# Stability and instability of the quasilinear Gross–Pitaevskii dark solitons

Erwan Le Quiniou111Univ. Lille, CNRS, Inria, UMR 8524 - Laboratoire Paul
Painlevé, Inria, F-59000 Lille, France.
E-mail<EMAIL_ADDRESS>
###### Abstract

We study a quasilinear Schrödinger equation with nonzero conditions at
infinity. In previous works, we obtained a continuous branch of traveling
waves, given by dark solitons indexed by their speed. Neglecting the
quasilinear term, one recovers the Gross–Pitaevskii equation, for which the
branch of dark solitons is stable. Moreover, Z. Lin showed that the
Vakhitov–Kolokolov (VK) stability criterion (in terms of the momentum of
solitons) holds for general semilinear equations with nonvanishing conditions
at infinity.

In the quasilinear case, we prove that the VK stability criterion still
applies, by generalizing Lin’s arguments. Therefore, we deduce that the branch
of dark solitons is stable for weak quasilinear interactions. For stronger
quasilinear interactions, a cusp appears in the energy-momentum diagram,
implying the stability of fast waves and the instability of slow waves.

Keywords: Quasilinear Schrödinger equation, Gross–Pitaevskii equation,
traveling waves, dark solitons, nonzero conditions at infinity, orbital
stability.

2020 MSC: 35Q55, 35C07, 35C08, 35J62, 35B35, 34L40, 34L05, 35Q60, 82D50,
34B40.

##  1 Introduction

We consider the following quasilinear Gross–Pitaevskii equation in one
dimension

$i\partial_{t}\Psi+\partial_{xx}\Psi+\Psi(1-|\Psi|^{2})+\kappa\Psi\partial_{xx}(1-|\Psi|^{2})=0,\quad\text{
in }\mathbb{R}\times\mathbb{R},$ (1)

where $\kappa\in\mathbb{R}$ is a parameter measuring the strength of the
quasilinear term, and $\Psi:\mathbb{R}\times\mathbb{R}\to\mathbb{C}$,
satisfies the nonzero conditions at infinity

$\lim_{|x|\to\infty}|\Psi(x,\cdot)|=1.$ (2)

Equation (1) was introduced as a generalization of the case $\kappa=0$ also
known as the Gross-Pitaevskii equation, which is a standard model in nonlinear
optics and quantum hydrodynamics [19, 14, 18]. For instance, in the context of
optical fibers in [21], they obtained (1) with $\kappa\geq 0$ to model weak
nonlocal interactions in the refracting index of the beam of light. We refer
to [20], for other relevant weakly nonlocal models. Equation (1) was also
obtained with $\kappa\leq 0$ in [22] to describe superfluid films with surface
tension.

Equation (1) is a particular case of the following quasilinear Schrödinger
equation in $\mathbb{R}^{d}$

$i\partial_{t}\Psi+\Delta\Psi+\Psi F(|\Psi|^{2})-\kappa\Psi
h^{\prime}(|\Psi|^{2})\Delta h(|\Psi|^{2})=0,\quad\text{ in
}\mathbb{R}^{d}\times\mathbb{R},$ (3)

with $F(y)=1-y$ and $h(y)=1-y$, $\kappa\in\mathbb{R}$. To our knowledge, (3)
was only considered with vanishing conditions by the mathematical community.
Studying (3) with $d=1$, I. D. Iliev and K. P. Kirchev obtained in [17] a
continuous branch of solitary-wave solutions of the form
$\Psi(x,t)=u_{\omega}(x)e^{i\omega t},$ and showed that the wave is stable
whenever $\partial_{\omega}||u_{\omega}||_{L^{2}(\mathbb{R})}^{2}>0$ and
unstable when $\partial_{\omega}||u_{\omega}||_{L^{2}(\mathbb{R})}^{2}<0$. In
arbitrary space dimension, M. Colin, L. Jeanjean, and M. Squassina obtained in
[10] the existence and stability or instability by blowup of the ground state
solitary-wave for (3) with power law nonlinearities. In the semilinear regime
($\kappa=0$ in (3)), we recover

$i\partial\Psi+\Delta\Psi+F(|\Psi|^{2})\Psi=0,\quad\text{ in
}\mathbb{R}^{d}\times\mathbb{R}.$ (4)

With vanishing conditions at infinity, M. I. Weinstein showed in [26] the
stability of the solitary wave solution to (4) satisfying a ground state
property using Lyapunov functionals. Assuming $F(\rho_{0})=0$ and that $\Psi$
satisfies the nonzero conditions at infinity

$\lim_{|x|\to\infty}|\Psi(x,t)|=\rho_{0},\quad\text{ for all }t\in\mathbb{R},$
(5)

Z. Lin investigated in [23] in one dimension of space, the stability of
nonvanishing traveling-wave solutions to (4) of the form,

$\Psi_{c}(x,t)=u_{c}(x-ct),$ (6)

by performing the analysis on the hydrodynamical variables
$\eta,\partial_{x}\theta$ with $\Psi=\sqrt{1-\eta}e^{i\theta}$. In particular,
he rigorously proves the VK criterion [25] using the framework of abstract
Hamiltonian systems presented in the celebrated work of M. Grillakis, J.
Shatah, and W. Strauss [15]. We also refer to the work of D. Chiron in [8]
which constitutes, to our knowledge, the most complete description of the
existing orbital stability results for the nontrivial traveling-waves
solutions of equation (4) and in particular deals with solutions vanishing at
some point. In this work, we show that the ideas of Lin can be generalized to
investigate the stability of traveling-wave solutions to the quasilinear
equation (1).

The quasilinear Gross–Pitaevskii equation (1) is Hamiltonian and formally
preserves the energy

$E_{\kappa}(\Psi(\cdot,t))=\int_{\mathbb{R}}|\partial_{x}\Psi(x,t)|^{2}dx+\frac{1}{2}\int_{\mathbb{R}}\left(1-|\Psi(x,t)|^{2}\right)^{2}dx-\frac{\kappa}{2}\int_{\mathbb{R}}\left(\partial_{x}|\Psi(x,t)|^{2}\right)^{2}dx,$
(7)

as well as the (renormalized) momentum, whenever
$\inf_{x\in\mathbb{R}}|\Psi(x,t)|>0$

$P(\Psi(\cdot,t))=-\int_{\mathbb{R}}\Re(i\partial_{x}\Psi(x,t)\overline{\Psi(x,t)})\left(1-\frac{1}{|\Psi(x,t)|^{2}}\right)dx.$
(8)

Because of the nonzero conditions at infinity (2), the natural energy space to
study (1) is (see [13])

$\mathcal{X}(\mathbb{R})=\\{u\in H^{1}_{\rm
loc}(\mathbb{R};\mathbb{C}):u^{\prime}\in L^{2}(\mathbb{R}),1-|u|^{2}\in
L^{2}(\mathbb{R})\\}.$ (9)

We endow $\mathcal{X}(\mathbb{R})$ with two metrics, the natural energy
distance, which writes, for $\psi$, $\tilde{\psi}\in\mathcal{X}(\mathbb{R})$

$d_{\mathcal{X}}(\psi,\tilde{\psi})=||\partial_{x}\psi-\partial_{x}\tilde{\psi}||_{L^{2}(\mathbb{R})}+|||\psi|-|\tilde{\psi}|||_{L^{2}(\mathbb{R})}+|\psi(0)-\tilde{\psi}(0)|,$
(10)

and the hydrodynamical distance defined for nonvanishing functions in
$\mathcal{X}(\mathbb{R})$ i.e. for $\psi=\sqrt{1-\eta}e^{i\theta}$ and
$\tilde{\psi}=\sqrt{1-\tilde{\eta}}e^{i\tilde{\theta}}$ in
$\mathcal{X}(\mathbb{R})$ with $\inf_{\mathbb{R}}|\psi|>0$ and
$\inf_{\mathbb{R}}|\tilde{\psi}|>0$, we define

$d_{\mathrm{hy}}(\psi,\tilde{\psi})=||\eta-\tilde{\eta}||_{H^{1}(\mathbb{R})}+||\partial_{x}\theta-\partial_{x}\tilde{\theta}||_{L^{2}(\mathbb{R})}+\mathrm{Arg}\left(\frac{\psi(0)}{\tilde{\psi}(0)}\right),$
(11)

where $\mathrm{Arg}$ is a continuous determination of the argument of complex
numbers defined near $1$. In a previous work with A. de Laire [13], we were
interested in the classification and stability properties of the so-called
finite energy traveling-wave solutions to (1). Namely, studying solutions to
(1) of the form (6) which represents a traveling wave with profile
$u_{c}=u_{c,\kappa}\in\mathcal{X}(\mathbb{R})$ propagating at speed
$c\in\mathbb{R}$. Hence, the profile $u_{c,\kappa}$ satisfies

$-icu_{c,\kappa}^{\prime}+u_{c,\kappa}^{\prime\prime}+u_{c,\kappa}(1-|u_{c,\kappa}|^{2})-\kappa
u_{c,\kappa}\left(|u_{c,\kappa}|^{2}\right)^{\prime\prime}=0.$
(TW$(c,\kappa)$)

Taking the complex conjugate of equation (TW$(c,\kappa)$), we can assume
without loss of generality that $c\geq 0$. We obtained a continuous branch of
$\mathcal{C}^{2}(\mathbb{R})$-solutions for every
$(c,\kappa)\in\mathcal{D}=[0,\sqrt{2})\times(-\infty,1/2)$. A (partial)
version of our result writes

######  Theorem 1.1 (Theorem 1, Proposition 3.8 in [13]).

Let $\kappa\in(-\infty,1/2)$ and $c\geq 0$.

