# Asymptotic stability of a finite sum of solitary waves for the Zakharov-
Kuznetsov equation

Didier Pilod  Department of Mathematics, University of Bergen, Postbox 7800,
5020 Bergen, Norway<EMAIL_ADDRESS>and  Frédéric Valet  CY Cergy Paris
Université. Laboratoire de recherche Analyse, Géométrie, Modélisation (UMR
CNRS 8088), 2 avenue Adolphe Chauvin, 95302 Cergy-Pontoise Cedex, France
<EMAIL_ADDRESS>
###### Abstract.

We prove the asymptotic stability of a finite sum of well-ordered solitary
waves for the Zakharov-Kuznetsov equation in dimensions two and three.
Moreover, we derive a qualitative version of the orbital stability result
which turns out to be useful for the study of the collision of two solitary
waves.

The proof extends the ideas of Martel, Merle and Tsai for the sub-critical
gKdV equation in dimension one to the higher-dimensional case. It relies on
monotonicity properties on oblique half-spaces and rigidity properties around
one solitary wave introduced by Côte, Muñoz, Pilod and Simpson in dimension
two, and by Farah, Holmer, Roudenko and Yang in dimension three.

##  1\. Introduction

###  1.1. The Zakharov-Kuznetsov equation

In this paper, we consider the Zakharov-Kuzentsov (ZK) equation

$\displaystyle\partial_{t}u+\partial_{x}\left(\Delta u+u^{2}\right)=0,$ (1.1)

where $u=u(t,\mathbf{x})$ is a real-valued function, $t\in\mathbb{R}$,
$\mathbf{x}=(x,\mathbf{y})\in\mathbb{R}\times\mathbb{R}^{d-1}$, $d\geq 2$ is
an integer, and $\Delta=\partial_{x}^{2}+\Delta_{\mathbf{y}}$ denotes the
Laplacian in $\mathbb{R}^{d}$. This equation is a natural high-dimensional
generalization of the Korteweg-de Vries (KdV) equation, which corresponds to
the case of spatial dimension $d=1$. In dimensions $d=2$ and $3$, it describes
the propagation of ionic-acoustic waves in a magnetized cold plasma (see
equation (15.175) in [1]). Equation (1.1) was introduced by Zakharov and
Kuznetsov [14] in the $3$-dimensional case. The rigorous derivation of the ZK
equation from the Euler-Poisson system with magnetized field in the long wave
regime was carried out by Lannes, Linares and Saut in [16]. We also refer to
the work of Han Kwan [10] for the derivation of ZK from the Vlasov-Poisson
system.

Contrary to the KdV equation in one dimension, the Zakharov-Kuznetsov equation
is not known to be completely integrable in dimensions $d\geq 2$.
Nevertheless, it enjoys a Hamiltonian structure, where the Hamiltonian (also
called energy) is given by

$E(u)(t):=\int_{\mathbb{R}^{d}}\left(\frac{|\nabla
u(t,\mathbf{x})|^{2}}{2}-\frac{u(t,\mathbf{x})^{3}}{3}\right)d\mathbf{x}.$
(1.2)

The functionals $\int_{\mathbb{R}^{d}}u(t,\mathbf{x})d\mathbf{x}$ and

$M(u)(t):=\int_{\mathbb{R}^{d}}u(t,\mathbf{x})^{2}d\mathbf{x}$ (1.3)

are also, at least formally, conserved by the flow of ZK. Moreover, note that
the set of solutions of (1.1) is invariant under scaling and translation. More
precisely, if $u$ is a solution of (1.1), then, for any $c>0$ and
$\mathbf{z}\in\mathbb{R}^{d}$, $u_{c,\mathbf{z}}$ is also a solution, where

$\displaystyle
u_{c,\mathbf{z}}(t,\mathbf{x}):=cu(c^{\frac{3}{2}}t,c^{\frac{1}{2}}(\mathbf{x}-\mathbf{z})).$

Here we will focus on the two and three dimensional cases which are
$L^{2}$-subcritical.

The Cauchy problem associated to the ZK equation has been extensively studied
in the recent years. In dimension $d=2$, global well-posedness has been proved
in the energy space $H^{1}(\mathbb{R}^{2})$ by Faminskiĭ [6]. We also refer to
[17, 9, 29, 12] for more results at lower regularity. To prove global well-
posedness in the energy space in dimension $d=3$ is more challenging and the
problem has only been solved recently by Herr and Kinoshita [11] (see [18, 31,
29] for former results). Below, we summarize the results in [6, 11]. Let $d=2$
or $d=3$. Then, for any $u_{0}\in H^{1}(\mathbb{R}^{d})$, there exists a
unique111The uniqueness only holds in a subspace of
$C(\mathbb{R}:H^{1}(\mathbb{R}^{d}))$ solution $u$ of (1.1) in
$C(\mathbb{R}:H^{1}(\mathbb{R}^{d}))$. Moreover, $M(u)(t)=M(u_{0})$ and
$E(u)(t)=E(u_{0})$, for all $t\in\mathbb{R}$.

###  1.2. Solitary waves

The Zakharov-Kuznetsov admits a family of solitary wave solutions propagating
at constant speed in the horizontal direction. These solutions, which play a
fundamental role in the study of the ZK dynamics, are of the form

$u(t,\mathbf{x})=Q_{c}(x-ct,\mathbf{y})\quad\text{with}\quad
Q_{c}(\mathbf{x})\underset{|\mathbf{x}|\to+\infty}{\longrightarrow}0,$

where

$\displaystyle
Q_{c}(\mathbf{x}):=cQ(c^{\frac{1}{2}}\mathbf{x}),\quad\text{for}\ c>0,$ (1.4)

and $Q$ is the _ground state_ of

$\displaystyle-\Delta Q+Q-Q^{2}=0.$ (1.5)

Indeed, it is well-known that the elliptic equation (1.5) has a unique
positive solution $Q$, called _ground state_ , which is smooth and radially
symmetric. We refer to [32, 2] for the existence and [15] for the uniqueness.
Note that, in contrast to the one dimensional case, $Q$ is not explicit in
dimension $d\geq 2$. However, it was proved in [8] that $Q$ and its derivative
decay exponentially at infinity, i.e. for any $\alpha\in\mathbb{N}^{d}$, there
exists $C=C(\alpha)>0$ such that

$\displaystyle\forall|\mathbf{x}|>1,\quad\left|\partial^{\alpha}Q(\mathbf{x})\right|\leq
C|\mathbf{x}|^{-\frac{d-1}{2}}e^{-|\mathbf{x}|}.$ (1.6)

The solitary waves $Q_{c}$ were proved to be orbitally stable in dimension
$d=2$ and $d=3$ by de Bouard [4]. More recently, the asymptotic stability of
the family $\\{Q_{c}\\}_{c>0}$ has been proved by Côte, Muñoz, Pilod and
Simpson [3] in dimension $d=2$ by following the strategy introduced by Martel
and Merle for the generalized KdV equation in dimension $1$ (see [21, 22,
23]). The proof relies on a Liouville property classifying the solutions
localized around a solitary wave. A solution close to a solitary wave
converges up to a subsequence to a limit object, whose emanating solution
satisfies good decay properties. Thanks to the rigidity result, this solution
has to be a solitary wave. The decay of the solution is shown by using
monotonicity results for portions of the mass and the energy away from the
solitary wave, exploiting the fact that solitary waves propagate to the right
while the radiation is ejected to the left.

In higher dimensions, the geometry of the problem is more involved.
Monotonicity properties on oblique half-planes were introduced in [3] in order
to reduce the convergence region to a compact set, where a classical
convergence result can be used. Another challenge that arises in higher
dimensions is the non-explicit nature of the solitary waves. This creates
additional problems in proving the rigidity result close to the solitary wave.
In particular the virial estimate in [3], which is shown through a dual
problem introduced by Martel [20] in the one dimensional case, relies on the
negativity of a scalar product which is checked numerically in dimension
$d=2$. However, this sign condition breaks down in dimension $d=3$. To bypass
this difficulty, Farah, Holmer, Roudenko and Yang derived in [7] a new virial
estimate in dimension $d=3$ based on different orthogonality conditions and
relying on the numerical analysis of the spectra of a linear operator. This
allowed them to extend the asymptotic stability result of the ZK solitary
waves to the dimension $d=3$.

Finally, let us also mention the work of Mendez, Muñoz, Poblete and Pozo [26]
where new virial estimates in dimension $2$ and $3$ are used to prove the
decay of the solutions in large time dependent spatial regions.

###  1.3. Finite sum of solitary waves

In dimension $1$, there exist explicit families of solitary waves called
$N$-solitons. These solutions behave asymptotically at time $-\infty$ and
$+\infty$ as the superposition of $N$ well-ordered solitary waves with
different speeds (see [27]). Since the solitary waves return to their original
shape after interacting, the collision is called _elastic_. The elasticity of
the collision is somehow related to the integrable nature of the KdV equation.

In higher dimensions, such explicit families of $N$-solitons are not known.
However, the study of solutions behaving at infinity as the sum of $N$
solitary waves is important, since these solutions are expected to be
universal attractors of the flow of (1.1). This is related to the _soliton
resolution conjecture_ for ZK, (see [13] for numerical simulations in
dimension $d=3$). The existence and uniqueness of such asymptotic $N$-solitary
waves was shown by Valet in the $2$ and $3$ dimensional setting [33], by using
the strategy introduced by Martel for the gKdV equation in the $1$ dimensional
case [19].

Another important question concerning the finite sum of solitary waves is that
of their stability. In [3], the orbital stability of a sum of $N$ well ordered
solitary waves was proved in $H^{1}$ by extending the fundamental work of
Martel, Merle and Tsai [25] for the gKdV equation to the $2$-dimensional case.
In this paper, we revisit the stability result of [3]. We first derive a
qualitative version of the orbital stability, which will be useful for the
study of the collision of solitary waves for ZK in a forthcoming work [30].
Then, we also prove the asymptotic stability of a finite sum of well ordered
solitary waves.

We state our main results in the theorem below. Without loss of generality, we
will work with two solitary waves.

######  Theorem 1.1.

Let $d=2$ or $d=3$ and let $0<\underline{c}<\bar{c}$ be fixed. There exist
positive constants $k=k(\underline{c},\bar{c})$, $K=K(\underline{c},\bar{c})$
and $A=A(\underline{c},\bar{c})$ such that the following is true. For two
velocities $c_{1}^{0}$ and $c_{2}^{0}$ satisfying
$\underline{c}<c_{2}^{0}<c_{1}^{0}<\bar{c}$ and
$\sigma=c_{1}^{0}-c_{2}^{0}<\min\\{1,\underline{c}\\}$, we define

$\displaystyle\alpha^{\star}:=k\sigma\quad\text{and}\quad
Z^{\star}:=K|\ln\sigma|.$

Let $\alpha<\alpha^{\star}$, $Z>Z^{\star}$,
$(z_{i}^{0},\omega_{i}^{0})\in\mathbb{R}\times\mathbb{R}^{d-1}$, $i=1,2$, and
$u_{0}\in H^{1}(\mathbb{R}^{d})$ satisfy

$\displaystyle\left\|u_{0}-\sum_{i=1}^{2}Q_{c_{i}^{0}}\left(\cdot-\left(z_{i}^{0},\omega_{i}^{0}\right)\right)\right\|_{H^{1}}<\alpha\quad\text{and}\quad
z_{1}^{0}-z_{2}^{0}>Z,$ (1.7)

and let $u\in C(\mathbb{R}:H^{1}(\mathbb{R}^{d}))$ be the solution of (1.1)
evolving from the initial condition $u(0)=u_{0}$. Then, there exist functions
$(z_{i},\omega_{i},c_{i})\in\mathcal{C}^{1}(\mathbb{R}_{+}:\mathbb{R}^{d}\times(0,+\infty))$,
$i=1,2,$ such that:

  * (i)

(Orbital stability) for all $t\geq 0$,

$\displaystyle\left\|u(t)-\sum_{i=1}^{2}Q_{c_{i}(t)}\left(\cdot-(z_{i}(t),\omega_{i}(t))\right)\right\|_{H^{1}}\leq
A\left(\alpha+e^{-\frac{1}{32}\sqrt{\underline{c}}Z}\right);$ (1.8)
$\displaystyle\sum_{i=1}^{2}\left(\left|\dot{z}_{i}(t)-c_{i}(t)\right|+\left|\dot{\omega}_{i}(t)\right|\right)\leq
A\left(\alpha+e^{-\frac{1}{32}\sqrt{\underline{c}}Z}\right);$ (1.9)
$\displaystyle z(t)=z_{1}(t)-z_{2}(t)\geq\frac{1}{2}(Z+\sigma t);$ (1.10)
$\displaystyle\sum_{i=1}^{2}\left|c_{i}(t)-c_{i}^{0}\right|\leq A\alpha.$
(1.11)

  * (ii)

(Asymptotic stability) the limits $c_{1}^{+}=\lim_{+\infty}c_{1}(t)$,
$c_{2}^{+}=\lim_{+\infty}c_{2}(t)$ exist, and

$\displaystyle\sum_{i=1}^{2}\left|c_{i}^{+}-c_{i}^{0}\right|\leq
A\left(\alpha+e^{-\frac{1}{8}\sqrt{\underline{c}}Z}\right);$ (1.12)
$\displaystyle\lim_{t\rightarrow+\infty}\left\|u(t)-\sum_{i=1}^{2}Q_{c_{i}(t)}\left(\cdot-\left(z_{i}(t),\omega_{i}(t)\right)\right)\right\|_{H^{1}(x>\frac{1}{100}\underline{c}t)}=0;$
(1.13)
$\displaystyle(\dot{z}_{i}(t),\dot{\omega}_{i}(t))\underset{t\rightarrow+\infty}{\longrightarrow}(c_{i}^{+},0),\quad
i=1,2.$

######  Remark 1.2.

The theorem can be generalized to $N$ solitary waves with velocities
$0<\underline{c}<c_{N}^{0}<\cdots<c_{1}^{0}<\bar{c}$ by defining

$\displaystyle\sigma:=\min\left(c_{N-1}^{0}-c_{N}^{0},\cdots,c_{1}^{0}-c_{2}^{0}\right).$

######  Remark 1.3.

Regarding the orbital stability results in [25, 3], we precise the dependence
of the constants $\alpha^{\star}$ and $Z^{\star}$ with respect to the
difference $\sigma$ between the velocities of the solitary waves. This
refinement will be useful to study the dynamics of the collision of nearly
equals solitary waves of the ZK equation in a forthcoming work [30].

For this reason and for the sake of simplicity, we choose to focus on the case
where $\sigma=c_{1}^{0}-c_{2}^{0}<\min\\{1,\underline{c}\\}$. Without this
restriction, a similar result would hold by redefining
$\alpha^{\star}=k\min\\{1,\sigma\\}$ and
$Z^{\star}=K\left(1+|\ln(\min\\{1,\sigma\\})|\right)$.

Finally, we refer to Proposition 3.2 in [24] for a different proof of a
similar result in one dimension based on a refined energy functional.

######  Remark 1.4.

It follows from the proof that the asymptotic stability result also holds on
half-spaces of the form

$\mathcal{AS}(t,\theta)=\left\\{\mathbf{x}=(x,\mathbf{y})\in\mathbb{R}^{d}:x-\frac{1}{100}\underline{c}t+(\tan{\theta})y_{j}>0\right\\},$

where $|\theta|<\frac{\pi}{3}$ and $j=1,2$ ($y_{j}=y$ if $d=2$). We refer to
Remark 1.1 in [3] for a similar property around one solitary wave.

Moreover, in dimension $d=3$, the convergence might be improved on conical
regions of the form

$\mathcal{CAS}(t,\theta)=\left\\{\mathbf{x}=(x,y_{1},y_{2})\in\mathbb{R}^{3}:x-\frac{1}{100}\underline{c}t+(\tan{\theta})\sqrt{1+y_{1}^{2}+y_{2}^{2}}>0\right\\},$

$|\theta|<\frac{\pi}{3}$, arguing as in [7].

###  1.4. Outline of the proof

The proof of the orbital stability follows the road map introduced in [25] in
the one dimensional case, and already used in [3] in the two dimensional case.
The main novelty here is that we quantify the range of values of the distance
$\alpha$ of the initial datum $u_{0}$ to the sum of the two solitary waves and
the initial distance of their centers $Z$ in the horizontal direction for
which the orbital stability holds in terms of a lower bound $\underline{c}$, a
higher bound $\bar{c}$ and the difference between the initial velocities
$\sigma:=c_{1}^{0}-c_{2}^{0}$.

The strategy consists in using a bootstrap argument in a tubular neighbourhood
of the sum of the two decoupled solitary waves and assuming for the sake of
contradiction that the maximal time interval $[0,t^{\star})$ for which the
solution stays in this neighbourhood is finite.

On this finite time interval, by using modulation theory, the geometrical
parameters representing the centers $\mathbf{z}_{i}(t)$ and the velocities
$c_{i}(t)$ of the waves are adjusted in such a way that the error
$\epsilon(t)$ between the solution and the sum of the solitary waves satisfies
suitable orthogonality relations. The conservation of the mass and the energy
(see (1.2) and (1.3)) provides a quadratic bound in the $H^{1}$-norm of the
error $\epsilon$ for the variation of $c_{1}(t)+c_{2}(t)$ and of
$c_{1}^{2}(t)+c_{2}^{2}(t)$. Furthermore, since the mass around the first
solitary wave is almost conserved, a monotonicity argument gives a signed
control of the evolution of $c_{1}(t)$. Then, an Abel resummation argument
yields a quadratic bound for the variation of the speeds $c_{1}(t)$ and
$c_{2}(t)$ in terms of the $H^{1}$ norm of the error.

Next, by refining and combining the previous bounds with a coercivity argument
for the linearized operator around each solitary wave, we obtain an estimate
of the error at time $t$ in terms of the initial error and the horizontal
distance between the solitary waves. Note that the coercivity is obtained up
to some orthogonality relations, which had been previously adjusted by
modulation theory. This allows to strictly improve the $H^{1}$-bound of the
error on the whole time interval $[0,t^{\star})$, which concludes the proof of
Theorem 1.1 (i) by contradiction.

The proof of the asymptotic stability of the sum of two solitary waves in
Theorem 1.1 (ii) extends the strategy of Martel and Merle for the generalized
KdV equation in one dimension (see [25, 23]) to the two and three dimensional
cases. It relies on the arguments used for the asymptotic stability around one
solitary wave (see see [21, 22, 23] in one dimension, [3] in two dimensions
and [7] in three dimensions). In particular, the rigidity results (Liouville
properties) around one solitary wave proved in the two dimensional case in [3]
and in the three dimensional case in [7] are the key ingredients.

In the orbital stability setting, the solution $u$ is close to a sum of two
well-ordered modulated solitary waves. We proceed by induction on the solitary
waves from right to left to prove the asymptotic stability.

First, we show that the solution retranslated around the first solitary wave
converges in a half-space to a rescaled ground state. By compactness, we
extract from the translated solution, a subsequence which converges on a
sequence of times tending to $\infty$ to a limit profile $\tilde{u}_{0,1}$,
whose emanating solution $\tilde{u}_{1}$ is orbitally close to the first
solitary wave and satisfies a suitable decay condition. Then, we conclude from
the Liouville property that $\tilde{u}_{1}$ has to be a rescaled ground state.
In order to prove the decay of the limit profile $\tilde{u}_{1}$, we make use
of monotonicity properties for portions of the mass and energy of $u$ on
oblique half spaces of angle $|\theta_{0}|<\frac{\pi}{3}$ around the
hyperplane $\\{x=0\\}$. This reduces the convergence region to a compact set
around the first solitary wave, where classical convergence results can be
used. By showing that the limit ground state is independent of the sequence of
times, we conclude the proof of the asymptotic stability on the left of the
first solitary wave.

Next, we show that the difference between the solution and the first solitary
wave, retranslated around the second solitary wave, converges in a half-space
to another rescaled ground-state. We follow the same strategy as the one
explained above around the first solitary wave and only highlight the main
difference. As in the one dimensional case (see [25]), the decay of the
solution $\tilde{u}_{2}$, emanating from the limit profile $\tilde{u}_{0,2}$,
is proved by using monotonicity arguments on the mass and the energy from the
right of the second solitary wave to the left of the first solitary wave. Note
however that in higher dimension, the geometry plays a significant role and
one has to carefully adapt the times and angles to ensure that the oblique
lines where the derivative of the weight function is localized always stay on
the right of the second solitary wave and on the left of the first one. This
problem does not arise in the case of one solitary wave [3, 7] or in the one
dimensional case in [25, 23].

Finally, we conclude the proof of the asymptotic stability result in a region
that goes slower than the two solitary waves using monotonicity properties as
previously.

The paper is organized as follows: the first sections are devoted to the case
of the dimension $d=2$; in Section 2, we introduce several properties which
will be used to prove the quantitative orbital stability in Section 3 and the
asymptotic stability in Section 4. Finally, in Section 5, we explain the main
modifications of the proof of Theorem 1.1 in dimension $d=3$.

###  1.5. Notation

  * •

The inequality $a\lesssim b$ means that there exists a constant $C$
independent of all the parameters such that $a\leq Cb$. This constant may
change from line to line.

  * •

The inequality $a\lesssim_{\underline{c},\bar{c}}b$ means that there exists a
function $C=C_{\underline{c},\bar{c}}$ that depends only on $\underline{c}$
and $\bar{c}$ such that $a\leq C_{\underline{c},\bar{c}}b$. Once again, the
constant may change from line to line.

  * •

In the case $d=2$, we will denote $\mathbf{x}=(x,y)\in\mathbb{R}^{2}$. Then,
$|\mathbf{x}|=(x^{2}+y^{2})^{\frac{1}{2}}$ and
$\langle\mathbf{x}\rangle=(1+|\mathbf{x}|^{2})^{\frac{1}{2}}.$ We specify the
notation for the $3$-dimensional case in Section 5.

