# On the transverse stability of smooth solitary waves in a two-dimensional
Camassa–Holm equation

Anna Geyer  Delft Institute of Applied Mathematics, Faculty Electrical
Engineering, Mathematics and Computer Science, Delft University of Technology,
Mekelweg 4, 2628 CD Delft, The Netherlands<EMAIL_ADDRESS>,  Yue Liu
Department of Mathematics, University of Texas at Arlington, Arlington, TX
76019, USA<EMAIL_ADDRESS>and  Dmitry E. Pelinovsky  Department of Mathematics
and Statistics, McMaster University, Hamilton, Ontario, Canada, L8S 4K1
<EMAIL_ADDRESS>
###### Abstract.

We consider the propagation of smooth solitary waves in a two-dimensional
generalization of the Camassa–Holm equation. We show that transverse
perturbations to one-dimensional solitary waves behave similarly to the KP-II
theory. This conclusion follows from our two main results: (i) the double
eigenvalue of the linearized equations related to the translational symmetry
breaks under a transverse perturbation into a pair of the asymptotically
stable resonances and (ii) small-amplitude solitary waves are linearly stable
with respect to transverse perturbations.

##  1\. Introduction

The Camassa–Holm equation, labelled as the CH equation,

$u_{t}-u_{txx}+3uu_{x}=2u_{x}u_{xx}+uu_{xxx},$ (1.1)

is a popular model for the dynamics of unidirectional shallow water waves [3,
20] which has been justified mathematically in [8]. It was originally
introduced in [13] as a deformation of the integrable KdV equations. The
equation models the behavior of shallow water waves both in the setting of
solitary and periodic waves. Global solutions exist for initial data with
sufficiently gradual slopes and wave breaking occurs in finite time for
initial data with steep slopes [6, 7]. There exist smooth and peaked traveling
waves both among the spatially solitary and periodic waves [16, 26]. The
smooth solitary waves were shown to be spectrally and orbitally stable in the
time evolution of the CH equation [11, 23]. Similar stability results were
obtained for the traveling periodic waves in [16, 27]. On the other hand,
although the peaked traveling waves (both solitary and periodic) are
energetically stable in the energy space $H^{1}$ [9, 10, 24, 25], the local
solutions are only defined in the function space $H^{1}\cap W^{1,\infty}$ [12,
28]. It was recently shown that the peaked traveling waves are both spectrally
and orbitally unstable in $H^{1}\cap W^{1,\infty}$ [22, 29, 34].

As a model for shallow water waves, the CH equation (1.1) is limited to two-
dimensional fluid motion confined by a one-dimensional time-dependent surface.
Transverse modulations on the water surface can be defined in terms of the two
spatial variables $(x,y)\in{\mathbb{R}}^{2}$. A generalization of the CH
equation with a two-dimensional time-dependent profile $u=u(x,y,t)$ has
appeared in the literature only recently. This equation can be written in its
simplest dimensionless form as

$(u_{t}-u_{txx}+3uu_{x}-2u_{x}u_{xx}-uu_{xxx})_{x}+u_{yy}=0.$ (1.2)

It was first derived in [4] as a model in the context of nonlinear elasticity
theory. More recently, it was obtained in [17] as a model in the context of
incompressible and irrotational shallow water wave theory. We refer to (1.2)
as the CH-KP equation because it generalizes the CH equation (1.1) in the same
way as the Kadomtsev–Petviashvili (KP) equation generalizes the classical
Korteweg–de Vries (KdV) equation [21].

In the following we review some mathematical results that have been obtained
for the CH-KP equation (1.2) so far. Local existence of solutions was obtained
in the space of functions $X^{s}({\mathbb{R}}^{2})$ with $s\geq 2$, where

$X^{s}({\mathbb{R}}^{2}):=\\{u\in
H^{s}({\mathbb{R}}^{2}):\quad\partial_{x}^{-1}u\in
H^{s}({\mathbb{R}}^{2}),\;\;\partial_{x}u\in H^{s}({\mathbb{R}}^{2})\\},$

see [17, Theorem 1.1]. The nonlocal operator $\partial_{x}^{-1}$ can be
formally defined as

$(\partial_{x}^{-1}f)(x):=\int_{+\infty}^{x}f(x^{\prime})\,dx^{\prime}$

for functions $f(x):{\mathbb{R}}\to{\mathbb{R}}$ that decay to zero as
$x\to+\infty$. This nonlocal operator can be used to rewrite (1.2) in the
evolution form

$u_{t}+(1-\partial_{x}^{2})^{-1}\left[3uu_{x}-2u_{x}u_{xx}-uu_{xxx}+\partial_{x}^{-1}u_{yy}\right]=0.$
(1.3)

The evolution equation (1.3) can be cast in Hamiltonian form

$u_{t}=-JF^{\prime}(u),$ (1.4)

with the skew-adjoint operator $J:=\partial_{x}(1-\partial_{x}^{2})^{-1}$ and
the conserved energy

$F(u):=\frac{1}{2}\int_{{\mathbb{R}}^{2}}\left[u^{3}+uu_{x}^{2}+(\partial_{x}^{-1}u_{y})^{2}\right]\,dx\,dy.$
(1.5)

It was shown in [17] that $F(u)$ is conserved in time for local solutions in
$X^{s}({\mathbb{R}}^{2})$ for $s\geq 2$, and so is the momentum

$E(u):=\frac{1}{2}\int_{{\mathbb{R}}^{2}}(u^{2}+u_{x}^{2})\,dx\,dy.$ (1.6)

In addition to $F(u)$ and $E(u)$, the mass

$M(u):=\int_{{\mathbb{R}}^{2}}u\,dx\,dy$ (1.7)

is formally conserved in the time evolution of the CH-KP equation (1.2).
Various wave breaking criteria were obtained in [17, Theorems 1.2–1.4]. A
recent work [39] explored numerical (Galerkin) methods for approximation of
solitary waves in the CH–KP equation.

The purpose of this work is to study the transverse stability of perturbed
solitary waves in the CH-KP equation (1.2). Line solitary waves are obtained
for functions of the form $u(x,y,t)=\phi(x+\gamma y-ct)$ with parameters
$\gamma,c\in{\mathbb{R}}$. In what follows, we will only consider the case
$\gamma=0$ for the traveling wave solutions of the CH equation (1.1).

It was the motivation of the pioneering work [21] to investigate the
transverse stability of solitary waves under small slowly varying
perturbations. It was discovered that the line solitary waves are transversely
unstable in one version of the KP equation and are transversely stable in
another version of the KP equation. These versions are now conventionally
referred to as the KP-I and KP-II equations, respectively. The CH-KP equation
(1.2) we are considering in the present work corresponds to KP-II.

A rigorous proof of transverse stability of traveling waves in the KP-II
equation was completed only recently. Linear and nonlinear stability of the
solitary waves have been proven for transversely periodic perturbations in
[31] and for decaying perturbations in $\mathbb{R}^{2}$ [30]. Linear stability
of traveling periodic waves was shown in [18] and the nonlinear stability of
periodic waves is still an open problem for the KP-II equation.

Asymptotic reductions of other nonlinear systems to the KP-II equation have
been explored in the literature. Mizumachi and Shimabukuro used the KP–II
equation as an approximation of the Benney–Luke system to prove linear and
nonlinear transverse stability of the line solitary waves of small amplitudes
[32, 33]. A justification of the asymptotic reduction to the KP-II equation
for the two-dimensional Boussinesq equation was done by Gallay and Schneider
[14]. In the recent series of papers [15, 19, 35], the KP-II equation was
justified as the leading model for a two-dimensional Fermi–Pasta–Ulam system
on a square lattice. See also [1, 2] for recent work on transverse stability
of line solitary waves in other generalizations of the KP equation.

We can formally obtain the asymptotic reduction of the CH–KP equation to the
KP-II equation. Let $k>0$ be a fixed parameter and consider the slowly varying
approximation of small-amplitude perturbations of a constant background in the
form

$u(x,y,t)=k+\varepsilon^{2}v(\varepsilon(x-3kt),\varepsilon^{2}y,\varepsilon^{3}t).$
(1.8)

By using the chain rule and the evolution form (1.3), we derive the following
evolution equation for the variable $v=v(X,Y,T)$ in scaled coordinates as

$v_{T}+(1-\varepsilon^{2}\partial_{X}^{2})^{-1}\left[2kv_{XXX}+3vv_{X}+\partial_{X}^{-1}v_{YY}-\varepsilon^{2}(2v_{X}v_{XX}+vv_{XXX})\right]=0.$

The formal truncation at $\varepsilon=0$ yields the KP-II equation in the form

$v_{T}+2kv_{XXX}+3vv_{X}+\partial_{X}^{-1}v_{YY}=0.$ (1.9)

For every fixed $k>0$, the line solitary waves are linearly and nonlinearly
stable in the KP-II equation (1.9) [30]. The main conclusion of this work is
that the smooth solitary waves are linearly transversely stable also in the
CH-KP equation (1.3). The nonlinear transverse stability is still an open
question, and our results on the linear transverse stability so far are
limited to two claims:

  * •

The transverse perturbation breaks the double zero eigenvalue of the
linearized equations into a pair of resonances located in the left half-plane.
This result is obtained for smooth solitary waves of arbitrary amplitude.

  * •

The line solitary waves are linearly stable with respect to transverse
perturbations if the wave amplitude is sufficiently small.

The precise statement of these two results will be given in Section 2, after
the traveling waves and their linear stability problems will be described.
Sections 3 and 4 contain the proofs of these two main results. Section 5
concludes the paper with a summary and a list of open questions for further
studies.

##  2\. Smooth solitary waves

We consider the traveling one-dimensional solitary waves described by
solutions to the CH-KP equation (1.2) of the form

$u(x,y,t)=\phi(x-ct),$

where $\phi(x)\to k$ as $|x|\to\infty$, for a fixed background parameter
$k>0$. It is well-known [16, 23], see also [11, 26] for earlier results, that
such solitary waves exist for $c>3k$ and have a smooth profile $\phi\in
C^{\infty}({\mathbb{R}})$. The following lemma formalizes the result.

######  Lemma 2.1.

Fix $k>0$. For every $c>3k$, there exists a traveling solitary wave solution
of the CH equation (1.1) with profile $\phi\in C^{\infty}({\mathbb{R}})$ of
the form $\phi(x)=k+\psi(x)$, where $\psi$ is found from the first-order
invariant

$(\psi^{\prime})^{2}=\psi^{2}\;\frac{c-3k-\psi}{c-k-\psi}.$ (2.1)

In particular, $\psi(x)>0$ for all $x\in{\mathbb{R}}$, $\psi(x)\to 0$ as
$|x|\to\infty$ exponentially fast, and $\psi(x)$ is monotonically decreasing
on both sides of its maximum at $\max\limits_{x\in\mathbb{R}}\psi(x)=c-3k$.

###### Proof.

The traveling wave of the CH equation (1.1) with profile $\phi$ satisfies the
third-order differential equation

$-c(\phi^{\prime}-\phi^{\prime\prime\prime})+3\phi\phi^{\prime}-2\phi^{\prime}\phi^{\prime\prime}-\phi\phi^{\prime\prime\prime}=0,$

which can either be integrated directly to give

$(c-\phi)(\phi-\phi^{\prime\prime})+\frac{1}{2}(\phi^{\prime})^{2}-\frac{1}{2}\phi^{2}=kc-\frac{3}{2}k^{2},$
(2.2)

or first multiplied by $(c-\phi)$ and then integrated to give

$-(c-\phi)^{2}(\phi^{\prime\prime}-\phi)=k(c-k)^{2}.$ (2.3)

In both cases, we have fixed the integration constant from the conditions
$\phi(x)\to k$ and $\phi^{\prime}(x),\phi^{\prime\prime}(x)\to 0$ as
$|x|\to\infty$. Multiplying (2.3) by $\phi^{\prime}$ and integrating again
gives

$\frac{1}{2}(\phi^{\prime})^{2}-\frac{1}{2}\phi^{2}+\frac{k(c-k)^{2}}{(c-\phi)}=kc-\frac{3}{2}k^{2}.$
(2.4)

Writing $\phi=k+\psi$, we obtain (2.1) from (2.4).

A solitary wave with $\psi(x)\to 0$ as $|x|\to\infty$ corresponds to a
homoclinic orbit on the phase plane $(\psi,\psi^{\prime})$ along the level
curve (2.1) to the saddle point $(0,0)$. The solitary wave exists if and only
if $c-3k>0$, because $(0,0)$ is a center point for $c-3k<0$ and no homoclinic
orbit exists for $c-3k=0$. Since $(0,0)$ is a saddle point for $c-3k>0$, the
convergence rate of $\psi(x)\to 0$ as $|x|\to\infty$ is exponential. The
stable and unstable curves at $(0,0)$ do not intersect if $\psi<0$ and
intersect if $\psi>0$. Hence $\psi(x)>0$ for all $x\in\mathbb{R}$ and the
turning point $x_{0}\in\mathbb{R}$ with $\psi^{\prime}(x_{0})=0$ exists if and
only if $\psi(x_{0})=c-3k$. Thus, the profile $\psi$ is monotonically
decreasing away from its maximum at
$\max\limits_{x\in\mathbb{R}}\psi(x)=c-3k$. ∎

######  Remark 2.2.

Due to the translational symmetry of the CH equation we may place the maximum
of $\psi$ at $x=0$ such that $\psi(0)=c-3k$.

######  Remark 2.3.

Since the scaling (1.8) suggests a reduction of the CH-KP equation (1.3) to
the KP-II equation (1.9), the traveling solitary wave of Lemma 2.1 must
converge to the traveling solitary wave of the KdV equation

$v_{T}+2kv_{XXX}+3vv_{X}=0.$ (2.5)

Indeed, solving the KdV equation (2.5) for the solitary wave profile with

$v(X,T)={\rm sech}^{2}\left(\frac{X-T}{2\sqrt{2k}}\right)$

gives the formal asymptotic expansion

$\phi(x)=k+\varepsilon^{2}{\rm sech}^{2}\left(\frac{\varepsilon
x}{2\sqrt{2k}}\right)+\mathcal{O}(\varepsilon^{4}),\quad
c=3k+\varepsilon^{2},$ (2.6)

where $\varepsilon>0$ is an arbitrary (small) parameter and $x$ stands for
$x-ct$. The asymptotic limit to the solitary wave of small amplitude
corresponds to the limit $c\to 3k$ for which $\varepsilon\to 0$. This
reduction is made rigorous in Lemma 4.1 below.

In order to set up the linear transverse stability problem for the smooth
solitary wave of Lemma 2.1, we consider the decomposition

$u(x,y,t)=\phi(x-ct)+v(x-ct,y,t)$

with the perturbation $v$ to the solitary wave profile $\phi\in
C^{\infty}({\mathbb{R}})$. After substitution of the decomposition into (1.3)
and neglecting the quadratic terms in $v$, we obtain the linearized equation

$v_{t}=J(L-\partial_{x}^{-2}\partial_{y}^{2})v,$ (2.7)

where $J:=\partial_{x}(1-\partial_{x}^{2})^{-1}$ as in (1.4) and

$L:=c-3\phi+\phi^{\prime\prime}-\partial_{x}(c-\phi)\partial_{x}.$ (2.8)

Separation of variables in the linearized equation (2.7) by using normal modes
of the form

$v(x,y,t)=e^{\lambda t}e^{i\eta y}\hat{v}(x),$

where $\lambda\in{\mathbb{C}}$ and $\eta\in{\mathbb{R}}$, yields the spectral
stability problem

$J(L+\eta^{2}\partial_{x}^{-2})\hat{v}=\lambda\hat{v}.$ (2.9)

The one-dimensional spectral stability problem is recovered for $\eta=0$. We
can now specify the following definition of transverse spectral stability.