  1. (i)

If $c\geq\sqrt{2}$ and
$u_{c,\kappa}\in\mathcal{X}(\mathbb{R})\cap\mathcal{C}^{2}(\mathbb{R})$
satisfies (TW$(c,\kappa)$), then $u_{c,\kappa}$ is trivial, i.e. there exists
$\varphi\in\mathbb{R}$ such that

$u_{c,\kappa}(x)=e^{i\varphi},\quad\text{ for all }x\in\mathbb{R}.$

  2. (ii)

If $0\leq c<\sqrt{2}$ so that $(c,\kappa)\in\mathcal{D}$, then there exists a
nonconstant function
$u_{c,\kappa}\in\mathcal{X}(\mathbb{R})\cap\mathcal{C}^{2}(\mathbb{R})$ called
dark soliton that satisfies (TW$(c,\kappa)$). Moreover, this solution is
unique up to translation and phase change, i.e. any other solution
$\phi\in\mathcal{X}(\mathbb{R})\cap\mathcal{C}^{2}(\mathbb{R})$ satisfies for
some $(\tau,\varphi)\in\mathbb{R}\times\mathbb{R}$

$\phi(x)=u_{c,\kappa}(x-\tau)e^{i\varphi},\quad\text{ for all
}x\in\mathbb{R}.$

Additionally, the function $(c,\kappa,x)\mapsto u_{c,\kappa}(x)$ is smooth in
$\mathcal{D}\times\mathbb{R}$ (where the derivative at $c=0$ is in the sense
of Dini) and satisfies the following decay estimate: for every multi-index
$\alpha=(\alpha_{1},\alpha_{2},\alpha_{3})\in\mathbb{N}^{3}$ and
$D^{\alpha}=\partial_{c}^{\alpha_{1}}\partial_{\kappa}^{\alpha_{2}}\partial_{x}^{\alpha_{3}}$,
there exist $A,C>0$ continuously depending on $(c,\kappa)$ such that

$|D^{\alpha}\partial_{x}u_{c,\kappa}|+|D^{\alpha}(1-|u|^{2})|\leq
Ae^{-Cx},\quad\text{ for all }x\in\mathbb{R}.$ (12)

Also $|u_{c,\kappa}|$ is even with $|u_{c,\kappa}|^{\prime}>0$ in
$(0,\infty)$. Finally we have the bounds $1>|u_{c,\kappa}(x)|\geq c/\sqrt{2}$
for all $x\in\mathbb{R}$ with equality at $x=0$, so that
$\inf_{\mathbb{R}}|u_{c,\kappa}|>0$ if and only if $c>0$.

In the case $c=0$, the traveling-wave $u_{0,\kappa}$ vanishes at $x=0$ and
$u_{0,\kappa}$ is also called black soliton or kink soliton [7, 4]. Showing
the stability of the black soliton requires other techniques (see [5, 8])
since the differentiable momentum (8) is not real-valued in that case [11].
Theorem 1 in [13] also show the existence of supersonic solution with
$(c,\kappa)\in(\sqrt{2},\infty)\times(1/2,\infty)$, but in that case, the
linearized Hamiltonian operator around the traveling wave has essential
spectrum in $(-\infty,\lambda)$ for some $\lambda>0$, thus the analysis
presented in this work, which is based on [23], is not suited to study the
stability of these waves. In its weak formulation, equation (TW$(c,\kappa)$)
can be recast as

$\displaystyle
c\frac{d}{ds}P(u+sh)\rvert_{s=0}=\frac{d}{ds}E_{\kappa}(u+sh)\rvert_{s=0},$

which are the Euler–Lagrange equations associated with the minimization
problem

$\mathcal{E}_{\kappa}(\mathfrak{q})=\inf\\{E_{\kappa}(u):u\in\mathcal{X}(\mathbb{R}),\inf_{x\in\mathbb{R}}|u(x)|>0,P(u)=\mathfrak{q}\\}.$
(13)

When $\kappa\leq 0$ so that (13) defines nonnegative function, we showed
stability and a ground state property for the dark soliton using a
concentration-compactness argument, generalizing the approach used for the
Gross–Pitaevski equation (equation (1) with $\kappa=0$) by F. Bethuel P.
Gravejat and J. C. Saut in [4] (see also [12]).

######  Theorem 1.2 (Theorem 1.10–1.11 and Proposition 6.12 in [13]).

Let $\kappa\in\mathbb{R}$ and $\mathcal{E}_{\kappa}$ be given by (13), then
the following statements hold.

  1. (i)

If $\kappa>0$, then $\mathcal{E}_{\kappa}(\mathfrak{q})=-\infty,$ for all
$\mathfrak{q}\in\mathbb{R}.$

  2. (ii)

If $\kappa\leq 0$, then there exist constants $\mathfrak{q}_{\kappa}^{*}>0$,
$c_{\kappa}^{*}\in[0,\sqrt{2})$ and a decreasing map
$\mathfrak{c}:[0,\mathfrak{q}_{\kappa}^{*})\to(c_{\kappa}^{*},\sqrt{2}]$
satisfying $P(u_{\mathfrak{c}(\mathfrak{q}),\kappa})=\mathfrak{q}$. Then also,
for every
$\mathfrak{q}\in(-\mathfrak{q}_{\kappa}^{*},\mathfrak{q}_{\kappa}^{*})$ the
infimum in (13) is attained, with
$\mathcal{E}_{\kappa}(\mathfrak{q})=E_{\kappa}(u_{\mathfrak{c}(|\mathfrak{q}|),\kappa})$,
the dark soliton $u_{\mathfrak{c}(|\mathfrak{q}|),\kappa}$ in the unique
minimizer up to invariances and for every
$|\mathfrak{q}|>\mathfrak{q}_{\kappa}^{*}$, we have
$\mathcal{E}_{\kappa}(\mathfrak{q})=E_{\kappa}(u_{0,\kappa})$ and this infimum
is not attained. Finally, for every $c\in(c_{\kappa}^{*},\sqrt{2})$ the dark
soliton $u_{c,\kappa}$ is orbitally stable for $d_{\mathcal{X}}$.

Theorem 1.2 implies that the dark solitons $u_{c,\kappa}$ with
$0<c<c_{\kappa}^{*}$ does not appear as a global minimum of the energy at
fixed momentum. Thus other methods must be employed to ensure stability. We
display the energy-momentum of the dark soliton parametrized by $c$ in the
left panel of Figure 1–2, it coincides with the function
$\mathcal{E}_{\kappa}(\cdot)$ up to $\mathfrak{q}_{\kappa}^{*}$. By orbital
stability for a distance $d$, we mean stability with respect to the orbit
described by translations and rotations of the dark soliton $u_{c,\kappa}$.

######  Definition 1.3.

  1. (i)

We say that $u_{c,\kappa}$ is orbitally stable for $d$ if for all
$\varepsilon>0$ if there exists $\delta>0$ with the following property: for
every $\phi_{0}\in H^{s}(\mathbb{R})$ with $s>2+1/2$, such that
$d(u_{c,\kappa},u_{c,\kappa}+\phi_{0})\leq\delta$, then, denoting
$T=T_{\phi_{0}}$ the maximal time of existence of the solution
$\Psi(x,t)=u_{c,\kappa}(x)+\phi(x,t)$ to (1) with
$\phi\in\mathcal{C}((-T,T);H^{s}(\mathbb{R}))$, we have

$\sup_{t\in(-T,T)}\inf_{(s,\varphi)\in\mathbb{R}^{2}}d(u_{c,\kappa}(\cdot-s)e^{i\varphi},u_{c,\kappa}(\cdot)+\phi(\cdot,t))\leq\varepsilon.$
(14)

  2. (ii)

Assuming furthermore that the solution is global, we say that $u_{c,\kappa}$
is orbitally unstable for $d$ if there exists $\varepsilon_{0}>0$ such that
for every $\delta>0$ we can find $\phi_{0}\in H^{s}(\mathbb{R})$ with
$s>2+1/2$ so that $d(u_{c,\kappa},u_{c,\kappa}+\phi_{0})\leq\delta$ and the
solution $\Psi(x,t)=u_{c,\kappa}(x)+\phi(x,t)$ to (1) with
$\phi\in\mathcal{C}(\mathbb{R};H^{s}(\mathbb{R}))$ satisfies the following:
there exists a constant $t=t(\delta,\epsilon_{0})$ such that

$\inf_{(s,\varphi)\in\mathbb{R}^{2}}d(u_{c,\kappa}(\cdot-s)e^{i\varphi},u_{c,\kappa}(\cdot)+\phi(\cdot,t))>\varepsilon_{0}.$
(15)

To our knowledge, the results concerning the well-posedness of (1) are limited
to local well-posedness (see Section 2). Therefore to make sense of orbital
instability, we place ourselves in the case where blowup does not occur in the
sense that the solution $\Psi$ to (1) starting at time $t=0$ close to the dark
soliton in Definition 1.3 (ii) exists for all $t\in\mathbb{R}$.

Our main result is the rigorous justification of the VK stability criterion
for equation (1).

######  Theorem 1.4.

Let $\kappa<1/2$ and $0<c<\sqrt{2}$. Assume that
$u_{c,\kappa}\in\mathcal{X}(\mathbb{R})\cap\mathcal{C}^{2}(\mathbb{R})$ is, up
to invariances, the unique solution to (TW$(c,\kappa)$) given by Theorem 1.1.
Then

$c\mapsto P(u_{c,\kappa})\in\mathcal{C}^{1}((0,\sqrt{2})),$

and the traveling-wave solution $u_{c,\kappa}$ is orbitally stable for
$d_{\mathcal{X}}$ if

$\frac{dP(u_{c,\kappa})}{dc}<0,$ (16)

whereas it is unstable for $d_{\mathcal{X}}$ if

$\frac{dP(u_{c,\kappa})}{dc}>0.$ (17)

As explained in Section 2, conditions (16) and (17) amount to
$d^{\prime\prime}(c)>0$ and $d^{\prime\prime}(c)<0$ respectively, where
$d:c\mapsto E_{\kappa}(u_{c,\kappa})-cP(u_{c,\kappa})$ is the scalar function
characterizing the stability in [15]. In Section 5 in [13], we were able to
compute explicitly the function $(c,\kappa)\mapsto P(u_{c,\kappa})$ and
analyze its variations. Hence, we can restate the stability result in Theorem
1.4 in terms of the propagation speed $c\in(0,\sqrt{2})$ of the dark soliton
follows.

