# Dynamics of two interacting kinks for the $\phi^{6}$ model

Abdon Moutinho Eletronic address<EMAIL_ADDRESS>Department of
Mathematics: LAGA, Université Sorbonne Paris Nord, 99 Avenue Jean Baptiste
Clément, 93430-Villetaneuse, France

###### Abstract

We consider the nonlinear wave equation known as the $\phi^{6}$ model in
dimension 1+1. We describe the long time behavior of all the solutions of this
model close to a sum of two kinks with energy slightly larger than twice the
minimum energy of non constant stationary solutions. We prove orbital
stability of two moving kinks. We show for low energy excess $\epsilon$ that
these solutions can be described for long time less o equivalent than
$-\ln{(\epsilon)}\epsilon^{-\frac{1}{2}}$ as the sum of two moving kinks such
that each kink’s center is close to an explicit function which is a solution
of an ordinary differential system. We give an optimal estimate in the energy
norm of the remainder $(g(t),\partial_{t}g(t))$ and we prove that this
estimate is achieved during a finite instant
$t=T\lesssim-\ln{(\epsilon)}\epsilon^{-\frac{1}{2}}.$

##  1 Introduction

We consider a nonlinear wave equation equation known as the $\phi^{6}$ model.
For the potential function $U(\phi)=\phi^{2}(1-\phi^{2})^{2}$ and
$\dot{U}(\phi)=2\phi-8\phi^{3}+6\phi^{5},$ the equation is written as

$\partial_{t}^{2}\phi(t,x)-\partial_{x}^{2}\phi(t,x)+\dot{U}(\phi(t,x))=0\text{,
$(t,x)\in\mathbb{R}\times\mathbb{R}$}.$ (1)

The potential energy $E_{pot},$ the kinetic energy $E_{kin}$ and total energy
$E_{total}$ associated to the equation (1) are given by

$\displaystyle
E_{pot}(\phi(t))=\frac{1}{2}\int_{\mathbb{R}}\partial_{x}\phi(t,x)^{2}\,dx+\int_{\mathbb{R}}\phi(t,x)^{2}(1-\phi(t,x)^{2})^{2}\,dx,$
$\displaystyle
E_{kin}(\phi(t))=\frac{1}{2}\int_{\mathbb{R}}\partial_{t}\phi(t,x)^{2}\,dx,$
$\displaystyle
E_{total}(\phi(t),\partial_{t}\phi(t))=\frac{1}{2}\int_{\mathbb{R}}\left[\partial_{x}\phi(t,x)^{2}+\partial_{t}\phi(t,x)^{2}\right]\,dx+\int_{\mathbb{R}}\phi(t,x)^{2}(1-\phi(t,x)^{2})^{2}\,dx.$

We say that if a solution $\phi(t,x)$ of the integral equation associated to
(1) has $E_{total}(\phi,\partial_{t}\phi)<+\infty,$ then it is in the energy
space. The solutions of (1) in the energy space have constant total energy
$E_{total}(\phi(t),\partial_{t}\phi(t)).$ By standard arguments, the Cauchy
Problem associated (1) is locally well-posed in the energy space, moreover is
globally well-posed since $U(\phi)=\phi^{2}(1-\phi^{2})^{2}$ satisfies
$\lim_{|\phi|\to\infty}U(\phi)=+\infty.$

The stationary solutions of (1) are the critical points of the potential
energy. The only non-constant stationary solutions in (1) are the topological
solitons called kinks and anti-kinks, for more details see chapter $5$ of
[21]. The kinks of (1) are given by

$H_{0,1}(x-a)=\frac{e^{\sqrt{2}(x-a)}}{(1+e^{2\sqrt{2}(x-a)})^{\frac{1}{2}}},\,H_{-1,0}(x-a)=-H_{0,1}(-x+a)$

for any real $a.$ The study of kink and multi kinks solitons solutions of
nonlinear wave equations has applications in many domains of mathematical
physics. More precisely, the model (1) that we study has applications in
condensed matter physics [2] and cosmology [32], [13], [10].

It is well known that the set of solutions in energy Space of (1) for any
potential $U$ is invariant under space translation, time translation and space
reflection. Also, for one of the stationary solutions $H$ of (1) and any
$-1<v<1$, we have that the following solitary wave

$H\Big{(}\frac{x-vt}{(1-v^{2})^{\frac{1}{2}}}\Big{)},$

which is the Lorentz transform of $H$ is a solution of (1).

The problem of stability of multi-kinks is of great interest in mathematical
physics, see for example [7], [9]. For the integrable model mKdV, Muñoz proved
in [24] the $H^{1}$ stability and asymptotic stability of multi-kinks.
However, for many non-integrable models as the $\phi^{6}$ nonlinear wave
equation, the asymptotic and long time dynamics of multi-kinks after the
instant where the collision or interaction happens are still unknown, even
though there are numerical studies of kink-kink collision for the $\phi^{6}$
model, see [9], which motivate our research on the topic of the description of
long time behavior of a kink-kink pair.

For nonlinear wave equation models in $1+1$ space dimension, results of
stability for a single kink were obtained, for example for the $\phi^{4}$
model it was obtained asymptotic stability for odd pertubations in [17] and
[6]. Also, it was recently proved in [18] by Martel, Muñoz, Kowalczyk and Van
Den Bosch asymptotic stability of a single kink for a general class of
nonlinear wave equations, including the model which we study here.

The main purpose of our material is to describe the long time behavior of
solutions $\phi(t,x)$ of (1) in the energy space such that

$\displaystyle\lim_{x\to+\infty}\phi(t,x)=1,$
$\displaystyle\lim_{x\to-\infty}\phi(t,x)=-1$

with total energy equals to $2E_{pot}(H_{01})+\epsilon,$ for $0<\epsilon\ll
1.$ More precisely, we proved orbital stability for a sum of two moving kinks
with total total energy $2E_{pot}(H_{0,1})+\epsilon$ and we verified that the
remainder has a better estimate during a long time interval which goes to
$\mathbb{R}$ as $\epsilon\to 0,$ indeed we proved that the estimate of the
remainder during this long time interval is optimal. Also, we prove that the
dynamics of the kinks movement is very close to two explicit functions
$d_{j}:\mathbb{R}\to\mathbb{R}$ defined in Theorem 1.11 during a long time
interval. This result is very important to understand the behavior of two
kinks after the instant of collision, which happens when the kinetic energy is
minimal, indeed, our main results Theorem 1.7 and Theorem 1.11 describe the
dynamics of the kinks before and after the collision instant for a long time
interval. The numerical study of interaction and collision between kinks for
the $\phi^{6}$ model was done in [9], in which it was verified that the
collision of kinks is close to an elastic collision when the speed of each
kink is low and smaller than a critical speed $v_{c}$.

For nonlinear wave equation models in dimension $2+1,$ there are similar
results obtained in the dynamics of topological multi-solitons. For the Higgs
Model, there are results in the description dynamics of multi-vortices in [28]
obtained by Stuart and in [12] obtained by Gustafson and Sigal. Indeed, we
took inspiration from the proof and statement of Theorem $2$ of [12] to
construct our main results. Also, in [29], Stuart described the dynamics of
monopole solutions for the Yang-Mills-Higgs equation. For more references, see
also [30], [8], [20] and [11].

In [1], Bethuel, Orlandi and Smets described the asymptotic behavior of
solutions of a parabolic Ginzburg-Landau equation closed to multi-vortices in
the initial instant. For mores references, see also [15] and [27].

There are also results in the dynamics in multi-vortices for nonlinear
Schrödinger equation, for example the description of the dynamics of multi-
vortices for the Gross-Pitaevski equation was obtained in [25] by Ovchinnikov
and Sigal and results in the dynamics of vortices for the Ginzburg-Landau-
Schrödinger equations were proved in [5] by Colliander and Jerrard, see also
[16] for more information about Gross-Pitaevski equation.

###  1.1 Main results

We recall that the objective of this paper is to show orbital stability for
the solutions of the equation (1) which are close to a sum of two interacting
kinks in an initial instant and estimate the size of the time interval where
better stability properties hold. The main techniques of the proof are
modulation techniques adapted from [19], [23] and [26] and a refined energy
estimate method to control the size of the remainder term.

######  Notation 1.1.

For any $D\subset\mathbb{R},$ any real function
$f:D\subset\mathbb{R}\to\mathbb{R},$ a real positive function $g$ with domain
$D$ is in $O(f(x))$ if and only if there is a uniform constant $C>0$ such that
$0<g(x)<C|f(x)|.$ We denote that two real non-negative functions
$f,g:D\subset\mathbb{R}\to\mathbb{R}_{\geq 0}$ satisfy

$f\lesssim g,$

if there is a constant $C>0$ such that

$|f(x)|\leq C|g(x)|\text{, for all $x\in D.$}$

If $f\lesssim g$ and $g\lesssim f,$ we denote that $f\cong g.$ We use the
notation $(x)_{+}\coloneqq\max(x,0)$. If $g(t,x)\in
C^{1}(\mathbb{R},L^{2}(\mathbb{R}))\cap C(\mathbb{R},H^{1}(\mathbb{R})),$ then
we define $\overrightarrow{g(t)}\in H^{1}(\mathbb{R})\times L^{2}(\mathbb{R})$
by

$\overrightarrow{g(t)}=(g(t),\partial_{t}g(t)),$

and we also denote the energy norm of the remainder $\overrightarrow{g(t)}$ as

$\left\lVert\overrightarrow{g(t)}\right\rVert=\left\lVert
g(t)\right\rVert_{H^{1}(\mathbb{R})}+\left\lVert\partial_{t}g(t)\right\rVert_{L^{2}(\mathbb{R})}$

to simplify our notation in the text.

######  Definition 1.2.

We define $S$ as the set $g\in L^{\infty}(\mathbb{R})$ such that

  1. 1.

$\frac{dg}{dx}\in L^{2}(\mathbb{R}),$

  2. 2.

$\int_{\mathbb{R}_{>0}}|g(x)-1|^{2}\,dx<\infty,$

  3. 3.

$\int_{\mathbb{R}_{<0}}|g(x)+1|^{2}\,dx<\infty.$

The partial differential equation (1) is locally well-posed in the affine
space $S\times L^{2}(\mathbb{R}).$ Motivated by the proof and computations
that we are going to present, we also consider:

######  Definition 1.3.

We define for $x_{1},\,x_{2}\in\mathbb{R}$

$H^{x_{2}}_{0,1}(x)\coloneqq H_{0,1}(x-x_{2})$ and
$H^{x_{1}}_{-1,0}(x)\coloneqq H_{-1,0}(x-x_{1}),$

and we say that $x_{2}$ is the kink center of $H^{x_{2}}_{0,1}(x)$ and $x_{1}$
is the kink center of $H^{x_{1}}_{-1,0}(x).$

######  Remark 1.4.

Indeed, $S=\\{g\in L^{\infty}(\mathbb{R})|\,g-H_{0,1}-H_{-1,0}\in
H^{1}(\mathbb{R})\\}.$

There are also non-stationary solutions $(\phi(t,x),\partial_{t}\phi(t,x))$ of
(1) with finite total energy $E_{total}(\phi(t),\partial_{t}\phi(t))$ that
satisfies for all $t\in\mathbb{R}$

$\lim_{x\to+\infty}\phi(t,x)=1,\,\lim_{x\to-\infty}\phi(t,x)=0.$ (2)

But, for any $a\in\mathbb{R},$ the kinks $H_{0,1}(x-a)$ are the unique
functions that minimize the Potential Energy in the set of functions $\phi(x)$
satisfying condition (2), the proof of this fact follows from the Bogomolny
identity, see [21] or section $2$ of [19]. By a similar reasoning, we can
verify that all functions $\phi(x)\in S$ have
$E_{pot}(\phi)>2E_{pot}(H_{0,1}).$

######  Definition 1.5.

We define the energy excess $\epsilon$ of a solution
$(\phi(t),\partial_{t}\phi(t))\in S\times L^{2}(\mathbb{R})$ as the following
value

$\epsilon=E_{total}(\phi(t),\partial_{t}\phi(t))-2E_{pot}(H_{0,1}).$

Also, for $\phi(t)$ solution of (1), we denote the Kinetic Energy of $\phi(t)$
by $E_{kin}(\phi(t))=E(\phi,\partial_{t}\phi)-E_{pot}(\phi(t))$. We recall the
notation $(x)_{+}\coloneqq\max(x,0)$. It’s not difficult to verify the
following inequalities

  1. (D1)

$|H_{0,1}(x)|\leq e^{-\sqrt{2}(-x)_{+}},$

  2. (D2)

$|H_{-1,0}(x)|\leq e^{-\sqrt{2}(x)_{+}},$

  3. (D3)

$|\dot{H}_{0,1}(x)|\leq\sqrt{2}e^{-\sqrt{2}(-x)_{+}},$

  4. (D4)

$|\dot{H}_{-1,0}(x)|\leq\sqrt{2}e^{-\sqrt{2}(x)_{+}}.$

Moreover, since

$\ddot{H}_{0,1}(x)=\dot{U}(H_{0,1}(x)),$ (3)

we can verify by induction the following estimate

$\Big{|}\frac{d^{k}H_{0,1}(x)}{dx^{k}}\Big{|}\lesssim_{k}\min\Big{(}e^{-2\sqrt{2}x},\,e^{\sqrt{2}x}\Big{)}$
(4)

for all $k\in\mathbb{N}\setminus\\{0\\}.$ The following result is crucial in
the framework of this material:

######  Lemma 1.6 (Modulation Lemma).

$\exists\,C_{0}\,,\delta_{0}>0$, such that if $0<\delta\leq\delta_{0}$,
$x_{2},\,x_{1}$ are real numbers with $x_{2}-x_{1}\geq\frac{1}{\delta}$ and
$g\in H^{1}(\mathbb{R})$ satisfies $\left\lVert
g\right\rVert_{H^{1}}\leq\delta$, then for
$\phi(x)=H_{-1,0}(x-x_{1})+H_{0,1}(x-x_{2})+g(x)$, $\exists!\,y_{1},\,y_{2}$
such that for

$g_{1}(x)=\phi(x)-H_{-1,0}(x-y_{1})-H_{0,1}(x-y_{2}),$

the four following statements are true

  1. 1

$\langle g_{1},\,\partial_{x}H_{-1,0}(x-y_{1})\rangle_{L^{2}}=0,$

  2. 2

$\langle g_{1},\,\partial_{x}H_{0,1}(x-y_{2})\rangle_{L^{2}}=0,$

  3. 3

$\left\lVert g_{1}\right\rVert_{H^{1}(\mathbb{R})}\leq C_{0}\delta,$

  4. 4

$|y_{2}-x_{2}|+|y_{1}-x_{1}|\leq C_{0}\delta.$

We will refer the first and second statements as the orthogonality conditions
of the Modulation Lemma.

###### Proof.

See the Appendix section A. ∎

Now, our main results are the following:

######  Theorem 1.7.

$\exists\,C,\delta_{0}>0$, such that if $\epsilon<\delta_{0}$ and

$(\phi(0),\partial_{t}\phi(0))\in S\times L^{2}(\mathbb{R})$

with $E_{total}(\phi(0),\partial_{t}\phi(0))=2E_{pot}(H_{0,1})+\epsilon$, then
there are $x_{2},x_{1}\in C^{2}(\mathbb{R})$ functions such that the unique
global time solution $\phi(t,x)$ of (1) is given by

$\phi(t)=H_{0,1}(x-x_{2}(t))+H_{-1,0}(x-x_{1}(t))+g(t),$ (5)

with $g(t)$ satisfying orthogonality conditions of the Modulation Lemma and

$e^{-\sqrt{2}(x_{2}(t)-x_{1}(t))}+\max_{j\in\\{1,2\\}}|\ddot{x}_{j}(t)|+\max_{j\in\\{1,2\\}}\dot{x}_{j}(t)^{2}+\left\lVert(g(t),\partial_{t}g(t))\right\rVert_{H^{1}\times
L^{2}}^{2}\lesssim\epsilon.$

Furthermore, we have that

$\left\lVert(g(t),\partial_{t}g(t))\right\rVert_{H^{1}\times L^{2}}^{2}\leq
C\Big{[}\left\lVert(g(0),\partial_{t}g(0))\right\rVert_{H^{1}\times
L^{2}}^{2}+\epsilon^{2}\Big{]}\exp\Big{(}\frac{C\epsilon^{\frac{1}{2}}|t|}{\ln{(\frac{1}{\epsilon})}}\Big{)}\text{
for all $t\in\mathbb{R}$}$ (6)

######  Remark 1.8.

In notation of the statement of Theorem 1.7, for any $\delta>0,$ there is
$0<K(\delta)<1$ such that if $0<\epsilon<K(\delta),$
$E_{total}(\phi(0),\partial_{t}\phi(0))=2E_{pot}(H_{0,1})+\epsilon,$ then we
have that $\left\lVert(g(0),\partial_{t}g(0))\right\rVert_{H^{1}\times
L^{2}}<\delta$ and $x_{2}(0)-x_{1}(0)>\frac{1}{\delta},$ for the proof see
Lemma A.3 and Corollary A.4 in the Appendix section A.

######  Remark 1.9 (Optimal decay.).

The result of Theorem 1.7 is optimal in the sense that for any function
$r:\mathbb{R}_{+}\to\mathbb{R}_{+}$ with $\lim_{h\to 0}r(h)=0,$ there is a
positive value $\delta(r)$ such that if $0<\epsilon<\delta(r)$ and
$\left\lVert\overrightarrow{g(0)}\right\rVert\leq r(\epsilon)\epsilon,$ then
$\epsilon\lesssim\left\lVert\overrightarrow{g(t)}\right\rVert$ for some
$0<t=O\Big{(}\frac{\ln{(\frac{1}{\epsilon})}}{\epsilon^{\frac{1}{2}}}\Big{)}.$
The proof of this fact is in the Appendix section B.

######  Remark 1.10.

From Remark 1.9, we obtain that there is an $0<\delta_{0}$ such that if
$0<\epsilon<\delta_{0},$ then for any $(\phi(0,x),\partial_{t}\phi(0,x))\in
S\times L^{2}(\mathbb{R})$ with $E_{total}(\phi(0),\partial_{t}\phi(0))$
equals to $2E_{pot}(H_{0,1})+\epsilon,\,g(t,x)$ defined in identity (5)
satisfies
$\epsilon\lesssim\limsup\limits_{t\to+\infty}\left\lVert\overrightarrow{g(t)}\right\rVert,$
similarly we have that
$\epsilon\lesssim\limsup\limits_{t\to-\infty}\left\lVert\overrightarrow{g(t)}\right\rVert.$
The proof of this fact is in the Appendix section B.

######  Theorem 1.11.

$\exists C,\delta_{0}>0,$ such that if
$0<\epsilon<\delta_{0},\,(\phi(0),\partial_{t}\phi(0))\in S\times
L^{2}(\mathbb{R}),$ and
$E_{total}(\phi(0),\partial_{t}\phi(0))=2E_{pot}(H_{0,1})+\epsilon,$ then
there are $v_{1},\,v_{2}\in\mathbb{R}$ such that

$\begin{pmatrix}\phi(0)\\\
\partial_{t}\phi(0)\end{pmatrix}=\begin{pmatrix}H_{0,1}(x-x_{2}(0))+H_{-1,0}(x-x_{1}(0))+g_{0}(x)\\\
v_{2}\partial_{x}H_{0,1}(x-x_{2}(0))+v_{1}\partial_{x}H_{-1,0}(x-x_{1}(0))+g_{1}(x)\end{pmatrix}$

with $g_{0}$ satisfying the orthogonality conditions of Modulation Lemma

$\displaystyle\left\langle\dot{H}_{0,1}(x-x_{2}(0)),\,g_{1}(x)\right\rangle_{L^{2}(\mathbb{R})}=-v_{2}\left\langle\ddot{H}_{0,1}(x-x_{2}(0)),\,g_{0}(x)\right\rangle_{L^{2}(\mathbb{R})},$
$\displaystyle\left\langle\dot{H}_{-1,0}(x-x_{1}(0)),\,g_{1}(x)\right\rangle_{L^{2}(\mathbb{R})}=-v_{1}\left\langle\ddot{H}_{-1,0}(x-x_{1}(0)),\,g_{0}(x)\right\rangle_{L^{2}(\mathbb{R})}$

and $\epsilon$ the energy excess of the solution
$(\phi(t,x),\partial_{t}\phi(t,x))$ of (1). Indeed, let the smooth functions
$d_{1}(t),\,d_{2}(t)$ be defined by

$\displaystyle
d_{1}(t)=a+bt-\frac{1}{2\sqrt{2}}\ln{\Big{(}\frac{8}{v^{2}}\cosh{\big{(}\sqrt{2}vt+c\big{)}}^{2}\Big{)}},$
(7) $\displaystyle
d_{2}(t)=a+bt+\frac{1}{2\sqrt{2}}\ln{\Big{(}\frac{8}{v^{2}}\cosh{\big{(}\sqrt{2}vt+c\big{)}}^{2}\Big{)}},$
(8)

such that $d_{j}(0)=x_{j}(0),\,\dot{d}_{j}(0)=-v_{j}$ for $j\in\\{1,\,2\\}.$
Let $d(t)=d_{2}(t)-d_{1}(t),$ then, for all $t\in\mathbb{R}$

$|z(t)-d(t)|\lesssim\min(\epsilon^{\frac{1}{2}}|t|,\epsilon
t^{2}),\,|\dot{z}(t)-\dot{d}(t)|\lesssim\epsilon|t|,$

moreover, we have the following estimates

$\displaystyle\epsilon\max_{j\in\\{1,\,2\\}}|d_{j}(t)-x_{j}(t)|=O\left(\max\Big{(}\left\lVert\overrightarrow{g(0)}\right\rVert,\epsilon\Big{)}^{2}\ln{\Big{(}\frac{1}{\epsilon}\Big{)}}^{11}\exp\Big{(}\frac{C\epsilon^{\frac{1}{2}}|t|}{\ln{(\frac{1}{\epsilon})}}\Big{)}\right),$
(9)
$\displaystyle\epsilon^{\frac{1}{2}}\max_{j\in\\{1,\,2\\}}|\dot{d}_{j}(t)-\dot{x}_{j}(t)|=O\left(\max\Big{(}\left\lVert\overrightarrow{g(0)}\right\rVert,\epsilon\Big{)}^{2}\ln{\Big{(}\frac{1}{\epsilon}\Big{)}}^{11}\exp\Big{(}\frac{C\epsilon^{\frac{1}{2}}|t|}{\ln{(\frac{1}{\epsilon})}}\Big{)}\right).$
(10)

######  Remark 1.12.

The proof of Theorem 1.7 and Theorem 1.11 for $t\leq 0$ is analogous to the
proof for $t\geq 0,$ so we will only prove them for $t\geq 0.$

Theorem 1.7 will be obtained as a consequence of Theorem 1.11. Clearly, from
Theorem 1.11, we can deduce the following corollary.