######  Definition 2.4.

We say that the solitary wave with profile $\phi\in C^{\infty}({\mathbb{R}})$
is transversely spectrally stable if for every $\eta\in\mathbb{R}$ there
exists no eigenvalue $\lambda\in{\mathbb{C}}$ with ${\rm Re}(\lambda)>0$ and
eigenfunction $\hat{v}\in{\rm Dom}(J(L+\eta^{2}\partial_{x}^{-2}))\subset
L^{2}({\mathbb{R}})$ of the spectral stability problem (2.9).

A common method to study the linear stability of solitary waves in the KdV
equation (2.5) is to use the exponentially weighted space $L_{\nu}^{2}$ with
fixed $\nu>0$ [5, 37], which is defined as

$L_{\nu}^{2}:=\\{f(x):{\mathbb{R}}\to{\mathbb{R}}:\quad e^{\nu\cdot}f\in
L^{2}({\mathbb{R}})\\}.$ (2.10)

If $f\in L_{\nu}^{2}$ with $\nu>0$, then $f(x)\to 0$ as $x\to+\infty$ and so
the nonlocal operator $\partial_{x}^{-1}$ is well-defined. Note however that
$f(x)$ does not have to decay and may even be slowly growing as $x\to-\infty$.
By using the exponentially weighted space $L^{2}_{\nu}$, we rephrase the
definition of the transverse spectral stability.

######  Definition 2.5.

We say that the solitary wave with profile $\phi\in C^{\infty}({\mathbb{R}})$
is transversely asymptotically stable in $L^{2}_{\nu}$ for some $\nu>0$ if for
every $\eta\in\mathbb{R}$, $\eta\neq 0$ there exists $b>0$ such that all
points $\lambda$ in the spectrum of the linear operator

$J(L+\eta^{2}\partial_{x}^{-2}):{\rm
Dom}(J(L+\eta^{2}\partial_{x}^{-2}))\subset L^{2}_{\nu}\to L^{2}_{\nu}$

satisfy ${\rm Re}(\lambda)\leq-b$.

The fact that $\phi(x)\to k$ as $|x|\to\infty$ exponentially fast greatly
simplifies the spectral analysis of our problem. As a result, Weyl’s theory
implies that the continuous spectrum of $J(L+\eta^{2}\partial_{x}^{-2})$ in
$L^{2}_{\nu}$ is uniquely determined by the purely continuous spectrum of
$J(L_{0}+\eta^{2}\partial_{x}^{-2})$, where

$L_{0}:=c-3k-(c-k)\partial_{x}^{2}.$ (2.11)

In addition, the point spectrum of $J(L+\eta^{2}\partial_{x}^{-2})$ in
$L^{2}_{\nu}$ may contain eigenvalues $\lambda\in{\mathbb{C}}$ with
eigenfunctions $\hat{v}\in{\rm Dom}(J(L+\eta^{2}\partial_{x}^{-2}))$.

The first result of this paper is to show that both the continuous spectrum
and the two eigenvalues near the origin in the complex plane satisfy the
transverse asymptotic stability condition of Definition 2.5 for some $\nu>0$.
The proof is developed in Section 3, where the continuous spectrum is computed
with the help of the Fourier transform and the two eigenvalues are computed by
using Puiseux expansions [38] in the small parameter $\eta$.

######  Theorem 2.6.

For every $c>3k$, $\eta\in\mathbb{R}$, and $\nu\in(0,\nu_{0})$ with
$\nu_{0}:=\sqrt{\frac{c-3k}{c-k}}$, there exists $b_{0}>0$ such that all
points $\lambda$ in the spectrum of the linear operator
$J(L_{0}+\eta_{2}\partial_{x}^{-2})$ in $L^{2}_{\nu}$ satisfy ${\rm
Re}(\lambda)\leq-b_{0}$. Furthermore, there exists $\eta_{0}>0$ such that the
spectrum of the linear operator $J(L+\eta^{2}\partial_{x}^{-2})$ in
$L^{2}_{\nu}$ with $\eta\in(-\eta_{0},\eta_{0})$ includes a pair of simple
eigenvalues $\lambda_{\pm}(\eta)$ such that for $\eta\neq 0$ we have

  * •

${\rm Re}(\lambda_{+}(\eta))={\rm Re}(\lambda_{-}(\eta))<0$,

  * •

${\rm Im}(\lambda_{+}(\eta))=-{\rm Im}(\lambda_{-}(\eta))>0$,

and $\lambda_{+}(0)=\lambda_{-}(0)=0$.

######  Remark 2.7.

The result of Theorem 2.6 is consistent with the transverse asymptotic
stability with respect to long transverse perturbations in the sense of
Definition 2.5 with small $\eta\neq 0$. However, the spectrum of $JL$ in
$L^{2}_{\nu}$ might include more than the continuous spectrum and the double
zero eigenvalue. There might exist additional embedded eigenvalues of $JL$ in
$L^{2}({\mathbb{R}})$ on the imaginary axis which could become isolated in
$L^{2}_{\nu}$ for $\nu>0$. The latter possibility has been ruled out for the
KdV equation (2.5), see [36, 37]. However, nothing is known about the
existence of additional embedded eigenvalues of $JL$ in $L^{2}({\mathbb{R}})$
on $i{\mathbb{R}}$ for the CH equation (1.1).

The second result of this paper explores the small-amplitude limit of the
solitary waves and provides transverse asymptotic stability for solitary waves
of small amplitudes in the sense of Definition 2.5. The proof is developed in
Section 4 based on estimates for the resolvent equation.

######  Theorem 2.8.

Let $\lambda_{\pm}(\eta)$ be the simple eigenvalues of
$J(L+\eta^{2}\partial_{x}^{-2})$ in $L^{2}_{\nu}$ for fixed
$\nu\in(0,\nu_{0})$ found in Theorem 2.6. There exists $\varepsilon_{0}>0$ and
$\beta_{0}>0$ such that for every $\varepsilon\in(0,\varepsilon_{0})$, where
$\varepsilon:=\sqrt{c-3k}$, and for every $\eta\in\mathbb{R}$, $\eta\neq 0$,
the spectrum of $J(L+\eta^{2}\partial_{x}^{-2})$ in $L^{2}_{\nu}$ is contained
in

$\mathcal{S}:=\\{\lambda\in\mathbb{C}:\quad{\rm
Re}(\lambda)\leq-\beta_{0}\varepsilon^{3}\\},$

with the exception of the two simple eigenvalues
$\lambda=\lambda_{\pm}(\eta)$.

######  Remark 2.9.

Since ${\rm Re}(\lambda_{\pm}(\eta))<0$ for $\eta\in(-\eta_{0},\eta_{0})$,
$\eta\neq 0$, the solitary waves of small amplitude are transversely
asymptotically stable in $L^{2}_{\nu}$. By using the Fourier transform in $y$,
the result of Theorem 2.8 also implies the transverse asymptotic stability of
these solitary waves with respect to perturbations in
$L^{2}_{\nu}(\mathbb{R}^{2})$, where the weight $\nu\in(0,\nu_{0})$ is only
applied in the direction of the solitary waves. This yields linear asymptotic
stability of solutions to the evolution equation (2.7) in
$L^{2}_{\nu}(\mathbb{R}^{2})$ by semi-group theory.

##  3\. Proof of Theorem 2.6

###  3.1. Preliminary results

The one-dimensional CH equation (1.1) has the following conserved quantities
which play a crucial role in the stability analysis of its traveling solitary
and periodic waves [11, 16]:

$\displaystyle\hat{F}(u):=\frac{1}{2}\int_{{\mathbb{R}}}(u^{3}+uu_{x}^{2}-k^{3})\,dx,$
$\displaystyle\hat{E}(u):=\frac{1}{2}\int_{{\mathbb{R}}}(u^{2}+u_{x}^{2}-k^{2})\,dx,$
$\displaystyle\hat{M}(u):=\int_{{\mathbb{R}}}(u-k)\,dx.$

The constant values have been subtracted from the integrands to ensure that
the integrals converge if $u(x)\to k$ as $|x|\to\infty$ sufficiently fast.
These quantities are the one-dimensional analogues of the conserved quantities
(1.5), (1.6), and (1.7) of the two-dimensional CH-KP equation (1.2). Using
$\hat{F}$, $\hat{E}$, and $\hat{M}$ we define the augmented energy

$\Lambda_{c}(u):=-\hat{F}(u)+c\hat{E}(u)-\left(ck-\frac{3}{2}k^{2}\right)\hat{M}(u).$

Smooth solutions to the second-order equation (2.2) with the profile $\phi\in
C^{\infty}({\mathbb{R}})$ are critical points of $\Lambda_{c}$ in the sense
that the first variation vanishes:

$\Lambda^{\prime}_{c}(\phi)=-\frac{3}{2}\phi^{2}+\frac{1}{2}(\phi^{\prime})^{2}+\phi\phi^{\prime\prime}+c\phi-c\phi^{\prime\prime}-ck+\frac{3}{2}k^{2}=0.$

The linear operator $L$ in (2.8) is the Hessian operator of $\Lambda_{c}$ at
the critical point with the profile $\phi\in C^{\infty}({\mathbb{R}})$. This
variational characterization of the traveling wave solutions was explored in
the stability analysis in [11, 16], see also [23] for alternative variational
characterizations of the traveling wave solutions in the CH equation (1.1).

######  Remark 3.1.

Since the linear operator $L$ in (2.8) is the Hessian operator
$\Lambda_{c}^{\prime\prime}(\phi)$ at the traveling solitary wave with the
profile $\phi\in C^{\infty}({\mathbb{R}})$ given by Lemma 2.1, it also arises
in the linearization of the CH equation (1.1) given by $v_{t}=JLv$.

If $\phi=k+\psi$, then

$\displaystyle E_{\rm 1D}(\psi)$ $\displaystyle:=\hat{E}(\phi)-k\hat{M}(\phi)$
$\displaystyle=\frac{1}{2}\int_{{\mathbb{R}}}[(k+\psi)^{2}+(\psi^{\prime})^{2}-k^{2}-2k\psi]dx$
$\displaystyle=\frac{1}{2}\int_{{\mathbb{R}}}[(\psi^{\prime})^{2}+\psi^{2}]dx$
(3.1)

and

$M_{\rm 1D}(\psi):=\hat{M}(\phi)=\int_{{\mathbb{R}}}\psi dx.$ (3.2)

The following lemma reports important monotonicity properties of $E_{\rm
1D}(\psi)$ and $M_{\rm 1D}(\psi)$ with respect to the parameter
$c\in(3k,\infty)$ for fixed $k>0$. The proof is based on direct computations.

######  Lemma 3.2.

For fixed $k>0$, let $\psi$ be the solitary wave defined by the first-order
invariant (2.1). Then, the mappings $c\mapsto M_{\rm 1D}(\psi)$ and $c\mapsto
E_{\rm 1D}(\psi)$ are monotonically increasing for every $c\in(3k,\infty)$.

###### Proof.

Without loss of generality, we place the maximum of $\psi$ at $x=0$ such that
$\psi(0)=c-3k$, see Remark 2.2. By Lemma 2.1, we have $\psi(x)=\psi(-x)>0$ for
every $x\in\mathbb{R}$ and $\psi^{\prime}(x)=-\psi^{\prime}(-x)<0$ for every
$x>0$. We obtain from (3.2) by explicit computations that

$\displaystyle M_{\rm 1D}(\psi)$ $\displaystyle=2\int_{0}^{\infty}\psi(x)dx$
$\displaystyle=2\int_{0}^{c-3k}\frac{\sqrt{c-k-\psi}}{\sqrt{c-3k-\psi}}d\psi$
$\displaystyle=2\int_{0}^{c-3k}\frac{\sqrt{2k+z}}{\sqrt{z}}dz$
$\displaystyle=8k\int_{0}^{\xi_{0}}\sqrt{1+\xi^{2}}d\xi,$

where we have made the substitutions $z=c-3k-\psi$ and

$\xi=\frac{\sqrt{z}}{\sqrt{2k}},\qquad\xi_{0}=\frac{\sqrt{c-3k}}{\sqrt{2k}}.$

The integral is evaluated explicitly to find that

$\displaystyle M_{\rm 1D}(\psi)=4k\left[\xi_{0}\sqrt{1+\xi_{0}^{2}}+{\rm
arcsinh}\xi_{0}\right],$

from which it follows that

$\frac{d}{dc}M_{\rm 1D}(\psi)=2\sqrt{\frac{c-k}{c-3k}}>0.$

Similarly, we find that

$\displaystyle E_{\rm 1D}(\psi)$
$\displaystyle=2\int_{0}^{c-3k}\frac{\psi(c-2k-\psi)}{\sqrt{(c-k-\psi)(c-3k-\psi)}}d\psi$
$\displaystyle=2\int_{0}^{c-3k}\frac{(c-3k-z)(z+k)}{\sqrt{z(z+2k)}}dz,$

from which we obtain that

$\displaystyle\frac{d}{dc}E_{\rm 1D}(\psi)$
$\displaystyle=2\int_{0}^{c-3k}\frac{z+k}{\sqrt{z(z+2k)}}dz$
$\displaystyle=2\sqrt{z(z+2k)}\biggr{|}_{z=0}^{z=c-3k}$
$\displaystyle=2\sqrt{(c-3k)(c-k)}>0.$

Thus, both mappings $c\mapsto M_{\rm 1D}(\psi)$ and $c\mapsto E_{\rm
1D}(\psi)$ are monotonically increasing for every $c\in(3k,\infty)$. ∎

######  Remark 3.3.

The monotonicity of $c\mapsto E_{\rm 1D}(\psi)$ plays a central role in the
proof of the orbital stability of smooth solitary wave in the CH equation
(1.1), see [11].

######  Remark 3.4.

For later reference, we also compute $\|\psi\|_{L^{2}}^{2}$ by using the same
idea as in the proof of Lemma 3.2:

$\displaystyle\|\psi\|_{L^{2}}^{2}$
$\displaystyle=2\int_{0}^{c-3k}\frac{\psi\sqrt{c-k-\psi}}{\sqrt{c-3k-\psi}}d\psi$
$\displaystyle=2\int_{0}^{c-3k}\frac{\sqrt{2k+z}(c-3k-z)}{\sqrt{z}}dz$
$\displaystyle=8k\int_{0}^{\xi_{0}}(c-3k-2k\xi^{2})\sqrt{1+\xi^{2}}d\xi,$

from which we obtain

$\displaystyle\|\psi\|_{L^{2}}^{2}$
$\displaystyle=4k(c-3k)\left[\xi_{0}\sqrt{1+\xi_{0}^{2}}+{\rm
arcsinh}\xi_{0}\right]$
$\displaystyle\qquad-2k^{2}\left[2\xi_{0}\sqrt{(1+\xi_{0}^{2})^{3}}-\xi_{0}\sqrt{1+\xi_{0}^{2}}-{\rm
arcsinh}\xi_{0}\right]$
$\displaystyle=2k(2c-5k)\left[\xi_{0}\sqrt{1+\xi_{0}^{2}}+{\rm
arcsinh}\xi_{0}\right]-4k^{2}\xi_{0}\sqrt{(1+\xi_{0}^{2})^{3}}.$

###  3.2. The continuous spectrum of the spectral problem (2.9)

We start by analyzing properties of $L$. First, $L$ is a self-adjoint Sturm-
Liouville operator in $L^{2}({\mathbb{R}})$ with dense domain in
$H^{2}({\mathbb{R}})$. The translational symmetry of the CH equation (1.1)
implies that

$L\phi^{\prime}=0,\quad\phi^{\prime}\in{\rm Dom}(L)\subset
L^{2}({\mathbb{R}}).$ (3.3)

Since $\phi^{\prime}$ has only one zero on $\mathbb{R}$, Sturm–Liouville
theory implies that the spectrum of $L$ in $L^{2}(\mathbb{R})$ consists of one
simple negative and a simple zero eigenvalue isolated from the strictly
positive part of the spectrum. Furthermore, since $\phi\in
C^{\infty}({\mathbb{R}})$ is smooth in $c$, we find by differentiating the
traveling wave equation (2.2) with respect to $c$ that

$L\partial_{c}\phi=k-\mu,\quad\partial_{c}\phi\in{\rm Dom}(L)\subset
L^{2}({\mathbb{R}}),$ (3.4)

where $\mu:=\phi-\phi^{\prime\prime}$. Based on these computations, the
following two lemmas specify properties of the linearized operator $JL$ in
$L^{2}({\mathbb{R}})$, included here for the sake of completeness, and in the
exponentially weighted space $L^{2}_{\nu}$ for small $\nu>0$.