######  Corollary 1.5.

There exists $\kappa_{0}<0$ with $\kappa_{0}\approx-3.636$ such that the two
following statements hold

  1. (i)

If $\kappa\in[\kappa_{0},1/2)$, then $u_{c,\kappa}$ is orbitally stable for
every $c\in(0,\sqrt{2})$.

  2. (ii)

If $\kappa<\kappa_{0}$, then there exists $\tilde{c}_{\kappa}\in(0,\sqrt{2})$
such that $u_{c,\kappa}$ is orbitally unstable if
$c\in(0,\tilde{c}_{\kappa})$, whereas it is orbitally stable if
$c\in(\tilde{c}_{\kappa},\sqrt{2})$.

When $\kappa_{0}\leq\kappa<1/2$ we recover that the whole branch is stable as
in the semilinear case $\kappa=0$ (see Theorem 4 in [4]). In the left and
center panels of Figure 1 and Figure 2 respectively, we displayed the momentum
of the dark soliton as a function of $c$ which yielded the stability
conjecture proven in Corollary 1.5. Figure 2 also depicts the value of
${c}_{\kappa}$ which satisfies
$\partial_{c}P(u_{\tilde{c}_{\kappa},\kappa})=0$ and that we approach using
Newton’s method.

\begin{overpic}[scale={0.4},trim=0.0pt 0.0pt 0.0pt 0.0pt,clip]{figures/EPDmin} \put(25.0,68.0){\vector(3,1){7.0}} \put(6.0,65.0){{$c\to 0$}} \put(20.0,10.0){\vector(-3,-1){10.0}} \put(22.0,10.0){{$c\to\sqrt{2}$}} \put(75.0,68.0){\vector(-3,1){7.0}} \put(78.0,66.0){{$c_{\kappa}^{*}$}} \dottedline{2}(68,70.5)(68,5.2) \put(65.0,-0.5){$\mathfrak{q}_{\kappa}^{*}$} \dottedline{2}(5,71.5)(70,71.5) \put(-12.0,70.0){3.748} \put(87.0,79.0){\vector(-1,0){7.0}} \put(90.0,78.0){$\tilde{c}_{\kappa}=0.473$} \put(92.0,8.0){$P(u_{c,\kappa})$} \put(2.0,84.0){$E_{\kappa}(u_{c,\kappa})$} \put(36.0,-1.0){$\frac{\pi}{2}$} \put(38.5,5.0){\line(0,1){2.0}} \end{overpic} |  \begin{overpic}[scale={0.4},trim=0.0pt 0.0pt 0.0pt 0.0pt,clip]{figures/PD2} \end{overpic}
---|---
Figure 1: Left panel depicts in grey the energy-momentum diagram of dark solitons $u_{c,\kappa}$ with $\kappa=-50$ it also displays in black the curve $\mathcal{E}_{\kappa}(\cdot)$. Right panel displays the momentum of the dark soliton $u_{c,\kappa}$ with $\kappa=-50$ as a function of the speed $c$. \begin{overpic}[scale={0.4},trim=0.0pt 0.0pt 0.0pt 0.0pt,clip]{figures/EPD2} \put(-14.0,80.0){1.347} \put(4.0,82.0){\line(1,0){3.0}} \dottedline{2}(4,82)(94.5,82) \dottedline{2}(94.5,0)(94.5,82) \put(84.0,74.0){\vector(3,2){9.0}} \put(74.0,68.0){{$c_{\kappa}^{*}=0$}} \put(18.0,9.0){\vector(-3,-1){10.0}} \put(20.0,9.0){{$c\to\sqrt{2}$}} \put(96.0,8.0){$P(u_{c,\kappa})$} \put(0.0,87.0){$E_{\kappa}(u_{c,\kappa})$} \put(93.0,-2.0){$\frac{\pi}{2}$} \put(94.5,4.0){\line(0,1){2.0}} \put(99.8,4.45){\line(1,0){10.0}} \end{overpic} |  \begin{overpic}[scale={0.6},trim=0.0pt 0.0pt 0.0pt 0.0pt,clip]{figures/PD1} \end{overpic} |  \begin{overpic}[scale={0.4},trim=20.075pt 0.0pt 0.0pt 0.0pt,clip]{figures/tildc} \put(0.0,60.0){$\tilde{c}_{\kappa}$} \put(85.0,15.0){$\kappa$} \end{overpic}
---|---|---
Figure 2: Left panel depicts the energy-momentum diagram of the dark soliton
$u_{c,\kappa}$ with $\kappa=-3$. The center panel represents the momentum of
the dark soliton $u_{c,\kappa}$ with $\kappa=-3$ as a function of the speed
$c$. The right panel displays the value of $\tilde{c}_{\kappa}$ depending on
$\kappa$, we put $\tilde{c}_{\kappa}=0$ for $\kappa\geq\kappa_{0}$.

From now on, we fix $\kappa<1/2$ and remove this variable from the subscripts
when no confusion can arise. The analysis will be conducted on the
hydrodynamical formulation of (1). Namely, writing
$\Psi=\sqrt{1-\eta}e^{i\theta}$ with $|\Psi|>0$ in $\mathbb{R}$, then $\eta$
and $v=\theta_{x}$ formally satisfy the following hydrodynamical system with
quantum pressure

$\left\\{\begin{aligned} &\eta_{t}=\partial_{x}(2v(1-\eta)),\\\
&v_{t}=-\partial_{x}\left(\frac{\eta_{xx}}{2(1-\eta)}+\frac{(\eta_{x})^{2}}{4(1-\eta)^{2}}+v^{2}-\eta-\kappa\eta_{xx}\right),\end{aligned}\right.$
(18)

We will check in Section 2 that in its weak form, we can recast (18) as the
Hamiltonian system

$\partial_{t}(\eta,v)=JE^{\prime}(\eta,v),\text{ where
}J=\begin{pmatrix}\partial_{x}&0\\\ 0&\partial_{x}\end{pmatrix},$ (19)

in particular $J$ is not onto as explained in Remark 2.5. Let us rewrite the
energy and momentum in terms of $(\eta,v)$ and recall other useful quantities
formally conserved in time by the solutions to (18)

$\displaystyle E_{\kappa}(\sqrt{1-\eta}e^{i\theta})=E((\eta,v))$
$\displaystyle=\int_{\mathbb{R}}\frac{(\eta_{x})^{2}}{4(1-\eta)}+v^{2}(1-\eta)+\frac{1}{2}(\eta^{2}+\kappa(\eta_{x})^{2}),\quad\text{with
}v=\theta_{x}\quad(\mathbf{energy}),$ (20) $\displaystyle
P(\sqrt{1-\eta}e^{i\theta})=P((\eta,v))$ $\displaystyle=-\int_{\mathbb{R}}\eta
v,\quad\text{with }v=\theta_{x}\quad(\mathbf{momentum}),$ (21) $\displaystyle
N((\eta,v))$
$\displaystyle=-\int_{\mathbb{R}}\eta,\quad(\mathbf{number~{}of~{}particles}),$
(22) $\displaystyle\Theta((\eta,v))$
$\displaystyle=\int_{\mathbb{R}}v,\quad(\mathbf{twisting~{}angle}).$ (23)

When $c>0,$ since the traveling wave
$u_{c}(x-ct)=\sqrt{1-\eta_{c}(x-ct)}e^{i\theta_{c}(x-ct)}$ is smooth and
rapidly decaying (in $(x,t)$), all the computations (18)–(23) become rigorous
and integrating in space (18), we get for all $x\in\mathbb{R}$

$\left\\{\begin{aligned} &-c\eta_{c}(x)=2v_{c}(x)(1-\eta_{c}(x)),\\\
&-cv_{c}(x)=-\left(\frac{\partial_{xx}\eta_{c}(x)}{2(1-\eta_{c}(x))}+\frac{(\partial_{x}\eta_{c}(x))^{2}}{4(1-\eta_{c}(x))^{2}}+v_{c}^{2}-\eta_{c}(x)-\kappa\partial_{xx}\eta_{c}(x)\right),\end{aligned}\right.$
(24)

where $v_{c}=\theta_{x}$. System (24) in its weak form can also be written as

$cP^{\prime}((\eta_{c},v_{c}))=E^{\prime}((\eta_{c},v_{c})),\quad\text{ in
}H^{{-1}}(\mathbb{R})\times L^{2}(\mathbb{R}),$ (25)

so that solving (24) amounts to find critical points of the Hamiltonian
$E-cP.$

The outline of this work follows that of [23]. In Section 2 we check that (18)
is a Hamiltonian system fitting the framework of [15] so that stability
criterion (16) holds for the hydrodynamical distance $d_{\mathrm{hy}}$ using
their result. We deal with the instability result in Section 3. Since $J$ is
not onto, the instability result of [15] does not apply, we instead follow the
construction of the energy-decreasing direction developped in [23]. In Section
4, we deduce the stability and instability for the energy distance
$d_{\mathcal{X}}$ in Theorem 1.4 and we show Corollary 1.5. In Appendix A, we
discuss some extensions of Theorem 1.4.