######  Corollary 1.13.

With the same hypotheses as in Theorem 1.11, we have that

$\max_{j\in\\{1,\,2\\}}|\ddot{d}_{j}(t)-\ddot{x}_{j}(t)|=O\left(\max\Big{(}\left\lVert\overrightarrow{g(0)}\right\rVert,\epsilon\Big{)}\epsilon^{\frac{1}{2}}\exp\Big{(}\frac{C\epsilon^{\frac{1}{2}}t}{\ln{(\frac{1}{\epsilon})}}\Big{)}+\max\Big{(}\left\lVert\overrightarrow{g(0)}\right\rVert,\epsilon\Big{)}^{2}\ln{\Big{(}\frac{1}{\epsilon}\Big{)}}^{11}\exp\Big{(}\frac{C\epsilon^{\frac{1}{2}}t}{\ln{(\frac{1}{\epsilon})}}\Big{)}\right).$

###### Proof of Corollary 1.13.

It follows directly from Theorem 1.11 and from Lemma A.1 presented in the
Appendix Section A. ∎

###  1.2 Resume of the proof

In this subsection, we present how the article is organized and explain
briefly the content of each section.
Section 2. In this section, we prove orbital stability of a perturbation of a
sum of two kinks. Moreover, we prove that if the initial data
$(\phi(0,x),\partial_{t}\phi(0,x))$ satisfies the hypothesis of Theorem 1.7,
then there are real functions $x_{1},\,x_{2}$ of class $C^{2}$ such that for
all $t\geq 0$

$\displaystyle\left\lVert\phi(t,x)-H^{x_{2}(t)}_{0,1}-H^{x_{1}(t)}_{-1,0}\right\rVert_{H^{1}(\mathbb{R})}\lesssim\epsilon^{\frac{1}{2}},$
$\displaystyle\left\lVert\partial_{t}\left(\phi(t,x)-H^{x_{2}(t)}_{0,1}-H^{x_{1}(t)}_{-1,0}\right)\right\rVert_{L^{2}(\mathbb{R})}\lesssim\epsilon^{\frac{1}{2}}.$

The proof of the orbital stability follows from studying the expression

$E_{pot}(H^{x_{2}(t)}_{0,1}+H^{x_{1}(t)}_{-1,0}+g)-E_{pot}(H^{x_{2}(t)}_{0,1}+H^{x_{1}(t)}_{-1,0}),$

which is bigger than $\left\lVert\overrightarrow{g(t)}\right\rVert^{2}$ less
some remaining terms from Taylor’s Expansion Theorem and the fact that the
kinks are critical points of $E_{pot}.$ But, from the modulation lemma, we
will introduce the functions $x_{2},\,x_{1}$ that will guarantee the following
coercitivity property

$\left\lVert\overrightarrow{g(t)}\right\rVert^{2}\lesssim
E_{pot}(H^{x_{2}(t)}_{0,1}+H^{x_{1}(t)}_{-1,0}+g)-E_{pot}(H^{x_{2}(t)}_{0,1}+H^{x_{1}(t)}_{-1,0}).$

From the orthogonality conditions of the modulation lemma and standard
ordinary differential equation techniques, we also obtain uniform bounds for
$\left\lVert\dot{x}_{j}(t)\right\rVert_{L^{\infty}(\mathbb{R})},\,\left\lVert\ddot{x}_{j}(t)\right\rVert_{L^{\infty}(\mathbb{R})}$
for $j\in\\{1,\,2\\}.$ The main techniques of this section are an adaption of
section 2 and 3 of [19].
Section 3. In this section, we study the long time behavior of
$\dot{x}_{j}(t),\,x_{j}(t)$ for $j\in\\{1,\,2\\}.$ More precisely, we
elaborate a Lemma similar to the Lemma $3.5$ of [19], but our estimates are
more precise, more precisely the errors of our estimate are written in
function of $z(t),\,\dot{x}_{j}(t),\,\dot{x}_{j}(t)$ and
$\left\lVert\overrightarrow{g(t)}\right\rVert.$
Section 4. In Section $4,$ we introduce a functional $F(t)$ with the objective
of controlling $\left\lVert\overrightarrow{g(t)}\right\rVert$ for a long time
interval. More precisely, we show that the function $F(t)$ satisfies for a
constant $K>0$ the global estimate
$\left\lVert\overrightarrow{g(t)}\right\rVert^{2}\lesssim F(t)+K\epsilon^{2}$
and we show that $|\dot{F}(t)|$ is small enough for a long time interval. We
start the functional from the quadratic part of the total energy of $\phi(t),$
more precisely with

$D(t)=\left\lVert\partial_{t}g(t,x)\right\rVert_{L^{2}(\mathbb{R})}^{2}+\left\lVert\partial_{x}g(t,x)\right\rVert_{L^{2}(\mathbb{R})}^{2}+\int_{\mathbb{R}}\ddot{U}(H^{x_{2}(t)}_{0,1}(x)+H^{x_{1}(t)}_{-1,0}(x))g(t,x)^{2}\,dx.$

However, we obtain that the terms of worst decay that appear in the
computation of $\dot{D}(t)$ are expressions similar to

$\int_{\mathbb{R}}\partial_{t}g(t,x)F(x_{1},x_{2},\dot{x}_{1},\dot{x}_{2},x)\,dx.$

But, we can cancel these bad terms after we add to the functional $D(t)$
correction terms similar to

$-\int_{\mathbb{R}}g(t,x)F(x_{1},x_{2},\dot{x}_{1},\dot{x}_{2},x)\,dx,$

and now in the time derivative of $D(t)$ plus the correction terms, we obtain
an expression with size smaller or equivalent to

$\left\lVert\overrightarrow{g(t)}\right\rVert\left\lVert\partial_{t}(F(x_{1},x_{2},\dot{x}_{1},\dot{x}_{2},x))\right\rVert_{L^{2}_{x}(\mathbb{R})}\max_{j\in{1,2}}|\dot{x}_{j}(t)|.$

Finally, based on the correction term described in the proof of Lemma 4.2 of
[19], we aggregate another kind of correction term such that its time
derivative cancels with

$-\int_{\mathbb{R}}U^{(3)}(H^{x_{2}(t)}_{0,1}(x)+H^{x_{1}(t)}_{-1,0}(x))(\dot{x}_{2}(t)\partial_{x}H^{x_{2}(t)}_{0,1}+\dot{x}_{1}(t)\partial_{x}H^{x_{1}(t)}_{-1,0})g(t,x)^{2},$

and then we evaluate the time derivative of the functional obtained from this
sum D(t) with all the corrections terms.
Remaining Sections. In the remaining part of this paper, we prove our main
results, Theorem 1.7 is a consequence of the energy estimate obtained in
Section 4 and the estimates with higher precision of the modulations
parameters $x_{1}(t),\,x_{2}(t)$ which are obtained in Section $5.$ In Section
5, we prove the result of Theorem 1.11, where we study the evolution of the
precision of the modulation parameters estimates by comparing it with a
solution of a system of ordinary differential equations. Complementary
information are given in Appendices A and B.

##  2 Global Stability of two moving kinks

Before the presentation of the proof of the main theorem, we define a
functional to study the potential energy of a sum of two kinks.

######  Definition 2.1.

The function $A:\mathbb{R}_{+}\to\mathbb{R}$ is defined by

$A(z)\coloneqq E_{pot}(H_{0,1}^{z}(x)+H_{-1,0}(x)).$ (11)

The study of the function $A$ is essential to obtain global in time control of
the norm of the remainder $g$ and the lower bound of $x_{2}(t)-x_{1}(t)$ in
Theorem 1.7.

######  Remark 2.2.

It’s easy to verify that
$E_{pot}(H_{0,1}(x-x_{2})+H_{-1,0}(x-x_{1}))=E_{pot}(H_{0,1}(x-(x_{2}-x_{1}))+H_{-1,0}(x)).$

We will use several times the following elementary estimate from the Lemma 2.5
of [19] given by:

######  Lemma 2.3.

For any real numbers $x_{2},x_{1}$, such that $x_{2}-x_{1}>0$ and
$\alpha,\,\beta>0$ with $\alpha\neq\beta$ the following bound holds:

$\int_{\mathbb{R}}e^{-\alpha(x-x_{1})_{+}}e^{-\beta(x_{2}-x)_{+}}\lesssim_{\alpha,\beta}e^{-\min(\alpha,\beta)(x_{2}-x_{1})},$

For any $\alpha>0$, the following bound holds

$\int_{\mathbb{R}}e^{-\alpha(x-x_{1})_{+}}e^{-\alpha(x_{2}-x)_{+}}\lesssim_{\alpha}(1+(x_{2}-x_{1}))e^{-\alpha(x_{2}-x_{1})}.$

The main result of this section is the following

######  Lemma 2.4.

The function $A$ is of class $C^{2}$ and there is a constant $C>0$, such that

  1. 1.

$|\ddot{A}(z)-4\sqrt{2}e^{-\sqrt{2}z}|\leq Cze^{-2\sqrt{2}z},$

  2. 2.

$|\dot{A}(z)+4e^{-\sqrt{2}z}|\leq Cze^{-2\sqrt{2}z},$

  3. 3.

$|A(z)-2E_{pot}(H_{0,1})-2\sqrt{2}e^{-\sqrt{2}z}|\leq Cze^{-2\sqrt{2}z}.$

###### Proof.

By definition of $A$, it’s clear that

$\displaystyle A(z)$
$\displaystyle=\frac{1}{2}\int_{\mathbb{R}}\Big{(}\partial_{x}\big{[}H^{z}_{0,1}(x)+H_{-1,0}(x)\big{]}\Big{)}^{2}\,dx+\int_{\mathbb{R}}U(H^{z}_{0,1}(x)+H_{-1,0}(x))\,dx$
$\displaystyle=\left\lVert\partial_{x}H_{0,1}\right\rVert_{L^{2}(\mathbb{R})}^{2}+\int_{\mathbb{R}}\partial_{x}H^{z}_{0,1}(x)\partial_{x}H_{-1,0}(x)\,dx+\int_{\mathbb{R}}U(H^{z}_{0,1}(x)+H_{-1,0}(x))\,dx.$

Since the functions $U$ and $H_{0,1}$ are smooth and $\partial_{x}H_{0,1}(x)$
has exponential decay when $|x|\to+\infty$, it’s possible to differentiate
$A(z)$ in $z.$ More precisely, we obtain

$\displaystyle\dot{A}(z)$
$\displaystyle=-\int_{\mathbb{R}}\partial^{2}_{x}H^{z}_{0,1}(x)\partial_{x}H_{-1,0}(x)\,dx-\int_{\mathbb{R}}\dot{U}(H^{z}_{0,1}(x)+H_{-1,0}(x))\partial_{x}H^{z}_{0,1}(x)\,dx$
(12)
$\displaystyle=\int_{\mathbb{R}}\partial_{x}H^{z}_{0,1}(x)\big{[}\dot{U}(H_{-1,0})(x)-\dot{U}(H_{-1,0}(x)+H^{z}_{0,1}(x))\big{]}\,dx.$
(13)

By similar reasons, it is always possible to differentiate $A(z)$ twice,
precisely, we obtain

$\ddot{A}(z)=\int_{\mathbb{R}}\partial_{x}H^{z}_{0,1}(x)^{2}\ddot{U}(H_{-1,0}(x)+H_{0,1}^{z}(x))-\partial^{2}_{x}H^{z}_{0,1}(x)\big{[}\dot{U}(H_{-1,0}(x))-\dot{U}\big{(}H_{-1,0}(x)+H^{z}_{0,1}(x))\big{]}\,dx.$

Then, integrating by parts, we obtain

$\ddot{A}(z)=\int_{\mathbb{R}}\partial_{x}H^{z}_{0,1}(x)\partial_{x}H_{-1,0}(x)\big{[}\ddot{U}(H_{-1,0}(x))-\ddot{U}(H_{-1,0}(x)+H^{z}_{0,1}(x))\big{]}\,dx.$
(14)

Now, consider the function

$B(z)=\int_{\mathbb{R}}\partial_{x}H_{0,1}(x)\partial_{x}H_{-1,0}(x+z)\big{[}\ddot{U}(0)-\ddot{U}(H_{0,1}(x))\big{]}\,dx.$
(15)

Then, we have

$|\ddot{A}(z)-B(z)|=\Big{|}\int_{\mathbb{R}}\partial_{x}H^{z}_{0,1}(x)\partial_{x}H_{-1,0}(x)\big{[}\big{(}\ddot{U}(H_{-1,0}(x))-\ddot{U}(H_{-1,0}(x)+H^{z}_{0,1}(x))\big{)}-\big{(}\ddot{U}(0)-\ddot{U}(H_{0,1}^{z}(x))\big{)}\big{]}\,dx\Big{|}.$
(16)

Also, it’s not difficult to verify the following identity

$\big{[}\ddot{U}(H_{-1,0}(x))-\ddot{U}(H_{-1,0}(x)+H^{z}_{0,1}(x))\big{]}-\big{[}\ddot{U}(0)-\ddot{U}(H^{z}_{0,1}(x))\big{]}=-\int_{0}^{H_{-1,0}(x)}\int_{0}^{H_{0,1}^{z}(x)}U^{(4)}(\omega_{1}+\omega_{2})\,d\omega_{1}\,d\omega_{2}.$
(17)

So, the identities (17) and (16) imply the following inequality

$|\ddot{A}(z)-B(z)|\leq\int_{\mathbb{R}}|\partial_{x}H^{z}_{0,1}(x)\partial_{x}H_{-1,0}(x)|\Big{|}\int_{0}^{H_{-1,0}(x)}\int_{0}^{H^{z}_{0,1}(x)}|U^{(4)}(\omega_{1}+\omega_{2})|\,d\omega_{1}\,d\omega_{2}\Big{|}\,dx.$
(18)

Since $U$ is smooth and $\left\lVert H_{0,1}\right\rVert_{L^{\infty}}=1$, we
have that there is a constant $C>0$ such that

$|\ddot{A}(z)-B(z)|\leq
C\int_{\mathbb{R}}|\partial_{x}H^{z}_{0,1}(x)\partial_{x}H_{-1,0}(x)H_{-1,0}(x)H^{z}_{0,1}(x)|\,dx.$
(19)

Now, Using the inequalities from (D1) to (D4) and Lemma 2.3 to the above
inequality (19), we obtain that exist a constant $C_{1}$ non dependent of $z$
such that

$|\ddot{A}(z)-B(z)|\leq C_{1}ze^{-2\sqrt{2}z}.$ (20)

Also, it’s not difficult to verify that the estimate

$\Big{|}\partial_{x}H_{-1,0}(x)-\sqrt{2}e^{-\sqrt{2}x}\Big{|}\leq
C\min(e^{-3\sqrt{2}x},e^{-\sqrt{2}x}).$ (21)

and the identity (15) imply the inequality

$\Big{|}B(z)-\sqrt{2}e^{-\sqrt{2}z}\int_{\mathbb{R}}e^{-\sqrt{2}x}\partial_{x}H_{0,1}(x)(\ddot{U}(0)-\ddot{U}(H_{0,1}(x)))\,dx\Big{|}\lesssim\int_{\mathbb{R}}|\partial_{x}H_{0,1}(x)|\min\big{(}e^{-3\sqrt{2}(x+z)},e^{-\sqrt{2}(x+z)}\big{)}\,dx\\\
\lesssim\int_{\mathbb{R}}e^{-2\sqrt{2}(-x)_{+}}\min\big{(}e^{-3\sqrt{2}(x+z)},e^{-\sqrt{2}(x+z)}\big{)}\,dx\lesssim\int_{-\infty}^{0}e^{-2\sqrt{2}(z-x)_{+}}e^{-\sqrt{2}x}\,dx+\int_{0}^{+\infty}e^{-2\sqrt{2}(z-x)_{+}}e^{-3\sqrt{2}(x)_{+}}\,dx.$
(22)

Since, we have the following identity and an estimate from Lemma 2.3

$\displaystyle\int_{-\infty}^{0}e^{-2\sqrt{2}(z-x)}e^{-\sqrt{2}x}\,dx=\frac{e^{-2\sqrt{2}z}}{\sqrt{2}},$
(23)
$\displaystyle\int_{0}^{+\infty}e^{-2\sqrt{2}(z-x)_{+}}e^{-3\sqrt{2}(x)_{+}}\lesssim
e^{-2\sqrt{2}z},$ (24)

we obtain, then:

$\Big{|}B(z)-\sqrt{2}e^{-\sqrt{2}z}\int_{\mathbb{R}}e^{-\sqrt{2}x}\partial_{x}H_{0,1}(x)\big{[}\ddot{U}(0)-\ddot{U}(H_{0,1}(x))\big{]}\,dx\Big{|}\lesssim
e^{-2\sqrt{2}z},$ (25)

which clearly implies with (20) the inequality

$\Big{|}\ddot{A}(z)-\sqrt{2}e^{-\sqrt{2}z}\int_{\mathbb{R}}e^{-\sqrt{2}x}\partial_{x}H_{0,1}(x)\big{[}\ddot{U}(0)-\ddot{U}(H_{0,1}(x))\big{]}\,dx\Big{|}\lesssim
ze^{-2\sqrt{2}z}.$ (26)

Also we have the identity

$\int_{\mathbb{R}}\big{(}8(H_{0,1}(x))^{3}-6(H_{0,1}(x))^{5}\big{)}e^{-\sqrt{2}x}\,dx=2\sqrt{2},$
(27)

for the proof consult the Appendix A. Also, since we have the identity
$\ddot{U}(0)-\ddot{U}(\phi)=24\phi^{2}-30\phi^{4}$, by integration by parts,
we obtain

$\int_{\mathbb{R}}\frac{e^{-\sqrt{2}x}}{\sqrt{2}}\partial_{x}H_{0,1}(x)\big{[}\ddot{U}(0)-\ddot{U}(H_{0,1}(x))\big{]}\,dx=\int_{\mathbb{R}}\big{(}8(H_{0,1}(x))^{3}-6(H_{0,1}(x))^{5}\big{)}e^{-\sqrt{2}x}\,dx.$
(28)

In conclusion, inequality (26) is equivalent to
$\Big{|}\ddot{A}(z)-4\sqrt{2}e^{-\sqrt{2}z}\Big{|}\lesssim ze^{-2\sqrt{2}z}.$
The identities

$\displaystyle\dot{U}(\phi)+\dot{U}(\theta)-\dot{U}(\phi+\theta)=24\phi\theta(\phi+\theta)-6\Big{(}\sum_{j=1}^{4}\begin{pmatrix}5\\\
j\end{pmatrix}\phi^{j}\omega^{5-j}\Big{)},$
$\displaystyle\dot{A}(z)=-\int_{\mathbb{R}}\partial_{x}H^{z}_{0,1}(x)\big{[}\dot{U}(H^{z}_{0,1}(x)+H_{-1,0}(x))+\dot{U}(H_{-1,0}(x))-\dot{U}(H^{z}_{0,1}(x))\big{]}\,dx$

and Lemma 2.3 imply the following estimate for $z>0$

$|\dot{A}(z)|\lesssim e^{-\sqrt{2}z},$

so $\lim_{|z|\to+\infty}|\dot{A}(z)|=0$. In conclusion, integrating inequality
$\Big{|}\ddot{A}(z)-4\sqrt{2}e^{-\sqrt{2}z}\Big{|}\lesssim ze^{-2\sqrt{2}z}$
from $z$ to $+\infty$ we obtain the second result of the lemma

$\Big{|}\dot{A}(z)+4e^{-\sqrt{2}z}\Big{|}\lesssim ze^{-2\sqrt{2}z}.$ (29)

Finally, from the fact that
$\lim_{z\to+\infty}E_{pot}(H_{-1,0}+H^{z}_{0,1}(x))=2E_{pot}(H_{0,1})$, we
obtain the last estimate integrating inequality (29) from $z$ to $+\infty$,
which is

$\Big{|}2E_{pot}(H_{0,1})+2\sqrt{2}e^{-\sqrt{2}z}-A(z)\Big{|}\lesssim
ze^{-2\sqrt{2}z}.$

∎

It is not difficult to verify that the Fréchet derivative of $E_{pot}$ as a
linear functional from $H^{1}(\mathbb{R})$ to $\mathbb{R}$ is given by

$(DE_{pot}(\phi))(v)\coloneqq\int_{\mathbb{R}}\partial_{x}\phi(x)\partial_{x}v(x)+\dot{U}(\phi(x))v(x)\,dx.$
(30)

Also, it is not difficult to verify that for any $v,\,w\in H^{1}(\mathbb{R})$,
we have

$\left\langle
D^{2}E_{pot}(\phi)v,\,w\right\rangle_{L^{2}(\mathbb{R})}=\int_{\mathbb{R}}\partial_{x}v(x)\partial_{x}w(x)\,dx+\int_{\mathbb{R}}\ddot{U}(\phi(x))v(x)w(x)\,dx.$
(31)

######  Lemma 2.5 (Coercitivity Lemma).

$\exists\,C,c,\,\delta>0,$ such that if $x_{2}-x_{1}\geq\frac{1}{\delta}$,
then for any $g\in H^{1}(\mathbb{R})$ we have

$\left\langle
D^{2}E_{pot}(H^{x_{2}}_{0,1}+H^{x_{1}}_{-1,0})g,\,g\right\rangle_{L^{2}(\mathbb{R})}\geq
c\left\lVert g\right\rVert_{H^{1}(\mathbb{R})}^{2}-C\left[\langle
g,\,\partial_{x}H^{x_{1}}_{-1,0}\rangle^{2}+\langle
g,\,\partial_{x}H^{x_{2}}_{0,1}\rangle^{2}\right].$ (32)

The proof of this lemma is based in the proof of Lemma 2.4 from [19]. To prove
the Coercitivity Lemma, we need the following result about the spectrum and
kernel of the operators
$D^{2}E_{pot}(H^{x_{1}}_{-1,0}),\,D^{2}E_{pot}(H^{x_{2}}_{0,1}).$

######  Lemma 2.6.

The operators $D^{2}E_{pot}(H^{x_{1}}_{-1,0}),\,D^{2}E_{pot}(H^{x_{2}}_{0,1})$
satisfy the following properties

  1. 1

$\ker
D^{2}E_{pot}(H^{x_{1}}_{-1,0})=\left\\{c\partial_{x}H^{x_{1}}_{-1,0},\,c\in\mathbb{C}\right\\},\,\ker
D^{2}E_{pot}(H^{x_{2}}_{0,1})=\left\\{c\partial_{x}H^{x_{2}}_{0,1},\,c\in\mathbb{C}\right\\}$

  2. 2

$\sigma(D^{2}E_{pot}(H^{x_{2}}_{0,1}))=\sigma(D^{2}E_{pot}(H^{x_{1}}_{-1,0}))\subset\\{0\\}\cup[\lambda_{1},+\infty)$,
with $\lambda_{1}>0.$

###### Proof.

Since the operators $D^{2}E_{pot}(H^{x_{1}}_{-1,0})$ and
$D^{2}E_{pot}(H^{x_{1}}_{-1,0})$ are equivalent by reflection and translation,
they have the same spectrum that $D^{2}E_{pot}(H_{-1,0}).$ So, we’ll just
analyse the spectrum of the operator

$D^{2}E_{pot}(H^{x_{1}}_{-1,0})=-\partial^{2}_{x}+\ddot{U}(H^{x_{1}}_{-1,0}).$
(33)

Also, we will only study the kernel of $D^{2}E_{pot}(H^{x_{1}}_{-1,0}),$ since
the kernel of the other operator can be found by similar reasoning.

If we derive the Bogomolny equation satisfied by $H_{-1,0}$

$\partial^{2}_{x}H^{x_{1}}_{-1,0}(x)=\dot{U}(H^{x_{1}}_{-1,0}(x))$ (34)

with respect to $x,$ we obtain the identity

$\partial^{2}_{x}(\partial_{x}H^{x_{1}}_{-1,0}(x))=\ddot{U}(H^{x_{1}}_{-1,0}(x))\partial_{x}H^{x_{1}}_{-1,0}(x),$
(35)

which implies that $\partial_{x}H^{x_{1}}_{-1,0}\in\ker
D^{2}E_{pot}(H^{x_{1}}_{-1,0})$. Also, $\partial_{x}H_{-1,0}(x)\neq 0$ for all
$x\in\mathbb{R}$, so the Sturm-Liouville Oscillation Theory implies indeed
that $0$ is the minimum element of the discrete spectrum of
$D^{2}E_{pot}(H^{x_{1}}_{-1,0})$ and

$\ker
D^{2}E_{pot}(H^{x_{1}}_{-1,0})=\left\\{c\partial_{x}H^{x_{1}}_{-1,0},\,c\in\mathbb{C}\right\\}.$
(36)

In conclusion, we have obtained that for some constant $\lambda_{1}>0$

$\sigma_{d}(D^{2}E_{pot}(H^{x_{1}}_{-1,0}))\subset\\{0\\}\cup[\lambda_{1},+\infty).$
(37)

By similar reasoning, we have

$\displaystyle\sigma_{d}(D^{2}E_{pot}(H^{x_{2}}_{0,1}))\subset\\{0\\}\cup[\lambda_{1},+\infty),$
(38)
$\displaystyle\ker(D^{2}E_{pot}(H^{x_{2}}_{0,1}))=\left\\{c\partial_{x}H^{x_{2}}_{0,1},\,c\in\mathbb{C}\right\\}.$
(39)

Now, it remains to estimate the lower bound of the essential spectrum of both
operators. The main tool used to estimate the essential spectrum is a theorem
of Spectral Theory written in the book [3].

######  Theorem 2.7.

Suppose $A$ and $B$ are self-adjoint operators on a Hilbert Space
$\mathcal{H}$. If $\exists z\in\mathbb{C}$, such that $(A-z)^{-1}-(B-z)^{-1}$
is compact, then $\sigma_{ess}(A)=\sigma_{ess}(B)$.

