zeta_{c}}$. We
thus need to use approximations of $\lambda_{\zeta_{c}}$ in conjunction with
appropriate perturbation arguments. As will be clear in the proof, here an
approximation $\lambda_{\zeta_{c}}^{app}$ is admissible if
$\mathrm{Op}(\lambda_{\zeta_{c}}-\lambda_{\zeta_{c}}^{app})$ is a bounded
operator from $H^{1/2}_{*}(\mathbb{R}^{2})$ to $L^{2}(\mathbb{R}^{2})$ with
operator norm sufficiently small compared to $\epsilon,$ for instance

$\|\mathrm{Op}(\lambda_{\zeta_{c}}-\lambda_{\zeta_{c}}^{app})\|_{B(H_{*}^{1/2},\,L^{2})}\leq
C\epsilon^{\alpha},\quad(\alpha>1).$

It turns out that the needed quality of the approximation, that is to say the
minimal $\alpha$ that we can allow, is different in different frequency
regions.

On the one hand, when focusing on uniform (in $\epsilon$) high transverse
frequencies $\\{|\eta|\geq 2\\}$, one may expect that the principal symbol
$\lambda_{\zeta_{c}}^{1}$ could be used as an admissible approximation.
Nevertheless, although $\mathrm{Op}(\lambda_{\zeta_{c}}^{1})$ approximates
well the operator $G_{a}[\zeta_{c}]$ in terms of regularity, it is far from
being a good approximation in terms of the size of the operator norm within
$B(H^{1/2}(\mathbb{R}^{2}),L^{2}(\mathbb{R}^{2})).$ Indeed, the operator norm
of the difference is only ${\mathcal{O}}(1).$ This is not enough to prove
resolvent bounds by perturbation arguments. Nevertheless, we will show in
Lemma C.2 that an admissible approximation can be obtained by using a
modification of the principal symbol

$\tilde{\lambda}_{\zeta_{c}}^{1}=\colon\lambda^{1}_{\zeta_{c}}\tanh\lambda^{1}_{\zeta_{c}}.$

The operator norm of the difference between $G_{a}[\zeta_{c}]$ and
$\mathrm{Op}(\tilde{\lambda}_{\zeta_{c}}^{1})$ is now of order
${\mathcal{O}}(\epsilon^{2}).$ We remark that unsurprisingly, such a
modification is needed due to the fact that we are considering water waves in
a domain with finite depth.

On the other hand, in the region of lower transverse frequencies
$\\{|\eta|\leq 2\\},$ the damping effect is quite weak, it thus necessitates a
more accurate approximation of the Dirichlet-Neumann operator. Here, we shall
adapt in the setting of weighted spaces its first order expansion in terms of
$\zeta_{c},$ which is a pseudo-differential operator with the symbol [34] :

${\lambda}_{\zeta_{c}}^{app}(x,\xi,\eta)=\mu_{a}\tanh\mu_{a}(\xi,\eta)+\zeta_{c}(x)\mu_{a}^{2}(\tanh^{2}\mu_{a}-1)(\xi,\eta),\quad\mu_{a}(\xi,\eta)=\sqrt{(\xi+ia)^{2}+\eta^{2}},$
(1.26)

the error belonging to $B(H_{*}^{1/2},\,L^{2})$ with norm of order
${\mathcal{O}}(\epsilon^{3}).$

We are now in position to outline the proof of uniform resolvent bounds
(1.25). We first use frequency cutoffs and different methods to establish the
resolvent bounds in different frequency regions. We divide the frequency space
$(\xi,\eta)\subset\mathbb{R}^{2}$ into three regions, corresponding to
(uniform) high, intermediate, and low transverse frequencies:

$\displaystyle R^{UH}=\big{\\{}(\xi,\eta)\big{|}\,|\eta|\geq 2\big{\\}},\quad
R^{I}=\big{\\{}(\xi,\eta)\big{|}\,A\epsilon^{2}\leq|\eta|\leq 2\big{\\}},\quad
R^{L}=\big{\\{}(\xi,\eta)\big{|}\,|\eta|\leq A\epsilon^{2}\big{\\}},$ (1.27)

where $A$ is a sufficiently large constant satisfying $A^{2}\epsilon\leq 1.$
We shall call the union of the uniform high frequency and intermediate
frequency regions the high frequency region. Note that at this stage we only
cut the transverse frequencies and the longitudinal frequency $\xi$ is just a
fixed parameter. We shall perform further localization in $\xi$ within the
different regions when needed.

$\bullet$ Uniform high transverse frequencies. In the uniform high transverse
frequency region $R^{UH},$ we define a modified operator $\tilde{L}_{a}$ via
replacing the Dirichlet Neumann operator in $L_{a}$ by
$\mathrm{Op}(\tilde{\lambda}_{\zeta_{c}}^{1}).$ We then diagonalize it by
using pseudo differential calculus and thus reduce the existence of the
resolvent problem as well as the uniform resolvent bounds to the study of
$\lambda-\lambda_{\pm}^{1}(x,\xi,\eta),$ where

$\lambda_{\pm}^{1}=i\bigg{(}d_{c}(x)(\xi+ia)\pm\sqrt{w_{c}}(x)\sqrt{\lambda^{1}_{\zeta_{c}}\tanh\lambda^{1}_{\zeta_{c}}}(x,\xi,\eta)\bigg{)}.$

It turns out that when $|\eta|\geq 2,$ for any $(x,\xi)\in\mathbb{R}^{2},$ any
$\lambda\in\Omega_{\beta,\,\epsilon}$ with $\beta$ small enough, it holds that
${\rm Re}\,(\lambda-\lambda_{\pm}^{1}(x,\xi,\eta))\gtrsim\epsilon.$ This,
together with Lemma C.2, allows us to establish the invertibility of
$(\lambda-L_{a})$ and the corresponding resolvent bounds in this region by
perturbation arguments.

$\bullet$ Intermediate transverse frequencies. The study in the intermediate
transverse frequency region $R^{I}$ is more involved due to a singularity in
the previous diagonalization process. Indeed, by using
$\mathrm{Op}({\lambda}_{\zeta_{c}}^{app})$ to approximate $G_{a}[\zeta_{c}],$
we would involve an operator of the form
$\mathrm{Op}\big{(}1/\sqrt{{\lambda}_{\zeta_{c}}^{app}}\big{)}$ which
unfortunately does not belong to $B(L^{2},H_{*}^{1/2}).$ This is due to the
singularity of the symbol $\mu_{a}(\xi,\eta)$ in the region

${\mathcal{S}}_{K,\delta}=\bigg{\\{}(\xi,\eta)\in\mathbb{R}^{2}\big{|}\,|\xi|\leq
K\epsilon,\,\delta\leq\frac{|\eta|}{\sqrt{\xi^{2}+a^{2}}}\leq
2\bigg{\\}},\qquad(\,0<\delta<1\text{ small },\,K\sim\sqrt{A}).$

However, since this region is indeed in low frequencies both in longitudinal
and transverse variables, we will show that we approximate $G_{a}[\zeta_{c}]$
by simply $-\Delta$ and perform directly energy estimates to obtain the
desired resolvent bounds. Since we would have to localize in the longitudinal
frequencies and study different sub-regions separately, we found more
convenient to adopt in this whole region $R^{I}$ an energy-based approach,
achieved through the design of appropriate energy functionals.

$\bullet$ Low transverse frequencies. In the low transverse frequency region
$R^{L},$ as in the previous case, in order not to lose derivatives, it
requires a further splitting involving frequencies dual to the $x$ variable:

$R_{H}^{L}=\\{|\xi|\geq K\epsilon,\,|\eta|\leq A\epsilon^{2}\\}\qquad
R_{L}^{L}=\\{|\xi|\leq K\epsilon,\,|\eta|\leq A\epsilon^{2}\\}.$

In the region $R_{H}^{L},$ we perform energy estimates by designing an energy
functional consistent with the symbol approximation (1.26) of the Dirichlet
Neumann operator. In the region $R_{L}^{L},$ we take benefits of the KP-II
approximation (see Subsection 2.1) and the resolvent bounds for the linearized
operator of KP-II equation around line KdV soliton proved in [43]. Due to the
weak damping effects in this region (which is of order $O(\epsilon^{3})$),
instead of using the Fourier projector as in the study of the region $R^{I},$
we use smooth cut-offs to split the longitudinal frequencies. Let us remark
finally that the arguments used in this region can provide a proof directly in
cartesian coordinates of the linear asymptotic stability of the 1d solitary
waves in the one-dimensional water waves system, which has been previously
obtained in [48] in holomorphic coordinates.

Organization of the paper: We establish the existence of continuous resonant
modes, as stated in Theorem 1.2, in Section 2. This section starts with a
formal derivation of the KP-II equation from the water waves in the long-wave
regime. Uniform resolvent estimates stated in Theorem 1.7 are then derived in
Sections 3 and 4, in the high (uniform high and intermediate) and low
transverse frequency regimes, respectively. The proof of spectral stability in
the unweighted space stated in Theorem 1.6 is performed in Section 5.
Additionally, various technical results used in the main proofs are presented
in the appendix. Specifically, in Appendix C, we prove several properties for
the transformed Dirichlet-Neumann operator $G_{a}[\zeta_{c}]$, which are of
independent interest.

##  2\. Spectral study in the regime of low transverse frequency

In this subsection, we prove Theorem 1.2. To be more specific, we will
establish that for sufficiently small values of $\eta_{0}$, there exists a
spectral curve $\lambda(\eta)\in C([-{\eta_{0}},{\eta_{0}}])$ and
corresponding eigenmodes $U(\cdot,\eta)\in
Y_{a,\eta}=L_{a}^{2}(\mathbb{R})\times H^{1/2}_{*,a,\eta}(\mathbb{R}),$ such
that

$L(\eta)\,U(\cdot,\eta)=\lambda(\eta)\,U(\cdot,\eta).$ (2.1)

Given that $L(\eta)=e^{-iy\eta}\,L\,e^{iy\eta}$, this yields a set of
continuous resonant modes

$\big{(}\lambda(\eta),e^{iy\,\eta}U(\cdot,\eta)\big{)}_{-\eta_{0}\leq\eta\leq\eta_{0}}$

for the linearized operator $L$.

###  2.1. KP-II approximation

As mentioned in the introduction, it is well known that the Korteweg-de Vries
(KdV) equation and the Kadomtsev-Petviashvili II (KP-II) equation are
effective approximate models for describing the water wave dynamics in the
long-wave region, in one and two dimensions respectively. We thus expect that
two dimensional water waves system linearized about the one dimensional
solitary water waves $(\zeta_{c},\varphi_{c})$ can be approximated by the
linearized KP-II equation about the line KdV soliton. This can be rigorously
shown by implementing the following scaling:

$\displaystyle\hat{t}=\epsilon^{3}t,\quad\,\hat{x}=\epsilon
x,\,\quad\hat{y}=\epsilon^{2}y,\qquad(\hat{\zeta},\hat{\phi})(\hat{t},\hat{x})=(\zeta,\phi)(t,x),$
(2.2)

Indeed, if $(\zeta,\phi)^{t}$ solves the linearized equation (1.11), then
$(\hat{\zeta},\hat{\phi})$ solves:

$\left\\{\begin{array}[]{l}\epsilon^{3}\partial_{\hat{t}}\hat{\zeta}-\epsilon\partial_{\hat{x}}(\hat{d}_{\epsilon}\hat{\zeta})-\hat{G}[\epsilon^{2}\hat{\zeta}_{\epsilon}]\hat{\phi}=0,\\\\[3.0pt]
\epsilon^{3}\partial_{\hat{t}}\hat{\phi}-\epsilon\hat{d}_{\epsilon}\partial_{\hat{x}}\hat{\phi}+\hat{w}_{\epsilon}\hat{\zeta}=0,\end{array}\right.$
(2.3)

where

$(\hat{v}_{\epsilon},\hat{Z}_{\epsilon},\hat{\zeta}_{\epsilon})(\hat{t},\hat{x})=\epsilon^{-2}(v_{c},Z_{c},\zeta_{c})(t,x),\quad\hat{d}_{\epsilon}=1-\epsilon^{2}\hat{v}_{\epsilon},\,\quad\hat{w}_{\epsilon}=\gamma-\epsilon^{3}\hat{d}_{\epsilon}\partial_{x}\hat{Z}_{\epsilon}.$
(2.4)

Moreover, the scaled Dirichlet-Neumann operator is defined as:

$\hat{G}[\epsilon^{2}\hat{\zeta}_{\epsilon}]f=\partial_{{z}}\hat{\Phi}-\epsilon^{4}\partial_{\hat{x}}\hat{\zeta}_{\epsilon}\partial_{\hat{x}}\hat{\Phi}$

where $\hat{\Phi}$ solves the elliptic problem in
$\Omega=\mathbb{R}\times[-1,\epsilon^{2}\hat{\zeta}_{\epsilon}]:$

$\big{(}\epsilon^{2}\partial_{\hat{x}}^{2}+\epsilon^{4}\partial_{\hat{y}}^{2}+\partial_{{z}}^{2}\big{)}\hat{\Phi}=0,\quad\hat{\Phi}|_{z=\epsilon^{2}\hat{\zeta}_{\epsilon}}=f,\quad\partial_{z}\hat{\Phi}|_{z=-1}=0.$

Let us now skip the subscript $\hat{}$ for the sake of notational convenience.
Plugging $\zeta=-w_{\epsilon}^{-1}\big{(}\epsilon^{3}\partial_{t}-\epsilon
d_{\epsilon}\partial_{x}\big{)}\phi$ into the first equation of (2.3), we find
that $\phi$ solves

$\displaystyle\bigg{(}\frac{\epsilon^{2}}{w_{\epsilon}}\partial_{t}^{2}-\frac{2d_{\epsilon}}{w_{\epsilon}}\partial_{t}\partial_{x}-\partial_{x}\big{(}\frac{d_{\epsilon}}{w_{\epsilon}}\big{)}\partial_{t}+\epsilon^{-2}\partial_{x}\big{(}\frac{d_{\epsilon}^{2}}{w_{\epsilon}}\partial_{x}\big{)}+\epsilon^{-4}\hat{G}[{\epsilon^{2}\zeta_{\epsilon}}]\bigg{)}\phi=0.$
(2.5)

We will do expansions with respect to $\epsilon$ in the above equation. The
expansion is usually performed by assuming that a high enough Sobolev norm of
the solution is bounded on the appropriate time scale. Here we shall assume
that the problem is localized in a bounded frequency region with respect to
both variables, so that we assume that all the differential operators with
positive order are bounded operators in $L_{\hat{a}}^{2}(\mathbb{R}^{2}).$ We
shall indeed perform this localization when we will use this approximation
argument.

It follows from the definition that

$\frac{d_{\epsilon}}{w_{\epsilon}}=-1+{\mathcal{O}}(\epsilon^{2}),\qquad\partial_{x}\big{(}\frac{d_{\epsilon}^{2}}{w_{\epsilon}}\partial_{x}\big{)}=\partial_{x}^{2}+\epsilon^{2}\partial_{x}((1-2v_{\epsilon})\partial_{x})+{\mathcal{O}}(\epsilon^{3}).$

Moreover, on the one hand, when the surface is flat, the Dirichlet-Neumann
operator $\hat{G}[0]$ has the form

$\displaystyle\hat{G}[0]=\sqrt{-\epsilon^{2}(\partial_{x}^{2}+\epsilon^{2}\partial_{y}^{2})}\tanh\sqrt{-\epsilon^{2}(\partial_{x}^{2}+\epsilon^{2}\partial_{y}^{2})}\,$

and thus has the expansion in a bounded frequency region:

$\displaystyle\hat{G}[0]$
$\displaystyle=-\epsilon^{2}(\partial_{x}^{2}+\epsilon^{2}\partial_{y}^{2})-\frac{1}{3}\epsilon^{4}(\partial_{x}^{2}+\epsilon^{2}\partial_{y}^{2})^{2}+{\mathcal{O}}(\epsilon^{6})$
$\displaystyle=-\epsilon^{2}\partial_{x}^{2}-\epsilon^{4}(\partial_{y}^{2}+\frac{1}{3}\partial_{x}^{4})+{\mathcal{O}}(\epsilon^{6}).$

On the other hand, by the expansion of the Dirichlet-Neumann operator in terms
of the small amplitude waves $\epsilon^{2}\zeta_{\epsilon}$ (refer to Section
3.6.2 of [34]), we can write, in a bounded frequency region,

$\displaystyle\hat{G}[{\epsilon^{2}\zeta_{\epsilon}}]$
$\displaystyle=\hat{G}[0]-\hat{G}[0](\epsilon^{2}\zeta_{\epsilon})\hat{G}[0]-\epsilon^{4}\partial_{x}(\zeta_{\epsilon}\partial_{x})+{\mathcal{O}}(\epsilon^{6})$
$\displaystyle=-\epsilon^{2}\partial_{x}^{2}-\epsilon^{4}(\partial_{y}^{2}+\frac{1}{3}\partial_{x}^{4})-\epsilon^{4}\partial_{x}(\zeta_{\epsilon}\partial_{x})+{\mathcal{O}}(\epsilon^{6}).$

Consequently, the leading order of the equation (2.5) is:

$-2\bigg{(}\partial_{t}-\partial_{x}\big{(}1-(2v_{0}+\zeta_{0})\cdot\big{)}+\frac{1}{3}\partial_{x}^{3}+\partial_{y}^{2}\partial_{x}^{-1}\big{)}\bigg{)}\partial_{x}\phi=0,$

where
$v_{0}=\zeta_{0}=\Psi_{\textnormal{\scalebox{0.8}{KdV}}}=\mathrm{sech}^{2}(\frac{\sqrt{3}}{2}\cdot)$
is the soliton profile of the KdV equation:

$\partial_{t}\Psi-\partial_{x}\Psi+3\Psi\partial_{x}\Psi+\frac{1}{3}\partial_{x}^{3}\Psi=0.$

We thus find the linear operator arising in the KP-II equation linearized
about the line soliton $\Psi_{\textnormal{\scalebox{0.8}{KdV}}}:$

$L_{\textnormal{\scalebox{0.8}{KP}}}=\partial_{x}\big{(}1-\frac{1}{3}\partial_{x}^{2}-3\Psi_{\textnormal{\scalebox{0.8}{KdV}}}\cdot\big{)}-\partial_{y}^{2}\partial_{x}^{-1}.$
(2.6)