##  2 Hamiltonian system framework and orbital stability

Let $X=H^{1}(\mathbb{R};\mathbb{R})\times L^{2}(\mathbb{R};\mathbb{R})$ with
the natural scalar product
$\langle(\eta_{1},v_{1}),(\eta_{2},v_{2})\rangle_{X}=\langle\eta_{1},\eta_{2}\rangle_{H^{1}(\mathbb{R})}+\langle
v_{1},v_{2}\rangle_{L^{2}(\mathbb{R})}$. We also introduce the dual space
$X^{*}=H^{{-1}}(\mathbb{R};\mathbb{R})\times L^{2}(\mathbb{R};\mathbb{R})$
with duality product

$\langle\Lambda((\eta_{1},v_{1}),(\eta_{2},v_{2})\rangle=\langle\eta_{1},v_{1}),(\eta_{2},v_{2})\rangle_{X},\quad\text{
for all }(\eta_{1},v_{1}),(\eta_{2},v_{2})\in X,$

where $\Lambda=(1-\partial_{xx},1)$ is an isometric isomorphism between $X$
and $X^{*}$, but we do not identify $X$ to $X^{*}$. Regarding (20), the space
$X$ should be interpreted as the hydrodynamical counterpart of the energy
space $\mathcal{X}(\mathbb{R})$. We have the variational triple $X\subset
L^{2}(\mathbb{R};\mathbb{R})^{2}\subset X^{*}$ and we denote by
$(\cdot,\cdot)_{L^{2}(\mathbb{R})}$ the natural scalar product on
$L^{2}(\mathbb{R})$. For notational convenience, we shall use these products
for $(\eta,v)\in X$ or solely in terms of $\eta\in H^{1}(\mathbb{R})$ or $v\in
L^{2}(\mathbb{R})$, assuming the other coordinate to be zero. In this setting,
straightforward computations implies that $E$ and $P$ are
$\mathcal{C}^{2}$-functional on $X$ with, in $X^{*}$,

$E^{\prime}(\eta,v)=\left(-\frac{\partial_{x}\eta_{x}}{2(1-\eta)}-\frac{\eta_{x}^{2}}{4(1-\eta)^{2}}-v^{2}+\eta+\kappa\partial_{x}\eta_{x},\quad
2v(1-\eta)\right),$ (26)

so that (18) can be recast as the Hamiltonian system (19). For every
$(\delta\eta,\delta v)\in H^{1}(\mathbb{R})\times L^{2}(\mathbb{R})$, we get
in $X^{*}$

$\displaystyle E^{\prime\prime}(\eta,v)(\delta\eta,\delta v)=2\Big{(}$
$\displaystyle-\partial_{x}\left(\frac{1-2\kappa+2\kappa\eta}{4(1-\eta)}(\delta\eta)_{x}\right)-\partial_{x}\left(\frac{\eta_{x}}{4(1-\eta)^{2}}\right)\delta\eta+\frac{\eta_{x}^{2}}{4(1-\eta)^{3}}\delta\eta-v\delta
v,$ (27) $\displaystyle(1-\eta)\delta v-v\delta\eta\Big{)},$ (28)
$P^{\prime}(\eta,v)=-(v,\eta),\quad\text{ and }\quad
P^{\prime\prime}(\eta,v)(\delta\eta,\delta v)=-(\delta v,\delta\eta).$ (29)

Let $\tau$ denote the unitary group of translation on $X$

$\tau(s)(\eta,v)=(\eta,v)(x-s),$ (30)

so that

$\tau^{\prime}(0)=\begin{pmatrix}-\partial_{x}&0\\\
0&-\partial_{x}\end{pmatrix},\quad\text{ and
}\quad\tau^{\prime}(s)=\tau(s)\tau^{\prime}(0)=\tau^{\prime}(0)\tau(s),$

are closed operator in $X$ with domain
$D(\tau^{\prime}(0))=H^{1}(\mathbb{R})\times H^{1}(\mathbb{R})$. Also, letting

$B=\begin{pmatrix}0&-1\\\ -1&0\end{pmatrix}\in\mathcal{L}(X;X^{*}),$ (31)

we get $P((\eta,v))=\langle B(\eta,v),(\eta,v)\rangle/2$ and $JB$ is an
extension of $\tau^{\prime}(0)$. We call bound state a solution to (18) of the
form

$(\eta,v)(x,t)=\tau(ct)(\eta_{c}(x),v_{c}(x)),\quad\text{ for all
}(x,t)\in\mathbb{R}\times\mathbb{R},$ (32)

such that $(\eta_{c},v_{c})\in X$ are critical critical points of $E-cP$ i.e.

$cP^{\prime}((\eta_{c},v_{c}))=E^{\prime}((\eta_{c},v_{c})).$

It will also be of importance to perform spectral analysis on the Hessian

$H_{c}=E^{\prime\prime}((\eta_{c},v_{c}))-cP^{\prime\prime}((\eta_{c},v_{c})),$
(33)

in this regard, a demonstration of every abstract result we use can be found
either in this textbook by C. Cheverry and N. Raymond [6] or in Appendix B of
the book by J.Angulo Pava [1]. Let us recall the three assumptions in [15]
that ensure their stability theorem.

Assumption 1 (Existence of solutions). For each $(\eta_{0},v_{0})\in
H^{s}(\mathbb{R})\times H^{{s-1}}(\mathbb{R})$, there exist $T>0$ and a
solution $(\eta,v)\in\mathcal{C}((-T,T);H^{s}(\mathbb{R})\times
H^{{s-1}}(\mathbb{R}))$ to (18) such that $(\eta,v)(\cdot,0)=(\eta_{0},v_{0})$
and the energy and momentum are conserved in time

$E_{\kappa}((\eta,v)(\cdot,t))=E_{\kappa}((\eta_{0},v_{0})),\quad\text{
and}\quad P((\eta,v)(\cdot,t))=P((\eta_{0},v_{0})).$ (34)

If $(\eta_{c},v_{c})$ is a bound state, it suffices to have the above for
$(\eta_{0},v_{0})$ in a neighborhood of $(\eta_{c},v_{c})$ in
$(H^{s}(\mathbb{R})\times H^{{s-1}}(\mathbb{R}),||\cdot||_{X})$

Assumption 2 (Existence of bound states). There exist constants $c_{1}<c_{2}$
and a $\mathcal{C}^{1}((c_{1},c_{2});X)$ map $c\mapsto(\eta_{c},v_{c})$ with
the following properties.

  1. (i)

$E^{\prime}(\eta_{c},v_{c})=cP^{\prime}(\eta_{c},v_{c})$.

  2. (ii)

$(\eta_{c},v_{c})\in D(\tau^{\prime}(0)^{3})\cap
D(J\Lambda\tau^{\prime}(0)^{2})$, where $\Lambda=(1-\partial_{xx},1)$ is the
isomorphism between $X$ and $X^{*}$.

  3. (iii)

$\tau^{\prime}(0)(\eta_{c},v_{c})\neq 0$.

Assumption 3 (Spectral decomposition of $H_{c}$). For every
$c\in(c_{1},c_{2})$ and $H_{c}$ given by (33), we assume

  1. (i)

There exists $\chi\in X$ such that

$\langle H_{c}\chi,\chi\rangle<0.$ (35)

  2. (ii)

There exists a closed subspace $P_{c}\subset X$, and $C>0$ such that

$\langle H_{c}p,p\rangle\geq C||p||^{2},\quad\text{ for all }p\in P_{c}.$ (36)

  3. (iii)

For every $u\in X$, there exist unique $(a,b)\in\mathbb{R}^{2}$ and $p\in
P_{c}$ such that

$u=a\chi+b\tau^{\prime}(0)(\eta_{c},v_{c})+p.$ (37)

######  Remark 2.1.

To our knowledge, when $\kappa\neq 0$, there are no result for the IVP
associated with (18) with initial data in $X$, thus Assumption 1 agrees with
our definition of orbital stability with respect to
$H^{s}(\mathbb{R})$-perturbations (14). Also, it is worth mentioning that
Assumption 3 corresponds Assumption 3B in [15] i.e. the decomposition (37) is
not necessarily the orthogonal decomposition; this enables them to handle
Hamiltonian systems in Banach spaces.

The system (18) can be rewritten as an Euler–Korteweg system for which the
local well-posedness was investigated by S. Benzoni-Gavage, R. Danchin, and S.
Descombes in [3]. Applying their result to (1) for $\kappa<1/2$ we deduce the
following result (see [13] for details), from which Assumption 1 follows in a
neighborhood of $(\eta_{c},v_{c})$ by Lemma 4.1.

######  Theorem 2.2 (Theorem 5.1 in [3]).

Let $s>1/2+2$, $\kappa<1/2$ and $c>0$ and

$J_{\kappa}=\begin{cases}(0,\infty)\text{ if }\kappa\leq 0,\\\
(0,1/\sqrt{2\kappa}),\text{ if }0<\kappa<1/2.\end{cases}$

If $\Psi_{0}\in u_{c,\kappa}+H^{s}(\mathbb{R})$ satisfies $|\Psi_{0}(x)|\in
J_{\kappa}$ for every $x\in\mathbb{R}$, then there exists $T_{\Psi_{0}}>0$,
the maximal time of existence such that for every $T\in(0,T_{\Psi_{0}})$, (1)
has a unique solution $\Psi$ on $\mathbb{R}\times[0,T]$, satisfying
$\Psi(\cdot,0)=\Psi_{0}$ and

$\left\\{\begin{aligned}
&\Psi\in\mathcal{C}([0,T];u_{c,\kappa}+H^{s}(\mathbb{R})),\\\
&|\Psi|(\mathbb{R}\times[0,T])\subset\subset J_{\kappa}.\end{aligned}\right.$

Moreover, the flow map is continuous in a neighborhood of $\Psi_{0}$ in
$u_{c,\kappa}+H^{s}(\mathbb{R})$, and the energy and momentum are conserved,
i.e. (34) holds for all $t\in[0,T].$ Also writing
$\Psi=\sqrt{1-\eta}e^{i\theta}$ we deduce that $\eta$ and
$v=\partial_{x}\theta$ satisfy (18) with
$(\eta,v)\in\mathcal{C}([0,T];H^{s}(\mathbb{R})\times H^{{s-1}}(\mathbb{R}))$.

