# Scattering between orthogonally wobbling kinks

A. Alonso-Izquierdo<EMAIL_ADDRESS>D. Miguélez-Caballero
<EMAIL_ADDRESS>L.M. Nieto<EMAIL_ADDRESS>Departamento
de Matématica Aplicada, Universidad de Salamanca, Casas del Parque 2, 37008,
Salamanca, Spain  IUFFyM, Universidad de Salamanca, Plaza de la Merced 1,
37008, Salamanca, Spain  Departamento de Física Teórica, Atómica y Óptica and
Laboratory for Disruptive Interdisciplinary Science (LaDIS),
Universidad de Valladolid, 47011 Valladolid, Spain

###### Abstract

The resonant energy transfer mechanism, responsible for the presence of
fractal patterns in the velocity diagrams of kink-antikink scattering, is
analyzed for a family of two-component scalar field theory models, in which
the kink solutions have two shape modes (one longitudinal and one orthogonal
to the kink orbit), in addition to the zero mode, and in which energy
redistribution can occur among these three discrete modes. We investigate the
scattering between wobbling kinks whose orthogonal shape mode is initially
excited, examining how the final velocities, amplitudes, and frequencies
depend on the initial excitation amplitude. The differences that this model
presents with respect to the $\phi^{4}$ model and its novel properties are
highlighted. This analysis sheds light on the intricate dynamics that arise
from the interplay between multiple degrees of freedom in kink scattering
processes, offering insights distinct from those observed in simpler models.

###### keywords:

Wobbling kinks, kink-antikink scattering, longitudinal and orthogonal modes,
two-component scalar field theories.

##  1 Introduction

The investigation of kink scattering has garnered significant attention in
recent years due to its remarkable characteristics such as the presence of
fractal structures in the velocity diagrams showing the final velocity of the
scattered kinks as a function of the colliding velocity. Initially
investigated in seminal references [1, 2, 3, 4, 5], the collision dynamics
between kinks and antikinks in the $\phi^{4}$ model and deformed sine-Gordon
models unveiled two scattering channels: bion formation and kink reflection.
Bion formation occurs when kinks collide and bounce back repeatedly, emitting
radiation with each impact, while kink reflection involves a finite number of
collisions before the kinks move apart. These channels dominate for different
ranges of initial collision velocity, with bion formation prevalent at low
velocities and kink reflection at higher velocities. A particularly intriguing
phenomenon arises during the transition between these regimes, where bion
formation and kink reflection intertwine infinitely, giving rise to a fractal
structure within the velocity diagram. Understanding these intricate dynamics
has significant implications for various physical applications, shedding light
on nonlinear phenomena fostered by the presence of such topological defects
[6, 7, 8, 9, 10, 11, 12, 13].

The emergence of a fractal structure in the velocity diagram depicting kink
scattering within the $\phi^{4}$ model stems from the presence of an internal
vibrational mode, known as the shape mode, associated with kink solutions.
This mode, together with the zero mode (or translational mode), allows the
existence of the resonant energy transfer mechanism, which facilitates the
redistribution of energy between kinetic and vibrational modes during kink
collisions. Normally, in a scattering event, kinks and antikinks approach each
other and collide, transferring some of the kinetic energy to the shape mode.
This turns them into wobblers (vibrationally excited kinks) when they try to
separate. If the kinetic energy of the wobblers is not enough, they can get
closer and collide again. This cycle can continue indefinitely or cease after
a finite number of collisions. In the latter scenario, enough vibrational
energy is converted back to kinetic energy in the zero mode, enabling the
wobblers to separate. This intricate mechanism and its associated phenomena
have been widely investigated through various models, revealing profound
complexity [14, 15, 16, 17, 18, 19, 20]. It is evident that the natural
scenario in a physical system where topological defects can be found is that
they are vibrating (i.e., with their shape modes excited), either due to
collisions between them or due to the formation of topological defects after a
phase transition. Therefore, from a physical point of view, the most
interesting scenario lies in the investigation of scattering processes between
wobbling kinks. The analysis of scattering between wobblers within the
$\phi^{4}$ model has been widely discussed in previous works [21, 22]. Through
the numerical analysis of the scattering solutions derived from the Klein-
Gordon partial differential equation, these investigations aim to shed light
on the resonant energy transfer mechanism. It is worth noting that studies on
wobblers scattering in the double sine-Gordon model have also been carried out
in [23, 24], contributing to a broader understanding of similar phenomena
across different models [25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37].
The resonant energy transfer mechanism has been recently demonstrated to occur
in vortex scattering within the Abelian-Higgs model in (2+1)-dimensions,
leading to the emergence of fractal patterns [38]. A collective coordinate
model explaining this behavior is detailed in [39]. This discovery underscores
the significance of investigating topological defects in (1+1)-dimensions, as
many phenomena observed in this lower-dimensional context persist in higher
dimensions. Notably, the phenomenon of spectral walls observed in kink
dynamics [19] has also been identified in vortex scattering [40].

In the present work, our aim is to investigate the scattering between wobblers
within a one-parametric family of relativistic (1+1)-dimensional two-component
scalar field theories known as the MSTB model, which is a natural
generalization of the $\phi^{4}$ model in this context. This system arises as
a deformation of the $O(2)$ linear sigma model where the term breaking the
symmetry includes the coupling constant $\sigma$. This model has been the
focus of study by many researchers for decades [41, 42, 43, 44, 45, 46, 51,
50, 49, 47, 48]. The search for two-component kink solutions has been explored
in references [25, 41, 42, 43, 44, 45, 46], while the generalization of such
models is considered in [47, 48, 49]. These works have proved the existence of
topological kinks and two families of non-topological kinks for $\sigma<1$.
However, when $\sigma\geq 1$, only topological kinks persist. The
investigation of the quantum corrections to these kinks is discussed in [51,
50], while the scattering behavior of these unexcited kinks is analyzed in
[25]. The existence of wobbling kinks in this model was described in [52],
where it was shown that the simplest kink solutions that arise in this model
have two shape modes, which allows their vibration in two different channels:
one longitudinal and another orthogonal to the kink orbit. Through
perturbation theory, the radiation emission channels and the attenuation of
the shape mode amplitudes were analytically identified. Therefore, the next
natural step in studying the MSTB model is to investigate scattering between
wobblers. In particular, we focus on the investigation of the scattering of
wobbling kinks or wobblers emerging in the MSTB model in the regime
$\sigma>1$. Several reasons justify the interest in carrying out this study.
First, the MSTB model has been widely addressed in the literature, as
mentioned earlier. Exploring the dynamics of topological defects in this model
carries an implicit interest. Second, the study of scattering between wobblers
has been limited to a few models, all of which involve a single scalar field.
This restriction poses a limited scenario, exemplified by cases like the
$\phi^{4}$ model, in which the kink possesses only one shape mode, causing the
wobblers to vibrate longitudinally along their trajectory in internal space.
However, the MSTB model stands out for its unique features: the kink exhibits
two shape modes, allowing the wobblers to vibrate both longitudinally and
orthogonally to their trajectory. This distinctive feature adds complexity and
richness to the scattering dynamics in the MSTB model, warranting further
exploration. This constitutes the fundamental reason for this research. Note
that now the resonant energy transfer mechanism can be activated, leading to
energy redistribution among three discrete modes: the zero (or translational)
mode (responsible for the kink motion), and the longitudinal and orthogonal
shape modes (responsible for the kink vibration in each component of the
field). This expansion in the number of modes involved introduces a new layer
of complexity to the scattering dynamics, potentially yielding novel phenomena
not observed in models with fewer degrees of freedom.