  * •

We will work with real-valued function
$f=f(\mathbf{x})=f(x,y):\mathbb{R}^{d}\to\mathbb{R}$. For $1\leq p\leq+\infty$
and $s\in\mathbb{R}$, $L^{p}(\mathbb{R}^{d})$, respectively
$W^{s,p}(\mathbb{R}^{d})$, denotes the standard Lebesgue spaces, respectively
Sobolev spaces. We also write $H^{s}(\mathbb{R}^{d})=W^{s,2}(\mathbb{R}^{d})$.
For $f,g\in L^{2}(\mathbb{R}^{d})$ two real-valued functions, we denote by
$\langle
f,g\rangle=\int_{\mathbb{R}^{d}}f(\mathbf{x})g(\mathbf{x})d\mathbf{x}$ their
scalar product222Note that there is no risk of confusion with the japanese
bracket defined above for an element $\mathbf{x}\in\mathbb{R}^{d}$..

  * •

For $c>0$, we define the generator of the scaling symmetry $\Lambda_{c}$ by

$\displaystyle\Lambda_{c}(f)(\mathbf{x})=\Lambda_{c}f(\mathbf{x}):=\frac{d}{d\tilde{c}}\left(\tilde{c}f(\sqrt{\tilde{c}}\cdot)\right)_{|\tilde{c}=c}(\mathbf{x})=\left(\left(1+\frac{1}{2}\mathbf{x}\cdot\nabla\right)f\right)\left(\sqrt{c}\mathbf{x}\right),$
(1.14)

In the case $c=1$, we denote

$\Lambda Q=\Lambda_{1}Q$ (1.15)

##  2\. Decomposition and Properties of a solution close to the sum of $2$
solitons

In this section, we state and prove several useful results that will be used
in the proof Theorem 4.1 (i). For simplicity, we focus in the case of
dimension $d=2$, but the same results hold also in dimension $d=3$.

###  2.1. Properties of the linearized operator around the ground state

We recall that the ground state $Q$ of (1.5) is positive, smooth and radially
symmetric. Moreover, it enjoys the exponential decay (1.6).

The linearized operator $L$ around the ground state $Q$ is defined by

$\displaystyle L:=-\Delta+1-2Q.$ (2.1)

The following proposition gathers some classical results on $L$ (see for
example [3] and the references therein).

######  Proposition 2.1.

The operator $L:H^{2}(\mathbb{R}^{2})\subseteq L^{2}(\mathbb{R}^{2})\to
L^{2}(\mathbb{R}^{2})$ satisfies the following properties.

  1. (i)

_Essential spectrum:_ $L$ is a self-adjoint operator and its essential
spectrum is given by $\sigma_{ess}(L)=[1,+\infty)$.

  2. (ii)

_Non-degeneracy of the spectrum:_
$ker(L)=\text{span}\,\\{\partial_{x}Q,\partial_{y}Q\\}$.

  3. (iii)

_Negative eigenvalue:_ $L$ has a unique negative eigenvalue $-\lambda_{0}<0$,
which is simple, and there exists a radially symmetric positive eigenfunction
$\chi_{0}$associated to $-\lambda_{0}$.

  4. (iv)

_Coercivity:_ there exists $C>0$ such that, for any $f\in H^{2}\subset L^{2}$
satisfying $\langle f,\partial_{x}Q\rangle=\langle
f,\partial_{y}Q\rangle=\langle f,Q\rangle=0$:

$\displaystyle\left\langle Lf,f\right\rangle\geq C\|f\|_{L^{2}}^{2}.$

  5. (v)

_Scaling:_ $L\Lambda Q=-Q$. Moreover, $\langle\Lambda
Q,Q\rangle=\frac{1}{2}\int Q>0$.

###  2.2. Modulation close to the sum of decoupled solitary waves

We consider a solution $v$ close to a sum of two decoupled and _well prepared_
solitary waves. In this subsection, we adjust the set of geometric parameters
of translation and velocities of the solitary waves
$\Gamma=\left(z_{1},z_{2},\omega_{1},\omega_{2},c_{1},c_{2}\right)$ to ensure
some important orthogonality relations for the difference between the solution
and the sum of the two solitary waves.

Let $\underline{c}$ and $\bar{c}$ be two positive numbers, such that
$0<\underline{c}<\bar{c}$, representing the lower bound and the upper bound
$\bar{c}$ of the set of velocities. For $Z>1$, $\alpha>0$ and two velocities
$\underline{c}<c_{2}^{0}<c_{1}^{0}<\bar{c}$, we define the sets of geometric
parameters

$\mathbb{R}^{4}_{Z}:=\left\\{(z_{1},z_{2},\omega_{1},\omega_{2})\in\mathbb{R}^{4}\
:\ z_{1}-z_{2}>Z,\right\\}.$ (2.2)

and the tubular neighbourhood of the sum of the two well-ordered solitary
waves $Q_{c_{1}^{0}}$ and $Q_{c_{2}^{0}}$ by

$\displaystyle\mathcal{U}_{c_{1}^{0},c_{2}^{0},Z,\alpha}:=\left\\{u\in
H^{1}(\mathbb{R}^{2}):\
\inf_{(z_{1},z_{2},\omega_{1},\omega_{2})\in\mathbb{R}^{4}_{Z}}\left\|u-\sum_{i=1}^{2}Q_{c_{i}^{0}}\left(\cdot-(z_{i},\omega_{i})\right)\right\|_{H^{1}}<\alpha\right\\}.$
(2.3)

######  Proposition 2.2 (Choice of modulation parameters).

Let $0<\underline{c}<\bar{c}$. There exist positive constants
$A_{1}=A_{1}(\underline{c},\bar{c})$,
$\alpha_{1}^{\star}=\alpha_{1}^{\star}(\underline{c},\bar{c})$, and
$Z_{1}^{\star}=Z_{1}^{\star}(\underline{c},\bar{c})$ such that the following
is true. For $0<\alpha<\alpha_{1}^{\star}$, $Z>Z_{1}^{\star}$ and
$\underline{c}<c_{2}^{0}<c_{1}^{0}<\bar{c}$, let $\mathcal{I}$ be a time
interval and $u\in\mathcal{C}(\mathcal{I}:H^{1}(\mathbb{R}^{2}))$ be a
solution of (1.1) satisfying
$u(t)\in\mathcal{U}_{c_{1}^{0},c_{2}^{0},Z,\alpha}$, for all
$t\in\mathcal{I}$. Then there exists a unique function
$\Gamma=\left(z_{1},z_{2},\omega_{1},\omega_{2},c_{1},c_{2}\right)\in\mathcal{C}^{1}\left(\mathcal{I}:\mathbb{R}^{4}\times(0,+\infty)^{2}\right)$
such that, by defining

$\displaystyle
R_{i}(t,\mathbf{x}):=Q_{c_{i}(t)}\left(\mathbf{x}-\left(z_{i}(t),\omega_{i}(t)\right)\right),\
i=1,2,\quad\text{and}\quad\epsilon(t):=u(t)-\sum_{i=1}^{2}R_{i}(t),$ (2.4)

we have, for any $t\in\mathcal{I}$ and $i\in\\{1,2\\}$,

$\displaystyle\int
R_{i}(t)\epsilon(t)=\int\partial_{x}R_{i}(t)\epsilon(t)=\int\partial_{y}R_{i}(t)\epsilon(t)=0,$
(2.5) $\displaystyle\left\|\epsilon(t)\right\|_{H^{1}}\leq A_{1}\alpha,$ (2.6)
$\displaystyle z(t)=z_{1}(t)-z_{2}(t)>Z-A_{1}\alpha>\frac{3}{4}Z,$ (2.7)
$\displaystyle|c_{i}(t)-c_{i}^{0}|\leq A_{1}\alpha,$ (2.8)
$\displaystyle\frac{14}{15}\underline{c}<c_{2}(t)\quad\text{and}\quad
c_{1}(t)<\frac{16}{15}\bar{c}.$ (2.9)

###### Proof.

The proof of the existence, uniqueness and continuity of $\Gamma$ relies on
classical arguments based on a qualitative version of the implicit function
theorem for functions independent of time (see for example Lemma 1 and Lemma 8
in [25], Proposition 2.2 in [5] or [30] and the references therein).

More precisely, for any $v\in\mathcal{U}_{c_{1}^{0},c_{2}^{0},Z,\alpha}$ and
$\Gamma=(z_{1},z_{2},\omega_{1},\omega_{2},c_{1},c_{2})\in\mathbb{R}^{4}_{Z}\times
I(c_{1}^{0};\gamma)\times I(c_{2}^{0};\gamma)$, where
$I(c_{j}^{0},\gamma)=(c_{j}^{0}-\gamma,c_{j}^{0}+\gamma)$ and $\gamma>0$ is
small enough, we define, for $i=1,2$,

$\displaystyle\bar{R}_{i}(\mathbf{x}):=Q_{c_{i}}(\mathbf{x}-(z_{i},\omega_{i})),\quad\bar{\epsilon}(\mathbf{x}):=v(\mathbf{x})-\sum_{i=1}^{2}\bar{R}_{i}(\mathbf{x})$

and
$\Phi:\mathcal{U}_{c_{1}^{0},c_{2}^{0},Z,\alpha}\times\mathbb{R}^{4}_{Z}\times
I(c_{1}^{0};\gamma)\times I(c_{2}^{0};\gamma)$ by

$\Phi(v,\Gamma)=\left(\int\partial_{x}\bar{R}_{1}\bar{\epsilon},\
\int\partial_{x}\bar{R}_{2}\bar{\epsilon},\
\int\partial_{y}\bar{R}_{1}\bar{\epsilon},\
\int\partial_{y}\bar{R}_{2}\bar{\epsilon},\ \int\bar{R}_{1}\bar{\epsilon},\
\int\bar{R}_{2}\bar{\epsilon}\right).$

The main point consists in proving that $d_{\Gamma}\Phi(v,\Gamma)$ is
invertible. This is ensured by decomposing $d_{\Gamma}\Phi(v,\Gamma)=D+A$,
where

$D=\text{diag}\left(c_{1}^{2}\int(\partial_{x}Q)^{2},c_{2}^{2}\int(\partial_{x}Q)^{2},c_{1}^{2}\int(\partial_{y}Q)^{2},c_{2}^{2}\int(\partial_{y}Q)^{2},\int
Q\Lambda Q,\int Q\Lambda Q\right)$

is invertible (see Proposition 2.1 (v) ) and, by choosing
$\gamma\lesssim\alpha$, $\alpha_{1}^{\star}$ small enough and $Z_{1}^{\star}$
large enough depending only on $\underline{c}$ and $\bar{c}$,
$\|A\|_{L^{\infty}}\leq
e^{-\frac{1}{2}\sqrt{\underline{c}}Z^{\star}}+\alpha^{\star}<\|D^{-1}\|_{L^{\infty}}^{-1}$
(see the computations in the proof of Lemma 2.3 in Appendix A). Thus, it
follows from the implicit function theorem that there exist a positive
constant $A_{1}=A_{1}(\underline{c},\bar{c})$ and a unique $C^{1}$-function
$\Gamma:\mathcal{U}_{c_{1}^{0},c_{2}^{0},Z,\alpha}\to\mathbb{R}^{4}_{Z-A_{1}\alpha}\times
I(c_{1}^{0};A_{1}\alpha)\times I(c_{2}^{0};A_{1}\alpha)$ such that
$\Phi(v,\Gamma(v))=0$, for all
$v\in\mathcal{U}_{\underline{c},\bar{c},Z,\alpha}$.

Now, let $u\in C(\mathcal{I}:\mathbb{R}^{2})$ be a solution of (1.1) such that
$u(t)\in\mathcal{U}_{c_{1}^{0},c_{2}^{0},Z,\alpha}$ for all $t\in\mathcal{I}$.
By abuse of notation, we define $\Gamma(t):=\Gamma(u(t))$, so that $\Gamma\in
C(\mathcal{I})$. The fact that $\Gamma$ is of class $C^{1}$ follows from the
Cauchy-Lipschitz theorem.

The orthogonality conditions (2.5) are equivalent to $\Phi(u(t),\Gamma(t))=0$,
$t\in\mathcal{I}$, while estimates (2.6)-(2.9) follow from the fact that
$(u(t),\Gamma(t))\in\mathcal{U}_{c_{1}^{0},c_{2}^{0},Z,\alpha}\to\mathbb{R}^{4}_{Z}\times
I(c_{1}^{0};A_{1}\alpha)\times I(c_{2}^{0};A_{1}\alpha)$ and the triangle
inequality. ∎

Below, we derive a useful lemma to estimate different integrals involving
$R_{1}$ and $R_{2}$, whose proof is given in Appendix A. With the notation
(2.4) in hand, we introduce, for $i=1,2$,

$\Lambda
R_{i}(t,\mathbf{x}):=\Lambda_{c_{i}(t)}Q(\mathbf{x}-(z_{i}(t),\omega_{i}(t))),$
(2.10)

where $\Lambda_{c_{i}}$ is defined in (1.14).

######  Lemma 2.3.

Under the assumptions of Proposition 2.2, we have, for $i=1,2$,

$\displaystyle\int R_{i}^{2}=c_{i}\int Q^{2},$ (2.11)
$\displaystyle\int(\partial_{x}R_{i})^{2}=\int(\partial_{y}R_{i})^{2}=c_{i}^{2}\int(\partial_{x}Q)^{2},$
(2.12) $\displaystyle\int(\Lambda_{i}R_{i})^{2}=c_{i}^{-1}\int(\Lambda
Q)^{2}.$ (2.13)

Moreover, if $z=z_{1}-z_{2}$, we also have, for $i,j=1,2$ with $i\neq j$,

$\displaystyle\left|\int
R_{i}R_{j}\right|\lesssim\bar{c}e^{-\frac{7}{8}\sqrt{\underline{c}}z},$ (2.14)
$\displaystyle\left|\int R_{i}\Lambda
R_{j}\right|\lesssim\bar{c}\underline{c}^{-1}e^{-\frac{7}{8}\sqrt{\underline{c}}z},$
(2.15)
$\displaystyle\left|\int\partial_{x}R_{i}R_{j}\right|+\left|\int\partial_{y}R_{i}R_{j}\right|\lesssim\bar{c}^{\frac{3}{2}}e^{-\frac{7}{8}\sqrt{\underline{c}}z},$
(2.16)
$\displaystyle\left|\int\partial_{x}R_{i}\partial_{x}R_{j}\right|+\left|\int\partial_{x}R_{i}\partial_{y}R_{j}\right|+\left|\int\partial_{y}R_{i}\partial_{y}R_{j}\right|\lesssim\bar{c}^{\frac{5}{2}}e^{-\frac{7}{8}\sqrt{\underline{c}}z},$
(2.17) $\displaystyle\left|\int\partial_{x}R_{i}\Lambda
R_{j}\right|+\left|\int\partial_{y}R_{i}\Lambda
R_{j}\right|\lesssim\bar{c}^{\frac{3}{2}}\underline{c}^{-1}e^{-\frac{7}{8}\sqrt{\underline{c}}z},$
(2.18)
$\displaystyle\left|\int\partial_{x}(R_{i}R_{j})R_{i}\right|\lesssim\bar{c}^{\frac{5}{2}}e^{-\frac{14}{15}\sqrt{\underline{c}}z}$
(2.19)
$\displaystyle\left|\int\partial_{x}(R_{i}R_{j})\partial_{x}R_{i}\right|\lesssim\bar{c}^{3}e^{-\frac{14}{15}\sqrt{\underline{c}}z}.$
(2.20)

If a solution $u$ falls within the scope of the previous proposition, then
additional assumptions on $\alpha$ and $Z$ provide a dynamical control on the
modulation parameters.

######  Proposition 2.4 (Evolution of the modulation parameters).

Let $0<\underline{c}<\bar{c}$ be fixed. There exist positive constants
$k_{2}=k_{2}(\underline{c},\bar{c})$, $K_{2}=K_{2}(\underline{c},\bar{c})$ and
$A_{2}=A_{2}(\underline{c},\bar{c})$ such that the following is true. For two
velocities $c_{1}^{0}$ and $c_{2}^{0}$ satisfying
$\underline{c}<c_{2}^{0}<c_{1}^{0}<\bar{c}$ and

$\sigma=c_{1}^{0}-c_{2}^{0}<\min\\{1,\underline{c}\\},$ (2.21)

we define

$\displaystyle\alpha^{\star}_{2}:=k_{2}\sigma\quad\text{and}\quad
Z^{\star}_{2}:=K_{2}|\ln\sigma|.$ (2.22)

Let $0<\alpha<\alpha_{2}^{\star}$, $Z>Z_{2}^{\star}$, $\mathcal{I}$ be a time
interval containing $0$ and $u\in C(\mathcal{I}:H^{1}(\mathbb{R}^{2}))$ be a
solution of (1.1) satisfying
$u(t)\in\mathcal{U}_{c_{1}^{0},c_{2}^{0},Z,\alpha}$ for all $t\in\mathcal{I}$.
Then, under the notation of Proposition 2.2, it holds, for all
$t\in\mathcal{I}$,

$\displaystyle\sum_{i=1}^{2}\left(\left|\dot{c}_{i}(t)\right|+\left|\dot{z}_{i}(t)-c_{i}(t)\right|+\left|\dot{\omega}_{i}(t)\right|\right)$
$\displaystyle\quad\quad\quad\quad\leq A_{2}\left(\sum_{i=1}^{2}\left(\int
e^{-\frac{1}{4}\sqrt{\underline{c}}\left|\mathbf{x}-\left(z_{i},\omega_{i}\right)\right|}\epsilon^{2}(\mathbf{x})d\mathbf{x}\right)^{\frac{1}{2}}+e^{-\frac{1}{2}\sqrt{\underline{c}}\left(Z+\sigma
t\right)}\right),$ (2.23) $\displaystyle
z(t)=z_{1}(t)-z_{2}(t)>\frac{3}{4}\left(Z+\sigma t\right),$ (2.24)
$\displaystyle\sum_{i=1}^{2}\left|c_{i}(t)-c_{i}^{0}\right|\leq\frac{1}{32}\sigma.$
(2.25)

###### Proof.

First, since $\sigma<1$, we choose $k_{2}$ and $K_{2}$ small enough to have
$\alpha_{2}^{\star}<\alpha_{1}^{\star}$ and $Z_{2}^{\star}>Z_{1}^{\star}$, so
that we are in the setting of Proposition 2.2.

By inserting (2.4) in (1.1) and using (1.5), we derive the equation satisfied
by $\epsilon$, i.e.

$\displaystyle\partial_{t}\epsilon+\partial_{x}\Delta\epsilon=-\sum_{i=1}^{2}\left(\dot{c}_{i}\Lambda_{i}R_{i}+(c_{i}-\dot{z}_{i})\partial_{x}R_{i}-\dot{\omega}_{i}\partial_{y}R_{i}\right)-\partial_{x}\left(2R_{1}R_{2}+2\epsilon(R_{1}+R_{2})+\epsilon^{2}\right).$
(2.26)

Now, by taking the time derivative of the orthogonality relations (2.5), using
the equation (2.26), estimates (2.6)-(2.7) and Lemma 2.3, we find that

$\displaystyle\sum_{i=1}^{2}\left(\left|\dot{c}_{i}\right|+\left|\dot{z}_{i}-c_{i}\right|+\left|\dot{\omega}_{i}\right|\right)$
$\displaystyle\lesssim_{\underline{c},\bar{c}}\sum_{i=1}^{2}\left(\int
e^{-\frac{1}{2}\sqrt{c_{i}}\left|\mathbf{x}-(z_{i},\omega_{i})\right|}\epsilon^{2}\right)^{\frac{1}{2}}+e^{-\frac{7}{8}\sqrt{\underline{c}}z}\lesssim_{\underline{c},\bar{c}}\left(\alpha+e^{-\frac{1}{2}\sqrt{\underline{c}}Z}\right).$
(2.27)

Therefore, by adjusting the constants $k_{2}$ and $K_{2}$ depending only on
$\underline{c}$ and $\bar{c}$, and by recalling (2.22), we deduce that
$|\dot{z}_{j}-c_{j}|\leq\frac{1}{32}\sigma$, for $j=1,2$. This, combined to
(2.8), yields

$\displaystyle\left|\dot{z}_{1}-\dot{z}_{2}-c_{1}^{0}+c_{2}^{0}\right|\leq\left|\dot{z}_{1}-\dot{z}_{2}-c_{1}+c_{2}\right|+\left|c_{1}-c_{2}-c_{1}^{0}+c_{2}^{0}\right|\leq\frac{1}{16}\sigma.$

An integration of the previous inequality from $0$ to $t$ implies

$\displaystyle\left|z(t)-z(0)-\sigma t\right|\leq\frac{1}{16}\sigma t,$ (2.28)

which, combined with (2.7), yields the bound (2.24). The proof of (2.23) is
then obtained by reinjecting (2.24) into (2.27), while the proof of (2.25) is
a direct consequence of (2.8) and the choice of $\alpha_{2}^{\star}$. ∎

###  2.3. Evolution of the energy

Recall the definition of the energy in (1.2). With $R_{1}$ and $R_{2}$ defined
in (2.4), we introduce

$\displaystyle R(t,\mathbf{x}):=R_{1}(t,\mathbf{x})+R_{2}(t,\mathbf{x}).$

######  Lemma 2.5.

Let $0<\underline{c}<\bar{c}$. There exists a positive constant
$A_{3}=A_{3}(\underline{c},\bar{c})$ such that the following holds. In the
setting of Propositions 2.2 and 2.4, we have, for any $t\in\mathcal{I}$,