###  2.2. Existence of resonant modes–proof of Theorem 1.2

Denote
$U(\cdot,\eta)=(\zeta,\phi)^{t}(\cdot,\eta),\,\,G_{\eta}[\zeta_{c}]=e^{-iy\eta}G_{\eta}[\zeta_{c}]e^{iy\eta},$
then $\eqref{resolnanteq-1}$ can be written as

$\left\\{\begin{array}[]{l}\lambda{\zeta}-\partial_{{x}}({d}_{c}{\zeta})-{G}_{\eta}[\zeta_{c}]{\phi}=0\,,\\\\[3.0pt]
\lambda{\phi}-d_{c}\partial_{{x}}{\phi}+{w}_{c}{\zeta}=0\,,\end{array}\right.$

or equivalently

$\big{(}\lambda-\partial_{{x}}({d}_{c}\cdot)\big{)}\big{(}w_{c}^{-1}\big{(}\lambda-{d}_{c}\partial_{{x}}\big{)}\big{)}\phi+G_{\eta}[\zeta_{c}]{\phi}=0\,.$

Let us define

$\lambda=\epsilon^{3}\Lambda,\,\,\quad\hat{\eta}=\epsilon^{-2}\eta,\,\,\quad\hat{\varphi}(\hat{x},\hat{\eta})=\varphi(x,\eta),\quad\hat{\psi}(\hat{x},\hat{\eta})=(\partial_{\hat{x}}\hat{\phi})(\hat{x},\hat{\eta}).$

By the change of variable and skipping the subscript $\hat{}$ for clarity, we
find in parallel to (2.5)

${\mathcal{T}}(\epsilon,\eta,\Lambda)\psi=\colon\big{[}L_{0}(\epsilon,\eta)+\Lambda(\eta)L_{1}(\epsilon)+\Lambda^{2}(\eta)L_{2}(\epsilon)\big{]}\psi=0\,,$
(2.7)

where

$\displaystyle\quad
L_{0}(\epsilon,\eta)=\epsilon^{-2}\partial_{x}\bigg{(}\frac{d_{\epsilon}^{2}}{w_{\epsilon}}\cdot\bigg{)}+\epsilon^{-4}\hat{G}_{\epsilon^{2}\eta}[{\epsilon^{2}\zeta_{\epsilon}}]\partial_{x}^{-1},$
(2.8) $\displaystyle
L_{1}(\epsilon)=-\partial_{x}\bigg{(}\frac{d_{\epsilon}}{w_{\epsilon}}\partial_{x}^{-1}\bigg{)}-\frac{d_{\epsilon}}{w_{\epsilon}},\qquad
L_{2}(\epsilon)=\epsilon^{2}w_{\epsilon}^{-1}\partial_{x}^{-1}.$

We refer to (2.4) for the definition of $d_{\epsilon}$ and $w_{\epsilon}.$ Let
us note that when $\epsilon=0,$

${\mathcal{T}}(0,\eta,\Lambda)=-2(\Lambda-
L_{\textnormal{\scalebox{0.8}{KP}}}),$

where
$L_{\textnormal{\scalebox{0.8}{KP}}}(\eta)=\partial_{x}\big{(}1-3\Psi_{\textnormal{\scalebox{0.8}{KdV}}}\cdot\big{)}-\frac{1}{3}\partial_{x}^{3}+\eta^{2}\partial_{x}^{-1}.$
In the following, we use as in [44] the Lyapunov-Schmidt method to prove the
existence of $(\Lambda,\psi)$ and an expansion in terms of $\eta$ under the
form:

$\displaystyle\Lambda(\eta)=i\eta\,\Lambda_{1\epsilon}-\eta^{2}\Lambda_{2\epsilon}+{\mathcal{O}}(\eta^{3}),$
(2.9)
$\displaystyle\psi(\cdot,\eta)=\psi_{0\epsilon}(\cdot)+i\eta\psi_{1\epsilon}(\cdot)+\eta^{2}\psi_{2\epsilon}(\cdot,\eta).$

Plugging the the above expansion into the equation (2.7) and sorting out in
term of $\eta^{j},$ we find:

$\displaystyle L_{0}({\epsilon,0})\,\psi_{0\epsilon}=0\,,$ $\displaystyle
L_{0}({\epsilon,0})\,\psi_{1\epsilon}+\Lambda_{1\epsilon}L_{1}({\epsilon})\,\psi_{0\epsilon}=0\,,$
$\displaystyle
L_{0}({\epsilon,0})\,\psi_{2\epsilon}-\Lambda_{1\epsilon}L_{1}({\epsilon})\,\psi_{1\epsilon}-\big{(}\Lambda_{1\epsilon}^{2}L_{2}(\epsilon)+\Lambda_{2\epsilon}L_{1}(\epsilon)-L_{0}^{1}(\epsilon)\big{)}\psi_{0\epsilon}=0\,,$
(2.10)

where we denote
$L_{0}^{1}(\epsilon)=\frac{L_{0}(\epsilon,\eta)-L_{0}(\epsilon,0)}{\eta^{2}}|_{\eta=0}\,.$

Since $L_{0}(\epsilon,0)$ and $L_{1}(\epsilon,0)$ are related to the
linearized operator of the one dimensional water wave system linearized about
the line soliton, it is shown in Appendix D (see (D.3)) that

$\psi_{0\epsilon}=\epsilon^{-3}v_{c}^{\prime}\bigg{(}\frac{\cdot}{\epsilon}\bigg{)},\qquad\psi_{1\epsilon}=-\Lambda_{1\epsilon}\varphi_{1c}^{\prime}\bigg{(}\frac{\cdot}{\epsilon}\bigg{)},$
(2.11)

where $v_{c}$ is defined in (1.10) and
$\varphi_{1c}=\partial_{c}(c\varphi_{c})-cZ_{c}\partial_{c}\zeta_{c}\,.$
Denote $L_{0}(\epsilon,0)^{*}\in B(L_{-\hat{a}}^{2}(\mathbb{R}))$ the adjoint
operator associated to $L_{0}(\epsilon,0)\in B(L_{\hat{a}}^{2}(\mathbb{R}))$
for the inner product
$\langle\cdot,\cdot\rangle_{L_{-\hat{a}}^{2}(\mathbb{R})\times
L_{\hat{a}}^{2}(\mathbb{R})}$. By using the explicit expression in (2.8), it
is direct to check that the operator $L_{0}(\epsilon,0)$ has the structure

$L_{0}(\epsilon,0)^{*}=-\partial_{x}^{-1}L_{0}(\epsilon,0)\partial_{x}\,,$
(2.12)

from which we obtain that $L_{0}(\epsilon,0)^{*}v_{\epsilon}=0$, with

$v_{\epsilon}(\cdot)=\colon\partial_{x}^{-1}\psi_{0,\epsilon}(\cdot)=\epsilon^{-2}v_{c}\big{(}{\cdot}/{\epsilon}\big{)}.$

Consequently, it holds that

$\langle L_{0}(\epsilon,0)\psi_{0\epsilon},v_{\epsilon}\rangle=0\,,$

where throughout this proof, we denote
$\langle\cdot,\cdot\rangle=\langle\cdot,\cdot\rangle_{(L_{-\hat{a}}^{2}(\mathbb{R}),L_{\hat{a}}^{2}(\mathbb{R}))}.$
Moreover, since
$d_{\epsilon}=1-\epsilon^{2}v_{\epsilon},\,w_{\epsilon}=\gamma-\epsilon^{3}Z_{\epsilon}\partial_{x}d_{\epsilon},\,v_{\epsilon}$
are all even functions, we get by integration by parts that

$\langle L_{1}(\epsilon)\psi_{0\epsilon},v_{\epsilon}\rangle=0\,.$

Therefore, it follows from (2.10) that

$\Lambda_{1\epsilon}^{2}\langle-
L_{1}(\epsilon)\partial_{x}\varphi_{1\epsilon}+L_{2}(\epsilon)\psi_{0\epsilon},v_{\epsilon}\rangle=\langle
L_{0}^{1}(\epsilon)\partial_{x}v_{\epsilon},v_{\epsilon}\rangle,$ (2.13)

where $\varphi_{1\epsilon}(\cdot)=\epsilon\varphi_{1c}(\cdot/\epsilon).$ We
first note that when restricting to $\epsilon=0,$

$L_{1}(0)=-2\,{\rm Id},\qquad
L_{0}^{1}(0)\,\partial_{x}=\epsilon^{-4}\frac{\hat{G}_{\epsilon^{2}\eta}[{\epsilon^{2}\zeta_{\epsilon}}]-\hat{G}_{0}[{\epsilon^{2}\zeta_{\epsilon}}]}{\eta^{2}}\big{|}_{\eta=\epsilon=0}={\rm
Id}.$ (2.14)

Moreover, thanks to (1.8), (1.9) and the relation
$\epsilon^{2}=1-\frac{gh}{c^{2}},$ it holds that:

$\displaystyle\qquad\qquad\qquad
v_{0}(x)=\Psi_{\textnormal{\scalebox{0.8}{KdV}}}(x)=\mathrm{sech}^{2}\bigg{(}\frac{\sqrt{3}}{2}x\bigg{)},$
$\displaystyle\partial_{x}\varphi_{10}\,(x)=c\partial_{c}\varphi_{c}^{\prime}\big{(}\frac{x}{\epsilon}\big{)}|_{\epsilon=0}=c\frac{\partial(\epsilon^{2})}{\partial
c}\partial_{\epsilon^{2}}\varphi_{c}^{\prime}\big{(}\frac{x}{\epsilon}\big{)}|_{\epsilon=0}=2\big{(}{\rm
Id}+\frac{x}{2}\partial_{x}\big{)}\Psi_{\textnormal{\scalebox{0.8}{KdV}}}(x).$

Consequently, since
$L_{2}(\epsilon)={\mathcal{O}}_{B(H^{1}_{\hat{a}}(\mathbb{R}),L_{\hat{a}}^{2}(\mathbb{R}))}(\epsilon^{2})\,,$

$\displaystyle\langle-
L_{1}(\epsilon)\partial_{x}\varphi_{1\epsilon}+L_{2}(\epsilon)\psi_{0\epsilon},v_{\epsilon}\rangle$
$\displaystyle=2\langle\partial_{x}\varphi_{10},v_{0}\rangle+{\mathcal{O}}(\epsilon)$
$\displaystyle=3\|v_{0}\|_{L^{2}}^{2}+{\mathcal{O}}(\epsilon)>0\,,$

provided that $\epsilon$ is small enough. It then follows from (2.13) that:

$\Lambda_{1\epsilon}^{2}=\frac{\langle
L_{0}^{1}(\epsilon)\partial_{x}v_{\epsilon},v_{\epsilon}\rangle}{\langle-
L_{1}(\epsilon)\partial_{x}\varphi_{1\epsilon}+L_{2}(\epsilon)\psi_{0\epsilon},v_{\epsilon}\rangle}=\frac{\|v_{0}\|_{L^{2}}^{2}}{3\|v_{0}\|_{L^{2}}^{2}}+{\mathcal{O}}(\epsilon)=\frac{1}{3}+{\mathcal{O}}(\epsilon)>0.$
(2.15)

It now remains to show the existence of the number $\Lambda_{2\epsilon}<0$ and
the function $\psi_{2\epsilon}(\cdot,\eta)\in L_{\hat{a}}^{2}(\mathbb{R}),$
such that (2.9) holds and

$\overline{\Lambda(\eta)}={\Lambda(-\eta)},\qquad\overline{\psi_{2\epsilon}(\cdot,\eta)}=\psi_{2\epsilon}(\cdot,-\eta).\,$

To use the Lyapunov-Schmidt method, it is more convenient to set
$\Lambda(\eta)=i\eta\,\tilde{\Lambda}(\eta)$ and look for
$\tilde{\Lambda}(\eta),\,\kappa(\eta),\tilde{\psi}_{2\epsilon}(\cdot,\eta)$
such that

$\psi(\cdot,\eta)=v_{\epsilon}^{\prime}-\big{(}i\eta\tilde{\Lambda}(\eta)-\kappa(\eta)\eta^{2}\big{)}\varphi_{1\epsilon}^{\prime}+\eta^{2}\tilde{\psi}_{2\epsilon}(\cdot,\eta)$
(2.16)

and

$\tilde{\psi}_{2\epsilon}(\cdot,\eta)\perp\mathrm{Span}\bigg{\\{}v_{\epsilon},\,\int_{-\infty}^{x}\tilde{\psi}_{1\epsilon}(x^{\prime})\,\mathrm{d}x^{\prime}\bigg{\\}}=\ker_{g}L_{0}^{*}(\epsilon,0),$

where $\tilde{\psi}_{1\epsilon}\in\ker_{g}L_{0}(\epsilon,0)$ satisfies
$L_{0}(\epsilon,0)\tilde{\psi}_{1\epsilon}=-v_{\epsilon}^{\prime},\,\tilde{\psi}_{1\epsilon}|_{\epsilon=0}=\frac{1}{2}\varphi^{\prime}_{1\epsilon}|_{\epsilon=0}.$
It is worth noting that the determination of $\tilde{\psi}_{1\epsilon}$ can be
achieved through the Lyapunov-Schmidt approach. That is, one starts from the
generalized kernel of $L_{0}(0,0),$ does expansions in $\epsilon,$ and
determines the remainders of order $\epsilon^{2}$ by the Implicit Function
Theorem. Since the procedure will be quite similar as in the subsequent
discussion, we omit the specific details for brevity.

Plugging the expansion (2.16) into the equation (2.7) and looking at the
${\mathcal{O}}(\eta^{2})$ term, we find the equation satisfied by
$\tilde{\psi}_{2\epsilon}:$

${\mathcal{T}}_{0}(\epsilon,\eta,\tilde{\Lambda})\,\tilde{\psi}_{2\epsilon}+{\mathcal{T}}_{1}(\epsilon,\eta,\tilde{\Lambda},\kappa)+i\eta\,{\mathcal{T}}_{2}(\epsilon,\eta,\tilde{\Lambda},\kappa)=0$
(2.17)

where

$\displaystyle{\mathcal{T}}_{0}(\epsilon,\eta,\tilde{\Lambda})=L_{0}(\epsilon,\eta)+i\eta\,\tilde{\Lambda}L_{1}(\epsilon)-\eta^{2}\tilde{\Lambda}^{2}L_{2}(\epsilon)\,,$
$\displaystyle{\mathcal{T}}_{1}(\epsilon,\eta,\tilde{\Lambda},\kappa)=\tilde{\Lambda}^{2}\big{(}L_{1}(\epsilon)\varphi_{1\epsilon}^{\prime}-L_{2}(\epsilon)v_{\epsilon}^{\prime}\big{)}+\kappa(\eta)L_{0}({\epsilon},\eta)\varphi_{1\epsilon}^{\prime}+\frac{L_{0}(\epsilon,\eta)-L_{0}(\epsilon,0)}{\eta^{2}}v_{\epsilon}^{\prime}\,,$
$\displaystyle{\mathcal{T}}_{2}(\epsilon,\eta,\tilde{\Lambda},\kappa)=\tilde{\Lambda}\bigg{(}\kappa(\eta)L_{1}(\epsilon)-\frac{L_{0}(\epsilon,\eta)-L_{0}(\epsilon,0)}{\eta^{2}}\bigg{)}\varphi_{1\epsilon}^{\prime}\,.$

Let $\mathbb{P}_{\epsilon}$ be the spectral projection in
$L_{\hat{\alpha}}^{2}(\mathbb{R})$ onto $\ker_{g}L_{0}(\epsilon,0)$ and
$\mathbb{Q}_{\epsilon}={\rm Id}-\mathbb{P}_{\epsilon},$ then the equation
(2.17) implies that

$\mathbb{Q}_{\epsilon}{\mathcal{T}}_{0}(\epsilon,\eta,\tilde{\Lambda})\mathbb{Q}_{\epsilon}\,\tilde{\psi}_{2\epsilon}+\mathbb{Q}_{\epsilon}{\mathcal{T}}_{1}(\epsilon,\eta,\tilde{\Lambda},\kappa)+i\eta\,\mathbb{Q}_{\epsilon}{\mathcal{T}}_{2}(\epsilon,\eta,\tilde{\Lambda},\kappa)=0.$

On the one hand, it holds by the definition of $\mathbb{Q}_{\epsilon}$ that

$\|\mathbb{Q}_{\epsilon}{\mathcal{T}}_{0}(\epsilon,0,\tilde{\Lambda})^{-1}\mathbb{Q}_{\epsilon}\|_{B(L_{\hat{\alpha}}^{2}(\mathbb{R}),H_{\hat{\alpha}}^{1}(\mathbb{R}))}<+\infty.$

On the other hand, in view of the expressions of
$L_{0}(\epsilon,\eta),L_{1}(\epsilon),L_{2}(\epsilon)$ in (2.8), one gets that

$\displaystyle\|{\mathcal{T}}_{0}(\epsilon,\eta,\tilde{\Lambda})-{\mathcal{T}}_{0}(\epsilon,0,\tilde{\Lambda})\|_{B(L_{a}^{2})}$
$\displaystyle\lesssim
a^{-1}(\epsilon\eta)^{2}(1+|\tilde{\Lambda}(\eta)|^{2})+\eta|\tilde{\Lambda}(\eta)|$
$\displaystyle\lesssim\eta_{0}^{2}(1+M^{2})+\eta_{0}M,$

where $M=\sup_{\eta\in[-\eta_{0},\eta_{0}]}|\tilde{\Lambda}(\eta)|.$ We thus
conclude that, upon choosing $\eta_{0}$ small enough,

$\|\mathbb{Q}_{\epsilon}{\mathcal{T}}_{0}(\epsilon,\eta,\tilde{\Lambda})^{-1}\mathbb{Q}_{\epsilon}\|_{B(L_{\hat{\alpha}}^{2}(\mathbb{R}),H_{\hat{\alpha}}^{1}(\mathbb{R}))}<+\infty.$

The existence of $\tilde{\psi}_{2\epsilon}$ is thus guaranteed upon
establishing the existence of the smooth curves
$\tilde{\Lambda}(\cdot),\kappa(\cdot):[-\eta_{0},\,\eta_{0}]\rightarrow\mathbb{C},$
which is the task of the following.