The organization of this paper is as follows. Section 2 introduces the MSTB
model along with its kink solutions. The stability of these solutions and
their vibrational modes are described. Section 3 outlines the numerical setup
used for simulating collisions of wobbling kinks. Section 4 presents detailed
numerical results, analyzing the behavior of the fractal patterns observed in
the velocity diagrams resulting from the scattering between wobbling kinks,
and exploring their dependency on both the coupling constant $\sigma$ and the
initial excitation amplitude. By analyzing the final amplitudes and
frequencies of the shape modes, the behavior of the resonant energy transfer
mechanism in such models is examined. Finally, Section 5 provides the
conclusions of the paper.

##  2 The MSTB model: kink solutions, linear stability and shape modes

In this paper we deal with a two-component scalar field theory model in a
(1+1)-Minkowskian space-time, whose dynamics is governed by the Lagrangian
density

$\mathcal{L}=\frac{1}{2}\,\partial_{\mu}\phi\,\partial^{\mu}\phi+\frac{1}{2}\,\partial_{\mu}\psi\,\partial^{\mu}\psi-U(\phi,\psi),$
(1)

where the potential $U(\phi,\psi)$ is given by the fourth degree algebraic
expression

$U(\phi,\psi)=\frac{1}{2}(\phi^{2}+\psi^{2}-1)^{2}+\frac{\sigma^{2}}{2}\psi^{2}.$
(2)

In (1) and (2), $\phi$ and $\psi$ denote real scalar fields, and the Minkowski
metric is represented as $g_{\mu,\nu}=\text{diag}\\{1,-1\\}$. Note that the
potential term (2) introduces a coupling constant $\sigma^{2}$, which is a
real positive parameter. This family of models is usually referred to as the
MSTB model and has been extensively investigated in the literature, as
indicated in the Introduction. Our focus here lies in analyzing the
interaction between discrete modes (longitudinal and orthogonal) during
collisions between topological defects in this model. The main objective of
this paper is to understand how the mechanism of resonant energy transfer is
influenced when the kinks include an orthogonal vibration channel. The reason
for choosing the MSTB model for this study is twofold. The first is because it
is a natural generalization of the $\phi^{4}$ model, in which the energy
transfer mechanism between modes was first studied. In fact, this two-
component scalar field theory incorporates the $\phi^{4}$ model, which implies
that the so-called $\phi^{4}$-kink arises as a solution of this model when the
second field is null. This can be directly observed from the field equations
of this model:

$\displaystyle\frac{\partial^{2}\phi}{\partial
t^{2}}-\frac{\partial^{2}\phi}{\partial
x^{2}}=2\phi\left(1-\phi^{2}-\psi^{2}\right),\qquad\frac{\partial^{2}\psi}{\partial
t^{2}}-\frac{\partial^{2}\psi}{\partial
x^{2}}=2\psi\left(1-\phi^{2}-\psi^{2}-\frac{\sigma^{2}}{2}\right).$ (3)

It can be verified that the following static kink/antikink solutions satisfy
the field equations (3) of the model:

$K^{(\pm)}(x)=(\phi(x),\psi(x))^{\intercal}=(\pm\tanh(x-x_{0}),0)^{\intercal}.$
(4)

Figure 1: Contour density plot of the potential (2) for $\sigma=1.5$. The
green line represents the orbit of the kink solutions (4).

In Figure 1, the kink orbit has been depicted on the MSTB potential (2). Note
that both kink solutions (4) interpolate between the two vacua or minimum
points of (2). For future use, it is worth noting that these solutions can be
transformed into traveling kinks

$K^{(\pm)}(x,t)=\left(\pm\tanh\frac{x-x_{0}-v_{0}t}{\sqrt{1-v_{0}^{2}}}\,,\,0\right)^{\intercal},$
(5)

simply by applying a Lorentz boost, where $v_{0}$ is the kink velocity and
$-1<v_{0}<1$. The second reason that makes the MSTB model a natural candidate
to develop the study we have embarked on is that the kink solution (4) can
vibrate both longitudinally and orthogonally with respect the kink orbit, and
these channels include only one eigenstate each. Specifically, the second-
order small kink fluctuation operator involves only one longitudinal vibration
mode (the so-called shape mode, already present in the $\phi^{4}$ model). But
we also have an orthogonal vibration mode, that is, as it will be explained
later, the second order fluctuation operator consists of two decoupled
Schrödinger-type equations whose corresponding resolution will provide us with
a single mode shape each. This scenario provides a natural framework to
analyze the energy transfer mechanism between longitudinal and orthogonal
modes. This statement is justified in the following way: substituting a
linearly perturbed kink

$\widetilde{K}(x,t;\omega,a)=K^{(\pm)}(x)+a\,e^{i\omega
t}F(x)=\begin{pmatrix}\pm\tanh(x)\\\ 0\end{pmatrix}+a\,e^{i\omega
t}\begin{pmatrix}\overline{\eta}(x)\\\ \widehat{\eta}(x)\end{pmatrix},$ (6)

into the field equations (3) leads to the spectral problem

$\mathcal{H}\begin{pmatrix}\overline{\eta}(x)\\\
\widehat{\eta}(x)\end{pmatrix}=\omega^{2}\begin{pmatrix}\overline{\eta}(x)\\\
\widehat{\eta}(x)\end{pmatrix},$ (7)

where ${\cal H}$ is given by the matrix differential operator

$\mathcal{H}=\begin{pmatrix}-\dfrac{d^{2}}{dx^{2}}+4-6\operatorname{sech}^{2}x&0\\\
0&-\dfrac{d^{2}}{dx^{2}}+\sigma^{2}-2\operatorname{sech}^{2}x\\\
\end{pmatrix}.$ (8)

From (8) it is clear that the longitudinal modes (vibrating along the kink
orbit)

$\overline{F}_{\omega}(x)=\begin{pmatrix}\overline{\eta}(x)\\\
0\end{pmatrix},$ (9)

are decoupled from the orthogonal modes (vibrating orthogonally with respect
the kink orbit)

$\widehat{F}_{\omega}(x)=\begin{pmatrix}0\\\ \widehat{\eta}(x)\end{pmatrix}.$
(10)

In addition to this, the longitudinal and orthogonal components
$\overline{\eta}(x)$ and $\widehat{\eta}(x)$ are respectively determined as
the eigenfunctions of the Schrödinger-like operators

$\mathcal{H}_{11}=-\dfrac{d^{2}}{dx^{2}}+4-6\operatorname{sech}^{2}x\hskip
14.22636pt\mbox{and}\hskip
14.22636pt\mathcal{H}_{22}=-\dfrac{d^{2}}{dx^{2}}+\sigma^{2}-2\operatorname{sech}^{2}x\,,$

with Pöschl-Teller potential wells [56, 57, 58]. Therefore, the eigenfunctions
associated with the operator (8) are described as:

  * •

Longitudinal eigenmodes: The fluctuations of the kink solution correspond to
those found in the $\phi^{4}$ model with a single component [10, 7, 53, 54,
55, 52]. Therefore, the discrete eigenfrequencies are given by $\omega=0$ and
$\overline{\omega}=\sqrt{3}$, whose eigenfunctions are respectively the zero
mode