For Assumption 2, take $c\in(0,\sqrt{2})$, and let $u_{c}$ be the dark soliton
given by Theorem 1.1 so that $\inf_{x\in\mathbb{R}}|u_{c}|>0$. Then we can
write $u_{c}=\sqrt{1-\eta_{c}}e^{i\theta_{c}}$ with
$(\eta_{c},\partial_{x}\theta_{c})$ satisfying (24) pointwisely and Assumption
2 (1) follows. By the smoothness of $u_{c}(x)$ with respect to $c$ and the
decay estimate (12) and using the identity

$u_{c}^{\prime}=\left(\frac{\eta_{c}^{\prime}}{2\sqrt{1-\eta}}+i\sqrt{1-\eta}\theta_{c}^{\prime}\right)e^{i\theta},$
(38)

it is easy to check that $c\mapsto(\eta_{c},\partial_{x}\theta_{c})$ belongs
in $\mathcal{C}^{\infty}((0,\sqrt{2})),X)$. Once again appealing to the
smoothness and decay of $u_{c}$, we recover
$(\eta_{c},\partial_{x}\theta_{c})\in H^{\infty}(\mathbb{R})^{2}$ which
implies Assumption 2 (2). Finally, since $u_{c}$ is nonconstant, Assumption 2
(3) follows.

To check Assumption 3, it remains to compute $\langle H_{c}(\delta\eta,\delta
v),(\delta\eta,\delta v)\rangle$. Using (27)–(29), we get

$\displaystyle\langle H_{c}(\delta\eta,\delta v),(\delta\eta,\delta
v)\rangle=$
$\displaystyle\langle-\partial_{x}\left(\frac{1-2\kappa+2\kappa\eta}{2(1-\eta)}\partial_{x}\right)\delta\eta,\delta\eta\rangle-\langle\partial_{x}\left(\frac{\eta_{x}}{2(1-\eta)^{2}}\right)\delta\eta,\delta\eta\rangle+\int_{\mathbb{R}}\frac{\eta_{x}^{2}}{2(1-\eta)^{3}}\delta\eta^{2}+\delta\eta^{2}$
$\displaystyle+\int_{\mathbb{R}}-4v\delta v\delta\eta+2(1-\eta)\delta
v^{2}+2c\delta v\delta\eta,$ $\displaystyle=$
$\displaystyle\langle-\partial_{x}\left(\frac{1-2\kappa+2\kappa\eta}{2(1-\eta)}\partial_{x}\right)\delta\eta,\delta\eta\rangle-\langle\partial_{x}\left(\frac{\eta_{x}}{2(1-\eta)^{2}}\right)\delta\eta,\delta\eta\rangle+\int_{\mathbb{R}}\frac{\eta_{x}^{2}}{2(1-\eta)^{3}}\delta\eta^{2}+\delta\eta^{2}$
$\displaystyle-\int_{\mathbb{R}}\frac{(c-2v)^{2}}{2(1-\eta)}\delta\eta^{2}+\int_{\mathbb{R}}2(1-\eta_{c})\left(\delta
v+\frac{c-2v}{2(1-\eta)}\delta\eta\right)^{2}.$ (39)

Using the first line in (24) and taking $(\eta,v)=(\eta_{c},v_{c})$ in (2), we
can rewrite the last line in (2) to obtain

$\langle H_{c}(\delta\eta,\delta v),(\delta\eta,\delta v)\rangle=\langle
L_{c}\delta\eta,\delta\eta\rangle+\int_{\mathbb{R}}2(1-\eta_{c})\left(\delta
v+\frac{c}{2(1-\eta_{c})^{2}}\delta\eta\right)^{2},$ (40)

where $L_{c}$ is an self-adjoint unbounded operator on
$L^{2}(\mathbb{R};\mathbb{R})$ with domain $H^{2}(\mathbb{R};\mathbb{R})$
given by

$L_{c}=-\partial_{x}\left(\frac{1-2\kappa+2\kappa\eta_{c}}{2(1-\eta_{c})}\partial_{x}\right)-\frac{\partial_{xx}\eta_{c}}{2(1-\eta_{c})^{2}}-\frac{(\partial_{x}\eta_{c})^{2}}{2(1-\eta)^{3}}+1-\frac{c^{2}}{2(1-\eta_{c})^{3}}.$
(41)

Since $L_{c}$ is semi-bounded from below, integrating by part the first term
in $\langle L_{c}\delta\eta,\delta\eta\rangle$, we infer that $L_{c}$ is
associated with a quadratic form on $H^{1}(\mathbb{R};\mathbb{R})$ by the
Lax–Milgram theorem (Theorem 2.89 in [6]).

Since $0<\eta_{c}\leq 1-c^{2}/2$ and $0<c<\sqrt{2}$, we have $1-\eta_{c}>0$
and $1-2\kappa+2\kappa\eta_{c}>0$ in $\mathbb{R}$. Then, using ideas similar
to the ones in section 6.3.2 of [6] and Lemma 6.13 in [6], we have in
$L^{2}(\mathbb{R};\mathbb{R})$

$\sigma_{\mathrm{ess}}\left(-\partial_{x}\left(\frac{1-2\kappa+2\kappa\eta_{c}}{2(1-\eta_{c})}\partial_{x}\right)\right)=[0,\infty).$
(42)

We infer that the other terms in $L_{c}-\lambda$ are compact operators for
some $\lambda>0$. Indeed, notice that

$\lim_{|x|\to\infty}-\frac{\partial_{xx}\eta_{c}(x)}{2(1-\eta_{c}(x))^{2}}-\frac{(\partial_{x}\eta_{c}(x))^{2}}{2(1-\eta_{c}(x))^{3}}+1-\frac{c^{2}}{2(1-\eta_{c}(x))^{3}}=1-\frac{c^{2}}{2}>0.$
(43)

######  Lemma 2.3.

Let $0<c<\sqrt{2}$ and fix $\lambda=1-c^{2}/2>0$. Let also
$T_{c}\in\mathcal{L}(H^{2}(\mathbb{R};\mathbb{R}),L^{2}(\mathbb{R};\mathbb{R}))$
be given by

$T_{c}=-\frac{\partial_{xx}\eta_{c}}{2(1-\eta_{c})^{2}}-\frac{(\partial_{x}\eta_{c})^{2}}{2(1-\eta_{c})^{3}}+1-\frac{c^{2}}{2(1-\eta_{c})^{3}}$
(44)

then $T_{c}-\lambda$ is compact i.e. for every sequence $(u_{n})\subset
H^{2}(\mathbb{R})$ with $u_{n}\rightharpoonup 0$ we can extract a subsequence
such that $(T_{c}-\lambda)(u_{n_{k}})\to 0$ in $L^{2}(\mathbb{R})$.

###### Proof.

Let $\varepsilon>0$ and $R=R_{\varepsilon}>0$. by Kato–Rellich theorem
(Theorem 4.20 in [6]) there exists $v\in H^{1}((-R,R))$ and a subsequence
$u_{n_{k}}\to v$ in $H^{1}((-R,R))$. Of course $v=0$ by the uniqueness of the
weak limit. Then denoting $I_{R}=\mathbb{R}\backslash(-R,R)$ we split
$||(T_{c}-\lambda)(u_{n_{k}})||_{L^{2}(\mathbb{R})}^{2}$ as

$||(T_{c}-\lambda)(u_{n_{k}})||_{L^{2}(\mathbb{R})}^{2}=||(T_{c}-\lambda)(u_{n_{k}})||_{L^{2}(I_{R})}^{2}+||(T_{c}-\lambda)(u_{n_{k}})||_{L^{2}((-R,R))}^{2}.$
(45)

We deduce, using (43) that taking $R>0$ big enough yield

$||T_{c}-\lambda||^{2}_{\mathcal{L}(H^{2}(I_{R});L^{2}(I_{R}))}\leq\varepsilon/(2\sup_{k\in\mathbb{N}}||u_{n_{k}}||^{2}_{H^{2}(\mathbb{R})}),$

letting us bound the first term in (45) by $\varepsilon/2$. Once $R$ is fixed,
we take $k$ big enough so that considering $T_{c}-\lambda$ as an operator in
$\mathcal{L}(H^{2}((-R,R));L^{2}((-R,R)))$, we have
$||(T_{c}-\lambda)(u_{n_{k}})||^{2}_{L^{2}(-R,R)}\leq\varepsilon/2$. This
shows that $(T_{c}-\lambda)(u_{n_{k}})\to 0$ in $L^{2}(\mathbb{R})$ ending the
proof. ∎

Lemma 2.3, enables us to compute
$\sigma_{\mathrm{ess}}(L_{c})=\sigma_{\mathrm{ess}}(\partial_{x}((1-2\kappa+2\kappa\eta_{c})/(2-2\eta_{c})\partial_{x})+\lambda+T_{c}-\lambda)$
which yields, using a generalized version of the Weyl essential spectrum
theorem (see Theorem B.48 in [1])

$\sigma_{\mathrm{ess}}(L_{c})=[\lambda,\infty)=[1-c^{2}/2,\infty).$ (46)