Since $\tilde{\psi}_{2\epsilon}\perp\ker_{g}L_{0}^{*}(\epsilon,0),$ we get
from (2.17) that

$\displaystyle{\mathcal{H}}_{1}(\epsilon,\eta,\tilde{\Lambda},\kappa)=\colon\big{\langle}{\mathcal{T}}_{1}(\epsilon,\eta,\tilde{\Lambda},\kappa)+i\eta\,{\mathcal{T}}_{2}(\epsilon,\eta,\tilde{\Lambda},\kappa)+\big{(}{\mathcal{T}}_{0}(\epsilon,\eta)-{\mathcal{T}}_{0}(\epsilon,0)\big{)}\tilde{\psi}_{2\epsilon},\,v_{\epsilon}\big{\rangle}=0\,,$
(2.18)
$\displaystyle{\mathcal{H}}_{2}(\epsilon,\eta,\tilde{\Lambda},\kappa)=\colon\big{\langle}{\mathcal{T}}_{1}(\epsilon,\eta,\tilde{\Lambda},\kappa)+i\eta\,{\mathcal{T}}_{2}(\epsilon,\eta,\tilde{\Lambda},\kappa)$
$\displaystyle\qquad\qquad\qquad\qquad+\big{(}{\mathcal{T}}_{0}(\epsilon,\eta)-{\mathcal{T}}_{0}(\epsilon,0)\big{)}\tilde{\psi}_{2\epsilon},\,\int_{-\infty}^{x}\tilde{\psi}_{1\epsilon}(x^{\prime})\,\mathrm{d}x^{\prime}\big{\rangle}=0\,.$

Looking at these two identities at $\eta=0$, we obtain the following relations

$\displaystyle
0={\mathcal{H}}_{1}|_{\eta=0}=\big{\langle}{\mathcal{T}}_{1}|_{\eta=0},\,v_{\epsilon}\big{\rangle}=\big{\langle}\tilde{\Lambda}^{2}(0)\big{(}L_{1}(\epsilon)\varphi_{1\epsilon}^{\prime}-L_{2}(\epsilon)v_{\epsilon}^{\prime}\big{)}+L_{0}^{1}(\epsilon)v^{\prime}_{\epsilon},\,v_{\epsilon}\big{\rangle},$
$\displaystyle
0={\mathcal{H}}_{2}|_{\eta=0}=\big{\langle}{\mathcal{T}}_{1}|_{\eta=0},\,\int_{-\infty}^{x}\tilde{\psi}_{1\epsilon}(x^{\prime})\,\mathrm{d}x^{\prime}\big{\rangle}$
$\displaystyle\qquad\qquad=\big{\langle}\tilde{\Lambda}^{2}(0)\big{(}L_{1}(\epsilon)\varphi_{1\epsilon}^{\prime}-L_{2}(\epsilon)v_{\epsilon}^{\prime}\big{)}+L_{0}^{1}(\epsilon)v^{\prime}_{\epsilon}+\kappa(0)L_{0}(\epsilon,0)\varphi_{1\epsilon}^{\prime},\,\int_{-\infty}^{x}\tilde{\psi}_{1\epsilon}(x^{\prime})\,\mathrm{d}x^{\prime}\big{\rangle},$

The first relation yields

$\tilde{\Lambda}^{2}(0)=\frac{\langle
L_{0}^{1}(\epsilon)v^{\prime}_{\epsilon},v_{\epsilon}\rangle}{\langle-
L_{1}(\epsilon)\varphi_{1\epsilon}^{\prime}+L_{2}(\epsilon)\psi_{0\epsilon},v_{\epsilon}\rangle}=\frac{1}{3}+{\mathcal{O}}(\epsilon)>0,$

which is consistent with (2.15). Next, since

$\varphi_{10}^{\prime}=2v_{0}+x\partial_{x}v_{0},\,\quad\tilde{\psi}_{10}=\frac{1}{2}\varphi_{10},\qquad
L_{0}^{*}(\epsilon,0)\int_{-\infty}^{x}\tilde{\psi}_{1\epsilon}(x^{\prime})\,\mathrm{d}x^{\prime}=v_{\epsilon},$

it holds that

$\big{\langle}L_{0}(\epsilon,0)\varphi_{1\epsilon}^{\prime}\,,\,\int_{-\infty}^{x}\tilde{\psi}_{1\epsilon}(x^{\prime})\,\mathrm{d}x^{\prime}\big{\rangle}=\big{\langle}\varphi_{1\epsilon}^{\prime},\,v_{\epsilon}\big{\rangle}=\big{\langle}\varphi_{10}^{\prime},\,v_{0}\big{\rangle}+{\mathcal{O}}(\epsilon)=\frac{3}{2}\,\|v_{0}\|_{L^{2}(\mathbb{R})}^{2}+{\mathcal{O}}(\epsilon)\neq
0.$

Moreover, in view of (2.14), we have that

$\displaystyle\big{\langle}\tilde{\Lambda}^{2}(0)\big{(}L_{1}(\epsilon)\varphi_{1\epsilon}^{\prime}-L_{2}(\epsilon)v_{\epsilon}^{\prime}\big{)}+L_{0}^{1}(\epsilon)v^{\prime}_{\epsilon},\,\int_{-\infty}^{x}\tilde{\psi}_{1\epsilon}(x^{\prime})\,\mathrm{d}x^{\prime}\big{\rangle}$
$\displaystyle=\big{\langle}v_{0}-\frac{2}{3}\,\varphi_{10}^{\prime},\,\int_{-\infty}^{x}\tilde{\psi}_{1\epsilon}(x^{\prime})\,\mathrm{d}x^{\prime}\big{\rangle}+{\mathcal{O}}(\epsilon)=\frac{1}{12}\,\|v_{0}\|_{L^{1}(\mathbb{R})}^{2}+{\mathcal{O}}(\epsilon).$

It thus follows from the second relation that

$\kappa(0)=-\frac{1}{18}\frac{\|v_{0}\|_{L^{1}(\mathbb{R})}^{2}}{\|v_{0}\|_{L^{2}(\mathbb{R})}^{2}}+{\mathcal{O}}(\epsilon)\neq
0.$ (2.19)

To determine the curves $\tilde{\Lambda}(\eta),\,\kappa(\eta)$ around
$\eta=0,$ we compute

$\displaystyle\partial_{\kappa}{\mathcal{H}}_{1}|_{\eta=0}=\big{\langle}L_{0}(\epsilon,0)\varphi_{1\epsilon}^{\prime},v_{\epsilon}\big{\rangle}=0,$
$\displaystyle\partial_{\kappa}{\mathcal{H}}_{2}|_{\eta=0}=\big{\langle}L_{0}(\epsilon,0)\varphi_{1\epsilon}^{\prime},\,-\int_{x}^{+\infty}\tilde{\psi}_{1\epsilon}(x^{\prime})\,\mathrm{d}x^{\prime}\big{\rangle}=\frac{3}{2}\,\|v_{0}\|_{L^{2}}^{2}+{\mathcal{O}}(\epsilon)\neq
0,$
$\displaystyle\partial_{\tilde{\Lambda}}{\mathcal{H}}_{2}|_{\eta=0}=2\tilde{\Lambda}(0)\big{\langle}L_{1}(\epsilon)\varphi_{1\epsilon}^{\prime}-L_{2}(\epsilon)v_{\epsilon}^{\prime},\,v_{\epsilon}\big{\rangle}=-6\tilde{\Lambda}(0)\|v_{0}\|_{L^{2}}^{2}+{\mathcal{O}}(\epsilon)\neq
0,$ (2.20)

which lead to

$\det\big{(}D_{\kappa,\tilde{\Lambda}}({\mathcal{H}}_{1},{\mathcal{H}}_{2})\big{)}|_{\eta=0}\neq
0.$

Consequently, by the Implicit Function Theorem, there exists $\eta_{0}$ and
two $C^{1}$ curves
$\tilde{\Lambda}(\eta),\,\kappa(\eta):[-\eta_{0},\,\eta_{0}]\rightarrow\mathbb{C},$
such that (2.18) holds true. Moreover, it holds that

$i\,\Lambda_{2\epsilon}=\colon\partial_{\eta}\tilde{\Lambda}\big{|}_{\eta=0}=-\frac{\partial_{\eta}{\mathcal{H}}_{1}}{\partial_{\tilde{\Lambda}}{\mathcal{H}}_{1}}\big{|}_{\eta=0}\,.$

In light of the expression of ${\mathcal{H}}_{1}$ in (2.18), we compute

$\displaystyle{i}\,\partial_{\eta}{\mathcal{H}}_{1}\big{|}_{\eta=0}$
$\displaystyle=-\tilde{\Lambda}(0)\big{\langle}\big{(}\kappa(0)L_{1}(\epsilon)-L_{0}^{1}(\epsilon)\big{)}\,\varphi_{1\epsilon}^{\prime}+L_{1}(\epsilon)\,\tilde{\psi}_{2\epsilon},\,v_{\epsilon}\big{\rangle}$
$\displaystyle=\tilde{\Lambda}(0)\big{\langle}(2\kappa(0)+\partial_{x}^{-1})\,\varphi_{10}^{\prime},\,v_{0}\big{\rangle}+{\mathcal{O}}(\epsilon)$
$\displaystyle=\tilde{\Lambda}(0)\bigg{(}3\kappa(0)\|v_{0}\|_{L^{2}(\mathbb{R})}^{2}-\frac{1}{2}\|v_{0}\|_{L^{1}(\mathbb{R})}^{2}\bigg{)}+{\mathcal{O}}(\epsilon).$

Therefore, it follows from (2.19) and (2.20) that for $\epsilon$ small enough,

$\Lambda_{2\epsilon}=\frac{1}{9}\frac{\|v_{0}\|_{L^{1}(\mathbb{R})}^{2}}{\|v_{0}\|_{L^{2}(\mathbb{R})}^{2}}+{\mathcal{O}}(\epsilon)>0\,.$

Finally, by the fact
$\overline{L_{0}(\epsilon,\eta)}={L_{0}(\epsilon,-\eta)},\,$ it holds that

$\overline{{\mathcal{H}}_{k}(\epsilon,\eta,\kappa,\tilde{\Lambda})}={\mathcal{H}}_{k}\big{(}\epsilon,-\eta,\overline{\kappa},\overline{\tilde{\Lambda}}\big{)},\qquad
k=1,2.$

We thus get from the uniqueness of the curves that:

$\overline{\tilde{\Lambda}(\eta)}=\tilde{\Lambda}(-\eta),\qquad\overline{\kappa(\eta)}=\kappa(-\eta),$

which further implies that

$\overline{{\Lambda}(\eta)}={\Lambda}(-\eta),\qquad\overline{\tilde{\psi}_{2\epsilon}(\cdot,\eta)}=\tilde{\psi}_{2\epsilon}(\cdot,-\eta).$

Let us summarize what we have obtained and their consequences, which lead to
Theorem 1.2.

######  Lemma 2.1.

There exists $\epsilon_{0},\hat{\eta}_{0}>0,$ such that for any
$\epsilon\leq\epsilon_{0},$

(1) For any $\hat{\eta}\leq\hat{\eta}_{0},$ one can find
$\Lambda(\hat{\eta}),\,{\psi}(\cdot,\hat{\eta})\in
H_{\hat{a},*,\hat{\eta}}^{1/2}$ which solve (2.7) and enjoy the expansion:

$\displaystyle\Lambda(\hat{\eta})=i\hat{\eta}\,\Lambda_{1\epsilon}-\hat{\eta}^{2}\Lambda_{2\epsilon}+{\mathcal{O}}(\hat{\eta}^{3}),$
(2.21)
$\displaystyle\psi(\cdot,\hat{\eta})=\partial_{x}\bigg{(}v_{\epsilon}-\big{(}i\,\Lambda_{1\epsilon}\hat{\eta}-\tilde{\Lambda}_{2\epsilon}\hat{\eta}^{2}\big{)}\,\varphi_{1\epsilon}\bigg{)}(\cdot)+\hat{\eta}^{2}\tilde{\psi}_{2\epsilon}(\cdot,\hat{\eta})$

where

$\displaystyle\Lambda_{1\epsilon}=\frac{1}{\sqrt{3}}+{\mathcal{O}}(\epsilon)\,,\qquad\Lambda_{2\epsilon}=\frac{1}{9}\frac{\|v_{0}\|_{L^{1}(\mathbb{R})}^{2}}{\|v_{0}\|_{L^{2}(\mathbb{R})}^{2}}+{\mathcal{O}}(\epsilon),\,\quad\tilde{\Lambda}_{2\epsilon}=\frac{1}{18}\frac{\|v_{0}\|_{L^{1}(\mathbb{R})}^{2}}{\|v_{0}\|_{L^{2}(\mathbb{R})}^{2}}+{\mathcal{O}}(\epsilon),$
$\displaystyle
v_{\epsilon}(\cdot)=\epsilon^{-2}v_{c}(\epsilon^{-1}\,\cdot)\,,\,\qquad\varphi_{1\epsilon}(\cdot)=\epsilon\,\big{(}\partial_{c}(c\varphi_{c})-cZ_{c}\partial_{c}\zeta_{c}\big{)}(\epsilon^{-1}\,\cdot)\,,\qquad\tilde{\psi}_{2\epsilon}(\cdot,\hat{\eta})\perp
v_{\epsilon}\,.$

(2) For any $\eta\in[-\epsilon^{2}\hat{\eta}_{0},\epsilon^{2}\hat{\eta}_{0}],$
the operator $L(\eta)$ has two pairs of eigenmodes
$\big{(}\lambda(\eta),U(\cdot,\eta)\big{)}$ and
$\big{(}\lambda(-\eta),U(\cdot,-\eta)\big{)}$ in the space $Y_{a,\eta}$ where

$\displaystyle\lambda(\eta)=\epsilon^{3}\Lambda(\epsilon^{-2}\eta)=i\epsilon\Lambda_{1\epsilon}\eta-\epsilon^{-1}\Lambda_{2\epsilon}\eta^{2}+{\mathcal{O}}(\eta^{3})\,,$
$\displaystyle U(\cdot,\eta)$
$\displaystyle=\left(\begin{array}[]{c}-{w}^{-1}_{c}\big{(}\lambda(\eta)+\epsilon\,d_{c}\,\partial_{\hat{x}}\big{)}(\partial_{\hat{x}}^{-1}\psi)\\\\[3.0pt]
\partial_{\hat{x}}^{-1}\psi\end{array}\right)(\epsilon\,\cdot,\,\epsilon^{-2}\eta)=\colon\left(\begin{array}[]{c}U_{1}(\cdot,\,\eta)\\\\[3.0pt]
U_{2}(\cdot,\,\eta)\end{array}\right)\,.$ (2.26)

(3) The eigenmode $(\lambda^{*}(\pm\eta),U^{*}(\cdot,\pm\eta))$ of
$L^{*}_{a}(\eta)$ in the space $Y_{a,\eta}^{*}$ are characterized by

$\displaystyle\lambda^{*}(\eta)=\lambda(-\eta),\qquad
U^{*}(\cdot,\eta)=(U_{2},U_{1})^{t}(-\,\cdot,-\eta)\,.$ (2.27)

Moreover, it holds that

$\overline{\lambda(\eta)}=\lambda(-\eta),\quad\overline{U(\cdot,\eta)}=U(\cdot,-\eta),\quad\big{\langle}U(\cdot,\eta),\,U^{*}(\cdot,-\eta)\big{\rangle}_{{}_{Y_{a,\eta}\times
Y_{a,\eta}^{*}}}=0,$

where the inner product is defined as follows: for $f=(f_{1},f_{2})^{t}\in
Y_{a,\eta},\,f^{*}=(f_{1}^{*},f_{2}^{*})^{t}\in Y_{a,\eta}^{*},$

$\big{\langle}f,f^{*}\rangle_{{}_{Y_{a,\eta}\times Y_{a,\eta}^{*}}}=:\langle
f_{1},f_{1}^{*}\rangle_{L_{a}^{2}(\mathbb{R})\times
L_{-a}^{2}(\mathbb{R})}+\left\langle\frac{(\partial_{x},\eta)}{\langle(D_{x},\eta)\rangle^{1/2}}f_{2},\frac{(\partial_{x},\eta)}{\langle(D_{x},\eta)\rangle^{1/2}}f^{*}_{2}\right\rangle_{L_{a}^{2}(\mathbb{R})\times
L_{-a}^{2}(\mathbb{R})}.$ (2.28)

###### Proof.

The first point and the existence of the eigenmode
$\big{(}\lambda(\eta),U(\cdot,\eta)\big{)}$ for $L(\eta)$ in the second point
have been proven. The existence of the other eigenmode follows from the
evenness of $L(\eta)$ in $\eta.$ Now only the last point requires some
clarification. In fact, as an operator taken on the space $Y_{a}^{*}(\eta),$
the dual operator $L^{\star}(\eta)$ takes the form

$L^{*}(\eta)=\left(\begin{array}[]{cc}-d_{c}\partial_{x}&-w_{c}\\\\[3.0pt]
G_{\eta}[\zeta_{c}]&-\partial_{x}(d_{c}\cdot)\end{array}\right).$

Note that we have used the fact that
$G^{*}_{a,\eta}[\zeta_{c}]=G_{-a,\eta}[\zeta_{c}].$ The identities in (2.27)
then follow from the evenness of $\zeta_{c},\,d_{c},\,w_{c}$ in $x$ and
$G_{\eta}[\zeta_{c}]$ in $\eta.$ ∎

We are now in position to define precisely the basis of $L(\eta).$ We already
know that $U(\cdot,\eta)$ and $\overline{U(\cdot,\eta)}$ are two
eigenfunctions. Nevertheless, their imaginary parts vanishes when $\eta=0$ and
thus they do not provide a basis of the two-dimensional algebraic kernel of
$L(0)$. To resolve this degeneracy, inspired by [44], we first introduce a
pair of functions with purely real inner product:

$\displaystyle g(\cdot,\eta)=(\alpha(\eta)-{i})\,U(\cdot,\eta),\quad
g^{*}(\cdot,\eta)=-\frac{\sqrt{3}}{4}\,U^{*}(\cdot,\eta)$

where $\alpha(\eta)=\frac{{\rm Re}\,\langle
U(\cdot,\eta),U^{*}(\cdot,\eta)\rangle}{{\rm Im}\,\langle
U(\cdot,\eta),U^{*}(\cdot,\eta)\rangle}.$ Note that

$\displaystyle\quad\overline{g(\cdot,\eta)}=-g(\cdot,-\eta),\qquad\overline{g^{*}(\cdot,\eta)}=g^{*}(\cdot,-\eta),$
(2.29) $\displaystyle\langle
g(\cdot,\eta),g^{*}(\cdot,-\eta)\rangle=0,\qquad{\rm Im}\,\langle
g(\cdot,\eta),g^{*}(\cdot,\eta)\rangle=0.$