$\overline{F}_{0}(x)=\begin{pmatrix}\overline{\eta}_{0}(x)\\\
0\end{pmatrix}=\begin{pmatrix}\operatorname{sech}^{2}x\\\ 0\end{pmatrix},$
(11)

and the longitudinal shape mode

$\overline{F}_{\sqrt{3}}(x)=\begin{pmatrix}\overline{\eta}(x)\\\
0\end{pmatrix}=\begin{pmatrix}\operatorname{sech}x\tanh x\\\ 0\end{pmatrix}.$
(12)

Additionally, a continuous spectrum arises on the threshold value
$\overline{\omega}_{0}^{c}=2$ with frequency
$\overline{\omega}_{\overline{q}}^{c}=\sqrt{4+\overline{q}^{2}}$, where
$\overline{q}$ is a real positive parameter. The corresponding eigenfunctions
for this case can be written as

$\overline{F}_{\sqrt{4+\bar{q}^{2}}}\,(x)=\begin{pmatrix}\overline{\eta}_{\bar{q}}(x)\\\
0\end{pmatrix}=\begin{pmatrix}(-1-\bar{q}^{2}+3\,\tanh^{2}x-3i\bar{q}\tanh
x)\,e^{i\bar{q}x}\\\ 0\end{pmatrix}.$ (13)

  * •

Orthogonal eigenmodes: Only one discrete mode appears, whose frequency
$\widehat{\omega}=\sqrt{\sigma^{2}-1}$ depends on the coupling constant
$\sigma$, while the now-called orthogonal shape mode is determined as

$\widehat{F}_{\sqrt{\sigma^{2}-1}}(x)=\begin{pmatrix}0\\\
\widehat{\eta}(x)\end{pmatrix}=\begin{pmatrix}0\\\
\operatorname{sech}x\end{pmatrix}.$ (14)

Note that these fluctuations describe perturbations which are orthogonal to
the kink orbit, in the direction of the second field $\psi$. It is clear from
the expression of the eigenvalue $\widehat{\omega}$ that the kink (4) is
unstable when $\sigma<1$. Consequently, a less energetic kink arises in the
theory in the same topological sector. In this regime the kink (4) decays into
this new kink, which takes the form [25, 52]

$K^{*}(x)=(\pm
q\tanh(\sigma(x-x_{0})),\lambda\sqrt{1-\sigma^{2}}\operatorname{sech}(\sigma(x-x_{0})))^{\intercal},$

where $q,\lambda=\pm 1$. However, for the regime $\sigma>1$, the kink is
stable and the orthogonal shape mode corresponds to a fluctuation in the
second field (which was initially zero), describing a vibration of the kink
orbit, which we can visualize like the movement of a violin string. For this
reason, the scattering of orthogonally wobbling kinks addressed in this
article will be restricted to the range $\sigma>1$. Along with the discrete
mode (14) there is also a continuous spectrum with eigenfunctions

$\widehat{F}_{\sqrt{\sigma^{2}-1}}(x)=\begin{pmatrix}0\\\
\widehat{\eta}_{\widehat{q}}(x)\end{pmatrix}=\begin{pmatrix}0\\\
(\,\widehat{q}+i\tanh
x)\,e^{i\widehat{q}x}\end{pmatrix},\qquad\widehat{\omega}_{\widehat{q}}^{c}=\sqrt{\sigma^{2}+\widehat{q}^{2}}.$
(15)

Figure 2: Spectrum of the longitudinal (left) and orthogonal (right)
fluctuations of the operator (8). The blue and red areas coincide with the
continuous eigenvalues for the first and second field components. The dashed
colored lines represent the radiation frequencies that arise when
$\widehat{\eta}$ is initially triggered, while the black dashed vertical lines
represent the values at which there is resonance between two frequencies.

The spectrum of the kink fluctuation operator (8) as a function of the
coupling constant $\sigma$ is shown in Figure 2, in which the graph on the
left represents the spectrum of longitudinal fluctuations, while the one on
the right corresponds to orthogonal fluctuations. The discrete
eigenfrequencies are represented by solid lines, while the continuous spectrum
appears as a shaded region. Note that the longitudinal spectrum (left plot)
does not depend on $\sigma$, exhibiting a single longitudinal eigenfrequency
denoted as $\overline{\omega}=\sqrt{3}$ and a continuous spectrum emerging in
the threshold value $\overline{\omega}_{0}^{c}=2$. On the other hand, the
orthogonal spectrum changes depending on the value of $\sigma$, presenting a
single orthogonal frequency $\widehat{\omega}=\sqrt{\sigma^{2}-1}$, and a
continuum that emerges from the value $\widehat{\omega}_{0}^{c}=\sigma$. As a
first conclusion, it should be noted that the orthogonal shape mode is easily
excited when the coupling constant is close to the value 1. In particular, it
can be observed that in the range $\sigma\in[1,2]$, the orthogonal
eigenfrequency is lower than the longitudinal one and, therefore, more prone
to excitation. When $\sigma>2$, the situation is reversed and it will be
easier to excite the longitudinal mode. Since these regimes are expected to
lead to different collision behavior of the wobblers, simulations with values
of $\sigma$ both lower and higher than $2$ (typically the values $\sigma=1.2$,
$\sqrt{7}/2$, $\sqrt{2}$, $1.5$ and $2.5$) will be carried out in Section 4.
Additionally, certain harmonics of the discrete eigenfrequencies are
represented by dashed curves in Figure 2. These frequencies can be excited
because of nonlinear terms of the theory at higher orders and play an
essential role in radiation phenomena. This type of process where a kink is
initially excited by the orthogonal shape mode $\widehat{\eta}$ has been
analyzed in [52] by means of a perturbation approach. It was found that the
orthogonally wobbling kink is able to emit radiation with frequency
$2\widehat{\omega}$ in the longitudinal channel. Note that the longitudinal
radiation captures the information coming from the orthogonal channel. On the
other hand, orthogonal radiation (traveling in the second field) is emitted
with frequencies $3\widehat{\omega}$ and $\widehat{\omega}+\overline{\omega}$.
Typically, the radiation emission remains small when the initial amplitude of
the orthogonal shape mode is reasonably small. However, for certain values of
the coupling constant $\sigma$, resonance phenomena occur in which this
radiation emission can increase by one or two orders of magnitude. These
special model parameter values can be visualized in Figure 2 by the
coincidence of certain eigenfrequencies, and are listed below:

  * •

$\mathbf{\sigma_{1}=\frac{\sqrt{7}}{2}}\approx 1.32288$: As seen in Figure 2
(left), a resonance occurs when twice the eigenfrequency of the orthogonal
shape mode $2\widehat{\omega}$ coincides with the longitudinal discrete
eigenfrequency $\overline{\omega}$ for the model parameter $\sigma_{1}$. In
this case, a second resonance coincidentally arises between
$3\widehat{\omega}$ and $\widehat{\omega}+\overline{\omega}$ in the orthogonal
fluctuation spectrum, see Figure 2 (right). In this scenario, there is a
significant loss of energy from the orthogonal shape mode in favor of the
longitudinal mode, accompanied by a pronounced emission of radiation in the
orthogonal channel with frequencies $\overline{\omega}+\widehat{\omega}$ and
$3\widehat{\omega}$ [52]. Obviously, this phenomenon can have relevant effects
on the resonant energy transfer mechanism, causing wobbler collisions to
exhibit singular behaviors within the values of the coupling constant
$\sigma$. In other words, as the wobblers approach each other, a significant
portion of the energy from the orthogonal shape mode is transferred to the
longitudinal shape mode and also radiated away. Consequently, for this
$\sigma$ value, the collision dynamics is predominantly governed by the
scattering of longitudinally wobbling kinks. Under these circumstances, we can
anticipate that behavior is similar to the wobbler scattering observed in the
$\phi^{4}$ model. Deviations from the patterns observed in this case provide
valuable information about the interaction between the orthogonal shape mode
and other modes within the resonant energy transfer mechanism.