$\displaystyle\left|\sum_{i=1}^{2}\left(E(R_{i}(t))-E(R_{i}(0))\right)+\frac{1}{2}\int\left(|\nabla\epsilon|^{2}-2R\epsilon^{2}\right)(t)\right|\leq
A_{3}\left(\|\epsilon(0)\|_{H^{1}}^{2}+\|\epsilon(t)\|_{H^{1}}^{3}+e^{-\frac{1}{2}\sqrt{\underline{c}}Z}\right).$

###### Proof.

Let us develop the energy of the solution at a time $t$. We have

$\displaystyle E(u)=E(R)+\int\nabla R\cdot\nabla\epsilon-\int
R^{2}\epsilon+\int\left(\frac{|\nabla\epsilon|^{2}}{2}-R\epsilon^{2}\right)-\int\frac{\epsilon^{3}}{3}.$

By using Lemma 2.3, we have

$\displaystyle\left|E(R)-E(R_{1})-E(R_{2})\right|\lesssim\int\left|\nabla
R_{1}\cdot\nabla
R_{2}\right|+\int\left(R_{1}^{2}R_{2}+R_{1}R_{2}^{2}\right)\lesssim_{\underline{c},\bar{c}}e^{-\frac{7}{8}\sqrt{\underline{c}}z}.$

In a similar way, we obtain

$\displaystyle\left|\int R^{2}\epsilon-\int R_{1}^{2}\epsilon-\int
R_{2}^{2}\epsilon\right|\leq\int
2R_{1}R_{2}|\epsilon|\lesssim_{\underline{c},\bar{c}}e^{-\frac{7}{8}\sqrt{\underline{c}}z}\|\epsilon\|_{L^{2}}.$

We deduce combining these estimates with (1.5), (2.5) and (2.24) that there
exists a constant $A_{3}=A_{3}(\underline{c},\bar{c})$ such that

$\displaystyle\left|E(u(t))-\sum_{i=1}^{2}E(R_{i}(t))-\int\left(\frac{|\nabla\epsilon|^{2}}{2}-R\epsilon^{2}\right)(t)\right|\leq
A_{3}\left(e^{-\frac{1}{2}\sqrt{\underline{c}}Z}+\|\epsilon(t)\|_{H^{1}}^{3}\right).$

Therefore, we conclude the proof of the lemma by taking the difference of the
previous inequality at times $t$ and $0$ and using the conservation of the
energy. ∎

###  2.4. Almost monotonicity of the mass on the right

In this part, we consider a solution that is close to the sum of two solitary
waves and study the evolution of a half portion of the mass of the solution.
To this aim, define the weight function $\psi$ by

$\displaystyle\psi(x):=\frac{2}{\pi}\arctan(e^{x}),$ (2.29)

so that $\psi$ is increasing on $\mathbb{R}$, $\lim_{x\to-\infty}\psi(x)=0$,
$\lim_{x\to-\infty}\psi(x)=1$ and $\psi(-x)=1-\psi(x)$. Note also that, for
any $x\in\mathbb{R}$,

$\psi^{\prime}(x)=\frac{1}{\pi\cosh(x)}\quad\text{and}\quad|\psi^{(3)}(x)|\leq\psi^{\prime}(x).$
(2.30)

We suppose that a solution of (1.1)
$u\in\mathcal{C}^{1}(\mathcal{I}:H^{1}(\mathbb{R}^{2}))$ satisfies
$u(t)\in\mathcal{U}_{c_{1}^{0},c_{2}^{0},Z,\alpha}$ on an interval
$\mathcal{I}=[0,t_{0}]$. As a consequence of Proposition 2.2, $u$ is
approximated by a sum of two ground states scaled by $c_{i}(t)$ and translated
by $(z_{i}(t),\omega_{i}(t))$.

We define the average function on the $x$-axis by

$\displaystyle m(t):=\frac{1}{2}\left(z_{1}(t)+z_{2}(t)\right).$ (2.31)

From the two fixed velocities $c_{1}^{0}>c_{2}^{0}$ we define, inspired by
[25, 3], the parameter $\gamma$, the scaled weight function $\psi_{s}$ and the
localized mass by

$\displaystyle\gamma:=\frac{1}{2}\left(c_{1}^{0}+c_{2}^{0}\right),\quad\psi_{\gamma}(x):=\psi\left(\frac{\sqrt{\gamma}}{2}x\right)\quad\text{and}\quad
I(t):=\int u^{2}(t,\mathbf{x})\psi_{\gamma}\left(x-m(t)\right)d\mathbf{x}.$
(2.32)

Note from (2.30) that the weight function $\psi_{s}$ satisfies

$\displaystyle\forall
x\in\mathbb{R},\quad\left|\psi_{\gamma}^{(3)}(x)\right|\leq\frac{\gamma}{4}\psi_{\gamma}^{\prime}(x).$
(2.33)

In Picture 1, the two solitary waves are represented by two disks in the
plane. The first solitary wave, represented by a black disk thus on the right,
is more concentrated and faster than the second solitary wave, that is
represented in grey, and is less concentrated and thus moves slower.

$x$$y$time $0$$x=m(0)$$x$$y$time $t_{0}>0$$x=m(t_{0})$ Figure 1. Scheme of the
localization of the two solitary waves and the line $x=m(t)$ at two different
times

######  Lemma 2.6 (Almost monotonicity of the mass).

There exist positive constants $k_{4}=k_{4}(\underline{c},\bar{c})$,
$K_{4}=K_{4}(\underline{c},\bar{c})$ and $A_{4}=A_{4}(\underline{c},\bar{c})$
such that the following is true. Define

$\displaystyle\alpha_{4}^{\star}:=k_{4}\sigma\quad\text{and}\quad
Z_{4}^{\star}:=K_{4}\left|\ln\sigma\right|,$ (2.34)

where we recall that $\sigma=c_{1}^{0}-c_{2}^{0}<\min\\{1,\underline{c}\\}$.
Assume that $Z>Z_{4}^{\star}$ and $\alpha<\alpha_{4}^{\star}$. Then, we have,
for any $t\in\mathcal{I}=[0,t_{0}]$,

$\displaystyle I(t)-I(0)\leq
A_{4}e^{-\frac{1}{16}\sqrt{\underline{c}}(Z+\sigma t))}.$ (2.35)

Before giving the proof of Lemma 2.6, we state some technical estimates for
the interactions of the weight function $\psi_{\gamma}$ with the solitary
waves $R_{i}$.

######  Lemma 2.7.

Under the assumptions of Lemma 2.6, we have

$\displaystyle\|R_{1}\psi_{\gamma}^{\prime}\left(\cdot-m(t)\right)\|_{L^{\infty}}+\|R_{2}\psi_{\gamma}^{\prime}\left(\cdot-m(t)\right)\|_{L^{\infty}}\lesssim\bar{c}e^{-\frac{1}{8}\sqrt{\underline{c}}\left(Z+\sigma
t\right)}$ (2.36)
$\displaystyle\left\|R_{1}\left(\psi_{\gamma}\left(\cdot-m\right)-1\right)\right\|_{L^{2}}+\left\|R_{2}\psi_{\gamma}\left(\cdot-m\right)\right\|_{L^{2}}\lesssim\bar{c}^{\frac{1}{2}}e^{-\frac{1}{16}\sqrt{\underline{c}}\left(Z+\sigma
t\right)}.$ (2.37)

The proof of Lemma 2.7 is given in Appendix A.

###### Proof of Lemma 2.6.

First, we choose $k_{4}\leq k_{2}$ and $Z_{4}\geq Z_{2}$ so that we are in the
setting of Propositions 2.2 and 2.4. Then, it follows from (2.8) and (2.23),
that $\gamma<2\dot{m}(t)$. From this and (2.33), we compute

$\displaystyle\frac{d}{dt}I(t)$
$\displaystyle=\int\left(-3(\partial_{x}u)^{2}-(\partial_{y}u)^{2}+\frac{4}{3}u^{3}-u^{2}\dot{m}\right)\psi_{\gamma}^{\prime}\left(\cdot-m(t)\right)+\int
u^{2}\psi_{\gamma}^{(3)}\left(\cdot-m(t)\right)$
$\displaystyle\quad\leq\int\left(-3(\partial_{x}u)^{2}-(\partial_{y}u)^{2}+\frac{4}{3}u^{3}-\frac{\gamma}{4}u^{2}\right)\psi_{\gamma}^{\prime}\left(\cdot-m(t)\right).$

We now split the cubic term into

$\displaystyle\int u^{3}\psi_{\gamma}^{\prime}\left(\cdot-m(t)\right)=\int
Ru^{2}\psi_{\gamma}^{\prime}\left(\cdot-m(t)\right)+\int
u^{2}(u-R)\psi_{\gamma}^{\prime}\left(\cdot-m(t)\right).$

From (2.36), the Sobolev embedding $H^{1}(\mathbb{R}^{2})\hookrightarrow
L^{3}(\mathbb{R}^{2})$ and (2.30), we obtain

$\displaystyle\int Ru^{2}\psi_{\gamma}^{\prime}\left(\cdot-m(t)\right)$
$\displaystyle\leq\|u\|_{L^{2}}^{2}\|R\psi_{\gamma}^{\prime}\left(\cdot-m(t)\right)\|_{L^{\infty}}\lesssim\left(\bar{c}+\|\epsilon\|_{L^{2}}^{2}\right)\bar{c}^{\frac{3}{2}}e^{-\frac{1}{8}\sqrt{\underline{c}}\left(Z+\sigma
t\right)},$ (2.38) $\displaystyle\int
u^{2}(u-R)\psi_{\gamma}^{\prime}\left(\cdot-m(t)\right)$
$\displaystyle\leq\|u\sqrt{\psi_{\gamma}^{\prime}\left(\cdot-m(t)\right)}\|_{L^{3}}^{2}\|\epsilon\|_{L^{3}}$
$\displaystyle\lesssim\left(\int
u^{2}\psi_{\gamma}^{\prime}\left(\cdot-m(t)\right)+\int|\nabla
u|^{2}\psi_{\gamma}^{\prime}\left(\cdot-m(t)\right)\right)\|\epsilon\|_{H^{1}}.$
(2.39)

Thus, we deduce by using (2.6) and by choosing $k_{4}$ small enough that there
exists a positive constant $C_{\underline{c},\overline{c}}$ such that

$\displaystyle\frac{d}{dt}I(t)\leq-\frac{\min(1,\gamma)}{8}\int\left(u^{2}+|\nabla
u|^{2}\right)\psi_{\gamma}^{\prime}\left(\cdot-m(t)\right)+C_{\underline{c},\overline{c}}e^{-\frac{1}{8}\sqrt{\underline{c}}\left(Z+\sigma
t\right)}.$ (2.40)

An integration from $0$ to $t$ concludes the proof of (2.35) by taking $K_{4}$
larger if necessary. ∎

##  3\. Quantified orbital stability

This section is dedicated to the proof of the orbital stability in Theorem 1.1
(i) oin the $2$-dimensional case.

###  3.1. Bootstrap setting and proof of Theorem 1.1 (i)

Let $\underline{c}$, $\bar{c}$ be such that $0<\underline{c}<\bar{c}$. For
$0<\alpha=\alpha(\underline{c},\bar{c})<\alpha_{1}^{\star}(\underline{c},\bar{c})$,
$Z=Z(\underline{c},\bar{c})>Z_{1}^{\star}(\underline{c},\bar{c})$ and two
velocities $c_{1}^{0}$, $c_{2}^{0}$ such that
$\underline{c}<c_{2}^{0}<c_{1}^{0}<\bar{c}$ and
$\sigma=c_{1}^{0}-c_{2}^{0}<\min\\{1,\underline{c}\\}$, let $u_{0}\in
H^{1}(\mathbb{R}^{2})$ and
$(z_{1}^{0},z_{2}^{0},\omega_{1}^{0},\omega_{2}^{0})\in\mathbb{R}^{4}$ satisfy

$\displaystyle\left\|u_{0}-\sum_{i=1}^{2}Q_{c_{i}^{0}}\left(\cdot-(z_{i}^{0},\omega_{i}^{0})\right)\right\|_{H^{1}}<\alpha\quad\text{and}\quad
z_{1}^{0}-z_{2}^{0}>Z.$ (3.1)

We denote by $u\in C(\mathbb{R}_{+}:H^{1}(\mathbb{R}^{2}))$ the solution of
(1.1) evolving from $u(0)=u_{0}$. As long as
$u(t)\in\mathcal{U}_{c_{1}^{0},c_{2}^{0},\tilde{Z},\tilde{\alpha}}$ defined in
(2.3), for some $0<\tilde{\alpha}<\alpha_{1}^{\star}$ and
$\tilde{Z}>Z_{1}^{\star}$, we consider the $C^{1}$ function
$\Gamma=(z_{1},z_{2},\omega_{1},\omega_{2},c_{1},c_{2})$ constructed in
Proposition 2.2. With the notation (2.4) in hand, we then introduce the
bootstrap estimate

$\displaystyle\|\epsilon(t)\|_{H^{1}}\leq
A\left(\alpha+e^{-\frac{1}{32}\sqrt{\underline{c}}Z}\right);$ (3.2)

for some constant $A=A(\underline{c},\bar{c})>1$ to be chosen later, and where
$\alpha$ and $Z$ are chosen so that
$A\left(\alpha+\sigma^{-\frac{1}{2}}e^{-\frac{1}{32}\sqrt{\underline{c}}Z}\right)<\alpha_{1}^{\star}$
and $\frac{1}{2}(Z+\sigma t)>Z_{1}^{\star}$ (see the statement of Proposition
3.1).

We define

$t^{\star}=\sup\left\\{\tilde{t}>0:\eqref{BS:eps}\ \text{is satisfied for
all}\ t\in[0,\tilde{t}]\right\\}.$ (3.3)

Note that by continuity of $u(t)$, there exists $\tau>0$ such that the
bootstrap estimate (3.2) holds on $[0,\tau]$, so that $t^{\star}$ is well
defined.

The main goal of the rest of the section is to prove that $t^{\star}=+\infty$.
The proof relies on the following key a priori estimate.

######  Proposition 3.1.

Let $0<\underline{c}<\bar{c}$ be fixed. There exist
$A(\underline{c},\bar{c})>1$, $K_{5}(\underline{c},\bar{c})>1$ and
$k_{5}(\underline{c},\bar{c})>0$ such that the following is true. For two
velocities $c_{1}^{0}$ and $c_{2}^{0}$ such that
$\underline{c}<c_{2}^{0}<c_{1}^{0}<\bar{c}$, let $\sigma=c_{1}^{0}-c_{2}^{0}$.
We define

$\displaystyle\alpha_{5}^{\star}:=k_{5}\sigma\quad\text{and}\quad
Z_{5}^{\star}=K_{5}|\ln\sigma|,$ (3.4)

For any $0<\alpha<\alpha_{5}^{\star}$ and $Z>Z_{5}^{\star}$, if a function
$u_{0}\in H^{1}(\mathbb{R}^{2})$ satisfies (3.1) then, with the above notation
in hand, we have, for all $t\in[0,t^{\star}]$,

$\displaystyle\|\epsilon(t)\|_{H^{1}}\leq\frac{1}{2}A\left(\alpha+e^{-\frac{1}{32}\sqrt{\underline{c}}Z}\right);$
(3.5)

Below we give the proof of Theorem 1.1 (i) by assuming Proposition 3.1.

###### Proof of Theorem 1.1 (i).

We assume for the sake of contradiction that $t^{\star}<\infty$. Then, by
Proposition 3.1, we see that (3.5) holds on $[0,t^{\star}]$. Hence, we infer
from a continuity argument that there exists $\tau>0$ such that, for all
$t\in[0,t^{\star}+\tau],$

$\displaystyle\|\epsilon(t)\|_{H^{1}}\leq\frac{3}{4}A\left(\alpha+e^{-\frac{1}{32}\sqrt{\underline{c}}Z}\right).$

This contradicts the definition of $t^{\star}$. Therefore, we deduce that
$t^{\star}=+\infty$, which concludes the proof of (1.8).

Observe as explained above that with $A=A(\underline{c},\bar{c})>0$ fixed by
Proposition 3.1, $k=k(\underline{c},\bar{c})>0$, respectively
$K=K(\underline{c},\bar{c})>0$, is chosen small enough, respectively large
enough, such that $k\leq k_{5}$, $K>K_{5}$ ($k_{5}$ and $K_{5}$ are given by
Proposition 3.1). If $\alpha<\alpha^{\star}=k\sigma$,
$Z>Z^{\star}=K|\ln\sigma|$, then
$A\left(\alpha^{\star}+e^{-\frac{1}{32}\sqrt{\underline{c}}Z^{\star}}\right)<\min\\{\alpha_{1}^{\star},\alpha_{2}^{\star}\\}$
and $\frac{1}{2}(Z^{\star}+\sigma t)>\max\\{Z_{1}^{\star},Z_{2}^{\star}\\}$.
In particular, thanks to (3.2), Propositions 2.2 and 2.4 hold on
$[0,+\infty)$. Therefore, we conclude from (2.8), (2.23) and (2.24), by taking
$A$ larger if necessary, that (1.9), (1.10) and (1.11) hold for all $t\geq 0$.
This concludes the proof of Theorem 1.1 (i). ∎

###  3.2. Proof of Proposition 3.1

Let $A=A(\underline{c},\bar{c})>1$ to be chosen later. First, we take
$k_{5}=k_{5}(\underline{c},\bar{c})>0$ small enough and
$K_{5}=K_{5}(\underline{c},\bar{c})>1$ large enough such that
$\alpha_{5}^{\star}$ and $Z_{5}^{\star}$ defined in (3.4) satisfy

$A\left(\alpha_{5}^{\star}+e^{-\frac{1}{32}\sqrt{\underline{c}}Z_{5}^{\star}}\right)<\min\\{\alpha_{1}^{\star},\alpha_{2}^{\star},\alpha_{4}^{\star}\\}\quad\text{and}\quad\frac{1}{2}(Z_{5}^{\star}+\sigma
t)>\max\\{Z_{1}^{\star},Z_{2}^{\star},Z_{4}^{\star}\\}.$ (3.6)

Thus the decomposition in Proposition 2.2 applies, so that the bootstrap
estimate (3.2) and the estimates in Proposition 2.4 and Lemmas 2.5 and 2.6
hold on $[0,t^{\star}]$.

We first prove a bound for the variation of the velocities $c_{i}(t)$ by a
quadratic quantity in $\|\epsilon(t)\|_{H^{1}}$.

######  Lemma 3.2 (Quadratic control of the variation of $c_{j}(t)$).

There exists a positive constant $A_{8}=A_{8}(\underline{c},\bar{c})$ such
that for any $t\in[0,t^{\star}]$,

$\displaystyle\sum_{i=1}^{2}\left|c_{i}(t)-c_{i}(0)\right|\leq
A_{8}\left(\|\epsilon(t)\|_{H_{1}}^{2}+\|\epsilon(0)\|_{H^{1}}^{2}+e^{-\frac{1}{16}\sqrt{\underline{c}}Z}\right).$
(3.7)

###### Proof.

First step: first order approximation of the localized mass. Recall the
definition of $I$ in (2.32). We claim that there exists
$A_{6}=A_{6}(\underline{c},\bar{c})>0$ such that

$\displaystyle\left|I(t)-I(0)-\left(c_{1}(t)-c_{1}(0)\right)\int
Q^{2}-\int\epsilon^{2}(t)\psi_{\gamma}(\cdot-m(t))\right|\leq
A_{6}\left(\|\epsilon(0)\|_{L^{2}}^{2}+e^{-\frac{1}{16}\sqrt{\underline{c}}Z}\right)$
(3.8)

and

$\displaystyle\left|\left(c_{2}(t)-c_{2}(0)+c_{1}(t)-c_{1}(0)\right)\int
Q^{2}+\int\epsilon^{2}(t)\right|\leq
A_{6}\left(\|\epsilon(0)\|_{L^{2}}^{2}+e^{-\frac{1}{8}\sqrt{\underline{c}}Z}\right).$
(3.9)

We expand the localized mass $I=I(t)$ as

$\displaystyle I=I_{1}+I_{2}+I_{3}+I_{4},$ (3.10)

with

$\displaystyle I_{1}$ $\displaystyle:=\int
R_{1}^{2}+\int\epsilon^{2}\psi_{\gamma}(x-m),$ $\displaystyle I_{2}$
$\displaystyle:=2\int R\epsilon\psi_{\gamma}(x-m),$ $\displaystyle I_{3}$
$\displaystyle:=\int R_{1}^{2}\left(\psi_{\gamma}(x-m)-1\right),$
$\displaystyle I_{4}$
$\displaystyle:=\int(R-R_{1})(R+R_{1})\psi_{\gamma}(x-m).$

We now estimate $I_{2}$, $I_{3}$ and $I_{4}$ separately. By using (2.37) and
the orthogonality relation (2.5), we have

$\displaystyle\left|I_{2}\right|$ $\displaystyle\lesssim\left|\int\epsilon
R_{1}\left(\psi_{\gamma}(\cdot-m)-1\right)\right|+\left|\int\epsilon
R_{2}\psi_{\gamma}(\cdot-m)\right|\lesssim\|\epsilon\|_{L^{2}}\bar{c}^{\frac{1}{2}}e^{-\frac{1}{16}\sqrt{\underline{c}}\left(Z+\sigma
t\right)};$ $\displaystyle\left|I_{3}\right|$
$\displaystyle\lesssim\|R_{1}\|_{L^{2}}\left\|R_{1}\left(\psi_{\gamma}\left(\cdot-
m_{1}\right)-1\right)\right\|_{L^{2}}\lesssim\bar{c}^{\frac{1}{2}}e^{-\frac{1}{16}\sqrt{\underline{c}}\left(Z+\sigma
t\right)};$ $\displaystyle\left|I_{4}\right|$
$\displaystyle\leq\left\|R_{2}\psi_{\gamma}\left(\cdot-
m_{1}\right)\right\|_{L^{2}}\|2R_{1}+R_{2}\|_{L^{2}}\lesssim\bar{c}^{\frac{1}{2}}e^{-\frac{1}{16}\sqrt{\underline{c}}\left(Z+\sigma
t\right)}.$

Taking the difference of (3.10) between the times $t$ and $0$, using (2.11)
and the previous bounds conclude the proof of (3.8).