We then define the basis and dual basis:

$\displaystyle
g_{1}(\cdot,\eta)=-\frac{1}{2i}\big{(}g(\cdot,\eta)+g(\cdot,-\eta)\big{)}=\big{(}{\rm
Re}\,U-\alpha(\eta)\,{\rm Im}\,U\big{)}(\cdot,\eta),$ (2.30) $\displaystyle
g_{2}(\cdot,\eta)=\frac{1}{2\kappa(\eta)}\big{(}g(\cdot,\eta)-g(\cdot,-\eta)\big{)}=\frac{1}{\kappa(\eta)}\big{(}{\rm
Im}\,U+\alpha(\eta){\rm Re}\,U\big{)}(\cdot,\eta),$ $\displaystyle
g^{*}_{1}(\cdot,\eta)=-\frac{1}{2i\kappa(\eta)}(g^{*}(\cdot,\eta)-g^{*}(\cdot,-\eta))=\frac{\sqrt{3}}{4\,\kappa(\eta)}{\rm
Im}\,U^{*}(\cdot,\eta),$ $\displaystyle
g_{2}^{*}(\cdot,\eta)=\frac{1}{2}(g^{*}(\cdot,\eta)+g^{*}(\cdot,-\eta))=\frac{\sqrt{3}}{4}\,{\rm
Re}\,U^{*}(\cdot,\eta),$

where

$\displaystyle\kappa(\eta)=\frac{1}{2}\,{\rm Re}\,\langle
g(\cdot,\eta),g^{*}(\cdot,\eta)\rangle_{{}_{Y_{a,\eta}\times
Y_{a,\eta}^{*}}}=-\frac{\sqrt{3}}{8}\bigg{(}{\rm Im}\,\langle
U,U^{*}\rangle+\frac{({\rm Re}\,\langle U,U^{*}\rangle)^{2}}{{\rm Im}\,\langle
U,U^{*}\rangle}\bigg{)}(\eta).$

Note that by (2.29), it holds that

$\langle g_{j}(\cdot,\eta),g_{k}^{*}(\cdot,\eta)\rangle_{{}_{Y_{a,\eta}\times
Y_{a,\eta}^{*}}}=\delta_{jk},\quad\forall\,1\leq j,k\leq
2;\,\eta\in[-\epsilon^{2}\hat{\eta}_{0},\epsilon^{2}\hat{\eta}_{0}]\,.$

##  3\.  Spectral stability in the uniform high and intermediate transverse
frequency regions.

In this section, we focus on the uniform high and intermediate transverse
frequency estimates that is the regions $R^{UH}$ and $R^{I}$ defined in 1.27.
Let $\mathbb{I}_{K}(\cdot):\mathbb{R}\rightarrow\mathbb{R}$ be the
characteristic function of $[-K,K]$ and
$\tilde{\mathbb{I}}_{K}=1-\mathbb{I}_{K}.$ For a large number $A$ such that
$A\epsilon^{2}<1,$ we define the spaces $X^{UH}$ and $X^{I}_{A}$ as:

$X^{UH}=\tilde{\mathbb{I}}_{2}(D_{y})X,\quad
X^{I}_{A}=\mathbb{I}_{2}\tilde{\mathbb{I}}_{A\epsilon^{2}}(D_{y})X,$

which are subspaces of $X$ that as introduced above, we refer to as the
uniform high transverse frequency region (when $\\{|\eta|\geq 2\\}$) and the
intermediate transverse frequency region (ie. $\\{A\epsilon^{2}\leq|\eta|\leq
2\\}$). Denote $X_{A}^{H}=X^{UH}\cup X_{A}^{I},$ our goal in this section is
to show the following proposition:

######  Proposition 3.1.

Assume that $0<\beta\leq\frac{1}{2}$ and $A$ sufficiently large. There exists
$\epsilon_{0}$ small enough, such that for any $\epsilon\in(0,\epsilon_{0}],$
it holds that

$\Omega_{\beta,\epsilon}=\colon\\{\lambda\in\mathbb{C}\,|\,{\rm
Re}\,\lambda\geq-\beta\epsilon^{3}\\}\subset\rho(L_{a};X_{A}^{H}).$

Moreover, for any $\lambda\in\Omega_{\beta,\epsilon},$

$\|\big{(}\lambda-L_{a}\big{)}^{-1}\|_{B(X_{A}^{H})}\lesssim
A^{-1}\epsilon^{-3},$

where $\lesssim\cdot$ denotes $\leq C\cdot$ for some constant $C$ that is
independent of $A,\epsilon\,.$

This proposition is the direct consequence of the following two propositions
concerning the resolvent estimates on the space $X^{UH}$ and $X_{A}^{I}$
respectively.

######  Proposition 3.2.

Suppose that $0<\beta\leq\frac{1}{2}$ and $\epsilon$ is sufficiently small.
For any $\lambda\in\Omega_{\beta,\epsilon},$ the operator $\big{(}\lambda-
L_{a}\big{)}$ is invertible on the space $X^{UH}$ and its inverse has the
bound:

$\|\big{(}\lambda-L_{a}\big{)}^{-1}\|_{B(X^{UH})}\leq C\epsilon^{-1}\,,$ (3.1)

where the constant $C>0$ is independent of $\epsilon.$

######  Proposition 3.3.

Suppose that $0<\beta\leq\frac{1}{2}.$ There exists a large number $A>0$ such
that for any $\epsilon$ sufficiently small so that $A\epsilon^{2}<1,$ the
following hold true: for any $\lambda\in\Omega_{\beta,\epsilon},$ the operator
$\big{(}\lambda-L_{a}\big{)}$ is invertible on the space $X_{A}^{I}$ and its
inverse has the bound:

$\displaystyle\|\big{(}\lambda-L_{a}\big{)}^{-1}\|_{B(X_{A}^{I})}\leq
CA^{-1}\epsilon^{-3},$ (3.2)

where $C$ is a constant independent of $A,\epsilon.$

We will prove Propositions 3.2 and 3.3 in the following two subsections.

###  3.1. Proof of Proposition 3.2

###### Proof of Proposition 3.2.

For any $f\in H^{1/2}_{*}(\mathbb{R}^{2})$ with transverse frequency
$|\eta|\geq 2,$ it holds that
$\|f\|_{H_{\star}^{{1}/{2}}(\mathbb{R}^{2})}\approx\|\sqrt{G_{a}[0]}f\|_{L^{2}(\mathbb{R}^{2})}$
since $a=\hat{a}\epsilon\leq 1/2.$ Therefore, to prove Proposition 3.2 , it
suffices to show that
$\Omega_{\beta,\epsilon}\subset\rho({\mathcal{L}}_{a};\mathcal{X}^{UH}),$ with
the bound

$\|\big{(}\lambda-{\mathcal{L}}_{a}\big{)}^{-1}\|_{B(\mathcal{X}^{UH})}\leq
C\epsilon^{-1},\quad\forall\,\lambda\in\Omega_{\beta,\epsilon},$ (3.3)

where
$\mathcal{X}^{UH}=\tilde{\mathbb{I}}_{2}(D_{y})(L^{2}(\mathbb{R}^{2})\times
L^{2}(\mathbb{R}^{2}))$ and

$\displaystyle{\mathcal{L}}_{a}$
$\displaystyle=\colon\textnormal{diag}(1,\sqrt{G_{a}[0]})\,L_{a}\,\textnormal{diag}\big{(}1,\sqrt{G_{a}[0]}^{-1}\big{)}$
(3.4)
$\displaystyle=\left(\begin{array}[]{cc}(\partial_{x}-a)(d_{c}\cdot)&G_{a}[\zeta_{c}]\sqrt{G_{a}[0]}^{-1}\\\\[5.0pt]
-\sqrt{G_{a}[0]}w_{c}&\qquad
d_{c}(\partial_{x}-a)+\big{[}\sqrt{G_{a}[0]},d_{c}\big{]}(\partial_{x}-a)\sqrt{G_{a}[0]}^{-1}\end{array}\right)$

where $\left[\cdot,\cdot\right]$ stands for the commutator of two operators.

For any $F\in\mathcal{X}^{UH},$ and $\lambda\in\Omega_{\beta,\epsilon},$
consider the resolvent problem:

$\big{(}\lambda-{\mathcal{L}}_{a}\big{)}U=F,$

we shall prove the existence of $U\in\mathcal{X}^{UH}$ and the resolvent
estimate simultaneously. Let us set
${\mathcal{L}}_{a}({\eta})=e^{-iy\eta}{\mathcal{L}}_{a}e^{iy\eta}.$ By taking
the Fourier transform in the transverse variable, we reduce the matter to
solving the resolvent problem

$\big{(}\lambda-{\mathcal{L}}_{a}(\eta)\big{)}u=f,\quad\,\forall\,|\eta|\geq
2$ (3.5)

in the space ${\mathcal{Y}}=L^{2}(\mathbb{R})\times L^{2}(\mathbb{R}),$ and
proving the resolvent bounds
$\|u\|_{{\mathcal{Y}}}\lesssim\epsilon^{-1}\|f\|_{{\mathcal{Y}}}$ uniformly
for $\eta$ such that $|\eta|\geq 2.$

From its definition, ${\mathcal{L}}_{a}(\eta)$ takes the same form as in (3.4)
with $G_{a}[\cdot]$ replaced by

$G_{a,\eta}[\cdot]=\colon e^{-iy\eta}\,G_{a}[\cdot]\,e^{iy\eta}.$

As $\zeta_{c}\in C^{\infty}(\mathbb{R})$ is sufficiently smooth, we could
translated in our framework the results of [5, 61] for example, to get an
approximation of the operator $G_{a,\eta}[\zeta_{c}]$ by isolating the
principal symbol

$\lambda^{1}_{\zeta_{c}}(x,\xi,\eta)=\sqrt{(\xi+ia)^{2}+\eta^{2}(1+(\partial_{x}\zeta_{c})^{2})}\,.$
(3.6)

Note that for any $|\eta|\geq 2$, we have that $\lambda_{\zeta_{c}}^{1}$ does
not vanish for any $(a,x,\xi)\in[0,\frac{1}{2}]\times\mathbb{R}^{2},$ upon
choosing $\epsilon$ small enough. Nevertheless, as mentioned in the
introduction, although $\mathrm{Op}(\lambda_{\zeta_{c}}^{1})$ approximates
well the operator $G_{a,\eta}[\zeta_{c}]$ in terms of the regularity, it is
far from being a good approximation in terms of the size of the operator norm
within $B(H^{1/2},L^{2}),$ since one only has:

$\|G_{a,\eta}[\zeta_{c}]-\mathrm{Op}(\lambda^{1}_{\zeta_{c}})\|_{B(L^{2}(\mathbb{R}))}\lesssim
1\,\,(\text{rather than }\epsilon^{2})\,,$

which prevents us to prove the resolvent bounds by perturbation arguments.
Nevertheless, as proved in Lemma C.2, the desired estimate holds true by
taking a modification of the principal symbol which is more accurate for low
frequencies:

$\|G_{a,\eta}[\zeta_{c}]-\mathrm{Op}(\lambda^{1}_{\zeta_{c}}\tanh\lambda^{1}_{\zeta_{c}})\|_{B(L^{2}(\mathbb{R}))}\lesssim\epsilon^{2}.$

We thus define an approximate operator $\tilde{{\mathcal{L}}}_{a}(\eta)$ by
replacing $G_{a,\eta}[\zeta_{c}]$ by
$\mathrm{Op}(\lambda_{\zeta_{c}}^{1}\tanh\lambda^{1}_{\zeta_{c}})$ in the
definition of ${{\mathcal{L}}}_{a}(\eta),$ that is:

$\tilde{{\mathcal{L}}}_{a}(\eta)=\colon\left(\begin{array}[]{cc}(\partial_{x}-a)(d_{c}\cdot)&\mathrm{Op}(\lambda^{1}_{\zeta_{c}}\tanh\lambda^{1}_{\zeta_{c}})\sqrt{G_{a,\eta}[0]}^{-1}\\\\[5.0pt]
-\sqrt{G_{a,\eta}[0]}w_{c}&\qquad
d_{c}(\partial_{x}-a)+\big{[}\sqrt{G_{a,\eta}[0]},d_{c}\big{]}(\partial_{x}-a)\sqrt{G_{a,\eta}[0]}^{-1}\end{array}\right).$
(3.7)

The strategy is thus to first prove the invertibility and the resolvent bounds
for $\lambda-\tilde{{\mathcal{L}}}_{a}(\eta)$ and then, in a second step to
extend these results to $\lambda-{{\mathcal{L}}}_{a}(\eta)$ through
perturbation arguments.

Define the function

$g_{\eta}(x,\xi)=\sqrt{\frac{w_{c}(x)(\mu_{a}\tanh\mu_{a})(\xi,\eta)}{\lambda^{1}_{\zeta_{c}}\tanh\lambda^{1}_{\zeta_{c}}(x,\xi,\eta)}},$
(3.8)

where $(\mu_{a}\tanh\mu_{a})(\xi,\eta)$ is the symbol of the operator
$G_{a,\eta}[0]$ and $\mu_{a}(\xi,\eta)=\sqrt{(\xi+ia)^{2}+\eta^{2}}.$ It is
shown in Lemma A.2 that for $\epsilon$ small enough, both $g_{\eta}$ and
${1}/{g_{\eta}}$ belong to the symbol class with zero order ($S^{0}$ as
defined in Appendix A). Denote $\mathrm{Op}(f)$ the operator associated to the
symbol $f(x,\xi)$ by the standard quantization (see Appendix A). Then by using
a standard result for the $L^{2}$ continuity of pseudodifferential operators
with $S^{0}$ symbols we get that for any $|\eta|\geq 2,$
$\mathrm{Op}(g_{\eta}),\mathrm{Op}(1/g_{\eta})$ are bounded operators on
$L^{2}(\mathbb{R})$ with norm uniformly bounded in $\eta$ and $\epsilon.$
Moreover, by using standard pseudo-differential calculus and the fact that
$\|\partial_{x}(\zeta_{c},\partial_{x}\varphi_{c})\|_{W_{x}^{k,\infty}(\mathbb{R})}\lesssim\epsilon^{3}$
for $k$ as large as we need to approximate the composition of pseudo-
differential operators [73], it holds that:

$\mathrm{Op}(g_{\eta})\circ\mathrm{Op}(1/g_{\eta})={\rm
Id}_{L^{2}(\mathbb{R})}+R_{1\eta}\,,\quad\mathrm{Op}(1/g_{\eta})\circ\mathrm{Op}(g_{\eta})={\rm
Id}_{L^{2}(\mathbb{R})}+R_{2\eta}\,,$

where $R_{1\eta},R_{2\eta}$ are two bounded operators in $L^{2}(\mathbb{R})$
with operator norms not larger than $C\epsilon^{3}$ (C being independent of
$\eta$).

Let the operator valued matrices $P_{1}(\eta)$ and $P_{2}(\eta)$ be defined
by:

$\displaystyle P_{1}(\eta)=\left(\begin{array}[]{cc}1&1\\\
\mathrm{Op}(g_{\eta})&-\mathrm{Op}(g_{\eta})\end{array}\right),\,\,P_{2}=\frac{1}{2}\left(\begin{array}[]{cc}1&\mathrm{Op}({1}/{g_{\eta}})\\\
1&-\mathrm{Op}(1/g_{\eta})\end{array}\right),$

we can then diagonalize $\tilde{{\mathcal{L}}}_{a}(\eta)$ as:

$\tilde{{\mathcal{L}}}_{a}(\eta)=P_{1}(\eta)\,\,\mathrm{Op}\left(\begin{array}[]{cc}\lambda_{-}^{1}(x,\xi,\eta)&\\\
&\lambda_{+}^{1}(x,\xi,\eta)\end{array}\right)P_{2}(\eta)+{\mathcal{R}}_{\eta}\,,$
(3.9)

where

$\lambda_{\pm}^{1}=i\bigg{(}d_{c}(x)(\xi+ia)\pm\sqrt{w_{c}}(x)\sqrt{\lambda^{1}_{\zeta_{c}}\tanh\lambda^{1}_{\zeta_{c}}}(x,\xi,\eta)\bigg{)}.$
(3.10)

Hereafter, ${\mathcal{R}}_{\eta}$ will always stand for a generic bounded
linear operator in $\mathcal{Y}=L^{2}(\mathbb{R})\times L^{2}(\mathbb{R})$
with norm proportional to $\epsilon^{3}$:

$\|{\mathcal{R}}_{\eta}\|_{B({\mathcal{Y}})}\leq
C\epsilon^{3},\quad\,\forall\,|\eta|\geq 2\,$

and which may change from line to line. Applying $P_{2}(\eta)$ to the
resolvent equation $(\lambda-\tilde{{\mathcal{L}}}_{a}(\eta))u=\tilde{f}$ and
noticing that $P_{2}(\eta)P_{1}(\eta)={\rm
Id}_{{\mathcal{Y}}}+{\mathcal{R}}_{\eta},$ we find:

$\mathrm{Op}\left(\begin{array}[]{cc}\lambda-\lambda_{-}^{1}(x,\xi,\eta)&\\\
&\lambda-\lambda_{+}^{1}(x,\xi,\eta)\end{array}\right)(P_{2}(\eta)u)=P_{2}(\eta)\tilde{f}+{\mathcal{R}}_{\eta}u\,.$

To conclude, it will be enough to show the following claim:

Claim 1: Let $\epsilon$ be sufficiently small. For any $|\eta|\geq 2,$ any
$\lambda\in\Omega_{\beta,\epsilon},$ the operators
$\mathrm{Op}\big{(}\lambda-\lambda_{\pm}^{1}(\cdot,\cdot,\eta)\big{)}$ are
invertible and their inverses have the bounds

$\displaystyle\|\big{(}\mathrm{Op}\big{(}\lambda-\lambda_{\pm}^{1}(\cdot,\cdot,\eta)\big{)}\big{)}^{-1}\|_{B(L^{2}(\mathbb{R}))}\lesssim\epsilon^{-1}.$
(3.11)

Indeed, once this claim is shown, we derive from (3.11) that

$\|P_{2}(\eta)u\|_{{\mathcal{Y}}}\lesssim\epsilon^{-1}\bigg{(}\|P_{2}(\eta)\tilde{f}\|_{{\mathcal{Y}}}+C\epsilon^{3}\|u\|_{{\mathcal{Y}}}\bigg{)}.$

Upon choosing $\epsilon$ sufficiently small and using the fact that
$P_{1}(\eta)P_{2}(\eta)={\rm Id}+{\mathcal{R}}_{\eta},$ we get that
$\|u\|_{{\mathcal{Y}}}\lesssim\epsilon^{-1}\|\tilde{f}\|_{{\mathcal{Y}}}\,.$
The existence also follows from (3.11) and the invertibility of $P_{2}(\eta).$
We thus proved the invertibility of $\lambda-\tilde{{\mathcal{L}}}_{a}(\eta)$
as well the resolvent bound
$\|(\lambda-\tilde{{\mathcal{L}}}_{a}(\eta))^{-1}\|_{{\mathcal{Y}}}\lesssim\epsilon^{-1}.$

To get similar results for the original operator, it suffices to show that
$\tilde{{\mathcal{L}}}_{a}(\eta)$ is a small perturbation of
${{\mathcal{L}}}_{a}(\eta)$ in the sense that,

$\|{{\mathcal{L}}}_{a}(\eta)-\tilde{{\mathcal{L}}}_{a}(\eta)\|_{{\mathcal{Y}}}\lesssim\epsilon^{2}$

uniformly for $\eta$ such that $|\eta|\geq 2.$ This is shown in Lemma C.2. The
proof of Proposition 3.2 is thus finished up to the proof of the above claim.