######  Lemma 3.5.

For every $c>3k$, the spectrum of $JL$ in $L^{2}({\mathbb{R}})$ covers
$i{\mathbb{R}}$ with $0$ being an embedded eigenvalue.

###### Proof.

It follows from (3.3) that $JL\phi^{\prime}=0$ with $\phi^{\prime}\in{\rm
Ker}(JL)\subset L^{2}({\mathbb{R}})$ so that $0\in\sigma(JL)$. Because
$\phi(x)\to k$ as $|x|\to\infty$ exponentially fast, Weyl’s theorem implies
that the continuous spectrum of $JL$ is given by the spectrum of $JL_{0}$ in
$L^{2}({\mathbb{R}})$, where $L_{0}$ is given by (2.11). By using the Fourier
transform in $L^{2}({\mathbb{R}})$, we obtain that

$\sigma(JL_{0})=\left\\{i\xi(1+\xi^{2})^{-1}[c-3k+(c-k)\xi^{2}],\quad\xi\in\mathbb{R}\right\\}=i{\mathbb{R}}\quad\mbox{\rm
in}\;\;L^{2}({\mathbb{R}}).$

Since $\phi$ is spectrally stable in the time evolution of the CH equation
(1.1) [11, 23], no other points of the spectrum of $JL$ in
$L^{2}({\mathbb{R}})$ exists outside $i{\mathbb{R}}$. Thus, the spectrum of
$JL$ in $L^{2}({\mathbb{R}})$ is $\sigma(JL)=i{\mathbb{R}}$ with $0$ being an
embedded eigenvalue. ∎

######  Lemma 3.6.

For every $c>3k$, there exists $\nu_{0}>0$ such that the continuous spectrum
of $JL$ in $L^{2}_{\nu}$ with $\nu\in(0,\nu_{0})$ is strictly negative and the
(isolated) zero eigenvalue in $L^{2}_{\nu}$ is algebraically double.

###### Proof.

By Weyl’s theorem, the continuous spectrum of $JL$ in $L^{2}_{\nu}$ is given
by the spectrum of $JL_{0}$ in $L^{2}_{\nu}$. Using the Fourier transform we
obtain that

$\sigma(JL_{0})=\left\\{(i\xi-\nu)[1-(i\xi-\nu)^{2}]^{-1}[c-3k-(c-k)(i\xi-\nu)^{2}],\quad\xi\in\mathbb{R}\right\\}\quad\mbox{\rm
in}\;\;L^{2}_{\nu}.$

We claim that if $0<\nu<\nu_{0}$ with $\nu_{0}=\sqrt{\frac{c-3k}{c-k}}$, then

$\text{Re}(\sigma(JL_{0}))<0\quad\mbox{\rm in}\;\;L^{2}_{\nu},$

where $\text{Re}(\sigma(JL_{0}))$ coincides with the range of the function
$\lambda_{r}(\xi):\mathbb{R}\to\mathbb{R}$ given by

$\displaystyle\lambda_{r}(\xi)$ $\displaystyle={\rm
Re}\left[(i\xi-\nu)[1-(i\xi-\nu)^{2}]^{-1}[c-3k-(c-k)(i\xi-\nu)^{2}]\right]$
$\displaystyle={\rm
Re}\left[(c-k)(i\xi-\nu)-2k(i\xi-\nu)[1-(i\xi-\nu)^{2}]^{-1}\right]$
$\displaystyle=-\nu(c-k)-\frac{2k\nu(\nu^{2}+\xi^{2}-1)}{(1-\nu^{2}+\xi^{2})^{2}+4\xi^{2}\nu^{2}}$

Expanding this quantity yields

$\displaystyle\lambda_{r}(\xi)$
$\displaystyle=-\frac{\nu}{(1-\nu^{2}+\xi^{2})^{2}+4\xi^{2}\nu^{2}}\left[c-3k+2c\xi^{2}-2(c-2k)\nu^{2}+(c-k)(\xi^{2}+\nu^{2})^{2}\right],$

which is strictly negative if $\nu>0$ and

$(c-k)\nu^{4}-2(c-2k)\nu^{2}+c-3k>0.$

The latter constraint is true if $\nu<\nu_{0}=\sqrt{\frac{c-3k}{c-k}}$. Note
that $\nu_{0}\in(0,1)$.

It remains to prove that $0\in\sigma(JL)$ is a double eigenvalue in
$L^{2}_{\nu}$. Since $\phi^{\prime}(x)\to 0$ as $|x|\to\infty$ exponentially
fast, we have $\phi^{\prime}\in L^{2}_{\nu}$ for sufficiently small $\nu>0$.
The Wronskian between two solutions $\\{f_{1},f_{2}\\}$ of $Lf=0$ is
asymptotically constant at infinity and nonzero since

$W(f_{1},f_{2})=\left|\begin{array}[]{cc}f_{1}&f_{2}\\\
f_{1}^{\prime}&f_{2}^{\prime}\end{array}\right|=\frac{W_{0}}{c-\phi},\quad
x\in\mathbb{R},$

where $W_{0}$ is a nonzero constant. If one solution $f_{1}:=\phi^{\prime}$
decays exponentially at infinity, the other (linearly independent) solution
$f_{2}$ grows exponentially at infinity. Hence

$\ker L={\rm span}(\phi^{\prime})\quad\mbox{\rm in}\;\;L^{2}_{\nu}.$

Furthermore, since $\phi$ is even, $L$ is parity preserving. There exists an
even solution $f_{0}$ to the inhomogeneous equation $Lf_{0}=1$ and since $L$
converges to $L_{0}$ at infinity, $f_{0}$ is non-decaying at infinity. Since
$JLf=0$ implies $Lf=C$ for some constant $C\in{\mathbb{R}}$ and
$f=Cf_{0}\notin L^{2}_{\nu}$ is non-decaying if $C\neq 0$, it follows that

$\ker(JL)=\ker(L)={\rm span}(\phi^{\prime})\quad\mbox{\rm in}\;\;L^{2}_{\nu}.$

In order to study the algebraic multiplicity of the zero eigenvalue, we
consider solutions of $JLf=\phi^{\prime}$. Since it follows from (3.4) that
$JL\partial_{c}\phi=-\phi^{\prime}$ and $\partial_{c}\phi\in L^{2}_{\nu}$, we
have

$\ker((JL)^{2})={\rm span}(\phi^{\prime},\partial_{c}\phi)\quad\mbox{\rm
in}\;\;L^{2}_{\nu}.$

The zero eigenvalue of $JL$ is algebraically double if and only if there
exists no $f\in L^{2}_{\nu}$ such that $JLf=\partial_{c}\phi$, or
equivalently,

$Lf=\partial_{x}^{-1}\partial_{c}\mu,$ (3.5)

where $\partial_{x}^{-1}\partial_{c}\mu\in L^{2}_{\nu}$. If the eigenfunctions
of $L$ are defined in $L^{2}_{\nu}$, then the adjoint eigenfunctions are
defined in $L^{2}_{-\nu}$ due to the transformation $L\mapsto L_{\nu}:=e^{\nu
x}Le^{-\nu x}$ for eigenfunctions in the weighted space $L^{2}_{\nu}$, see [5,
37]. As a result, the inner product in $L^{2}_{\mu}$ is equivalent to the
inner product in $L^{2}$, i.e.

$\forall f\in L^{2}_{\nu},\;\;\forall g\in L^{2}_{-\nu}:\quad\langle
f,g\rangle_{L^{2}_{\nu}}:=\langle e^{\nu x}f,e^{-\nu
x}g\rangle_{L^{2}}=\langle f,g\rangle_{L^{2}}.$ (3.6)

In what follows, we drop the subscript $L^{2}$ for the inner product in
$L^{2}$. To provide the existence of solutions $f\in L^{2}_{\nu}$ of the
linear inhomogeneous equation (3.5), we check the Fredholm condition given by

$\langle\phi^{\prime},\partial_{x}^{-1}\partial_{c}\mu\rangle=-\langle(\phi-k),\partial_{c}\mu\rangle=-\frac{d}{dc}E_{\rm
1D}(\psi),$ (3.7)

where $E_{\rm 1D}(\psi)$ is given by (3.1) and integration by parts gives no
contribution at infinity since $\phi(x)\to k$ as $|x|\to\infty$ exponentially
fast. By Lemma 3.2, the right-hand side is strictly negative so that no $f\in
L^{2}_{\nu}$ exists such that $JLf=\partial_{c}\phi$. Hence, $0\in\sigma(JL)$
is a double eigenvalue in $L^{2}_{\nu}$. ∎

Figure 3.1. A plot of $\lambda(\xi)$, $\xi\in{\mathbb{R}}$ in the complex
plane for $k=1$, $c=4$, $\eta=0.01$, and $\nu=0.1$

Based on Lemma 3.6, we can study properties of the spectral stability problem
(2.9) with transverse wave number $\eta\in\mathbb{R}$, $\eta\neq 0$. The
continuous spectrum of $J(L+\eta^{2}\partial_{x}^{-2})$ in $L^{2}_{\nu}$
coincides with the purely continuous spectrum of
$J(L_{0}+\eta^{2}\partial_{x}^{-2})$ in $L^{2}_{\nu}$, which can be obtained
by using the Fourier transform in $x$. The spectrum
$\sigma(L_{0}+\eta^{2}\partial_{x}^{-2})$ in $L^{2}_{\nu}$ is defined by the
range of the function $\lambda(\xi):\mathbb{R}\to\mathbb{C}$ given by

$\lambda(\xi)=(i\xi-\nu)[1-(i\xi-\nu)^{2}]^{-1}\left[c-3k-(c-k)(i\xi-\nu)^{2}+\eta^{2}(i\xi-\nu)^{-2}\right].$
(3.8)

Figure 3.1 gives a plot of $\lambda(\xi)$ for specific values of $k$, $c$,
$\eta$, and $\nu$. The plot suggests that
$\sigma(L_{0}+\eta^{2}\partial_{x}^{-2})$ in $L^{2}_{\nu}$ is located in the
left half-plane bounded away from zero. The following lemma proves this
property.

######  Lemma 3.7.

For every $c>3k$, $\eta\in\mathbb{R}$ and $\nu\in(0,\nu_{0})$, where
$\nu_{0}:=\sqrt{\frac{c-3k}{c-k}}$, we have $\text{\rm Re}(\lambda(\xi))<0$
for all $\xi\in{\mathbb{R}}$.

###### Proof.

The expression (3.8) can be simplified in the form:

$\displaystyle\lambda(\xi)=$
$\displaystyle(c-k)(i\xi-\nu)-2k(i\xi-\nu)[1-(i\xi-\nu)^{2}]^{-1}+\eta^{2}(i\xi-\nu)^{-1}[1-(i\xi-\nu)^{2}]^{-1}.$

Computing the real part and using $\lambda_{r}(\xi)$ from the proof of Lemma
3.6, we obtain

$\displaystyle\text{\rm Re}(\lambda(\xi))$
$\displaystyle=\lambda_{r}(\xi)-\frac{\eta^{2}\nu(1-\nu^{2}+3\xi^{2})}{(\xi^{2}+\nu^{2})[(1-\nu^{2}+\xi^{2})^{2}+4\xi^{2}\nu^{2}]}.$

Since $\lambda_{r}(\xi)<0$ for $\nu\in(0,\nu_{0})$ with
$\nu_{0}:=\sqrt{\frac{c-3k}{c-k}}$ and $\nu_{0}\in(0,1)$, we have $\text{\rm
Re}(\lambda(\xi))<0$ for all $\xi\in{\mathbb{R}}$. ∎

###  3.3. Splitting of the double zero eigenvalue in $L^{2}_{\nu}$ for
$\eta\neq 0$

By Lemma 3.6, $0$ is a double (isolated) eigenvalue of $JL$ in $L^{2}_{\nu}$
for small $\nu>0$. When $\eta\neq 0$ in (2.9), the translational symmetry is
broken and the double zero eigenvalue may split into two complex eigenvalues
of $J(L+\eta^{2}\partial_{x}^{-2})$. Since it is isolated away from the
continuous spectrum of $J(L+\eta^{2}\partial_{x}^{-2})$ for every
$\eta\in{\mathbb{R}}$ and small $\nu>0$ by Lemma 3.7, the splitting can be
studied by using perturbative methods in powers of $\eta$.

The following lemma states that when $\eta\neq 0$ the double zero eigenvalue
of $JL$ in $L^{2}_{\nu}$ for small $\nu>0$ splits into a pair of eigenvalues
of $J(L+\eta^{2}\partial_{x}^{-2})$ located in the left half of the complex
plane. The result holds for solitary waves of arbitrary amplitude and is
derived by means of Puiseux expansions in $\eta$. Together with Lemma 3.7,
this proves the result of Theorem 2.6.

######  Lemma 3.8.

There exists $\nu_{0}>0$ such that for every fixed $\nu\in(0,\nu_{0})$ there
exists $\eta_{0}>0$ such that the spectrum of $J(L+\eta^{2}\partial_{x}^{-2})$
in $L^{2}_{\nu}$ for $\eta\in(-\eta_{0},\eta_{0})$ contains a pair of simple
eigenvalues $\lambda_{\pm}(\eta)$ such that for $\eta\neq 0$ we have

  * •

${\rm Re}(\lambda_{+}(\eta))={\rm Re}(\lambda_{-}(\eta))<0$,

  * •

${\rm Im}(\lambda_{+}(\eta))=-{\rm Im}(\lambda_{-}(\eta))>0$,

and $\lambda_{+}(0)=\lambda_{-}(0)=0$.

###### Proof.

By Lemma 3.7, there exists $\nu_{0}>0$ such that for every fixed
$\nu\in(0,\nu_{0})$, the double zero eigenvalue of $JL$ in $L^{2}_{\nu}$ is
isolated from its continuous spectrum of $J(L+\eta^{2}\partial_{x}^{-2})$ in
$L^{2}_{\nu}$. Since $\eta^{2}\partial_{x}^{-2}$ is a bounded analytic
perturbation to the unbounded operator $L$ in $L^{2}_{\nu}$ for $\nu>0$, the
eigenvalues of $J(L+\eta^{2}\partial_{x}^{-2})$ in $L^{2}_{\nu}$ are
continuous functions of $\eta$.