  * •

$\mathbf{\sigma_{2}=\sqrt{2}}\approx 1.41421$: In this case, the frequency
$2\widehat{\omega}$ begins to form part of the continuous longitudinal
spectrum (see Figure 2 left). In fact, this value is characterized by the
coincidence between the values $2\widehat{\omega}$ and
$\overline{\omega}_{0}^{c}$. This implies that the quadratic longitudinal
radiation source is activated from this point. As a consequence, the radiation
emission in the longitudinal channel with frequency $2\widehat{\omega}$ is
very significant (relative to other values of $\sigma$). This implies that a
large amount of energy is extracted from the orthogonal shape mode in the form
of longitudinal radiation. Again, in this situation some peculiar behaviors
could appear in the scattering between wobblers. However, despite this
phenomenon, the energy loss resulting from radiation emission is not
considered significant enough to alter the wobbler scattering characteristics
for this particular value.

In the analysis of kink scattering in this model, we will take into account
previous observations regarding the relevant values of the coupling constant
$\sigma$, with the aim of performing simulations at various values of the
model parameter that can lead to different behaviors. We have already
mentioned the values $\sigma=1.5$ and $\sigma=2.5$ as representative of
possible different behaviors. To these two values we will also add
$\sigma=1.2$, for which the emission of radiation in the longitudinal channel
with frequency $2\widehat{\omega}$ is suppressed. Additionally, studies of the
scattering of orthogonally wobbling kinks will be carried out in the special
cases $\sigma_{1}=\sqrt{7}/2$ and $\sigma_{2}=\sqrt{2}$ mentioned above.

##  3 Numerical setup for the scattering between wobblers in the MSTB model

In this paper we are analyzing collisions between a kink and antikink (4),
with their orthogonal shape modes initially activated. It is well-known that
the scattering between wobbling kinks in the $\phi^{4}$ model exhibits complex
fractal structures depending on the vibration amplitude of its only shape mode
[21, 22]. These fractal structures arise due to the resonant energy transfer
mechanism between the zero and shape discrete modes. In this model both of
these eigenfluctuations are embedded in the longitudinal channel. Our goal now
is to numerically investigate whether this is a necessary condition to trigger
the resonant energy transfer mechanism or whether it can also be initiated by
orthogonal eigenmodes. The initial numerical setup will consist of a well-
separated orthogonally wobbling kink/antikink configuration whose centers are
initially located at points $x_{0}$ and $-x_{0}$ (with $x_{0}\gg 1$) and
travel respectively with velocities $v_{0}$ and $-v_{0}$. Therefore, the
initial configuration for the numerical simulations can be characterized as

$K^{(\pm)}_{WK}(x,t,v_{0},x_{0},\sigma,a_{0})\cup
K^{(\mp)}_{WK}(x,t,-v_{0},-x_{0},\sigma,a_{0})=\left\\{\begin{array}[]{cc}K^{(\pm)}\left(\tilde{x}_{K^{\pm}}\right)+\widehat{a}_{0}\,\sin(\widehat{\omega}t)\,\widehat{F}_{\widehat{\omega}}\left(\tilde{x}_{K^{\pm}}\right),\quad\mathrm{if}\,x<0,\vspace{0.3cm}\\\
K^{(\mp)}\left(\tilde{x}_{K^{\mp}}\right)+\widehat{a}_{0}\,\sin(\widehat{\omega}t)\,\widehat{F}_{\widehat{\omega}}\left(\tilde{x}_{K^{\mp}}\right),\quad\mathrm{if}\,x>0,\end{array}\right.$
(16)

where

$\tilde{x}_{K^{\pm}}=\frac{x+x_{0}-v_{0}t}{\sqrt{1-v_{0}^{2}}},\qquad\tilde{x}_{K^{\mp}}=\frac{x-x_{0}+v_{0}t}{\sqrt{1-v_{0}^{2}}}.$

Here, $\widehat{a}_{0}$ is the initial amplitude of the orthogonal shape mode,
which measures the excitation of the wobbling kinks. Note that (16) determines
well-behaved initial conditions because the single kinks in this model are
exponentially localized. It is expected that during the scattering process
there will be an energy transfer between the orthogonal, longitudinal and
translational modes, together with a small radiation emission in both field
components. This energy loss causes the wobbling amplitude $\widehat{a}$ to
decay following the theoretical law

$\widehat{a}(t)^{2}\approx\frac{\widehat{a}(0)^{2}}{1+\widehat{\omega}\,\,\widehat{c}\,\,\widehat{a}(0)^{2}\,t},$
(17)

where $\widehat{c}$ is a constant that depends on the radiation amplitudes and
$\widehat{\omega}$ and, therefore, also on $\sigma$ [52]. Generally, this
phenomenon only plays an important role when $\widehat{a}(0)>0.1$. In a
scattering event, once the collision occurs, a part of the energy could be
transferred to the longitudinal eigenmode. In this case, the resulting
longitudinally wobbling kinks behave approximately like those of $\phi^{4}$,
that is, the kink would emit radiation in the longitudinal channel whose
frequency would be $2\overline{\omega}$. If this happens, the wobbling
amplitude of this internal mode will also decay following the formula

$\overline{a}(t)^{2}\approx\frac{\overline{a}(0)^{2}}{1+\overline{\omega}\,\,\overline{c}\,\,\overline{a}(0)^{2}\,t},$
(18)

where $\overline{c}$ is a constant related to the radiation amplitude [53, 54,
55]. As before, this decrease is only relevant only for amplitudes
$\overline{a}(0)>0.1$. This phenomenon implies that the initial amplitudes of
the shape modes of the colliding kinks will decrease as they approach each
other. When the initial amplitudes of the shape modes involved are small, the
described phenomenon can be considered negligible and these magnitudes are
suitable parameters to describe the scattering processes described in this
article.