Since
$D^{2}E_{pot}(H^{x_{1}}_{-1,0})=-\partial^{2}_{x}+(2-24(H^{x_{1}}_{-1,0})^{2}+30(H^{x_{1}}_{-1,0})^{4})$,
we can rewrite this operator as

$D^{2}E_{pot}(H^{x_{1}}_{-1,0})=\Big{(}-\partial^{2}_{x}+\big{[}2-24(H^{x_{1}}_{-1,0})^{2}+30(H^{x_{1}}_{-1,0})^{4}-2\chi_{[0,+\infty)}(x)-8\chi_{(-\infty,0)}(x)\big{]}\\\
+\big{[}2\chi_{[0,+\infty)}(x)+8\chi_{(-\infty,0)}(x)\big{]}\Big{)}.$

Now, we consider

$T_{1}=-\partial^{2}_{x}+\big{[}2\chi_{[0,+\infty)}(x)+8\chi_{(-\infty,0)}(x)\big{]}.$

The next step is to check that for the self-adjoint operators
$A=D^{2}E_{pot}(H^{x_{1}}_{-1,0})$, $B=T_{1}$ and for $z=-i$ all the
hypothesis of Theorem 2.7 are fulfilled, which would imply that
$\sigma_{ess}(D^{2}E_{pot}(H^{x_{1}}_{-1,0}))=\sigma_{ess}(T_{1}).$

Since we have the identity

$(D^{2}E_{pot}(H^{x_{1}}_{-1,0})+i)^{-1}-(T_{1}+i)^{-1}=-(D^{2}E_{pot}(H^{x_{1}}_{-1,0})+i)^{-1}\circ\Big{(}2-24(H^{x_{1}}_{-1,0})^{2}\\\
+30(H^{x_{1}}_{-1,0})^{4}-2\chi_{[0,+\infty)}(x)-8\chi_{(-\infty,0)}(x)\Big{)}\circ(T_{1}+i)^{-1},$
(40)

to prove that $D^{2}E_{pot}(H^{x_{1}}_{-1,0})$ and $T_{1}$ have same essential
spectrum, we only need to verify that

$T_{2}=\Big{(}2-24(H^{x_{1}}_{-1,0})^{2}+30(H^{x_{1}}_{-1,0})^{4}-2\chi_{[0,+\infty)}(x)-8\chi_{(-\infty,0)}(x)\Big{)}\circ(T_{1}+i)^{-1}$
(41)

is a compact operator on $L^{2}(\mathbb{R})$. By asymptotic properties of
$H_{-1,0}$, it is not difficult to verify that

$Y=\Big{(}2-24(H^{x_{1}}_{-1,0})^{2}+30(H^{x_{1}}_{-1,0})^{4}-2\chi_{[0,+\infty)}(x)-8\chi_{(-\infty,0)}(x)\Big{)}$

decays exponentially when $|x|$ goes to $+\infty$. Also, it is not difficult
to verify that $(T_{1}+i)^{-1}$ is a bounded map from $L^{2}(\mathbb{R})$ to
$H^{1}(\mathbb{R})\subset L^{\infty}(\mathbb{R})$. The last information and
the Banach-Alaoglu Theorem imply that for any bounded sequence $(v_{n})\subset
L^{2}(\mathbb{R})$, $\exists\,w\in H^{1}(\mathbb{R})$ and a subsequence that
for simplicity we’ll still denote by $(v_{n})$ such that

$(T_{1}+i)^{-1}(v_{n})\underset{H^{1}}{\rightharpoonup}w.$ (42)

Also, from the fact that $(T_{1}+i)^{-1}(v_{n})$ is uniformly bounded in
$H^{1}(\mathbb{R})$, it can be verified that for any compact interval
$K\subset\mathbb{R}$ that

$(T_{1}+i)^{-1}(v_{n})\underset{L^{\infty}(K)}{\rightarrow}w,$

and this fact with the exponential decay of $Y$ and $H^{1}(\mathbb{R})\subset
L^{\infty}(\mathbb{R})$ implies directly the following convergence

$T_{2}(v_{n})\underset{L^{2}}{\rightarrow}Y(w),$ (43)

which implies that $T_{1}$ and $D^{2}E_{pot}(H^{x_{1}}_{-1,0})$ have the same
essential spectrum, more precisely

$\sigma_{ess}\left(D^{2}E_{pot}(H^{x_{1}}_{-1,0})\right)\subset[2,+\infty),$
(44)

and so,

$\sigma_{ess}\left(D^{2}E_{pot}(H^{x_{2}}_{0,1})\right)\subset[2,+\infty).$
(45)

This finishes the proof of Lemma 2.6. ∎

Before starting the demonstration of the Coercitivity Lemma, let’s consider
from now on the function $0\leq\zeta\leq 1$ to be a smooth function
satisfying:

$\zeta(x)=\begin{cases}1,\text{ if $x\leq\frac{3}{4}$,}\\\ 0,\text{ if
$x\geq\frac{4}{5}$.}\end{cases}$ (46)

###### Proof of Coercitivity Lemma.

Our proof follows the scheme of proof of Lemma $2.4$ of [19]. Here we denote
$\langle,\,\rangle$ to be the scalar product on $L^{2}(\mathbb{R})$. First,
because of Lemma 2.6, there is a $\lambda>0$ such that for any $v\in
H^{1}(\mathbb{R})$

$\left\langle
D^{2}E_{pot}(H^{x_{1}}_{-1,0})v,\,v\right\rangle\geq\lambda(\left\lVert
v\right\rVert_{L^{2}(\mathbb{R})}^{2}-\left\lVert\partial_{x}H_{-1,0}\right\rVert_{L^{2}}^{-2}\langle
v,\,\partial_{x}H_{-1,0}^{x_{1}}\rangle^{2}).$ (47)

Also, because of the identity (33), we have

$\left\langle
D^{2}E_{pot}(H^{x_{1}}_{-1,0})v,\,v\right\rangle=\left\lVert\partial_{x}v\right\rVert_{L^{2}(\mathbb{R})}^{2}+\int_{\mathbb{R}}\ddot{U}(H^{x_{1}}_{-1,0}(x))(v(x))^{2}\,dx.$
(48)

Then, inequalities (47) and (48) imply for any $0<\theta<1$ that

$\left\langle
D^{2}E_{pot}(H^{x_{1}}_{-1,0})v,\,v\right\rangle\geq\theta\lambda\big{[}\left\lVert
v\right\rVert_{L^{2}(\mathbb{R})}^{2}-\left\lVert\partial_{x}H_{-1,0}\right\rVert_{L^{2}}^{-2}\langle
v,\,\partial_{x}H_{-1,0}^{x_{1}}\rangle^{2}\big{]}+(1-\theta)\big{[}\left\lVert\partial_{x}v\right\rVert_{L^{2}(\mathbb{R})}^{2}\\\
+\int_{\mathbb{R}}\ddot{U}(H^{x_{2}}_{0,1}(x))(v(x))^{2}\,dx\big{]}.$ (49)

Since $\left\lVert\ddot{U}(H_{-1,0}(x))\right\rVert_{L^{\infty}}<\infty$, we
can choose $\theta$ close enough to 1, and obtain from (49) the following
inequality for a positive constant $c>0$ such that

$\left\langle D^{2}E_{pot}(H^{x_{1}}_{-1,0})v,\,v\right\rangle\geq
c\big{[}\left\lVert
v\right\rVert_{H^{1}(\mathbb{R})}^{2}-\left\lVert\partial_{x}H_{-1,0}\right\rVert_{L^{2}}^{-2}\langle
v,\,\partial_{x}H_{-1,0}^{x_{1}}\rangle^{2}\big{]}.$ (50)

By similar reasoning, we also have

$\langle D^{2}E_{pot}(H^{x_{2}}_{0,1})v,\,v\rangle\geq c\big{[}\left\lVert
v\right\rVert_{H^{1}(\mathbb{R})}^{2}-\left\lVert\partial_{x}H_{-1,0}\right\rVert_{L^{2}}^{-2}\langle
v,\,\partial_{x}H_{0,1}^{x_{2}}\rangle^{2}\big{]}.$ (51)

Now to study the operator $D^{2}E_{pot}(H^{x_{2}}_{0,1}+H^{x_{1}}_{-1,0})$,
consider the function $\zeta_{1}(x)=\zeta(\frac{x-x_{1}}{x_{2}-x_{1}})$ and
also

$\displaystyle
V\coloneqq(\ddot{U}(H^{x_{1}}_{-1,0}+H^{x_{2}}_{0,1})-\ddot{U}(H^{x_{2}}_{0,1})),$
(52) $\displaystyle
D^{2}E_{pot}(H^{x_{2}}_{0,1}+H^{x_{1}}_{-1,0})=-\partial^{2}_{x}+\ddot{U}(H^{x_{2}}_{0,1})+V.$
(53)

It can be verified that the support of $(1-\zeta_{1}(x))$ is included in
$\\{x\in\mathbb{R}\,|\frac{x-x_{1}}{x_{2}-x_{1}}\geq\frac{3}{4}\\}$, and so

$\displaystyle(1-\zeta_{1}(x))|H^{x_{1}}_{-1,0}(x)|\leq
e^{-\frac{3\sqrt{2}(x_{2}-x_{1})}{4}},$ (54) $\displaystyle\Big{|}\langle
V(x)(1-\zeta_{1}(x))v(x),\,(1-\zeta_{1}(x))v(x)\rangle\Big{|}\lesssim
e^{-\frac{3\sqrt{2}(x_{2}-x_{1})}{4}}\left\lVert v\right\rVert_{L^{2}}^{2}.$
(55)

Therefore, if $\delta>0$ is small enough, from
$x_{2}-x_{1}\geq\frac{1}{\delta}$, we obtain from the inequality (55) that

$\Big{|}\langle
V(x)(1-\zeta_{1}(x))v(x),\,(1-\zeta_{1}(x))v(x)\rangle\Big{|}\leq
O(\delta)\left\lVert v\right\rVert_{L^{2}}^{2},$ (56)

so the inequalities (56) and (51) imply for a $c>0$

$\left\langle
D^{2}E_{pot}(H^{x_{1}}_{-1,0}+H^{x_{2}}_{0,1})((1-\zeta_{1}(x))v),\,(1-\zeta_{1}(x))v\right\rangle\geq
c\big{[}\left\lVert(1-\zeta_{1})v\right\rVert_{H^{1}(\mathbb{R})}^{2}-\left\lVert\partial_{x}H_{0,1}\right\rVert_{L^{2}}^{-2}\langle(1-\zeta_{1})v,\,\partial_{x}H_{0,1}^{x_{2}}\rangle^{2}\big{]}\\\
-O(\delta)\left\lVert v\right\rVert_{L^{2}(\mathbb{R})}^{2}.$ (57)

Also the support of $\zeta_{1}(x)$ is included
$\\{x|\,\frac{x-x_{2}}{x_{2}-x_{1}}\leq{-\frac{1}{5}}\\}.$ So, by similar
arguments, we can verify the analogous inequality

$\left\langle
D^{2}E_{pot}(H^{x_{1}}_{-1,0}+H^{x_{2}}_{0,1})(\zeta_{1}(x)v),\,\zeta_{1}(x)v\right\rangle\geq
c\big{[}\left\lVert\zeta_{1}v\right\rVert_{H^{1}(\mathbb{R})}^{2}-\left\lVert\partial_{x}H_{-1,0}\right\rVert_{L^{2}}^{-2}\langle\zeta_{1}v,\,\partial_{x}H_{-1,0}^{x_{1}}\rangle^{2}\big{]}-O(\delta)\left\lVert
v\right\rVert_{L^{2}(\mathbb{R})}^{2}.$ (58)

Also, we obtain that there is a uniform constant $C>0$ such that if $\delta>0$
is small enough, then we obtain the following estimate for all $v\in
H^{1}(\mathbb{R})$

$\left\langle\partial_{x}(\zeta_{1}(x)v(x)),\,\partial_{x}\left[(1-\zeta_{1}(x))v(x)\right]\right\rangle\geq-C\delta\left\lVert
v\right\rVert_{H^{1}(\mathbb{R})}^{2}.$ (59)

Also, if $\delta>0$ is small enough, we have that $\ddot{U}(H_{-1,0}(x))>1$
for $\frac{3(x_{2}-x_{1})}{4}\leq x-x_{1}\leq\frac{4(x_{2}-x_{1})}{5}$. In
conclusion, since the support of $(1-\zeta_{1}(x))\zeta_{1}(x)$ is included in
$\\{x-x_{1}\in[\frac{3(x_{2}-x_{1})}{4},\,\frac{4(x_{2}-x_{1})}{5}]\\}$, we
have the following inequality

$\displaystyle\int_{\mathbb{R}}\ddot{U}(H^{x_{1}}_{-1,0}(x))\zeta_{1}(x)(1-\zeta_{1}(x))(v(x))^{2}\,dx\geq
0.$ (60)

Finally, from the mean value theorem, the knowledge of the support of
$\zeta_{1}$ and the exponential decay of $H_{0,1}(x)$, we have that

$\displaystyle\Big{|}\int_{\mathbb{R}}\big{[}\ddot{U}(H^{x_{1}}_{-1,0}(x)+H^{x_{2}}_{0,1}(x))-\ddot{U}(H^{x_{1}}_{-1,0}(x))\big{]}\zeta_{1}(x)(1-\zeta_{1}(x))v(x)^{2}\,dx\Big{|}\leq
o(1)\left\lVert v\right\rVert_{L^{2}}^{2}.$ (61)

Therefore, the inequalities (59), (60) and (61) imply for a uniform constant
$C>0$ that

$\langle
D^{2}E_{pot}(H^{x_{1}}_{-1,0}+H^{x_{2}}_{0,1})(\zeta_{1}v),\,(1-\zeta_{1})v\rangle\geq-C\delta\left\lVert
v\right\rVert_{L^{2}(\mathbb{R})}^{2}.$ (62)

Since we know that support of $\zeta_{1}$ is included in
$\\{x|\,\frac{x-x_{2}}{x_{2}-x_{1}}\leq{-\frac{1}{5}}\\}$, we can deduce the
estimate

$\Big{|}\langle\partial_{x}H^{x_{2}}_{0,1},\,(1-\zeta_{1})v\rangle^{2}-\langle\partial_{x}H^{x_{2}}_{0,1},\,v\rangle^{2}\Big{|}=\Big{|}\langle\partial_{x}H^{x_{2}}_{0,1},\,\zeta_{1}v\rangle\langle\partial_{x}H^{x_{2}}_{0,1},\,(2-\zeta_{1})v\rangle\Big{|}\leq
O(\delta)\left\lVert v\right\rVert_{L^{2}}^{2},$ (63)

and similarly,

$\Big{|}\langle\partial_{x}H^{x_{1}}_{-1,0},\,\zeta_{1}v\rangle^{2}-\langle\partial_{x}H^{x_{1}}_{-1,0},\,v\rangle^{2}\Big{|}\leq
O(\delta)\left\lVert v\right\rVert_{L^{2}}^{2}.$ (64)

Therefore, we have that (57), (58), (62), (63) and (64) imply the inequality
(32) of the statement. ∎

######  Lemma 2.8.

There is a constant $C_{2}$, such that if $x_{2}-x_{1}>0$, then

$\left\lVert
DE_{pot}(H^{x_{2}}_{0,1}+H^{x_{1}}_{-1,0})\right\rVert_{L^{2}(\mathbb{R})}\leq
C_{2}e^{-\sqrt{2}(x_{2}-x_{1})}.$ (65)

###### Proof.

By the definition of the potential energy, the equation (3) and the
exponential decay of the two kinks functions, we have that

$DE_{pot}(H^{x_{2}}_{0,1}+H^{x_{1}}_{-1,0})=\dot{U}(H^{x_{2}}_{0,1}+H^{x_{1}}_{-1,0})-\dot{U}(H^{x_{2}}_{0,1})-\dot{U}(H^{x_{1}}_{-1,0})$

as a bounded linear operator from $L^{2}(\mathbb{R})$ to $\mathbb{C}$. So, we
have that

$DE_{pot}(H^{x_{2}}_{0,1}+H^{x_{1}}_{-1,0})=-24H^{x_{2}}_{0,1}H^{x_{1}}_{-1,0}\big{[}H^{x_{2}}_{0,1}+H^{x_{1}}_{-1,0}\big{]}+6\Big{[}\sum_{j=1}^{4}\begin{pmatrix}5\\\
j\end{pmatrix}(H^{x_{1}}_{-1,0})^{j}(H^{x_{2}}_{0,1})^{5-j}\Big{]},$

and, then, the conclusion follows directly from Lemma 2.3, $(D1)$ and $(D2).$
∎

######  Theorem 2.9 (Orbital Stability of a sum of two moving kinks).

$\exists\delta_{0}>0$ such that if the solution $\phi$ of (1) satisfies
$(\phi(0),\partial_{t}\phi(0))\in S\times L^{2}(\mathbb{R})$ and the energy
excess $\epsilon=E(\phi)-2E_{pot}(H_{0,1})$ is smaller than $\delta_{0}$, then
$\exists x_{1},\,x_{2}:\mathbb{R}\to\mathbb{R}$ functions of class $C^{2}$,
such that for all $t\in\mathbb{R}$ denoting
$g(t)=\phi(t)-H_{0,1}(x-x_{2}(t))+H_{-1,0}(x-x_{1}(t))$ and
$z(t)=x_{2}(t)-x_{1}(t)$, we have:

  1. 1.

$\left\lVert g(t)\right\rVert_{H^{1}(\mathbb{R})}=O(\epsilon^{\frac{1}{2}}),$

  2. 2.

$z(t)\geq\frac{1}{\sqrt{2}}[\ln{(\frac{1}{\epsilon})}+\ln{2}],$

  3. 3.

$\left\lVert\partial_{t}\phi(t)\right\rVert_{L^{2}(\mathbb{R})}^{2}\leq
2\epsilon,$

  4. 4.

$\max_{j\in\\{1,2\\}}|\dot{x}_{j}(t)|^{2}+\max_{j\in\\{1,2\\}}|\ddot{x}_{j}(t)|=O(\epsilon).$

###### Proof.

First, from the fact that $E_{total}(\phi(x))>2E_{pot}(H_{0,1}),$ we deduce,
from the conservation of total energy, the estimate

$\left\lVert\partial_{t}\phi(t)\right\rVert_{L^{2}}^{2}\leq 2\epsilon.$ (66)

From Remark 1.8, we can assume if $\epsilon\ll 1$ that there are
$p_{1},\,p_{2}\in\mathbb{R}$ such that

$\phi(0,x)=H_{0,1}(x-p_{2})+H_{-1,0}(x-p_{1})+g_{1}(x),$

such that

$\left\lVert
g_{1}\right\rVert_{H^{1}(\mathbb{R})}<\delta,\,p_{2}-p_{1}>\frac{1}{\delta},$

for a small constant $\delta>0$. Since the equation 1 is locally well-posed in
the space $S\times L^{2}(\mathbb{R}),$ we conclude that there is a
$\delta_{1}>0$ depending only on $\delta$ and $\epsilon$ such that if
$-\delta_{1}\leq t\leq\delta_{1},$ then

$\left\lVert\phi(t,x)-H_{0,1}(x-p_{2})-H_{-1,0}(x-p_{1})\right\rVert_{H^{1}(\mathbb{R})}\leq
2\delta.$ (67)

If $\delta,\epsilon>0$ are small enough, then, from the inequality (67) and
the Modulation Lemma, we obtain in the time interval
$[-\delta_{1},\delta_{1}]$ the existence of modulations parameters
$x_{1}(t),\,x_{2}(t)$ such that for

$g(t)=\phi(t)-H_{0,1}(x-x_{2}(t))-H_{-1,0}(x-x_{1}(t)),$

we have

$\displaystyle\left\langle
g(t),\,\partial_{x}H_{0,1}(x-x_{2}(t))\right\rangle_{L^{2}}=\left\langle
g(t),\,\partial_{x}H_{-1,0}(x-x_{1}(t))\right\rangle_{L^{2}}=0,$ (68)
$\displaystyle\frac{1}{|x_{2}(t)-x_{1}(t)|}+\left\lVert
g(t)\right\rVert_{H^{1}}\lesssim\delta.$ (69)

From now on, we denote $z(t)=x_{2}(t)-x_{1}(t).$ From the Energy Conservation
Law, we have for $-\delta_{1}\leq t\leq\delta_{1}$ that

$E(\phi(t))=2E_{pot}(H_{0,1})+\epsilon=\frac{\left\lVert\partial_{t}\phi(t)\right\rVert_{L^{2}(\mathbb{R})}^{2}}{2}+E_{pot}\big{(}H^{x_{2}(t)}_{0,1}+H^{x_{1}(t)}_{-1,0}\big{)}+\big{\langle}DE_{pot}\big{(}H^{x_{2}(t)}_{0,1}+H^{x_{1}(t)}_{-1,0}\big{)},\,g(t)\big{\rangle}_{L^{2}(\mathbb{R})}\\\
+\frac{\big{\langle}D^{2}E_{pot}\big{(}H^{x_{2}(t)}_{0,1}+H^{x_{1}(t)}_{-1,0}\big{)}g(t),\,g(t)\big{\rangle}_{L^{2}(\mathbb{R})}}{2}+O(\left\lVert
g(t)\right\rVert_{H^{1}}^{3}).$

From Lemma 2.4 and (69), the above identity implies that

$\epsilon=\frac{\left\lVert\partial_{t}\phi(t)\right\rVert_{L^{2}(\mathbb{R})}^{2}}{2}+2\sqrt{2}e^{-\sqrt{2}z(t)}+\big{\langle}DE_{pot}\big{(}H^{x_{2}(t)}_{0,1}+H^{x_{1}(t)}_{-1,0}\big{)},\,g(t)\big{\rangle}_{L^{2}(\mathbb{R})}+\frac{\big{\langle}D^{2}E_{pot}\big{(}H^{x_{2}(t)}_{0,1}+H^{x_{1}(t)}_{-1,0}\big{)}g(t),\,g(t)\big{\rangle}_{L^{2}(\mathbb{R})}}{2}\\\
+O\Big{(}\left\lVert
g(t)\right\rVert_{H^{1}}^{3}+z(t)e^{-2\sqrt{2}z(t)}\Big{)}$ (70)

for $-\delta_{1}\leq t\leq\delta_{1}.$ From (65), it is not difficult to
verify that $|\langle
DE_{pot}(H^{x_{2}(t)}_{0,1}+H^{x_{1}(t)}_{-1,0}),\,g(t)\rangle|\leq
C_{2}e^{-\sqrt{2}z(t)}\left\lVert g(t)\right\rVert_{H^{1}(\mathbb{R})}.$ So,
the equation (70) and the Coercitivity Lemma imply, while $-\delta_{1}\leq
t\leq\delta_{1}$, the following inequality

$\epsilon+C_{2}e^{-\sqrt{2}z(t)}\left\lVert
g(t)\right\rVert_{H^{1}(\mathbb{R})}\geq\frac{\left\lVert\partial_{t}\phi(t)\right\rVert_{L^{2}}^{2}}{2}+2\sqrt{2}e^{-\sqrt{2}z(t)}+\frac{c\left\lVert
g(t)\right\rVert_{H^{1}(\mathbb{R})}^{2}}{2}+O\big{(}\left\lVert
g(t)\right\rVert_{H^{1}}^{3}+z(t)e^{-2\sqrt{2}z(t)}\big{)}.$ (71)

Finally, applying the Young Inequality in the term
$C_{2}e^{-\sqrt{2}z(t)}\left\lVert g(t)\right\rVert_{H^{1}(\mathbb{R})}$, we
obtain that the inequality (71) can be rewritten in the form

$\epsilon\geq\frac{\left\lVert\partial_{t}\phi(t)\right\rVert_{L^{2}}^{2}}{2}+2\sqrt{2}e^{-\sqrt{2}z(t)}+\frac{c\left\lVert
g(t)\right\rVert_{H^{1}(\mathbb{R})}^{2}}{4}+O\big{(}\left\lVert
g(t)\right\rVert_{H^{1}}^{3}+z(t)e^{-2\sqrt{2}z(t)}+e^{-2\sqrt{2}z(t)}\big{)}.$
(72)

Then, the estimates (72), (69) imply for $\delta>0$ small enough the following
inequality

$\epsilon\geq\frac{\left\lVert\partial_{t}\phi(t)\right\rVert_{L^{2}}^{2}}{2}+2e^{-\sqrt{2}z(t)}+\frac{c\left\lVert
g(t)\right\rVert_{H^{1}(\mathbb{R})}^{2}}{8}.$ (73)

So, the inequality (73) implies the estimates

$\displaystyle e^{-\sqrt{2}z(t)}<\frac{\epsilon}{2},$ (74)
$\displaystyle\left\lVert
g(t)\right\rVert_{H^{1}(\mathbb{R})}^{2}\lesssim\epsilon,$ (75)

for $t\in[-\delta_{1},\delta_{1}]$. In conclusion, if
$\frac{1}{\delta}\lesssim\ln{(\frac{1}{\epsilon})}^{\frac{1}{2}},$ we can
conclude by a bootstrap argument that the inequalities (66), (74), (75) are
true for all $t\in\mathbb{R}.$ More precisely, we study the set

$C=\left\\{b\in\mathbb{R}_{>0}|\,\epsilon\geq\frac{\left\lVert\partial_{t}\phi(t)\right\rVert_{L^{2}}^{2}}{2}+2e^{-\sqrt{2}z(t)}+\frac{c\left\lVert
g(t)\right\rVert_{H^{1}(\mathbb{R})}^{2}}{8}\text{, if $|t|\leq b.$}\right\\}$

and prove that $M=\sup_{b\in C}b=+\infty.$ We already have checked that $C$ is
not empty, also $C$ is closed by its definition. Now from the previous
argument, we can verify that the set where inequality (73) holds is open. So,
by connectivity, we obtain that $C=\mathbb{R}_{>0}.$

In conclusion, it remains to prove that the modulation parameters
$x_{1}(t),\,x_{2}(t)$ are of class $C^{2}$ and that the fourth item of the
statement of Theorem 2.9 is true.
(Proof of the $C^{2}$ regularity of $x_{1},\,x_{2},$ and of the fourth item.)

For $\delta_{0}>0$ small enough, we denote $(y_{1}(t),\,y_{2}(t))$ to be the
solution of the following system of ordinary differential equations, with the
function $g_{1}(t)=\phi(t,x)-H^{y_{2}(t)}_{0,1}(x)-H^{y_{1}(t)}_{-1,0}(x)$,

$\Big{(}\left\lVert\partial_{x}H_{0,1}\right\rVert_{L^{2}}^{2}-\left\langle
g_{1}(t),\,\partial^{2}_{x}H^{y_{1}(t)}_{-1,0}\right\rangle\Big{)}\dot{y}_{1}(t)+\Big{(}\left\langle\partial_{x}H^{y_{2}(t)}_{0,1},\,\partial_{x}H^{y_{1}(t)}_{-1,0}\right\rangle\Big{)}\dot{y}_{2}(t)=-\left\langle\partial_{t}\phi(t),\,\partial_{x}H^{y_{1}(t)}_{-1,0}(x)\right\rangle,$
(76)
$\Big{(}\left\langle\partial_{x}H^{y_{2}(t)}_{0,1},\,\partial_{x}H^{y_{1}(t)}_{-1,0}\right\rangle\Big{)}\dot{y}_{1}(t)+\Big{(}\left\lVert\partial_{x}H_{0,1}(t)\right\rVert_{L^{2}}^{2}-\left\langle
g_{1}(t),\,\partial^{2}_{x}H^{y_{2}}_{0,1}\right\rangle\Big{)}\dot{y}_{2}(t)=-\left\langle\partial_{t}\phi(t),\,\partial_{x}H^{y_{2}(t)}_{0,1}(x)\right\rangle,$
(77)

with initial condition $(y_{2}(0),y_{1}(0))=(x_{2}(0),x_{1}(0))$. This
ordinary differential equation system is motivated from the time derivative of
the orthogonality conditions of the Modulation Lemma.