The operator $L_{c}$ being self-adjoint, the rest of its spectrum
$\sigma_{\mathrm{fred}}(L_{c})=\sigma(L_{c})\backslash\sigma_{\mathrm{ess}}(L_{c})$
lies in $(-\infty,1-c^{2}/2)$ and every $\mu\in\sigma_{\mathrm{fred}}(L_{c})$
is an isolated eigenvalue of finite multiplicity (see Section 6.3 in [6]).
Using the first line in (24) we can rewrite second equation in (24) only in
terms of $\eta_{c}$

$-\left(\frac{1-2\kappa+2\kappa\eta_{c}}{2(1-\eta_{c})}\partial_{x}\partial_{x}\eta_{c}\right)-\frac{(\partial_{x}\eta_{c})^{2}}{4(1-\eta)^{2}}+\eta_{c}-c^{2}\frac{\eta^{2}-2\eta}{4(1-\eta)^{2}},\quad\text{
in }\mathbb{R}.$ (47)

Differentiating (47) with respect to $x$ yields
$\partial_{x}\eta_{c}\in\ker(L_{c})$. Since $|u_{c}|^{\prime}>0$ in
$(0,\infty)$ and $|u_{c}|$ is even, we deduce that
$\partial_{x}\eta_{c}=\partial_{x}(1-|u_{c}|^{2})$ vanishes once at $x=0$, so
that, by the Sturm-Liouville oscillation theorem (see Appendix B.5 in [1]),
there exists one and only one negative eigenvalue $\mu_{-}<0$ of multiplicity
one with eigenvector $\chi_{1}\in H^{2}(\mathbb{R})$. The eigenvalues
$\mu_{-}$ and $0$ being simple and isolated, by the min-max principle (see
Section 6.4 in [6]), we infer that there exists $\mu_{1}>0$ such that for
every $z\in H^{1}(\mathbb{R})$ satisfying

$(z,\chi_{1})_{L^{2}(\mathbb{R})}=(z,\partial_{x}\eta_{c})_{L^{2}(\mathbb{R})}=0,$
(48)

we have $\langle L_{c}z,z\rangle\geq\mu_{1}||z||_{L^{2}(\mathbb{R})}^{2}.$ It
is also clear that $L_{c}$ satisfies some kind of
G$\mathring{\mathrm{a}}$rding inequality so that proceeding along the same
lines as in Lemma 4.1 in [23] yields the existence of $\mu>0$ such that
$\langle L_{c}z,z\rangle\geq\mu||z||_{H^{1}(\mathbb{R})}^{2}.$ Setting

$\chi_{-}=\left(\chi_{1},-\frac{c}{2(1-\eta)^{2}}\chi_{1}\right),$ (49)

and using (40), we get $\langle H_{c}\chi_{-},\chi_{-}\rangle=\langle
L_{c}\chi_{1},\chi_{1}\rangle<0$. Letting

$P_{c}=\\{(p_{1},p_{2})\in
X:(p_{1},\chi_{1})_{L^{2}(\mathbb{R})}=(p_{1},\partial_{x}\eta_{c})_{L^{2}(\mathbb{R})}=0\\},$

and considering the $L^{2}(\mathbb{R})$-orthogonal decomposition of
$\eta=a\chi_{1}+b\partial_{x}\eta+p_{1}$, we perform the following
decomposition of $(\eta,v)\in X$:
$(\eta,v)=a\chi_{-}+b\partial_{x}(\eta_{c},v_{c})+(p_{1},p_{2}),$ with
$p_{2}=v-ac\chi_{1}/(2-2\eta)-b\partial_{x}v_{c}$. The uniqueness of the
decomposition of $(\eta,v)$ follows from the uniqueness of the orthogonal
decomposition of $\eta$. Using Lemma 4.2 in [23] (see also [16]) we deduce
that there exists $C>0$ satisfying $\langle H_{c}p,p\rangle>C||p||^{2}_{X}$
for all $p\in P_{c}$ so that Assumption 3 follows.

Let $d(c)=E((\eta_{c},v_{c}))-cP((\eta_{c},v_{c}))$ since
$c\mapsto(\eta_{c},v_{c})$ in $\mathcal{C}^{1}((0,\sqrt{2}))$ we can compute

$d^{\prime\prime}(c)=\left(\langle
E^{\prime}((\eta_{c},v_{c}))-cP^{\prime}((\eta_{c},v_{c})),\partial_{c}(\eta_{c},v_{c})\rangle-P((\eta_{c},v_{c}))\right)^{\prime}=-\frac{d}{dc}P((\eta_{c},v_{c})).$
(50)

Assuming (16), we get $d^{\prime\prime}(c)>0$ and we can apply Theorem 3 in
[15]. The only remaining thing to check is that for any sequence
$(\eta^{n},v^{n})=(\eta,v)_{n}\subset X$ with $E((\eta,v)_{n})\to
E((\eta_{c},v_{c}))$ and $P((\eta,v)_{n})\to P((\eta_{c},v_{c})),$ we can
construct $(u_{n})\subset X$ such that, up to extraction, we have
$||u_{n}-(\eta,v)_{n}||_{X}\to 0$ and $P(u_{n})=P((\eta_{c},v_{c}))$. By the
first line in (24), we have $P((\eta_{c},v_{c}))>0$ for all
$c\in(0,\sqrt{2})$, thus, up to a subsequence extraction, we assume
$P((\eta,v)_{n})>0$ for all $n\in\mathbb{N}$. Hence,
$\alpha_{n}=P((\eta_{c},v_{c}))/P((\eta,v)_{n})$ is well defined with
$\alpha_{n}\to 1$ so that $u_{n}=(\eta_{n},\alpha_{n}v_{n})$ satisfies
$P(u_{n})=P((\eta_{c},v_{c}))$ and we can check
$||u_{n}-(\eta_{n},v_{n})||_{X}\to 0$. For $\psi=\sqrt{1-\eta}e^{i\theta}$
with $\inf_{\mathbb{R}}|\psi|>0$, notice that we can always choose
$\varphi\in\mathbb{R}$ so that $\mathrm{Arg}(\psi(0)/u_{c}(0))=0$ thus

$\inf_{\varphi\in\mathbb{R}}d_{\mathrm{hy}}(\psi,u_{c}e^{i\varphi})=||(\eta,v)-(\eta_{c},v_{c})||_{X},$

so that the stability of $u_{c}$ for $d_{\mathrm{hy}}$ follows from the
orbital stability of the bound state $(\eta_{c},v_{c})$ in $X$.

######  Theorem 2.4 (Theorem 3 in [15]).

If Assumptions 1–3 hold, then the following statement is satisfied. If
$d^{\prime\prime}(c)>0$ then $E(\mathfrak{u})$ is minimized at
$u=(\eta_{c},v_{c})$ for $\mathfrak{u}$ in a neighborhood of
$(\eta_{c},v_{c})$ in $X$ with $P(\mathfrak{u})=P((\eta_{c},v_{c}))$. Moreover
the dark soliton $u_{c}$ is orbitally stable for $d_{\mathrm{hy}}$.

######  Remark 2.5.

Theorem 3 in [15] is stated as an equivalence between the variation of
$d(\cdot)$ and the stability. However, we cannot apply the converse statement
since in our case the skew-symmetric operator $J$ in (19) is not onto. For
instance $(\eta_{c},v_{c})\notin\mathrm{Range}(J)$ because $\eta_{c}>0$ in
$\mathbb{R}$ and hence $C+\int^{x}_{-\infty}\eta_{c}\notin
H^{{-1}}(\mathbb{R})$ for every $C\in\mathbb{R}$.

##  3 Instability

Regarding the instability, arguments similar to the ones used by Lin in [23]
yield the result. We explain how to modify the first steps that consist of
creating an energy-decreasing direction in $X$ while referring to [23] for
detailed demonstrations. From now on, we assume that $c\in(0,\sqrt{2})$ is
such that (17) holds and we denote by $(\eta_{c},v_{c})\in X$ the dark soliton
satisfying _Assumption 2_ we also fix $s>2+1/2$ the regularity of the
perturbations in Definition 14 (ii). Firstly, we have $\chi_{1}\in
H^{\infty}(\mathbb{R})$ by a bootstrap argument since it satisfies
$\chi_{1}\in H^{2}(\mathbb{R})$ and $L_{c}(\chi_{1})=\mu_{-}\chi_{1}$, with
$L_{c}$ given by (41). Thus as in [23], we can assume that $\chi_{-}\in
X_{s}\subset X$ with

$X_{s}=(H^{s}(\mathbb{R})\times
H^{{s}}(\mathbb{R}))\cap(L^{1}(\mathbb{R},(1+|x|)dx)\times
L^{1}(\mathbb{R},(1+|x|)dx)),$ (51)

where $\chi_{-}$ is the negative direction of $H_{c}$ given by (49). We also
get $(\eta_{c},v_{c})\in X_{s}$ using the smoothness and rapid decay of the
dark solitons established in Theorem 1.1. Using the implicit function theorem
on $(q,l)\mapsto(\eta_{q},v_{q})+l\chi_{-}$, we can find a
$\mathcal{C}^{1}$-map, $l(q)$ and a curve $\Psi(q)$ defined in the
neighborhood of $c$ with

$\Psi(q)=(\eta_{q},v_{q})+l(q)\chi_{-},$ (52)

satisfies $P(\Psi(q))\equiv P((\eta_{c},v_{c}))$. This allows us to build a
negative direction for the Hessian $H_{c}$ given by (33) with the following
properties.

######  Lemma 3.1 (Corollary 5.1 in [23]).

There exists $y_{0}\in X_{s}$ such that $\langle
By_{0},(\eta_{c},v_{c})\rangle=0$ and $\langle H_{c}y_{0},y_{0}\rangle<0$,
where $B$ is given by (31).