By Lemma 3.6, the zero eigenvalue of $JL$ in $L^{2}_{\nu}$ is geometrically
simple and algebraically double. Hence we use Puiseux expansions [38] in order
to trace the eigenvalues $\lambda_{\pm}(\eta)$ satisfying
$\lambda_{\pm}(\eta)\to 0$ as $\eta\to 0$ with respect to small but nonzero
$\eta$. Solutions of the spectral stability problem (2.9) with
$\lambda=\lambda(\eta)$ are expanded as

$\displaystyle\hat{v}=v_{0}+v_{1}\eta+v_{2}\eta^{2}+v_{3}\eta^{3}+\mathcal{O}(\eta^{4}),$
$\displaystyle\lambda(\eta)=\lambda_{1}\eta+\lambda_{2}\eta^{2}+\lambda_{3}\eta^{3}+\mathcal{O}(\eta^{4}).$

where $v_{0},v_{1},v_{2},v_{3}\in L^{2}_{\nu}$ and
$\lambda_{1},\lambda_{2},\lambda_{3}\in{\mathbb{C}}$ are to be determined. We
obtain at different orders in powers of $\eta$ that

$\displaystyle\mathcal{O}(1):$ $\displaystyle\quad JLv_{0}=0,$
$\displaystyle\mathcal{O}(\eta):$ $\displaystyle\quad
JLv_{1}=\lambda_{1}v_{0},$ $\displaystyle\mathcal{O}(\eta^{2}):$
$\displaystyle\quad
JLv_{2}=\lambda_{2}v_{0}+\lambda_{1}v_{1}-(1-\partial_{x}^{2})^{-1}\partial_{x}^{-1}v_{0}$
$\displaystyle\mathcal{O}(\eta^{3}):$ $\displaystyle\quad
JLv_{3}=\lambda_{3}v_{0}+\lambda_{2}v_{1}+\lambda_{1}v_{2}-(1-\partial_{x}^{2})^{-1}\partial_{x}^{-1}v_{1}.$

With arbitrary normalization, we can set $v_{0}=\phi^{\prime}$ and
$v_{1}=-\lambda_{1}\partial_{c}\phi$ due to computations in the proof of Lemma
3.6. Then, at the order of $\mathcal{O}(\eta^{2})$, we write
$v_{2}=-\lambda_{2}\partial_{c}\phi+\hat{v}_{2}$, where $\hat{v}_{2}$
satisfies

$JL\hat{v}_{2}=-\lambda_{1}^{2}\partial_{c}\phi-(1-\partial_{x}^{2})^{-1}(\phi-k).$

After inverting $J$ in $L^{2}_{\nu}$ with $\nu>0$ we rewrite this linear
inhomogeneous equation in the equivalent form

$L\hat{v}_{2}=-\lambda_{1}^{2}\partial_{x}^{-1}\partial_{c}\mu-\partial_{x}^{-1}(\phi-k).$

By using (3.6) we check the Fredholm condition for the existence of solutions
$\hat{v}_{2}\in L^{2}_{\nu}$:

$\lambda_{1}^{2}\langle\phi^{\prime},\partial_{x}^{-1}\partial_{c}\mu\rangle+\langle\phi^{\prime},\partial_{x}^{-1}(\phi-k)\rangle=0.$

Note that $\partial_{x}^{-1}(\phi-k)=\int_{+\infty}^{x}(\phi-k)$, so the
second term gives after integration by parts

$\displaystyle\langle\phi^{\prime},\partial_{x}^{-1}(\phi-k)\rangle$
$\displaystyle=(\phi-k)\int_{+\infty}^{x}(\phi-k)dx^{\prime}\biggr{|}_{x\to-\infty}^{x\to+\infty}-\int_{-\infty}^{\infty}(\phi-k)^{2}dx$
$\displaystyle=-\|\phi-k\|_{L^{2}}^{2}=-\|\psi\|^{2}_{L^{2}}.$

On the other hand, the first term is evaluated with the help of (3.7). Since
$\frac{d}{dc}E_{\rm 1D}(\psi)>0$ by Lemma 3.2, we obtain that

$\lambda_{1}^{2}=-\frac{\langle\phi^{\prime},\partial_{x}^{-1}(\phi-k)\rangle}{\langle\phi^{\prime},\partial_{x}^{-1}\partial_{c}\mu\rangle}\\\
=-\frac{\|\psi\|_{L^{2}}^{2}}{\frac{d}{dc}E_{\rm 1D}(\psi)}<0.$ (3.9)

Thus, we have two roots for $\lambda_{1}\in i{\mathbb{R}}$, which determine
two simple eigenvalues $\lambda=\lambda_{\pm}(\eta)$. At the leading order, we
have ${\rm Im}(\lambda_{+}(\eta))=-{\rm Im}(\lambda_{-}(\eta))>0$ and the
complex-conjugate symmetry of eigenvalues is preserved since $J$ and $L$ are
real-valued.

At the next order $\mathcal{O}(\eta^{3})$ we write
$v_{3}=-\lambda_{3}\partial_{c}\phi+\hat{\phi}_{3}$, where $\hat{v}_{3}$
satisfies

$JL\hat{v}_{3}=\lambda_{1}\left[\hat{v}_{2}+(1-\partial_{x}^{2})^{-1}\partial_{x}^{-1}\partial_{c}\phi-2\lambda_{2}\partial_{c}\phi\right],$

which, after inverting $J$ in $L^{2}_{\nu}$ with $\nu>0$, gives

$L\hat{v}_{3}=\lambda_{1}\left[(1-\partial_{x}^{2})\partial_{x}^{-1}\hat{v}_{2}+\partial_{x}^{-2}\partial_{c}\phi-2\lambda_{2}\partial_{x}^{-1}\partial_{c}\mu\right].$

By using (3.6) we check the Fredholm condition for the existence of solutions
$\hat{v}_{3}\in L^{2}_{\nu}$:

$\displaystyle 2\lambda_{2}$
$\displaystyle=\frac{\langle\phi^{\prime},\partial_{x}^{-1}\left[(1-\partial_{x}^{2})\hat{v}_{2}+\partial_{x}^{-1}\partial_{c}\phi\right]\rangle}{\langle\phi^{\prime},\partial_{x}^{-1}\partial_{c}\mu\rangle}$
$\displaystyle=\frac{\langle\phi-k,(1-\partial_{x}^{2})\hat{v}_{2}\rangle+\langle\phi-k,\partial_{x}^{-1}\partial_{c}\phi\rangle}{\frac{d}{dc}E_{\rm
1D}(\psi)}.$

For the first term in the numerator, we use (3.4) and obtain

$\displaystyle\langle\phi-k,(1-\partial_{x}^{2})\hat{v}_{2}\rangle$
$\displaystyle=\langle\mu-k,\hat{v}_{2}\rangle=-\langle
L\partial_{c}\phi,\hat{v}_{2}\rangle=-\langle\partial_{c}\phi,L\hat{v}_{2}\rangle$
$\displaystyle=\lambda_{1}^{2}\langle\partial_{c}\phi,\partial_{x}^{-1}\partial_{c}\mu\rangle+\langle\partial_{c}\phi,\partial_{x}^{-1}(\phi-k)\rangle.$

We use the even parity of $\phi$ for which
$\int_{+\infty}^{x}(\phi-k)dx^{\prime}=-\frac{1}{2}\int_{-\infty}^{\infty}(\phi-k)dx^{\prime}+\int_{0}^{x}(\phi-k)dx^{\prime}$,
where the second term is odd, and obtain

$\displaystyle\langle\phi-k,\partial_{x}^{-1}\partial_{c}\phi\rangle$
$\displaystyle=-\frac{1}{2}\int_{-\infty}^{\infty}(\phi-k)dx\left(\int_{-\infty}^{\infty}\partial_{c}\phi
dx\right)=-\frac{1}{2}M_{\rm 1D}(\psi)\frac{d}{dc}M_{\rm 1D}(\psi),$
$\displaystyle\langle\partial_{c}\phi,\partial_{x}^{-1}(\phi-k)\rangle$
$\displaystyle=-\frac{1}{2}\left(\int_{-\infty}^{\infty}\partial_{c}\phi
dx\right)\int_{-\infty}^{\infty}(\phi-k)dx=-\frac{1}{2}M_{\rm
1D}(\psi)\frac{d}{dc}M_{\rm 1D}(\psi),$
$\displaystyle\langle\partial_{c}\phi,\partial_{x}^{-1}\partial_{c}\mu\rangle$
$\displaystyle=-\frac{1}{2}\left(\int_{-\infty}^{\infty}\partial_{c}\phi
dx\right)\int_{-\infty}^{\infty}\partial_{c}\mu
dx=-\frac{1}{2}\left(\frac{d}{dc}M_{\rm 1D}(\psi)\right)^{2},$

which then yields

$\displaystyle 2\lambda_{2}$ $\displaystyle=\frac{\frac{d}{dc}M_{\rm
1D}(\psi)}{\frac{d}{dc}E_{\rm
1D}(\psi)}\left[\frac{\|\psi\|_{L^{2}}^{2}}{2\frac{d}{dc}E_{\rm
1D}(\psi)}\frac{d}{dc}M_{\rm 1D}(\psi)-M_{\rm 1D}(\psi)\right]$
$\displaystyle=\frac{\frac{d}{dc}M_{\rm 1D}(\psi)}{2\left(\frac{d}{dc}E_{\rm
1D}(\psi)\right)^{2}}\left[\|\psi\|_{L^{2}}^{2}\frac{d}{dc}M_{\rm
1D}(\psi)-2M_{\rm 1D}(\psi)\frac{d}{dc}E_{\rm 1D}(\psi)\right],$ (3.10)

where we have used (3.9) for $\lambda_{1}^{2}$.

In order to identify the sign of $\lambda_{2}$, we recall from Lemma 3.2 that
the mappings $c\mapsto M_{\rm 1D}(\psi)$ and $c\mapsto E_{\rm 1D}(\psi)$ are
monotonically increasing. Hence, the sign of $\lambda_{2}$ is equivalent to
the sign of

$\displaystyle\|\psi\|_{L^{2}}^{2}\frac{d}{dc}M_{\rm 1D}(\psi)-2M_{\rm
1D}(\psi)\frac{d}{dc}E_{\rm 1D}(\psi)$
$\displaystyle=\frac{4k\sqrt{c-k}}{\sqrt{c-3k}}\left[(7k-2c)(\xi_{0}\sqrt{1+\xi_{0}^{2}}+{\rm
arcsinh}\xi_{0})-2k\xi_{0}\sqrt{(1+\xi_{0}^{2})^{3}}\right],\quad\xi_{0}:=\frac{\sqrt{c-3k}}{\sqrt{2k}},$

where we have substituted explicit expressions from Lemma 3.2 and Remark 3.4.
Since $c>3k$, we obtain

$\displaystyle\|\psi\|_{L^{2}}^{2}\frac{d}{dc}M_{\rm 1D}(\psi)-2M_{\rm
1D}(\psi)\frac{d}{dc}E_{\rm 1D}(\psi)$
$\displaystyle\qquad\leq\frac{4k^{2}\sqrt{c-k}}{\sqrt{c-3k}}\left[\xi_{0}\sqrt{1+\xi_{0}^{2}}+{\rm
arcsinh}\xi_{0}-2\xi_{0}\sqrt{(1+\xi_{0}^{2})^{3}}\right],$
$\displaystyle\qquad=-\frac{4k^{2}\sqrt{c-k}}{\sqrt{c-3k}}\xi_{0}\sqrt{1+\xi_{0}^{2}}\left[1+2\xi_{0}^{2}-\frac{\log(\xi_{0}+\sqrt{1+\xi_{0}^{2}})}{\xi_{0}\sqrt{1+\xi_{0}}^{2}}\right],$

where we have used ${\rm arcsinh}\xi_{0}=\log(\xi_{0}+\sqrt{1+\xi_{0}^{2}})$.
Since $\log(\xi_{0}+\sqrt{1+\xi_{0}^{2}})<\xi_{0}\sqrt{1+\xi_{0}^{2}}$ for
every $\xi_{0}>0$, the expression in the bracket is positive so that
$\lambda_{2}<0$. This yields ${\rm Re}(\lambda_{+}(\eta))={\rm
Re}(\lambda_{-}(\eta))<0$ at the leading order and hence for sufficiently
small $\eta\neq 0$. ∎

######  Remark 3.9.

In the KdV limit (2.6) as $c\to 3k$, we can simplify the expressions (3.9) and
(3.10) for $\lambda_{1}$ and $\lambda_{2}$ to obtain

$\displaystyle\lambda_{1}^{2}$
$\displaystyle=-\frac{\sqrt{2k}}{2\sqrt{c-3k}}\left[4(c-3k)\xi_{0}-\frac{8}{3}k\xi_{0}^{3}+\mathcal{O}(\xi_{0}^{5})\right]\sim-\frac{4}{3}(c-3k)$

and

$\displaystyle 2\lambda_{2}$
$\displaystyle=\frac{k}{(c-3k)^{2}}\left[4(3k-c)\xi_{0}-\frac{8}{3}k\xi_{0}^{3}+\mathcal{O}(\xi_{0}^{5})\right]\sim-\frac{8\sqrt{2k}}{3\sqrt{c-3k}},$

where we have used the explicit expressions in the proof of Lemma 3.2 and the
asymptotic limit $\xi_{0}\to 0$. Extracting the postive square root for
$\lambda_{1}$ yields the expansion for $\lambda_{\pm}(\eta)$ in the form

$\lambda_{\pm}(\eta)=\pm\frac{2i}{\sqrt{3}}\sqrt{c-3k}\eta-\frac{4}{3}\frac{\sqrt{2k}}{\sqrt{c-3k}}\eta^{2}+\mathcal{O}(\eta^{3}).$

Using the KP-II scaling (1.8) and (2.6) with $\eta=\varepsilon^{2}\Upsilon$
and $c-3k=\varepsilon^{2}$, we obtain

$\varepsilon^{-3}\lambda_{\pm}(\varepsilon^{2}\Upsilon)=\pm\frac{2i}{\sqrt{3}}\Upsilon-\frac{4}{3}\sqrt{2k}\Upsilon^{2}+\mathcal{O}(\Upsilon^{3}),$

which is the asymptotic expansion of the exact expression of the pair of
eigenvalues $\Lambda_{\pm}(\Upsilon)$ of the corresponding linearized operator
for the KP-II equation (1.9),

$\Lambda_{\pm}(\Upsilon)=\pm\frac{2i}{\sqrt{3}}\Upsilon\sqrt{1\pm\frac{4i}{\sqrt{3}}\sqrt{2k}\Upsilon},$
(3.11)

see [30].

######  Remark 3.10.

The continuous spectrum of $J(L+\eta^{2}\partial_{x}^{-2})$ in $L^{2}_{\nu}$
deforms to $i{\mathbb{R}}$ as $\nu\to 0$, which can be seen by taking the
limit $\nu\to 0$ in equation (3.8). On the other hand, the location of the
simple eigenvalues $\lambda_{\pm}(\eta)$ is independent of $\nu$ for
$\eta\in(-\eta_{0},\eta_{0})$ and $\nu\in(0,\nu_{0})$ as follows from (3.9)
and (3.10). As a result, the continuous spectrum crosses the location of the
simple eigenvalues for some $\nu_{1}\in(0,\nu_{0})$ that depends on $\eta\neq
0$. Consequently, as is shown in [37], the simple eigenvalues of
$J(L+\eta^{2}\partial_{x}^{-2})$ in $L^{2}_{\nu}$ for
$\nu\in(\nu_{1},\nu_{0})$ are no longer eigenvalues of
$J(L+\eta^{2}\partial_{x}^{-2})$ in $L^{2}_{\nu}$ for $\nu\in(0,\nu_{1})$ and
in $L^{2}({\mathbb{R}})$, because they are associated with the eigenfunctions
growing exponentially as $x\to-\infty$. Such points are referred to as
resonances of the linear operator $J(L+\eta^{2}\partial_{x}^{-2})$, see [37].