Similarly, the proof of (3.9) follows from the conservation of the mass
between the times $t$ and $0$, (2.5), (2.11) and (2.14).

Second step: control of the variation of the quadratic speed. Since
$E(Q_{c})=c^{2}E(Q)$, it follows from Lemma 2.5 and (2.24) that

$\displaystyle\left|E(Q)\sum_{i=1}^{2}\left(c_{i}(t)^{2}-c_{i}(0)^{2}\right)+\frac{1}{2}\int\left(|\nabla\epsilon|^{2}-2R\epsilon^{2}\right)(t)\right|\leq
A_{3}\left(\|\epsilon(0)\|_{H^{1}}^{2}+\|\epsilon(t)\|_{H^{1}}^{3}+e^{-\frac{1}{2}\sqrt{\underline{c}}Z}\right).$
(3.11)

Third step: Abel resummation argument. Let us define the quantity

$D(t)=(c_{1}(0)-c_{2}(0))\left(c_{1}(t)-c_{1}(0)\right)+c_{2}(0)\left(c_{1}(t)-c_{1}(0)+c_{2}(t)-c_{2}(0)\right).$
(3.12)

Then, we claim that there exists a positive constant
$A_{7}=A_{7}(\underline{c},\bar{c})$ such that

$\displaystyle\left|D(t)\right|-\frac{1}{2}\sum_{i=1}^{2}|c_{i}(t)-c_{i}(0)|^{2}\leq
A_{7}\left(\|\epsilon(0)\|_{H^{1}}^{2}+\|\epsilon(t)\|_{H^{1}}^{2}+e^{-\frac{1}{8}\sqrt{\underline{c}}Z}\right).$
(3.13)

Indeed, by using a resummation argument, we have
$D(t)=\sum_{i=1}^{2}c_{i}(0)\left(c_{i}(t)-c_{i}(0)\right)$, which combined
with the triangle inequality and (3.11) implies

$\displaystyle\left|D(t)\right|$
$\displaystyle=\frac{1}{2}\left|\sum_{i=1}^{2}\left(c_{i}(t)^{2}-c_{i}(0)^{2}\right)-\sum_{i=1}^{2}(c_{i}(t)-c_{i}(0))^{2}\right|$
(3.14)
$\displaystyle\leq\frac{A_{3}}{2E(Q)}\left(\|\epsilon(0)\|_{H^{1}}^{2}+\|\epsilon(t)\|_{H^{1}}^{3}+e^{-\frac{1}{2}\sqrt{\underline{c}}Z}\right)+\frac{(1+\bar{c})}{2E(Q)}\|\epsilon(t)\|_{H^{1}}^{2}+\frac{1}{2}\sum_{i=1}^{2}|c_{i}(t)-c_{i}(0)|^{2}$

Then, we conclude the proof of (3.13) by using
$\|\epsilon(t)\|_{H^{1}}\leq\alpha_{5}^{\star}<1$.

Fourth step: proof of (3.7). Note that $\sigma\leq 2(c_{1}(0)-c_{2}(0))$
(see(2.25)) and $\underline{c}<c_{2}^{0}$. Combining the statement that if
$a\leq b$ with $b\geq 0$, then $|a|\leq-a+2b$ with (2.35) (3.8), (3.9) and
(3.13) yields

$\displaystyle\frac{1}{2}\left(\sigma\left|c_{1}(t)-c_{1}(0)\right|+\underline{c}\left|c_{1}(t)-c_{1}(0)+c_{2}(t)-c_{2}(0)\right|\right)$
$\displaystyle\leq\left(c_{1}(0)-c_{2}(0)\right)\left|c_{1}(t)-c_{1}(0)\right|+c_{2}(0)\left|c_{1}(t)-c_{1}(0)+c_{2}(t)-c_{2}(0)\right|$
$\displaystyle\leq-D(t)+4\bar{c}\frac{A_{6}}{\int
Q^{2}}\|\epsilon(0)\|_{H^{1}}^{2}+\bar{c}\left(2A_{4}+3\frac{A_{6}}{\int
Q^{2}}\right)e^{-\frac{1}{16}\sqrt{\underline{c}}Z}$
$\displaystyle\leq\left(4\bar{c}\frac{A_{6}}{\int
Q^{2}}+A_{7}\right)\left(\|\epsilon(t)\|_{H^{1}}^{2}+\|\epsilon(0)\|_{H^{1}}^{2}\right)+\left(2A_{4}+4\frac{A_{6}}{\int
Q^{2}}+A_{7}\right)e^{-\frac{1}{16}\sqrt{\underline{c}}Z}$
$\displaystyle\quad+\frac{1}{2}\sum_{i=1}^{2}|c_{i}(t)-c_{i}(0)|^{2}.$

Thus, we deduce from the previous inequality, (2.25) and
$\sigma<\underline{c}$ that

$\displaystyle\frac{1}{4}\sigma\left[|c_{1}(t)-c_{1}(0)|+|c_{2}(t)-c_{2}(0)|\right]$
$\displaystyle\leq\frac{1}{2}\left(\sigma\left|c_{1}(t)-c_{1}(0)\right|+\underline{c}\left|c_{1}(t)-c_{1}(0)+c_{2}(t)-c_{2}(0)\right|\right)$
$\displaystyle\leq 2\left(4\bar{c}\frac{A_{6}}{\int
Q^{2}}+A_{7}\right)\left(\|\epsilon(t)\|_{H^{1}}^{2}+\|\epsilon(0)\|_{H^{1}}^{2}\right)+2\left(2A_{4}+4\frac{A_{6}}{\int
Q^{2}}+A_{7}\right)e^{-\frac{1}{16}\sqrt{\underline{c}}Z},$

which concludes the proof of Lemma 3.2. ∎

Next, we give a bound for the error term $\|\epsilon(t)\|_{H^{1}}$.

######  Lemma 3.3.

There exist positive constants $k_{9}=k_{9}(\underline{c},\bar{c})$,
$K_{9}=K_{9}(\underline{c},\bar{c})$ and $A_{9}=A_{9}(\underline{c},\bar{c})$
such that if

$\displaystyle\alpha_{9}^{\star}:=k_{9}\sigma\quad\text{and}\quad
Z_{9}^{\star}=K_{9}|\ln\sigma|,$ (3.15)

if $0<\alpha<\alpha_{9}^{\star}$ and $Z>Z_{9}^{\star}$, it holds on
$[0,t^{\star}]$,

$\displaystyle\left\|\epsilon(t)\right\|_{H^{1}}^{2}\leq
A_{9}\left(\|\epsilon(0)\|_{H^{1}}^{2}+e^{-\frac{1}{16}\sqrt{\underline{c}}Z}\right).$
(3.16)

Before giving the proof of Lemma 3.3, we state a coercivity lemma whose proof
is obtained arguing as in Lemma 4 [25] with Proposition 2.1 in hand.

######  Lemma 3.4 (Positivity of the quadratic form [25]).

Let $\frac{14}{15}\underline{c}\leq c_{2}(t)\leq
c_{1}(t)\leq\frac{16}{15}\bar{c}$. There exists
$\lambda=\lambda(\underline{c})$ and
$Z_{10}^{\star}=Z_{10}^{\star}(\underline{c})$ such that, if for any
$t\in[0,t^{\star}]$, $z(t)=z_{1}(t)-z_{2}(t)>Z_{10}^{\star}$, then

$\displaystyle\int\left(|\nabla\epsilon|^{2}-2R(t)\epsilon^{2}(t)+c(t,x)\epsilon^{2}(t)\right)\geq\lambda\|\epsilon(t)\|_{H^{1}}^{2}$
(3.17)

where $c(t,x):=c_{2}(t)+(c_{1}(t)-c_{2}(t))\psi_{\gamma}(x-m(t))$.

###### Proof of Lemma 3.3.

On the one hand, we obtain from the energy bound (3.11), the identity (3.14)
and the control of the variation of the velocities (3.7) that

$\displaystyle\frac{1}{2}\int\left(|\nabla\epsilon|^{2}-2R\epsilon^{2}\right)(t)$
$\displaystyle\leq-2E(Q)D(t)+A_{3}\left(\|\epsilon(0)\|_{H^{1}}^{2}+\|\epsilon(t)\|_{H^{1}}^{3}+e^{-\frac{1}{2}\sqrt{\underline{c}}Z}\right)$
$\displaystyle\quad+\left|E(Q)\right|A_{8}^{2}\left(\|\epsilon(t)\|_{H^{1}}^{2}+\|\epsilon(0)\|_{H^{1}}^{2}+e^{-\frac{1}{16}\sqrt{\underline{c}}Z}\right)^{2}.$
(3.18)

On the other hand, by using the identity $-2E(Q)=\frac{1}{2}\int
Q^{2}=\frac{1}{2}M(Q)>0$ (see Appendix B in [3]),
$c_{1}(0)-c_{2}(0)>\frac{1}{2}\sigma>0$ from (2.25), and combining (3.8),
(3.9) and (2.35), we deduce

$\displaystyle-2E(Q)D(t)$
$\displaystyle=\frac{1}{2}M(Q)\left(c_{1}(t)-c_{1}(0)\right)\left(c_{1}(0)-c_{2}(0)\right)+\frac{1}{2}M(Q)\left(c_{1}(t)-c_{1}(0)+c_{2}(t)-c_{2}(0)\right)c_{2}(0)$
$\displaystyle\leq-\frac{1}{2}\int\epsilon^{2}(t)\left((c_{1}(0)-c_{2}(0))\psi_{\delta}(\cdot-m(t))+c_{2}(0)\right)+A_{4}\bar{c}e^{-\frac{1}{16}\sqrt{\underline{c}}Z}$
$\displaystyle\quad+A_{6}\bar{c}\left(\|\epsilon(0)\|_{L^{2}}^{2}+e^{-\frac{1}{16}\sqrt{\underline{c}}Z}\right).$
(3.19)

Moreover, by Lemma 3.2, it holds

$\displaystyle\left|\int\epsilon^{2}(t)\left[\left(\left(c_{1}(0)-c_{2}(0)\right)\psi_{s}\left(\cdot-
m_{1}(t)\right)+c_{2}(0)\right)-\left(\left(c_{1}(t)-c_{2}(t)\right)\psi_{s}\left(\cdot-
m_{1}(t)\right)+c_{2}(t)\right)\right]\right|$
$\displaystyle\leq\|\epsilon(t)\|_{L^{2}}^{2}A_{8}\left(\|\epsilon(t)\|_{H^{1}}^{2}+\|\epsilon(0)\|_{H^{1}}^{2}+e^{-\frac{1}{16}\sqrt{\underline{c}}Z}\right).$
(3.20)

Therefore, we deduce by choosing $k_{9}=k_{9}(\underline{c},\bar{c})$ small
enough and $K_{9}=K_{9}(\underline{c},\bar{c})$ large enough, gathering
(3.17), (3.18), (3.19), (3.20) and recalling that
$\|\epsilon(t)\|_{H^{1}}<\alpha_{9}^{\star}=k_{9}\sigma$ and
$Z>Z_{9}^{\star}=K_{9}|\ln\sigma|$ that, for all $t\in[0,t^{\star}]$,

$\lambda\|\epsilon(t)\|_{H^{1}}^{2}\lesssim_{\underline{c},\bar{c}}\left(\|\epsilon(t)\|_{H^{1}}^{3}+\|\epsilon(0)\|_{H^{1}}^{2}+e^{-\frac{1}{16}\sqrt{\underline{c}}Z}\right),$

which concludes the proof of (3.16) by a continuity argument. ∎

We now conclude the proof of Proposition 3.1. We choose
$A(\underline{c},\bar{c})=4\sqrt{A_{9}(\underline{c},\bar{c})}$ and
$k_{5}=k_{5}(\underline{c},\bar{c})$, $K_{5}=K_{5}(\underline{c},\bar{c})$
such that (3.6) holds and $k_{5}<k_{9}$, $K_{5}>K_{9}$. Then, (3.5) follows
directly from (3.16).

##  4\. Asymptotic stability

###  4.1. Proof of Theorem 1.1 (ii)

In this section, we prove the asymptotic stability part of Theorem 1.1 (ii) in
the $2$-dimensional case.

Suppose that the initial condition $u_{0}$ satisfies (1.7). By using the
orbital stability in Theorem 1.1 (i), there exist functions
$(z_{i},\omega_{i},c_{i})\in\mathcal{C}^{1}(\mathbb{R}_{+}:\mathbb{R}^{2}\times(0,\infty))$,
$i\in(1,2)$ such that, by recalling that $R_{i}$ is defined in (2.4),

$\displaystyle\left\|u(t)-\sum_{i=1}^{2}R_{i}(t)\right\|_{H^{1}}\leq\beta.$
(4.1)

where

$\displaystyle\beta=\beta(\alpha,Z):=A\left(\alpha+e^{-\frac{1}{32}\sqrt{\underline{c}}Z}\right)$
(4.2)

and $\mathbf{z}_{i}:=(z_{i},\omega_{i})$. We also denote by $\mathbf{z}$ the
function $\mathbf{z}=\mathbf{z}_{1}-\mathbf{z}_{2}=(z,\omega)$. Along the
proof, $\beta$ can be chosen as small as desired by increasing
$K=K(\underline{c},\bar{c})$ and decreasing $k=k(\underline{c},\bar{c})$.

Following [25], the strategy is to prove inductively that $u$ converges on a
half-plane containing each solitary wave. To achieve this goal, two important
results will be needed. The first one is the classical orbital stability
result around one solitary wave proved in [4].

######  Theorem 4.1 (Orbital stability of a solitary wave [4]).

Let $0<\underline{c}<\tilde{c}<\bar{c}$. There exist two positive constants
$\alpha_{10}=\alpha_{10}(\underline{c},\bar{c})$ and
$A_{10}=A_{10}(\underline{c},\bar{c})$ such that if $\tilde{u}_{0}\in
H^{1}(\mathbb{R}^{2})$ satisfies
$\left\|\tilde{u}_{0}-Q_{\tilde{c}}\right\|_{H^{1}}\leq\alpha<\alpha_{10}$,
the solution $\tilde{u}$ of 1.1 with $\tilde{u}(0)=\tilde{u}_{0}$ satisfies

$\displaystyle\sup_{t\in\mathbb{R}}\inf_{\mathbf{z}\in\mathbb{R}^{2}}\left\|\tilde{u}(t)-Q_{\tilde{c}}\left(\cdot-\mathbf{z}\right)\right\|_{H^{1}}\leq
A_{10}\alpha.$

The second one is a rigidity property classifying the $L^{2}$-compact solution
around a solitary wave, which is obtained by a direct rescaling of Theorem 1.2
in [3].

######  Theorem 4.2 (Nonlinear Liouville property around $Q$ [3]).

Let $0<\underline{c}<\tilde{c}<\bar{c}$. There exist two positive constants
$\alpha_{11}=\alpha_{11}(\underline{c},\bar{c})$ and
$A_{11}=A_{11}(\underline{c},\bar{c})$ such that the following is true. If
$0<\alpha<\alpha_{11}$,
$\tilde{v}\in\mathcal{C}\left(\mathbb{R}:H^{1}(\mathbb{R}^{2})\right)$ is a
solution to 1.1 satisfying for some function
$(z_{\tilde{v}}(t),\omega_{\tilde{v}}(t))=\mathbf{z}_{\tilde{v}}(t)$,

$\displaystyle\forall
t\in\mathbb{R},\quad\left\|\tilde{v}(\cdot+\mathbf{z}_{\tilde{v}}(t))-Q_{\tilde{c}}\right\|_{H^{1}(\mathbb{R}^{2})}\leq\alpha,$

and for any $\delta>0$, there exists $B>0$ such that

$\displaystyle\sup_{t\in\mathbb{R}}\int_{|x|>B}\tilde{v}^{2}(t,\mathbf{x}+\mathbf{z}_{\tilde{v}}(t))dxdy\leq\delta,$
(4.3)

then there exists $\breve{c}>0$ satisfying
$\left|\breve{c}-\tilde{c}\right|\leq A_{11}\alpha$ and
$(\breve{z},\breve{\omega})\in\mathbb{R}^{2}$ such that

$\displaystyle\tilde{v}(t,x,y)=Q_{\breve{c}}\left(x-\breve{c}t-\breve{z},y-\breve{\omega}\right).$

We first prove the asymptotic stability on a half-plane containing the first
solitary wave.

######  Proposition 4.3.

Under the assumptions of Theorem 1.1, by taking $k=k(\underline{c},\bar{c})>0$
smaller, $K=K(\underline{c},\bar{c})>0$ and $A=A(\underline{c},\bar{c})>0$
larger if necessary, the limit $c_{1}^{+}=\lim_{+\infty}c_{1}(t)$ exists and

$\displaystyle|c_{1}^{+}-c_{1}^{0}|<A\left(\alpha+e^{-\frac{1}{32}\sqrt{\underline{c}}Z}\right),$
(4.4) $\displaystyle
u\left(t,\cdot+\mathbf{z}_{1}(t)\right)\underset{t\rightarrow+\infty}{\longrightarrow}Q_{c_{1}^{+}}\quad\text{in}\quad
H^{1}(x>-B),$ (4.5)
$\displaystyle(\dot{z}_{1}(t),\dot{\omega}_{1}(t))\underset{t\rightarrow+\infty}{\longrightarrow}(c_{1}^{+},0),$
(4.6)

where
$(z_{1},\omega_{1},c_{1})\in\mathcal{C}^{1}(\mathbb{R}_{+}:\mathbb{R}^{2}\times(0,+\infty))$
are the modulation parameters defined in Theorem 1.1 (i).

The proof of Proposition 4.3 relies on the following key result.

######  Proposition 4.4.

Under the assumptions of Theorem 1.1, by taking $k=k(\underline{c},\bar{c})>0$
smaller and $K=K(\underline{c},\bar{c})>0$ larger if necessary, for any
increasing sequence $\\{t_{n}\\}_{n}\rightarrow+\infty$, there exists a
subsequence $\\{t_{n_{k}}\\}_{k}$ and $\tilde{u}_{0,1}\in
H^{1}(\mathbb{R}^{2})$ such that, for any $B>0$,

$\displaystyle
u\left(t_{n_{k}},\cdot+\mathbf{z}_{1}(t_{n_{k}})\right)\underset{k\rightarrow+\infty}{\rightarrow}\tilde{u}_{0,1}\quad\text{in}\quad
H^{1}(x>-B).$

Moreover, the solution $\tilde{u}_{1}$ of 1.1 with initial condition
$\tilde{u}_{1}(0)=\tilde{u}_{0,1}$ satisfies for any $t\in\mathbb{R}$

$\displaystyle\left\|\tilde{u}_{1}\left(t,\cdot+\tilde{\mathbf{z}}_{1}(t)\right)-Q_{c_{1}^{0}}\right\|_{H^{1}}\lesssim\beta$
(4.7)

with $\beta$ defined in (4.2) and for any $(t,x)\in\mathbb{R}^{2}$

$\displaystyle\int_{y}\tilde{u}_{1}^{2}\left(t,\mathbf{x}+\tilde{\mathbf{z}}_{1}(t)\right)dy\lesssim
e^{-\tilde{\kappa}_{1}|x|}$ (4.8)

for some positive constant $\tilde{\kappa}_{1}$, where
$\tilde{\mathbf{z}}_{1}=\left(\tilde{z}_{1},\tilde{\omega}_{1}\right)$
corresponds to the function given by the modulation theory around one solitary
wave.

We give the proof of Proposition 4.3 assuming that Proposition 4.4 holds.

###### Proof of Proposition 4.3.

We deduce from Proposition 4.4 and compactness that, for any sequence
$\\{t_{n}\\}_{n}\rightarrow+\infty$, there exist a subsequence
$\\{t_{n_{k}}\\}_{k}$, a function $\tilde{u}_{0,1}\in H^{1}(\mathbb{R}^{2})$
and a constant $\tilde{c}_{0,1}>0$ such, that for any $B>0$,

$\displaystyle c_{1}(t_{n_{k}})\rightarrow\tilde{c}_{0,1}\quad\text{and}\quad
u\left(t_{n_{k}},\cdot+\mathbf{z}_{1}(t_{n_{k}})\right)\rightarrow\tilde{u}_{0,1}\quad\text{in}\quad
H^{1}(x>-B).$ (4.9)

Moreover, the solution $\tilde{u}_{1}$ to 1.1 with initial condition
$\tilde{u}_{1}(0)=\tilde{u}_{1,0}$ satisfies (4.7) and (4.8).

By applying modulation theory around one solitary wave, (see Lemma 3.1 in
[3]), there exists a unique function
$\left(\tilde{\mathbf{z}}_{1},\tilde{c}_{1}\right)=(\tilde{z}_{1},\tilde{\omega}_{1},\tilde{c}_{1})\in\mathcal{C}^{1}(\mathbb{R}:\mathbb{R}^{2}\times(0,+\infty))$
such that, by denoting
$\tilde{R}_{1}(t,\mathbf{x}):=Q_{\tilde{c}_{1}(t)}\left(\mathbf{x}-\tilde{\mathbf{z}}_{1}(t)\right)$,
it holds for any $t\in\mathbb{R}$

$\displaystyle\begin{cases}&\left\|\tilde{u}_{1}(t)-\tilde{R}_{1}(t)\right\|_{H^{1}}\lesssim\beta,\\\
&\int\left(\tilde{u}_{1}(t)-\tilde{R}_{1}(t)\right)\tilde{R}_{1}(t)=\int\tilde{u}_{1}(t)\partial_{x}\tilde{R}_{1}(t)=\int\tilde{u}_{1}(t)\partial_{y}\tilde{R}_{1}(t)=0.\end{cases}$

Moreover, we infer from the uniqueness of the decomposition around one
solitary wave and passing to the limit when $k\to\infty$ in (2.5) that
$\tilde{\mathbf{z}}_{1}(0)=0$ and $\tilde{c}_{1}(0)=\tilde{c}_{0,1}$.