We want to build a negative direction as in Lemma 3.1 for which the number of
particles (22) and twisting angle (23) are zero. We use the following.

######  Lemma 3.2.

For all $\eta\in H^{2}(\mathbb{R})$ and $\nu\in\mathbb{R}$, there exists a
sequence $(u_{n})\subset X_{s}$ such that $\int_{\mathbb{R}}u_{n}=\nu$,
$u_{n}\to 0$ in $H^{1}(\mathbb{R})$ and $\langle u_{n},r\rangle_{X}=0$.

###### Proof.

The proof proceeds as in [23]; let
$\varphi\in\mathcal{C}^{\infty}_{c}(\mathbb{R})$ with
$\int_{\mathbb{R}}\varphi=\nu$. First if $\eta\equiv 0$, then taking
$u_{n}(\cdot)=\varphi(\cdot/n)/n$ for any $\varphi$ defined above suffices.
Assuming that $\eta\neq 0$, then by a duality and density argument there
exists $\psi\in\mathcal{C}^{\infty}_{c}(\mathbb{R})$ such that
$\langle\eta_{x},\psi\rangle_{H^{1}(\mathbb{R})}\neq 0$. Indeed, otherwise we
would have $\eta_{x}=0$ but since $\eta\in L^{2}(\mathbb{R})$ this contradicts
$\eta\neq 0$. Letting $u_{n}(\cdot)=\varphi(\cdot/n)/n+\alpha_{n}\psi_{x}$, we
readily get $u_{n}\in X_{s}$ and $\int_{\mathbb{R}}u_{n}=\nu$. Then, setting
$\alpha_{n}=\langle\varphi(\cdot/n)/n,\eta\rangle_{H^{1}(\mathbb{R})}/\langle\eta_{x},\psi\rangle_{H^{1}(\mathbb{R})},$
we obtain $\langle\eta,u_{n}\rangle_{H^{1}(\mathbb{R})}=0$. Finally, we deduce
after rescaling that $\alpha_{n}\to 0$ using
$|\alpha_{n}||\langle\eta_{x},\psi\rangle_{H^{1}(\mathbb{R})}|\leq||\eta||_{H^{1}(\mathbb{R})}||\varphi(\cdot/n)/n||_{H^{1}(\mathbb{R})}$
and also $u_{n}\to 0$ in $H^{1}(\mathbb{R})$. ∎

Since $s>2+1/2$ we can apply Lemma 3.2 to each coordinate of
$y_{0}=(\eta_{0},v_{0})$ where $y_{0}\in X_{s}$ is given by Lemma 3.1.

######  Lemma 3.3 (Lemma 5.3 in [23]).

For all $s>2+1/2$, there exists $y=(\eta_{1},v_{1})\in X_{s}$ such that
$\langle By,(\eta_{c},v_{c})\rangle=0,$ $\langle H_{c}y,y\rangle<0$ and

$\int_{\mathbb{R}}\eta_{1}=\int_{\mathbb{R}}v_{1}=0.$

Following along Section 5 in [23] we can deduce that $(\eta_{c},v_{c})$ is not
a minimizer of the energy at fixed momentum and that it is unstable. More
precisely, let $(\eta,v)=\Psi(q)$ where $\Psi(\cdot)$ is given by (52) and $q$
is in the vicinity of $c$. Taking $\psi=\sqrt{1-\eta}e^{i\theta}$ with
$\theta(x)=\theta_{c}(0)+\int_{0}^{x}v$, we can prove that $\psi$ is
arbitrarily close to $u_{c,\kappa}$ for $d_{\mathrm{hy}}$ as $q\to c$ with
$q\neq c$. This implies that eventually as $q\to c$ we have
$|\psi|(\mathbb{R})\subset\subset J_{\kappa}$ and $\psi\in u_{c}+H^{{s}}$ so
that there exists a solution to (1) starting at $\psi$ at $t=0$. The
instability of $u_{c}$ for $d_{\mathrm{hy}}$ follows as in [23].

######  Theorem 3.4 (Theorem 5.1 in [23]).

Let $c\in(0,\sqrt{2})$, $s>2+1/2$ and assume that $(\eta_{c},v_{c})\in X_{s}$
is the dark soliton. If $\partial_{c}P((\eta_{c},v_{c}))>0,$ then the curve
$\Psi(\cdot)$ given by (52) defined in the vicinity of $c$ satisfies
$P(\Psi(\cdot))\equiv P((\eta_{c},v_{c}))$ and
$E(\Psi(q))<E((\eta_{c},v_{c})),$ when $q\neq c$. Moreover the dark soliton
$u_{c}$ is unstable for $d_{\mathrm{hy}}$.

##  4 Stability in the energy distance $d_{\mathcal{X}}$

Let $\kappa<1/2$ and $c\in(0,\sqrt{2})$, we first show that the stability and
instability results stated in terms of $d_{\mathrm{hy}}$ given by (11) in
Theorem 2.4 and in Theorem 3.4 respectively, are equivalent to the stability
and instability result in Theorem 1.4 in terms of the distance
$d_{\mathcal{X}}$ given by (10) respectively; thus proving Theorem 1.4. For
deeper insights on this equivalence we refer to Lemma 9–10 in [8]).

######  Lemma 4.1.

Let $\kappa<1/2$, $c\in(0,\sqrt{2})$ and $u_{c,\kappa}$ be a dark soliton
given by Theorem 1.1. Then there exists $K$ and $\varepsilon>0$ depending only
on $u_{c,\kappa}$ such that for every $\Psi\in\mathcal{X}(\mathbb{R})$, if
$\Psi$ satisfies $d_{\mathcal{X}}(\Psi,u_{c,\kappa})<\varepsilon$, then we can
write $\Psi=\sqrt{1-\eta}e^{i\theta}$ with $(\eta,\partial_{x}\theta)\in X$
and we have

$\frac{1}{K}d_{\mathrm{hy}}(\Psi,u_{c,\kappa})\leq
d_{\mathcal{X}}(\Psi,u_{c,\kappa})\leq Kd_{\mathrm{hy}}(\Psi,u_{c,\kappa}).$
(53)

###### Proof.

For clarity’s sake, we recall the main arguments of the proof in [9]. We
remove $\kappa$ in the subscript of $u_{c,\kappa}=u_{c}$ and write
$u_{c}=\sqrt{1-\eta_{c}}e^{i\theta_{c}}$. Since $u_{c}$ does not vanish,
taking $\varepsilon>0$ small enough then, for any
$\Psi\in\mathcal{X}(\mathbb{R})$ satisfying
$d_{\mathcal{X}}(\Psi,u_{c})\leq\varepsilon$, we get
$|||\Psi|-|u_{c}|||_{L^{\infty}(\mathbb{R})}\leq\inf_{x\in\mathbb{R}}|u_{c}(x)|/2$
so that

$\inf_{x\in\mathbb{R}}|\Psi(x)|\geq\inf_{x\in\mathbb{R}}|u_{c}(x)|/2.$ (54)

We can express $\partial_{x}\Psi$ in terms of $\eta$ and $\theta$,

$\partial_{x}\Psi=e^{i\theta}\left(\frac{\eta_{x}}{2\sqrt{1-\eta}}+i\sqrt{1-\eta}\theta_{x}\right),$
(55)

and similarly for $u_{c}$. We also need to bound
$||\Psi||_{L^{\infty}(\mathbb{R})}$ in terms of $\varepsilon$,
$||u_{c}||_{L^{\infty}(\mathbb{R})}$ and
$||\partial_{x}u_{c}||_{L^{2}(\mathbb{R})}$. To do so, notice that for any
constants $y<x$, we have

$|\Psi(x)|\leq|\Psi(y)|+\int_{y}^{x}|\partial_{x}\Psi|\leq|u_{c}(y)|+||\Psi(y)|-|u_{c}(y)||+\sqrt{x-y}||\partial_{x}\Psi||_{L^{2}(\mathbb{R})},$
(56)

where we used Cauchy–Schwarz inequality to bound the integral. Taking
$0<x-y<1$ then integrating (56) with respect to $y$ from $x-1$ to $x$ we get
$|\Psi(x)|\leq|u_{c}|+\int_{\mathbb{R}}||u_{c}|-|\Psi||+||\partial_{x}\Psi||_{L^{2}(\mathbb{R})},$
so that

$||\Psi||_{L^{\infty}(\mathbb{R})}\leq K_{0}(u_{c},\varepsilon).$ (57)

We show that $d_{\mathrm{hy}}(\Psi,u_{c})\leq Kd_{\mathcal{X}}(\Psi,u_{c}).$
First notice that $|\Psi|^{2}\theta_{x}=\Im(\Psi_{x}\bar{\Psi})$, thus

$\partial_{x}\theta-\partial_{x}\theta_{c}=\Im(\partial_{x}u_{c}\bar{u_{c}})\left(\frac{1}{|\Psi|^{2}}-\frac{1}{|u_{c}|^{2}}\right)+\frac{1}{|\Psi|^{2}}(\Im((\partial_{x}\Psi-\partial_{x}u_{c})\bar{\Psi})-\Im(\partial_{x}u_{c}(\bar{u_{c}}-\bar{\Psi}))),$

hence

$\displaystyle||\partial_{x}\theta-\partial_{x}\theta_{c}||_{L^{2}(\mathbb{R})}\leq$
$\displaystyle\frac{\Im(\partial_{x}u_{c}\bar{u_{c}})(||\Psi||_{L^{\infty}(\mathbb{R})}+||u_{c}||_{L^{\infty}(\mathbb{R})})}{\inf_{\mathbb{R}}|\Psi|^{2}\inf_{\mathbb{R}}|u_{c}|^{2}}|||\Psi|-|u_{c}|||_{L^{2}(\mathbb{R})}$
$\displaystyle+\frac{1}{\inf_{\mathbb{R}}|\Psi|^{2}}\left(||\Psi||_{L^{\infty}(\mathbb{R})}||\partial_{x}\Psi-\partial_{x}u||_{L^{2}(\mathbb{R})}+||\partial_{x}u_{c}(\bar{u_{c}}-\bar{\Psi})||_{L^{2}(\mathbb{R})}\right).$
(58)