∎

We now come back to the proof of the Claim 1. It will be a consequence of the
following lemma:

######  Lemma 3.4.

Assume $\epsilon$ to be sufficiently small. Then for any $\eta$ such that
$|\eta|\geq 2$, any $(x,\xi)\in\mathbb{R}^{2},$ it holds that:

${\rm
Re}\,\lambda_{\pm}^{1}(x,\xi,\eta)\leq-\frac{a}{4}=-\frac{\hat{a}\epsilon}{4}.$
(3.12)

Let us postpone the proof of this lemma and first finish the proof of Claim 1.
By Lemma 3.4, for any $\lambda$ such that ${\rm
Re}\,\lambda>-\hat{a}\epsilon^{3},$ we have for $\epsilon$ small enough,
$|\lambda-\lambda_{\pm}|\geq|{\rm
Re}\,(\lambda-\lambda_{\pm}^{1})|>\frac{\hat{a}}{8}\epsilon.$ By applying
Lemma A.1 in the appendix, we have, upon choosing $\epsilon$ smaller if
necessary, that for any $|\eta|\geq 2,$ the operators
$\mathrm{Op}\big{(}(\lambda-\lambda_{\pm}^{1})(\cdot,\cdot,\eta)\big{)}$ are
invertible and

$\|\big{(}\mathrm{Op}\big{(}\lambda-\lambda_{\pm}^{1}(\cdot,\cdot,\eta)\big{)}\big{)}^{-1}\|_{B(L^{2}(\mathbb{R}))}\leq\frac{16}{\hat{a}\epsilon}\,.$

###### Proof of Lemma 3.4.

It suffices to show that for any $|\eta|\geq 2,$ it holds that

$\big{|}{\rm
Im}\,(\sqrt{\lambda_{\zeta_{c}}^{1}\tanh\lambda_{\zeta_{c}}^{1}})\big{|}\leq\frac{3}{5}a\,.$
(3.13)

Indeed, once this is shown, we have

$\displaystyle{\rm Re}\,\lambda^{1}_{\pm}$
$\displaystyle=-a\,d_{c}(x)\mp\sqrt{w_{c}}\,{\rm
Im}\,\big{(}\sqrt{\lambda_{\zeta_{c}}^{1}\tanh\lambda_{\zeta_{c}}^{1}}\big{)}$
$\displaystyle=-a(1-v_{c})\mp\sqrt{\gamma-(1-v_{c})\partial_{x}Z_{c}}\,{\rm
Im}\,\big{(}\sqrt{\lambda_{\zeta_{c}}^{1}\tanh\lambda_{\zeta_{c}}^{1}}\big{)}$
$\displaystyle=-a\mp{\rm
Im}\,\big{(}\sqrt{\lambda_{\zeta_{c}}^{1}\tanh\lambda_{\zeta_{c}}^{1}}\big{)}+{\mathcal{O}}(\epsilon^{3})\leq-\frac{a}{4}$

upon choosing $\epsilon$ small enough. Note that in the above estimate we have
used that as consequences of Theorem 1.1:

$\|v_{c}\|_{L_{x}^{\infty}}\lesssim\epsilon^{2},\quad\|\partial_{x}Z_{c}\|_{L_{x}^{\infty}}\lesssim\epsilon^{3},\quad\gamma={1-\epsilon^{2}}.$

We now come back to the proof of (3.13) which follows from algebraic
computations. By the definition of $\lambda_{\zeta_{c}}^{1}$ in (3.6), one
has:

${\lambda_{\zeta_{c}}^{1}}=\sqrt{R+iI}=\colon u+iv,\,\text{ }\,$

with $R=\xi^{2}+\eta^{2}(1+(\partial_{x}\zeta_{c})^{2})-a^{2},\,I=2a\xi$ and

$u=\sqrt{\frac{R+\sqrt{R^{2}+I^{2}}}{2}},\quad v=\frac{I}{2u}.$ (3.14)

Without loss of generality, we assume $\xi>0$ so that $I>0,v>0.$ By choosing
$\epsilon$ small enough, so that
$\|\partial_{x}\zeta_{c}\|_{L^{\infty}_{x}}\leq 1/4,\,a=\hat{a}\epsilon\leq
1/2,$ one has that $R\geq\xi^{2}+\frac{3}{4}\eta^{2}\geq\xi^{2}+3$ for any
$|\eta|\geq 2.$ Therefore $u\geq\sqrt{R}\geq\sqrt{\xi^{2}+3},\,0<v\leq a.$

Since

$\tanh(u+iv)=\frac{\sinh u\cos v+i\cosh u\sin v}{\cosh u\cos v+i\sinh u\sin
v}=\frac{\sinh(2u)+i\sin(2v)}{2\,(\cosh^{2}u-\sin^{2}v)},$

we have that
$\lambda_{\zeta_{c}}^{1}\tanh\lambda_{\zeta_{c}}^{1}=\tilde{u}+i\tilde{v},$
with

$\tilde{u}=\frac{u\sinh
2u-v\sin(2v)}{2\,(\cosh^{2}u-\sin^{2}v)},\qquad\tilde{v}=\frac{v\sinh(2u)+u\sin(2v)}{2\,(\cosh^{2}u-\sin^{2}v)}.$

As $a\leq\frac{1}{2}<\frac{\pi}{4},$ and $0<v\leq a$, one has that
$2\,(\cosh^{2}u-\sin^{2}v)=\cosh 2u+(1-2\sin^{2}v)>\cosh 2u.$ which leads to

$\tilde{v}\leq v\big{(}\tanh 2u+\frac{2u}{\cosh 2u}\big{)}\leq 2v.$ (3.15)

Moreover, by using the fact that $\frac{\sinh x}{1+\cosh x}$ is an increasing
function on $\mathbb{R}_{+}$, it holds that

$\tilde{u}\geq\frac{u\sinh 2u-v\sin 2v}{1+\cosh 2u}\geq\frac{\sinh
2\sqrt{3}}{1+\cosh 2\sqrt{3}}u+{\mathcal{O}}(\epsilon^{2}).$ (3.16)

Let
$\sqrt{{\lambda_{\zeta_{c}}^{1}}\tanh\lambda_{\zeta_{c}}^{1}}=\alpha+i\beta,$
then

$\alpha=\sqrt{\frac{\tilde{u}+\sqrt{\tilde{u}^{2}+\tilde{v}^{2}}}{2}},\quad\beta=\frac{\tilde{v}}{2\alpha}\,.$

In view of the identity (3.14) and the estimates (3.15), (3.16), we have that

$\beta\leq\frac{I}{2\alpha{u}}\leq\sqrt{\frac{1+\cosh 2\sqrt{3}}{\sinh
2\sqrt{3}}}\frac{I}{2u^{3/2}}+{\mathcal{O}}(\epsilon^{2})\leq\sqrt{\frac{1+\cosh
2\sqrt{3}}{\sinh
2\sqrt{3}}}\frac{|\xi|}{(3+\xi^{2})^{{3}/{4}}}a+{\mathcal{O}}(\epsilon^{2}).$

Since $\cosh 2\sqrt{3}\approx 16,\sinh 2\sqrt{3}\approx 16$ and
$\frac{|\xi|}{(3+\xi^{2})^{{3}/{4}}}\leq\frac{\sqrt{6}}{9^{3/4}}\approx 0.47,$
one sees that, upon choosing $\epsilon$ smaller if necessary,
$\beta\leq\frac{3}{5}a,$ which is the desired result. ∎

###  3.2. Proof of Proposition 3.3

###### Proof of Proposition 3.3.

When focusing on the intermediate transverse frequencies
$\\{A\epsilon^{2}\leq|\eta|\leq 2\\},$ there are two new issues. The symbol of
$\sqrt{G_{a}[0]}$ may vanish and the diagonalization like (3.9) does not work
anymore since $P_{2}(\eta)$ is not a bounded operator on
$L^{2}(\mathbb{R})\times L^{2}(\mathbb{R}).$ Moreover, it is not realistic to
first prove the invertibility of the operator with constant coefficients and
then use perturbation arguments. Indeed, due to the appearance of
$v_{c}\partial_{x}$ terms in the diagonal entries of $L_{a},$ there would be a
loss of derivatives. In order to prove the resolvent bounds (3.2), we shall
use an energy-based approach, achieved through appropriate energy functionals.

For any $\lambda\in\Omega_{\beta,\epsilon},$ $F\in X_{A}^{I},$ consider the
resolvent equation

$(\lambda I-L_{a})\,U=F\,.$ (3.17)

Let us first assume the existence of $U\in X_{A}^{I}$ and prove the resolvent
estimates:

$\|U\|_{X}\lesssim A^{-1}\epsilon^{-3}\,\|F\|_{X}.$ (3.18)

To get this estimate, we shall perform energy estimates which control
quantities that are equivalent to $\|\cdot\|_{X},$ but are different according
to frequency regions (both in the longitudinal frequency $\xi$ and the
transverse frequency $\eta$). On the one hand, when handling the region (most
of cases) where the the operator
$|\frac{\nabla_{a}}{\langle\nabla\rangle^{1/2}}|$
$(\nabla_{a}=(\partial_{x}-a,\partial_{y})^{t})$ is equivalent to
$|\sqrt{G_{a}}[0]|=|\sqrt{\mu_{a}\tanh\mu_{a}}|,$ it is convenient to use a
norm which is equivalent to
$\|\cdot\|_{L^{2}}\times\|\sqrt{G_{a}}[0]\cdot\|_{L^{2}}.$ On the other hand,
when focusing on the region where $\mu_{a}(\xi,\eta)$ may vanish, for instance
the set

${\mathcal{S}}_{K,\delta}=\bigg{\\{}(\xi,\eta)\in\mathbb{R}^{2}\big{|}\,|\xi|\leq
K\epsilon,\,\delta\leq\frac{|\eta|}{\sqrt{\xi^{2}+a^{2}}}\leq
2\bigg{\\}},\quad(\,0<\delta<1\text{ small, }K(K+1)\in[A/2,A]),$ (3.19)

we have to instead introduce an alternative norm. Indeed, in this region, we
have

$(G_{a}[\zeta_{c}]-G_{a}[0])\sqrt{G_{a}[0]}^{-1}=-\partial_{x}\zeta_{c}\frac{\partial_{x}-a}{\sqrt{G_{a}[0]}}+\text{good
operator},$

which is not a bounded operator on $L^{2}(\mathbb{R}^{2}).$ Nevertheless,
since the singularity happens only in a low $\xi$ frequency region, we can
simply use the norm $\|\cdot\|_{L^{2}}\times\|\nabla_{a}\cdot\|_{L^{2}}.$

Let $\tilde{\mathbb{I}}_{\delta}:\mathbb{R}\rightarrow\mathbb{R},$
$\pi_{s}(\cdot),\,\pi_{r}(\cdot):\mathbb{R}^{2}\rightarrow\mathbb{R}$ be
respectively the characteristic functions of the sets

$\mathbb{R}\backslash[-\delta,\delta]\,,\qquad{\mathcal{S}}_{K,\delta}\,,\qquad{\mathcal{S}}_{K,\delta}^{c}=\big{\\{}(\xi,\eta)\in\mathbb{R}^{2}\big{|}\,|\xi|\leq\delta,\,A\epsilon^{2}\leq{|\eta|}\leq
2\big{\\}}\backslash{\mathcal{S}}_{K,\delta}$ (3.20)

respectively. We split the solution $U$ to the problem (3.17) into three parts
$U=U_{h}+U_{s}+U_{r},$ where

$U_{h}=\tilde{\mathbb{I}}_{\delta}(D_{x})U,\quad U_{s}=\pi_{s}(D)U,\quad
U_{r}=\pi_{r}(D)U$

denote a high longitudinal frequency part, a low frequency ‘singular’ part and
a low frequency ‘regular’ part. In order to prove (3.18), it is enough to show
the following estimates on $U_{h},U_{s}$ and $U_{r}:$

$\displaystyle\|U_{h}\|_{X}$
$\displaystyle\lesssim_{\delta}\epsilon^{-1}\|F\|_{X}+\epsilon\|(U_{s},U_{r})\|_{X},$
(3.21) $\displaystyle\|U_{s}\|_{X}$
$\displaystyle\lesssim_{\delta}\epsilon^{-1}\|F\|_{X}+K^{3}\epsilon^{2}\|(U_{h},U_{r})\|_{X},$
(3.22) $\displaystyle\|U_{r}\|_{X}$ $\displaystyle\lesssim
K^{-2}\epsilon^{-3}\big{(}\|F\|_{X}+\epsilon^{2}\|(U_{h},U_{s})\|_{X}\big{)}.$
(3.23)

Indeed, we derive from the above estimates that

$\|U\|_{X}\leq CK^{-2}\epsilon^{-3}\|F\|_{X}+C(K\epsilon+K^{-2})\|U\|_{X}$

where $C>0$ is independent of $K$ and $\epsilon.$ The estimate (3.18) then
follows by choosing first $K$ large enough, and then $\epsilon$ sufficiently
small. We will prove (3.21)-(3.23) in the following subsections.

Once the above estimates are obtained, to finish the proof of the Proposition
3.3, we just need to prove the existence for the resolvent equation (3.17),
that is that any element of $\Omega_{\beta,\epsilon}$ lies in the resolvent
set of $L_{a}$ in the space $X_{A}^{I}.$ As in [48], we can invoke an abstract
result from Spectral Theory (Theorem III.6.7, [30]) in order to reduce the
problem to the proof of the existence of one element of the resolvent set in
$\Omega_{\beta,\epsilon}$. We shall thus prove that $\lambda=1$ lies in the
resolvent set of $L_{a}.$ We split the operator $L_{a}$ into two parts:
$L_{a}=L_{a}^{0}+L_{a}^{1},$ where

${L}_{a}^{0}=\left(\begin{array}[]{cc}\partial_{x}-a&G_{a}[0]\\\\[5.0pt]
-\gamma&\partial_{x}-a\end{array}\right),\,\,L_{a}^{1}=\left(\begin{array}[]{cc}-(\partial_{x}-a)(v_{c}\cdot)&G_{a}[\zeta_{c}]-G_{a}[0]\\\\[5.0pt]
d_{c}\partial_{x}Z_{c}&-v_{c}(\partial_{x}-a)\end{array}\right).$ (3.24)

It is then sufficient to prove on the one hand that ${\rm Id}-L_{a}^{0}$ is
invertible in $X_{A}^{I}$ with the norm of the inverse uniformly bounded in
$\epsilon$ and on the other hand that $({\rm Id}-L_{a}^{0})^{-1}L_{a}^{1}\in
B(X_{A}^{I})$ with the norm

$\|({\rm Id}-L_{a}^{0})^{-1}L_{a}^{1}\|_{B(X_{A}^{I})}\lesssim\epsilon^{2}.$
(3.25)

Since $L_{a}^{0}$ is an operator with constant coefficient, the invertibility
of the operator ${\rm Id}-L_{a}^{0}$ is equivalent to the invertibility of the
complex valued matrix ${\rm Id}_{2}-L_{a}^{0}(\xi,\eta),$
$L_{a}^{0}(\xi,\eta)$ being the symbol of $L_{a}^{0}(D).$

Denote $\lambda_{0}(\xi,\eta)=(\mu_{a}\tanh\mu_{a})(\xi,\eta)$ the symbol
associated to $G_{a}[0],$ and

$\lambda_{\pm}^{0}(\xi,\eta)=i(\xi-ia)\pm\sqrt{-\gamma\lambda_{0}(\xi,\eta)}$

the eigenvalues of $L_{a}^{0}(\xi,\eta).$ By (B.2) in Appendix B, it holds
that ${\rm Re}\,\lambda_{\pm}^{0}\leq a,$ for any $(\xi,\eta)\in\mathbb{R},$
provided $0<a<\frac{\pi}{6}.$ Consequently, we find that ${\rm
Re}\,(1-\lambda_{\pm}^{0})$ never vanishes, which ensures the invertibility of
${\rm Id}_{2}-L_{a}^{0}(\xi,\eta).$ Moreover, straightforward computation show
that

$\displaystyle\big{(}{\rm Id}_{2}-L_{a}^{0}(\xi,\eta)\big{)}^{-1}$
$\displaystyle=\frac{1}{(1-\lambda^{0}_{+})(1-\lambda^{0}_{-})}\left(\begin{array}[]{cc}\frac{(1-\lambda^{0}_{+})+(1-\lambda^{0}_{-})}{2}&\frac{(1-\lambda^{0}_{-})-(1-\lambda^{0}_{+})}{2}\sqrt{-\lambda_{0}/\gamma}\\\\[5.0pt]
-\gamma&\frac{(1-\lambda^{0}_{+})+(1-\lambda^{0}_{-})}{2}\end{array}\right)$
(3.26) $\displaystyle=\colon\left(\begin{array}[]{cc}A_{11},&A_{12}\\\
A_{21}&A_{22}\end{array}\right).$

In light of this explicit expression, we have, by remembering
$X_{A}^{I}=\mathbb{I}_{2}\tilde{\mathbb{I}}_{A\epsilon^{2}}(D_{y})(L^{2}(\mathbb{R}^{2})\times
H_{\star}^{\frac{1}{2}}(\mathbb{R}^{2})),$ that