##  4\. Proof of Theorem 2.8

###  4.1. Preliminary results

We consider the spectral stability problem in the form (2.9). Writing
$\phi=k+\psi$ and $c=3k+\gamma$, we can rewrite the spectral problem (2.9) in
the equivalent form

$\partial_{x}(1-\partial_{x}^{2})^{-1}\left(\gamma-3\psi+\psi^{\prime\prime}-\partial_{x}(\gamma-\psi)\partial_{x}-2k\partial_{x}^{2}+\eta^{2}\partial_{x}^{-2}\right)\hat{v}=\lambda\hat{v}.$
(4.1)

In order to analyze the spectral problem (4.1) in the limit of small-amplitude
solitary waves, we give a rigorous proof of the approximation result in Remark
2.3 and justify the asymptotic approximation (2.6). The following lemma
presents this asymptotic result.

######  Lemma 4.1.

There exists $\varepsilon_{0}>0$ and $C_{0}>0$ such that for every
$\varepsilon\in(0,\varepsilon_{0})$ the solitary wave solution of Lemma 2.1
satisfying $\psi(0)=c-3k$ and $\psi^{\prime}(0)=0$ can be written in the form

$\psi(x)=\varepsilon^{2}\Psi_{\rm KdV}(X)+\varepsilon^{4}\tilde{\Psi}(X),\quad
X=\varepsilon x,\quad c=3k+\varepsilon^{2},$ (4.2)

where

$\Psi_{\rm KdV}(X):={\rm
sech}^{2}\left(\frac{X}{2\sqrt{2k}}\right)\quad\mbox{\rm
and}\quad\|\tilde{\Psi}\|_{L^{\infty}}\leq C_{0}.$

###### Proof.

Substituting $\psi(x)=\varepsilon^{2}\Psi(X)$, $X=\varepsilon x$, and
$c=3k+\varepsilon^{2}$ into (2.1) yields the first-order invariant

$(\Psi^{\prime})^{2}=\Psi^{2}\frac{1-\Psi}{2k+\varepsilon^{2}(1-\Psi)},$

for some $\Psi\in H^{2}({\mathbb{R}})$. The function $\Psi_{\rm KdV}$ is a
solution of the above equation in the limit $\varepsilon\to 0$. To prove (4.2)
we differentiate the first-order invariant and obtain the second-order
equation in the form $F(\Psi,\varepsilon^{2})=0$, where
$F(\Psi,\varepsilon^{2}):H^{2}({\mathbb{R}})\times{\mathbb{R}}\to
L^{2}({\mathbb{R}})$ is the operator function given by

$F(\Psi,\varepsilon^{2}):=-\Psi^{\prime\prime}+\Psi\frac{k(2-3\Psi)+\varepsilon^{2}(1-\Psi)^{2}}{(2k+\varepsilon^{2}(1-\Psi))^{2}}.$

It is clear that $F$ is a $C^{1}$ function near $(\Psi_{\rm KdV},0)$
satisfying

$F(\Psi_{\rm KdV},0)=-\Psi_{\rm KdV}^{\prime\prime}+\frac{1}{4k}\Psi_{\rm
KdV}(2-3\Psi_{\rm KdV})=0$

and

$D_{\Psi}F(\Psi_{\rm KdV},0)=-\partial_{x}^{2}+\frac{1}{2k}(1-3\Psi_{\rm
KdV}).$

Since $0$ is a simple eigenvalue of $D_{\Psi}F(\Psi_{\rm KdV},0)$ with odd
eigenfunction $\Psi_{\rm KdV}^{\prime}$, and the rest of its spectrum is
bounded away from $0$, the operator $D_{\Psi}F(\Psi_{\rm KdV},0)$ is
invertible in the subspace of even functions in $H^{2}({\mathbb{R}})$. By the
implicit function theorem, there exists a unique $C^{1}$ mapping
$\varepsilon^{2}\mapsto\Psi(\cdot,\varepsilon^{2})\in H^{2}({\mathbb{R}})$
which yields the unique even solution of
$F(\Psi(\cdot,\varepsilon^{2}),\varepsilon^{2})=0$ for small $\varepsilon^{2}$
such that $\Psi(\cdot,\varepsilon^{2})\to\Psi_{\rm KdV}$ as
$\varepsilon^{2}\to 0$. The decomposition (4.2) follows from the $C^{1}$
property of this mapping and the continuous embedding of $H^{2}({\mathbb{R}})$
into $L^{\infty}({\mathbb{R}})$. ∎

The KP-II scaling (1.8) and (2.6) corresponds to

$\lambda=\varepsilon^{3}\Lambda,\quad\eta=\varepsilon^{2}\Upsilon,\quad\gamma=\varepsilon^{2},\quad
x=\varepsilon^{-1}X,\quad\hat{v}(x)=\hat{V}(X).$ (4.3)

By Lemma 4.1, we can also write

$\psi(x)=\varepsilon^{2}\Psi(X),\quad\Psi:=\Psi_{\rm
KdV}+\varepsilon^{2}\tilde{\Psi},\quad c=3k+\varepsilon^{2}.$ (4.4)

The spectral problem (4.1) can then be rewritten as

$\partial_{X}(1-\varepsilon^{2}\partial_{X}^{2})^{-1}\left(L_{\rm
KdV}+\varepsilon^{2}L_{\rm
pert}+\Upsilon^{2}\partial_{X}^{-2}\right)\hat{V}=\Lambda\hat{V},$ (4.5)

where

$L_{\rm KdV}:=1-3\Psi_{\rm KdV}-2k\partial_{X}^{2},\quad L_{\rm
pert}:=\Psi^{\prime\prime}-\partial_{X}(1-\Psi)\partial_{X}-3\tilde{\Psi}.$

Since

$\nu_{0}=\frac{\sqrt{c-3k}}{\sqrt{c-k}}=\frac{\varepsilon}{\sqrt{2k+\varepsilon^{2}}}$

in Lemma 3.7, we need to rescale the exponential weight $\nu$ as
$\nu=\varepsilon\rho$ and replace the weighted space (2.10) by

$L_{\rho}^{2}:=\\{F(X):{\mathbb{R}}\to{\mathbb{R}}:\quad e^{\rho\cdot}F\in
L^{2}({\mathbb{R}})\\}.$

The parameter $\rho$ is fixed in $(0,\rho_{0})$, where
$\rho_{0}:=1/\sqrt{2k}$. In order to prove Theorem 2.8, we consider the
resolvent equations obtained from the spectral stability problem (4.1) in the
original variables and (4.5) in the scaled variables. The two resolvent
equations are used in two different regions:

  * •

the high-frequency region with $|\eta|\geq K^{2}_{0}\varepsilon^{2}$ for
sufficiently large $K_{0}>0$;

  * •

the low-frequency region with $|\eta|\leq K^{2}\varepsilon^{2}$ for every
fixed $K>0$.

Combining the two regions covers the entire range of $\eta$ values since $K$
can be taken to be greater than $K_{0}$. Estimates in Lemma 4.6 and Lemma 4.8
below prove the result of Theorem 2.8.

###  4.2. The high-frequency region

We start with the following result, which is a generalization of [32, Lemma
3.1] obtained for the linearized KP-II equation and extended here for the
spectral problem (4.5).

######  Proposition 4.2.

For every $\rho\in(0,\rho_{0})$ there exist $\varepsilon_{0}>0$ and
$\beta_{0}>0$ such that for every $\varepsilon\in(0,\varepsilon_{0})$,
$\Upsilon\in\mathbb{R}$, and every $\Lambda\in\mathbb{C}$ satisfying ${\rm
Re}(\Lambda)>-\beta_{0}$, we have

$\|\left(\Lambda-\partial_{X}(1-\varepsilon^{2}\partial_{X}^{2})^{-1}(1-(2k+\varepsilon^{2})\partial_{X}^{2}+\Upsilon^{2}\partial_{X}^{-2})\right)^{-1}\|_{L_{\rho}^{2}\to
L_{\rho}^{2}}\leq({\rm Re}(\Lambda)+\beta_{0})^{-1}.$ (4.6)

Moreover, there exists $C>0$ such that

$\displaystyle\|\partial_{X}(1-\varepsilon^{2}\partial_{X}^{2})^{-1}\left(\Lambda-\partial_{X}(1-\varepsilon^{2}\partial_{X}^{2})^{-1}(1-(2k+\varepsilon^{2})\partial_{X}^{2}+\Upsilon^{2}\partial_{X}^{-2})\right)^{-1}\|_{L_{\rho}^{2}\to
L_{\rho}^{2}}$ $\displaystyle\qquad\leq C\left({\rm
Re}(\Lambda)+\beta_{0}\right)^{-1/2}.$ (4.7)

if ${\rm Re}(\Lambda)>-\frac{1}{2}\beta_{0}$.

###### Proof.

Since the operators in the estimates (4.6) and (4.7) have constant
coefficients, we can use the Fourier transform in $X$ and introduce the
spectral function

$\Lambda(\Xi):=(i\Xi-\rho)[1-\varepsilon^{2}(i\Xi-\rho)^{2}]^{-1}[1-(2k+\varepsilon^{2})(i\Xi-\rho)^{2}+\Upsilon^{2}(i\Xi-\rho)^{-2}],$

for $\Upsilon\in{\mathbb{R}}$. The function $\Lambda(\Upsilon)$ is a scaled
version of the function $\lambda(\xi)$ in (3.8). We deduce the explicit
expression as in the proof of Lemmas 3.6 and 3.7:

$\displaystyle{\rm Re}\left(\Lambda(\Xi)\right)$
$\displaystyle=-\rho\Big{[}1+\frac{2k(3\Xi^{2}-\rho^{2}+\varepsilon^{2}(\Xi^{2}-\rho^{2})^{2})}{1+2\varepsilon^{2}(\Xi^{2}-\rho^{2})+\varepsilon^{4}(\Xi^{2}+\rho^{2})^{2}}$
$\displaystyle\qquad+\frac{\Upsilon^{2}(1+3\varepsilon^{2}\Xi^{2}-\varepsilon^{2}\rho^{2})}{(\Xi^{2}+\rho^{2})[1+2\varepsilon^{2}(\Xi^{2}-\rho^{2})+\varepsilon^{4}(\Xi^{2}+\rho^{2})^{2}]}\Big{]}.$
(4.8)

Since

$1-2\varepsilon^{2}\rho^{2}\leq
1+2\varepsilon^{2}(\Xi^{2}-\rho^{2})+\varepsilon^{4}(\Xi^{2}+\rho^{2})^{2}\leq[1+\varepsilon^{2}(\Xi^{2}+\rho^{2})]^{2},$

we have

$\displaystyle-{\rm Re}(\Lambda(\Xi))$
$\displaystyle\geq\rho\left[1-2k\rho^{2}+\frac{2k(-\varepsilon^{2}\rho^{4}+3\Xi^{2}+\varepsilon^{2}\Xi^{4}+\varepsilon^{4}\rho^{2}(\Xi^{2}+\rho^{2})^{2})}{1+2\varepsilon^{2}(\Xi^{2}-\rho^{2})+\varepsilon^{4}(\Xi^{2}+\rho^{2})^{2}}\right]$
$\displaystyle\geq\rho\left[1-2k\rho^{2}-\frac{2k\varepsilon^{2}\rho^{4}}{1-2\varepsilon^{2}\rho^{2}}+\frac{2k[\Xi^{2}(3+\varepsilon^{2}\Xi^{2})+\varepsilon^{4}\rho^{2}(\Xi^{2}+\rho^{2})^{2}]}{[1+\varepsilon^{2}(\Xi^{2}+\rho^{2})]^{2}}\right]$
(4.9)

uniformly for all $\Upsilon\in{\mathbb{R}}$. Therefore, there exists
$\rho_{0}=1/\sqrt{2k}$ such that for every $\rho\in(0,\rho_{0})$ there exists
$\varepsilon_{0}>0$ and $\beta_{0}>0$ such that $-{\rm
Re}\Lambda(\Xi)\geq\beta_{0}$ for every $\varepsilon\in(0,\varepsilon_{0})$
uniformly for all $\Xi\in{\mathbb{R}}$. For instance, we can choose

$\beta_{0}:=\rho\left[1-2k\rho^{2}-\frac{2k\varepsilon_{0}^{2}\rho^{4}}{1-2\varepsilon_{0}^{2}\rho^{2}}\right]>0$

for a suitable choice of $\varepsilon_{0}>0$. Hence, for every
$\Lambda\in\mathbb{C}$ satisfying ${\rm Re}(\Lambda)>-\beta_{0}$, we have

$|\Lambda-\Lambda(\Xi)|\geq({\rm Re}(\Lambda)+\beta_{0})$

and the bound (4.6) holds from standard Fourier estimates.

For the bound (4.7), we obtain from (4.9) that there exists $\gamma_{0}>0$
such that

$\displaystyle-{\rm Re}(\Lambda(\Xi))$
$\displaystyle\geq\beta_{0}+\frac{\gamma_{0}\Xi^{2}}{1+\varepsilon^{2}(\Xi^{2}+\rho^{2})}.$

For instance, we can choose $\gamma_{0}:=2k\rho$ since
$\varepsilon_{0}\rho_{0}<1$. Hence for every $\Lambda\in\mathbb{C}$ satisfying
${\rm Re}(\Lambda)>-\frac{1}{2}\beta_{0}$, we have

$|\Lambda-\Lambda(\Xi)|\geq\frac{1}{2}\beta_{0}+\frac{\gamma_{0}\Xi^{2}}{1+\varepsilon^{2}(\Xi^{2}+\rho^{2})}.$
(4.10)

Since there exists $C_{0}\in(0,1)$ such that

$1+2\varepsilon^{2}(\Xi^{2}-\rho^{2})+\varepsilon^{4}(\Xi^{2}+\rho^{2})^{2}\geq
C_{0}[1+\varepsilon^{2}(\Xi^{2}+\rho^{2})]^{2},$

we obtain

$\displaystyle\frac{|i\Xi-\rho|}{|1-\varepsilon^{2}(i\Xi-\rho)^{2}||\Lambda-\Lambda(\Xi)|}$
$\displaystyle\leq\frac{C\sqrt{\Xi^{2}+\rho^{2}}}{|1+\varepsilon^{2}(\Xi^{2}+\rho^{2})||\Lambda-\Lambda(\Xi)|}$
$\displaystyle\leq\frac{C}{\sqrt{1+\varepsilon^{2}(\Xi^{2}+\rho^{2})}\sqrt{|\Lambda-\Lambda(\Xi)|}}$
$\displaystyle\leq C({\rm Re}(\Lambda)+\beta_{0})^{-1/2},$ (4.11)

for some generic constants $C>0$ uniformly in $\Xi\in\mathbb{R}$. The bound
(4.7) follows again from Fourier theory. ∎

In order to complete the estimates in the high-frequency region, we obtain a
modified version of Proposition 4.2.