Based on all the results presented above, after the collision of the wobbling
kinks and in the case that they are reflected, it is expected that the kinks
will move away with a certain final velocity $v_{f}$, with both longitudinal
and orthogonal shape modes excited, that is, with a configuration of the form:

$K^{(\pm)}_{WK,f}(x,t,v_{0},x_{0},\sigma,a_{0})\cup
K^{(\mp)}_{WK,f}(x,t,-v_{0},-x_{0},\sigma,a_{0})=\left\\{\\!\\!\begin{array}[]{cc}K^{(\pm)}\left(\tilde{x}_{K^{\pm},f}\right)+\overline{a}_{f}\sin(\overline{\omega}t+\delta)\,\overline{F}_{\overline{\omega}}\left(\tilde{x}_{K^{\pm},f}\right)+\widehat{a}_{f}\sin(\widehat{\omega}t)\,\widehat{F}_{\widehat{\omega}}\left(\tilde{x}_{K^{\pm},f}\right),\,\mathrm{if}\,x<0,\vspace{0.3cm}\\\
K^{(\mp)}\left(\tilde{x}_{K^{\mp},f}\right)+\overline{a}_{f}\sin(\overline{\omega}t+\delta)\,\overline{F}_{\overline{\omega}}\left(\tilde{x}_{K^{\mp},f}\right)+\widehat{a}_{f}\sin(\widehat{\omega}t)\,\widehat{F}_{\widehat{\omega}}\left(\tilde{x}_{K^{\mp},f}\right),\,\mathrm{if}\,x>0,\end{array}\right.$
(19)

where

$\tilde{x}_{K^{\pm},f}=\frac{x+v_{f}t}{\sqrt{1-v_{f}^{2}}},\qquad\tilde{x}_{K^{\mp},f}=\frac{x-v_{f}t}{\sqrt{1-v_{f}^{2}}}.$
(20)

Once the initial conditions of our problem have been described, the kink
scattering processes are simulated by discretizing equations (3) using a
fourth-order algorithm designed specifically to handle general nonlinear
systems of two coupled Klein-Gordon partial differential equations, as
described in [32]. To prevent the radiation emitted by the system from being
reflected from the spatial grid boundaries, Mur absorbing boundary conditions
were implemented at $x=100$ and $x=-100$. Simulations were performed for
various values of the coupling constant $\sigma\in[1.2,2.5]$ over a range of
initial velocities $v_{0}\in\left[0.1,0.9\right]$ with increments of $\Delta
v_{0}=10^{-5}$. For velocity ranges where the kinks experience only a rebound,
the step size was increased to $\Delta v_{0}=10^{-3}$. In each of these cases
the simulations have been carried out changing the initial orthogonal
amplitude $\widehat{a}_{0}$ with steps $\Delta\widehat{a}_{0}=0.02$ in the
range $\widehat{a}_{0}\in(0,0.2)$. The total number of simulations carried out
to obtain the results presented in this article amounts to approximately
$10^{6}$, which have been carried out by using the Caléndula Supercomputer
integrated into the high-performance computing resources of the Castilla y
León Supercomputing Center (SCAYLE). All simulations were performed within the
spatial interval $x\in[-100,100]$ with a step size of $\Delta x=0.005$.
Initially, the kink/antikink centers are separated by a distance
$d=2x_{0}=15$. When kinks are reflected, their final velocities are measured
numerically. In addition to this, the amplitudes of the orthogonal and
longitudinal shape modes are also estimated by using a fast Fourier transform
algorithm, implemented at the comoving kink centers $\widehat{x}_{M}=x_{C}$
and the respective points

$\overline{x}_{M}=x_{C}\pm\sqrt{1-v_{0}^{2}}\,\operatorname{arccosh}\sqrt{2},$

where the discrete longitudinal fluctuations present their maximum elongation.

It is also worth mentioning that the initial conditions proposed in (16) are
symmetric with respect to the origin of the spatial coordinate. That is, the
kink and the antikink approach each other at the same velocity (this does not
imply a loss of generality since it is simply assumed that we fix the
reference system with the origin located at the center of mass) and their
shape modes have been excited with the same amplitude and phase. As
demonstrated in [52, 59], this scenario provides the most extreme phenomena
that can arise in kink collisions, since at the moment of impact a
constructive interference occurs, amplifying the redistribution of energy
between the vibrational and translational modes. Since the numerical algorithm
used to discretize the field equations (3) also preserves this symmetry, the
scattering results can be extracted directly from one of the traveling
wobblers involved in the simulations.

##  4 Numerical results for the scattering between wobblers in the MSTB model

Now that we have detailed the implementation of the numerical simulations, we
proceed to present the main results derived from the data obtained from the
simulations. The parameter $\sigma$ emerges as pivotal to understanding the
energy transfer between internal modes and significantly influences the
velocity diagrams of the scattering process. In fact, $\sigma$ plays a crucial
role in the appearance of new single-bounce windows in these diagrams. Indeed,
as it will be explained later, increasing the value of $\sigma$ is equivalent
to increasing the energy needed to trigger the orthogonal eigenmode. This
implies that, for example, for the same value of $\widehat{a}_{0}$ the
critical velocity where the one-bounce tail arises will be smaller as $\sigma$
is increased, since more energy can be transferred to the translational
eigenmode. Furthermore, the amplitude of the orthogonal shape mode holds
considerable importance: a larger $\widehat{a}_{0}$ facilitates greater energy
transfer to the longitudinal shape mode and the translational one,
facilitating separation of the wobblers. The main characteristics of these
scattering processes are delineated in the following subsections. Firstly,
Section 4.1 delves into the analysis of the velocity diagrams, elucidating
their behavior in each of the cases. Next, in Section 4.2, we investigate the
resonant energy transfer mechanism by examining the final wobbling amplitudes
(both longitudinal and orthogonal), which provides information on the strength
of shape mode excitations at the end of the process.

###  4.1 Properties of the velocity diagrams for the scattering between
wobblers in the MSTB model

In this subsection we will describe the characteristics of the velocity
diagrams observed in the scattering between wobblers within the MSTB model,
when the orthogonal shape mode is initially excited. We will classify these
scenarios according to the strength of the initial excitation into three
distinct regimes. We will first examine the regime in which wobblers are
weakly excited, typically for $\widehat{a}_{0}<0.05$. Next, we will
investigate the regime characterized by moderate excitations, within the range
$0.05<\widehat{a}_{0}<0.1$. Finally, we will explore the strong excitation
regime, identified by $\widehat{a}_{0}>0.1$. Through this analysis we aim to
provide a comprehensive understanding of the behavior exhibited by the
velocity diagrams (and the fractal structures included in them) when varying
the value of the initial excitation and also the coupling constant $\sigma$,
as we have already indicated in Section 2.

Figure 3: Final velocity $v_{f}$ of the scattered kinks as a function of the
initial velocity $v_{0}$. The color code shown in the graphs indicates the
number of bounces suffered by the kink-antikink pair before moving appart. In
initial velocity ranges where no final velocity is shown, a bion is assumed to
form. The resonance window where various bounces can be observed has been
expanded inside each graphics to better show the fractal pattern. For the sake
of comparison the dashed grey line indicates the elastic scenario
$v_{0}=v_{f}$.

####  4.1.1 Velocity diagrams for the scattering between weakly wobbling
kinks.

In Figure 3, the velocity diagram for a low initial wobbling amplitude
$\widehat{a}_{0}=0.04$ for the orthogonal mode is displayed for different
values of $\sigma$. Specifically, the values chosen are $\sigma=1.2$,
$\sqrt{7}/2$, $\sqrt{2}$, $1.5$ and $2.5$, which allows us to describe the
entire range of behaviors of the velocity diagrams in this scenario. When the
values of $\sigma$ are close to $1$, the resulting fractal structure appears
relatively simple, with only a few windows where the wobblers collide more
than once. The resonant energy transfer mechanism redistributes energy between
three modes: two vibrational modes (longitudinal and orthogonal) and the zero
mode. The chances of gaining energy by the zero mode are less than in the case
where there is only one shape mode. Consequently, the number of velocity
windows in the resonance interval where the kinetic energy of the wobblers is
large enough to allow them to escape must be smaller compared to the
$\phi^{4}$ model, where only one of these massive modes is present. It is
important to note that, for the same reason, the one-bounce tail (represented
in blue in the upper plots of Figure 3) exhibits small oscillations,
indicating that part of the kinetic energy has been trapped by one of the
shape modes.