Since we have the estimate $\ln{(\frac{1}{\epsilon})}\lesssim
x_{2}(0)-x_{1}(0)$ and $g_{1}(0)=g(0),$ Lemma 2.3 and the inequality (75)
imply that the matrix

$\begin{bmatrix}\left\lVert\partial_{x}H_{0,1}\right\rVert_{L^{2}}^{2}-\left\langle
g_{1}(0),\,\partial^{2}_{x}H^{y_{1}(0)}_{-1,0}\right\rangle&\left\langle\partial_{x}H^{y_{2}(0)}_{0,1},\,\partial_{x}H^{y_{1}(0)}_{-1,0}\right\rangle\\\
\left\langle\partial_{x}H^{y_{2}(0)}_{0,1},\,\partial_{x}H^{y_{1}(0)}_{-1,0}\right\rangle&\left\lVert\partial_{x}H_{0,1}\right\rVert_{L^{2}}^{2}-\left\langle
g_{1}(0),\,\partial^{2}_{x}H^{y_{2}}_{0,1}\right\rangle\end{bmatrix}$ (78)

is positive, so we have from Picard-Lindelöf Theorem that
$(y_{2}(t),y_{1}(t))$ are of class $C^{1}$ for some interval
$[-\delta,\delta],$ with $\delta>0$ depending on $|x_{2}(0)-x_{1}(0)|$ and
$\epsilon.$ From the fact that $(y_{2}(0),y_{1}(0))=(x_{2}(0),x_{1}(0)),$ we
obtain, from the equations (76) and (77), that $(y_{2}(t),y_{1}(t))$ also
satisfies the orthogonality conditions of Modulation Lemma for
$t\in[-\delta,\delta].$ In conclusion, the uniqueness of Modulation Lemma
implies that $(y_{2}(t),y_{1}(t))=(x_{2}(t),x_{1}(t))$ for
$t\in[-\delta,\delta].$ From this argument, we also have for
$t\in[-\delta,\delta]$ that
$e^{-\sqrt{2}(y_{2}(t)-y_{1}(t))}\leq\frac{\epsilon}{2\sqrt{2}}.$ By
bootstrap, we can show, repeating the argument above, that

$\sup\left\\{C>0|\,(y_{2}(t),y_{1}(t))=(x_{2}(t),x_{1}(t)),\text{for
$t\in[-C,C]$}\right\\}=+\infty.$ (79)

Also, the argument above implies that if
$(y_{1}(t),y_{2}(t))=(x_{1}(t),x_{2}(t))$ in an instant $t,$ then
$y_{1},\,y_{2}$ are of class $C^{1}$ in a neighborhood of $t.$ In conclusion,
$x_{1},\,x_{2}$ are functions in $C^{1}(\mathbb{R}).$ Finally, since
$\left\lVert g(t)\right\rVert_{H^{1}}=O(\epsilon^{\frac{1}{2}})$ and
$e^{-\sqrt{2}z(t)}=O(\epsilon),$ the following matrix

$M(t)\coloneqq\begin{bmatrix}\left\lVert\partial_{x}H_{0,1}\right\rVert_{L^{2}}^{2}-\left\langle
g(t),\,\partial^{2}_{x}H^{x_{1}(t)}_{-1,0}\right\rangle&\left\langle\partial_{x}H^{x_{2}(t)}_{0,1},\,\partial_{x}H^{x_{1}(t)}_{-1,0}\right\rangle\\\
\left\langle\partial_{x}H^{x_{2}(t)}_{0,1},\,\partial_{x}H^{x_{1}(t)}_{-1,0}\right\rangle&\left\lVert\partial_{x}H_{0,1}\right\rVert_{L^{2}}^{2}-\left\langle
g(t),\,\partial^{2}_{x}H^{x_{2}(t)}_{0,1}\right\rangle\end{bmatrix}$ (80)

is uniformly positive for all $t\in\mathbb{R}.$ So, from the estimate
$\left\lVert\partial_{t}\phi(t)\right\rVert_{L^{2}(\mathbb{R})}=O(\epsilon^{\frac{1}{2}}),$
the identities $x_{j}(t)=y_{j}(t)$ for $j=1,2$ and the equations (76) and
(77), we obtain

$\max_{j\in\\{1,2\\}}|\dot{x}_{j}(t)|=O(\epsilon^{\frac{1}{2}}).$ (81)

Since the matrix $M(t)$ is invertible for any $t\in\mathbb{R},$ we can obtain
from the equations (76), (77) that the functions
$\dot{x}_{1}(t),\dot{x}_{2}(t)$ are given by

$\begin{bmatrix}\dot{x}_{1}(t)\\\
\dot{x}_{2}(t)\end{bmatrix}=M(t)^{-1}\begin{bmatrix}-\left\langle\partial_{t}\phi(t),\,\partial_{x}H^{x_{1}(t)}_{-1,0}(x)\right\rangle\\\
-\left\langle\partial_{t}\phi(t),\,\partial_{x}H^{x_{2}(t)}_{0,1}(x)\right\rangle\end{bmatrix}.$
(82)

Now, since we have that $(\phi(t),\partial_{t}\phi(t))\in C(\mathbb{R},S\times
L^{2}(\mathbb{R}))$ and $x_{1}(t),\,x_{2}(t)$ are of class $C^{1},$ we can
deduce that $(g(t),\partial_{t}g(t))\in C(\mathbb{R},H^{1}(\mathbb{R})\times
L^{2}(\mathbb{R})).$ So, by definition, we can verify that $M(t)\in
C^{1}(\mathbb{R},\mathbb{R}^{4}).$

Also, since $\phi(t,x)$ is the solution in distributional sense of (1), we
have that for any $y_{1},\,y_{2}\in\mathbb{R}$ the following identities hold

$\displaystyle\big{\langle}\partial_{x}H^{y_{2}}_{0,1},\,\partial^{2}_{t}\phi(t)\big{\rangle}$
$\displaystyle=\big{\langle}\partial_{x}H^{y_{2}}_{0,1},\,\partial^{2}_{x}\phi(t)-\dot{U}(\phi(t))\big{\rangle}=-\big{\langle}\partial^{2}_{x}H^{y_{2}}_{0,1},\,\partial_{x}\phi(t)\big{\rangle}-\big{\langle}\partial_{x}H^{y_{2}(t)}_{0,1},\,\dot{U}(\phi(t))\big{\rangle},$
$\displaystyle\big{\langle}\partial_{x}H^{y_{1}}_{-1,0},\,\partial^{2}_{t}\phi(t)\big{\rangle}$
$\displaystyle=\big{\langle}\partial_{x}H^{y_{1}}_{-1,0},\,\partial^{2}_{x}\phi(t)-\dot{U}(\phi(t))\big{\rangle}=-\big{\langle}\partial^{2}_{x}H^{y_{1}}_{-1,0},\,\partial_{x}\phi(t)\big{\rangle}-\big{\langle}\partial_{x}H^{y_{1}}_{-1,0},\,\dot{U}(\phi(t))\big{\rangle}.$

Since (1) is locally well-posed in $S\times L^{2}(\mathbb{R}),$ we obtain from
the identities above that the following functions
$h(t,y)\coloneqq\big{\langle}\partial_{x}H^{y}_{0,1},\,\partial^{2}_{t}\phi(t)\big{\rangle}$
and
$l(t,y)\coloneqq\big{\langle}\partial_{x}H^{y}_{-1,0},\,\partial^{2}_{t}\phi(t)\big{\rangle}$
are continuous in the domain $\mathbb{R}\times\mathbb{R}.$

So, from the continuity of the functions $h(t,y),\,l(t,y)$ and from the fact
that $x_{1},\,x_{2}\in C^{1}(\mathbb{R}),$ we obtain that the functions

$h_{1}(t)\coloneqq-\langle\partial_{t}\phi(t),\,\partial_{x}H^{x_{1}(t)}_{-1,0}(x)\rangle,\,h_{2}(t)\coloneqq-\langle\partial_{t}\phi(t),\,\partial_{x}H^{x_{2}(t)}_{0,1}(x)\rangle$

are of class $C^{1}.$ In conclusion, from the equation (82), by chain rule and
product rule, we verify that $x_{1},\,x_{2}$ are in $C^{2}(\mathbb{R}).$

Now, since $x_{1},\,x_{2}\in C^{2}(\mathbb{R})$ and
$\dot{x}_{1},\,\dot{x}_{2}$ satisfy (82), we deduce after derive at time the
function

$M(t)\begin{bmatrix}\dot{x}_{1}(t)\\\ \dot{x}_{2}(t)\end{bmatrix},$

the following equations

$\ddot{x}_{1}(t)\Big{(}\left\lVert\partial_{x}H_{0,1}\right\rVert_{L^{2}}^{2}+\left\langle\partial_{x}g(t),\,\partial_{x}H^{x_{1}(t)}_{-1,0}\right\rangle\Big{)}+\ddot{x}_{2}(t)\Big{(}\left\langle\partial_{x}H^{x_{1}(t)}_{-1,0},\,\partial_{x}H^{x_{2}(t)}_{0,1}\right\rangle\Big{)}=\dot{x}_{1}(t)^{2}\Big{(}\left\langle\partial^{2}_{x}H^{x_{1}(t)}_{-1,0},\,\partial_{x}g(t)\right\rangle\Big{)}\\\
+\dot{x}_{1}(t)\left\langle\partial^{2}_{x}H^{x_{1}(t)}_{-1,0},\,\partial_{t}g(t)\right\rangle+\dot{x}_{1}(t)\dot{x}_{2}(t)\left\langle\partial^{2}_{x}H^{x_{1}(t)}_{-1,0},\,\partial_{x}H^{x_{2}(t)}_{0,1}\right\rangle+\dot{x}_{2}(t)^{2}\left\langle\partial_{x}H^{x_{1}(t)}_{-1,0},\,\partial^{2}_{x}H^{x_{2}(t)}_{0,1}\right\rangle+\dot{x}_{1}(t)\left\langle\partial^{2}_{x}H^{x_{1}(t)}_{-1,0},\,\partial_{t}\phi(t)\right\rangle\\\
-\left\langle\partial_{x}H^{x_{1}(t)}_{-1,0},\,\partial^{2}_{t}\phi(t)\right\rangle,$
(83)
$\ddot{x}_{2}(t)\Big{(}\left\lVert\partial_{x}H_{0,1}\right\rVert_{L^{2}}^{2}+\left\langle\partial_{x}g(t),\,\partial_{x}H^{x_{2}(t)}_{0,1}\right\rangle\Big{)}+\ddot{x}_{1}(t)\Big{(}\big{\langle}\partial_{x}H^{x_{1}(t)}_{-1,0},\,\partial_{x}H^{x_{2}(t)}_{0,1}\big{\rangle}\Big{)}=\dot{x}_{2}(t)^{2}\Big{(}\left\langle\partial^{2}_{x}H^{x_{2}(t)}_{0,1},\,\partial_{x}g(t)\right\rangle\Big{)}\\\
+\dot{x}_{2}(t)\Big{(}\left\langle\partial^{2}_{x}H^{x_{2}(t)}_{0,1},\,\partial_{t}g(t)\right\rangle\Big{)}+\dot{x}_{1}(t)\dot{x}_{2}(t)\left\langle\partial_{x}H^{x_{1}(t)}_{-1,0},\,\partial^{2}_{x}H^{x_{2}(t)}_{0,1}\right\rangle+(\dot{x}_{1}(t))^{2}\left\langle\partial_{x}H^{x_{2}(t)}_{0,1},\,\partial^{2}_{x}H^{x_{1}(t)}_{-1,0}\right\rangle+\dot{x}_{2}(t)\left\langle\partial^{2}_{x}H^{x_{2}(t)}_{0,1},\,\partial_{t}\phi(t)\right\rangle\\\
-\left\langle\partial_{x}H^{x_{2}(t)}_{0,1},\,\partial^{2}_{t}\phi(t)\right\rangle.$
(84)

Also, from the identity $g(t)=\phi(t)-H^{x_{1}(t)}_{-1,0}-H^{x_{2}(t)}_{0,1},$
we obtain that
$\partial_{t}g(t)=\partial_{t}\phi(t,x)+\dot{x}_{1}(t)\partial_{x}H^{x_{1}(t)}_{-1,0}+\dot{x}_{2}(t)\partial_{x}H^{x_{2}(t)}_{0,1},$
so, from the estimates (66) and (81), we obtain that

$\left\lVert\partial_{t}g(t)\right\rVert_{L^{2}}=O(\epsilon^{\frac{1}{2}}).$
(85)

Now, since $\phi(t)$ is a distributional solution of (1), we also have, from
the global equality $\phi(t)=H^{x_{1}(t)}_{-1,0}+H^{x_{2}(t)}_{0,1}+g(t),$ the
following identity

$\left\langle\partial_{x}H^{x_{1}(t)}_{-1,0},\,\partial^{2}_{t}\phi(t)\right\rangle=\left\langle\partial_{x}H^{x_{1}(t)}_{-1,0},\,\partial^{2}_{x}g(t)-\ddot{U}\left(H^{x_{1}(t)}_{-1,0}\right)g(t)\right\rangle-\left\langle\partial_{x}H^{x_{1}(t)}_{-1,0},\left[\ddot{U}\left(H^{x_{1}(t)}_{-1,0}+H^{x_{2}(t)}_{0,1}\right)-\ddot{U}\left(H^{x_{1}(t)}_{-1,0}\right)\right]g(t)\right\rangle\\\
+\left\langle\partial_{x}H^{x_{1}(t)}_{-1,0},\,\dot{U}\left(H^{x_{1}(t)}_{-1,0}\right)+\dot{U}\left(H^{x_{2}(t)}_{0,1}\right)-\dot{U}\left(H^{x_{1}(t)}_{-1,0}+H^{x_{2}(t)}_{0,1}\right)\right\rangle\\\
-\left\langle\partial_{x}H^{x_{1}(t)}_{-1,0},\dot{U}\left(H^{x_{1}(t)}_{-1,0}+H^{x_{2}(t)}_{0,1}+g(t)\right)-\dot{U}\left(H^{x_{1}(t)}_{-1,0}+H^{x_{2}(t)}_{0,1}\right)-\ddot{U}\left(H^{x_{1}(t)}_{-1,0}+H^{x_{2}(t)}_{0,1}\right)g(t)\right\rangle$
(86)

Since $\partial_{x}H^{x_{1}(t)}_{-1,0}\in
kerD^{2}E_{pot}\left(H^{x_{1}(t)}_{-1,0}\right)$, we have by integration by
parts that
$\left\langle\partial_{x}H^{x_{1}(t)}_{-1,0},\,\partial^{2}_{x}g(t)-\ddot{U}\left(H^{x_{1}(t)}_{-1,0}\right)g(t)\right\rangle=0.$
Since, we have

$\dot{U}\left(H^{x_{1}(t)}_{-1,0}\right)+\dot{U}\left(H^{x_{2}(t)}_{0,1}\right)-\dot{U}\left(H^{x_{1}(t)}_{-1,0}+H^{x_{2}(t)}_{0,1}\right)=24H^{x_{1}(t)}_{-1,0}H^{x_{2}(t)}_{0,1}(H^{x_{1}(t)}_{-1,0}+H^{x_{2}(t)}_{0,1})-6\sum_{j=1}^{4}\begin{pmatrix}5\\\
j\end{pmatrix}\left(H^{x_{1}(t)}_{-1,0}\right)^{j}\left(H^{x_{2}(t)}_{0,1}\right)^{5-j},$
(87)

Lemma 2.3 implies that
$\left\langle\partial_{x}H^{x_{1}(t)}_{-1,0},\,\dot{U}\left(H^{x_{1}(t)}_{-1,0}\right)+\dot{U}\left(H^{x_{2}(t)}_{0,1}\right)-\dot{U}\left(H^{x_{1}(t)}_{-1,0}+H^{x_{2}(t)}_{0,1}\right)\right\rangle=O\Big{(}e^{-\sqrt{2}(z(t))}\Big{)}.$
Also, we have from Taylor’s Expansion Theorem the estimate

$\left\langle\partial_{x}H^{x_{1}(t)}_{-1,0},\dot{U}\left(H^{x_{1}(t)}_{-1,0}+H^{x_{2}(t)}_{0,1}+g(t)\right)-\dot{U}\left(H^{x_{1}(t)}_{-1,0}+H^{x_{2}(t)}_{0,1}\right)-\ddot{U}\left(H^{x_{1}(t)}_{-1,0}+H^{x_{2}(t)}_{0,1}\right)g(t)\right\rangle=O(\left\lVert
g(t)\right\rVert_{H^{1}}^{2}).$

From Lemma 2.3, the fact that $U$ is a smooth function and $H_{0,1}\in
L^{\infty}(\mathbb{R})$, we can obtain

$\left\langle\partial_{x}H^{x_{1}(t)}_{-1,0},\left[\ddot{U}\left(H^{x_{1}(t)}_{-1,0}+H^{x_{2}(t)}_{0,1}\right)-\ddot{U}\left(H^{x_{1}(t)}_{-1,0}\right)\right]g(t)\right\rangle=O\Big{(}\int_{\mathbb{R}}\partial_{x}H^{x_{1}(t)}_{-1,0}H^{x_{2}(t)}_{0,1}|g(t)|\,dx\Big{)}=O\Big{(}e^{-\sqrt{2}z(t)}\left\lVert
g(t)\right\rVert_{H^{1}}z(t)^{\frac{1}{2}}\Big{)}.$

In conclusion, we have

$\left\langle\partial_{x}H^{x_{1}(t)}_{-1,0},\,\partial^{2}_{t}\phi(t)\right\rangle=O\Big{(}\left\lVert
g(t)\right\rVert_{H^{1}}^{2}+e^{-\sqrt{2}z(t)}\Big{)},$ (88)

and by similar arguments, we have

$\left\langle\partial_{x}H^{x_{2}(t)}_{0,1},\,\partial^{2}_{t}\phi(t)\right\rangle=O\Big{(}\left\lVert
g(t)\right\rVert_{H^{1}}^{2}+e^{-\sqrt{2}z(t)}\Big{)}.$ (89)

Also, the equations (83) and (84) form a linear system with
$\ddot{x}_{1}(t),\,\ddot{x}_{2}(t).$ Recalling that the Matrix $M(t)$ is
uniformly positive, we obtain from the estimates (75), (81), (85), (88) and
(89) that

$\max_{j\in\\{1,2\\}}|\ddot{x}_{j}(t)|=O(\epsilon).$ (90)

∎

The Theorem 2.9 can also be improved when the kinetic energy of the solution
is included in the computation and additional conditions are added, more
precisely:

######  Theorem 2.10.

$\exists\delta_{0}>0,$ such that if
$0<\epsilon\leq\delta_{0},\,(\phi(0,x),\partial_{t}\phi(0,x))\in S\times
L^{2}(\mathbb{R})$ and
$E_{total}((\phi(0,x),\partial_{t}\phi(0,x)))=2E_{pot}(H_{0,1})+\epsilon,$
then there are $x_{2},\,x_{1}\in C^{2}(\mathbb{R})$ such that
$g(t,x)=\phi(t,x)-H^{x_{2}(t)}_{0,1}(x)-H^{x_{1}(t)}_{-1,0}$ satisfies

$\left\langle
g(t,x),\partial_{x}H^{x_{2}(t)}_{0,1}(x)\right\rangle=0,\,\left\langle
g(t,x),\partial_{x}H^{x_{1}(t)}_{-1,0}(x)\right\rangle=0,$

and

$\epsilon\cong
e^{-\sqrt{2}(x_{2}(t)-x_{1}(t))}+\left\lVert(g(t),\partial_{t}g(t))\right\rVert_{H^{1},L^{2}}^{2}+|\dot{x}_{1}(t)|^{2}+|\dot{x}_{2}(t)|^{2},$
(91)

for all $t\in\mathbb{R},$ which means the existence of positive constants
$C,c$ independent on $\epsilon$, such that for all $t\in\mathbb{R}$

$c\epsilon\leq
e^{-\sqrt{2}(x_{2}(t)-x_{1}(t))}+\left\lVert(g(t),\partial_{t}g(t))\right\rVert_{H^{1},L^{2}}^{2}+|\dot{x}_{1}(t)|^{2}+|\dot{x}_{2}(t)|^{2}\leq
C\epsilon.$ (92)

###### Proof.

In this proof, $L^{2},\,H^{1}$ mean, respectively,
$L^{2}(\mathbb{R}),\,H^{1}(\mathbb{R})$. From Modulation Lemma and Theorem
2.9, we can rewrite the solution $\phi(t)$ in the form

$\phi(t,x)=H^{x_{1}(t)}_{-1,0}(x)+H^{x_{2}(t)}_{0,1}(x)+g(t,x)$

with $x_{1}(t),\,x_{2}(t),\,g(t)$ satisfying the conclusion of Theorem 2.9.
First we denote

$\phi_{\sigma}(t)=\left(H^{x_{1}(t)}_{-1,0}(x)+H^{x_{2}(t)}_{0,1}(x),-\dot{x}_{1}(t)\partial_{x}H^{x_{1}(t)}_{-1,0}-\dot{x}_{2}(t)\partial_{x}H^{x_{2}(t)}_{0,1}\right)\in
S\times L^{2}(\mathbb{R}),$ (93)

then we apply Taylor’s Expansion Theorem in $E(\phi(t))$ around
$\phi_{\sigma}(t)$, more precisely for $R_{\sigma}(t)$ the residue of second
order of Energy’s Taylor Expansion of $E(\phi(t),\partial_{t}\phi(t))$ around
$\phi_{\sigma}(t)$, we have:

$2E_{pot}(H_{0,1})+\epsilon=E(\phi_{\sigma}(t))+\left\langle
DE(\phi_{\sigma}(t)),\,\left(g(t),\partial_{t}g(t)\right)\right\rangle_{L^{2}\times
L^{2}}+\frac{\left\langle
D^{2}E(\phi_{\sigma}(t))\big{(}g(t),\partial_{t}g(t)\big{)},\,\big{(}g(t),\partial_{t}g(t)\big{)}\right\rangle_{L^{2}\times
L^{2}}}{2}\\\ +R_{\sigma}(t),$ (94)

such that for $(w_{1},w_{2})\in S\times L^{2}(\mathbb{R})$ and
$(v_{1},v_{2})\in H^{1}(\mathbb{R})\times L^{2}(\mathbb{R})$, we have the
identities

$E(w_{1},w_{2})=\frac{\left\lVert\partial_{x}w_{1}\right\rVert_{L^{2}}^{2}+\left\lVert
w_{2}\right\rVert_{L^{2}}^{2}}{2}+\int_{\mathbb{R}}U(w_{1}(x))\,dx,$
$\displaystyle\left\langle
DE(w_{1},w_{2}),(v_{1},v_{2})\right\rangle_{L^{2}\times
L^{2}}=\int_{\mathbb{R}}\partial_{x}w_{1}(x)\partial_{x}v_{1}(x)+\dot{U}(w_{1})v_{1}+w_{2}(x)v_{2}(x)\,dx,$
(95) $\displaystyle
D^{2}E(w_{1},w_{2})=\begin{bmatrix}-\partial^{2}_{x}+\ddot{U}(w_{1})&0\\\
0&\mathbb{I}\end{bmatrix}$ (96)

with $D^{2}E(w_{1},w_{2})$ defined as a bilinear operator from $H^{1}\times
L^{2}$ to $\mathbb{C}$. So, from identities (95) and (96), it is not difficult
to verify that

$R_{\sigma}(t)=\int_{\mathbb{R}}U\left(H^{x_{1}(t)}_{-1,0}(x)+H^{x_{2}(t)}_{0,1}(x)+g(t,x)\right)-U\left(H^{x_{1}(t)}_{-1,0}(x)+H^{x_{2}(t)}_{0,1}(x)\right)-\dot{U}\left(H^{x_{1}(t)}_{-1,0}(x)+H^{x_{2}(t)}_{0,1}(x)\right)g(t,x)\\\
-\frac{\ddot{U}\left(H^{x_{1}(t)}_{-1,0}(x)+H^{x_{2}(t)}_{0,1}(x)\right)g(t,x)^{2}}{2}\,dx,$

and, so,

$|R_{\sigma}(t)|=O(\left\lVert g(t)\right\rVert_{H^{1}}^{3}).$ (97)

Also, we have

$\left\langle
DE(\phi_{\sigma}(t)),\,(g(t),\partial_{t}g(t))\right\rangle_{L^{2}\times
L^{2}}=\left\langle
DE_{pot}\left(H^{x_{1}(t)}_{-1,0}+H^{x_{2}(t)}_{0,1}\right),\,g(t)\right\rangle+\left\langle-\dot{x}_{1}(t)\partial_{x}H^{x_{1}(t)}_{-1,0}-\dot{x}_{2}(t)\partial_{x}H^{x_{2}(t)}_{0,1},\,\partial_{t}g(t)\right\rangle.$
(98)

The orthogonality conditions satisfied by $g(t)$ also imply for all
$t\in\mathbb{R}$ that

$\displaystyle\left\langle\partial_{t}g(t),\,\partial_{x}H^{x_{1}(t)}_{-1,0}\right\rangle_{L^{2}}=\dot{x}_{1}(t)\langle
g(t),\,\partial^{2}_{x}H^{x_{1}(t)}_{-1,0}\rangle_{L^{2}},$ (99)
$\displaystyle\left\langle\partial_{t}g(t),\,\partial_{x}H^{x_{2}(t)}_{0,1}\right\rangle_{L^{2}}=\dot{x}_{2}(t)\langle
g(t),\,\partial^{2}_{x}H^{x_{2}(t)}_{0,1}\rangle_{L^{2}}.$ (100)

So, the inequality (65) and the identities (98), (99), (100) imply that

$|\langle DE(\phi_{\sigma}(t)),\,(g(t),\partial_{t}g(t))\rangle_{L^{2}\times
L^{2}}|=O\Big{(}\sup_{j\in\\{1,2\\}}|\dot{x}_{j}(t)|^{2}\left\lVert
g(t)\right\rVert_{H^{1}}+e^{-\sqrt{2}z(t)}\left\lVert
g(t)\right\rVert_{H^{1}}\Big{)}.$ (101)

From the Coercitivity Lemma and the definition of $D^{2}E(\phi_{\sigma}(t))$,
we have that

$\left\langle
D^{2}E(\phi_{\sigma}(t))(g(t),\partial_{t}g(t)),\,(g(t),\partial_{t}g(t))\right\rangle_{L^{2}\times
L^{2}}\cong\left\lVert(g(t),\partial_{t}g(t))\right\rVert_{H^{1}\times
L^{2}}^{2}.$ (102)

Finally, there is the identity

$\left\lVert\dot{x}_{1}(t)\partial_{x}H_{-1,0}^{x_{1}(t)}(x)+\dot{x}_{2}(t)\partial_{x}H_{0,1}^{x_{2}(t)}(x)\right\rVert_{L^{2}}^{2}=2\dot{x}_{1}(t)\dot{x}_{2}(t)\left\langle\partial_{x}H^{z(t)}_{0,1},\,\partial_{x}H_{-1,0}\right\rangle_{L^{2}}\\\
+|\dot{x}_{1}(t)|^{2}\left\lVert\partial_{x}H_{0,1}\right\rVert_{L^{2}}^{2}+|\dot{x}_{2}(t)|^{2}\left\lVert\partial_{x}H_{0,1}\right\rVert_{L^{2}}^{2}.$
(103)

From Lemma 2.3, we have that
$|\langle\partial_{x}H^{z}_{0,1},\,\partial_{x}H_{-1,0}\rangle_{L^{2}}|=O\big{(}ze^{-\sqrt{2}z}\big{)}$
for $z$ big enough. Then, it is not difficult to verify that Lemma 2.4, (97),
(101), (102) and (103) imply directly the statement of the Theorem 2.10 which
finishes the proof. ∎

######  Remark 2.11.

Theorem 2.10 implies that it is possible to have a solution $\phi$ of the
equation (1) with energy excess $\epsilon>0$ small enough satisfying all the
hypotheses of Theorem 1.7. More precisely, in notation of Theorem 1.7, if
$\left\lVert(g(0,x),\partial_{t}g(0,x))\right\rVert_{H^{1}\times
L^{2}}\ll\epsilon^{\frac{1}{2}},$ then, from Theorem 2.10, we have that

$e^{-\sqrt{2}z(0)}+v_{1}^{2}+v_{2}^{2}\cong\epsilon.$

In conclusion, we obtain that
$E(\phi(0),\partial_{t}\phi(0))-2E_{pot}(H_{0,1})\cong\epsilon.$

##  3 Long Time Behavior of Modulation Parameters

Even though Theorem 2.9 proves the orbital stability of a sum of two kinks
with low energy excess, this theorem doesn’t explain the movement of the
kinks’ centers $x_{2}(t),\,x_{1}(t)$ and their speed for long time. More
precisely, we still don’t know if there is a explicit smooth real function
$d(t),$ such that $(z(t),\dot{z}(t))$ is close to $(d(t),\dot{d}(t))$ in a
large time interval.