Therefore, we deduce from Theorem 4.2 that there exists $c_{1}^{+}$ satisfying
$|c_{1}^{+}-c_{1}^{0}|\lesssim\beta$ and
$\mathbf{z}^{+}_{1}=(z_{1}^{+},\omega_{1}^{+})\in\mathbb{R}^{2}$ such that

$\displaystyle\tilde{u}_{1}(t,\mathbf{x})=Q_{c_{1}^{+}}\left(x-c_{1}^{+}t-z_{1}^{+},y-\omega_{1}^{+}\right).$
(4.10)

Notice that the speed $c_{1}^{+}$ depends on the sequence
$\\{t_{n_{k}}\\}_{k}$. The uniqueness of the decomposition provides that
$\mathbf{z}_{1}^{+}=\tilde{\mathbf{z}}_{1}(0)=0$ and
$c_{1}^{+}=\tilde{c}_{1}(0)=\tilde{c}_{0,1}$. Thus for any $B>0$, the sequence
$u\left(t_{n_{k}},\cdot+\mathbf{z}_{1}(t_{n_{k}})\right)-Q_{c_{1}(t_{n_{k}})}$
goes to $0$ in $H^{1}(x>-B)$ as $k$ goes to $\infty$. Since it is true for any
sequence $\\{t_{n}\\}_{n}$ that tends to $+\infty$, we conclude that, for any
$B>0$,

$\displaystyle
u\left(t,\cdot+\mathbf{z}_{1}(t)\right)-Q_{c_{1}(t)}\underset{t\to+\infty}{\longrightarrow}0\quad\text{in}\quad
H^{1}(x>-B).$ (4.11)

We now prove the convergence of the scaling parameter $c_{1}(t)$ as
$t\rightarrow+\infty$. Let us define the function
$\psi_{\underline{c}}(x):=\psi(\frac{\sqrt{\underline{c}}}{2}x)$, where $\psi$
is defined in (2.29). For any $\gamma>0$, there exist a time $T=T(\gamma)>0$
and a distance $x_{1}=x_{1}(\gamma)>0$ such that, arguing as in the proof of
Lemma 4.9, we have, for any $t>t^{\prime}>T$ and $x_{0}>x_{1}$,

$\displaystyle\int
u^{2}(t,\mathbf{x})\psi_{\underline{c}}(x-z_{1}(t)+x_{0})d\mathbf{x}\leq\int
u^{2}(t^{\prime},\mathbf{x})\psi_{\underline{c}}(x-z_{1}(t^{\prime})+x_{0})d\mathbf{x}+\gamma.$

Furthermore, increasing $T$ and $x_{1}$ if necessary, we get from (4.11) and
(1.6), that for any $t>T$,

$\displaystyle\left|\int
u^{2}(t,\mathbf{x})\psi_{\underline{c}}(x-z_{1}(t)+x_{0})d\mathbf{x}-\int
Q_{c_{1}(t)}^{2}(\mathbf{x})d\mathbf{x}\right|\leq\gamma.$

Gathering the previous inequalities at times $t$ and $t^{\prime}$ with
$t>t^{\prime}>T$, we obtain

$\displaystyle\int Q_{c_{1}(t)}^{2}\leq\int
Q_{c_{1}(t^{\prime})}^{2}+3\gamma.$

Since $\int Q_{c}^{2}=c\int Q^{2}$, we conclude that there exists
$c_{1}^{+}>0$ such that

$\displaystyle\left|c_{1}^{+}-c_{1}^{0}\right|\lesssim\beta\quad\text{and}\quad
c_{1}(t)\rightarrow c_{1}^{+}.$

Finally, the proof of (4.6) follows from the modulation estimate around one
solitary wave (see (3.5) in [3]) and (4.11). This concludes the proof of
Proposition 4.3. ∎

We continue with the proof of the asymptotic stability on a half-plane
containing the two solitary waves. Recall that $\mathbf{z}$ is defined at the
beginning of this section.

######  Proposition 4.5.

Under the assumptions of Theorem 1.1, by taking $k=k(\underline{c},\bar{c})>0$
smaller, $K=K(\underline{c},\bar{c})>0$ and $A=A(\underline{c},\bar{c})>0$
larger if necessary, the limit $c_{2}^{+}=\lim_{+\infty}c_{2}(t)$ exists and

$\displaystyle|c_{2}^{+}-c_{2}^{0}|<A\left(\alpha+e^{-\frac{1}{32}\sqrt{\underline{c}}Z}\right),$
(4.12)
$\displaystyle\left(u-R_{1}\right)(t,\cdot+\mathbf{z}_{2}(t))\underset{t\rightarrow+\infty}{\rightarrow}Q_{c_{2}^{+}}\quad\text{in}\quad
H^{1}(x>-B),$ (4.13)
$\displaystyle(\dot{z}_{2}(t),\dot{\omega}_{2}(t))\underset{t\rightarrow+\infty}{\longrightarrow}(c_{2}^{+},0),$
(4.14)

where
$(z_{2},\omega_{2},c_{2})\in\mathcal{C}^{1}(\mathbb{R}_{+}:\mathbb{R}^{2}\times(0,+\infty))$
are the modulation parameters defined in Theorem 1.1 (i).

In order to prove Proposition 4.5, we use the following key result.

######  Proposition 4.6.

Under the assumptions of Theorem 1.1, by taking $k=k(\underline{c},\bar{c})>0$
smaller and $K=K(\underline{c},\bar{c})>0$ larger if necessary, for any
increasing sequence $\\{t_{n}\\}_{n}\rightarrow+\infty$, there exists a
subsequence $\\{t_{n_{k}}\\}_{k}$ and $\tilde{u}_{0,1}\in
H^{1}(\mathbb{R}^{2})$ such that, for any $B>0$,

$\displaystyle\left(u-R_{1}\right)\left(t_{n_{k}},\cdot+\mathbf{z}_{2}(t_{n_{k}})\right)-\tilde{u}_{0,2}\rightarrow
0\quad\text{in}\quad H^{1}(x>-B).$ (4.15)

Moreover, $\tilde{u}_{2}$ solution to (1.1) with initial condition
$\tilde{u}_{2}(0)=\tilde{u}_{0,2}$ satisfies for any $t\in\mathbb{R}$

$\displaystyle\left\|\tilde{u}_{2}\left(t,\cdot+\tilde{\mathbf{z}}_{2}(t)\right)-Q_{c_{2}^{0}}\right\|_{H^{1}}\lesssim\beta$

with $\beta$ defined in (4.2) and for any $(t,x)\in\mathbb{R}^{2}$

$\displaystyle\int_{y}\tilde{u}_{2}^{2}\left(t,\mathbf{x}+\tilde{\mathbf{z}}_{2}(t)\right)dy\lesssim
e^{-\tilde{\kappa}_{2}|x|}$

for some positive constant $\tilde{\kappa}_{2}$, where
$\tilde{\mathbf{z}}_{2}$ corresponds to the function given by modulation
theory around one solitary wave.

Assuming that Proposition 4.6 holds, we prove Proposition 4.5.

###### Proof of Proposition 4.5.

For any increasing sequence $\\{t_{n}\\}_{n}\rightarrow+\infty$, there exist a
subsequence $\\{t_{n_{k}}\\}_{k}$, $\tilde{c}_{0,2}>0$ and $\tilde{u}_{0,2}\in
H^{1}(\mathbb{R}^{2})$ such that for any $B>0$, (4.15) holds and
$\\{c_{2}(t_{n_{k}})\\}_{k}$ converges to $\tilde{c}_{0,2}$. By using the same
arguments as in the proof of Proposition 4.3, we obtain that

$\displaystyle\left(u-R_{1}\right)(t,\cdot+\mathbf{z}_{2}(t))-Q_{c_{2}(t)}\rightarrow
0\quad\text{in}\quad H^{1}(x>-B).$ (4.16)

The convergence of $c_{2}(t)$ to a speed $c_{2}^{+}$ is obtained proceeding as
for $c_{1}^{+}$. For any $\gamma>0$ there exist $T$ and $x_{1}$ large enough
such that for any $t>t^{\prime}>T$ and $x_{0}>x_{1}$, it holds

$\displaystyle\int
u^{2}(t,\mathbf{x})\psi_{\underline{c}}\left(x-z_{2}(t)+x_{0}\right)d\mathbf{x}\leq\int
u^{2}(t^{\prime},\mathbf{x})\psi_{\underline{c}}(x-z_{2}(t^{\prime})+x_{0})d\mathbf{x}+\gamma.$

Moreover, by using (4.16), we have, by taking $T$ and $x_{1}$ larger if
necessary,

$\displaystyle\left|\int
u^{2}(t,\mathbf{x})\psi_{\underline{c}}\left(x-z_{2}(t)+x_{0}\right)d\mathbf{x}-\int
Q_{c_{1}(t)}^{2}-\int Q^{2}_{c_{2}(t)}\right|\leq\gamma,$

which yields the existence of the limit $c_{2}^{+}$ of $c_{2}(t)$ satisfying

$\displaystyle\left|c_{2}^{+}-c_{2}^{0}\right|\lesssim\beta.$

∎

We are now in position to finish the proof of Theorem 1.1 (ii).

###### Proof of Theorem 1.1 (ii).

From (4.13), and the convergence of $c_{2}(t)$ to $c_{2}^{+}$, we have the
convergence

$\displaystyle\left(u-R_{1}-R_{2}\right)(t)\underset{t\rightarrow+\infty}{\rightarrow}0\quad\text{in}\quad
H^{1}(x>z_{2}(t)-B).$ (4.17)

To conclude the proof of Theorem 1.1 (ii), it remains to extend the
convergence above to the region $\\{x>\frac{1}{100}\underline{c}t\\}$.

We first prove the following claim

######  Claim 4.7.

Let $x_{0}>0$ and define the time

$\displaystyle T=T(x_{0}):=\inf\left\\{t>0:\
\frac{1}{2}\frac{1}{100}\underline{c}t>z_{2}(0)-x_{0}\ \text{and}\
z_{2}(0)-x_{0}+(\frac{1}{2}-\frac{1}{100})\underline{c}t>0\right\\}.$

There exists a constant $\kappa>0$ such that, for any $t>T$, there exists a
unique time $t^{\prime}$ satisfying
$z_{2}(t^{\prime})+\frac{1}{200}\underline{c}(t-t^{\prime})-x_{0}=\frac{1}{100}\underline{c}t$.
Furthermore, it holds $t^{\prime}>\kappa t$.

###### Proof of Claim 4.7.

Consider, for a fixed $t>T$, the function
$f(t^{\prime})=z_{2}(t^{\prime})+\frac{1}{2}\frac{1}{100}\underline{c}(t-t^{\prime})-x_{0}-\frac{1}{100}\underline{c}t$
in $\mathcal{C}^{1}(\mathbb{R})$. It satisfies
$\left|f^{\prime}(t^{\prime})-c_{2}(t^{\prime})+\frac{1}{2}\frac{1}{100}\underline{c}\right|\lesssim\alpha\leq\frac{1}{8}\underline{c}$
and thus $\frac{1}{2}\underline{c}<f^{\prime}(t^{\prime})<2\underline{c}$,
which shows that $f$ is an increasing function.

Moreover, it holds
$f(0)=z_{2}(0)-\frac{1}{2}\frac{1}{100}\underline{c}t-x_{0}<0$. On the other
hand, with similar computations to (2.28), it holds
$z_{2}(t)>z_{2}(0)+\frac{1}{2}\underline{c}t$, and by definition of $T$,
$f(t)=z_{2}(t)-x_{0}-\frac{1}{100}\underline{c}t>z_{2}(0)-x_{0}+(\frac{1}{2}-\frac{1}{100})\underline{c}t>0$.
By the mean value theorem, there exists a unique time $t^{\prime}\in(0,t)$
such that $f(t^{\prime})=0$. Integrating $f^{\prime}$ from $0$ to $t^{\prime}$
provides

$\displaystyle
0<-z_{2}(0)+\frac{1}{2}\frac{1}{100}\underline{c}t+x_{0}=-f(0)<2\underline{c}t^{\prime},$

which concludes the lower bound on $t^{\prime}$. ∎

We introduce the weight function

$\displaystyle\phi(x):=\psi\left(\frac{\sqrt{\underline{c}}}{4}x\right)\quad\text{satisfying}\quad\forall
x\in\mathbb{R},\quad\left|\phi^{(3)}(x)\right|\leq\frac{\underline{c}}{16}\phi^{\prime}(x),$
(4.18)

where $\psi$ is defined in (2.29). Let $x_{0}>0$ and define the function

$\displaystyle\check{x}(t)=\check{x}(x_{0},t_{1},t,\mathbf{x}):=x-z_{2}(t_{1})-\frac{1}{200}\underline{c}(t-t_{1})+x_{0}$

and the localized mass

$\displaystyle\check{I}_{x_{0},t_{1}}(t):=\int_{\mathbb{R}^{2}}u(t,\mathbf{x})^{2}\phi(\check{x}(t))d\mathbf{x}.$

Let $t>T(x_{0})$ with $T(x_{0})$ defined in Claim 4.7 and define $t_{1}$ as
the unique time such that
$z_{2}(t)+\frac{1}{200}\underline{c}(t-t_{1})-x_{0}=\frac{1}{100}\underline{c}t$.
Following the almost monotonicity in Lemma 4.9 away from the two solitary
waves, we infer, decreasing $k=k(\underline{c},\bar{c})$ if necessary,

$\displaystyle\check{I}_{x_{0},t_{1}}(t)-\check{I}_{x_{0},t_{1}}(t_{1})\lesssim
e^{-\frac{1}{8}\sqrt{\underline{c}}x_{0}}.$

Furthermore, recalling the definition of $\epsilon$ in (2.4) and using the
orthogonality relations (2.5), it holds for $i\in\\{1,2\\}$ at time $t$ and
$t_{1}$

$\displaystyle\left|\int\epsilon(t,\mathbf{x})R_{i}(t,\mathbf{x})\phi(\check{x})d\mathbf{x}\right|=\left|\int\epsilon(t,\mathbf{x})R_{i}(t,\mathbf{x})\left(1-\phi(\check{x})\right)d\mathbf{x}\right|\lesssim
e^{-\frac{1}{8}\sqrt{\underline{c}}x_{0}}.$

Hence, with the decomposition
$\epsilon^{2}=u^{2}-2\epsilon(R_{1}+R_{2})-(R_{1}+R_{2})^{2}$, we obtain

$\displaystyle\int\epsilon^{2}(t,\mathbf{x})\phi(\check{x}(t))d\mathbf{x}-\int
u^{2}(t,\mathbf{x})\phi(\check{x}(t))d\mathbf{x}+\int\left(R_{1}(t,\mathbf{x})+R_{2}(t,\mathbf{x})\right)^{2}\phi(\check{x}(t))d\mathbf{x}\lesssim
e^{-\frac{1}{8}\sqrt{\underline{c}}x_{0}},$
$\displaystyle\int\epsilon^{2}(t,\mathbf{x})\phi(\check{x}(t))d\mathbf{x}-\int\epsilon^{2}(t_{1},\mathbf{x})\phi(\check{x}(t_{1}))d\mathbf{x}\lesssim
e^{-\frac{1}{8}\sqrt{\underline{c}}x_{0}}+\sum_{i}\left|c_{i}(t)-c_{i}(t_{1})\right|+e^{-\frac{1}{32}\sqrt{\underline{c}}(Z+\sigma
t_{1})}.$

Passing to the limit as $t$ tends to $+\infty$, recalling that
$t_{1}\geq\kappa t$, $c_{1}(t)$ and $c_{2}(t)$ have finite limits and using
the convergence on the left of $R_{2}$ in (4.17), we deduce

$\displaystyle\limsup_{t\rightarrow+\infty}\int\epsilon^{2}(t,\mathbf{x})\phi(\check{x}(t))d\mathbf{x}\lesssim
e^{-\frac{1}{8}\sqrt{\underline{c}}x_{0}}.$

Since the bound holds for any $x_{0}$ and
$\check{x}(t)=x-\frac{1}{100}\underline{c}t$, we obtain

$\displaystyle\lim_{t\rightarrow+\infty}\int\epsilon^{2}(t,\mathbf{x})\phi(x-\frac{1}{100}\underline{c}t)d\mathbf{x}=0.$

Arguing similarly for the derivatives of $\epsilon$, we infer

$\displaystyle\lim_{t\rightarrow+\infty}\int|\nabla\epsilon|^{2}(t,\mathbf{x})\phi(x-\frac{1}{100}\underline{c}t)d\mathbf{x}=0,$

which concludes (1.13).

∎

###  4.2. Proof of Proposition 4.4

We recall that the definition of the modulated parameters
$(\mathbf{z}_{1},c_{1})=(z_{1},\omega_{1},c_{1})$ and
$(\mathbf{z}_{2},c_{2})=(z_{2},\omega_{2},c_{2})$ in Proposition 2.4.

For any increasing sequence $(t_{n})_{n}\rightarrow+\infty$, since the
function $u(t,\cdot+\mathbf{z}_{1}(t))$ is bounded in $H^{1}$ and $c_{1}(t)$
is bounded by $\bar{c}+K_{1}\alpha$ (see (2.8)), there exists a subsequence
$\\{t_{n_{k}}\\}_{k}$, a function $\tilde{u}_{0,1}$ and a constant
$\tilde{c}_{0,1}>0$ such that

$\displaystyle
u\left(t_{n_{k}},\cdot+\mathbf{z}_{1}(t_{n_{k}})\right)\underset{k\rightarrow+\infty}{\rightharpoonup}\tilde{u}_{0,1}\quad\text{in}\quad
H^{1}\quad\text{and}\quad c_{1}(t_{n_{k}})\rightarrow\tilde{c}_{0,1}.$ (4.19)

Moreover, since $z(t)\to+\infty$ as $t\to+\infty$ (see (1.10)), we have
$Q_{c_{2}(t_{n_{k}})}(\cdot+\mathbf{z}(t_{n_{k}}))\rightharpoonup 0$ in
$H^{1}$ as $k\to\infty$, which combined with (4.1) yields

$\displaystyle\left\|\tilde{u}_{0,1}-Q_{\tilde{c}_{0,1}}\right\|_{H^{1}}$
$\displaystyle\leq\liminf_{k\to+\infty}\left\|\left(u-R_{1}-R_{2}\right)\left(t_{n_{k}},\cdot+\mathbf{z}_{1}(t_{n_{k}})\right)\right\|_{H^{1}}\leq\beta.$

Let us denote by $\tilde{u}_{1}$ the solution to 1.1 with initial condition
$\tilde{u}_{1}(0)=\tilde{u}_{0,1}$. From the global well-posedness result in
[6] of 1.1 in $H^{1}$ and the orbital stability result in Theorem 4.1,
$\tilde{u}_{1}$ is defined in $\mathcal{C}(\mathbb{R}:H^{1}(\mathbb{R}^{2}))$
and satisfies

$\displaystyle\left\|\tilde{u}_{1}\left(t,\cdot-\tilde{\mathbf{z}}_{1}(t)\right)-Q_{\tilde{c}_{0,1}}\right\|_{H^{1}}\lesssim\beta,$

where $(\tilde{z}_{1},\tilde{\omega}_{1})$ corresponds to the modulation
function around one solitary wave, see for instance Lemma 3.1 of [3]. Since
the ideas to prove Proposition 4.4 are similar to the proof of Proposition 4.1
of [3], we split the proof of Proposition 4.4 into several steps, summarize
each of the steps and underline the differences with [3].

First step: Monotonicity property of $u$ on the right. To deal with the mass
around the first scaled solitary wave, recall that the function $\phi$ is
defined in (4.18).

######  Lemma 4.8.

For any $x_{0}>0$, we have that

$\displaystyle\limsup_{t\rightarrow+\infty}\int\left(|\nabla
u|^{2}+u^{2}\right)\left(t,\mathbf{x}+\mathbf{z}_{1}(t)\right)\phi\left(x-x_{0}\right)d\mathbf{x}\lesssim
e^{-\frac{1}{4}\sqrt{\underline{c}}x_{0}}.$

The proof is based on the monotonicity property of a weighted mass on the
right of the first solitary wave at two different times, $t_{0}$ and $t$ with
$t_{0}>t$, as shown in Figure 2. The detailed proof can be found in Lemma 4.2
in [3] where the weight function is chosen to be parallel to the vertical
axis.

$\mathbf{x}$$t$$t_{0}$$\psi$$\psi$$\mathbf{z}_{2}(t)$$\mathbf{z}_{1}(t)$$\mathbf{z}_{2}(t_{0})$$\mathbf{z}_{1}(t_{0})$
Figure 2. Evolution of the position of the weight function from the time $t$
to $0$.

Instead of recalling the proof of Lemma 4.8, we introduce the next more
general lemma as in Lemma 4.3 in [3] for which the weight function is based on
a non-vertical line. We thus introduce the following notations. Recall that
$\phi$ is defined in (4.18). For $x_{0}>0$, $t_{0}>0$,
$|\theta_{0}|<\frac{\pi}{3}$, $t\in\mathbb{R}_{+}$ and $x\in\mathbb{R}$ we
define

$\displaystyle\tilde{x}:=\tilde{x}(x_{0},\theta_{0},t_{0},t,\mathbf{x})=x-z_{1}(t_{0})-x_{0}+\frac{\underline{c}}{2}(t_{0}-t)+\tan(\theta_{0})(y-\omega_{1}(t_{0}))$

and

$\displaystyle\tilde{I}_{x_{0},t_{0},\theta_{0}}(t):=\int
u^{2}(t,\mathbf{x})\phi(\tilde{x})d\mathbf{x},\quad\tilde{J}_{x_{0},t_{0},\theta_{0}}(t):=\int\left(\left|\nabla
u\right|^{2}(t,\mathbf{x})-\frac{2}{3}u^{3}(t,\mathbf{x})\right)\phi(\tilde{x})d\mathbf{x}.$
(4.20)

Notice that $\theta_{0}$ corresponds to the angle between the vertical axis
and the line $\tilde{x}(\mathbf{x})=0$, see Figure 3.