The previous effect is diminished in the singular case
$\sigma_{1}=\sqrt{7}/2$. As mentioned above, the resonance occurring under the
present circumstances implies that the energy accumulated by the orthogonal
shape mode is partially discharged into the longitudinal mode before the
collision occurs. Scattering can therefore be described in this case as the
collision between two longitudinally wobbling kinks. Thus, the velocity
diagrams found are similar to those observed in the collision between wobblers
in the $\phi^{4}$ model. Note, for example, that the oscillations of the one-
bounce tail are more pronounced than in the case $\sigma=1.2$.

As $\sigma$ increases, the fractal structure undergoes substantial changes,
becoming increasingly complex. For sufficiently large values of this coupling
constant $\sigma$, the diagrams stabilize and resemble the velocity diagram
observed for kink/antikink scattering in the $\phi^{4}$ field theory.
Furthermore, the critical velocity at which the one-bounce tail emerges is
approximately the same, $v_{cr}\approx 0.26$ [1, 21]. This suggests that when
the initial orthogonal amplitude $\widehat{a}_{0}$ is very small, the shape
mode retains its energy without significant transfer to other modes. As a
result, the resulting scattering process resembles that of unexcited kinks.
This phenomenon can be attributed to the fact that the value of
$\widehat{\omega}$ increases as the model parameter $\sigma$ increases.
Consequently, exciting the orthogonal mode during collisions becomes much more
challenging. Certainly, this behavior changes when the amplitude
$\widehat{a}_{0}$ reaches a certain threshold. This suggests that the
orthogonal shape mode must be sufficiently excited to initiate the resonant
energy transfer mechanism. More discussion on this point will be provided in
Section 4.2 when the behavior of the amplitude of the internal modes after the
collision will be discussed.

Another remarkable phenomenon is that the critical velocity $v_{cr}$, at which
the one-bounce tail arises, decreases as $\sigma$ increases. As the energy of
the orthogonal mode increases with $\sigma$ this also implies that more energy
can be transferred to the zero mode allowing the kink-antinkink pair to escape
more easily. As it will be shown in Section 4.1.2 and Section 4.1.3, this
phenomenon can also be observed for higher values of $\widehat{a}_{0}$.

Figure 4: Final velocity $v_{f}$ of the scattered kinks as a function of the
initial velocity. The color code shown in the graphs indicates the number of
bounces suffered by the kink-antikink pair before moving apart. In initial
velocity ranges where no final velocity is shown, a bion is assumed to form.
The resonance window where various bounces can be observed has been expanded.
For the sake of comparison the dashed grey line indicates the elastic scenario
$v_{0}=v_{f}$.

####  4.1.2 Velocity diagrams for the scattering between moderate wobbling
kinks.

When the initial amplitude of the orthogonal shape mode is increased, the
behavior of the velocity diagrams and the fractal pattern that appears in them
exhibit novel characteristics with respect to the previous situation, which in
turn depends on the value of $\sigma$. It is worth noting that in the case of
the $\phi^{4}$ model, the study of wobbling kink collisions led to the
identification of new phenomena such as the splitting of the one-bounce tail
into isolated one-bounce windows, along with dividing $n$-bounce windows into
those with a higher number of bounces. These phenomena are also present in the
MSTB model with distinctive characteristics. Recall that at the critical value
$\sigma_{1}=\sqrt{7}/2$, the collision of wobblers occurs with a maximum
longitudinal wobbling amplitude compared to other $\sigma$ values.
Consequently, the influence of the orthogonal shape mode decreases, allowing
the characteristic features observed in the scattering of wobblers in the
$\phi^{4}$ model to emerge more prominently and distinctively. For other
values of $\sigma$, the orthogonal shape mode assumes a more important role in
the dynamics. For example, when $\sigma<\sigma_{1}$, the orthogonal frequency
is low, which makes the orthogonal shape mode easily excited. This implies
that the appearance of $n$-bounce windows within the fractal structure
decreases. On the other hand, when $\sigma>\sigma_{1}$, the number of isolated
one-bounce windows decreases, although the complexity of the fractal pattern
involving $n$-windows grows and the resonance interval reduces with increasing
values of $\sigma$. All of these characteristics can be observed in Figure 4,
where the velocity diagrams are shown for an initial amplitude
$\widehat{a}_{0}=0.08$ for the values of the model parameter $\sigma=1.2$,
$\sigma=\sqrt{7}/2$, $\sigma=\sqrt{2}$ and $\sigma=2.5$. One of the most
notable behaviors occurs in the resonance interval, where the splitting of the
$n$-bounce windows into others of the same type occurs, the gap being filled
by windows with a greater number of bounces. Additionally, isolated one-bounce
windows appear that were not present before. Another surprising behavior is
that the one-bounce tail begins to exhibit oscillations that, when the initial
vibration amplitude is large enough, lead to the reappearance of isolated one-
bounce windows. These processes were also present in the scattering of
wobbling kinks in the $\phi^{4}$ model, although, as we will see in the
present scenario, these phenomena show novel features, which we describe in
more detail in the following points:

  * •

Emergence of one-bounce windows in the resonance regime: As a general rule,
when $\sigma$ is not close to the critical value $\sigma_{1}=\sqrt{7}/{2}$,
only a few one-bounce windows arising via this mechanism will be observed in
the velocity diagrams. In Figure 5, the velocity plots for one-bounce
scattering events are depicted as a function of the initial amplitude of the
orthogonal shape mode for $\sigma=1.2$ within the range
$\widehat{a}_{0}\in[0,0.2]$ using a color-coded scheme. In it, red represents
the case in which the kinks are not excited, while blue corresponds to the
maximum value of the amplitude within the selected range
$\widehat{a}_{0}\in[0,0.20]$. From this figure it is clear that there are only
four windows present under the critical velocity ${v_{0}\approx 0.27}$
(associated with unexcited kink-antikink scattering) for relatively high
values of $\widehat{a}_{0}$, and their heights are relatively small.

Figure 5:  Velocity diagram for one-bounce scattering events for different
values of the initial wobbling amplitude, $\widehat{a}_{0}\in[0,0.2]$.

The presence of these windows can be explained by considering that the larger
$\widehat{a}_{0}$ is, the more energy the kinks possess before the collision.
Consequently, more energy can be transferred to the translational mode,
allowing the kinks to escape more easily after a collision for relatively low
values of $v_{0}$. This phenomenon can be better understood by referring to
Figure 6, which represents a sequence of velocity diagrams for closely spaced
initial amplitude values $\widehat{a}_{0}$ for the case of $\sigma=1.2$. It
can be seen that the fractal structure around $v_{0}\approx 0.255$ is
compressed when the amplitude of the orthogonal mode increases. When this
amplitude is high enough, a new one-bounce window emerges in the same place
where the two-bounce windows existed.