But, the global estimates on the modulus of the first and second derivatives
of $x_{1}(t),\,x_{2}(t)$ obtained in Theorem 2.9 will be very useful to
estimate with high precision the functions $x_{1}(t),\,x_{2}(t)$ during a very
large time interval. Moreover, we first have the following auxiliary lemma.

######  Lemma 3.1.

Let $0<\theta,\,\gamma<1.$ We recall the function

$A(z)=E_{pot}(H^{z}_{0,1}+H_{-1,0})$

for any $z>0.$ If the same hypothesis of Theorem 2.9 are true and let
$\chi(x)$ be a smooth function such that

$\chi(x)=\begin{cases}1,\text{ if $x\leq\theta(1-\gamma)$,}\\\ 0,\text{ if
$x\geq\theta$.}\end{cases}$ (104)

and $0\leq\chi(x)\leq 1$ for all $x\in\mathbb{R}.$ In notation of Theorem 2.9,
we denote

$\chi_{0}(t,x)=\chi\Big{(}\frac{x-x_{1}(t)}{z(t)}\Big{)},\,\overrightarrow{g(t)}=(g(t),\partial_{t}g(t))\in
H^{1}(\mathbb{R})\times L^{2}(\mathbb{R})$

and
$\left\lVert\overrightarrow{g(t)}\right\rVert=\left\lVert(g(t),\partial_{t}g(t))\right\rVert_{H^{1}(\mathbb{R})\times
L^{2}(\mathbb{R})},$

$\alpha(t)=\max_{j\in\\{1,2\\}}|\dot{x}_{j}(t)|^{2}z(t)e^{-\sqrt{2}z(t)}+\frac{\max_{j\in\\{1,\,2\\}}|\dot{x}_{j}(t)|^{2}}{z(t)\gamma}\Big{(}e^{-2\sqrt{2}z(t)(\frac{1-\gamma}{2-\gamma})}\Big{)}\\\
+\left\lVert\overrightarrow{g(t)}\right\rVert\max_{j\in\\{1,\,2\\}}|\dot{x}_{j}(t)|\Big{[}1+\frac{1}{z(t)\gamma}+\frac{1}{z(t)^{2}\gamma^{2}}\max_{j\in\\{1,\,2\\}}|\dot{x}_{j}(t)|\Big{]}\Big{(}e^{-\sqrt{2}z(t)(\frac{1-\gamma}{2-\gamma})}\Big{)}+\left\lVert\overrightarrow{g(t)}\right\rVert^{2}\Big{[}\frac{1}{\gamma^{2}z(t)^{2}}+\frac{1}{\gamma
z(t)}+\Big{(}e^{-\sqrt{2}z(t)(\frac{1-\gamma}{2-\gamma})}\Big{)}\Big{]}.$
(105)

Then, for $\theta=\frac{1-\gamma}{2-\gamma}$ and the correction terms

$\displaystyle
p_{1}(t)=-\frac{\big{\langle}\partial_{t}\phi(t),\,\partial_{x}H^{x_{1}(t)}_{-1,0}(x)+\partial_{x}(\chi_{0}(t,x)g(t))\big{\rangle}}{\left\lVert\partial_{x}H_{0,1}\right\rVert_{L^{2}}^{2}},$
$\displaystyle
p_{2}(t)=-\frac{\big{\langle}\partial_{t}\phi(t),\,\partial_{x}H^{x_{2}(t)}_{0,1}(x)+\partial_{x}([1-\chi_{0}(t,x)]g(t))\big{\rangle}}{\left\lVert\partial_{x}H_{0,1}\right\rVert_{L^{2}}^{2}},$

we have the estimates, for $j\in\\{1,2\\},$

$|\dot{x}_{j}(t)-p_{j}(t)|\lesssim\Big{[}1+\frac{\left\lVert\dot{\chi}\right\rVert_{L^{\infty}}}{z(t)}\Big{]}\Big{(}\max_{j\in\\{1,2\\}}|\dot{x}_{j}(t)|\left\lVert\overrightarrow{g(t)}\right\rVert+\left\lVert\overrightarrow{g(t)}\right\rVert^{2}\Big{)}+\max_{j\in\\{1,2\\}}|\dot{x}_{j}(t)|z(t)e^{-\sqrt{2}z(t)},$
(106)
$\Big{|}\dot{p}_{j}(t)+(-1)^{j}\frac{\dot{A}(z(t))}{\left\lVert\partial_{x}H_{0,1}\right\rVert_{L^{2}}^{2}}\Big{|}\lesssim\alpha(t).$
(107)

######  Remark 3.2.

We will take
$\gamma=\frac{\ln{\ln{(\frac{1}{\epsilon})}}}{\ln{(\frac{1}{\epsilon})}}.$
With this value of $\gamma$ and the estimates of Theorem 2.9, we will see in
Lemma 5.1 that $\exists C>0$ such that

$\alpha(t)\lesssim\frac{\Big{(}\left\lVert(g_{0},g_{1})\right\rVert_{H^{1}\times
L^{2}}+\epsilon\ln{(\frac{1}{\epsilon})}\Big{)}^{2}}{\ln{\ln{(\frac{1}{\epsilon})}}}\exp\Big{(}\frac{2C|t|\epsilon^{\frac{1}{2}}}{\ln{(\frac{1}{\epsilon})}}\Big{)}.$

###### Proof.

For $\gamma\ll 1$ enough and from the definition of $\chi(x),$ it is not
difficult to verify that

$\left\lVert\dot{\chi}\right\rVert_{L^{\infty}(\mathbb{R})}\lesssim\frac{1}{\gamma},\,\left\lVert\ddot{\chi}\right\rVert_{L^{\infty}(\mathbb{R})}\lesssim\frac{1}{\gamma^{2}}.$
(108)

We will only do the proof of the estimates (106) and (107) for $j=1,$ the
proof for the case $j=2$ is completely analogous. From the proof of Theorem
2.9, we know that $\dot{x}_{1}(t),\,\dot{x}_{2}(t)$ solve the linear system

$M(t)\begin{bmatrix}\dot{x}_{1}(t)\\\
\dot{x}_{2}(t)\end{bmatrix}=\begin{bmatrix}-\langle\partial_{t}\phi(t),\,\partial_{x}H^{x_{1}(t)}_{-1,0}\rangle\\\
-\langle\partial_{t}\phi(t),\,\partial_{x}H^{x_{2}(t)}_{0,1}\rangle\end{bmatrix},$

where $M(t)$ is the matrix defined by(80). Then, from Cramer’s rule, we obtain
that

$\dot{x}_{1}(t)=\frac{-\left\langle\partial_{t}\phi(t),\,\partial_{x}H^{x_{1}(t)}_{-1,0}\right\rangle\Big{(}\left\langle\partial_{x}H^{x_{2}(t)}_{0,1},\,\partial_{x}g(t)\right\rangle+\left\lVert\partial_{x}H_{0,1}\right\rVert_{L^{2}}^{2}\Big{)}}{\det(M(t))}+\frac{\left\langle\partial_{t}\phi(t),\,\partial_{x}H^{x_{2}(t)}_{0,1}\right\rangle\left\langle\partial_{x}H^{x_{2}(t)}_{0,1},\,\partial_{x}H^{x_{1}(t)}_{-1,0}\right\rangle}{\det(M(t))}.$
(109)

Using the definition (80) of the matrix $M(t)$,
$\left\lVert\overrightarrow{g(t)}\right\rVert=O(\epsilon^{\frac{1}{2}})$ and
Lemma 2.3 which implies the following estimate

$\left\langle\partial_{x}H^{x_{2}(t)}_{0,1},\,\partial_{x}H^{x_{1}(t)}_{-1,0}\right\rangle=O(z(t)e^{-\sqrt{2}z(t)}),$
(110)

we obtain that

$\Big{|}\det(M(t))-\left\lVert\partial_{x}H_{0,1}\right\rVert_{L^{2}}^{4}\Big{|}=O\Big{(}\left\lVert\overrightarrow{g(t)}\right\rVert+z(t)^{2}e^{-2\sqrt{2}z(t)}\Big{)}=O(\epsilon^{\frac{1}{2}}).$
(111)

So, from the estimate (111) and the identity (109), we obtain that

$\Big{|}\dot{x}_{1}(t)+\frac{1}{\left\lVert\partial_{x}H_{0,1}\right\rVert_{L^{2}(\mathbb{R})}^{2}}\left\langle\partial_{t}\phi(t),\,\partial_{x}H^{x_{1}(t)}_{-1,0}\right\rangle\Big{|}=O\Big{(}\Big{|}\left\langle\partial_{x}H^{x_{1}(t)}_{-1,0},\,\partial_{x}g(t)\right\rangle\left\langle\partial_{t}\phi(t),\,\partial_{x}H^{x_{1}(t)}_{-1,0}\right\rangle\Big{|}\Big{)}\\\
+O\Big{(}\Big{|}\left\langle\partial_{x}H^{x_{1}(t)}_{-1,0},\,\partial_{x}H^{x_{2}(t)}_{0,1}\right\rangle\left\langle\partial_{t}\phi(t),\,\partial_{x}H^{x_{2}(t)}_{0,1}\right\rangle\Big{|}\Big{)}+O\Big{(}\Big{|}\left\langle\partial_{t}\phi(t),\,\partial_{x}H^{x_{1}(t)}_{-1,0}(x)\right\rangle\Big{|}\Big{[}\left\lVert\overrightarrow{g(t)}\right\rVert+z(t)^{2}e^{-2\sqrt{2}z(t)}\Big{]}\Big{)}.$
(112)

Finally, from the definition of $g(t,x)$ in Theorem 2.9 we know that

$\partial_{t}\phi(t,x)=-\dot{x}_{1}(t)\partial_{x}H^{x_{1}(t)}_{-1,0}(x)-\dot{x}_{2}(t)\partial_{x}H^{x_{2}(t)}_{0,1}(x)+\partial_{t}g(t,x),$

from the Modulation Lemma we also have verified that

$\left\langle\partial_{t}g(t),\,\partial_{x}H^{x_{1}(t)}_{-1,0}\right\rangle=O\Big{(}\left\lVert\overrightarrow{g(t)}\right\rVert|\dot{x}_{1}(t)|\Big{)},\,\left\langle\partial_{t}g(t),\,\partial_{x}H^{x_{2}(t)}_{0,1}\right\rangle=O\Big{(}\left\lVert\overrightarrow{g(t)}\right\rVert|\dot{x}_{2}(t)|\Big{)}$

and from Theorem 2.9 we have that
$\left\lVert\overrightarrow{g(t)}\right\rVert+\max_{j\in\\{1,2\\}}|\dot{x}_{j}(t)|\ll
1.$ In conclusion, we can rewrite the estimate (112) as

$\Big{|}\dot{x}_{1}(t)+\frac{1}{\left\lVert\partial_{x}H_{0,1}\right\rVert_{L^{2}(\mathbb{R})}^{2}}\left\langle\partial_{t}\phi(t),\,\partial_{x}H^{x_{1}(t)}_{-1,0}\right\rangle\Big{|}=O\Big{(}\max_{j\in\\{1,2\\}}|\dot{x}_{j}(t)|\left\lVert\overrightarrow{g(t)}\right\rVert+\left\lVert\overrightarrow{g(t)}\right\rVert^{2}+z(t)e^{-\sqrt{2}z(t)}\max_{j\in\\{1,2\\}}|\dot{x}_{j}(t)|\Big{)}.$
(113)

By a similar reasoning, we can also deduce that

$\Big{|}\dot{x}_{2}(t)+\frac{1}{\left\lVert\partial_{x}H_{0,1}\right\rVert_{L^{2}(\mathbb{R})}^{2}}\left\langle\partial_{t}\phi(t),\,\partial_{x}H^{x_{2}(t)}_{0,1}\right\rangle\Big{|}=O\Big{(}\max_{j\in\\{1,2\\}}|\dot{x}_{j}(t)|\left\lVert\overrightarrow{g(t)}\right\rVert+\left\lVert\overrightarrow{g(t)}\right\rVert^{2}+z(t)e^{-\sqrt{2}z(t)}\max_{j\in\\{1,2\\}}|\dot{x}_{j}(t)|\Big{)}.$
(114)

Following the reasoning of Lemma 3.5 of [19], we will use the terms
$p_{1}(t),\,p_{2}(t)$ with the objective of obtaining the estimates (107),
which have high precision and will be useful later to approximate
$x_{j}(t),\,\dot{x}_{j}(t)$ by explicit smooth functions during a long time
interval.

First, it is not difficult to verify that

$\left\langle\partial_{t}\phi(t),\,\partial_{x}(\chi_{0}(t)g(t))\right\rangle=O\Big{(}\Big{[}1+\frac{\left\lVert\dot{\chi}\right\rVert_{L^{\infty}}}{z(t)}\Big{]}\left\lVert\overrightarrow{g(t)}\right\rVert^{2}+\max_{j\in\\{1,2\\}}|\dot{x}_{j}(t)|\left\lVert\overrightarrow{g(t)}\right\rVert\Big{)},$

which clearly implies with estimate (113) the inequality (106) for $j=1.$ The
proof of inequality (106) for $j=2$ is completely analog.

Now, the demonstration of the inequality (107) is similar to the proof of the
second inequality of Lemma 3.5 of [19]. First, we have

$\dot{p}_{1}(t)=-\frac{\left\langle\partial_{t}\phi(t),\,\partial_{t}\big{(}\partial_{x}H^{x_{1}(t)}_{-1,0}(x)\big{)}\right\rangle}{\left\lVert\partial_{x}H_{0,1}\right\rVert_{L^{2}}^{2}}-\frac{\left\langle\partial_{t}\phi(t),\,\partial_{x}\big{(}\partial_{t}\chi_{0}(t)g(t)\big{)}\right\rangle}{\left\lVert\partial_{x}H_{0,1}\right\rVert_{L^{2}}^{2}}-\frac{\left\langle\partial_{x}\big{(}\chi_{0}(t)\partial_{t}g(t)\big{)},\,\partial_{t}\phi(t)\right\rangle}{\left\lVert\partial_{x}H_{0,1}\right\rVert_{L^{2}}^{2}}-\frac{\left\langle\partial_{x}H^{x_{1}(t)}_{-1,0},\,\partial^{2}_{t}\phi(t)\right\rangle}{\left\lVert\partial_{x}H_{0,1}\right\rVert_{L^{2}}^{2}}\\\
-\frac{\left\langle\partial_{x}\chi_{0}(t)g(t),\,\partial^{2}_{t}\phi(t)\right\rangle}{\left\lVert\partial_{x}H_{0,1}\right\rVert_{L^{2}}^{2}}-\frac{\left\langle\chi_{0}(t)\partial_{x}g(t),\,\partial^{2}_{t}\phi(t)\right\rangle}{\left\lVert\partial_{x}H_{0,1}\right\rVert_{L^{2}}^{2}}=I+II+III+IV+V+VI,$
(115)

and we will estimate each term one by one. More precisely, from now on, we
will work with a general cut function $\chi(x),$ that is a smooth function
$0\leq\chi\leq 1$ satisfying

$\chi(x)=\begin{cases}1,\text{ if $x\leq\theta(1-\gamma)$,}\\\ 0,\text{ if
$x\geq\theta$.}\end{cases}$ (116)

with $0<\theta,\,\gamma<1$ and

$\chi_{0}(t,x)=\chi\Big{(}\frac{x-x_{1}(t)}{z(t)}\Big{)}.$ (117)

The reason for this notation is to improve the precision of the estimate of
$\dot{p}_{1}(t)$ by the searching of the $\gamma,\,\theta$ which minimize
$\alpha(t)$.
Step 1.(Estimate of $I$) We will only use the identity
$I=\dot{x}_{1}(t)\frac{\big{\langle}\partial_{t}\phi(t),\,\partial^{2}_{x}H^{x_{1}(t)}_{-1,0}\big{\rangle}}{\left\lVert\partial_{x}H_{0,1}\right\rVert_{L^{2}}^{2}}.$
Step 2.(Estimate of $II.$) We have, by chain rule and definition of
$\chi_{0},$ that

$\displaystyle II$
$\displaystyle=-\frac{\big{\langle}\partial_{t}\phi(t),\,\partial_{x}\big{(}\partial_{t}\chi_{0}(t)g(t)\big{)}\big{\rangle}}{\left\lVert\partial_{x}H_{0,1}\right\rVert_{L^{2}}^{2}}=-\frac{\Big{\langle}\partial_{t}\phi(t),\,\partial_{x}\Big{[}\frac{d}{dt}\Big{[}\chi\Big{(}\frac{x-x_{1}(t)}{z(t)}\Big{)}\Big{]}g(t,x)\Big{]}\Big{\rangle}}{\left\lVert\partial_{x}H_{0,1}\right\rVert_{L^{2}}^{2}}$
$\displaystyle=-\frac{\Big{\langle}\partial_{t}\phi(t),\,\partial_{x}\Big{(}\dot{\chi}\Big{(}\frac{x-x_{1}(t)}{z(t)}\Big{)}\frac{d}{dt}\Big{[}\frac{x-x_{1}(t)}{z(t)}\Big{]}g(t)\Big{)}\Big{\rangle}}{\left\lVert\partial_{x}H_{0,1}\right\rVert_{L^{2}}^{2}}=\frac{\Big{\langle}\partial_{t}\phi(t),\,\partial_{x}\Big{(}\dot{\chi}\Big{(}\frac{x-x_{1}(t)}{z(t)}\Big{)}\Big{[}\frac{\dot{x}_{1}(t)z(t)+(x-x_{1}(t))\dot{z}(t)}{z(t)^{2}}\Big{]}g(t)\Big{)}\Big{\rangle}}{\left\lVert\partial_{x}H_{0,1}\right\rVert_{L^{2}}^{2}}.$

So, we obtain that

$II=\frac{\Big{\langle}\partial_{t}\phi(t),\,\ddot{\chi}\Big{(}\frac{x-x_{1}(t)}{z(t)}\Big{)}\Big{[}\frac{\dot{x}_{1}(t)}{z(t)}+\frac{(x-x_{1}(t))\dot{z}(t)}{z(t)^{2}}\Big{]}g(t)\Big{\rangle}}{z(t)\left\lVert\partial_{x}H_{0,1}\right\rVert_{L^{2}}^{2}}+\frac{\Big{\langle}\partial_{t}\phi(t),\,\dot{\chi}\Big{(}\frac{x-x_{1}(t)}{z(t)}\Big{)}\frac{\dot{z}(t)}{z(t)^{2}}g(t)\Big{\rangle}}{\left\lVert\partial_{x}H_{0,1}\right\rVert_{L^{2}}^{2}}\\\
+\frac{\Big{\langle}\partial_{t}\phi(t),\,\dot{\chi}\Big{(}\frac{x-x_{1}(t)}{z(t)}\Big{)}\Big{[}\frac{\dot{x}_{1}(t)}{z(t)}+\frac{(x-x_{1}(t))\dot{z}(t)}{z(t)^{2}}\Big{]}\partial_{x}g(t)\Big{\rangle}}{\left\lVert\partial_{x}H_{0,1}\right\rVert_{L^{2}}^{2}}.$
(118)

First, note that since the support of $\dot{\chi}$ is contained in
$[\theta(1-\gamma),\,\theta],$ from the estimates (D3) and (D4) we obtain that

$\displaystyle\left\lVert\partial_{x}H^{x_{1}(t)}_{-1,0}\right\rVert_{L^{2}_{x}\Big{(}\operatorname{supp}\partial_{x}\chi_{0}(t,x)\Big{)}}^{2}=O\Big{(}e^{-2\sqrt{2}\theta(1-\gamma)z(t)}\Big{)},$
(119)
$\displaystyle\left\lVert\partial_{x}H^{x_{2}(t)}_{0,1}\right\rVert_{L^{2}_{x}\Big{(}\operatorname{supp}\partial_{x}\chi_{0}(t,x)\Big{)}}^{2}=O\Big{(}e^{-2\sqrt{2}(1-\theta)z(t)}\Big{)},$
(120)

Now, we recall the identity
$\partial_{t}\phi(t,x)=-\dot{x}_{1}(t)\partial_{x}H^{x_{1}(t)}_{-1,0}-\dot{x}_{2}(t)\partial_{x}H^{x_{2}(t)}_{0,1}+\partial_{t}g(t),$
by using the estimates (119), (120) in the identity (118), we deduce that

$II=O\Bigg{(}\left\lVert\dot{\chi}\right\rVert_{L^{\infty}(\mathbb{R})}\frac{\max_{j\in\\{1,\,2\\}}|\dot{x}_{j}(t)|}{z(t)}\left\lVert\overrightarrow{g(t)}\right\rVert^{2}+\left\lVert\ddot{\chi}\right\rVert_{L^{\infty}(\mathbb{R})}\left\lVert\overrightarrow{g(t)}\right\rVert^{2}\frac{\max_{j\in\\{1,\,2\\}}|\dot{x}_{j}(t)|}{z(t)^{2}}\\\
+(e^{-\sqrt{2}\theta(1-\gamma)z(t)}+e^{-\sqrt{z(t)}(1-\theta)})\left\lVert\ddot{\chi}\right\rVert_{L^{\infty}(\mathbb{R})}\frac{\max_{j\in\\{1,\,2\\}}\dot{x}_{j}(t)^{2}}{z(t)^{2}}\left\lVert\overrightarrow{g(t)}\right\rVert\\\
+\left\lVert\overrightarrow{g(t)}\right\rVert(e^{-\sqrt{2}z(t)(1-\theta)}+e^{-\sqrt{2}\theta(1-\gamma)z(t)})\left[\frac{\left\lVert\ddot{\chi}\right\rVert_{L^{\infty}(\mathbb{R})}}{z(t)^{2}}+\frac{\left\lVert\dot{\chi}\right\rVert_{L^{\infty}(\mathbb{R})}}{z(t)}\right]\max_{j\in\\{1,\,2\\}}\dot{x}_{j}(t)^{2}\Bigg{)}.$
(121)

Since $\frac{1-\gamma}{2-\gamma}\leq\max((1-\theta),\theta(1-\gamma))$ for
$0<\gamma,\theta<1,$ we have that the estimate (121) is minimal when
$\theta=\frac{1-\gamma}{2-\gamma}.$ So, from now on, we consider

$\theta=\frac{1-\gamma}{2-\gamma},$ (122)

which with (108) and (121) imply that $II=O(\alpha(t)).$
Step 3.(Estimate of $III.$) We deduce from the identity

$III=-\frac{\langle\partial_{x}(\chi_{0}(t)\partial_{t}g(t)),\,\partial_{t}\phi(t)\rangle}{\left\lVert\partial_{x}H_{0,1}\right\rVert_{L^{2}}^{2}}$

that

$III=-\frac{\Big{\langle}\dot{\chi}\Big{(}\frac{x-x_{1}(t)}{z(t)}\Big{)}\partial_{t}g(t),\,-\dot{x}_{1}(t)\partial_{x}H^{x_{1}(t)}_{-1,0}-\dot{x}_{2}(t)\partial_{x}H^{x_{2}(t)}_{0,1}+\partial_{t}g(t,x)\Big{\rangle}}{z(t)\left\lVert\partial_{x}H_{0,1}\right\rVert_{L^{2}}^{2}}\\\
-\frac{\Big{\langle}\chi_{0}(t,x)\partial^{2}_{t,x}g(t,x),\,-\dot{x}_{1}(t)\partial_{x}H^{x_{1}(t)}_{-1,0}-\dot{x}_{2}(t)\partial_{x}H^{x_{2}(t)}_{0,1}+\partial_{t}g(t,x)\Big{\rangle}}{\left\lVert\partial_{x}H_{0,1}\right\rVert_{L^{2}}^{2}}=III.1+III.2.$
(123)

The identity (122) and the estimates (108), (119) and (120) imply by Cauchy-
Schwarz inequality that