Since
$\Psi(x)-u_{c}(x)=\Psi(0)-u_{c}(0)+\int_{0}^{x}\partial_{x}\Psi-\partial_{x}u_{c}$
for all $x\in\mathbb{R}$, appealing the Cauchy–Schwarz inequality, we get

$|\partial_{x}u_{c}(x)||\Psi(x)-u_{c}(x)|\leq|\partial_{x}u(x)||\Psi(0)-u_{c}(0)|+|\partial_{x}u_{c}(x)\sqrt{|x|}|||\partial_{x}\Psi-\partial_{x}u_{c}||_{L^{2}(\mathbb{R})},\quad\text{
for all }x\in\mathbb{R}.$ (59)

From the decay estimate (12) satisfied by $\partial_{x}u_{c}$, we can bound
the last term in (4), therefore using also (54) and (57), we deduce that there
exists $K_{1}=K_{1}(\varepsilon,u_{c})$ such that

$||\partial_{x}\theta-\partial_{x}\theta_{c}||\leq
K_{1}d_{\mathrm{hy}}(\Psi,u_{c}).$ (60)

Similarly, we have $2\Re(\Psi_{x}\bar{\Psi})=\eta_{x}$, thus

$\displaystyle||\partial_{x}\eta-\partial_{x}\eta_{c}||_{L^{2}(\mathbb{R})}$
$\displaystyle\leq
2||\Psi||_{L^{\infty}(\mathbb{R})}||\partial_{x}\Psi-\partial_{x}u_{c}||_{L^{2}(\mathbb{R})}+2||\partial_{x}{u_{c}}(\bar{u_{c}}-\bar{\Psi})||_{L^{2}(\mathbb{R})},$
$\displaystyle\leq
2K_{0}d_{\mathrm{hy}}(\Psi,u_{c})+2||\partial_{x}u_{c}||_{L^{2}(\mathbb{R})}||\Psi(0)|-|u_{c}(0)||+||\sqrt{x}\partial_{x}u_{c}||_{L^{2}(\mathbb{R})}||\partial_{x}\Psi-\partial_{x}u_{c}||_{L^{2}(\mathbb{R})},$
$\displaystyle\leq K_{2}d_{\mathrm{hy}}(\Psi,u_{c}).$ (61)

Then, we also have
$||\eta-\eta_{c}||_{L^{2}(\mathbb{R})}=||(|\Psi|+|u_{c}|)(|\Psi|-|u_{c}|)||_{L^{2}(\mathbb{R})}\leq(||\Psi||_{L^{\infty}(\mathbb{R})}+||u_{c}||_{L^{\infty}(\mathbb{R})})|||\Psi|-|u_{c}|||_{L^{2}(\mathbb{R})}$.
We bound the last term $|\mathrm{Arg}(\Psi(0)/u_{c}(0))|$. Letting
$\mathrm{Log}$ be the principal determination of the logarithm defined in
$\mathbb{C}\backslash(-\infty,0]$ and taking $\varepsilon>0$ small enough, we
get

$|\mathrm{Arg}(\Psi(0)/u_{c}(0))|=|\Im(\mathrm{Log}(\Psi(0)/u_{c}(0)))|\leq\sup_{|z-1|\leq\varepsilon/u_{c}(0)}|1/z|\left|\frac{\Psi(0)-u_{c}(0))}{u_{c}(0)}\right|,$

which ends to prove that $d_{\mathrm{hy}}(\Psi,u_{c})\leq
Kd_{\mathcal{X}}(\Psi,u_{c})$. Ideas along the same lines as in Lemma 10 in
[8] show that $d_{\mathcal{X}}(\Psi,u_{c})\leq Kd_{\mathrm{hy}}(\Psi,u_{c})$.
∎

To prove Corollary 1.5, we shall recall some properties of the function

$P_{\kappa}(c)=P(u_{c,\kappa}),\quad\text{ for all }c\in(0,\sqrt{2}).$ (62)

######  Lemma 4.2 (Proposition 5.6 and Lemma 5.7 in [13]).

There exists $\kappa_{0}<0$ with $\kappa_{0}\approx-3.636$ such that the two
following statements hold.

  1. (i)

If $\kappa\in[\kappa_{0},1/2)$, then $P_{\kappa}^{\prime}<0$ in
$(0,\sqrt{2})$.

  2. (ii)

If $\kappa<\kappa_{0}$, then there exists $\tilde{c}_{\kappa}$ such that
$P^{\prime}_{\kappa}>0$ in $(0,\tilde{c}_{\kappa})$ while
$P^{\prime}_{\kappa}<0$ in $(\tilde{c}_{\kappa},\sqrt{2})$.

##  Appendix A Extensions of the stability result

Finally, we briefly describe the Lyapunov functional approach to the stability
of the dark solitons (see [8]). For the sake of conciseness, we chose not to
use this method in this work to avoid tedious computations. As pointed out by
C. A. Stuart in Section 4–5 in [24], in the Hamiltonian system framework of
[15], the stability criterion (16) together with Assumptions 1–3 allow us to
construct a Lyapunov function invariant by translation by taking $K$ large
enough in

$V(\eta,v)=E(\eta,v)-cP(\eta,v)-E(u_{c})+cP(u_{c})+K(P(\eta,v)-P(u_{c}))^{2},$
(63)

giving a quantitative measure of stability. The heuristic is the following,
since $V$ is constant on the flow of solutions, to ensure the stability, we
want $V$ to satisfy, for some $\alpha>0$

$V(\eta,v)\geq\alpha||(\eta,v)-(\eta_{c},v_{c})||^{2}_{X},$

for some $\varepsilon>0$ and all $\eta$, $v$ satisfying
$\inf_{s\in\mathbb{R}}||(\eta,v)-(\eta_{c},v_{c})(\cdot-s)||_{X}<\varepsilon$.
This problem reduces to the positive definiteness of
$V^{\prime\prime}(\eta_{c},v_{c})$ which writes

$\langle V^{\prime\prime}(\eta_{c},v_{c})(\delta\eta,\delta
v),(\delta\eta,\delta v)\rangle=\langle H_{c}(\delta\eta,\delta
v),(\delta\eta,\delta v)\rangle+2K(\langle
P^{\prime}(u_{c}),(\delta\eta,\delta v)\rangle)^{2}\geq C||(\delta\eta,\delta
v)||^{2}_{X},$

for some $C>0$. We readily compute
$-P^{\prime}(u_{c})=\partial_{c}(E^{\prime}-cP^{\prime})(u_{c})$ and since
$E^{\prime}(u_{c})-cP^{\prime}(u_{c})=0$ for every $c\in(0,\sqrt{2})$, we get
$\partial_{c}(E^{\prime}(u_{c})-cP^{\prime}(u_{c}))=H_{c}\partial_{c}u_{c}-P^{\prime}(u_{c})=0$
by the chain rule. Testing this linear form against
$\partial_{c}(\eta_{c},v_{c})\in X$, if (16) holds, we deduce that,

$\langle
H_{c}\partial_{c}(\eta_{c},v_{c}),\partial_{c}(\eta_{c},v_{c})\rangle=\frac{d}{dc}P(\eta_{c},v_{c})<0.$
(64)

By Lemma 5.3 in [24], considering Assumption 3, we can assume the
decomposition to be the orthogonal decomposition

$u=a\partial_{c}(\eta_{c},v_{c})+b\tau^{\prime}(0)(\eta_{c},v_{c})+p,\quad\text{
for all }u\in X,$ (65)

where
$P=\\{\partial_{c}(\eta_{c},v_{c}),\tau^{\prime}(0)(\eta_{c},v_{c})\\}^{\perp}$.
Then we can show that taking $K$ big enough in (63) yield the coercivity of
the quadratic form $V^{\prime\prime}$ on
$\mathrm{span}\\{\partial_{c}(\eta_{c},v_{c})\\}$ which upon further
computations leads to (see Lemma 4.3 in [24])

$\langle V^{\prime\prime}(\eta_{c},v_{c})(\delta\eta\delta
v),(\delta\eta,\delta v)\rangle\geq C||\Pi(\delta\eta,\delta v)||_{X}^{2},$
(66)

for some $C>0$ and where $\Pi$ is the orthogonal projection on
$\\{\tau^{\prime}(0)(\eta_{c},v_{c})\\}^{\perp}$. Finally if
$(\delta\eta,\delta v)=(\eta-\eta_{c},v-v_{c})$ then we can show that (see
Lemma 2.1 in [24])

$||\Pi(\delta\eta,\delta
v)||=\inf_{s\in\mathbb{R}}||(\eta,v)-(\eta_{c},v_{c})(\cdot-s)||_{X}.$ (67)

For any solution $(\eta,v)$ to (18), one can readily bound from above
$V((\eta,v))(t)$ for all $t$ by the quantity (67) at $t=0$. Therefore a
solution starting close to the orbit of traveling waves cannot leave it,
ensuring stability. This approach is implemented for the cubic-quintic
nonlinear Schrödinger equation with nonzero conditions at infinity by I. V.
Barashenkov in [2].

######  Acknowledgments.

The author is grateful to André de Laire for helpful discussions and his
interest in the publication of this paper. The author acknowledge support from
the Labex CEMPI (ANR-11-LABX-0007-01) and from the Région Hauts-de-France.

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