  * •

One-bounce reflection tail splitting: As shown in Figures 4 and 5, the one-
bounce tail (plotted in blue) begins to oscillate. In fact, for weakly excited
wobbling kinks, this phenomenon is already noticeable in the graph associated
with $\sigma=1.2$ in Figure 3. With an increase in the amplitude of the
orthogonal mode, these oscillations can extend to the $v_{f}=0$ axis,
resulting in isolated one-bounce windows. This behavior is more pronounced for
values of $\sigma$ close to 1, as seen in Figure 4. This process is clearly
illustrated in Figure 6, where the one-bounce tail is recurrently split into
multiple isolated one-bounce windows. In the sequence represented, one can
observe the formation of such a window around $v_{0}\approx 0.28$ in the
second graph, while a new window appears in the third graph for $v_{0}\approx
0.31$. Furthermore, as $\widehat{a}_{0}$ increases, these new windows begin to
shift to the right side of the graph. This phenomenon was not observed in the
study of scattering between wobblers in the $\phi^{4}$ model [21]. For
example, considering the velocity diagram for $\sigma=1.2$ shown in Figure 6,
it is evident that the initial point of the second one-bounce window is
shifted from $v_{0}\approx 0.277$ when $\widehat{a}_{0}=0.04$ to $v_{0}\approx
0.279$ when $\widehat{a}_{0}=0.08$. Indeed, this phenomenon is similar to the
aforementioned squeezing of the fractal structure that gave as a result new
one-bounce windows in the resonant interval of the diagram. The main
difference between both phenomena is that now no new one-bounce windows are
created; instead, the existing window is compressed to fill the gap between
one-bounce windows with $n$-bounce windows. Note in Figure 4 that splitting
the one-bounce tail results in significantly more isolated one-bounce windows
for small values of $\sigma$ than for higher values of $\sigma$.

Figure 6: Velocity diagrams for the case $\sigma=1.2$ in the initial velocity
range $v_{0}\in[0.25,0.34]$ for several values of excitation amplitude
$\widehat{a}_{0}$. This graphics illustrates the window splitting mechanism.
The color code used to indicate the total number of bounces is the same as in
Figure 3.

  * •

$n$-bounce window splitting: Another mechanism that contributes to the
increasing complexity of the fractal structure of the velocity diagrams as the
initial amplitude increases is the splitting of $n$-bounce windows, where one
of these windows is divided into others of the same type. This phenomenon
already appeared in the $\phi^{4}$ model. In this case, increasing the
amplitude of the single shape mode results in the generation of more two-
bounce windows, which appear progressively from left to right in the
corresponding velocity diagram. Figure 7 illustrates the evolution of a single
two-bounce window as $\widehat{a}_{0}$ goes from $0.02$ to $0.12$ for the
kink-antikink scattering in the MSTB model for $\sigma=2.5$. It can be seen
that as the orthogonal wobbling amplitude increases, a new two-bounce window
emerges on the left side of the original window between $\widehat{a}_{0}=0.06$
and $\widehat{a}_{0}=0.08$. The gap between these two regions is filled with
scattering events exhibiting three and four bounces. Subsequently, for
$0.10<\widehat{a}_{0}<0.12$, another window is created, this time on the right
side of the plot. This phenomenon can be explained by considering that, for
large values of $\sigma$, the frequency of the orthogonal mode is greater than
that corresponding to the longitudinal mode. Consequently, the vibrational
energy of the wobblers increases, providing more energy to transfer to
translational mode. This allows the wobbling kink-antikink pair to generate
new windows on either side of the original ones as $\widehat{a}_{0}$
increases.

Figure 7: Evolution of a two-bounce window found in the velocity diagram for
$\sigma=2.5$ as the value of $\widehat{a}_{0}$ increases from
$\widehat{a}_{0}=0.02$ to $\widehat{a}_{0}=0.12$. The color code used to
indicate the total number of bounces is the same as in Figure 3.

####  4.1.3 Velocity diagrams for the scattering between strongly wobbling
kinks.

Let us analyze the case where the scattering of orthogonally wobbling kinks
occurs with a large initial amplitude $\widehat{a}_{0}$. In Figure 8, the
velocity diagrams for the case in which $\widehat{a}_{0}=0.16$ are depicted.
Now, the behaviors described above are much more pronounced. For example, for
small values of $\sigma$, the prevalence of isolated one-bounce windows is
observed, which for the case of $\sigma=1.2$ occupies the range between
$v_{0}\in[0.19,0.405]$, and the one-bounce tail presents significant
oscillations.

Figure 8: Final velocity $v_{f}$ of the scattered kinks as a function of the
initial velocity. The color code shown in the graphs indicates the number of
bounces suffered by the kink-antikink pair before moving apart. In initial
velocities ranges where no final velocity is shown, a bion is assumed to form.
The resonance window where various bounces can be observed has been expanded.
For the sake of comparison tThe dashed grey line indicates the elastic
scenario $v_{0}=v_{f}$.

In the singular case $\sigma_{1}={\sqrt{7}}/{2}$, we observe a highly
organized structure in the fractal pattern within the velocity diagram.
Isolated one-bounce windows emerge in a repeating pattern, decreasing in both
height and width as the initial velocity $v_{0}$ decreases. As previously
mentioned, this is related to the fact that the resonant energy mechanism is
practically governed in this case by only two modes (the zero mode and the
longitudinal one), resembling the behavior observed in the $\phi^{4}$ model.
It is worth noting that the peak of these windows corresponds closely to the
velocity associated with the elastic scenario.

For the cases where $\sigma=\sqrt{2}$ and $1.5$, the fractal structure is
highly complex, also showing isolated one-bounce windows along with others
with a greater number of bounces, all densely packed. Note that the regularity
seen in the case of $\sigma=\sigma_{1}$ is completely lost, replaced by a
sequence of isolated one-bounce windows that alternate between higher and
lower heights. This suggests a higher level of chaotic behavior compared to
that observed in the $\phi^{4}$ model, attributable to the presence of an
additional shape mode. Similarly, it can be seen in Figure 8 how the
oscillations of the one-bounce tail decrease as the value of $\sigma$
increases, as seen in the cases of $\sigma=1.5$ and $\sigma=2.5$. Finally, in
the case where $\sigma=2.5$, the presence of isolated one-bounce windows is
practically suppressed, and the fractal structure is compressed into a much
smaller region than that presented in the previously mentioned cases.
Additionally, it is worth mentioning that the behavior observed in the
scattering of wobbling kinks in the $\phi^{4}$ model, where the kinks
transferred a significant amount of energy from the vibrational modes to the
zero modes, allowing final velocities of the kinks greater than the initial
ones, does not usually appear in the present case. The reason is clearly due
to the presence of two shape modes in this case, both accumulating energy, and
it is difficult for both to simultaneously transfer their energy to the zero
mode to recreate the situation discussed above.

###  4.2 Analysis of the shape mode amplitudes

Figure 9: Final wobbling amplitudes of the scattered kinks after the last
collision as a function of $v_{0}$ for various values of $\sigma$ and initial
wobbling amplitude $\widehat{a}_{0}=0.04$. The blue line represents
$\overline{a}_{f}$ (the final longitudinal wobbling amplitude) and the red
line represents $\widehat{a}_{f}$ (the final orthogonal wobbling amplitude).