$III.1=O\Bigg{(}\frac{\max_{j\in\\{1,\,2\\}}|\dot{x}_{j}(t)|e^{-\sqrt{2}z(t)(\frac{1-\gamma}{2-\gamma})}}{\gamma
z(t)}\left\lVert\overrightarrow{g(t)}\right\rVert+\frac{1}{z(t)\gamma}\left\lVert\overrightarrow{g(t)}\right\rVert^{2}\Bigg{)}.$
(124)

In conclusion, we have the estimate that $III.1=O(\alpha(t)).$ Also, from
condition (116) and the estimate (4), we can deduce that

$\left\lVert(1-\chi_{0}(t))\partial^{2}_{x}H^{x_{1}(t)}_{-1,0}\right\rVert_{L^{2}_{x}(\mathbb{R})}+\left\lVert\chi_{0}(t)\partial^{2}_{x}H^{x_{2}(t)}_{0,1}\right\rVert_{L^{2}_{x}(\mathbb{R})}=O\Big{(}e^{-\sqrt{2}z(t)(\frac{1-\gamma}{2-\gamma})}\Big{)}.$
(125)

Also, we have that

$III.2=-\frac{\Big{\langle}\chi_{0}(t,x)\Big{[}\partial^{2}_{t,x}\phi(t)+\dot{x}_{1}(t)\partial^{2}_{x}H^{x_{1}(t)}_{-1,0}+\dot{x}_{2}(t)\partial^{2}_{x}H^{x_{2}(t)}_{0,1}\Big{]},\,\partial_{t}\phi(t)\Big{\rangle}}{\left\lVert\partial_{x}H_{0,1}\right\rVert_{L^{2}}^{2}}.$
(126)

By integration by parts, we have that

$\Big{|}\Big{\langle}\chi\Big{(}\frac{x-x_{1}(t)}{z(t)}\Big{)}\partial^{2}_{t,x}\phi(t,x),\,\partial_{t}\phi(t,x)\Big{\rangle}\Big{|}=O\Big{(}\frac{1}{\gamma
z(t)}\left\lVert\partial_{t}\phi(t)\right\rVert_{L^{2}_{x}(\operatorname{supp}\partial_{x}\chi_{0}(t))}^{2}\Big{)}.$

In conclusion, from the estimates (108), (119), (120) and identity (122), we
obtain that

$\Big{|}\Big{\langle}\chi\Big{(}\frac{x-x_{1}(t)}{z(t)}\Big{)}\partial^{2}_{t,x}\phi(t,x),\,\partial_{t}\phi(t,x)\Big{\rangle}\Big{|}=O\Big{(}\frac{1}{\gamma
z(t)}\left\lVert\overrightarrow{g(t)}\right\rVert^{2}+\max_{j\in\\{1,\,2\\}}\dot{x}_{j}(t)^{2}\frac{1}{\gamma
z(t)}\Big{[}e^{-2\sqrt{2}z(t)(\frac{1-\gamma}{2-\gamma})}\Big{]}\Big{)}.$
(127)

Also, from Lemma (2.3), the estimate (4) and the fact of $0\leq\chi_{0}\leq
1,$ we deduce that

$\displaystyle\Big{|}\Big{\langle}\chi_{0}(t,x)\partial^{2}_{x}H^{x_{2}(t)}_{0,1},\,\partial_{x}H^{x_{1}(t)}_{-1,0}\Big{\rangle}\Big{|}=O\Big{(}z(t)e^{-\sqrt{2}z(t)}\Big{)},$
(128)
$\displaystyle\Big{|}\Big{\langle}(1-\chi_{0}(t,x))\partial^{2}_{x}H^{x_{1}(t)}_{-1,0},\,\partial_{x}H^{x_{2}(t)}_{0,1}\Big{\rangle}\Big{|}=O\Big{(}z(t)e^{-\sqrt{2}z(t)}\Big{)}.$
(129)

From the estimates (119), (120) and identity (122), we can verify by
integration by parts the following estimates

$\Big{\langle}(1-\chi_{0}(t))\dot{x}_{1}(t)\partial^{2}_{x}H^{x_{1}(t)}_{-1,0},\,\dot{x}_{1}(t)\partial_{x}H^{x_{1}(t)}_{-1,0}\Big{\rangle}=O\Big{(}\frac{\dot{x}_{1}(t)^{2}}{\gamma
z(t)}e^{-2\sqrt{2}z(t)(\frac{1-\gamma}{2-\gamma})}\Big{)},$ (130)
$\Big{\langle}\chi_{0}(t)\dot{x}_{2}(t)\partial^{2}_{x}H^{x_{2}(t)}_{0,1},\,\dot{x}_{2}(t)\partial_{x}H^{x_{2}(t)}_{0,1}\Big{\rangle}=O\Big{(}\frac{\dot{x}_{2}(t)^{2}}{\gamma
z(t)}e^{-2\sqrt{2}z(t)(\frac{1-\gamma}{2-\gamma})}\Big{)}.$ (131)

Finally, from Cauchy-Schwarz inequality and the estimate (125) we obtain that

$\displaystyle\Big{\langle}(1-\chi_{0}(t))\dot{x}_{1}(t)\partial^{2}_{x}H^{x_{1}(t)}_{-1,0},\,\partial_{t}g(t)\Big{\rangle}=O\Big{(}|\dot{x}_{1}(t)|\left\lVert\overrightarrow{g(t)}\right\rVert
e^{-\sqrt{2}z(t)(\frac{1-\gamma}{2-\gamma})}\Big{)},$ (132)
$\displaystyle\Big{\langle}\chi_{0}(t)\dot{x}_{1}(t)\partial^{2}_{x}H^{x_{2}(t)}_{0,1},\,\partial_{t}g(t)\Big{\rangle}=O\Big{(}|\dot{x}_{2}(t)|\left\lVert\overrightarrow{g(t)}\right\rVert
e^{-\sqrt{2}z(t)(\frac{1-\gamma}{2-\gamma})}\Big{)}.$ (133)

In conclusion, we obtain from the estimates (128), (129), (130), (131) (132)
and (133) that

$III.2=-\dot{x}_{1}(t)\frac{\Big{\langle}\partial^{2}_{x}H^{x_{1}(t)}_{-1,0},\,\partial_{t}\phi(t)\Big{\rangle}}{\left\lVert\partial_{x}H_{0,1}\right\rVert^{2}}+O(\alpha(t)).$
(134)

This estimate of $III.2$ and the estimate (124) of $III.1$ imply

$III=-\dot{x}_{1}(t)\frac{\Big{\langle}\partial^{2}_{x}H^{x_{1}(t)}_{-1,0},\,\partial_{t}\phi(t)\Big{\rangle}}{\left\lVert\partial_{x}H_{0,1}\right\rVert^{2}}+O(\alpha(t)).$
(135)

In conclusion, from the estimates $II=O(\alpha(t)),$ (135) and the definition
of $I,$ we have that $I+II+III=O(\alpha(t)).$
Step 4.(Estimate of $V.$) We recall that
$V=-\frac{\langle\partial_{x}\chi_{0}(t)g(t),\,\partial^{2}_{t}\phi(t)\rangle}{\left\lVert\partial_{x}H_{0,1}\right\rVert_{L^{2}}^{2}},$
and that

$\partial^{2}_{t}\phi(t)=\partial^{2}_{x}g(t)+\Big{[}\dot{U}\left(H^{x_{1}(t)}_{-1,0}\right)+\dot{U}\left(H^{x_{2}(t)}_{0,1}\right)-\dot{U}\left(H^{x_{1}(t)}_{-1,0}+H^{x_{2}(t)}_{0,1}\right)\Big{]}+\Big{[}\dot{U}\left(H^{x_{1}(t)}_{-1,0}+H^{x_{2}(t)}_{0,1}\right)-\dot{U}\left(H^{x_{1}(t)}_{-1,0}+H^{x_{2}(t)}_{0,1}+g(t)\right)\Big{]}.$
(136)

First, by integration by parts, using estimate (108), we have the following
estimate

$-\frac{1}{\left\lVert\partial_{x}H_{0,1}\right\rVert_{L^{2}}^{2}}\langle\partial_{x}\chi_{0}(t)\partial^{2}_{x}g(t),\,g(t)\rangle=O\Big{(}\Big{[}\frac{1}{\gamma
z(t)}+\frac{1}{\gamma^{2}z(t)^{2}}\Big{]}\left\lVert\overrightarrow{g(t)}\right\rVert^{2}\Big{)}=O(\alpha(t)).$
(137)

Second, since $U$ is smooth and $\left\lVert
g(t)\right\rVert_{L^{\infty}}=O\big{(}\epsilon^{\frac{1}{2}}\big{)}$ for all
$t\in\mathbb{R},$ we deduce that

$\left\langle\dot{U}\left(H^{x_{1}(t)}_{-1,0}+H^{x_{2}(t)}_{0,1}\right)-\dot{U}\left(H^{x_{1}(t)}_{-1,0}+H^{x_{2}(t)}_{0,1}+g(t)\right),\partial_{x}\chi_{0}(t)g(t)\right\rangle=O\Big{(}\frac{1}{z(t)\gamma}\left\lVert\overrightarrow{g(t)}\right\rVert^{2}\Big{)}=O(\alpha(t)).$
(138)

Next, from equation (87) and Lemma 2.3, we have that

$\left\lVert\dot{U}\left(H^{x_{1}(t)}_{-1,0}\right)+\dot{U}\left(H^{x_{2}(t)}_{0,1}\right)-\dot{U}\left(H^{x_{1}(t)}_{-1,0}+H^{x_{2}(t)}_{0,1}\right)\right\rVert_{L^{2}(\mathbb{R})}=O(e^{-\sqrt{2}z(t)}),$
(139)

then, by Hölder inequality we have that

$\left\langle\dot{U}\left(H^{x_{1}(t)}_{-1,0}\right)+\dot{U}\left(H^{x_{2}(t)}_{0,1}\right)-\dot{U}\left(H^{x_{1}(t)}_{-1,0}+H^{x_{2}(t)}_{0,1}\right),\partial_{x}\chi_{0}(t)\partial_{x}g(t)\right\rangle=O\Big{(}\frac{1}{\gamma
z(t)}\left\lVert\overrightarrow{g(t)}\right\rVert
e^{-\sqrt{2}z(t)}\Big{)}=O(\alpha(t)).$ (140)

Clearly, the estimates (137), (138) and (140) imply that $V=O(\alpha(t)).$
Step 5.(Estimate of $VI.$) We know that

$VI=-\frac{\big{\langle}\partial_{x}g(t)\chi_{0}(t),\,\partial^{2}_{t}\phi(t)\big{\rangle}}{\left\lVert\partial_{x}H_{0,1}\right\rVert_{L^{2}}^{2}}.$

We recall the equation (136) which implies that

$\left\lVert\partial_{x}H_{0,1}\right\rVert_{L^{2}}^{2}VI=-\left\langle\partial_{x}g(t)\chi_{0}(t),\,\partial^{2}_{x}g(t)\right\rangle+\left\langle\partial_{x}g(t)\chi_{0}(t),\,\dot{U}\left(H^{x_{1}(t)}_{-1,0}+H^{x_{2}(t)}_{0,1}+g(t)\right)-\dot{U}\left(H^{x_{1}(t)}_{-1,0}+H^{x_{2}(t)}_{0,1}\right)\right\rangle\\\
+\left\langle\partial_{x}g(t)\chi_{0}(t),\,\dot{U}\left(H^{x_{1}(t)}_{-1,0}+H^{x_{2}(t)}_{0,1}\right)-\dot{U}\left(H^{x_{1}(t)}_{-1,0}\right)-\dot{U}\left(H^{x_{2}(t)}_{0,1}\right)\right\rangle.$

By integration by parts, we have from estimate (108) that

$\langle\partial_{x}g(t,x)\chi_{0}(t,x),\,\partial^{2}_{x}g(t,x)\rangle=O\Big{(}\frac{1}{\gamma
z(t)}\left\lVert\overrightarrow{g(t)}\right\rVert^{2}\Big{)}.$ (141)

From the estimate (139) and Cauchy-Schwarz inequality, we can obtain the
following estimate

$\left\langle\partial_{x}g(t)\chi_{0}(t),\,\dot{U}\left(H^{x_{1}(t)}_{-1,0}+H^{x_{2}(t)}_{0,1}\right)-\dot{U}\left(H^{x_{1}(t)}_{-1,0}\right)-\dot{U}\left(H^{x_{2}(t)}_{0,1}\right)\right\rangle=O\Big{(}e^{-\sqrt{2}z(t)}\left\lVert\overrightarrow{g(t)}\right\rVert\Big{)}.$
(142)

Then, to conclude the estimate of $VI$ we just need to study the following
term
$C(t)\coloneqq\big{\langle}\partial_{x}g(t)\chi_{0}(t),\,\dot{U}(H^{x_{1}(t)}_{-1,0}+H^{x_{2}(t)}_{0,1}+g(t))-\dot{U}(H^{x_{1}(t)}_{-1,0}+H^{x_{2}(t)}_{0,1})\big{\rangle}.$
Since we have from the Taylor’s theorem that

$\dot{U}\left(H^{x_{1}(t)}_{-1,0}+H^{x_{2}(t)}_{0,1}+g(t)\right)-\dot{U}\left(H^{x_{1}(t)}_{-1,0}+H^{x_{2}(t)}_{0,1}\right)=\sum_{k=2}^{6}U^{(k)}\left(H^{x_{1}(t)}_{-1,0}+H^{x_{2}(t)}_{0,1}\right)\frac{g(t)^{k-1}}{(k-1)!},$

from estimate (108), we can deduce by integration by parts that

$C(t)=-\Big{\langle}\chi_{0}(t)\partial_{x}\left(H^{x_{1}(t)}_{-1,0}+H^{x_{2}(t)}_{0,1}\right),\,\sum_{k=3}^{6}U^{(k)}\left(H^{x_{1}(t)}_{-1,0}+H^{x_{2}(t)}_{0,1}\right)\frac{g(t)^{k-1}}{(k-1)!}\Big{\rangle}+O\left(\frac{1}{\gamma
z(t)}\left\lVert\overrightarrow{g(t)}\right\rVert^{2}\right).$

Since

$\left\lVert\chi_{0}(t)\partial_{x}H^{x_{2}(t)}_{0,1}\right\rVert_{L^{\infty}}+\left\lVert(1-\chi_{0}(t))\partial_{x}H^{x_{1}(t)}_{-1,0}\right\rVert_{L^{\infty}}=O\Big{(}e^{-\sqrt{2}z(t)(\frac{1-\gamma}{2-\gamma})}\Big{)},$

we obtain that

$C(t)=-\Big{\langle}\partial_{x}H^{x_{1}(t)}_{-1,0},\,\sum_{k=3}^{6}U^{(k)}\left(H^{x_{1}(t)}_{-1,0}+H^{x_{2}(t)}_{0,1}\right)\frac{g(t)^{k-1}}{(k-1)!}\Big{\rangle}+O\left(\frac{1}{\gamma
z(t)}\left\lVert\overrightarrow{g(t)}\right\rVert^{2}+e^{-\sqrt{2}z(t)(\frac{1-\gamma}{2-\gamma})}\left\lVert\overrightarrow{g(t)}\right\rVert^{2}\right).$

Also, from Lemma 2.3 and the fact that $\left\lVert
g(t)\right\rVert_{L^{\infty}}\lesssim\left\lVert\overrightarrow{g(t)}\right\rVert$,
we deduce that

$\Big{\langle}\partial_{x}H^{x_{1}}_{-1,0},\Big{[}\ddot{U}\left(H^{x_{1}(t)}_{-1,0}\right)-\ddot{U}\left(H^{x_{1}(t)}_{-1,0}+H^{x_{2}(t)}_{0,1}\right)\Big{]}g(t)\Big{\rangle}=O\Big{(}e^{-\sqrt{2}z(t)}\left\lVert\overrightarrow{g(t)}\right\rVert\Big{)}.$
(143)

In conclusion, we obtain that

$C(t)=-\int_{\mathbb{R}}\partial_{x}H^{x_{1}(t)}_{-1,0}\Big{(}\dot{U}\left(H^{x_{1}(t)}_{-1,0}+H^{x_{2}(t)}_{0,1}+g(t)\right)-\dot{U}\left(H^{x_{1}(t)}_{-1,0}+H^{x_{2}(t)}_{0,1}\right)\Big{)}\,dx+\int_{\mathbb{R}}\partial_{x}H^{x_{1}(t)}_{-1,0}\ddot{U}\left(H^{x_{1}(t)}_{-1,0}\right)g(t,x)\,dx\\\
+O(\alpha(t)).$ (144)

So

$VI=\frac{-\int_{\mathbb{R}}\partial_{x}H^{x_{1}(t)}_{-1,0}\Big{(}\dot{U}\left(H^{x_{1}(t)}_{-1,0}+H^{x_{2}(t)}_{0,1}+g(t)\right)-\dot{U}(H^{x_{1}(t)}_{-1,0}+H^{x_{2}(t)}_{0,1})\Big{)}\,dx}{\left\lVert\partial_{x}H_{0,1}\right\rVert_{L^{2}}^{2}}\\\
+\frac{\int_{\mathbb{R}}\partial_{x}H^{x_{1}(t)}_{-1,0}\ddot{U}\left(H^{x_{1}(t)}_{-1,0}\right)g(t,x)\,dx}{\left\lVert\partial_{x}H_{0,1}\right\rVert_{L^{2}}^{2}}+O(\alpha(t)).$
(145)

Step 6.(Sum of $IV,\,VI.$) From the identities (136) and

$IV=-\frac{\big{\langle}\partial_{x}H^{x_{1}(t)}_{-1,0},\,\partial^{2}_{t}\phi(t)\big{\rangle}}{\left\lVert\partial_{x}H_{0,1}\right\rVert_{L^{2}}^{2}},$

we obtain that

$IV=-\frac{\Big{\langle}\partial^{2}_{x}g(t)-\Big{(}\dot{U}\left(H^{x_{1}(t)}_{-1,0}+H^{x_{2}(t)}_{0,1}+g(t)\right)-\dot{U}\left(H^{x_{1}(t)}_{-1,0}+H^{x_{2}(t)}_{0,1}\right)\Big{)},\,\partial_{x}H^{x_{1}(t)}_{-1,0}\Big{\rangle}}{\left\lVert\partial_{x}H_{0,1}\right\rVert_{L^{2}}^{2}}\\\
-\frac{\Big{\langle}\dot{U}\left(H^{x_{1}(t)}_{-1,0}\right)+\dot{U}\left(H^{x_{2}(t)}_{0,1}\right)-U\left(H^{x_{1}(t)}_{-1,0}+H^{x_{2}(t)}_{0,1}\right),\,\partial_{x}H^{x_{1}(t)}_{-1,0}\Big{\rangle}}{\left\lVert\partial_{x}H_{0,1}\right\rVert_{L^{2}}^{2}}.$
(146)

In conclusion, from the identity

$\Big{[}\partial_{x}^{2}-\ddot{U}\left(H^{x_{1}(t)}_{-1,0}\right)\Big{]}\partial_{x}H^{x_{1}(t)}_{-1,0}=0$

and by integration by parts we have that

$IV+VI=-\frac{\Big{\langle}\dot{U}\left(H^{x_{1}(t)}_{-1,0}\right)+\dot{U}\left(H^{x_{2}(t)}_{0,1}\right)-U\left(H^{x_{1}(t)}_{-1,0}+H^{x_{2}(t)}_{0,1}\right),\,\partial_{x}H^{x_{1}(t)}_{-1,0}\Big{\rangle}}{\left\lVert\partial_{x}H_{0,1}\right\rVert_{L^{2}}^{2}}+O(\alpha(t)).$
(147)

From our previous results, we conclude that

$I+II+III+IV+V+VI=\\\
-\frac{\Big{\langle}\dot{U}\left(H^{x_{1}(t)}_{-1,0}\right)+\dot{U}\left(H^{x_{2}(t)}_{0,1}\right)-\dot{U}\left(H^{x_{1}(t)}_{-1,0}+H^{x_{2}(t)}_{0,1}\right),\,\partial_{x}H^{x_{1}(t)}_{-1,0}\Big{\rangle}}{\left\lVert\partial_{x}H_{0,1}\right\rVert_{L^{2}}^{2}}+O(\alpha(t)).$
(148)

The conclusion of the lemma follows from estimate (148) with identity

$\dot{A}(z(t))=-\Big{\langle}\dot{U}\left(H_{-1,0}\right)+\dot{U}\left(H^{z(t)}_{0,1}\right)-\dot{U}\left(H_{-1,0}+H^{z(t)}_{0,1}\right),\,\partial_{x}H_{-1,0}\Big{\rangle},$

which can be obtained from (13) by integration by parts with the fact that

$\left\langle\dot{U}\left(H_{-1,0}+H^{z(t)}_{0,1}\right),\,\partial_{x}H_{-1,0}+\partial_{x}H^{z(t)}_{0,1}\right\rangle=0.$

∎

######  Remark 3.3.

Since, we know from Lemma 2.3 that

$\left|\dot{A}(z(t))+4e^{-\sqrt{2}z(t)}\right|\lesssim
z(t)e^{-2\sqrt{2}z(t)},$

and, by elementary calculus with change of variables, that
$\left\lVert\partial_{x}H_{0,1}\right\rVert_{L^{2}}^{2}=\frac{1}{2\sqrt{2}},$
then the estimates (106) and (107) obtained in Lemma 3.1 motivate us to study
the following ordinary differential equation

$\ddot{z}(t)=16\sqrt{2}e^{-\sqrt{2}z(t)}.$ (149)

Clearly, the solution of (149) satisfies the equation

$\frac{d}{dt}\Big{[}\frac{\dot{z}(t)^{2}}{4}+8e^{-\sqrt{2}z(t)}\Big{]}=0.$
(150)

As a consequence, it can be verified that if $z(t_{0})>0$ for some
$t_{0}\in\mathbb{R},$ then there are real constants $v>0,\,c$ such that

$z(t)=\frac{1}{\sqrt{2}}\ln{\Big{(}\frac{8}{v^{2}}\cosh{\big{(}\sqrt{2}vt+c\big{)}}^{2}\Big{)}}\text{
for all $t\in\mathbb{R}.$}$ (151)

In conclusion, the solution of the equations

$\displaystyle\ddot{d}_{1}(t)=-8\sqrt{2}e^{-\sqrt{2}z(t)},$
$\displaystyle\ddot{d}_{2}(t)=8\sqrt{2}e^{-\sqrt{2}z(t)},$ $\displaystyle
d_{2}(t)-d_{1}(t)=z(t)>0,$

are given by

$\displaystyle
d_{2}(t)=a+bt+\frac{1}{2\sqrt{2}}\ln{\Big{(}\frac{8}{v^{2}}\cosh{\big{(}\sqrt{2}vt+c\big{)}}^{2}\Big{)}},$
(152) $\displaystyle
d_{1}(t)=a+bt-\frac{1}{2\sqrt{2}}\ln{\Big{(}\frac{8}{v^{2}}\cosh{\big{(}\sqrt{2}vt+c\big{)}}^{2}\Big{)}},$
(153)

for $a,\,b$ real constants. So, we now are motivated to study how close the
modulations parameters $x_{1},\,x_{2}$ of Theorem 2.9 can be to functions
$d_{1},\,d_{2}$ satisfying, respectively the identities (153) and (152) for
constants $v\neq 0,\,a,\,b$.

At first view, the statement of the Lemma 3.1 seems too complex and
unnecessary for use and that a simplified version should be more useful for
our objectives. However, we will show later that for a suitable choice of
$\gamma$ depending on the energy excess of the solution $\phi(t),$ we can get
a high precision in the approximation of the modulation parameters
$x_{1},\,x_{2}$ by smooth functions $d_{1},\,d_{2}$ satisfying (153) and (152)
for a large time interval.

##  4 Energy Estimate Method

Before applying Lemma 3.1, we need to construct a functional $F(t)$ to get
lower estimate on the value of
$\left\lVert(g(t),\partial_{t}g(t))\right\rVert_{H^{1}\times L^{2}}$ than that
obtained in Theorem 2.9.