$x$$y$$\tilde{x}(t=0)=0$$\tilde{d}_{2}(t_{0},0)$$\tilde{d}_{1}(t_{0},0)$$x$$y$$\tilde{x}(t=t_{0})=0$$\theta_{0}$$\tilde{d}_{2}(t_{0},t_{0})$$\tilde{d}_{1}(t_{0},t_{0})$
Figure 3. Evolution of the position of the line $\tilde{x}=0$ at two different
times : $t=0$ on the left and $t=t_{0}>0$ on the right.

We next compare the weighted mass at two different times, as in [25, 22, 3].

######  Lemma 4.9.

Let $|\theta_{0}|<\frac{\pi}{3}$, there exists a constant $C=C(\theta_{0})$
such that for any $x_{0}>0$ the following holds. There exists a time
$T_{0}=T_{0}(\theta_{0},x_{0})$ such that for any $t_{0}>T_{0}$ and
$t\in[0,t_{0}]$, it holds

$\displaystyle\tilde{I}_{x_{0},t_{0},\theta_{0}}(t_{0})-\tilde{I}_{x_{0},t_{0},\theta_{0}}(t)\leq
Ce^{-\frac{1}{4}\sqrt{\underline{c}}x_{0}},\quad\tilde{J}_{x_{0},t_{0},\theta_{0}}(t_{0})-\tilde{J}_{x_{0},t_{0},\theta_{0}}(t)\leq
Ce^{-\frac{1}{4}\sqrt{\underline{c}}x_{0}}$ (4.21)

and

$\displaystyle\limsup_{t\rightarrow+\infty}\int\left(|\nabla
u|^{2}+u^{2}\right)\left(t,\mathbf{x}+\mathbf{z}_{1}(t)\right)\phi(x+\tan(\theta_{0})y-x_{0})d\mathbf{x}\leq
Ce^{-\frac{1}{4}\sqrt{\underline{c}}x_{0}}.$ (4.22)

###### Proof of Lemma 4.9.

We first explain the need of $T_{0}$. Since $\mathbf{z}_{i}(t)$ stands for the
center of the $i^{\text{th}}$-solitary wave, we expect that the line
$\tilde{x}=0$ at $y=\omega_{i}(t)$ is at the right of $z_{i}(t)$ and at a
distance larger than $x_{0}$. We thus define for fixed $\theta_{0}$ and
$x_{0}$,

$\displaystyle\tilde{d}_{1}(t_{0},t)$
$\displaystyle=\left(z_{1}(t_{0})+x_{0}-\frac{1}{2}\underline{c}(t_{0}-t)-\tan(\theta_{0})\left(\omega_{1}(t)-\omega_{1}(t_{0})\right)\right)-z_{1}(t)$
$\displaystyle\tilde{d}_{2}(t_{0},t)$
$\displaystyle=\left(z_{1}(t_{0})+x_{0}-\frac{1}{2}\underline{c}(t_{0}-t)-\tan(\theta_{0})\left(\omega_{2}(t)-\omega_{1}(t_{0})\right)\right)-z_{2}(t).$

We define the time

$\displaystyle
T_{0}:=\inf\left\\{t_{0}\in\mathbb{R}_{+};\quad\tilde{d}_{1}(t_{0},t_{0})\geq
x_{0}\ \text{and}\ \tilde{d}_{2}(t_{0},t_{0})\geq x_{0}\right\\}.$

First observe that $T_{0}$ is well-defined. Indeed,
$\tilde{d}_{1}(t_{0},t_{0})=x_{0}$ and the function
$f(t_{0}):=\tilde{d}_{2}(t_{0},t_{0})$ is increasing on $\mathbb{R}_{+}$ and
tends to $+\infty$, since
$\left|f^{\prime}(t_{0})-\sigma\right|\leq\frac{1}{4}\sigma$.

We claim that for any $t_{0}>T_{0}$ and $t\in[0,t_{0}]$, the distances are
larger than $x_{0}$, i.e. $\tilde{d}_{1}(t_{0},t)\geq x_{0}$ and
$\tilde{d}_{2}(t_{0},t)\geq x_{0}$. Indeed, by (1.9) and (1.11), for
$i\in\\{1,2\\}$, it holds
$\left|\partial_{t}\tilde{d}_{i}(t_{0},t)-\frac{1}{2}\underline{c}+c_{i}^{0}\right|\lesssim\alpha\leq\frac{1}{4}\underline{c}$,
so that $:t\mapsto\partial_{t}\tilde{d}_{i}(t_{0},t)$ decreases on
$[0,t_{0}]$. From the definition of $T_{0}$, we obtain that
$\tilde{d}_{i}(t_{0},t)\geq x_{0}$ for any $t\in[0,t_{0}]$.

We continue with the proof of (4.21) following [3]. We detail the estimate for
$\tilde{I}_{x_{0},t_{0},\theta_{0}}$ and combine the same reasoning with the
one in the proof of Lemma 3.4 in [3] for the estimate on
$\tilde{J}_{x_{0},t_{0},\theta_{0}}$. By (4.18), we have

$\displaystyle\frac{d}{dt}\tilde{I}_{x_{0},t_{0},\theta_{0}}(t)$
$\displaystyle=-\int\left(3(\partial_{x}u)^{2}+(\partial_{y}u)^{2}+2\partial_{x}u\partial_{y}u\tan(\theta_{0})+\frac{\underline{c}}{2}u^{2}+\frac{4}{3}u^{3}\right)\phi^{\prime}(\tilde{x})$
$\displaystyle\quad+\int
u^{2}\left(1+\tan^{2}(\theta_{0})\right)\phi^{(3)}(\tilde{x}).$

Using Young inequality with a coefficient
$\kappa_{0}^{2}\in(1,3\tan(\theta_{0})^{-2})$, with
$|\theta_{0}|<\frac{\pi}{3}$, we have

$\displaystyle\left|\int
2\partial_{x}u\partial_{y}u\tan(\theta_{0})\phi^{\prime}(\tilde{x})\right|\leq\kappa_{0}^{2}\tan^{2}(\theta_{0})\int(\partial_{x}u)^{2}\phi^{\prime}(\tilde{x})+\frac{1}{\kappa_{0}^{2}}\int(\partial_{y}u)^{2}\phi^{\prime}(\tilde{x})$

and thus for a constant $C=C(\theta_{0})$,

$\displaystyle-\int\left(3(\partial_{x}u)^{2}+(\partial_{y}u)^{2}+2\tan(\theta_{0})\partial_{x}u\partial_{y}u\right)\phi^{\prime}(\tilde{x})\leq-C\int|\nabla
u|^{2}\phi^{\prime}(\tilde{x}).$

From $\tan(\theta_{0})^{2}<3$ and (4.18), we have

$\displaystyle-\int\frac{\underline{c}}{2}u^{2}\phi^{\prime}(\tilde{x})+\int
u^{2}(1+\tan^{2}(\theta_{0}))\phi^{(3)}(\tilde{x})\leq-\frac{\underline{c}}{4}\int
u^{2}\phi^{\prime}(\tilde{x}).$

To deal with the non-linear term, we proceed as in (2.38)-(2.39). First,
notice that an integration of
$\partial_{t}\tilde{d}_{i}(t_{0},t)\leq-\frac{1}{4}\underline{c}$ from $t$ to
$t_{0}$ provides $\tilde{d}_{i}(t_{0},t)\geq
x_{0}+\frac{1}{4}\underline{c}(t_{0}-t)$. Since $Q$ is rotationally invariant
and by performing the change of variable
$(x^{\prime},y^{\prime})=(1+\tan^{2}(\theta_{0}))^{-\frac{1}{2}}\left(x+\tan(\theta_{0})y,\tan(\theta_{0})x-y\right)$,
it holds using similar arguments as in the proof of estimate (2.36),

$\displaystyle\left\|R_{i}(t)\phi^{\prime}(\tilde{x})\right\|_{L^{\infty}}$
$\displaystyle=\frac{\sqrt{\underline{c}}}{4}\left\|c_{i}(t)Q(\sqrt{c_{i}(t)}\mathbf{x})\psi^{\prime}\left(\frac{\sqrt{\underline{c}}}{4}\left(\sqrt{1+\tan^{2}(\theta_{0})}x+\tilde{d_{i}}(t_{0},t)\right)\right)\right\|_{L^{\infty}}$
$\displaystyle\lesssim\exp\left(-\frac{1}{4}\sqrt{\underline{c}}\left(x_{0}+\frac{\underline{c}}{4}(t_{0}-t)\right)\right).$
(4.23)

We obtain the estimate for the non-linear terms

$\displaystyle\left|\int
u^{3}\phi^{\prime}(\tilde{x})\right|\lesssim\|\epsilon\|_{H^{1}}\left(\int\left(|\nabla
u|^{2}+u^{2}\right)\phi^{\prime}(\tilde{x})\right)+e^{-\frac{1}{4}\sqrt{\underline{c}}\left(x_{0}+\frac{1}{4}\underline{c}(t_{0}-t)\right)}.$
(4.24)

Gathering the previous inequalities and (2.24) we get the bound

$\displaystyle\frac{d}{dt}\tilde{I}_{x_{0},t_{0},\theta_{0}}(t)\lesssim
e^{-\frac{1}{4}\sqrt{\underline{c}}\left(x_{0}+\frac{1}{4}\underline{c}(t_{0}-t)\right)}.$

An integration from $t$ to $t_{0}$ concludes (4.21).

The proof of (4.22) relies on estimate (4.21) and
$\lim_{t_{0}\rightarrow+\infty}\tilde{I}_{x_{0},t_{0},\theta_{0}}(t)=0$. ∎

Second step : Strong $L^{2}$ convergence of
$u(t_{n_{k}},\cdot+\mathbf{z}_{1}(t_{n_{k}}))$ to $\tilde{u}_{0,1}$ on the
right.

######  Lemma 4.10.

It holds, for any $B>0$,

$\displaystyle
v_{k,0,1}:=u\left(t_{n_{k}},\cdot+\mathbf{z}_{1}(t_{n_{k}})\right)-\tilde{u}_{0,1}\underset{k\rightarrow+\infty}{\rightarrow}0\quad\text{in}\quad
L^{2}(x>-B).$

The proof of this lemma is the same as the one of Lemma 4.4 in [3]. It relies
on the covering of the half-space ${x>-B}$ by three regions. For any
$\varepsilon>0$, the half-plane is covered by the three (overlapping)
following regions. By choosing $x_{0}$ large enough and
$\theta_{0}\in(0,\frac{\pi}{3})$, we obtain the existence of
$R=R_{\theta_{0},x_{0}}$ such that by the first step and the exponential decay
of $\tilde{u}_{1,0}$, the mass of $v_{k,1,0}$ on each half-plane $\left\\{x\pm
y\geq R\right\\}$ is lower than $\frac{\varepsilon}{3}$. On the remaining
triangle $\left\\{\mathbf{x}\in\mathbb{R}^{2}:x+y\leq R,x-y\leq
R,x>-B\right\\}$, the error $v_{k,1,0}$ tends to $0$ as $k\rightarrow+\infty$
by the Rellich-Kondrachov theorem. Choosing $k$ large enough concludes the
proof of the lemma.

Third step : Exponential decay of $\tilde{u}_{0,1}$ on the right on finite
time intervals.

######  Lemma 4.11.

For any $x_{0}>0$,

$\displaystyle\int\left(|\nabla\tilde{u}_{0,1}|^{2}+\tilde{u}_{0,1}^{2}\right)(\mathbf{x})\phi(x-x_{0})d\mathbf{x}\lesssim
e^{-\frac{1}{4}\sqrt{\underline{c}}x_{0}}.$ (4.25)

Moreover, for all $t_{0}\geq 0$, there exists $K(t_{0})>0$ such that

$\displaystyle\sup_{t\in[0,t_{0}]}\int\left(|\nabla\tilde{u}_{1}|^{2}+\tilde{u}_{1}^{2}\right)\left(t,\mathbf{x}\right)e^{\frac{1}{4}\sqrt{\underline{c}}x}dx\leq
K(t_{0}).$ (4.26)

By weak convergence (4.19), it holds for any $x_{0}>0$

$\displaystyle\left\|\tilde{u}_{0,1}\sqrt{\phi(\cdot-
x_{0})}\right\|_{H^{1}}\leq\liminf_{k}\left\|u\left(t_{n_{k}},\cdot+\mathbf{z}_{1}(t_{n_{k}})\right)\sqrt{\phi(\cdot-
x_{0})}\right\|_{H^{1}},$

which combined to Lemma 2 concludes the proof of (4.25).

The proof of (4.26) is similar to the one of Lemma 4.5 in [3]. The time
derivative of the mass and the energy provides an estimate which can be
integrated by Grönwall’s inequality. In particular, the constant of
integration depends on the size of the interval $t_{0}$.

Fourth step: Strong $L^{2}$ convergence of
$u\left(t_{n_{k}}+t,\cdot+\mathbf{z}_{1}(t_{n_{k}})\right)$ to
$\tilde{u}_{1}(t)$ on the right.

######  Lemma 4.12.

For any $B>0$ and $t\in\mathbb{R}$, it holds

$\displaystyle
u\left(t_{n_{k}}+t,\cdot+\mathbf{z}_{1}(t_{n_{k}})\right)-\tilde{u}_{1}(t)\underset{k\rightarrow+\infty}{\rightarrow}0\quad\text{in}\quad
L^{2}(x>-B)$ (4.27)

and for any $t\in\mathbb{R}$

$\displaystyle
u\left(t_{n_{k}}+t,\cdot+\mathbf{z}_{1}(t_{n_{k}})\right)-\tilde{u}_{1}(t)\underset{k\rightarrow+\infty}{\rightharpoonup}0\quad\text{in}\quad
H^{1}(\mathbb{R}^{2}).$ (4.28)

Moreover, with $\tilde{\mathbf{z}}_{1}=(\tilde{z}_{1},\tilde{\omega}_{1})$ the
$\mathcal{C}^{1}$-function associated to the modulation parameters of
$\tilde{u}_{1}$, it holds

$\displaystyle\mathbf{z}_{1}(t+t_{n_{k}})-\mathbf{z}_{1}(t_{n_{k}})\underset{k\to+\infty}{\rightarrow}\tilde{\mathbf{z}}_{1}(t)\quad\text{and}\quad\tilde{\mathbf{z}}_{1}(0)=0.$
(4.29)

The proof is similar to the proof of Lemma 4.6 in [3]. Let us define

$\displaystyle
v_{k,1}(t,\mathbf{x}):=u\left(t+t_{n_{k}},\mathbf{x}+\mathbf{z}_{1}(t_{n_{k}})\right)-\tilde{u}_{1}(t,\mathbf{x}).$
(4.30)

Computing the time derivative of the mass of the error yields

$\displaystyle\frac{d}{dt}\int
v_{k,1}^{2}(t,\mathbf{x})\phi(x)d\mathbf{x}+\int|\nabla
v_{k,1}(t,\mathbf{x})|^{2}\phi^{\prime}(x)d\mathbf{x}\leq\tilde{K}(t_{0})\int
v_{k,1}(t,\mathbf{x})^{2}\phi(x)d\mathbf{x}.$ (4.31)

By a Gronwall’s inequality on any interval $[0,t_{0}]$ and (4.26), we obtain a
bound on the weighted mass

$\displaystyle\sup_{t\in[0,t_{0}]}\int_{\mathbb{R}^{2}}v_{k,1}(t,\mathbf{x})^{2}\phi(x)d\mathbf{x}\leq\tilde{K}(t_{0})\int_{\mathbb{R}^{2}}v_{k,1}(0,\mathbf{x})^{2}\phi(x)d\mathbf{x}.$

This inequality combined with Lemma 4.10 provides (4.27) for $t_{0}\geq 0$,
whereas (4.28) is deduced by uniqueness of the weak limit. The result for the
negative interval $[\tilde{t}_{1};0]$ is obtained by applying the same
arguments to the sequence of functions
$u\left(t_{n_{k}}+\tilde{t}_{1},\cdot+\mathbf{z}_{1}(t_{n_{k}})\right)$ and
conclude by uniqueness of the Cauchy problem.

The convergence result (4.29) is deduced by passing to the limit in the
orthogonality relations and the uniqueness of the modulation parameters.

Fifth step: Exponential decay of $\tilde{u}_{1}$ on the right.

######  Lemma 4.13.

For any $t\in\mathbb{R}$, $x_{0}>0$

$\displaystyle\int\left(|\nabla\tilde{u}_{1}|^{2}+\tilde{u}_{1}^{2}\right)\left(t,\mathbf{x}+\tilde{\mathbf{z}}_{1}(t)\right)\phi(x-x_{0})d\mathbf{x}\lesssim
e^{-\frac{1}{4}\sqrt{\underline{c}}x_{0}}.$ (4.32)

Furthermore, for any $t\in\mathbb{R}$ and $x>0$,

$\displaystyle\int_{y}\tilde{u}_{1}^{2}\left(t,\mathbf{x}+\tilde{\mathbf{z}}_{1}(t)\right)dy\lesssim
e^{-\frac{1}{4}\sqrt{\underline{c}}x}.$ (4.33)

The exponential decay in (4.22) and the weak convergence in (4.28) provide
(4.32). Then, (4.33) is a consequence of the Sobolev embedding
$H^{1}(\mathbb{R})\hookrightarrow L^{\infty}(\mathbb{R})$ and the Cauchy-
Schwarz inequality applied to (4.32) (see the proof of Lemma 4.7 of [3]).

Sixth step: Strong $H^{1}$-convergence of
$u\left(t_{n_{k}}+t,\cdot+\mathbf{z}_{1}(t_{n_{k}})\right)$ to
$\tilde{u}_{1}(t)$ on the right.

######  Lemma 4.14.

For any $t\in\mathbb{R}$ and $B>0$, it holds

$\displaystyle
u\left(t+t_{n_{k}},\cdot+\mathbf{z}_{1}(t_{n_{k}})\right)-\tilde{u}_{1}(t)\underset{n\rightarrow+\infty}{\rightarrow}0\quad\text{in}\quad
H^{1}(x>-B).$

Let us paint the proof with a broad brush, the details may be found in Lemma
4.8 of [3]. As in the fourth step, it suffices to prove the convergence for a
fixed positive time $t_{0}$. First, an integration of the time derivative of
the mass of the error term as in (4.31) gives

$\displaystyle\int_{t_{0}-1}^{t_{0}}\int|\nabla
v_{k,1}|^{2}(t,\mathbf{x})\phi(x)d\mathbf{x}\underset{k\rightarrow+\infty}{\rightarrow}0.$
(4.34)

Then, computing the time derivative of the energy of the error yields

$\displaystyle\frac{d}{dt}\int\left(\frac{1}{2}|\nabla
v_{k,1}|^{2}+\frac{1}{3}v_{k,1}^{3}\right)(t,\mathbf{x})\phi(x)d\mathbf{x}\leq
K_{2}(t_{0})\int\left(|\nabla
v_{k,1}|^{2}+v_{k,1}\right)(t,\mathbf{x})\phi(x)d\mathbf{x}.$

A first integration from $t\in[t_{0}-1,t_{0}]$ to $t_{0}$ provides

$\displaystyle\int|\nabla
v_{k,1}|^{2}(t_{0},\mathbf{x})\phi(x)d\mathbf{x}\lesssim_{t_{0}}\int|\nabla
v_{k,1}|^{2}(t,\mathbf{x})\phi(x)d\mathbf{x}$
$\displaystyle+\int_{t_{0}-1}^{t_{0}}\int
v_{k,1}(t^{\prime},\mathbf{x})\phi(x)d\mathbf{x}dt^{\prime}+\sup_{t^{\prime}\in[t_{0}-1,t_{0}]}\int
v_{k,1}(t^{\prime},\mathbf{x})^{2}\phi(x)d\mathbf{x}$

and a second integration from $t_{0}-1$ to $t_{0}$ combined with (4.34)
concludes the proof of the lemma.

Seventh step: Exponential decay of $\tilde{u}_{1}$ on the left.

######  Lemma 4.15.

We have that, for any $t\in\mathbb{R}$ and $x_{0}>0$,

$\displaystyle\int\tilde{u}_{1}^{2}\left(t,\mathbf{x}+\tilde{\mathbf{z}}_{1}(t)\right)\left(1-\phi(x+x_{0})\right)d\mathbf{x}\lesssim
e^{-\frac{1}{4}\sqrt{\underline{c}}x_{0}}.$

Furthermore, for any $t\in\mathbb{R}$ and $x<0$,

$\displaystyle\int_{y}\tilde{u}_{1}^{2}\left(t,\mathbf{x}+\tilde{\mathbf{z}}_{1}(t)\right)dy\lesssim
e^{-\frac{1}{4}\sqrt{\underline{c}}x}.$

Notice that in (4.21), the bound was obtained by using the monotonicity of the
mass on the right of the first solitary wave. A similar monotonicity result
holds for the mass of $u$ on the left of the first solitary wave with a weight
going further from the first solitary wave forward in time but always staying
on the right of the second wave (see Picture (4)). We refer to the proof of
Lemma 2.6 with a weight $\psi_{\gamma}(x-z_{1}(t)+\gamma(t-t_{0})+x_{0})$.

$\mathbf{x}$$t_{0}$$t$$\mathbf{z}_{2}(t_{0})$$\mathbf{z}_{1}(t_{0})$$\mathbf{z}_{2}(t)$$\mathbf{z}_{1}(t)$
Figure 4. Evolution of the localization of the weighted mass on the left of
the first solitary wave.