######  Proposition 4.3.

Let $\varepsilon_{0}>0$ and $\beta_{0}>0$ be the same as in Proposition 4.2.
There are $K_{0}>0$ and $C_{0}>0$ such that for every $\Lambda\in\mathbb{C}$
satisfying ${\rm Re}(\Lambda)>-\frac{1}{2}\beta_{0}$ and every
$\Upsilon\in{\mathbb{R}}$ satisfying $|\Upsilon|\geq K_{0}^{2}$, we have

$\displaystyle\|\partial_{X}(1-\varepsilon^{2}\partial_{X}^{2})^{-1}\left(\Lambda-\partial_{X}(1-\varepsilon^{2}\partial_{X}^{2})^{-1}(1-(2k+\varepsilon^{2})\partial_{X}^{2}+\Upsilon^{2}\partial_{X}^{-2})\right)^{-1}\|_{L_{\rho}^{2}\to
L_{\rho}^{2}}$ $\displaystyle\qquad\leq CK_{0}^{-1}\left({\rm
Re}(\Lambda)+\beta_{0}\right)^{-1/2}.$ (4.12)

###### Proof.

This follows from the bounds on $\Lambda(\Xi)$ obtained in the proof of
Proposition 4.2. If $|\Xi|\geq K_{0}$ and $K_{0}>0$ is sufficiently large,
then it follows from (4.10) that for every $\Upsilon\in\mathbb{R}$, we have

$|\Lambda-\Lambda(\Xi)|\geq\frac{\gamma_{0}K_{0}^{2}}{1+\varepsilon^{2}(\Xi^{2}+\rho^{2})}.$

On the other hand, if $|\Xi+i\rho|\leq K_{0}$ and $|\Upsilon|\geq
K_{0}^{2}\geq K_{0}|\Xi+i\rho|$, then it follows from (4.8) that

$|\Lambda-\Lambda(\Xi)|\geq\frac{\rho\Upsilon^{2}(1+3\varepsilon^{2}\Xi^{2}-\varepsilon^{2}\rho^{2})}{(\Xi^{2}+\rho^{2})[1+\varepsilon^{2}(\Xi^{2}-\rho^{2})]^{2}}\geq\frac{\rho
K_{0}^{2}}{1+\varepsilon^{2}(\Xi^{2}+\rho^{2})}.$

Then, similarly to (4.11), we obtain

$\frac{|i\Xi-\rho|}{|1-\varepsilon^{2}(i\Xi-\rho)^{2}||\Lambda-\Lambda(\Xi)|}\leq\frac{C}{\sqrt{1+\varepsilon^{2}(\Xi^{2}+\rho^{2})}\sqrt{|\Lambda-\Lambda(\Xi)|}}\leq
CK_{0}^{-1}({\rm Re}(\Lambda)+\beta_{0})^{-1/2},$

for some generic constant $C>0$ uniformly in $\Xi\in{\mathbb{R}}$. This
justifies the bound (4.16). ∎

The resolvent equation in the original variables is obtained from the spectral
problem (4.1) with $\gamma=\varepsilon^{2}$ in the form:

$\left(\lambda-A_{0}-A_{1}-A_{2}\right)u=f,\qquad f\in L_{\nu}^{2},$ (4.13)

where

$\displaystyle A_{0}$
$\displaystyle:=\partial_{x}(1-\partial_{x}^{2})^{-1}(\varepsilon^{2}-(2k+\varepsilon^{2})\partial_{x}^{2}+\eta^{2}\partial_{x}^{-2}),$
$\displaystyle A_{1}$
$\displaystyle:=\partial_{x}(1-\partial_{x}^{2})^{-1}\partial_{x}\psi\partial_{x},$
$\displaystyle A_{2}$
$\displaystyle:=\partial_{x}(1-\partial_{x}^{2})^{-1}(-3\psi+\psi^{\prime\prime}).$

Using this notation we obtain the following corollary of Proposition 4.3 which
gives the bounds in original variables.

######  Corollary 4.4.

For every $\lambda\in\mathbb{C}$ satisfying ${\rm
Re}(\lambda)>-\frac{1}{2}\beta_{0}\varepsilon^{3}$ with some $\beta_{0}>0$ and
every $\eta\in\mathbb{R}$ satisfying $|\eta|\geq K_{0}^{2}\varepsilon^{2}$
with sufficiently large $K_{0}>0$ we find that

$\|(\lambda-A_{0})^{-1}\|_{L^{2}_{\varepsilon\rho}\to
L^{2}_{\varepsilon\rho}}\leq C\varepsilon^{-3},$ (4.14)
$\|\partial_{x}(1-\partial_{x}^{2})^{-1}(\lambda-
A_{0})^{-1}\|_{L^{2}_{\varepsilon\rho}\to L^{2}_{\varepsilon\rho}}\leq
C\varepsilon^{-2},$ (4.15)

and

$\|\partial_{x}(1-\partial_{x}^{2})^{-1}(\lambda-
A_{0})^{-1}\|_{L^{2}_{\varepsilon\rho}\to L^{2}_{\varepsilon\rho}}\leq
CK_{0}^{-1}\varepsilon^{-2}.$ (4.16)

######  Remark 4.5.

Since the continuous spectrum of $\varepsilon^{-3}A_{0}$ in $L^{2}_{\rho}$ is
bounded away from $i{\mathbb{R}}$ by the $\varepsilon$-independent constant
$\beta_{0}$, and $\varepsilon^{-3}A_{1}$ is a relatively bounded perturbation
to $\varepsilon^{-3}A_{0}$ of order $\mathcal{O}(\varepsilon^{2})$ due to the
scaling (4.4), the estimates (4.14), (4.15), and (4.16) apply also for
$(\lambda-A_{0}-A_{1})^{-1}$ instead of $(\lambda-A_{0})^{-1}$. Hence, we will
use

$\|(\lambda-A_{0}-A_{1})^{-1}\|_{L^{2}_{\varepsilon\rho}\to
L^{2}_{\varepsilon\rho}}\leq C\varepsilon^{-3},$ (4.17)
$\|\partial_{x}(1-\partial_{x}^{2})^{-1}(\lambda-
A_{0}-A_{1})^{-1}\|_{L^{2}_{\varepsilon\rho}\to L^{2}_{\varepsilon\rho}}\leq
C\varepsilon^{-2},$ (4.18)

and

$\|\partial_{x}(1-\partial_{x}^{2})^{-1}(\lambda-
A_{0}-A_{1})^{-1}\|_{L^{2}_{\varepsilon\rho}\to L^{2}_{\varepsilon\rho}}\leq
CK_{0}^{-1}\varepsilon^{-2}.$ (4.19)

instead of (4.14), (4.15), and (4.16).

The following lemma uses the fact that the operator $A_{2}$ in (4.13) is small
compared to the operator $A_{0}+A_{1}$ in $L^{2}_{\varepsilon\rho}$ due to the
KP-II scaling (4.3) and (4.4), see the estimate (4.22) below. As a result, we
obtain the following resolvent estimate in the high-frequency region.

######  Lemma 4.6.

For every $\rho\in(0,\rho_{0})$ there exists $\varepsilon_{0}>0$,
$\beta_{0}>0$, and $K_{0}>0$ such that for every
$\varepsilon\in(0,\varepsilon_{0})$, $\eta\in\mathbb{R}$ satisfying
$|\eta|\geq K_{0}^{2}\varepsilon^{2}$, and $\lambda\in\mathbb{C}$ satisfying
${\rm Re}(\lambda)>-\beta_{0}\varepsilon^{3}$, there exists a unique solution
$u\in{\rm Dom}(A_{0})\subset L^{2}_{\varepsilon\rho}$ to the resolvent
equation (4.13) with $f\in L^{2}_{\varepsilon\rho}$ such that

$\|u\|_{L^{2}_{\varepsilon\rho}}\leq
C\varepsilon^{-3}\|f\|_{L^{2}_{\varepsilon\rho}},$ (4.20)

for some $C>0$ independently of $f\in L^{2}_{\varepsilon\rho}$ and
$\varepsilon$.

###### Proof.

We use the resolvent identity

$(\lambda-A_{0}-A_{1}-A_{2})^{-1}=[I-(\lambda-
A_{0}-A_{1})^{-1}A_{2}]^{-1}(\lambda-A_{0}-A_{1})^{-1}.$

It follows from the bound (4.17) that we only need to show that the operator

$I-(\lambda-A_{0}-A_{1})^{-1}A_{2}$

is invertible with a bounded inverse in $L^{2}_{\varepsilon\rho}$, which is
true if $\|(\lambda-A_{0}-A_{1})^{-1}A_{2}\|_{L^{2}_{\varepsilon\rho}\to
L^{2}_{\varepsilon\rho}}$ is small. Since the decomposition (4.4) implies that

$\|(-3\psi+\psi^{\prime\prime})f\|_{L^{2}_{\varepsilon\rho}}\leq
C\varepsilon^{2}\|f\|_{L^{2}_{\varepsilon\rho}},$ (4.21)

it follows from the bound (4.18) that the smallness of $\|(\lambda-
A_{0}-A_{1})^{-1}A_{2}\|_{L^{2}_{\varepsilon\rho}\to L^{2}_{\varepsilon\rho}}$
cannot be deduced from smallness of $\varepsilon$. Nevertheless, if we use the
estimates (4.19) and (4.21), then we obtain

$\displaystyle\|(\lambda-A_{0}-A_{1})^{-1}A_{2}\|_{L^{2}_{\varepsilon\rho}\to
L^{2}_{\varepsilon\rho}}\leq C_{0}K_{0}^{-1}$ (4.22)

for some $C_{0}>0$. If $K_{0}>0$ is sufficiently large, the norm is small and
the operator $I-(\lambda-A_{0}-A_{1})^{-1}A_{2}$ is invertible with a bounded
inverse in $L^{2}_{\varepsilon\rho}$. The bound (4.20) follows from (4.17). ∎

###  4.3. The low-frequency region

We first consider the two eigenvalues $\lambda_{\pm}(\eta)$ of the spectral
problem (4.1) in $L^{2}_{\nu}$ for small $\eta\neq 0$, see Lemma 3.8. By
Remark 3.9, the expansion of
$\varepsilon^{-3}\lambda_{\pm}(\varepsilon^{2}\Upsilon)$ in $\Upsilon$ agrees
with the exact expression (3.11) known for the KP-II equation (1.9). The
following lemma states that the same correspondence holds for every $\Upsilon$
if $\varepsilon$ is sufficiently small.

######  Lemma 4.7.

Let $\Lambda_{\pm}(\Upsilon)$ be given by (3.11) for every
$\Upsilon\in{\mathbb{R}}$. For every $\rho\in(0,\rho_{0})$, there exists
$\varepsilon_{0}>0$ and $C_{0}>0$ such that for every
$\varepsilon\in(0,\varepsilon_{0})$ the spectral problem (4.5) admits
eigenvalues $\varepsilon^{-3}\lambda_{\pm}(\varepsilon^{2}\Upsilon)$ in
$L^{2}_{\rho}$ such that

$|\varepsilon^{-3}\lambda_{\pm}(\varepsilon^{2}\Upsilon)-\Lambda_{\pm}(\Upsilon)|\leq
C_{0}\varepsilon^{2}.$

###### Proof.

By bootstrapping arguments, an eigenfunction $\hat{V}$ of the spectral problem
(4.5) in $L^{2}_{\rho}$ satisfies that

$\hat{V}\in{\rm
Dom}(\partial_{X}(1-\varepsilon^{2}\partial_{X}^{2})^{-1}(L_{\rm
KdV}+\Upsilon^{2}\partial_{X}^{-2}))\subset L^{2}_{\rho}$

if and only if

$\hat{V}\in{\rm Dom}(\partial_{X}(L_{\rm
KdV}+\Upsilon^{2}\partial_{X}^{-2}))\subset L^{2}_{\rho}.$

Hence we can rewrite the spectral problem (4.5) for the eigenfunction
$\hat{V}$ in $L^{2}_{\rho}$ in the equivalent form

$\partial_{X}\left(L_{\rm KdV}+\varepsilon^{2}L_{\rm
pert}+\Upsilon^{2}\partial_{X}^{-2}\right)\hat{V}=\Lambda(1-\varepsilon^{2}\partial_{X}^{2})\hat{V}.$
(4.23)

Since $(\Lambda_{\pm}(\Upsilon),U_{\pm})\in\mathbb{C}\times L^{2}_{\rho}$ are
solutions of the truncated problem

$\partial_{X}\left(L_{\rm
KdV}+\Upsilon^{2}\partial_{X}^{-2}\right)U_{\pm}=\Lambda_{\pm}(\Upsilon)U_{\pm},$
(4.24)

we can write the decomposition
$\Lambda=\Lambda_{\pm}(\Upsilon)+\varepsilon^{2}\tilde{\Lambda}$,
$\hat{V}=U_{\pm}+\varepsilon\tilde{U}$ and obtain the perturbed problem for
$(\tilde{\Lambda},\tilde{U})$ given by

$\displaystyle\partial_{X}\left(L_{\rm KdV}+\varepsilon^{2}L_{\rm
pert}+\Upsilon^{2}\partial_{X}^{-2}\right)\tilde{U}-(\Lambda_{\pm}(\Upsilon)+\varepsilon^{2}\tilde{\Lambda})(1-\varepsilon^{2}\partial_{X}^{2})\tilde{U}$
$\displaystyle\quad=-L_{\rm
pert}U_{\pm}-\Lambda_{\pm}\partial_{X}^{2}U_{\pm}+\tilde{\Lambda}(1-\varepsilon^{2}\partial_{X}^{2})U_{\pm}.$

This equation is routinely solved by using the method of Lyapunov–Schmidt
reduction with $\tilde{\Lambda}$ being uniquely defined from the condition
that $\tilde{U}\in{\rm Dom}(\partial_{X}(L_{\rm
KdV}+\Upsilon^{2}\partial_{X}^{-2}))\subset L^{2}_{\rho}$ satisfy the
orthogonality condition to the adjoint eigenfunction for the eigenvalue
$\Lambda_{\pm}(\Upsilon)$. See Lemma 3.4 and Corollary 3.5 in [32] for
details. ∎

The resolvent equation in the scaled variables is obtained from the spectral
stability problem (4.5) in the form

$\left(\Lambda-\partial_{X}(1-\varepsilon^{2}\partial_{X}^{2})^{-1}\left(L_{\rm
KdV}+\varepsilon^{2}L_{\rm
pert}+\Upsilon^{2}\partial_{X}^{-2}\right)\right)U=F,\qquad F\in
L_{\rho}^{2}.$ (4.25)

The following lemma uses the smallness of $\varepsilon^{2}L_{\rm pert}$ and
the formalism from [32] in order to obtain the resolvent estimate in the low-
frequency region.

######  Lemma 4.8.

For every $\rho\in(0,\rho_{0})$ there exists $\varepsilon_{0}>0$,
$\beta_{0}>0$ such that for every $\varepsilon\in(0,\varepsilon_{0})$,
$\Upsilon\in\mathbb{R}$ and $\Lambda\in\mathbb{C}$ satisfying ${\rm
Re}(\Lambda)>-\beta_{0}$ and
$\Lambda\neq\varepsilon^{-3}\lambda_{\pm}(\varepsilon^{2}\Upsilon)$, there
exists a unique solution

$U\in{\rm Dom}(\partial_{X}(1-\varepsilon^{2}\partial_{X}^{2})^{-1}(L_{\rm
KdV}+\Upsilon^{2}\partial_{X}^{-2}))\subset L^{2}_{\rho}$

of the resolvent equation (4.25) for every $F\in L^{2}_{\rho}$ satisfying

$\|U\|_{L^{2}_{\rho}}\leq C\|F\|_{L^{2}_{\rho}}$ (4.26)

for $C>0$.