$\displaystyle\|\big{(}{\rm
Id}_{2}-L_{a}^{0}(\xi,\eta)\big{)}^{-1}\|_{B(X_{A}^{I})}\lesssim\max\sup_{\xi\in\mathbb{R},|\eta|\leq
2}\bigg{\\{}\frac{1}{|1-\lambda^{0}_{+}|}\,,\frac{1}{|1-\lambda^{0}_{-}|}\,,\frac{(1+\xi^{2})^{1/4}}{|(1-\lambda^{0}_{+})(1-\lambda^{0}_{-})|}\bigg{\\}}<+\infty.$
(3.27)

Note that we have used the facts
$\|\sqrt{G_{a}[0]}f\|_{L^{2}}\lesssim\|f\|_{H_{*}^{1/2}},$ and $|{\rm
Im}\,(1-\lambda^{0}_{\pm})|\thickapprox(1+\xi^{2}+\eta^{2})^{\frac{1}{2}}$ for
$|\xi|\gg 1.$

To show (3.25), it suffices to prove the following estimates

$\displaystyle\|A_{11}(\partial_{x}-a)+A_{12}(d_{c}\partial_{x}Z_{c})\|_{B(L^{2},L^{2})}\lesssim\epsilon^{2},$
$\displaystyle\|A_{11}(G_{a}[\zeta_{c}]-G_{a}[0])+A_{12}(v_{c}(\partial_{x}-a))\|_{B(H_{*}^{\frac{1}{2}},L^{2})}\lesssim\epsilon^{2},$
$\displaystyle\|A_{21}(\partial_{x}-a)+A_{22}(d_{c}\partial_{x}Z_{c})\|_{B(L^{2},H_{*}^{\frac{1}{2}})}\lesssim\epsilon^{2},$
$\displaystyle\|A_{21}(G_{a}[\zeta_{c}]-G_{a}[0])+A_{22}(v_{c}(\partial_{x}-a))\|_{B(H_{*}^{\frac{1}{2}},H_{*}^{\frac{1}{2}})}\lesssim\epsilon^{2}.$

Thanks to the the explicit expressions (3.26) and the estimates

$\|v_{c}\|_{W_{x}^{1,\infty}}\lesssim\epsilon^{2},\quad\|d_{c}\partial_{x}Z_{c}\|_{{W_{x}^{1,\infty}}}\lesssim\epsilon^{3},$

as well as the estimate

$\displaystyle\qquad\|\mathbb{I}_{2}(D_{y})(G_{a}[\zeta_{c}]-G_{a}[0])\|_{B(H_{*}^{\frac{1}{2}}(\mathbb{R}^{2}),L^{2}(\mathbb{R}^{2}))}\lesssim\epsilon^{2}$

whose proof is detailed in Lemma C.3, the above four estimates are the
consequences of the following algebraic computation:

$\displaystyle\sup_{\xi\in\mathbb{R},|\eta|\leq
2}\big{(}\langle\xi\rangle+\sqrt{\lambda_{0}}\langle\xi\rangle^{1/2}\big{)}\max\bigg{\\{}\frac{1}{|1-\lambda^{0}_{+}|}\,,\frac{1}{|1-\lambda^{0}_{-}|}\,,\frac{1}{|(1-\lambda^{0}_{+})(1-\lambda^{0}_{-})|}\bigg{\\}}<+\infty.$

To summarize, we have shown that $1\in\rho(L_{a};X_{A}^{I})$ and thus the
proof of Proposition 3.3 is complete. ∎

Before going into the details for the proof of (3.21)-(3.23) in the following
three subsections, we first state some algebraic properties for
$\lambda_{\pm}^{0}$ whose proof is given in the appendix. Since they depend on
the localization in frequencies, we split the frequency domain
$R_{\eta}^{L}=\\{(\xi,\eta)\in\mathbb{R}^{2}\,||\eta|\leq 2\\}$ into the
following subsets:

$\displaystyle
R_{\xi}^{H}=\\{(\xi,\eta)\in\mathbb{R}^{2}\,\big{|}|\xi|\geq\delta,\,|\eta|\leq
2\\},$ $\displaystyle
R_{\xi}^{I}=\\{(\xi,\eta)\in\mathbb{R}^{2}\,\big{|}K\epsilon\leq|\xi|\leq\delta,\,|\eta|\leq
2\\},$ $\displaystyle
R_{\eta,1}^{I}=\\{(\xi,\eta)\in\mathbb{R}^{2}\,\big{|}|\xi|\leq
K\epsilon,\,\delta|\xi+ia|\leq|\eta|\leq 2\\},$ $\displaystyle
R_{\eta,2}^{I}=\\{(\xi,\eta)\in\mathbb{R}^{2}\,\big{|}|\xi|\leq
K\epsilon,\,A\epsilon^{2}\leq|\eta|\leq\delta|\xi+ia|\\},$

where $K(K+1)\in[{A}/{2},A].$ We begin with a lemma about the properties of
the symbol of $\sqrt{-G_{a}[0]}\,.$

######  Lemma 3.5.

Let $\mu_{a}(\xi,\eta)=\sqrt{(\xi+ia)^{2}+\eta^{2}}.$ It holds that:

$\displaystyle 0\leq{\rm
Re}\,\sqrt{-\mu_{a}\tanh\mu_{a}}\leq\left\\{\begin{array}[]{c}a(1-C\delta),\qquad\forall\,(\xi,\eta)\in
R_{\xi}^{H}\cup R_{\eta,1}^{I}\,,\\\\[5.0pt]
a(1-CA\epsilon^{2}),\qquad\forall\,(\xi,\eta)\in R_{\xi}^{I}\cup
R_{\eta,2}^{I}\,.\end{array}\right.$ (3.30)

As a direct consequence of the above lemma, we have the following:

######  Proposition 3.6.

Let
$\lambda_{\pm}^{0}(\xi,\eta)=i(\xi+ia)\pm\sqrt{1-\epsilon^{2}}\sqrt{-\mu_{a}\tanh\mu_{a}}(\xi,\eta),$
for any $\lambda\in\Omega_{\beta,\epsilon},$ such that ${\rm
Re}\,\lambda>-\beta\epsilon^{3},$ it holds that,

${\rm Re}\,(\lambda-\lambda_{-}^{0})\geq
a-\beta\epsilon^{3},\qquad\forall\,(\xi,\eta)\in R_{\eta}^{L}.$ (3.31)
$\displaystyle{\rm
Re}\,(\lambda-\lambda_{+}^{0})\gtrsim\left\\{\begin{array}[]{c}\delta
a,\qquad\forall\,(\xi,\eta)\in R_{\xi}^{H}\cup R_{\eta,1}^{I},\\\\[4.0pt]
A\epsilon^{3},\qquad\forall\,(\xi,\eta)\in R_{\xi}^{I}\cup
R_{\eta,2}^{I}\,.\end{array}\right.$ (3.32)

###### Proof.

It follows from Lemma 3.5 and the identity

${\rm Re}\,(\lambda-\lambda_{\pm})={\rm
Re}\,\lambda-\bigg{(}-a\pm\sqrt{1-\epsilon^{2}\,}\,{\rm
Re}\,\sqrt{-\mu_{a}\tanh\mu_{a}}\bigg{)}.$

∎

####  3.2.1. Proof of (3.21)

We focus in this subsection on the uniform high longitudinal frequency region
$\\{|\xi|\geq\delta\\}$ on which
$m\big{(}\sqrt{G_{a}[0]}\big{)}=\sqrt{\mu_{a}\tanh\mu_{a}}(\xi)\approx_{\delta}m\big{(}\frac{\nabla_{a}}{\langle\nabla\rangle^{1/2}}\big{)}$
(we denote $m(\cdot)$ the symbol of a Fourier multiplier). Therefore, for any
$f\in\tilde{\mathbb{I}}_{\delta}H_{*}^{1/2},$ it holds that
$\|f\|_{H_{*}^{1/2}}\approx\|\sqrt{G_{a}[0]}f\|_{L^{2}}.$ We thus define
$(\tilde{U},\tilde{F})=\textnormal{diag}\,(1,\sqrt{G_{a}[0]}\,)(U,F)$ and aim
to show the following estimate for
$\tilde{U}_{h}=:\tilde{\mathbb{I}}_{\delta}\,\tilde{U}:$

$\|\tilde{U}_{h}\|_{\mathcal{X}}\lesssim_{\delta}\epsilon^{-1}\|\tilde{F}\|_{\mathcal{X}}+\epsilon\|{U}\|_{X},\quad(\mathcal{X}=L^{2}(\mathbb{R}^{2})\times
L^{2}(\mathbb{R}^{2})).$ (3.33)

Taking
$\tilde{\mathbb{I}}_{\delta}(D_{x})\,\textnormal{diag}\,(1,\sqrt{G_{a}[0]}\,)$
on both sides of (3.17) and using
$\tilde{\mathbb{I}}_{\delta}\circ\tilde{\mathbb{I}}_{\delta}=\tilde{\mathbb{I}}_{\delta},$
we find that $\tilde{U}^{h}$ satisfies

$\big{(}\lambda-\tilde{\mathbb{I}}_{\delta}\,{\mathcal{L}}_{a}^{0}\,\tilde{\mathbb{I}}_{\delta}\big{)}\,\tilde{U}_{h}=\tilde{F}+H(U)$
(3.34)

where

$H(U)=\tilde{\mathbb{I}}_{\delta}\,{\mathcal{L}}_{a}^{1}\,{U}+\tilde{\mathbb{I}}_{\delta}\,\textnormal{diag}\,(1,\sqrt{G_{a}[0]})\,{\mathcal{L}}_{a}^{2}\,{\mathbb{I}}_{\delta}\,{U}$

and

$\small{\mathcal{L}}_{a}^{0}=\left(\begin{array}[]{cc}(\partial_{x}-a)(d_{c}\cdot)&\sqrt{G_{a}[0]}\\\\[5.0pt]
-\sqrt{G_{a}[0]}(w_{c}\cdot)&d_{c}(\partial_{x}-a)\end{array}\right),\quad{\mathcal{L}}_{a}^{1}=\left(\begin{array}[]{cc}0&G_{a}[\zeta_{c}]-G_{a}[0]\\\\[5.0pt]
0&0\end{array}\right),\quad{\mathcal{L}}_{a}^{2}=\left(\begin{array}[]{cc}-(\partial_{x}-a)(v_{c}\cdot)&0\\\\[5.0pt]
d_{c}\partial_{x}Z_{c}&-v_{c}(\partial_{x}-a)\end{array}\right).$ (3.35)

Denote $\tilde{U}_{h}=(\tilde{\zeta}_{h},\tilde{\varphi}_{h}).$ Taking the
real part of the inner product between the equation (3.34) and
$(w_{c}\,\tilde{\zeta}_{h},\tilde{\varphi}_{h})$ in $\mathcal{X},$ we find the
identity

$\displaystyle{\rm Re}\,\lambda\int
w_{c}|\tilde{\zeta}_{h}|^{2}+|\tilde{\varphi}_{h}|^{2}\,\mathrm{d}x\mathrm{d}y+a\int
d_{c}\big{(}w_{c}|\tilde{\zeta}_{h}|^{2}+|\tilde{\varphi}_{h}|^{2}\big{)}\,\mathrm{d}x\mathrm{d}y$
(3.36) $\displaystyle\qquad+{\rm
Re}\,\int\big{(}\sqrt{G_{a}[0]}-\sqrt{G_{-a}[0]}\big{)}(w_{c}\tilde{\zeta}_{h})\,\overline{\tilde{\varphi}_{h}}\,\mathrm{d}x\mathrm{d}y$
$\displaystyle={\mathcal{J}}_{1}+{\mathcal{J}}_{2}+{\mathcal{J}}_{3}$

where

$\displaystyle{\mathcal{J}}_{1}$ $\displaystyle=\frac{1}{2}\,{\rm
Re}\,\int\big{(}2w_{c}\partial_{x}d_{c}-\partial_{x}({d_{c}w_{c}})\big{)}|\tilde{\zeta}_{h}|^{2}-\partial_{x}d_{c}|\tilde{\varphi}_{h}|^{2}\,\mathrm{d}x\mathrm{d}y,$
$\displaystyle{\mathcal{J}}_{2}$ $\displaystyle={\rm
Re}\,\int{\tilde{\varphi}_{h}}\,\,\overline{\sqrt{G_{-a}[0]}\big{[}\tilde{\mathbb{I}}_{\delta},w_{c}\big{]}\tilde{\mathbb{I}}_{\delta}\tilde{\zeta}_{h}}-d_{c}\tilde{\varphi}_{h}\,\,\partial_{x}\big{[}\tilde{\mathbb{I}}_{\delta},w_{c}\big{]}\tilde{\mathbb{I}}_{\delta}\overline{\tilde{\zeta}_{h}}\,\mathrm{d}x\mathrm{d}y,$
$\displaystyle{\mathcal{J}}_{3}$ $\displaystyle={\rm
Re}\,\int(\tilde{F}+H(U))\cdot\overline{(w_{c}\tilde{\zeta}_{h},\tilde{\varphi}_{h})}\,\mathrm{d}x\mathrm{d}y\,.$

By using the Cauchy-Schwarz inequality, the real part of the left hand side of
(3.36) is larger than

$\displaystyle\big{(}{\rm
Re}\,\lambda+a-\sup_{x\in\mathbb{R}}\sqrt{w_{c}}(x)\sup_{|\xi|\geq\delta,\,|\eta|\leq
2}\,{\rm
Im}\,\sqrt{\mu_{a}\tanh\mu_{a}}(\xi,\eta)\big{)}\big{(}\|\sqrt{w_{c}}\tilde{\zeta}_{h}\|_{L^{2}}^{2}+\|\tilde{\varphi}_{h}\|_{L^{2}}^{2}\big{)},$

which thanks to Proposition 3.6, is greater than

$a(1-C\delta)\big{(}\|\sqrt{w_{c}}\tilde{\zeta}_{h}\|_{L^{2}}^{2}+\|\tilde{\varphi}_{h}\|_{L^{2}}^{2}\big{)}\geq\frac{\hat{a}\epsilon}{2}\big{(}\|\sqrt{w_{c}}\tilde{\zeta}_{h}\|_{L^{2}}^{2}+\|\tilde{\varphi}_{h}\|_{L^{2}}^{2}\big{)}$

for any $\lambda\in\Omega_{\beta,\epsilon}.$ We now control the terms
${\mathcal{J}}_{1}-{\mathcal{J}}_{3}$ in the right hand side of (3.36). Since
$\|\partial_{x}(d_{c},w_{c})\|_{L_{x}^{\infty}}\lesssim\epsilon^{3},$ it holds
that for sufficiently small $\epsilon,$

$w_{c}(x)=\gamma-
d_{c}\partial_{x}Z_{c}={1-\epsilon^{2}}+{\mathcal{O}}(\epsilon^{3})\in[1/2,1],\quad\forall\,x\in\mathbb{R}\,,$

we thus have that

${\mathcal{J}}_{1}\lesssim\epsilon^{3}\big{(}\|\sqrt{w_{c}}\tilde{\zeta}_{h}\|_{L^{2}}^{2}+\|\tilde{\varphi}_{h}\|_{L^{2}}^{2}\big{)}.$
(3.37)

Next, it follows from the commutator estimate (3.38) stated in Lemma 3.7
below, that we use for $s=0$, that

$\|\big{(}\mathbb{I}_{2}(D_{y})\sqrt{G_{-a}[0]},\partial_{x}\big{)}\big{[}\tilde{\mathbb{I}}_{\delta},w_{c}\big{]}\tilde{\mathbb{I}}_{\delta}\|_{B(L^{2})}\leq
C\|{\mathcal{F}}_{x\rightarrow\xi}(w_{c})\|_{L_{\xi}^{1}}\lesssim\epsilon^{2},$

where $C=\delta+\sup_{|\xi|\leq\delta,|\eta|\leq
2}|\sqrt{\mu_{a}\tanh\mu_{a}}(\xi,\eta)|.$ Consequently, by again the Cauchy-
Schwarz inequality, it holds that

${\mathcal{J}}_{2}\lesssim\epsilon^{2}\big{(}\|\sqrt{w_{c}}\tilde{\zeta}_{h}\|_{L^{2}}^{2}+\|\tilde{\varphi}_{h}\|_{L^{2}}^{2}\big{)}.$

Finally, thanks to Lemma C.3 and Lemma C.9 in the appendix, we have

$\|H(U)\|_{\mathcal{X}}\lesssim\epsilon^{2}\|U\|_{X}.$

The Cauchy-Schwarz inequality then stems:

${\mathcal{J}}_{3}\lesssim(\|\tilde{F}\|_{\mathcal{X}}+\epsilon^{2}\|U\|_{X})\big{(}\|\sqrt{w_{c}}\tilde{\zeta}_{h}\|_{L^{2}}+\|\tilde{\varphi}_{h}\|_{L^{2}}\big{)}.$

To summarize, it holds that

${\mathcal{J}}_{1}+{\mathcal{J}}_{2}+{\mathcal{J}}_{3}\lesssim\epsilon^{2}\big{(}\|\sqrt{w_{c}}\tilde{\zeta}_{h}\|_{L^{2}}^{2}+\|\tilde{\varphi}_{h}\|_{L^{2}}^{2}\big{)}+(\|\tilde{F}\|_{\mathcal{X}}+\epsilon^{2}\|U\|_{X})\big{(}\|\sqrt{w_{c}}\tilde{\zeta}_{h}\|_{L^{2}}+\|\tilde{\varphi}_{h}\|_{L^{2}}\big{)}.$

Since
$\|\tilde{U}_{h}\|_{\mathcal{X}}^{2}\approx\|\sqrt{w_{c}}\tilde{\zeta}_{h}\|_{L^{2}}^{2}+\|\tilde{\varphi}_{h}\|_{L^{2}}^{2},$
we find eventually that

$\|\tilde{U}_{h}\|_{\mathcal{X}}\lesssim\epsilon\|\tilde{U}_{h}\|_{\mathcal{X}}+\epsilon^{-1}\|\tilde{F}\|_{\mathcal{X}}+\epsilon\|U\|_{X},$

which yields (3.33) by assuming that $\epsilon$ is sufficiently small.