From now on, we consider
$\phi(t)=H_{0,1}(x-x_{2}(t))+H_{-1,0}(x-x_{1}(t))+g(t,x)$, with
$x_{1}(t),\,x_{2}(t)$ satisfying the orthogonality conditions of the
Modulation Lemma and $x_{1},\,x_{2}$, $(g(t),\partial_{t}g(t))$ and
$\epsilon>0$ satisfying all the properties of Theorem 2.9. Before the
enunciation of the main theorem of this section, to simplify the notation in
computations, we denote:

$D^{2}E(H^{x_{2}(t)}_{0,1}+H^{x_{1}(t)}_{-1,0})=\begin{bmatrix}-\partial^{2}_{x}+\ddot{U}(H^{x_{2}(t)}_{0,1}+H^{x_{1}(t)}_{-1,0})&0\\\
0&\mathbb{I}\end{bmatrix}$

as a bilinear operator from $H^{1}(\mathbb{R})\times L^{2}(\mathbb{R})$ to
$\mathbb{C}.$ We also denote
$\omega_{1}(t,x)=\omega\big{(}\frac{x-x_{1}(t)}{x_{2}(t)-x_{1}(t)}\big{)}$ for
$\omega$ a smooth cut function with image contained in the interval $[0,1]$,
satisfying the following condition

$\omega(x)=\begin{cases}1,\text{ if $x\leq\frac{3}{4}$,}\\\ 0,\text{ if
$x\geq\frac{4}{5}$.}\end{cases}$

We consider now the following functional

$F(t)=\big{\langle}D^{2}E(H^{x_{2}(t)}_{0,1}+H^{x_{1}(t)}_{-1,0})\overrightarrow{g(t)},\,\overrightarrow{g(t)}\big{\rangle}_{L^{2}\times
L^{2}}+2\int_{\mathbb{R}}\partial_{t}g(t,x)\partial_{x}g(t,x)\Big{[}\dot{x}_{1}(t)\omega_{1}(t,x)+\dot{x}_{2}(t)(1-\omega_{1}(t,x))\Big{]}\,dx\\\
-2\int_{\mathbb{R}}g(t,x)\Big{(}\dot{U}(H^{x_{1}(t)}_{-1,0}(x))+\dot{U}(H^{x_{2}(t)}_{0,1}(x))-\dot{U}(H^{x_{2}(t)}_{0,1}(x)+H^{x_{1}(t)}_{-1,0}(x))\Big{)}\,dx\\\
+2\int_{\mathbb{R}}g(t,x)\Big{[}(\dot{x}_{1}(t))^{2}\partial^{2}_{x}H^{x_{1}(t)}_{-1,0}(x)+(\dot{x}_{2}(t))^{2}\partial^{2}_{x}H^{x_{2}(t)}_{0,1}(x)\Big{]}\,dx+\frac{1}{3}\int_{\mathbb{R}}U^{(3)}(H^{x_{2}(t)}_{0,1}(x)+H^{x_{1}(t)}_{-1,0}(x))g(t,x)^{3}\,dx.$
(154)

Since $x_{1},\,x_{2}$ are functions of class $C^{2}$, is not difficult to
verify that $(g(t),\partial_{t}g(t))$ solves the integral equation associated
to the following partial differential equation

$\partial^{2}_{t}g(t,x)-\partial^{2}_{x}g(t,x)+\ddot{U}(H^{x_{2}(t)}_{0,1}(x)+H^{x_{1}(t)}_{-1,0}(x))g(t,x)=\\\
-\left[\dot{U}(H^{x_{2}(t)}_{0,1}(x)+H^{x_{1}(t)}_{-1,0}(x)+g(t,x))-\dot{U}(H^{x_{2}(t)}_{0,1}(x)+H^{x_{1}(t)}_{-1,0}(x))-\ddot{U}(H^{x_{2}(t)}_{0,1}(x)+H^{x_{1}(t)}_{-1,0}(x))g(t,x)\right]\\\
+\dot{U}(H^{x_{1}(t)}_{-1,0}(x))+\dot{U}(H^{x_{2}(t)}_{0,1}(x))-\dot{U}(H^{x_{2}(t)}_{0,1}(x)+H^{x_{1}(t)}_{-1,0}(x))\\\
-\dot{x}_{1}(t)^{2}\partial^{2}_{x}H^{x_{1}(t)}_{-1,0}(x)-\dot{x}_{2}(t)^{2}\partial^{2}_{x}H^{x_{2}(t)}_{0,1}(x)+\ddot{x}_{1}(t)\partial_{x}H^{x_{1}(t)}_{-1,0}(x)+\ddot{x}_{2}(t)\partial_{x}H^{x_{2}(t)}_{0,1}(x)$
(155)

in the space $H^{1}(\mathbb{R})\times L^{2}(\mathbb{R}).$

######  Theorem 4.1.

Assuming the hypotheses of Theorem 2.9 and recalling its notation, let
$\delta(t)$ be the following quantity

$\delta(t)=\left\lVert\overrightarrow{g(t)}\right\rVert\Big{(}e^{-\sqrt{2}z(t)}\max_{j\in\\{1,2\\}}|\dot{x}_{j}(t)|+\max_{j\in\\{1,2\\}}|\dot{x}_{j}(t)|^{3}e^{-\frac{\sqrt{2}z(t)}{5}}+\max_{j\in\\{1,2\\}}|\dot{x}_{j}(t)||\ddot{x}_{j}(t)|\Big{)}\\\
+\left\lVert\overrightarrow{g(t)}\right\rVert^{2}\Big{(}\frac{\max_{j\in\\{1,\,2\\}}|\dot{x}_{j}(t)|}{z(t)}+\max_{j\in\\{1,\,2\\}}\dot{x}_{j}(t)^{2}+\max_{j\in\\{1,\,2\\}}|\ddot{x}_{j}(t)|\Big{)}+\left\lVert\overrightarrow{g(t)}\right\rVert^{4}.$

Then, $\exists$ positive constants $A_{1},\,A_{2},\,A_{3}$ such that the
functional $F(t)$ satisfies the inequalities

$F(t)+A_{1}\epsilon^{2}\geq
A_{2}\left\lVert\overrightarrow{g(t)}\right\rVert^{2},\,|\dot{F}(t)|\leq
A_{3}\delta(t).$

######  Remark 4.2.

Theorem 2.9 and Theorem 4.1 imply

$|\dot{F}(t)|\lesssim\frac{\epsilon^{\frac{1}{2}}}{\ln{(\frac{1}{\epsilon})}}\left\lVert\overrightarrow{g(t)}\right\rVert^{2}+\left\lVert\overrightarrow{g(t)}\right\rVert\epsilon^{\frac{3}{2}}.$

###### Proof.

Since the formula defining function $F(t)$ is very large, we decompose the
function in a sum of five terms $F_{1},\,F_{2},\,F_{3},\,F_{4}$ and $F_{5}$.
More specifically:

$\displaystyle
F_{1}(t)=\int_{\mathbb{R}}\partial_{t}g(t,x)^{2}+\partial_{x}g(t,x)^{2}+\ddot{U}(H^{x_{1}(t)}_{-1,0}(x)+H^{x_{2}(t)}_{0,1}(x))g(t,x)^{2}\,dx,$
$\displaystyle
F_{2}(t)=-2\int_{\mathbb{R}}g(t,x)\big{[}\dot{U}(H^{x_{1}(t)}_{-1,0}(x))+\dot{U}(H^{x_{2}(t)}_{0,1}(x))-\dot{U}(H^{x_{2}(t)}_{0,1}(x)+H^{x_{1}(t)}_{-1,0}(x))\big{]}\,dx,$
$\displaystyle
F_{3}(t)=2\int_{\mathbb{R}}g(t,x)\big{[}\dot{x}_{1}(t)^{2}\partial^{2}_{x}H^{x_{1}(t)}_{-1,0}(x)+\dot{x}_{2}(t)^{2}\partial^{2}_{x}H^{x_{2}(t)}_{0,1}(x)\big{]}\,dx,$
$\displaystyle
F_{4}(t)=2\int_{\mathbb{R}}\partial_{t}g(t,x)\partial_{x}g(t,x)(\dot{x}_{1}(t)\omega_{1}(t,x)+\dot{x}_{2}(t)(1-\omega_{1}(t,x)))\,dx,$
$\displaystyle
F_{5}(t)=\frac{1}{3}\int_{\mathbb{R}}U^{(3)}(H^{x_{2}(t)}_{0,1}(x)+H^{x_{1}(t)}_{-1,0}(x))g(t,x)^{3}\,dx.$

First, We prove that $|\dot{F}(t)|\lesssim\delta(t).$ The main idea of the
proof of this item is to estimate each derivative $\frac{dF_{j}(t)}{dt},$ for
$1\leq j\leq 5,$ with an error of size $O(\delta(t)),$ then we will check that
the sum of these estimates are going to be a value of order $O(\delta(t)),$
which means that the estimates of these derivatives cancel.
Step 1.(The derivative of $F_{1}(t).$) By definition of $F_{1}(t),$ we have
that

$\frac{dF_{1}(t)}{dt}=2\int_{\mathbb{R}}\left(\partial^{2}_{t}g(t,x)-\partial^{2}_{x}g(t,x)+\ddot{U}(H^{x_{2}(t)}_{0,1}(x)+H^{x_{1}(t)}_{-1,0}(x))g(t,x)\right)\partial_{t}g(t,x)\,dx\\\
-\int_{\mathbb{R}}\left(\dot{x}_{1}(t)\partial_{x}H^{x_{1}(t)}_{-1,0}(x)+\dot{x}_{2}(t)\partial_{x}H^{x_{2}(t)}_{0,1}(x)\right)U^{(3)}\left(H^{x_{2}(t)}_{0,1}(x)+H^{x_{1}(t)}_{-1,0}(x)\right)g(t,x)^{2}\,dx.$
(156)

Moreover, from the identity (155) satisfied by $g(t,x)$, we can rewrite the
value of $\frac{dF_{1}(t)}{dt}$ as

$\frac{dF_{1}(t)}{dt}=2\int_{\mathbb{R}}\left[\dot{U}\left(H^{x_{1}(t)}_{-1,0}(x)\right)+\dot{U}\left(H^{x_{2}(t)}_{0,1}(x)\right)-\dot{U}\left(H^{x_{1}(t)}_{-1,0}(x)+H^{x_{2}(t)}_{0,1}(x)\right)\right]\partial_{t}g(t,x)\,dx\\\
-2\int_{\mathbb{R}}\left[U\left(H^{x_{2}(t)}_{0,1}(x)+H^{x_{1}(t)}_{-1,0}(x)+g(t,x)\right)-\dot{U}\left(H^{x_{1}(t)}_{-1,0}(x)+H^{x_{2}(t)}_{0,1}(x)\right)-\ddot{U}\left(H^{x_{2}(t)}_{0,1}(x)+H^{x_{1}(t)}_{-1,0}(x)\right)g(t,x)\right]\partial_{t}g(t,x)\,dx\\\
-2\int_{\mathbb{R}}\Big{[}\dot{x}_{1}(t)^{2}\partial^{2}_{x}H^{x_{1}(t)}_{-1,0}(x)+\dot{x}_{2}(t)^{2}\partial^{2}_{x}H^{x_{2}(t)}_{0,1}(x)\Big{]}\partial_{t}g(t,x)\,dx+2\int_{\mathbb{R}}\Big{[}\ddot{x}_{1}(t)\partial_{x}H^{x_{1}(t)}_{-1,0}(x)+\ddot{x}_{2}(t)\partial_{x}H^{x_{2}(t)}_{0,1}(x)\Big{]}\partial_{t}g(t,x)\,dx\\\
-\int_{\mathbb{R}}\Big{[}\dot{x}_{1}(t)\partial_{x}H^{x_{1}(t)}_{-1,0}(x)+\dot{x}_{2}(t)\partial_{x}H^{x_{2}(t)}_{0,1}(x)\Big{]}U^{(3)}(H^{x_{2}(t)}_{0,1}(x)+H^{x_{1}(t)}_{-1,0}(x))g(t,x)^{2}\,dx,$

and, from the orthogonality conditions of the Modulation Lemma, we obtain

$\frac{dF_{1}(t)}{dt}=2\int_{\mathbb{R}}\left[\dot{U}\left(H^{x_{1}(t)}_{-1,0}(x)\right)+\dot{U}\left(H^{x_{2}(t)}_{0,1}(x)\right)-\dot{U}\left(H^{x_{1}(t)}_{-1,0}(x)+H^{x_{2}(t)}_{0,1}(x)\right)\right]\partial_{t}g(t,x)\,dx\\\
-2\int_{\mathbb{R}}\left[U\left(H^{x_{2}(t)}_{0,1}(x)+H^{x_{1}(t)}_{-1,0}(x)+g(t,x)\right)-\dot{U}\left(H^{x_{1}(t)}_{-1,0}(x)+H^{x_{2}(t)}_{0,1}(x)\right)-\ddot{U}\left(H^{x_{2}(t)}_{0,1}(x)+H^{x_{1}(t)}_{-1,0}(x)\right)g(t,x)\right]\partial_{t}g(t,x)\,dx\\\
-2\int_{\mathbb{R}}\Big{[}\dot{x}_{1}(t)^{2}\partial^{2}_{x}H^{x_{1}(t)}_{-1,0}(x)+\dot{x}_{2}(t)^{2}\partial^{2}_{x}H^{x_{2}(t)}_{0,1}(x)\Big{]}\partial_{t}g(t,x)\,dx+2\int_{\mathbb{R}}\Big{[}\ddot{x}_{1}(t)\dot{x}_{1}(t)\partial^{2}_{x}H^{x_{1}(t)}_{-1,0}(x)+\ddot{x}_{2}(t)\dot{x}_{2}(t)\partial^{2}_{x}H^{x_{2}(t)}_{0,1}(x)\Big{]}g(t,x)\,dx\\\
-\int_{\mathbb{R}}\Big{[}\dot{x}_{1}(t)\partial_{x}H^{x_{1}(t)}_{-1,0}(x)+\dot{x}_{2}(t)\partial_{x}H^{x_{2}(t)}_{0,1}(x)\Big{]}U^{(3)}\left(H^{x_{2}(t)}_{0,1}(x)+H^{x_{1}(t)}_{-1,0}(x)\right)g(t,x)^{2}\,dx,$

which implies

$\frac{dF_{1}(t)}{dt}=2\int_{\mathbb{R}}\left[\dot{U}\left(H^{x_{1}(t)}_{-1,0}(x)\right)+\dot{U}\left(H^{x_{2}(t)}_{0,1}(x)\right)-\dot{U}\left(H^{x_{1}(t)}_{-1,0}(x)+H^{x_{2}(t)}_{0,1}(x)\right)\right]\partial_{t}g(t,x)\,dx\\\
-2\int_{\mathbb{R}}\left[U\left(H^{x_{2}(t)}_{0,1}(x)+H^{x_{1}(t)}_{-1,0}(x)+g(t,x)\right)-\dot{U}\left(H^{x_{1}(t)}_{-1,0}(x)+H^{x_{2}(t)}_{0,1}(x)\right)-\ddot{U}\left(H^{x_{2}(t)}_{0,1}(x)+H^{x_{1}(t)}_{-1,0}(x)\right)g(t,x)\right]\partial_{t}g(t,x)\,dx\\\
-2\int_{\mathbb{R}}\Big{[}\dot{x}_{1}(t)^{2}\partial^{2}_{x}H^{x_{1}(t)}_{-1,0}(x)+\dot{x}_{2}(t)^{2}\partial^{2}_{x}H^{x_{2}(t)}_{0,1}(x)\Big{]}\partial_{t}g(t,x)\,dx\\\
-\int_{\mathbb{R}}\Big{[}\dot{x}_{1}(t)\partial_{x}H^{x_{1}(t)}_{-1,0}(x)+\dot{x}_{2}(t)\partial_{x}H^{x_{2}(t)}_{0,1}(x)\Big{]}U^{(3)}\left(H^{x_{2}(t)}_{0,1}(x)+H^{x_{1}(t)}_{-1,0}(x)\right)g(t,x)^{2}\,dx+O(\delta(t)).$
(157)

Step 2.(The derivative of $F_{2}(t).$) It is not difficult to verify that

$\frac{dF_{2}(t)}{dt}=-2\int_{\mathbb{R}}\partial_{t}g(t,x)\left[\dot{U}\left(H^{x_{1}(t)}_{-1,0}(x)\right)+\dot{U}\left(H^{x_{2}(t)}_{0,1}(x)\right)-\dot{U}\left(H^{x_{2}(t)}_{0,1}(x)+H^{x_{1}(t)}_{-1,0}(x)\right)\right]\,dx\\\
+2\int_{\mathbb{R}}g(t,x)\left[\ddot{U}\left(H^{x_{1}(t)}_{-1,0}(x)\right)\partial_{x}H^{x_{1}(t)}_{-1,0}(x)\dot{x}_{1}(t)+\ddot{U}\left(H^{x_{2}(t)}_{0,1}(x)\right)\partial_{x}H^{x_{2}(t)}_{0,1}(x)\dot{x}_{2}(t)\right]\,dx\\\
-2\int_{\mathbb{R}}\ddot{U}\left(H^{x_{2}(t)}_{0,1}(x)+H^{x_{1}(t)}_{-1,0}(x)\right)\left[\partial_{x}H^{x_{1}(t)}_{-1,0}(x)\dot{x}_{1}(t)+\partial_{x}H^{x_{2}(t)}_{0,1}(x)\dot{x}_{2}(t)\right]g(t,x)\,dx.$

Since from the definition of the function $U$, we can deduce that

$\displaystyle\left|\ddot{U}\left(H^{x_{2}(t)}_{0,1}(x)+H^{x_{1}(t)}_{-1,0}(x)\right)-\ddot{U}\left(H^{x_{1}(t)}_{-1,0}(x)\right)\right|=O\Big{(}\left|H^{x_{1}(t)}_{-1,0}(x)H^{x_{2}(t)}_{0,1}(x)\right|+\left|H^{x_{2}(t)}_{0,1}(x)\right|^{2}\Big{)},$
$\displaystyle|\ddot{U}\left(H^{x_{2}(t)}_{0,1}(x)+H^{x_{1}(t)}_{-1,0}(x)\right)-\ddot{U}\left(H^{x_{2}(t)}_{0,1}(x)\right)|=O\Big{(}\left|H^{x_{1}(t)}_{-1,0}(x)H^{x_{2}(t)}_{0,1}(x)\right|+\left|H^{x_{1}(t)}_{-1,0}(x)\right|^{2}\Big{)},$

we obtain from Lemma 2.3 and Cauchy-Schwarz Inequality that

$\displaystyle\left|\int_{\mathbb{R}}\left[\ddot{U}\left(H^{x_{2}(t)}_{0,1}(x)\right)-\ddot{U}\left(H^{x_{2}(t)}_{0,1}(x)+H^{x_{1}(t)}_{-1,0}(x)\right)\right]\partial_{x}H^{x_{2}(t)}_{0,1}(x)g(t,x)\,dx\right|=O\Big{(}\left\lVert\overrightarrow{g(t)}\right\rVert
e^{-\sqrt{2}z(t)}\Big{)},$
$\displaystyle\left|\int_{\mathbb{R}}\left[\ddot{U}\left(H^{x_{1}(t)}_{-1,0}(x)\right)-\ddot{U}\left(H^{x_{2}(t)}_{0,1}(x)+H^{x_{1}(t)}_{-1,0}(x)\right)\right]\partial_{x}H^{x_{1}(t)}_{-1,0}(x)g(t,x)\,dx\right|=O\Big{(}\left\lVert\overrightarrow{g(t)}\right\rVert
e^{-\sqrt{2}z(t)}\Big{)}.$

In conclusion, we obtain from the identity satisfied by $\frac{dF_{2}(t)}{dt}$
that

$\frac{dF_{2}(t)}{dt}=-2\int_{\mathbb{R}}\partial_{t}g(t,x)\left[\dot{U}\left(H^{x_{1}(t)}_{-1,0}(x)\right)+\dot{U}\left(H^{x_{2}(t)}_{0,1}(x)\right)-\dot{U}\left(H^{x_{2}(t)}_{0,1}(x)+H^{x_{1}(t)}_{-1,0}(x)\right)\right]\,dx+O(\delta(t)).$
(158)

Step 3.(The derivative of $F_{3}(t).$) From the definition of $F_{3}(t),$ we
obtain that

$\frac{dF_{3}(t)}{dt}=2\int_{\mathbb{R}}\partial_{t}g(t,x)\big{[}\dot{x}_{1}(t)^{2}\partial^{2}_{x}H^{x_{1}(t)}_{-1,0}(x)+\dot{x}_{2}(t)^{2}\partial^{2}_{x}H^{x_{2}(t)}_{0,1}(x)\big{]}\,dx\\\
-2\int_{\mathbb{R}}g(t,x)\big{[}\dot{x}_{1}(t)^{3}\partial^{3}_{x}H^{x_{1}(t)}_{-1,0}(x)+\dot{x}_{2}(t)^{3}\partial^{3}_{x}H^{x_{2}(t)}_{0,1}(x)\big{]}\,dx+4\int_{\mathbb{R}}g(t,x)\big{[}\dot{x}_{1}(t)\ddot{x}_{1}(t)\partial^{2}_{x}H^{x_{1}(t)}_{-1,0}(x)+\dot{x}_{2}(t)\ddot{x}_{2}(t)\partial^{2}_{x}H^{x_{2}(t)}_{0,1}(x)\big{]}\,dx,$

which can be rewritten as

$\frac{dF_{3}(t)}{dt}=2\int_{\mathbb{R}}\partial_{t}g(t,x)\big{[}\dot{x}_{1}(t)^{2}\partial^{2}_{x}H^{x_{1}(t)}_{-1,0}(x)+\dot{x}_{2}(t)^{2}\partial^{2}_{x}H^{x_{2}(t)}_{0,1}(x)\big{]}\,dx-2\int_{\mathbb{R}}g(t,x)\big{[}\dot{x}_{1}(t)^{3}\partial^{3}_{x}H^{x_{1}(t)}_{-1,0}(x)+\dot{x}_{2}(t)^{3}\partial^{3}_{x}H^{x_{2}(t)}_{0,1}(x)\big{]}\,dx\\\
+O(\delta(t)).$ (159)

Step 4.(Sum of $\frac{dF_{1}}{dt},\frac{dF_{2}}{dt},\frac{dF_{3}}{dt}.$) If we
sum the estimates (157), (158) and (159), we obtain that

$\sum_{i=1}^{3}\frac{dF_{i}(t)}{dt}=-2\int_{\mathbb{R}}\Big{[}\dot{U}\left(H^{x_{2}(t)}_{0,1}(x)+H^{x_{1}(t)}_{-1,0}(x)+g(t,x)\right)-\dot{U}\left(H^{x_{2}(t)}_{0,1}(x)+H^{x_{1}(t)}_{-1,0}(x)\right)\\\
-\ddot{U}\left(H^{x_{2}(t)}_{0,1}(x)+H^{x_{1}(t)}_{-1,0}(x)\right)g(t,x)\Big{]}\partial_{t}g(t,x)\,dx\\\
-\int_{\mathbb{R}}\left[\dot{x}_{1}(t)\partial_{x}H^{x_{1}(t)}_{-1,0}(x)+\dot{x}_{2}(t)\partial_{x}H^{x_{2}(t)}_{0,1}(x)\right]U^{(3)}\left(H^{x_{2}(t)}_{0,1}(x)+H^{x_{1}(t)}_{-1,0}(x)\right)g(t,x)^{2}\,dx\\\
-2\int_{\mathbb{R}}g(t,x)\left[\dot{x}_{1}(t)^{3}\partial^{3}_{x}H^{x_{1}(t)}_{-1,0}(x)+\dot{x}_{2}(t)^{3}\partial^{3}_{x}H^{x_{2}(t)}_{0,1}(x)\right]\,dx+O(\delta(t)).$
(160)

More precisely, from Taylor’s Expansion Theorem and since
$\left\lVert\overrightarrow{g(t)}\right\rVert^{4}\leq\delta(t),$

$\sum_{i=1}^{3}\frac{dF_{i}(t)}{dt}=-\int_{\mathbb{R}}\left[U^{(3)}\left(H^{x_{2}(t)}_{0,1}(x)+H^{x_{1}(t)}_{-1,0}(x)\right)g(t,x)^{2}\right]\partial_{t}g(t,x)\,dx\\\
-\int_{\mathbb{R}}\left[\dot{x}_{1}(t)\partial_{x}H^{x_{1}(t)}_{-1,0}(x)+\dot{x}_{2}(t)\partial_{x}H^{x_{2}(t)}_{0,1}(x)\right]U^{(3)}\left(H^{x_{2}(t)}_{0,1}(x)+H^{x_{1}(t)}_{-1,0}(x)\right)g(t,x)^{2}\,dx\\\
-2\int_{\mathbb{R}}g(t,x)\left[\dot{x}_{1}(t)^{3}\partial^{3}_{x}H^{x_{1}(t)}_{-1,0}(x)+\dot{x}_{2}(t)^{3}\partial^{3}_{x}H^{x_{2}(t)}_{0,1}(x)\right]\,dx+O(\delta(t)).$
(161)

Step 5.(The derivative of $F_{4}(t).$) The computation of the derivative of
$F_{4}(t)$ will be more careful, since the motivation for the addition of this
term is to cancel with the expression

$-\int_{\mathbb{R}}\left[\dot{x}_{1}(t)\partial_{x}H^{x_{1}(t)}_{-1,0}(x)+\dot{x}_{2}(t)\partial_{x}H^{x_{2}(t)}_{0,1}(x)\right]U^{(3)}\left(H^{x_{2}(t)}_{0,1}(x)+H^{x_{1}(t)}_{-1,0}(x)\right)g(t,x)^{2}\,dx$

of (161). The construction of functional $F_{4}(t)$ is based on the momentum
correction term of Lemma 4.2 of [19]. To estimate $\frac{dF_{4}(t)}{dt}$ with
precision of $O(\delta(t))$, it is just necessary to study the time derivative
of

$2\int_{\mathbb{R}}\partial_{t}g(t,x)\partial_{x}g(t,x)\dot{x}_{1}(t)\omega_{1}(t,x)\,dx,$
(162)

since the estimate of the other term in $F_{4}(t)$ is completely analogous.
First, we have the identity

$\frac{d}{dt}\Big{[}2\int_{\mathbb{R}}\partial_{t}g(t,x)\partial_{x}g(t,x)\dot{x}_{1}(t)\omega_{1}(t,x)\,dx\Big{]}=2\ddot{x}_{1}(t)\int_{\mathbb{R}}\omega_{1}(t,x)\partial_{t}g(t,x)\partial_{x}g(t,x)\,dx\\\
+2\dot{x}_{1}(t)\int_{\mathbb{R}}\omega_{1}(t,x)\partial^{2}_{t}g(t,x)\partial_{x}g(t,x)\,dx+2\dot{x}_{1}(t)\int_{\mathbb{R}}\partial_{t}\omega_{1}(t,x)\partial_{t}g(t,x)\partial_{x}g(t,x)\,dx+2\dot{x}_{1}(t)\int_{\mathbb{R}}\omega_{1}(t,x)\partial^{2}_{t,x}g(t,x)\partial_{t}g(t,x)\,dx.$
(163)

From the definition of
$\omega_{1}(t,x)=\omega\big{(}\frac{x-x_{1}(t)}{x_{2}(t)-x_{1}(t)}\big{)}$, we
have

$\partial_{t}\omega_{1}(t,x)=\dot{\omega}\Big{(}\frac{x-x_{1}(t)}{x_{2}(t)-x_{1}(t)}\Big{)}\Big{(}\frac{-\dot{x}_{1}(t)z(t)-\dot{z}(t)(x-x_{1}(t))}{z(t)^{2}}\Big{)}.$
(164)