With this monotonicity result in hand, we conclude the proof of Lemma (4.15)
by using (4.27)and arguing exactly as in the proof of Lemma 4.10 in [3].

We conclude the proof of Proposition 4.4 by gathering the seven intermediate
steps.

###  4.3. Proof of Proposition 4.6

The proof of this proposition follows the same ideas of the proof of
Proposition 4.4. We paint the proof in broad brush detailing the differences.

Let $\\{t_{n}\\}_{n}$ a sequence which tends to $+\infty$. Since
$\left(u-R_{1}\right)(t,\cdot+\mathbf{z}_{2}(t))$ is bounded in $H^{1}$ and
$c_{2}(t)$ is bounded in $\mathbb{R}$, there exist a subsequence
$\\{t_{n}\\}_{n_{k}}$, a limit object $\tilde{u}_{0,2}$ and a speed
$\tilde{c}_{0,2}$ such that

$\displaystyle\left(u-R_{1}\right)\left(t_{n_{k}},\cdot+\mathbf{z}_{2}(t_{n_{k}})\right)-\tilde{u}_{0,2}\underset{H^{1}}{\rightharpoonup}0\quad\text{and}\quad
c_{2}(t_{n_{k}})\rightarrow\tilde{c}_{0,2}.$

Furthermore, by using the orbital stability (see Theorem 1.1 (i)), it follows
that

$\displaystyle\left\|\tilde{u}_{0,2}-Q_{\tilde{c}_{0,2}}\right\|_{H^{1}}\leq\liminf_{k\to+\infty}\left\|\left(u-R_{1}-R_{2}\right)\left(t_{n_{k}},\cdot+\mathbf{z}_{2}(t_{n_{k}})\right)\right\|_{H^{1}}\leq\beta.$

Let us denote by $\tilde{u}_{2}$ the solution to 1.1 with initial condition
$\tilde{u}_{2}(0)=\tilde{u}_{0,2}$. By the local well-posedness
$\tilde{u}_{2}\in\mathcal{C}\left(\mathbb{R}:H^{1}(\mathbb{R}^{2})\right)$ and
by the orbital stability result around one solitary wave in Theorem 4.1, it
holds

$\displaystyle\sup_{t\in\mathbb{R}}\left\|\tilde{u}_{2}(t,\cdot+\tilde{\mathbf{z}}_{2}(t))-Q_{\tilde{c}_{0,2}}\right\|_{H^{1}}\lesssim\beta,$

where $\tilde{\mathbf{z}}_{2}=(\tilde{z}_{2},\tilde{\omega}_{2})$ the
modulation parameter of translation.

As before, we split the proof into different steps.

First step: Monotonicity property of $u$ on the right of the second solitary
wave. We state a lemma on the decay of the $H^{1}$-norm on the right of the
second solitary wave, using the function $\phi$ defined in (4.18).

######  Lemma 4.16.

Let $x_{0}>0$ and $|\theta_{1}|<\frac{\pi}{3}$. It holds

$\displaystyle\limsup_{t\rightarrow+\infty}\int\left[\left|\nabla\left(u-R_{1}\right)\right|^{2}+(u-R_{1})^{2}\right]\left(t,\mathbf{x}+\mathbf{z}_{2}(t)\right)\phi(x+\tan(\theta_{1})y-x_{0})dx\lesssim
e^{-\frac{1}{4}\sqrt{\underline{c}}x_{0}}.$

When working on the right of the second solitary wave, we have to take into
account the mass of the first solitary wave (observe the difference with Lemma
4.8).

To prove Lemma 4.16, we state a monotonicity property on the right of the
second solitary wave (see also Section 6 in [28] for an alternative definition
of the lines between two peakons).

$\mathbf{x}$$T_{1}$$t_{0}$$\mathbf{z}_{2}(T_{1})$$\mathbf{z}_{1}(T_{1})$$\mathbf{z}_{2}(t_{0})$$\mathbf{z}_{1}(t_{0})$
Figure 5. Evolution of the weight function at times $t_{0}$ and $T_{1}$

We define

$\displaystyle\hat{x}=\hat{x}\left(x_{0},\theta_{1},t_{0},t,\mathbf{x}\right):=x-z_{2}(t_{0})-x_{0}+\frac{\underline{c}}{2}(t_{0}-t)+\tan(\theta_{1})(y-\omega_{2}(t_{0})),$

and

$\displaystyle\hat{I}_{x_{0},t_{0},\theta_{1}}(t):=\int
u^{2}(t,\mathbf{x})\phi(\hat{x})d\mathbf{x},\quad\hat{J}_{x_{0},t_{0},\theta_{1}}(t):=\int\left(\left|\nabla
u\right|^{2}(t,\mathbf{x})-\frac{2}{3}u^{3}(t,\mathbf{x})\right)\phi(\hat{x})d\mathbf{x}.$
(4.35)

Inspired from the proof of Lemma 4.9, we define the following distances

$\displaystyle\hat{d}_{1}(t_{0},t)$
$\displaystyle:=z_{1}(t)-z_{2}(t_{0})-x_{0}+\frac{\underline{c}}{2}(t_{0}-t)+\tan(\theta_{1})\left(\omega_{1}(t)-\omega_{2}(t_{0})\right),$
$\displaystyle\hat{d}_{2}(t_{0},t)$
$\displaystyle:=-z_{2}(t)+z_{2}(t_{0})+x_{0}-\frac{\underline{c}}{2}\left(t_{0}-t\right)-\tan(\theta_{1})\left(\omega_{2}(t)-\omega_{2}(t_{0})\right).$

######  Lemma 4.17.

Let $\left|\theta_{1}\right|<\frac{\pi}{3}$. There exist $C=C(\theta_{1})$ and
$\kappa>0$ such that, for any $x_{0}>0$ the following holds. There exists
$T_{0}=T_{0}(\theta_{1},x_{0})$ such that for any $t_{0}>T_{0}$, by defining
$T_{1}$ the unique time such that $\hat{d}_{1}(t_{0},T_{1})=x_{0}$, it holds
$T_{1}\geq\kappa t_{0}$ and for any $t\in[T_{1},t_{0}]$,

$\displaystyle\hat{I}_{x_{0},t_{0},\theta_{1}}(t_{0})-\hat{I}_{x_{0},t_{0},\theta_{1}}(t)\leq
Ce^{-\frac{1}{4}\sqrt{\underline{c}}x_{0}},\quad\hat{J}_{x_{0},t_{0},\theta_{1}}(t_{0})-\hat{J}_{x_{0},t_{0},\theta_{1}}(t)\leq
Ce^{-\frac{1}{4}\sqrt{\underline{c}}x_{0}}.$ (4.36)

###### Proof.

Let us fix $\left|\theta_{1}\right|<\frac{\pi}{3}$ and $x_{0}>0$. First, for
$t_{0}>T_{0}$ to be fixed later, we define a positive time $T_{1}$ such that
on a time interval $[T_{1},t_{0}]$, the weight function is located between the
centers $\mathbf{z}_{i}(t)$ of the two scaled solitary waves and at least at a
distance $x_{0}$, see Figure 7.

at time $t_{0}$$x$$y$$R_{2}$$R_{1}$$\theta_{1}$$x_{0}$at time
$T_{1}(t_{0})$$x$$y$$R_{2}$$R_{1}$$\theta_{1}$$x_{0}$ Figure 6. Localization
of the weight function at times $t_{0}$ and $T_{1}$ with respect to the two
solitary waves

$R_{1}$$\mathcal{D}_{-}$$\theta_{0}$$\mathcal{D}_{+}$$\theta_{0}$$\mathcal{D}$$\theta_{1}$$B$$\mathcal{T}$a$\times$
Figure 7. Positions of $\mathcal{D}$, $\mathcal{D}_{-}$ and $\mathcal{D}_{+}$
with respect to the first solitary wave.

First, we want that the line $\hat{x}=0$ remains located between the two
solitary waves with distances $\hat{d}_{1}$ and $\hat{d}_{2}$ larger or equal
to $x_{0}$. Since $\hat{d}_{2}(t_{0},t_{0})=x_{0}$, we define
$\tilde{T_{0}}:=\inf\big{\\{}t_{0}\in\mathbb{R}_{+}:\hat{d}_{1}(t_{0},t_{0})>x_{0}\big{\\}}$.
Observe that $g(t_{0}):=\hat{d}_{1}(t_{0},t_{0})$ satisfies
$\left|g^{\prime}(t_{0})-\sigma\right|\lesssim\alpha$, thus $g$ is increasing
and tends to $+\infty$ by (2.24), which justifies the definition of
$\tilde{T}_{0}$.

We now define the time $T_{0}:=\inf\\{t_{0}>\tilde{T}_{0}:\
\hat{d}_{1}(t_{0},0)<0\\}$. By (2.23) and (2.25), computing
$\frac{d}{dt_{0}}\hat{d}_{1}(t_{0},0)\leq-
c_{2}^{0}+\frac{1}{4}\sigma+\frac{1}{2}\underline{c}<-\frac{1}{4}\underline{c}$,
one obtains that $\hat{d}_{1}(t_{0},0)<0$ for any $t_{0}$ large enough, which
justifies the definition of $T_{0}$.

Next, for $t_{0}>T_{0}$, we define $T_{1}=T_{1}(t_{0})$ such that
$\hat{d}_{1}(t_{0},T_{1})=x_{0}$. Then, it follows that, for any
$t\in[T_{1},t_{0}]$,

$\displaystyle\hat{d}_{1}(t_{0},t)\geq
x_{0}\quad\text{and}\quad\hat{d}_{2}(t_{0},t)\geq x_{0}.$ (4.37)

Indeed, from (1.9) and $\beta\leq\kappa\underline{c}$ with an adequate
$\kappa$, the two distance functions are monotonic

$\displaystyle\left|\partial_{t}\hat{d}_{1}(t_{0},t)-\left(c_{1}^{0}-\frac{\underline{c}}{2}\right)\right|\leq\frac{\underline{c}}{16}\quad\text{and}\quad\left|\partial_{t}\hat{d}_{2}(t_{0},t)+c_{2}^{0}-\frac{\underline{c}}{2}\right|\leq\frac{\underline{c}}{16},$
(4.38)

which implies (4.37).

To prove the lower bound on $T_{1}$, an integration of (4.38) from $T_{1}$ to
$t_{0}$ gives

$\displaystyle\left|\hat{d}_{1}(t_{0},t_{0})-x_{0}-\left(c_{1}^{0}-\frac{\underline{c}}{2}\right)(t_{0}-T_{1})\right|\leq\frac{\underline{c}}{16}(t_{0}-T_{1})$

and since $\hat{d}_{1}(t_{0},t_{0})\leq Z+(1+\frac{1}{16})\sigma t_{0}$ by
(2.28), we obtain, recalling that $\sigma<\underline{c}$, that
$T_{1}\geq\kappa t_{0}$ for some $\kappa>0$.

We prove the monotonocity formula (4.36) for
$\hat{I}_{x_{0},t_{0},\theta_{1}}$. The computations are similar to the ones
in the proof of (4.21), the only difference relies in estimating the cubic
term. Observe that an integration of
$\partial_{t}\hat{d}_{1}(t_{0},t)>\frac{1}{4}\underline{c}$ (see (4.38)) from
$T_{1}$ to $t$ provides
$\hat{d}_{1}(t_{0},t)>x_{0}+\frac{1}{4}\underline{c}(t-T_{1})$ and an
integration of $\partial_{t}\hat{d}_{2}(t_{0},t)<\frac{1}{4}\underline{c}$
from $t$ to $t_{0}$ provides
$\hat{d}_{2}(t_{0},t)>x_{0}-\frac{1}{4}\underline{c}(t_{0}-t)$. Then, inspired
from (4.23), we have

$\displaystyle\left\|R_{1}(t)\phi^{\prime}(\hat{x})\right\|_{L^{\infty}}\lesssim
e^{-\frac{1}{4}\sqrt{\underline{c}}\left(x_{0}+\frac{\underline{c}}{4}(t-T_{1})\right)}\
\text{and}\ \left\|R_{2}(t)\phi^{\prime}(\hat{x})\right\|_{L^{\infty}}\lesssim
e^{-\frac{1}{4}\sqrt{\underline{c}}\left(x_{0}-\frac{\underline{c}}{4}(t_{0}-t)\right)}.$

This implies, arguing as in (4.24), the following bound on the non-linear term

$\displaystyle\left|\frac{4}{3}\int
u^{3}\phi^{\prime}(\hat{x})\right|\lesssim\|\epsilon\|_{H^{1}}\left(\int\left(|\nabla
u|^{2}+u^{2}\right)\phi^{\prime}(\hat{x})\right)+e^{-\frac{1}{4}\sqrt{\underline{c}}\left(x_{0}+\frac{1}{4}\underline{c}(t_{0}-t)\right)}+e^{-\frac{1}{4}\sqrt{\underline{c}}\left(x_{0}+\frac{1}{4}\underline{c}(t-T_{1})\right)}.$

The rest of the proof of (4.36) is similar to the one of (2.35).

The proof of the formula for $\hat{J}_{x_{0},t_{0},\theta_{1}}$ is obtained by
combining the ideas explained above with the computations in Lemma 3.4 in [3].
∎

###### Proof of Lemma 4.16.

We only deal with the $L^{2}$-weighted norm, the case of the gradient is
similar. We first claim that, for $t_{0}>T_{0}$ large enough,

$\displaystyle\int\left(u-R_{1}\right)^{2}\left(t_{0},\mathbf{x}+\mathbf{z}_{2}(t_{0})\right)\phi(x-\tan\theta_{1}y-x_{0})d\mathbf{x}$
$\displaystyle\lesssim\int
u^{2}(T_{1},\mathbf{x}+\mathbf{z}_{1}(T_{1}))\phi(x-\tan\theta_{1}y+x_{0})d\mathbf{x}-\int
R_{1}^{2}(T_{1},\mathbf{x})d\mathbf{x}+e^{-\frac{1}{4}\sqrt{\underline{c}}x_{0}}.$
(4.39)

Indeed, it holds

$\displaystyle\int\left[\left(u-R_{1}\right)^{2}-u^{2}+R_{1}^{2}\right](t_{0},\mathbf{x}+\mathbf{z}_{2}(t_{0}))\phi\left(x-\tan(\theta_{1})y-x_{0}\right)d\mathbf{x}$
$\displaystyle=2\int\left[(R_{1}-u)R_{1}\right](t_{0},\mathbf{x}+\mathbf{z}_{2}(t_{0}))\phi(x-\tan(\theta_{1})y-x_{0})d\mathbf{x}.$
(4.40)

Since $Q_{c_{1}(t_{0})}$ decays exponentially and
$u(t_{0},\cdot+\mathbf{z}_{1}(t_{0}))-Q_{c_{1}(t_{0})}$ converges to $0$ in
$L^{2}(x>-B)$ by (4.5), the right-hand side of (4.40) can be chosen for
$t_{0}$ large enough lower than $e^{-\frac{1}{4}\sqrt{\underline{c}}x_{0}}$.
Next, we notice as in (2.37) that

$\displaystyle\left|\int
R_{1}^{2}(t_{0},\mathbf{x}+\mathbf{z}_{2}(t_{0}))\phi\left(x-\tan(\theta_{1})y-x_{0}\right)-\int
R_{1}^{2}(t_{0})\right|\lesssim e^{-\frac{1}{4}\sqrt{\underline{c}}x_{0}}.$
(4.41)

On the other, by recalling $\hat{d}_{1}(t_{0},T_{1})=x_{0}$, (4.36) yields

$\displaystyle\int
u^{2}(t_{0},\mathbf{x}+\mathbf{z}_{2}(t_{0}))\phi\left(x-\tan(\theta_{1})y-x_{0}\right)d\mathbf{x}-\int
u^{2}(T_{1},\mathbf{x}+\mathbf{z}_{1}(T_{1}))\phi(x-\tan(\theta_{1})y+x_{0})d\mathbf{x}$
$\displaystyle\lesssim e^{-\frac{1}{4}\sqrt{\underline{c}}x_{0}}.$ (4.42)

Gathering (4.40), (4.41) and (4.42) and using that $c_{1}(t)$ converges at
infinity conclude the proof of (4.39).

To conclude the proof, it suffices to prove that the $\limsup$ as $t_{0}$ (and
thus $T_{1}$) goes to $\infty$ of the right-hand side of (4.39) is bounded by
$Ce^{-\frac{1}{8}\sqrt{\underline{c}}x_{0}}$. Let
$|\theta_{0}|\in(|\theta_{1}|;\frac{\pi}{3})$ and define the three lines
$\mathcal{D}$, $\mathcal{D}_{+}$, $\mathcal{D}_{-}$ at time $T_{1}(t_{0})$
around the first solitary wave (see Figure 7) by

$\displaystyle\mathcal{D}_{\pm}:=\left\\{x-z_{1}(T_{1})\pm\tan(\theta_{0})\left(y-\omega_{1}(T_{1})\right)-x_{0}=0\right\\},$
$\displaystyle\mathcal{D}:=\left\\{x-z_{1}(T_{1})-\tan(\theta_{1})\left(y-\omega_{1}(T_{1})\right)-x_{0}=0\right\\}.$

Now, as explained in Figure 7, choose $B=B(x_{0},\theta_{0},\theta_{1})>0$
such that any point ${\bf a}=(a,b)$ below $\mathcal{D}_{-}$ and above
$\mathcal{D}_{+}$ with $a<-B$ is closer to $\mathcal{D}_{-}$ or
$\mathcal{D}_{+}$ than to $\mathcal{D}$ and denote the triangle

$\mathcal{T}=\left\\{\mathbf{x}:x>-B;x<z_{1}(t)\pm\tan(\theta_{1})\left(y-\omega_{1}(t)\right)+x_{0}\right\\}.$

Since $\phi$ is strictly increasing, it follows from this construction that

$\displaystyle\int
u^{2}\left(T_{1},\mathbf{x}+\mathbf{z}_{1}(T_{1})\right)\phi(x-\tan(\theta_{1})y+x_{0})d\mathbf{x}\leq\int_{\mathcal{T}}u^{2}\left(T_{1},\mathbf{x}+\mathbf{z}_{1}(T_{1})\right)d\mathbf{x}$
$\displaystyle\quad+\int
u^{2}\left(T_{1},\mathbf{x}+\mathbf{z}_{1}(T_{1})\right)\left(\phi\left(x+\tan(\theta_{0})y-x_{0}\right)+\phi\left(x-\tan(\theta_{0})y-x_{0}\right)\right)d\mathbf{x}.$

Therefore, we conclude the proof of Lemma 4.16 gathering (4.39), (4.22) and
the strong convergence in (4.5), and recalling the lower bound on
$T_{1}\geq\kappa t_{0}$. ∎

Second step: Strong $L^{2}$ convergence of
$\left(u-R_{1}\right)(t_{n_{k}},\cdot+\mathbf{z}_{2}(t_{n_{k}}))$ to
$\tilde{u}_{0,2}$ on the right.

######  Lemma 4.18.

It holds, for any $B>0$,

$\displaystyle
v_{k,2,0}:=\left(u-R_{1}\right)(t_{n_{k}},\cdot+\mathbf{z}_{2}(t_{n_{k}}))-\tilde{u}_{0,2}\underset{k\rightarrow+\infty}{\rightarrow}0\quad\text{in}\quad
L^{2}(x>-B).$

The proof is the same as in Lemma 4.10 by using Lemma 4.16 instead of Lemma
4.8.

Third step: Exponential decay of $\tilde{u}_{0,2}$ on the right on finite time
intervals.

######  Lemma 4.19.

For any $x_{0}>0$, it holds

$\displaystyle\int\left(|\nabla\tilde{u}_{0,2}|^{2}+\tilde{u}_{0,2}^{2}\right)(\mathbf{x})\phi(x-x_{0})d\mathbf{x}\lesssim
e^{-\frac{1}{4}\sqrt{\underline{c}}x_{0}}.$

Moreover, for all $t_{0}\geq 0$, there exists $K(t_{0})>0$ such that

$\displaystyle\sup_{t\in[0,t_{0}]}\int\left(|\nabla\tilde{u}_{2}|^{2}+\tilde{u}_{2}^{2}\right)\left(t,\mathbf{x}\right)e^{\frac{1}{4}\sqrt{\underline{c}}x}dx\leq
K(t_{0}).$

By weak convergence (4.19), it holds for any $x_{0}>0$

$\displaystyle\left\|\tilde{u}_{0,2}\sqrt{\phi(\cdot-
x_{0})}\right\|_{H^{1}}\leq\liminf_{k}\left\|\left(u-R_{1}\right)\left(t_{n_{k}},\cdot+\mathbf{z}_{2}(t_{n_{k}})\right)\sqrt{\phi(\cdot-
x_{0})}\right\|_{H^{1}}.$

We conclude the first part of the lemma gathering the previous inequality with
Lemma 4.16.

The second estimate in Lemma 4.19 is obtained by a Gronwall argument arguing
exactly as in Lemma 4.11.

Fourth step: Strong $L^{2}$ convergence of
$\left(u-R_{1}\right)(t_{n_{k}}+t,\cdot+\mathbf{z}_{2}(t_{n_{k}}))$ to
$\tilde{u}_{2}(t)$ on the right.