###### Proof.

Let $Q_{\rm KP}$ be the projection operator for the spectral problem (4.24)
which reduces $L^{2}_{\rho}$ to the subspace orthogonal to the two adjoint
eigenfunctions for the eigenvalues $\Lambda_{\pm}(\Upsilon)$. It follows from
Proposition 3.2 in [32] (proven in [30]) that there exists $\beta_{0}>0$ and
$C_{0}>0$ such that for every $\Lambda\in{\mathbb{C}}$ satisfying ${\rm
Re}(\Lambda)>-\beta_{0}$ and every $F\in L^{2}_{\rho}$, we have

$\|(\Lambda-\partial_{X}(L_{\rm
KdV}+\Upsilon^{2}\partial_{X}^{-2}))^{-1}Q_{\rm KP}F\|_{L_{\rho}^{2}}\leq
C_{0}\|F\|_{L^{2}_{\rho}}.$ (4.27)

By the proximity result of Lemma 4.7, we can introduce $\mathcal{Q}$, the
projection operator for the spectral problem (4.23) which reduces
$L^{2}_{\rho}$ to the subspace orthogonal to the two adjoint eigenfunctions
for the eigenvalues $\varepsilon^{-3}\lambda_{\pm}(\varepsilon^{2}\Upsilon)$.
The bound (4.27) and the proximity result suggest that there exists
$\beta_{0}>0$ and $C_{0}>0$ such that for every $\Lambda\in{\mathbb{C}}$
satisfying ${\rm Re}(\Lambda)>-\beta_{0}$ and every $F\in L^{2}_{\rho}$, we
have

$\|(\Lambda-\mathcal{M})^{-1}\mathcal{Q}F\|_{L_{\rho}^{2}}\leq
C_{0}\|F\|_{L^{2}_{\rho}},$ (4.28)

where

$\mathcal{M}:=\partial_{X}(1-\varepsilon^{2}\partial_{X}^{2})^{-1}(L_{\rm
KdV}-\varepsilon^{2}\partial_{X}(1-\Psi)\partial_{X}+\Upsilon^{2}\partial_{X}^{-2}).$

Writing again the resolvent identity as

$\displaystyle(\Lambda-\partial_{X}(1-\varepsilon^{2}\partial_{X}^{2})^{-1}(L_{\rm
KdV}+\varepsilon^{2}L_{\rm pert}+\Upsilon^{2}\partial_{X}^{-2}))^{-1})^{-1}$
$\displaystyle\qquad=[I-\varepsilon^{2}(\Lambda-\mathcal{M})(\Psi^{\prime\prime}-3\tilde{\Psi})]^{-1}(\Lambda-\mathcal{M})^{-1}$

and using smallness of $\varepsilon^{2}$, we obtain the invertibility of the
near-identity operator

$[I-\varepsilon^{2}(\Lambda-\mathcal{M})(\Psi^{\prime\prime}-3\tilde{\Psi})]:L^{2}_{\rho}\to
L^{2}_{\rho}$

for every $\Lambda\in{\mathbb{C}}$ satisfying ${\rm Re}(\Lambda)>-\beta_{0}$.
The bound (4.26) on the unique solution $U$ to the resolvent equation (4.25)
follows from the bound (4.28). ∎

##  5\. Conclusion

We have derived two results, which suggest that the transverse perturbations
to the one-dimensional solitary waves of the CH equation (1.1) are stable in
the time evolution of the CH-KP equation (1.2), similar to the KP-II theory.
First, we proved that the double zero eigenvalue of the linearized equation
related to the translational symmetry breaks under a transverse perturbation
into a pair of the asymptotically stable resonances, which are isolated
eigenvalues in the exponentially weighted $L^{2}$ space. Second, we considered
the small-amplitude solitary waves governed by the perturbed KP-II equation
and proved their linear stability under transverse perturbations.

We conclude the paper with a list of further questions. First, nonlinear
stability of small-amplitude solitary waves of CH-KP is an open question, see
[33] for such analysis in the Benney–Luke equation. Second, peaked traveling
waves of the CH equation (1.1) exist but they are linearly and nonlinearly
unstable in the time evolution in $H^{1}({\mathbb{R}})\cap
W^{1,\infty}({\mathbb{R}})$, see [22, 34]. It would be interesting to see how
the peaked profile of the solitary waves breaks under transverse perturbations
and whether cusps (waves with infinite slopes at their maximum) would form in
finite time. Third, transverse stability of smooth periodic waves and
transverse instability of peaked periodic waves can be studied based on the
stability analysis of the periodic waves in the one-dimensional model, see
[16] and [29]. Finally, hydrodynamical applications of the obtained results
are interesting in their own right within modeling of shallow water waves in
seas and oceans [17].

Acknowledgement. This project was started in June 2022 during a Research in
Teams stay at the Erwin Schrödinger Institute, Vienna. The authors thank
members of stuff of the ESI for support during this work. D. E. Pelinovsky
acknowledges the funding of this study provided by Grants No. FSWE-2020-0007
and No. NSH-70.2022.1.5.

## References

  * [1] Bhavna, A. K. Pandey, S. Singh, “Transverse spectral instabilities in Konopelchenko–Dubrovsky equation”, Stud. Appl. Math. (2023) in print.
  * [2] H. Borluk, G. Bruell, and D. Nilsson, “Traveling waves and transverse instability forthe fractional Kadomtsev–Petviashvili equation”, Stud. Appl. Math. 149 (2022) 95–123.
  * [3] R. Camassa and D.D. Holm, “An integrable shallow water equation with peaked solitons”, Phys. Rev. Lett. 71 (1993), 1661–1664.
  * [4] R.M. Chen, “Some nonlinear dispersive waves arising in compressible hyperelastic plates”, Int. J. Eng. Sci. 44 (2006) 1188–1204.
  * [5] A. Comech, S. Cuccagna, and D. Pelinovsky, “Nonlinear instability of a critical traveling wave in the generalized Korteweg-de Vries equation”, SIAM J. Math. Anal. 39 (2007) 1–33.
  * [6] A. Constantin and J. Escher, “Wave breaking for nonlinear nonlocal shallow water equations”, Acta Math. 181 (1998), 229–243.
  * [7] A. Constantin and J. Escher, “Well-posedness, global existence, and blowup phenomena for a periodic quasi-linear hyperbolic equation”, Comm. Pure Appl. Math. 51 (1998), 475–504.
  * [8] A. Constantin and D. Lannes, “The hydrodynamical relevance of the Camassa–Holm and Degasperis–Procesi equations”, Arch. Ration. Mech. Anal. 192 (2009), 165–186.
  * [9] A. Constantin and L. Molinet, “Orbital stability of solitary waves for a shallow water equation”, Physica D 157 (2001) 75–89.
  * [10] A. Constantin and W.A. Strauss, “Stability of peakons”, Comm. Pure Appl. Math. 53 (2000), 603–610.
  * [11] A. Constantin and W.A. Strauss, “Stability of the Camassa–Holm solitons”, J. Nonlinear Sci. 12 (2002), 415–422.
  * [12] C. De Lellis, T. Kappeler, and P. Topalov, “Low-regularity solutions of the periodic Camassa–Holm equation”, Comm. PDEs 32 (2007), 87–126.
  * [13] A. Fokas and B. Fuchssteiner, “Symmpletic structures, their Backlund transform and hereditary symmetries”, Physica D 4 (1981), 47–66
  * [14] T. Gallay and G. Schneider, “KP description of unidirectional long waves. The model case,” Proc. R. Soc. Edinburgh A 131 (2001), 885–898.
  * [15] M. Gallone and S. Pasquali, “Metastability phenomena in two-dimensional rectangular lattices with nearest-neighbour interaction”, Nonlinearity 34 (2021), 4983–5044.
  * [16] A. Geyer, R.H. Martins, F. Natali, and D.E. Pelinovsky, “Stability of smooth periodic traveling waves in the Camassa-Holm equation”, Stud. Appl. Math. 148 (2022) 27–61.
  * [17] G. Gui, Y. Liu, W. Luo, and Z. Yin, “On a two-dimensional nonlocal shallow-water model”, Adv. Math. 392 (2021) 108021 (44 pages).
  * [18] M. Haragus, J. Li, and D.E. Pelinovsky, “Counting unstable eigenvalues in Hamiltonian spectral problems via commuting operators”, Comm. Math. Phys. 354 (2017) 247–268.
  * [19] N. Hristov and D. E. Pelinovsky, “Justification of the KP-II approximation in dynamics fo two-dimensional FPU systems”, ZAMP 73 (2022) 213 (26 pages).
  * [20] R.S. Johnson, “Camassa–Holm, Korteweg–de Vries and related models for water waves”, J. Fluid Mech. 455 (2002) 63–82.
  * [21] B.B. Kadomtsev and V.I. Petviashvili, “On the stability of solitary waves in weakly dispersing media”, Sov. Phys. Dokl. 15 (1970) 539–541.
  * [22] S. Lafortune and D.E. Pelinovsky, “Spectral instability of peakons in the b-family of the Camassa-Holm equations”, SIAM J. Math. Anal. 54 (2022) 4572–4590
  * [23] S. Lafortune and D.E. Pelinovsky, “Stability of smooth solitary waves in the $b$-Camassa–Holm equations”, Physica D 440 (2022) 133477 (10 pages).
  * [24] J. Lenells, “Stability of periodic peakons”, Int. Math. Res. Not. 2004 (2004), 485–499.
  * [25] J. Lenells, “A variational approach to the stability of periodic peakons”, J. Nonlinear Math. Phys. 11 (2004) 151–163.
  * [26] J. Lenells, “Traveling wave solutions of the Camassa-Holm equation”, J. Diff. Eq. 217(2) (2005) 393–430.
  * [27] J. Lenells, “Stability for the periodic Camassa–Holm equation”, Math. Scand. 97 (2005) 188–200.
  * [28] F. Linares, G. Ponce, and Th. C. Sideris, “Properties of solutions to the Camassa–Holm equation on the line in a class containing the peakons”, Advanced Studies in Pure Mathematics 81 (2019), 196–245.
  * [29] A. Madiyeva and D.E. Pelinovsky, “Growth of perturbations to the peaked periodic waves in the Camassa-Holm equation”, SIAM J. Math. Anal. 53 (2021), 3016–3039.
  * [30] T. Mizumachi, “Stability of line solitons for the KP-II equation in $\mathbb{R}^{2}$”, Mem. Amer. Math. Soc. 238 (2015), no. 1125 (95 pages).
  * [31] T. Mizumachi and N. Tzvetkov, “Stability of the line soliton of the KP-II equation under periodic transverse perturbations”, Math. Ann. 352 (2012) 659–690.
  * [32] T. Mizumachi and Y. Shimabukuro, “Asymptotic linear stability of Benney–Luke line solitary waves in 2D”, Nonlinearity 30 (2017) 3419–3465
  * [33] T. Mizumachi and Y. Shimabukuro, “Stability of Benney–Luke line solitary waves in 2 dimensions”, SIAM J. Math. Anal. 52 (2020) 4238–4283
  * [34] F. Natali and D.E. Pelinovsky, “Instability of $H^{1}$-stable peakons in the Camassa–Holm equation”, J. Diff. Eqs. 268 (2020), 7342–7363.
  * [35] D.E. Pelinovsky and G. Schneider, “KP-II approximation for a scalar FPU system on a 2D square lattice”, SIAM J. Appl. Math. 83 (2023) 79–98.
  * [36] R. Pego and M. I. Weinstein, “Eigenvalues, and instabilities of solitary waves”, Philos. Trans. Roy. Soc. London Ser. A 340 (1992) 47–94.
  * [37] R. Pego and M. I. Weinstein, “Asymptotic stability of solitary waves”, Comm. Math. Phys. 164 (1994) 305–349
  * [38] A. Welters, “On explicit recursive formulas in the spectral perturbation analysis of a Jordan block”, SIAM J. Matrix Anal. Appl. 32 (2011) 1–22
  * [39] Q. Zhang, Y. Xu, and Y. Liu, “A discontinuous Galerkin method for the Camassa–Holm–Kadomtsev–Petviashvili type equations”, Numer. Methods PDE (2023)


# Benefits and Limitations of Web3

Collin Connors  Department of Computer Science
University of Miami
Coral Gable, FL - 33124
<EMAIL_ADDRESS>Dilip Sarkar  Department of Computer Science
University of Miami
Coral Gable, FL - 33124
<EMAIL_ADDRESS>
###### Abstract

Web3 provides users and service providers several benefits not found in Web2.
However, despite the benefits provided, Web3 faces several obstacles that
prevent the paradigm from gaining widespread adoption. Developers should
understand the benefits and limitations of the technology in order to create
more accessible Web3 smart applications.

###### Index Terms:

Web3, Blockchain, Privacy, Security, Semantic Web

##  I Introduction

Web3 combines older notions of a semantic web (Web3.0) with blockchain
technology to allow for a more dynamic distributed web architecture. Using the
Web3 architecture, end-users and service providers could see substantial
benefits to data security, privacy, and overall user experience.

Our motivation for this work is to highlight for developers the benefits of
Web3 while also emphasizing challenges the developers may face when
implementing and utilizing a Web3 architecture in hopes of guiding developers
in creating more accessible Web3 smart applications. We first discuss how a
blockchain-based Web3 can vastly benefit all parties involved in creating and
maintaining the World Wide Web. We then describe some obstacles that Web3
architectures currently face so that developers are aware of the limitations
of the technology. We hope developers can use this knowledge to design their
Web3 architectures and smart applications to be more accessible so that more
end-users can reap the benefits of Web3.

In the following sections, we will briefly discuss the history of the World
Wide Web and how it has evolved since its inception [11] [7]. We then discuss
the semantic web (Web3.0), a precursor to the present Web3. We analyze the
limitations of semantic web models and how blockchain has been proposed to
address these challenges, leading to the current iteration of Web3. We
highlight the benefits the Web3 paradigm can bring to both users and service
providers. We then give an overview of the challenges Web3 architecture faces.
Lastly, we conclude by providing guidance on overcoming the current
limitations of the technology.

##  II Background

Since Web3 builds off previous generations of the World Wide Web, to best
understand the advantages of Web3 it is critical to briefly review the history
of the World Wide Web. Figure 1 provides a timeline of important events in Web
technologies and blockchain technologies as they relate to Web3 [1].

Figure 1: A timeline of key events in the history of the World Wide Web and
the rise in popularity of Web3

###  II-A Web1

In 1989, Tim Burners-Lee sent a proposal to management at CERN where he
pointed out that the hierarchical system of keeping records that was in use
made it difficult and time-consuming to search for related documents [3]. To
solve this problem, Burners-Lee suggested a new paradigm based on hypertext,
which links records to each other through hyperlinks. Later, Burners-Lee
combined hypertext with the internet to create the first version of the World
Wide Web.

This first version of the World Wide Web later became known as Web1. Features
first introduced in Web1 have become ubiquitous with the modern web. The
defining technology behind Web1 was Hypertext. Developers used HyperText
Markup Language (HTML) to create content for web pages; web servers would then
host the content created and allow end-users to find the content using
Universal Record Locators (URLs); the end-users (will also be referred to as
users) could utilize web browsers to access the web pages using the HyperText
Transport Protocol (HTTP). While these technologies have been augmented and
improved, they remain core to the modern World Wide Web.

In Web1, web pages were static pages that utilized Hyperlinks to connect to
other websites. These early web pages did not have the capability for end-
users to generate new content. Instead, the early WWW acted as a directory
where end-users could find HTML files from worldwide web servers.

This model of a Read-Only web provided end-users with an easy way to traverse
the internet. Figure 2 shows a high-level view of Web1. In this paradigm, end-
users can read data from web pages that are hyperlinked to each other. Various
service providers host these web pages. Because there is no user
interactivity, this era of the web was focused on companies providing content
to end-users.

Figure 2: A high level overview of Web1’s Read Only model.