In the next lemma, we state a commutator estimate that is used in the above
proof. It is a simple generalization of the Proposition 5.1 of [48] in one
dimension:

######  Lemma 3.7.

Let $\mathcal{A},\mathcal{B},\mathcal{W}$ be three Fourier multipliers on the
space $L^{2}(\mathbb{R}^{2})$ with symbols $A,B,W$ and let $f\in
C^{\infty}(\mathbb{R})$ and ${\mathcal{F}}(f)$ be sufficiently localized, for
any $s\in\mathbb{R}$ such that $C_{s}$ and $C_{f,s}$ below are finite, it
holds that

$\|\mathcal{A}[\mathcal{B},f]\mathcal{W}\|_{B(L^{2}(\mathbb{R}^{2}))}\lesssim
C_{s}\,C_{f,s}$ (3.38)

where

$C_{s}=\sup_{(\xi,\xi^{\prime},\eta)\in\mathbb{R}^{3}}\frac{A(\xi,\eta)\big{(}B(\xi,\eta)-B(\xi^{\prime},\eta)\big{)}W(\xi^{\prime},\eta)}{|\xi-\xi^{\prime}|^{s}},\qquad
C_{f,s}=\int_{\mathbb{R}}|\xi|^{s}|{\mathcal{F}}(f)|(\xi)\,\mathrm{d}\xi.$

####  3.2.2. Proof of (3.22)

In this subsection, we prove the estimate (3.22) for the singular part
$U_{s}=\pi_{s}({D})U.$ Since $\pi_{s}$ localizes in the low frequency region
and $\|\pi_{s}(D)f\|_{H_{*}^{1/2}}\approx\|\nabla_{a}\pi_{s}(D)f\|_{L^{2}},$
it is convenient to split $L_{a}$ into two parts:

${L}_{a}=\left(\begin{array}[]{cc}\partial_{x}-a&-\Delta_{a}\\\\[5.0pt]
-\gamma&\partial_{x}-a\end{array}\right)+\left(\begin{array}[]{cc}-(\partial_{x}-a)(v_{c}\cdot)&G_{a}[\zeta_{c}]+\Delta_{a}\\\\[5.0pt]
-\partial_{x}Z_{c}&-v_{c}(\partial_{x}-a)\end{array}\right)=\colon\tilde{L}_{a}^{0}+\tilde{L}_{a}^{1}\,,$

where $\Delta_{a}=(\partial_{x}-a)^{2}+\partial_{y}^{2}.$ Applying the Fourier
multiplier $\pi_{s}(D)$ on both sides of (3.17), we find that $U_{s}$ solves
the equation:

$\big{(}\lambda-\tilde{L}_{a}^{0}\big{)}U_{s}=\pi_{s}(D)(F+\tilde{L}_{a}^{1}U)\,.$

Denote $(U_{\lambda},F_{\lambda})=e^{\lambda t}(U,F)\in C([0,\infty);X)^{2}.$
Since $e^{\lambda t}U_{s}$ solves the evolution problem

$\big{(}\partial_{t}-\tilde{L}_{a}^{0}\big{)}V=\pi_{s}(D)(F_{\lambda}+\tilde{L}_{a}^{1}U_{\lambda})\,,$
(3.39)

we first prove some a priori estimates for the equation (3.39). Note that we
could perform the energy estimates directly on the eigenvalue problem, but the
structure is somewhat more clear if we go through the time evolution problem.

Let us set $V=(V_{1},V_{2})^{t}$ and define the energy functional

${\mathcal{E}}_{s}(V(t))=\colon\frac{1}{2}\int\big{(}\gamma|V_{1}(t)|^{2}+|\nabla_{a}V_{2}(t)|^{2}\big{)}\,\,\mathrm{d}x\mathrm{d}y\,.$

We get, after performing the straightforward energy estimate for the equation
(3.39) in the space $L^{2}\times\nabla_{a}^{-1}L^{2},$ that

$\displaystyle\partial_{t}\,{\mathcal{E}}_{s}(V)$
$\displaystyle={\mathcal{I}}_{1}+{\mathcal{I}}_{2}+{\mathcal{I}}_{3}$ (3.40)

where

$\displaystyle{\mathcal{I}}_{1}$
$\displaystyle=-2a\,{\mathcal{E}}_{s}(V)+2a\gamma\,{\rm Re}\,\int
V_{1}\,(\partial_{x}-a)\overline{V_{2}}\,\mathrm{d}x\mathrm{d}y,$
$\displaystyle{\mathcal{I}}_{2}$ $\displaystyle={\rm
Re}\,\int\big{(}F_{\lambda 1}\overline{V_{1}}+\nabla_{a}F_{\lambda
2}\cdot\nabla_{a}\overline{V_{2}}\big{)}\,\mathrm{d}x\mathrm{d}y,$
$\displaystyle{\mathcal{I}}_{3}$ $\displaystyle={\rm
Re}\,\int\big{(}S_{1}\overline{V_{1}}+\nabla_{a}S_{2}\cdot\nabla_{a}\overline{V_{2}}\big{)}\,\mathrm{d}x\mathrm{d}y,$

and

$S_{1}=\pi_{s}({D})\big{(}-(\partial_{x}-a)(v_{c}U_{\lambda
1})+(G_{a}[\zeta_{c}]+\Delta_{a})U_{\lambda 2}\big{)};\quad
S_{2}=-\pi_{s}({D})\big{(}d_{c}\partial_{x}Z_{c}U_{\lambda
1}+v_{c}(\partial_{x}-a)U_{\lambda 2}\big{)}.$

Since $\pi_{s}(\xi,\eta)$ is supported in
$\\{|\eta|\geq\frac{\delta}{2}|\xi+ia|\\},$ we have

$\|(\partial_{x}-a)V_{2}\|_{L^{2}}\leq\frac{1}{\sqrt{1+{\delta^{2}}/{4}}}\|\nabla_{a}V_{2}\|_{L^{2}}.$

It then follows from the Cauchy-Schwarz inequality that

${\mathcal{I}}_{1}\leq-2a\bigg{(}1-\sqrt{\frac{\gamma}{1+\delta^{2}/4}}\bigg{)}{\mathcal{E}}_{s}(V)\leq-\frac{\delta^{2}a}{4}{\mathcal{E}}_{s}(V).$
(3.41)

Next, by using again the Cauchy-Schwarz inequality

${\mathcal{I}}_{2}\lesssim\|(F_{\lambda 1},\nabla_{a}F_{\lambda
2})\|_{L^{2}}\|(V^{\ell},\nabla_{a}V_{2}^{\ell})\|_{L^{2}}\lesssim_{\gamma}\|F_{\lambda}\|_{X}\sqrt{{\mathcal{E}}_{a}}(V).$
(3.42)

We now estimate the term $I_{3}$ which could be bounded as
$I_{3}\leq\frac{1}{\gamma}\|(S_{1},\nabla_{a}S_{2})\|_{L^{2}}\sqrt{{\mathcal{E}}_{s}}(V).$
Recalling that
$v_{c}={\mathcal{O}}_{L_{x}^{\infty}}(\epsilon^{2}),\,\partial_{x}Z_{c}={\mathcal{O}}_{L_{x}^{\infty}}(\epsilon^{3})$
and that $\pi_{s}(\xi,\eta)$ is supported on the region
${\mathcal{S}}_{K,\,\delta}=\\{|\xi|\leq K\epsilon,\,\delta|\xi|\leq|\eta|\leq
2|\xi+ia|\\},$ one readily gets that:

$\displaystyle\|\pi_{s}({D})(\partial_{x}-a)(v_{c}U_{\lambda
1})\|_{L^{2}}+\|\pi_{s}({D})\nabla_{a}\big{(}d_{c}\partial_{x}Z_{c}U_{\lambda
1}\big{)}\|_{L^{2}}\lesssim K\epsilon^{3}\|U_{\lambda 1}\|_{L^{2}},$ (3.43)
$\displaystyle\quad\|\pi_{s}({D})\nabla_{a}(v_{c}(\partial_{x}-a)U_{\lambda
2})\|_{L^{2}}$
$\displaystyle\lesssim\|\pi_{s}({D})\nabla_{a}\partial_{x}\big{(}v_{c}U_{\lambda
2}\big{)}\|_{L^{2}}+\|\pi_{s}({D})\nabla_{a}(\partial_{x}v_{c}U_{\lambda
2})\|_{L^{2}}+K\epsilon^{2}\|v_{c}U_{\lambda 2}\|_{L^{2}}$
$\displaystyle\lesssim(K^{2}\epsilon^{2}\|v_{c}\|_{L_{x}^{\infty}}+K\epsilon\|\partial_{x}v_{c}\|_{L_{x}^{\infty}})\|U_{\lambda
2}\|_{L^{2}}\lesssim K^{2}\epsilon^{3}\|U_{\lambda
2}\|_{H_{*}^{\frac{1}{2}}}.$

Moreover, we prove in Lemma C.6 that:

$\|\pi_{s}({D})(G_{a}[\zeta_{c}]+\Delta_{a})\|_{B(L^{2})}\lesssim(K\epsilon)^{3},$

which, combined with (3.43), yields

$\|(S_{1},\nabla_{a}S_{2})\|_{L^{2}}\lesssim(K\epsilon)^{3}\|U_{\lambda}\|_{X}.$

We thus conclude that

${\mathcal{I}}_{3}\lesssim(K\epsilon)^{3}\|U_{\lambda}\|_{X}\sqrt{{\mathcal{E}}_{s}}(V).$
(3.44)

Plugging the estimates (3.41)-(3.44) into (3.40) and applying the Grönwall
inequality, we find

$\sqrt{{\mathcal{E}}_{s}}(V(t))\leq
e^{-\delta^{2}at/4}\sqrt{{\mathcal{E}}_{a}}(V(0))+C\int_{0}^{t}e^{-\delta^{2}a(t-s)/4}\big{(}\|F_{\lambda}(s)\|_{X}+(K\epsilon)^{3}\|U_{\lambda}(s)\|_{X}\big{)}\,\mathrm{d}s.$
(3.45)

As $e^{\lambda t}U_{s}$ solves (3.39) with
$(U_{\lambda},F_{\lambda})=e^{\lambda t}(U,F),$ we then derive that

$\sqrt{{\mathcal{E}}_{s}}(U_{s})\leq
e^{-\alpha_{\lambda}t}\sqrt{{\mathcal{E}}_{s}}(U_{s})+C\big{(}\|F\|_{X}+K^{3}\epsilon^{3}\|U\|_{X}\big{)}\int_{0}^{t}e^{-\alpha_{\lambda}(t-s)}\,\mathrm{d}s\,$

where $\alpha_{\lambda}={\rm Re}\,\lambda+\delta^{2}a/4.$ As
$a=\hat{a}\epsilon$ and ${\rm Re}\,\lambda>-\beta\epsilon^{3},$ one has that
$\alpha_{\lambda}\geq\delta^{2}a/8>0$ as long as $\epsilon$ being small
enough. Consequently, by taking the limit $t\rightarrow+\infty$, we find

$\sqrt{{\mathcal{E}}_{s}}(U_{s})\lesssim\epsilon^{-1}\|F\|_{X}+K^{3}\epsilon^{2}\|U\|_{X},$

which leads to (3.22) since
$\sqrt{{\mathcal{E}}_{s}}(U_{s})\approx\|U_{s}\|_{X}.$

####  3.2.3. Proof of (3.23)

In this subsection, we estimate $U_{r}=\pi_{r}(D)U$ which is localized in
$\\{(\xi,\eta)\in\mathbb{R}\big{|}\,|\xi|\leq\delta,A\epsilon^{2}\leq|\eta|\leq
2\\}\backslash{\mathcal{S}}_{K,\delta}$ and
$\|U_{r}\|_{X}\approx\|\textnormal{diag}\big{(}1,\sqrt{G_{a}[0]}\big{)}U_{r}\|_{\mathcal{X}}.$
Since the damping mechanism is weaker in this region (proportional to
$\epsilon^{3}$), we need to use an energy functional which involves the
background surface wave $\zeta_{c}$ and is equivalent to the norm
$\|\textnormal{diag}\big{(}1,\sqrt{G_{a}[0]}\big{)}U_{r}\|_{\mathcal{X}}.$

Taking the Fourier multiplier $\pi_{r}(D)$ on both sides of (3.17), we find
that $U_{r}$ solves the equation

$\big{(}\lambda-\pi_{r}(D)L_{a}\pi_{r}(D)\big{)}U_{r}=\pi_{r}(D)\big{(}F+{L}_{a}^{1}(U_{h}+U_{s})\big{)},$

where $L_{a}^{1}$ is defined in (3.24). Denote
$W_{\lambda}=\pi_{r}(D)e^{\lambda t}\big{(}F+{L}_{a}^{1}(U_{h}+U_{s})\big{)}.$
As in the estimate for $U_{s},$ we consider the evolution problem:

$\big{(}\partial_{t}-\pi_{r}(D)L_{a}\pi_{r}(D)\big{)}V=W_{\lambda}.$ (3.46)

We shall use the multiplier $Q_{a}(D)=\sqrt{1-\tanh^{2}\mu_{a}(D)}$ and the
energy functional

${\mathcal{E}}_{r}(V(t))=\colon\frac{1}{2}\int\bigg{(}\gamma\,|V_{1}(t)|^{2}+|\sqrt{G_{a}[0]}V_{2}(t)|^{2}+\zeta_{c}\big{|}\nabla_{a}Q_{a}(D)V_{2}(t)\big{|}^{2}\bigg{)}\,\,\mathrm{d}x\mathrm{d}y.$

By computing the time derivative of the above functional and then by using the
equation (3.46), we find the energy identity

$\displaystyle\partial_{t}\,{\mathcal{E}}_{r}(V)={\mathcal{K}}_{1}+\cdots{\mathcal{K}}_{6}\,,$

where

$\displaystyle{\mathcal{K}}_{1}$
$\displaystyle=-a\int(1-v_{c}(x))\bigg{(}\gamma\,|V_{1}(t)|^{2}+|\sqrt{G_{a}[0]}V_{2}(t)|^{2}+\zeta_{c}\big{|}\nabla_{a}Q_{a}(D)V_{2}(t)\big{|}^{2}\bigg{)}\,\,\mathrm{d}x\mathrm{d}y$
$\displaystyle\qquad\qquad-\gamma\,{\rm
Re}\,\int\big{(}\sqrt{G_{a}[0]}-\sqrt{G_{-a}[0]}\big{)}V_{1}\,\overline{\sqrt{G_{a}[0]}V_{2}}\,\mathrm{d}x\mathrm{d}y,$
$\displaystyle{\mathcal{K}}_{2}$ $\displaystyle=\gamma\,{\rm
Re}\,\int\pi_{r}(D)\big{(}G_{a}[\zeta_{c}]-G_{a}[0]+Q_{a}(D)\,\mathrm{div}_{a}(\zeta_{c}\nabla_{a}Q_{a}(D))\big{)}\pi_{r}(D)V_{2}\,\,\overline{V_{1}}\,\mathrm{d}x\mathrm{d}y,$
$\displaystyle{\mathcal{K}}_{3}$ $\displaystyle=\gamma\,{\rm
Re}\,\int\,\mathrm{div}_{a}(\zeta_{c}\nabla_{a}Q_{a}(D)V_{2})\,\,\overline{(Q_{a}(D)-Q_{-a}(D))V_{1}}\,\mathrm{d}x\mathrm{d}y,$
$\displaystyle{\mathcal{K}}_{4}$ $\displaystyle=2a\gamma\,{\rm
Re}\,\int\,\zeta_{c}(\partial_{x}-a)Q_{a}(D)V_{2}\,\,\overline{Q_{a}(D)V_{1}}\,\mathrm{d}x\mathrm{d}y,$
$\displaystyle{\mathcal{K}}_{5}$
$\displaystyle=\frac{1}{2}\int\partial_{x}d_{c}\big{(}\gamma|V_{1}|^{2}-|\sqrt{G_{a}[0]}V_{2}|^{2}\big{)}-\partial_{x}(d_{c}\,\zeta_{c})|\nabla_{a}Q_{a}(D)V_{2}|^{2}\,\mathrm{d}x\mathrm{d}y,$
$\displaystyle{\mathcal{K}}_{6}$ $\displaystyle={\rm
Re}\,\int\pi_{r}(D)\sqrt{G_{a}[0]}(d_{c}\partial_{x}Z_{c}V_{1})\overline{\sqrt{G_{a}[0]}V_{2}}+\zeta_{c}\pi_{r}(D)\nabla_{a}Q_{a}(D)(d_{c}\partial_{x}Z_{c}V_{1})\overline{\nabla_{a}Q_{a}(D)V_{2}}\,\mathrm{d}x\mathrm{d}y,$
$\displaystyle{\mathcal{K}}_{7}$ $\displaystyle={\rm
Re}\,\int\big{(}W_{\lambda 1}\overline{\gamma V_{1}}+\sqrt{G_{a}[0]}W_{\lambda
2}\,\overline{\sqrt{G_{a}[0]}V_{2}}+\zeta_{c}\nabla_{a}Q_{a}(D)W_{\lambda
2}\cdot\overline{\nabla_{a}Q_{a}(D)V_{2}}\big{)}\,\mathrm{d}x\mathrm{d}y.$

We now control $K_{1}-K_{7}$ term by term. At first, by the Cauchy-Schwarz
inequality along with Lemma 3.5 and the fact that
$\|v_{c}\|_{L^{\infty}_{x}}\lesssim\epsilon^{2},$ we can find a constant $C>0$
such that, for $A$ large enough and $\epsilon$ sufficiently small,

${\mathcal{K}}_{1}\leq-2\bigg{(}a\inf_{x\in\mathbb{R}}(1-v_{c}(x))-\gamma\sup_{|\xi|\leq\delta,\,A\epsilon^{2}\leq|\eta|\leq
2}\big{|}{\rm
Im}\,\sqrt{\mu_{a}\tanh\mu_{a}}(\xi,\eta)\big{|}\bigg{)}{\mathcal{E}}_{r}(V(t))\leq-
CA\epsilon^{3}{\mathcal{E}}_{r}(V(t)).$

Next, we apply Lemma C.7 and the Cauchy-Schwarz inequality to find that

${\mathcal{K}}_{2}\lesssim\epsilon^{3}\,{\mathcal{E}}_{r}(V(t))\,.$

Hereafter, we denote $\lesssim$ as $\leq C$ for some constant that is
independent of $A$ and $\epsilon.$ Moreover, let us notice that $Q_{a}(D)\in
S^{\infty}$ and by (B.1),