######  Lemma 4.20.

For any $B>0$ and $t\in\mathbb{R}$, it holds

$\displaystyle\left(u-R_{1}\right)\left(t_{n_{k}}+t,\cdot+\mathbf{z}_{2}(t_{n_{k}})\right)-\tilde{u}_{2}(t)\underset{k\rightarrow+\infty}{\rightarrow}0\quad\text{in}\quad
L^{2}(x>-B)$

and for any $t\in\mathbb{R}$

$\displaystyle\left(u-R_{1}\right)\left(t_{n_{k}}+t,\cdot+\mathbf{z}_{2}(t_{n_{k}})\right)-\tilde{u}_{2}(t)\underset{k\rightarrow+\infty}{\rightharpoonup}0\quad\text{in}\quad
H^{1}(\mathbb{R}^{2}).$

Moreover, with $\tilde{\mathbf{z}}_{2}=(\tilde{z}_{2},\tilde{\omega}_{2})$ the
$\mathcal{C}^{1}$-function associated to the modulation parameters of
$\tilde{u}_{2}$, it holds

$\displaystyle\mathbf{z}_{2}(t+t_{n_{k}})-\mathbf{z}_{2}(t_{n_{k}})\underset{k\to+\infty}{\rightarrow}\tilde{\mathbf{z}}_{2}(t)\quad\text{and}\quad\tilde{\mathbf{z}}_{2}(0)=0.$

Let us define

$\displaystyle
v_{k,2}(t,\mathbf{x}):=\left(u-R_{1}\right)\left(t+t_{n_{k}},\mathbf{x}+\mathbf{z}_{2}(t_{n_{k}})\right)-\tilde{u}_{2}(t,\mathbf{x}).$
(4.43)

We argue by using a Gronwall argument as in the proof of Theorem 3 (b) in [23]
to obtain that

$\displaystyle\sup_{t\in[0,t_{0}]}\int_{\mathbb{R}^{2}}v_{k,2}(t,\mathbf{x})^{2}\phi(x)d\mathbf{x}\leq\tilde{K}(t_{0})\int_{\mathbb{R}^{2}}v_{k,2}(0,\mathbf{x})^{2}\phi(x)d\mathbf{x}.$

The rest of the proof is then similar to the proof of Lemma 4.12.

Since the next steps are similar to the ones around the first rescaled limit
object, we state the lemmas around the second limit object without giving the
proofs.

Fifth step: Exponential decay of $\tilde{u}_{2}$ on the right.

######  Lemma 4.21.

For all $t\in\mathbb{R}$ and $x_{0}>0$, it holds

$\displaystyle\int\left(|\nabla\tilde{u}_{2}|^{2}+\tilde{u}_{2}^{2}\right)\left(t,\mathbf{x}+\tilde{\mathbf{z}}_{2}(t)\right)\phi(x-x_{0})d\mathbf{x}\lesssim
e^{-\frac{1}{4}\sqrt{\underline{c}}x_{0}}.$

Furthermore, for any $t>0$ and $x>0$,

$\displaystyle\int_{y}\tilde{u}_{2}\left(t,\mathbf{x}+\tilde{\mathbf{z}}_{2}(t)\right)^{2}dy\lesssim
e^{-\frac{1}{4}\sqrt{\underline{c}}x}.$

Sixth step: Strong $H^{1}$-convergence of
$\left(u-R_{1}\right)\left(t_{n_{k}}+t,\cdot+\mathbf{z}_{2}(t_{n_{k}})\right)$
to $\tilde{u}_{2}(t)$ on the right.

######  Lemma 4.22.

For any $t\in\mathbb{R}$ and $B>0$, it holds

$\displaystyle\left(u-R_{1}\right)\left(t_{n_{k}}+t,\cdot+\mathbf{z}_{2}(t_{n_{k}})\right)-\tilde{u}_{2}(t)\underset{n\rightarrow+\infty}{\rightarrow}0\quad\text{in}\quad
H^{1}(x>-B).$

Seventh step: Exponential decay of $\tilde{u}_{2}$ on the left.

######  Lemma 4.23.

We have that, for any $t\in\mathbb{R}$ and $x_{0}>0$,

$\displaystyle\int\tilde{u}_{2}^{2}\left(t,\mathbf{x}+\tilde{\mathbf{z}}_{2}(t)\right)\left(1-\phi(x+x_{0})\right)d\mathbf{x}\lesssim
e^{-\frac{1}{4}\sqrt{\underline{c}}x_{0}}.$

Furthermore, for any $t\in\mathbb{R}$ and $x<0$,

$\displaystyle\int_{y}\tilde{u}_{2}^{2}\left(t,\mathbf{x}+\tilde{\mathbf{z}}_{2}(t)\right)dy\lesssim
e^{-\frac{1}{4}\sqrt{\underline{c}}x}.$

##  5\. The three dimensional case

In this section, we explain how to prove Theorem 4.1 in the case of the
dimension $d=3$. The proof of the orbital stability is exactly the same as in
the $2$-dimensional. Hence we will focus on the proof of the asymptotic
stability. We introduce the notation
$\mathbf{x}=(x,\mathbf{y})=(x,y_{1},y_{2})\in\mathbb{R}^{3}$.

The proof is similar to the one in dimension $d=2$ in the former section. The
main difference is that the Liouville theorem proved in [7] requires a
slightly stronger decay assumption than the one in the $2$-dimensional case
stated in Theorem 4.2. Let us recall the rigidity result of Proposition 2.7 in
[7].

######  Theorem 5.1.

Let $0<\underline{c}<\tilde{c}<\bar{c}$. There exists
$\alpha_{12}=\alpha_{12}(\underline{c},\bar{c})$ and
$A_{12}=A_{12}(\underline{c},\bar{c})$ such that the following is true.
Suppose that for $0<\alpha<\alpha_{12}$,
$\tilde{v}\in\mathcal{C}(\mathbb{R}:H^{1}(\mathbb{R}^{3}))$ is a solution to
1.1d=3 satisfying for some function
$\mathbf{z}_{\tilde{v}}(t)\in\mathbb{R}^{3}$,

$\displaystyle\forall
t\in\mathbb{R},\quad\left\|\tilde{v}\left(\cdot+\mathbf{z}_{\tilde{v}}(t)\right)-Q_{\tilde{c}}\right\|_{H^{1}(\mathbb{R}^{3})}<\alpha.$

By denoting $\tilde{c},\tilde{\mathbf{z}}$ the modulation parameters, if
$\tilde{v}$ satisfies the spatial decay property: for each $k\geq 0$,

$\displaystyle\sup_{t\in\mathbb{R}}\int_{\mathbf{x}\in\mathbb{R}^{3}}\langle\mathbf{x}\rangle^{2k}\tilde{v}^{2}(t,\mathbf{x}+\tilde{\mathbf{z}}(t))d\mathbf{x}<\infty,$
(5.1)

then there exists $\left|\check{c}-\tilde{c}\right|<A_{12}\alpha$ and
$\check{\mathbf{z}}\in\mathbb{R}^{3}$ such that

$\displaystyle\tilde{v}(t,\mathbf{x})=Q_{\check{c}}\left(\mathbf{x}-\check{\mathbf{z}}-\check{c}t\mathbf{e}_{1}\right).$

The condition (5.1) is a slightly stronger than the condition (4.3) of Theorem
4.2. Thus, when taking the limit profiles around the two solitary waves, we
have to ensure that they verify condition (5.1). We explain how to ascertain
this condition for the profile $\tilde{u}_{1}$ around the first solitary wave.
We state an equivalent of Proposition 4.4 in the $3$-dimensional case.

######  Proposition 5.2.

Under the assumptions of Theorem 1.1, by taking $k=k(\underline{c},\bar{c})>0$
smaller and $K=K(\underline{c},\bar{c})>0$ larger if necessary, for any
increasing sequence $\\{t_{n}\\}_{n}\rightarrow+\infty$, there exists a
subsequence $\\{t_{n_{k}}\\}_{k}$ and $\tilde{u}_{0,1}\in
H^{1}(\mathbb{R}^{3})$ such that, for any $B>0$,

$\displaystyle
u\left(t_{n_{k}},\cdot+\mathbf{z}_{1}(t_{n_{k}})\right)\underset{k\rightarrow+\infty}{\rightarrow}\tilde{u}_{0,1}\quad\text{in}\quad
H^{1}(x>-B).$

Moreover, the solution $\tilde{u}_{1}$ of 1.1 with initial condition
$\tilde{u}_{1}(0)=\tilde{u}_{0,1}$ satisfies for any $t\in\mathbb{R}$

$\displaystyle\left\|\tilde{u}_{1}\left(t,\cdot+\tilde{\mathbf{z}}_{1}(t)\right)-Q_{c_{1}^{0}}\right\|_{H^{1}}\lesssim\beta$
(5.2)

with $\beta$ defined in (4.2) and for any $t\in\mathbb{R}$,
$|\theta_{0}|<\frac{\pi}{3}$ and $j=1,2$,

$\displaystyle\int\tilde{u}_{1}^{2}\left(t,\mathbf{x}+\tilde{\mathbf{z}}_{1}(t)\right)e^{\frac{1}{4}\sqrt{\underline{c}}|x+\tan{\theta_{0}}y_{j}|}d\mathbf{x}\lesssim
1.$ (5.3)

where $\tilde{\mathbf{z}}_{1}$ corresponds to the function given by the
modulation theory around one solitary wave.

###### Proof.

The proof follows the same steps than the one of Proposition 4.4. The
monotonicity result in Lemma 4.9 extends to the $3$-dimensional case by
replacing $y$ by $y_{j}$ for $j=1,2$. Then one deduce that, for any $x_{0}>0$,

$\displaystyle\limsup_{t\rightarrow+\infty}\int\left(|\nabla
u|^{2}+u^{2}\right)\left(t,\mathbf{x}+\mathbf{z}_{1}(t)\right)\phi\left(x+\tan{\theta_{0}}y_{j}-x_{0}\right)d\mathbf{x}\lesssim
e^{-\frac{1}{4}\sqrt{\underline{c}}x_{0}}.$

This decay propagates to the limit profile $\tilde{u}_{0,1}$ and to its
emanating solution $\tilde{u}_{1}$ as in the third, fourth and fifth steps. We
deduce that, for any $t\in\mathbb{R}$, $|\theta_{0}|<\frac{\pi}{3}$, $j=1,2$,
and $x_{0}>0$,

$\displaystyle\int\left(|\nabla\tilde{u}_{1}|^{2}+\tilde{u}_{1}^{2}\right)\left(t,\mathbf{x}+\tilde{\mathbf{z}}_{1}(t)\right)\phi\left(x+\tan{\theta_{0}}y_{j}-x_{0}\right)d\mathbf{x}\lesssim
e^{-\frac{1}{4}\sqrt{\underline{c}}x_{0}},$

so that

$\displaystyle\int\tilde{u}_{1}^{2}\left(t,\mathbf{x}+\tilde{\mathbf{z}}_{1}(t)\right)e^{\frac{1}{4}\sqrt{\underline{c}}(x+\tan{\theta_{0}}y_{j})}d\mathbf{x}\lesssim
1,$

by using that
$e^{\frac{1}{4}\sqrt{\underline{c}}(x+\tan{\theta_{0}}y_{j}-x_{0})}\leq\phi(x+\tan{\theta_{0}}y_{j}-x_{0})$
for $x+\tan{\theta_{0}}y_{j}\leq x_{0}$ and then letting $x_{0}\to+\infty$.
Finally, we deduce the decay on the left in (5.3) arguing as in the sixth and
seventh steps. ∎

With Proposition 5.2 in hand, we prove the $3$-dimensional version of
Proposition 4.3. The only difference with the $2$-dimensional case consists in
proving that $\tilde{u}_{1}$ satisfies (5.1). We proceed as follows: observe
that for $\mathbf{x}\in\mathbb{R}^{3}$,
$|\mathbf{x}|\lesssim|x|+|x+y_{1}|+|x+y_{2}|$. Then an application of (5.3)
with $\theta_{0}=0$ and $\theta_{0}=\frac{\pi}{4}$ for $j=1,2$ yields (5.1).

The $3$-dimensional version of Proposition 4.5 is proved similarly. Then the
conclusion of the proof of Theorem 4.1 (ii) in the $3$-dimensional case
follows as in the $2$-dimensional case.

##  Appendix A Proof of Lemmas 2.3 and 2.7

Before proving Lemma 2.3, we state a technical lemma.

######  Lemma A.1.

Let $f,g\in C^{\infty}(\mathbb{R}^{d})$ be two functions satisfying, for some
$n\in\mathbb{N}$,

$|f(\mathbf{x})|+|g(\mathbf{x})|\lesssim\langle\mathbf{x}\rangle^{n}e^{-|\mathbf{x}|}.$
(A.1)

Then, there exists a positive constant $c=c(n,d)>1$, such that for all
$(z,\omega)\in\mathbb{R}^{d}$ with $z>c(n,d)$,

$\displaystyle\left|f(\mathbf{x}-(z,\omega))g(\mathbf{x})\right|$
$\displaystyle\lesssim
e^{-z}(z^{n}+\langle(x,y-\omega)\rangle^{n})\langle(x,y)\rangle^{n},$ (A.2)
$\displaystyle\left|f(\mathbf{x}-(z,\omega))g(\mathbf{x})\right|$
$\displaystyle\lesssim e^{-\frac{31}{32}z}\left({\bf
1}_{\\{x<0\\}}e^{-\frac{1}{64}|(x,y-\omega)|}+{\bf
1}_{\\{0<x<z\\}}(e^{-\frac{1}{64}|y|}+e^{-\frac{1}{64}|y-\omega|})\right.$
$\displaystyle\left.\quad\quad\quad\quad+{\bf
1}_{\\{x>z\\}}e^{-\frac{1}{64}|(x,y)|}\right)$ (A.3)

and, for all $1\leq p<\infty$,

$\|f(\cdot-(z,\omega))g\|_{L^{p}}\lesssim e^{-\frac{15}{16}z}.$ (A.4)

###### Proof.

Observe that (A.4) is a direct consequence of (A.3). We turn to the proof of
(A.2) and (A.3). In the region $x<0$, $|(x-z,y-\omega)|\geq|x-z|>z$ and
$|(x-z,y-\omega))|\geq\frac{1}{\sqrt{2}}(|x|+|y-\omega|)\geq\frac{1}{\sqrt{2}}|(x,y-\omega)|$.
Thus it follows that

$\displaystyle\left|f(\mathbf{x}-(z,\omega))g(\mathbf{x})\right|$
$\displaystyle\lesssim e^{-|(x-z,y-\omega)|}\langle(x-z,y-\omega)\rangle^{n}$
$\displaystyle\lesssim\min\\{e^{-z}(z^{n}+\langle(x,y-\omega)\rangle^{n}),e^{-\frac{31}{32}z}e^{-\frac{1}{64}|(x,y-\omega))|}\\}.$

In the region $x>z$, we argue similarly and use $|x|>z$, so that

$\left|f(\mathbf{x}-(z,\omega))g(\mathbf{x})\right|\lesssim
e^{-|(x,y)|}\langle(x,y)\rangle^{n}\lesssim\min\\{e^{-z}(z^{n}+\langle(x,y)\rangle^{n}),e^{-\frac{31}{32}z}e^{-\frac{1}{64}|(x,y)|}\\}.$

Finally, in the region $0<x<z$, we have $|(x-z,y-\omega)|\geq z-x$ and
$|(x,y)|\geq x$, so that

$\left|f(\mathbf{x}-(z,\omega))g(\mathbf{x})\right|\lesssim\min\\{e^{-z}(z^{n}+\langle(x,y-\omega)\rangle^{n})\langle(x,y)\rangle^{n},e^{-\frac{31}{32}z}(e^{-\frac{1}{64}|y|}+e^{-\frac{1}{64}|y+\omega|})\\}.$

Therefore, gathering these estimates conclude the proof of (A.2) and (A.3). ∎

###### Proof of Lemma 2.3.

The first set of identities follow directly from a rescaling argument.

Let us prove the first inequality in the second set of estimates. By using the
bound (1.6) and the notation $(z,\omega)=(z_{1}-z_{2},\omega_{1}-\omega_{2})$,
we find

$\displaystyle\left|\int R_{1}R_{2}\right|$ $\displaystyle\lesssim
c_{1}c_{2}\int
e^{-\sqrt{c_{1}}|\mathbf{x}-(z_{1},\omega_{1})|}e^{-\sqrt{c_{2}}|\mathbf{x}-(z_{2},\omega_{2})|}d\mathbf{x}$
$\displaystyle=c_{1}\int
e^{-\sqrt{c_{1}c_{2}^{-1}}|\mathbf{x}-\sqrt{c_{2}}(z,\omega)|}e^{-|\mathbf{x}|}d\mathbf{x}$
$\displaystyle\leq c_{1}\int
e^{-|\mathbf{x}-\sqrt{c_{2}}(z,\omega)|}e^{-|\mathbf{x}|}d\mathbf{x},$

so that (A.4) and (2.9) conclude the proof of the first estimate on the right-
hand side of (2.14).

The proof of (2.14)-(2.18) follows arguing similarly.

Finally we estimate (2.19). By using the bound (1.6) and (A.2), we deduce that

$\displaystyle\left|\int\partial_{x}(R_{1}R_{2})R_{1}\right|$
$\displaystyle\lesssim c_{1}^{2}c_{2}\int
e^{-2\sqrt{c_{1}}|\mathbf{x}-(z_{1},\omega_{1})|}e^{-\sqrt{c_{2}}|\mathbf{x}-(z_{2},\omega_{2})|}d\mathbf{x}\leq
c_{1}^{3}\int
e^{-2|\mathbf{x}-\sqrt{c_{2}}(z,\omega)|}e^{-|\mathbf{x}|}d\mathbf{x}\lesssim
c_{1}^{3}e^{-\sqrt{c_{2}}z}.$

Estimate (2.20) follows by arguing in a similar way. ∎

###### Proof of Lemma 2.7.

We first start by proving (2.36). Observe from (1.6) and (2.30) that

$\displaystyle\left|R_{1}(\mathbf{x})\psi_{\gamma}^{\prime}\left(x-m(t)\right)\right|\lesssim\bar{c}e^{-\frac{1}{2}\sqrt{\underline{c}}\left(\left|x-z_{1}(t)\right|+\left|x-m(t)\right|\right)}.$

In the case where $x<m(t)$, we have $|x-z_{1}(t)|=z_{1}(t)-x>\frac{1}{2}z$,
while in the case $x>z_{1}(t)$, we have $|x-m(t)|=x-m(t)>\frac{1}{2}z$.
Finally, in the case $x\in[m(t),z_{1}(t)]$ we have
$|x-z_{1}(t)|+|x-m(t)|=z_{1}(t)-x+x-m(t)=\frac{1}{2}z(t)$. Thus, we conclude
from the above estimates and (2.24) that

$\displaystyle\|R_{1}\psi_{\gamma}^{\prime}\left(\cdot-m(t)\right)\|_{L^{\infty}}\lesssim\bar{c}e^{-\frac{1}{4}\sqrt{\underline{c}}z}\lesssim
e^{-\frac{1}{4}\sqrt{\underline{c}}\left(Z+\sigma t\right)}$

The estimate for the second term on the left-hand side of (2.36) is obtained
arguing similarly.

Next, we turn to the proof of (2.37). Observe that, for $z>0$,
$\omega\in\mathbb{R}$ and $c>0$, it holds

$\displaystyle\left\|Q_{c}\left(\cdot+z,\cdot-\omega\right)\psi\right\|_{L^{2}}\lesssim
c^{\frac{1}{2}}e^{-\frac{1}{2}\min\\{1,\sqrt{c}\\}z}.$ (A.5)

Indeed, on the one hand, in the region where $x\leq-\frac{z}{2}$, we have
$\psi(x)\lesssim e^{-\frac{z}{2}}$ (see (2.29)), and thus, it follows from
(1.6) that

$\displaystyle\left\|Q_{c}\left(\cdot+z,\cdot-\omega\right)\psi(\cdot)\right\|_{L^{2}(x<-\frac{z}{2})}^{2}\lesssim
c^{2}\iint_{x\leq-\frac{z}{2}}e^{-2\sqrt{c}\left|(x+z,y-\omega)\right|}e^{-z}dxdy\lesssim
ce^{-z}.$

On the other hand, in the region $x>-\frac{z}{2}$, we use $0\leq\psi\leq 1$, a
polar coordinate change and the decay (1.6) to deduce

$\displaystyle\left\|Q_{c}\left(\cdot+z,\cdot-\omega\right)\psi\right\|_{L^{2}(x>-\frac{z}{2})}^{2}$
$\displaystyle\lesssim
c^{2}\int_{r>\frac{z}{2}}\frac{1}{r}e^{-2\sqrt{c}r}rdr\lesssim
ce^{-\sqrt{c}z}.$

We now prove (2.37). By using $\psi(-x)=1-\psi(x)$, (A.5) and $4c_{1}>\gamma$
(see (2.25)), we have

$\displaystyle\left\|R_{1}\left(\psi_{\gamma}(\cdot-m)-1\right)\right\|_{L^{2}}=\frac{\sqrt{\gamma}}{2}\left\|Q_{\frac{4c_{1}}{\gamma}}\left(\cdot+\frac{\sqrt{\gamma}}{4}z,\cdot-\frac{\sqrt{\gamma}}{2}\omega_{1}\right)\psi\right\|_{L^{2}}\lesssim\bar{c}^{\frac{1}{2}}e^{-\frac{1}{8}\sqrt{\gamma}z}.$

Similarly, the same estimate holds for $\|R_{2}\psi_{s}(\cdot-m)\|_{L^{2}}$ in
the case $4c_{2}\geq\gamma$. In the case where $4c_{2}<\gamma$, we obtain from
(A.5) that

$\displaystyle\left\|R_{2}\psi_{\gamma}\right\|_{L^{2}}=\frac{\sqrt{\gamma}}{2}\left\|Q_{\frac{4c_{2}}{\gamma}}\left(\cdot+\frac{\sqrt{\gamma}}{4}z,\cdot-\frac{\sqrt{\gamma}}{4}\omega_{2}\right)\psi\right\|_{L^{2}}\lesssim\underline{c}^{\frac{1}{2}}e^{-\frac{1}{4}\sqrt{c_{2}}z}.$

Therefore, we conclude the proof of (2.37) by combining these estimates with
(2.9) and (2.25). ∎

## Acknowledgements

The authors were supported by a Trond Mohn Foundation grant.

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