While the new technology developed to support Web1 was revolutionary, users
quickly realized the limitations of a Read-Only web. This model requires
content creators to host their web pages, leading to only those with
specialized knowledge being able to create content for the World Wide Web.
This meant that creating content for Web1, where businesses created most
content, was highly inaccessible. Furthermore, the lack of communication
between a content creator and the end-users consuming the content made it
difficult for the end-users to engage with the content meaningfully. To
address these limitations, Web2 was introduced to make the web more dynamic
and interactive.

###  II-B Web2

Web2 differentiates itself from Web1 by allowing for dynamic user-generated
content. By introducing new technologies such as databases, Web2 allows for
developing more complex web applications compared to the static web pages from
Web1 [10]. These dynamic applications allowed users to interact with content
on the web in new ways.

Figure 3: A high level overview of Web2’s Read-Write data model

Figure 3 gives a high-level view of the Web2 model. This model builds on Web1,
keeping ideas such as hypertext; however, it now allows users to store data in
databases controlled by the web applications, allowing users to create their
content and interact with the web on a deeper level.

Because Web2 allows for interactivity, this paradigm is called the Read-Write
web, where users can read the data just like in Web1 but can also write their
data for web applications, a bidirectional data flow; users can send and
receive data from applications. To handle the bidirectional data flow, more
complex forms of storing data, such as XML and RSS, were created to support
Web2. Web2 changes the focus of the web away from companies or site publishers
and onto communities. With the rise of Web2, we begin to see new types of
applications such as blogs, wikis, and social media.

While Web2 improved the user experience over Web1 [11] [7] [10], it came with
new security challenges. Web applications were now storing users’ data,
putting these applications at risk of cyberattacks. This also decreased
privacy as users now shared personal data with Web2 applications. Users also
quickly noticed that the siloed nature of Web2 applications meant that data
was often replicated across multiple applications, increasing a user’s attack
surface and making updating information across platforms inefficient. While
many of these problems were identified in the early days of Web2, they still
exist in modern Web2 applications. Researchers began exploring new models to
make the web even more versatile and mitigate the limitations presented in the
Web2 paradigm.

##  III Web3

###  III-A What is Web3

As early as 2006, researchers noticed the limitations of Web2. To solve these
problems, they began to propose a new paradigm of a semantic web[2] [9]. The
semantic web expands on Tim Burners-Lee’s original ideas of linking documents
to each other through hyperlinks, but rather than linking documents,
individual pieces of data are linked to each other.

To highlight the benefits of a semantic web, take an end-user of multiple Web2
social media applications. In each of these applications, the user has a short
bio describing their interests. If the user finds a new interest, they must
log in to each social media application and update each bio. This is time-
consuming and inefficient since users must update the same data across
multiple sites. The semantic web allows the user to update the data once and
have that change reflected across all platforms on the web.

This new paradigm reduced the data replication in Web2 and gave users more
control over their data. However, Web2 applications are currently centralized
and siloed off from one another, making it difficult to share data across
platforms. Thus, implementing a large-scale semantic web requires
decentralization.

As cryptocurrency gained popularity and evolved, its underlying blockchain
technology evolved in such a way that blockchain could provide a decentralized
backbone for a large-scale semantic web [13] [8]. Cryptocurrency, through the
use of blockchain, allows for a trustless, decentralized exchange of data. The
same principle required by semantic web, making blockchain a viable candidate
to support the semantic web.

Like the web, blockchain has evolved, adding more dynamic features [6]. Early
blockchain projects such as Bitcoin were only focused on allowing users to
transact cryptocurrencies. The next generation of blockchain, such as
Ethereum, allowed users to create smart contracts, code that could be executed
on a distributed state machine. With the dawn of generation 2 blockchains,
developers began to combine semantic web ideas with decentralized smart
contracts. Gavin Wood, one of the co-founders of Ethereum, first referred to
this new paradigm as Web3 in 2014.

In Web 2, users relied on various third-party service providers to store and
maintain their data. For example, each platform must keep a copy of the user’s
bio in the social media use case. With Web3, users can store their bio on a
public blockchain, such as Ethereum, via a smart contract, allowing each
social media application to access this data. Thus, the user only needs to
update their bio on the blockchain once and have that change reflected across
all the social media platforms. In this Read-Write-Own model, the user
controls their data and stores it on a blockchain, unlike Web2, where the
platforms own the user’s data and are responsible for storage. Figure 4
highlights the Read-Write-Own model.

Figure 4: A high level overview of the Web3 Read-Write-Own data model.

Table I provides a summary comparison of the various generations of the web;
more detailed comparisons can be found in [11] [7] [4].

TABLE I: Comparison of Web1, Web2, and Web3

| Web1 | Web2 | Web3
---|---|---|---
Data Model | Read | Read Write | Read Write Own
Key Technologies | URL, HTTP, Web Server, Web Browser | Databases | Semantic Web, Blockchain
Data Formats | HTML | XML, RSS | RDF, RDFS, OWLS
Content | Static Web Pages | Web Applications | Smart Applications
Context | Centralized | Centralized | Decentralized
Core Focus | Company | Community | Individual
Information Flow | None | User-Application (Bidirectional) | Multi-directional
Data Owner | Site Publisher | Web Application Platform | Users

###  III-B Benefits of Web3

One of the immediate benefits of Web3’s Read-Write-Own paradigm is that users
are in control of their data. Thus, users can restrict who has access to their
data and when access is granted. This is in contrast to Web2, where once users
give their data to a platform, the platform is free to use that data in any
manner, including selling the user’s data to data brokers.

Likewise, in Web2, if a user shares their data with an application, there is
no guarantee that the user will be able to remove their data from the
application. Users must trust that third-party application providers securely
handle their data and will remove it when requested. In contrast, in Web3, the
user controls the data and can grant or revoke an application’s access to
their data anytime. Furthermore, users can utilize blockchain to record which
applications have access to their data and what data these applications have
access to, allowing them to understand their attack surface and risk better.

In Web3, users no longer need to trust third-party applications to keep their
data secure. In Web2, a security-conscious user may review SOC audits or other
security reports to ensure that an application is taking steps to keep data
secure. However, in Web3, since the user owns their data, they can implement
their controls to ensure that the data is secured. While this may be daunting
for individual users, security-conscious businesses can greatly benefit from
this paradigm.

Another advantage of the Web3 paradigm is that users no longer need to
replicate data across multiple applications. In Web2, a user may need to enter
the same data into multiple applications, for example, updating their bio in
multiple social media applications. With Web3, the user only needs to update
this data one time. This makes interacting with multiple platforms on the web
more streamlined, as data no longer needs to be replicated. For enterprises
that need to manage complex relationships between applications, simplifying
how data is entered can reduce costs and increase efficiencies.

Likewise, since users no longer need to replicate their data in multiple
places, this reduces a user’s attack surface. In Web2, attackers targeting a
user must find one weak application where a user has stored sensitive
information to successfully steal that user’s data. In Web3, the user only
needs to worry about securing one data source.

Since applications no longer store data, the user and the provider’s security
risk is reduced. In Web2, an application accumulates data from many users,
centralizing this data into one storage system, such as a SQL database. Since
this storage system contains the data for many users of a centralized
application, it makes it a high-value target for attackers. They only need to
breach the storage system once to get the data for many users. In contrast, in
Web3, the data is decentralized; thus, attackers no longer have a single point
of attack, making it more difficult to breach all of the users’ data.
Application providers can reduce threat levels and liability by switching to
Web3 and decentralizing users’ data. It is critical to note that Web3 must be
implemented correctly and securely for applications to experience a risk
reduction; if implemented incorrectly, applications may expose themselves to
higher levels of risk.

Likewise, in a Web3 model, service providers no longer need to store users’
data, which allows applications to spend less on storage. This cost reduction
allows developers to focus their resources elsewhere to improve the
application. In addition, since the storage is passed off to users, Web3 smart
applications no longer need to worry about data storage when scaling the
application.

Overall, if implemented properly, Web3 can provide several benefits to both
users and applications. These benefits include:

  * •

Giving users control of their data

  * •

increasing users’ privacy and security,

  * •

reducing data replication,

  * •

reducing the risk of cyberattacks on platforms,

  * •

and improving the developer experience.

While Web3 offers many benefits, it has struggled to become mainstream due to
several drawbacks of implementing the technology.

##  IV Limitations of Web3

Just as Web2’s improvements to Web1 came with a new set of challenges, Web3
presents its own unique set of issues, including scalability, technological
barriers, a desire for centralization, and an unwillingness to adopt the
paradigm by large applications.

A consistent challenge with blockchain technology has been the inability to
scale effectively. Even more modern generation 3 blockchains fail to scale to
the size necessary to support a global Web3 fully. However, for smaller
enterprises, the web’s modern blockchains can handle a more limited number of
transactions, making Web3 valuable for businesses with local intranets. Web3
designers need to work to implement blockchain in a scalable manner to
accommodate the entire web. Similarly, Web3 developers must know the scale
limitations and design their applications to account for this.

In addition to limited transaction throughput, blockchains often add cost per
transaction. This additional cost adds an extra layer of decision-making to
application developers. In some applications, the cost per transaction may
offset or even eclipse the savings the developer experienced by reducing
storage costs. Developers should carefully decide how to manage this cost to
make more accessible Web3 applications, and developers should avoid passing
this cost to users when possible.

Web3 also proves difficult for non-technical users to utilize securely. In
Web2, users trust that the application securely stores and maintains data.
However, Web3 moves this burden to the users. Web2 development teams have
security specialists who understand the risk to the user’s data and have the
resources to implement controls to mitigate risk. In Web3, the average user
does not understand security to the depth necessary to secure sensitive data.
This lack of knowledge may put users’ data at risk, whereas a Web2 application
would have provided some protection to the user. Since businesses have the
resources to hire security experts, this challenge of Web3 can be overcome in
an enterprise environment. Web3 developers should be aware of the security
maturity of their users and design safeguards to guide users in utilizing
smart applications securely.

While technologists often hailed decentralization as a revolutionary idea,
users can drift towards a centralized model in practice. In 2019, Raman et al.
[12] examined the decentralized application Mastodon. Mastodon is a social
media app like Twitter that relies on decentralized instances to host content.
In this study, the researchers found that users, the infrastructure, and the
content drove the application towards a more centralized model where only a
few instances hosted most of the data—the pressure for applications to become
centralized challenges developers trying to create a fully decentralized Web3.
It is important to remember that currently most web users are only familiar
with the centralization of Web2. Developers should understand this tendency of
users to gravitate towards centralization and design systems to guide users
towards a decentralized model.

A final challenge to Web3 is the unwillingness of large applications to adopt
a decentralized web. Tech giants such as Google or Amazon enjoy many benefits
by being a large, centralized data source. They can aggregate and sell users’
data and perform analytics to understand their user base better. It is
unlikely that these companies will want to switch to Web3 since they will lose
control of the data. For Web3 to become viable, users must pressure these
companies into switching to a decentralized model. Because aggregating user
data can drive profits, these centralized applications would only give up
control of the user’s data with extreme pressure from their user base.

While Web3 offers benefits over Web2, this model comes with its challenges.

  * •

Since it relies on a blockchain, Web3 is difficult to scale and can be cost-
inefficient.

  * •

Web3 can be challenging for non-technical users to implement and utilize
securely.

  * •

Internal pressures from Web3 applications and their users can lead toward
centralization.

  * •

The unwillingness of existing centralized applications to decentralize makes
achieving a fully distributed Web3 challenging.

While many of these limitations can be overcome in a business environment with
proper resources, developers must work to make Web3 available to all users.

##  V Conclusion

Since the dawn of the World Wide Web, developers have changed the architecture
to improve the user experience. Developers continue to innovate, starting with
simple web pages in Web1 and evolving to dynamic web applications in Web2 and
then to smart applications in Web3. Each new generation of the web
incrementally improves on the previous generation. However, as the web
evolves, new challenges arise in each generation. While many of the challenges
of Web2 have yet to be overcome, the benefits of Web2 greatly outweigh the
limitations that have allowed Web2 to flourish. Currently, the impediments to
Web3 have prevented it from gaining widespread adoption.

In future work, we plan to expand on our existing blockchain system [5] to
address the limitations that Web3 presents. If implemented carefully, Web3 can
offer users substantial privacy benefits while reducing the burden of storing
and maintaining users’ data on service providers.

With this work, we want to clarify Web3 by highlighting its history and
evolution. This work outlines the benefits Web3 systems can provide users
while showing the challenges developers face when implementing such a system.
Web3’s Read-Write-Own data paradigm is the next step in the evolution of the
web, and while this model comes with its obstacles, developers can work to
mitigate the risks and make Web3 more accessible.

## References

  * [1] Google trends.
  * [2] C. Bizer.  The emerging web of linked data.  IEEE Intelligent Systems, 24(5):87–92, 2009.
  * [3] Tim Burners-Lee.  Information management: A proposal.
  * [4] Bin Cao, Zheng Yan, and Xu Xia.  Web3.  IEEE Communications Magazine, 61(8):18–19, 2023.
  * [5] Collin Connors and Dilip Sarkar.  Pbl: System for creating and maintaining personal blockchain ledgers.  Arxiv, 5 2023.
  * [6] Collin Connors and Dilip Sarkar.  Survey of prominent blockchain development platforms.  Journal of Network and Computer Applications, page 103650, 2023\.
  * [7] S. Filipcic.  Web3 and daos: An overview of the development and possibilities for the implementation in research and education.  2022 45th Jubilee International Convention on Information, Communication and Electronic Technology (MIPRO), 2022.
  * [8] Shaoyong Guo, Fan Zhang, Song Guo, Siya Xu, and Feng Qi.  Blockchain-assisted privacy-preserving data computing architecture for web3.  IEEE Communications Magazine, 61(8):28–34, 2023.
  * [9] James Hendler.  Web 3.0: The dawn of semantic search.  Computer, 43(1):77–80, 2010.
  * [10] S. Murugesan.  Understanding web 2.0.  IT Professional, 9(4):34–41, 2007.
  * [11] Keshab Nath, Sourish Dhar, and Subhash Basishtha.  Web 1.0 to web 3.0 - evolution of the web and its various challenges.  2014 International Conference on Reliability Optimization and Information Technology (ICROIT), 2014.
  * [12] Aravindh Raman, Sagar Joglekar, Emiliano De Cristofaro, Nishanth Sastry, and Gareth Tyson.  Challenges in the decentralised web.  Proceedings of the Internet Measurement Conference, 2019.
  * [13] Javad Zarrin, Hao Wen Phang, Lakshmi Babu Saheer, and Bahram Zarrin.  Blockchain for decentralization of internet: Prospects, trends, and challenges.  Cluster Computing, 24(4):2841–2866, 2021.

|  Collin Connors received B.Sc. degrees in computer science and mathematics
from the University of Miami, FL, in 2020. He is currently pursuing a Ph.D.
degree with the Computer Science Department. His research interests include
blockchain technologies, cybersecurity, and machine learning.
---|---
|  Dilip Sarkar (SM’96) received the B.Tech.(Hons.) degree in electronics and
electrical communication engineering from the Indian Institute ofTechnology at
Kharagpur, Kharagpur, India, in 1983, the M.S. degree in computer science from
the Indian Institute of Science, Bengaluru, India, in 1984, and the Ph.D.
degree in computer science from the University of Central Florida, Orlando, in
1988. He is an Associate Professor of Computer Science at the University of
Miami, Coral Gables, FL, USA. His research interests include concurrent
transport protocols, parallel and distributed processing, neural networks,
wireless sensor networks and their applications, and the security of virtual
machines. In these areas, he has guided several theses and has authored
numerous papers. He served as a Guest Editor of a Special Issue on Computer
Communications on Concurrent Multipath Transport. For many years, he has
served on ICC, Globecom, ICME, ICCCN, and INFOCOM program committees.
---|---