$|Q_{a}(\xi,\eta)-Q_{-a}(\xi,\eta)|\lesssim|{\rm Im}\,\mu_{a}|\lesssim
a\lesssim\epsilon,\qquad\forall\,(\xi,\eta)\in{\rm Supp}\,\pi_{r}\,,$

which, together with the fact
$\|\zeta_{c}\|_{L_{x}^{\infty}}\lesssim\epsilon^{2},\,$ yield:

${\mathcal{K}}_{3}\lesssim\epsilon^{3}\,{\mathcal{E}}_{r}(V(t))\,.$

In addition, thanks to the facts $a=\hat{a}\epsilon$ and
$\|\partial_{x}(\zeta_{c},d_{c})\|_{L_{x}^{\infty}}\lesssim\epsilon^{3},$ one
readily gets that:

${\mathcal{K}}_{4}+{\mathcal{K}}_{5}\lesssim\epsilon^{3}\,{\mathcal{E}}_{r}(V(t))\,.$

Furthermore, as $\|\pi_{r}(D)\sqrt{G_{a}[0]}\|_{B(L^{2})}\lesssim 1,$ and
$\|\partial_{x}Z_{c}\|_{L_{x}^{\infty}}\lesssim\epsilon^{3},$ we find also

${\mathcal{K}}_{6}\lesssim\epsilon^{3}{\mathcal{E}}_{r}(V(t))\,.$

Finally, it stems from the Cauchy-Schwarz inequality and the fact that

$\|\pi_{r}(D)\sqrt{G_{a}[0]}W_{\lambda
2}\|_{L^{2}}\approx\|\pi_{r}(D)\nabla_{a}W_{\lambda
2}\|_{L^{2}}\approx\|\pi_{r}(D)W_{\lambda 2}\|_{H_{*}^{1/2}}$

that

${\mathcal{K}}_{7}\lesssim\|W_{\lambda}\|_{X}\sqrt{{\mathcal{E}}_{r}(V(t))}\,.$

Therefore, upon choosing $A$ large enough, it holds that:

$\partial_{t}\,{\mathcal{E}}_{r}(V(t))\leq-\frac{C}{2}A\epsilon^{3}{\mathcal{E}}_{r}(V(t))+C_{1}\|W_{\lambda}\|_{X}\sqrt{{\mathcal{E}}_{r}(V(t))}\,.$

It then follows from the Grönwall inequality that:

$\sqrt{{\mathcal{E}}_{r}}(V(t))\leq
e^{-CA\epsilon^{3}t/2}\sqrt{{\mathcal{E}}_{a}}(V(0))+C_{1}\int_{0}^{t}e^{-CA\epsilon^{3}(t-s)/2}\|W_{\lambda}(s)\|_{X}\,\mathrm{d}s\,.$

Plugging $V(t)=e^{\lambda t}U_{r}$ into the above estimate and since we had
set $W_{\lambda}=\pi_{r}(D)e^{\lambda
t}\big{(}F+\tilde{L}_{a}(U_{h}+U_{s})\big{)},$ we find the inequality

$\sqrt{{\mathcal{E}}_{r}}(U_{r})\leq
e^{-\beta_{\lambda}t}\sqrt{{\mathcal{E}}_{r}}(U_{r})+C_{1}\big{(}\|F\|_{X}+\|\pi_{r}(D){L}_{a}^{1}(U_{h}+U_{s})\|_{X}\big{)}\int_{0}^{t}e^{-\beta_{\lambda}(t-s)}\,\mathrm{d}s\,,$

where $\beta_{\lambda}={\rm Re}\,\lambda+CA\epsilon^{3}/2>CA\epsilon^{3}/4$
for any $\lambda\in\Omega_{\beta,\epsilon}$ provided that $A$ is large enough.
As a result, we deduce

$\sqrt{{\mathcal{E}}_{r}}(U_{r})\lesssim(A\epsilon^{3})^{-1}\big{(}\|F\|_{X}+\|\pi_{r}(D){L}_{a}^{1}(U_{h}+U_{s})\|_{X}\big{)}\,.$

The estimate (3.23) then follows by applying estimate (C.13).

##  4\. Spectral stability for low transverse frequencies

In this section, we focus on the low transverse frequency region $R^{L}$
defined in (1.27). Let us set
$Z=\mathbb{Q}_{a}(\epsilon^{2}\eta_{0})\mathbb{I}_{A\epsilon^{2}}(D_{y})X,$ we
want to show that for $\epsilon$ small enough,
$\Omega_{\beta,\epsilon}\subset\rho(L_{a};Z)$ and that there is a constant
$C>0,$ such that for any $\lambda\in\Omega_{\beta,\epsilon},$

$\|(\lambda I-L_{a})^{-1}\|_{B(Z)}\leq C.$ (4.1)

We begin with a lemma for the resolvent estimate in the high longitudinal
frequency region $\\{|\xi|\geq K\epsilon\\}:$

######  Lemma 4.1.

Let $U\in\tilde{\mathbb{I}}_{K\epsilon}(D_{x})X,\,F\in X$ which satisfy
$\big{(}\lambda I-L_{a}\big{)}U=F,$ then it holds that, for $K>1$ sufficiently
large and $\epsilon$ small such that $K^{4}\epsilon\leq 1,$

$\displaystyle\|U\|_{X}\lesssim K^{-2}\epsilon^{-3}\,\|F\|_{X}.$ (4.2)

###### Proof.

The proof is close to the one of Proposition 3.3, detailed in Subsection 3.2.
The general idea is to split $U$ into two parts $U=U^{h}+U^{m},$ where
$U^{h},U^{m}$ are localized respectively on the (uniform) high and
intermediate longitudinal frequencies :

$U^{h}=\tilde{\mathbb{I}}_{\delta}(D_{x})U,\qquad
U^{m}=(\tilde{\mathbb{I}}_{K\epsilon}{\mathbb{I}}_{\delta})(D_{x})U.$

By the definition, $U^{h},U^{m}$ solves the equations:

$\displaystyle\left\\{\begin{array}[]{l}\tilde{\mathbb{I}}_{\delta}(D_{x})\,(\lambda
I-L_{a})\,U^{h}=\tilde{\mathbb{I}}_{\delta}(D_{x})\,(L_{a}^{1}U^{m}+F),\\\\[4.0pt]
(\tilde{\mathbb{I}}_{K\epsilon}{\mathbb{I}}_{\delta})(D_{x})\,(\lambda
I-L_{a})\,U^{m}=(\tilde{\mathbb{I}}_{K\epsilon}{\mathbb{I}}_{\delta})(D_{x})\,(L_{a}^{1}U^{h}+F)\,,\end{array}\right.$

where $L_{a}^{1}$ is defined in (3.24). As in the treatment of $U_{h}$ and
$U_{r}$ in subsections 3.2.1, 3.2.3, we introduce the functionals:

$\displaystyle{\mathcal{E}}_{h}(U^{h})=\int|\sqrt{w_{c}}U_{1}^{h}|^{2}+\|\sqrt{G_{a}[0]}U_{2}^{h}\|\,\mathrm{d}x\mathrm{d}y\,,$
$\displaystyle{\mathcal{E}}_{r}(U^{m})=\colon\frac{1}{2}\int\bigg{(}\gamma\,|U_{1}^{m}|^{2}+|\sqrt{G_{a}[0]}U_{2}^{m}|^{2}+\zeta_{c}\big{|}\nabla_{a}Q_{a}(D)U_{2}^{m}\big{|}^{2}\bigg{)}\,\,\mathrm{d}x\mathrm{d}y\,,$

and then find by performing energy estimates that:

$\displaystyle\sqrt{{\mathcal{E}}_{h}(U^{h})}\lesssim\epsilon^{-1}\big{(}\|F\|_{X}+\|L_{a}^{1}U^{m}\|_{X}\big{)},$
$\displaystyle\sqrt{{\mathcal{E}}_{m}(U^{m})}\lesssim
K^{-2}\epsilon^{-3}\big{(}\|F\|_{X}+\|{\mathbb{I}}_{\delta}(D_{x})L_{a}^{1}U^{h}\|_{X}\big{)}.$

Note that we have used the algebraic property of ${\rm
Re}\,\sqrt{-\mu_{a}\tanh\mu_{a}}$ in the region $R_{\xi}^{I}$ stated in Lemma
3.5. The estimate (4.2) can then be derived from the fact
$\|{\mathbb{I}}_{\delta}(D_{x})L_{a}^{1}\|_{B(X)}+\|L_{a}^{1}{\mathbb{I}}_{\delta}(D_{x})\|_{B(X)}\lesssim\epsilon^{2}$
which is shown in Lemma C.10 . ∎

We are now in position to prove the resolvent bound (4.1):

######  Proposition 4.2.

Assume that $\epsilon$ and $\eta_{0}$ are small enough. For any
$\lambda\in\Omega_{\beta,\epsilon},\,$ any $U,F\in
Z=\colon\mathbb{Q}_{a}(\epsilon^{2}\eta_{0})\mathbb{I}_{A\epsilon^{2}}(D_{y})X,$
such that $(\lambda I-L_{a})U=F,$ it holds that

$\|U\|_{X}\lesssim\epsilon^{-3}\|F\|_{X}.$ (4.3)

###### Proof.

For the proof, we shall perform different estimates for low longitudinal
frequencies (${|\xi|\leq K\epsilon}$) and high longitudinal frequencies
(${|\xi|\geq K\epsilon}$). In the high-frequency regime, we apply Lemma 4.1,
while in the low-frequency regime, we use the KP-II approximation. Due to the
excessively large resolvent bound in the low longitudinal frequency regime,
which is of order $\epsilon^{-3}$, we need to use a smooth cutoff function to
localize in frequency.

Let $\chi,\chi_{1},\chi_{2}:\mathbb{R}\rightarrow[0,1]$ be two cut-off
functions with the properties:

$\displaystyle{\rm
Supp}\,\chi\subset[-2K\epsilon,2K\epsilon]\,,\quad\chi_{1}\equiv 1\text{ on
}[-K\epsilon,K\epsilon]\,,$ (4.4) $\displaystyle{\rm
Supp}\,\chi_{1}\subset[-4K\epsilon,4K\epsilon]\,,\quad\chi\equiv 1\text{ on
}[-3K\epsilon,3K\epsilon]\,,$ $\displaystyle{\rm
Supp}\,\chi_{2}\subset[-8K\epsilon,8K\epsilon]\,,\quad\chi_{2}\equiv 1\text{
on }[-4K\epsilon,4K\epsilon]\,,$

where $K\geq 1$ is large and $\epsilon$ is small, such that $K^{4}\epsilon\leq
1.$

We define then

$U^{H}=(1-{\chi})(D_{x})\,U,\,\qquad U^{L}=\chi_{1}(D_{x})\,U,$

which solve the equations

$\displaystyle\left\\{\begin{array}[]{l}\big{(}\lambda
I-L_{a}\big{)}\,U^{H}=(1-\chi)(D_{x})F-[{\chi}(D_{x}),\,L_{a}^{1}]\,U,\\\\[5.0pt]
\big{(}\lambda-
L_{a}\big{)}\,{\chi}_{2}(D_{x})\,U^{L}=[{\chi}_{1}(D_{x}),\,L_{a}^{1}]\,U+{\chi}(D_{x})F\,.\end{array}\right.$
(4.7)

We refer to (3.24) for the definition of $L_{a}^{1}.$

Applying Lemma 4.1 and Lemma C.8, we first have that:

$\displaystyle\|U^{H}\|_{X}$ $\displaystyle\lesssim
K^{-2}\epsilon^{-3}\big{(}\big{\|}[{\chi}(D_{x}),\,L_{a}^{1}]\,U\big{\|}_{X}+\|F\|_{X}\big{)}$
$\displaystyle\lesssim
K^{-2}\epsilon^{-3}\big{(}(K\epsilon^{3}+K^{4}\epsilon^{5})\|U\|_{X}+\|F\|_{X}\big{)}$
$\displaystyle\lesssim(K^{-1}+\epsilon)\,\|U\|_{X}+K^{-2}\epsilon^{-3}\|F\|_{X}\,.$

Note that we have used the assumption $K^{4}\epsilon\leq 1$ for the final
inequality. It follows from the fact $(1-\chi)(1-\chi_{1})=(1-\chi_{1})$ that
$\|U\|_{X}\lesssim\|U^{L}\|_{X}+\|U^{H}\|_{X}.$ Therefore, upon choosing $K$
large enough and $\epsilon$ sufficiently small,

$\|U^{H}\|_{X}\lesssim(K^{-1}+\epsilon)\,\|U^{L}\|_{X}+K^{-2}\epsilon^{-3}\|F\|_{X}\,.$
(4.8)

It now remains to bound $U^{L}.$ Since it is localized in low frequencies in
both variables, we can diagonalize smoothly the constant matrix

$\displaystyle{L}_{a}^{0}=\left(\begin{array}[]{cc}\partial_{x}-a&G_{a}[0]\\\\[5.0pt]
-\gamma&\partial_{x}-a\end{array}\right)=P\left(\begin{array}[]{cc}\lambda_{+}^{0}&0\\\\[5.0pt]
0&\lambda_{-}^{0}\end{array}\right)P^{-1},\quad
P^{-1}=\frac{1}{2}\left(\begin{array}[]{cc}1&-\sqrt{-G_{a}[0]/\gamma}\\\\[5.0pt]
1&\sqrt{-G_{a}[0]/\gamma}\end{array}\right).$ (4.15)

where

$\lambda_{\pm}^{0}(\xi,\eta)=i(\xi-
ia)\pm\sqrt{-\gamma\mu_{a}\tanh\mu_{a}(\xi,\eta)}\,.$

Note that we have,

$\|\mathbb{I}_{A\epsilon^{2}}(D_{y})f\|_{X}\approx\|\mathbb{I}_{A\epsilon^{2}}(D_{y})P^{-1}f\|_{\mathcal{X}}\,,\quad(\mathcal{X}=L^{2}(\mathbb{R}^{2})\times
L^{2}(\mathbb{R}^{2})),$ (4.16)

and it thus suffices to get an estimate for $\tilde{U}^{L}=\colon P^{-1}U^{L}$
in the space $\mathcal{X}.$

Let us set

$R(x,D)=P^{-1}L_{a}^{1}\,P=\big{(}R_{ij}\big{)}_{1\leq i,j\leq
2}\,,\qquad\tilde{F}=P^{-1}\big{(}[{\chi}(D_{x}),\,L_{a}^{1}]\,U+{\chi}(D_{x})F\big{)}.$

In particular, $R_{11}$ takes the form

$\displaystyle R_{11}(x,D)$
$\displaystyle=-\frac{1}{2}\bigg{(}(\partial_{x}-a)(v_{c}\,\cdot)+\sqrt{-G_{a}[0]}\,v_{c}\,(\partial_{x}-a)\sqrt{-G_{a}[0]}^{-1}$
(4.17)
$\displaystyle\qquad\qquad+\sqrt{\gamma\,}\big{(}G_{a}[\zeta_{c}]-G_{a}[0]\big{)}\sqrt{-G_{a}[0]}^{-1}-\sqrt{\gamma}^{-1}\sqrt{-G_{a}[0]}(d_{c}\,\partial_{x}Z_{c})\bigg{)}.$

Moreover, thanks to (4.16) and (C.14), we have that

$\|\,R\,\|_{B(\mathcal{X})}\lesssim(K\epsilon^{3}+K^{4}\epsilon^{5})\lesssim
K\epsilon^{3}.$ (4.18)

We take $P^{-1}$ on both sides of $\eqref{eqULUH}_{2}$ and translate it into
the following system:

$\displaystyle\left\\{\begin{array}[]{l}\big{(}\lambda-\lambda_{+}^{0}(D)+R_{11}(x,D)\big{)}\,{\chi}_{2}(D_{x})\,\tilde{U}_{1}^{L}=-R_{12}\,\tilde{U}_{2}^{L}+\tilde{F}_{1}\,,\\\\[5.0pt]
\big{(}\lambda-\lambda_{-}^{0}(D)\big{)}\,\,\tilde{U}_{2}^{L}=-\big{(}R_{21}\,\tilde{U}_{1}^{L}+R_{22}\,\tilde{U}_{2}^{L}\big{)}+\tilde{F}_{2}\,.\end{array}\right.$
(4.19)

Thanks to (3.31) and (4.18), we get by energy estimates that

$\|\tilde{U}_{2}^{L}\|_{L^{2}}\lesssim\epsilon^{-1}\big{(}\|\tilde{F}_{2}\|_{L^{2}}+K\epsilon^{3}\|\tilde{U}^{L}\|_{\mathcal{X}}\big{)}.$
(4.20)

To estimate $U_{1}^{L},$ we need to use the KP-II approximation. As a
preparation, we introduce the scaling operator
$S_{\epsilon}:L^{2}(\mathbb{R}^{2})\rightarrow L^{2}(\mathbb{R}^{2})$ as
$S_{\epsilon}f(x,y)=\epsilon^{-3/2}f(\epsilon^{-1}x,\epsilon^{-2}y).$ Define
then

${\mathcal{P}}=S_{\epsilon}^{-1}\,\mathbb{P}^{\hat{a}}_{\textnormal{\scalebox{0.8}{KP}}}(\hat{\eta}_{0})\,S_{\epsilon}\,,\qquad{\mathcal{Q}}=S_{\epsilon}^{-1}\,\mathbb{Q}_{\textnormal{\scalebox{0.8}{KP}}}^{\hat{a}}(\hat{\eta}_{0})\,S_{\epsilon}\,.$

where $\mathbb{P}^{\hat{a}}_{\textnormal{\scalebox{0.8}{KP}}}$ is the
projection (see (4.23) for precise definition) to the resonant space
associated to the operator $L_{\textnormal{\scalebox{0.8}{KP}}}$ and
$\mathbb{Q}^{\hat{a}}_{\textnormal{\scalebox{0.8}{KP}}}={\rm
Id}-\mathbb{P}^{\hat{a}}_{\textnormal{\scalebox{0.8}{KP}}}.$ We first claim
that it suffices to prove the following two properties:

$\displaystyle\|{\mathcal{P}}\,\tilde{U}_{1}^{L}\|_{L^{2}}\lesssim(\hat{\eta}_{0}^{2}+\epsilon+K^{3}\epsilon^{2}+K^{-3})\,\|\tilde{U}_{1}^{L}\|_{L^{2}}+\|\tilde{U}^{L}_{2}\|_{L^{2}}+\|U^{H}\|_{X}\,,$