Since in the support of $\dot{\omega}(x)$ is contained in the set
$\frac{3}{4}\leq x\leq\frac{4}{5}$, we obtain the following estimate:

$2\dot{x}_{1}(t)\int_{\mathbb{R}}\partial_{t}\omega_{1}(t,x)\partial_{t}g(t,x)\partial_{x}g(t,x)\,dx=O\Big{(}\max_{j\in\\{1,2\\}}\frac{|\dot{x}_{j}(t)|}{z(t)}\left\lVert\overrightarrow{g(t)}\right\rVert^{2}\Big{)}=O(\delta(t)).$
(165)

Clearly from integration by parts, we deduce that

$2\dot{x}_{1}(t)\int_{\mathbb{R}}\omega_{1}(t,x)\partial^{2}_{t,x}g(t,x)\partial_{t}g(t,x)\,dx=O\Big{(}\max_{j\in\\{1,2\\}}\frac{|\dot{x}_{j}(t)|}{z(t)}\left\lVert\overrightarrow{g(t)}\right\rVert^{2}\Big{)}=O(\delta(t)).$
(166)

Also, we have

$2\ddot{x}_{1}(t)\int_{\mathbb{R}}\omega_{1}(t,x)\partial_{t}g(t,x)\partial_{x}g(t,x)\,dx=O\Big{(}\max_{j\in\\{1,2\\}}|\ddot{x}_{j}(t)|\left\lVert\overrightarrow{g(t)}\right\rVert^{2}\Big{)}=O(\delta(t)).$
(167)

So, to estimate the time derivative of (162) with precision $O(\delta(t)),$ it
is enough to estimate

$2\dot{x}_{1}(t)\int_{\mathbb{R}}\omega_{1}(t,x)\partial^{2}_{t}g(t,x)\partial_{x}g(t,x)\,dx.$

We have that

$2\dot{x}_{1}(t)\int_{\mathbb{R}}\omega_{1}(t,x)\partial^{2}_{t}g(t,x)\partial_{x}g(t,x)\,dx=2\dot{x}_{1}(t)\int_{\mathbb{R}}\omega_{1}(t,x)\partial^{2}_{x}g(t,x)\partial_{x}g(t,x)\,dx\\\
-2\dot{x}_{1}(t)\int_{\mathbb{R}}\omega_{1}(t,x)\ddot{U}\left(H^{x_{1}(t)}_{-1,0}(x)+H^{x_{2}(t)}_{0,1}(x)\right)g(t,x)\partial_{x}g(t,x)\,dx\\\
+2\dot{x}_{1}(t)\int_{\mathbb{R}}\omega_{1}(t,x)\left[\partial^{2}_{t}g(t,x)-\partial^{2}_{x}g(t,x)+\ddot{U}\left(H^{x_{1}(t)}_{-1,0}(x)+H^{x_{2}(t)}_{0,1}(x)\right)g(t,x)\right]\partial_{x}g(t,x)\,dx.$
(168)

From integration by parts, the first term of the equation (168) satisfies

$2\dot{x}_{1}(t)\int_{\mathbb{R}}\omega_{1}(t,x)\partial^{2}_{x}g(t,x)\partial_{x}g(t,x)\,dx=O\Big{(}\max_{j\in\\{1,2\\}}\frac{|\dot{x}_{j}(t)|}{z(t)}\left\lVert\overrightarrow{g(t)}\right\rVert^{2}\Big{)}=O(\delta(t)).$
(169)

From Taylor’s Expansion Theorem, we have that

$\left\lVert\dot{U}\left(H^{x_{2}(t)}_{0,1}+H^{x_{1}(t)}_{-1,0}+g(t)\right)-\dot{U}\left(H^{x_{2}(t)}_{0,1}+H^{x_{1}(t)}_{-1,0}\right)-\ddot{U}\left(H^{x_{2}(t)}_{0,1}+H^{x_{1}(t)}_{-1,0}\right)g(t)-U^{(3)}\left(H^{x_{2}(t)}_{0,1}+H^{x_{1}(t)}_{-1,0}\right)\frac{g(t)^{2}}{2}\right\rVert_{L^{2}(\mathbb{R})}\\\
=O\Big{(}\left\lVert\overrightarrow{g(t)}\right\rVert^{3}\Big{)}.$ (170)

Also, we have verified the identity

$\dot{U}(\phi)+\dot{U}(\theta)-\dot{U}(\phi+\theta)=24\phi\theta(\phi+\theta)-6\Big{(}\sum_{j=1}^{4}\begin{pmatrix}5\\\
j\end{pmatrix}\phi^{j}\theta^{5-j}\Big{)},$

which clearly with the inequalities (D1), (D2) and Lemma 2.3 imply the
estimate

$\left\lVert\dot{U}\left(H^{x_{2}(t)}_{0,1}\right)+\dot{U}\left(H^{x_{1}(t)}_{-1,0}\right)-\dot{U}\left(H^{x_{2}(t)}_{0,1}+H^{x_{1}(t)}_{-1,0}\right)\right\rVert_{L^{2}(\mathbb{R})}=O(e^{-\sqrt{2}z(t)}).$
(171)

Finally, is not difficult to verify that

$\left\lVert-\dot{x}_{1}(t)^{2}\partial^{2}_{x}H^{x_{1}(t)}_{-1,0}-\dot{x}_{2}(t)^{2}\partial^{2}_{x}H^{x_{2}(t)}_{0,1}+\ddot{x}_{1}(t)\partial_{x}H^{x_{1}(t)}_{-1,0}+\ddot{x}_{2}(t)\partial_{x}H^{x_{2}(t)}_{0,1}\right\rVert_{L^{2}(\mathbb{R})}=O\big{(}\max_{j\in\\{1,2\\}}|\dot{x}_{j}(t)|^{2}+|\ddot{x}_{j}(t)|\big{)}.$
(172)

Then, from estimates (170), (171) and (172) and the Partial Differential
Equation (155) satisfied by $g(t,x)$, we can obtain the estimate

$2\dot{x}_{1}(t)\int_{\mathbb{R}}\omega_{1}(t,x)\left[\partial^{2}_{t}g(t,x)-\partial^{2}_{x}g(t,x)+\ddot{U}\left(H^{x_{1}(t)}_{-1,0}(x)+H^{x_{2}(t)}_{0,1}(x)\right)g(t,x)\right]\partial_{x}g(t,x)\,dx=\\\
-\dot{x}_{1}(t)\int_{\mathbb{R}}\omega_{1}(t,x)U^{(3)}\left(H^{x_{1}(t)}_{-1,0}(x)+H^{x_{2}(t)}_{0,1}(x)\right)g(t,x)^{2}\partial_{x}g(t,x)\,dx-2\dot{x}_{1}(t)^{3}\int_{\mathbb{R}}\partial^{2}_{x}H^{x_{1}(t)}_{-1,0}(x)\partial_{x}g(t,x)\,dx\\\
-2\dot{x}_{1}(t)^{3}\int_{\mathbb{R}}(\omega_{1}(t,x)-1)\partial^{2}_{x}H^{x_{1}(t)}_{-1,0}(x)\partial_{x}g(t,x)\,dx-2\dot{x}_{1}(t)\dot{x}_{2}(t)^{2}\int_{\mathbb{R}}\omega_{1}(t,x)\partial^{2}_{x}H^{x_{2}(t)}_{0,1}(x)\partial_{x}g(t,x)\,dx\\\
+O\Big{(}\max_{j\in\\{1,2\\}}|\ddot{x}_{j}(t)\dot{x}_{j}(t)|\left\lVert\overrightarrow{g(t)}\right\rVert+e^{-\sqrt{2}z(t)}\max_{j\in\\{1,2\\}}|\dot{x}_{j}(t)|\left\lVert\overrightarrow{g(t)}\right\rVert+\left\lVert\overrightarrow{g(t)}\right\rVert^{4}\max_{j\in\\{1,2\\}}|\dot{x}_{j}(t)|\Big{)},$

which, by integration by parts and by Cauchy-Schwarz inequality using the
estimate (125) for $\omega_{1},$ we obtain that

$2\dot{x}_{1}(t)\int_{\mathbb{R}}\omega_{1}(t,x)\left[\partial^{2}_{t}g(t,x)-\partial^{2}_{x}g(t,x)+\ddot{U}\left(H^{x_{1}(t)}_{-1,0}(x)+H^{x_{2}(t)}_{0,1}(x)\right)g(t,x)\right]\partial_{x}g(t,x)\,dx=\\\
\frac{\dot{x}_{1}(t)}{3}\int_{\mathbb{R}}\omega_{1}(t)U^{(4)}\left(H^{x_{1}(t)}_{-1,0}+H^{x_{2}(t)}_{0,1}\right)\left[\partial_{x}H^{x_{1}(t)}_{-1,0}+\partial_{x}H^{x_{2}(t)}_{0,1}\right]g(t)^{3}\,dx+O\Big{(}\max_{j\in\\{1,2\\}}\frac{|\dot{x}_{j}(t)|}{z(t)}\left\lVert\overrightarrow{g(t)}\right\rVert^{3}\Big{)}\\\
-2\dot{x}_{1}(t)^{3}\int_{\mathbb{R}}\partial^{2}_{x}H^{x_{1}(t)}_{-1,0}(x)\partial_{x}g(t,x)\,dx+O\Big{(}\max_{j\in\\{1,2\\}}|\dot{x}_{j}(t)|^{3}e^{-\frac{\sqrt{2}z(t)}{5}}\left\lVert\overrightarrow{g(t)}\right\rVert\Big{)}+O(\delta(t)).$
(173)

Now, to finish the estimate of
$2\dot{x}_{1}(t)\int_{\mathbb{R}}\omega_{1}(t,x)\partial^{2}_{t}g(t,x)\partial_{x}g(t,x)\,dx,$
it remains to study the integral given by

$-2\dot{x}_{1}(t)\int_{\mathbb{R}}\omega_{1}(t,x)\ddot{U}\left(H^{x_{1}(t)}_{-1,0}(x)+H^{x_{2}(t)}_{0,1}(x)\right)g(t,x)\partial_{x}g(t,x)\,dx,$
(174)

which by integration by parts is equal to

$\dot{x}_{1}(t)\int_{\mathbb{R}}\omega_{1}(t,x)U^{(3)}\left(H^{x_{1}(t)}_{-1,0}(x)+H^{x_{2}(t)}_{0,1}(x)\right)\big{[}\partial_{x}H^{x_{1}(t)}_{-1,0}(x)+\partial_{x}H^{x_{2}(t)}_{0,1}(x)\big{]}g(t,x)^{2}\,dx+O(\delta(t)).$
(175)

Since the support of $\omega_{1}(t,x)$ is included in
$\\{x|\,(x-x_{2}(t))\leq-\frac{z(t)}{5}\\}$ and the support of
$1-\omega_{1}(t,x)$ is included in
$\\{x|\,(x-x_{1}(t))\geq\frac{3z(t)}{4}\\},$ from the exponential decay
properties of the kink solutions in (D1), (D2), (D3), (D4) we obtain the
estimates

$\displaystyle\Big{|}\dot{x}_{1}(t)\int_{\mathbb{R}}(\omega_{1}(t,x)-1)U^{(3)}\left(H^{x_{1}(t)}_{-1,0}(x)+H^{x_{2}(t)}_{0,1}(x)\right)(\partial_{x}H^{x_{1}(t)}_{-1,0}(x))g(t,x)^{2}\,dx\Big{|}=O(\delta(t)),$
(176)
$\displaystyle\Big{|}\dot{x}_{2}(t)\int_{\mathbb{R}}\omega_{1}(t,x)U^{(3)}\left(H^{x_{1}(t)}_{-1,0}(x)+H^{x_{2}(t)}_{0,1}(x)\right)\partial_{x}H^{x_{2}(t)}_{0,1}(x)g(t,x)^{2}\,dx\Big{|}=O(\delta(t)),$
(177)
$\displaystyle\Big{|}\frac{1}{3}\dot{x}_{1}(t)\int_{\mathbb{R}}(1-\omega_{1}(t))U^{(4)}(H^{x_{1}(t)}_{-1,0}+H^{x_{2}(t)}_{0,1})\partial_{x}H^{x_{1}(t)}_{-1,0}g(t)^{3}\,dt\Big{|}=O(\delta(t)),$
(178)
$\displaystyle\Big{|}\frac{1}{3}\dot{x}_{2}(t)\int_{\mathbb{R}}(\omega_{1}(t))U^{(4)}\left(H^{x_{1}(t)}_{-1,0}+H^{x_{2}(t)}_{0,1}\right)\partial_{x}H^{x_{2}(t)}_{0,1}g(t)^{3}\,dt\Big{|}=O(\delta(t)).$
(179)

In conclusion, we obtain that the estimates (176), (177) imply the following
estimate

$-2\dot{x}_{1}(t)\int_{\mathbb{R}}\omega_{1}(t,x)\ddot{U}\left(H^{x_{1}(t)}_{-1,0}(x)+H^{x_{2}(t)}_{0,1}(x)\right)g(t,x)\partial_{x}g(t,x)\,dx=\int_{\mathbb{R}}\dot{x}_{1}(t)\partial_{x}H^{x_{1}(t)}_{-1,0}(x)U^{(3)}\left(H^{x_{2}(t)}_{0,1}+H^{x_{1}(t)}_{-1,0}\right)g(t)^{2}\,dx\\\
+O(\delta(t)).$ (180)

Then, the estimates (168), (173), (178), (179) and (180) imply that

$\displaystyle
2\frac{d}{dt}\left(\int_{\mathbb{R}}\partial_{t}g(t,x)\partial_{x}g(t,x)\dot{x}_{1}(t)\omega_{1}(t,x)\,dx\right)=\frac{1}{3}\int_{\mathbb{R}}U^{(4)}\left(H^{x_{1}(t)}_{-1,0}+H^{x_{2}(t)}_{0,1}\right)\big{(}\dot{x}_{1}(t)\partial_{x}H^{x_{1}(t)}_{-1,0}\big{)}g(t)^{3}\,dx$
$\displaystyle+\int_{\mathbb{R}}\big{(}\dot{x}_{1}(t)\partial_{x}H^{x_{1}(t)}_{-1,0})U^{(3)}(H^{x_{2}(t)}_{0,1}+H^{x_{1}(t)}_{-1,0})g(t)^{2}\,dx-2\dot{x}_{1}(t)^{3}\int_{\mathbb{R}}\partial^{2}_{x}H^{x_{1}(t)}_{-1,0}(x)\partial_{x}g(t,x)\,dx.+O(\delta(t)).$

By an analogous argument, we deduce that

$\displaystyle
2\frac{d}{dt}\left(\int_{\mathbb{R}}\partial_{t}g(t,x)\partial_{x}g(t,x)\dot{x}_{2}(t)(1-\omega_{1}(t,x))\,dx\right)=\frac{1}{3}\int_{\mathbb{R}}U^{(4)}\left(H^{x_{1}(t)}_{-1,0}+H^{x_{2}(t)}_{0,1}\right)\big{(}\dot{x}_{2}(t)\partial_{x}H^{x_{2}(t)}_{0,1}\big{)}g(t)^{3}\,dx$
$\displaystyle+\int_{\mathbb{R}}\big{(}\dot{x}_{2}(t)\partial_{x}H^{x_{2}(t)}_{0,1})U^{(3)}\left(H^{x_{2}(t)}_{0,1}+H^{x_{1}(t)}_{-1,0}\right)g(t)^{2}\,dx-2\dot{x}_{2}(t)^{3}\int_{\mathbb{R}}\partial^{2}_{x}H^{x_{2}(t)}_{0,1}(x)\partial_{x}g(t,x)\,dx+O(\delta(t)).$

In conclusion, we have that

$\frac{dF_{4}(t)}{dt}=\int_{\mathbb{R}}\left[\dot{x}_{1}(t)\partial_{x}H^{x_{1}(t)}_{-1,0}+\dot{x}_{2}(t)\partial_{x}H^{x_{2}(t)}_{0,1}\right]U^{(3)}\left(H^{x_{2}(t)}_{0,1}+H^{x_{1}(t)}_{-1,0}\right)g(t)^{2}\,dx-2\dot{x}_{2}(t)^{3}\int_{\mathbb{R}}\partial^{2}_{x}H^{x_{2}(t)}_{0,1}(x)\partial_{x}g(t,x)\,dx\\\
+\frac{1}{3}\int_{\mathbb{R}}U^{(4)}\left(H^{x_{1}(t)}_{-1,0}+H^{x_{2}(t)}_{0,1}\right)\left[\dot{x}_{1}(t)\partial_{x}H^{x_{1}(t)}_{-1,0}+\dot{x}_{2}(t)\partial_{x}H^{x_{2}(t)}_{0,1}\right]g(t)^{3}\,dx-2\dot{x}_{1}(t)^{3}\int_{\mathbb{R}}\partial^{2}_{x}H^{x_{1}(t)}_{-1,0}(x)\partial_{x}g(t,x)\,dx+O(\delta(t)).$
(181)

Step 6.(The derivative of $F_{5}(t).$) We have that

$\frac{dF_{5}(t)}{dt}=\int_{\mathbb{R}}U^{(3)}\left(H^{x_{1}(t)}_{-1,0}+H^{x_{2}(t)}_{0,1}\right)g(t)^{2}\partial_{t}g(t)\,dx-\frac{1}{3}\int_{\mathbb{R}}U^{(4)}\left(H^{x_{1}(t)}_{-1,0}+H^{x_{2}(t)}_{0,1}\right)\big{(}\dot{x}_{1}(t)\partial_{x}H^{x_{1}(t)}_{-1,0}+\dot{x}_{2}(t)\partial_{x}H^{x_{2}(t)}_{0,1}\big{)}g(t)^{3}\,dx.$
(182)

Step 7.(Conclusion of estimate of $|\dot{F}(t)|$) From the identities (182)
and (181), we obtain that

$\frac{dF_{4}(t)}{dt}+\frac{dF_{5}(t)}{dt}=\int_{\mathbb{R}}\big{(}\dot{x}_{1}(t)\partial_{x}H^{x_{1}(t)}_{-1,0}+\dot{x}_{2}(t)\partial_{x}H^{x_{2}(t)}_{0,1}\big{)}U^{(3)}\left(H^{x_{2}(t)}_{0,1}+H^{x_{1}(t)}_{-1,0}\right)g(t)^{2}\,dx-2\dot{x}_{1}(t)^{3}\int_{\mathbb{R}}\partial^{2}_{x}H^{x_{1}(t)}_{-1,0}(x)\partial_{x}g(t,x)\,dx\\\
-2\dot{x}_{2}(t)^{3}\int_{\mathbb{R}}\partial^{2}_{x}H^{x_{2}(t)}_{0,1}(x)\partial_{x}g(t,x)\,dx+\int_{\mathbb{R}}U^{(3)}\left(H^{x_{1}(t)}_{-1,0}+H^{x_{2}(t)}_{0,1}\right)g(t)^{2}\partial_{t}g(t)\,dx+O(\delta(t)).$
(183)

Then, the sum of identities (161) and (183) implies
$\sum_{i=1}^{5}\frac{dF_{i}(t)}{dt}=O(\delta(t)),$ this finishes the proof of
inequality $|\dot{F}(t)|=O(\delta(t)).\\\ $ Proof of
$F(t)+A_{1}\epsilon^{2}\geq A_{2}\epsilon^{2}.$ The Coercitivity Lemma implies
that $\exists\,c>0,$ such that $F_{1}(t)\geq
c\left\lVert\overrightarrow{g(t)}\right\rVert^{2}.$ Also, from Theorem 2.9, we
have the global estimate

$\max_{j\in\\{1,2\\}}|\dot{x}_{j}(t)|^{2}+|\ddot{x}_{j}(t)|+e^{-\sqrt{2}z(t)}+\left\lVert\overrightarrow{g(t)}\right\rVert^{2}=O(\epsilon)$
(184)

that implies that
$|F_{3}(t)|=O\Big{(}\left\lVert\overrightarrow{g(t)}\right\rVert\epsilon\Big{)},\,|F_{4}(t)|=O\Big{(}\left\lVert\overrightarrow{g(t)}\right\rVert^{2}\epsilon^{\frac{1}{2}}\Big{)},\,|F_{5}(t)|=O\Big{(}\left\lVert\overrightarrow{g(t)}\right\rVert^{2}\epsilon^{\frac{1}{2}}\Big{)}.$
Also, since

$\left|U\left(H^{x_{1}(t)}_{-1,0}(x)\right)+\dot{U}\left(H^{x_{2}(t)}_{0,1}(x)\right)-\dot{U}\left(H^{x_{2}(t)}_{0,1}(x)+H^{x_{1}(t)}_{-1,0}(x)\right)\right|=O\Big{(}\Big{|}H^{x_{1}(t)}_{-1,0}(x)H^{x_{2}(t)}_{0,1}(x)\big{[}H^{x_{2}(t)}_{0,1}(x)+H^{x_{1}(t)}_{-1,0}(x)\big{]}\Big{|}\Big{)},$

the Lemma 2.3 and Cauchy-Schwarz Inequality imply that

$|F_{2}(t)|=O\Big{(}\left\lVert\overrightarrow{g(t)}\right\rVert
e^{-\sqrt{2}z(t)}\Big{)}.$

Then, the conclusion of $F(t)+A_{1}\epsilon^{2}\geq
A_{2}\left\lVert\overrightarrow{g(t)}\right\rVert^{2}$ follows from Young
Inequality for $\epsilon$ small enough. ∎

######  Remark 4.3.

In the proof of Theorem 4.1, from Theorem 2.9 we have
$|F_{2}(t)|+|F_{3}(t)|=O\left(\left\lVert\overrightarrow{g(t)}\right\rVert\epsilon\right).$
Since
$|F_{4}(t)|+|F_{5}(t)|=O\left(\left\lVert\overrightarrow{g(t)}\right\rVert^{2}\epsilon^{\frac{1}{2}}\right)$
and $|F_{1}(t)|\lesssim\left\lVert\overrightarrow{g(t)}\right\rVert^{2},$ then
Young Inequality implies that

$|F(t)|\lesssim\left\lVert\overrightarrow{g(t)}\right\rVert^{2}+\epsilon^{2}.$

######  Remark 4.4 (General Energy Estimate).

For any $0<\theta,\gamma<1,$ we can create a smooth cut function
$0\leq\chi(x)\leq 1$ such that

$\chi(x)=\begin{cases}0,\text{ if $x\leq\theta(1-\gamma),$}\\\ 1,\text{if
$x\geq\theta.$}\end{cases}$

We define

$\chi_{0}(t,x)=\chi\left(\frac{x-x_{1}(t)}{x_{2}(t)-x_{1}(t)}\right).$

If we consider the following functional

$L(t)=\big{\langle}D^{2}E(H^{x_{2}(t)}_{0,1}+H^{x_{1}(t)}_{-1,0})\overrightarrow{g(t)},\,\overrightarrow{g(t)}\big{\rangle}_{L^{2}\times
L^{2}}+2\int_{\mathbb{R}}\partial_{t}g(t,x)\partial_{x}g(t,x)\Big{[}\dot{x}_{1}(t)\chi_{0}(t,x)+\dot{x}_{2}(t)(1-\chi_{0}(t,x))\Big{]}\,dx\\\
-2\int_{\mathbb{R}}g(t,x)\Big{(}\dot{U}(H^{x_{1}(t)}_{-1,0}(x))+\dot{U}(H^{x_{2}(t)}_{0,1}(x))-\dot{U}(H^{x_{2}(t)}_{0,1}(x)+H^{x_{1}(t)}_{-1,0}(x))\Big{)}\,dx\\\
+2\int_{\mathbb{R}}g(t,x)\Big{[}(\dot{x}_{1}(t))^{2}\partial^{2}_{x}H^{x_{1}(t)}_{-1,0}(x)+(\dot{x}_{2}(t))^{2}\partial^{2}_{x}H^{x_{2}(t)}_{0,1}(x)\Big{]}\,dx+\frac{1}{3}\int_{\mathbb{R}}U^{(3)}(H^{x_{2}(t)}_{0,1}(x)+H^{x_{1}(t)}_{-1,0}(x))g(t,x)^{3}\,dx,$
(185)

then, by a similar proof to the Theorem 4.1, we obtain that if $0<\epsilon\ll
1$ and

$\delta_{1}(t)=\delta(t)+\max_{j\in\\{1,2\\}}|\dot{x}_{j}(t)|^{3}\max(e^{-\sqrt{2}z(t)(1-\theta)},e^{-\sqrt{2}z(t)\theta(1-\gamma)})\left\lVert\overrightarrow{g(t)}\right\rVert-\max_{j\in\\{1,2\\}}|\dot{x}_{j}(t)|^{3}e^{-\frac{\sqrt{2}}{5}z(t)}\left\lVert\overrightarrow{g(t)}\right\rVert,$
(186)

then there are positive constants $A_{1},\,A_{2}>0$ such that

$|\dot{L}(t)|=O(\delta_{1}(t)),\,L(t)+A_{1}\epsilon^{2}\geq
A_{2}\epsilon^{2}.$

Our first application of Theorem 4.1 is to estimate the size of the remainder
$\left\lVert\overrightarrow{g(t)}\right\rVert$ during a long time interval.
More precisely, this corresponds to the following theorem, which is a weaker
version of Theorem 1.7.

######  Theorem 4.5.

There is $\delta>0,$ such that if $0<\epsilon<\delta$ enough,
$(\phi(0),\partial_{t}\phi(0))\in S\times L^{2}(\mathbb{R})$ and
$E_{total}(\phi(0),\partial_{t}\phi(0))=2E_{pot}(H_{0,1})+\epsilon,$ then
there are $x_{2},x_{1}\in C^{2}(\mathbb{R})$ functions such that the unique
global time solution of (1) is given, for

$\phi(t)=H_{0,1}(x-x_{2}(t))+H_{-1,0}(x-x_{1}(t))+g(t),$ (187)