In this section, we will analyze the behavior of the final vibration
amplitudes in both the longitudinal and orthogonal channels. This analysis
provides valuable information on how the resonant energy transfer mechanism
works in this model. To accomplish this task, the final amplitudes of both
shape modes (longitudinal and orthogonal) have been plotted as a function of
initial velocity for various values of the amplitude $\widehat{a}_{0}$ of the
orthogonal shape mode and the coupling constant $\sigma$. Figure 9 illustrates
the behavior of these amplitudes with the initial amplitude value set to
$\widehat{a}_{0}=0.04$, while the model parameter $\sigma$ takes various
values: $\sigma=1.2,\sqrt{7}/2,\sqrt{2},1.5$, and $2.5$. In the graphs shown
in Figure 9, the amplitude of the longitudinal shape mode is represented in
blue, while that of the orthogonal shape mode is represented in red. The
behavior shown in this figure is similar for other values of the amplitude
$\widehat{a}_{0}$.

It is evident that the dispersion process depends largely on the value of the
constant $\sigma$. Analogous to the results obtained previously, we can
identify two significant regimes. If the value of $\sigma$ is close to 1 and
below the resonance value associated with $\sigma=\sigma_{1}$, the effect of
the resonant energy transfer mechanism becomes particularly pronounced. There
are values of the collision velocity $v_{0}$ for which the final amplitude of
the orthogonal shape mode is almost suppressed, making its value almost
disappear. These values are aligned with the peaks of the longitudinal shape
mode amplitude. On the other hand, the minima of the longitudinal shape mode
amplitude correspond to the maxima of the orthogonal mode amplitude. This
clearly indicates the energy transfer between these two modes. It is worth
noting that the oscillations of the former mode are smaller than those of the
latter. This difference arises because the eigenvalue of the orthogonal shape
mode is smaller than that of the longitudinal mode, which means that the
orthogonal shape mode is less energetic than the longitudinal mode.

Figure 10: Final amplitudes of both shape modes after the scattering as a
function of the initial velocity $v_{0}$ for various values of
$\widehat{a}_{0}\in[0,0.20]$ and for $\sigma=1.2$ and $2.5$. The first legend
located in the upper right corner of the figure corresponds to the amplitude
of the longitudinal shape mode while the second corresponds to the amplitude
associated with the orthogonal mode.

In the other regime that we mentioned, for values of $\sigma$ greater than
$\sigma_{1}=\sqrt{7}/2$, we observe a similar behavior in the amplitude of the
orthogonal shape mode, which oscillates as a function of the initial velocity.
However, these oscillations have a smaller amplitude and decrease as the
coupling constant $\sigma$ increases. In fact, for $\sigma=2.5$, these
fluctuations are almost imperceptible. Additionally, it should be noted that,
as a general trend, the frequency of these oscillations increases as the value
of $\sigma$ increases, which is related to the fact that the frequency of the
orthogonal shape mode also increases (see [52]). On the other hand, the
amplitude of the longitudinal shape mode is not influenced by the orthogonal
shape mode. In this case, the excitation of this mode can be attributed almost
entirely to the collision of the kinks, or more specifically, to the
interaction with the zero modes of these solutions. This behavior is best
observed in Figure 10, where the evolution of the amplitudes corresponding to
both shape modes for $\sigma=1.2$ and $\sigma=2.5$ as a function of $v_{0}$
and $\widehat{a}_{0}$ can be observed. As shown in the top graph of Figure 10,
for low values of $\sigma$ increasing the value of the initial amplitude
$\widehat{a}_{0}$ results in a greater final amplitude of orthogonal shape
mode $\widehat{\eta}$ after the last collision. Simultaneously, as the maxima
corresponding to $\widehat{a}_{f}$ increase, the minima of $\overline{a}_{f}$
decrease. Note that there is an energy loss in the longitudinal shape mode
compared to the case where the orthogonal shape mode is not initially excited.
Another important aspect to highlight is that the nodes of the oscillations
exhibited by the shape modes depend on the initial amplitude. As seen in
Figure 10, the nodes shift slightly to the right as the value of
$\widehat{a}_{0}$ increases. On the other hand, as shown in the lower graph of
Figure 10, the behavior of the longitudinal shape mode amplitude does not
depend on the initial wobbling amplitude $\widehat{a}_{0}$ when $\sigma$ is
high enough. In fact, the wobbling amplitude of the orthogonal mode stops
oscillating and its value decreases as the kinks collide with greater
velocity.

Figure 11: Final wobbling amplitudes for the scattered kinks after the last
collision as a function of $v_{0}$ for $\sigma=\sqrt{7}/2$ and wobbling
amplitude $\widehat{a}_{0}=0.04$. The blue line represents $\overline{a}_{f}$
(the final longitudinal wobbling amplitude) and the red line represents
$\widehat{a}_{f}$ (the final orthogonal wobbling amplitude).

However, the behavior of the system is completely different for the special
value $\sigma_{1}=\sqrt{7}/2$, where a resonance occurs between the
frequencies $\overline{\omega}$ and $2\widehat{\omega}$. In this case, the
graphical representation of the amplitudes is more irregular than in the other
cases and, surprisingly, the final amplitude of both shape modes does not
depend on the initial wobbling amplitude $\widehat{a}_{0}$, being
$\widehat{a}_{f}$ almost always slightly larger than the amplitude
corresponding to the longitudinal eigenmode (see Figure 11). The mechanism
that seems to explain the scattering of the initially excited orthogonally
wobbling kinks in this case is the following: when these kinks approach each
other, a significant energy transfer from the orthogonal shape mode to the
longitudinal mode occurs due to the aforementioned resonance [52]. This
results in the orthogonal shape mode being weakly excited immediately before
the collision of the wobblers, while the longitudinal mode has gained
considerable energy. This leads, for example, to velocity diagrams similar to
those found in [21] for the $\phi^{4}$ model, where there is no second
component. After the collision, the longitudinal mode is assumed to be quite
excited, but a channel exists that recharges the orthogonal mode through
higher-order nonlinear couplings. Since both modes have been excited, the
energy transfer channel from the orthogonal to the longitudinal mode is
suppressed. However, it should be noted that for this critical value of
$\sigma$, there is also strong radiation emission in the orthogonal channel
with frequencies $\omega=\overline{\omega}+\widehat{\omega}$ and
$3\widehat{\omega}$, resulting in an increase in decay in the shape mode
chain. It is important to consider that this process is quite complex, and its
description through numerical schemes may depend on the $t_{\rm max}$ used in
the simulations, which justifies the irregular behavior of the graphs shown
for this value. Furthermore, the reason behind
$\widehat{a}_{f}>\overline{a}_{f}$ can be explained by considering that the
longitudinal eigenvalue is larger than the orthogonal one, which makes the
orthogonal mode easier to excite. In fact, it can be seen in Figure 11 that in
the one bounce tail regime
$\frac{\overline{a}_{f}}{\widehat{a}_{f}}\approx\frac{\overline{\omega}}{\widehat{\omega}}$.

##  5 Concluding remarks

In this paper, we have investigated the interaction between the translational
mode and the shape modes in a two-component scalar field theory in the context
of kink-antikink scattering processes when only the mode corresponding to the
second field component is initially activated. The energy transfer mechanism
between modes has already been discussed, both analytically and numerically in
[52], but only for a static kink whose orthogonal mode had initially been
triggered. The results presented in this work are therefore a natural
continuation of the aforementioned paper, shedding light on how the resonant
energy mechanism affects the fractal structure found in the velocity diagrams.
The presence of two different shape modes makes the energy transfer mechanism
more complex than when wobbler collisions were studied in the $\phi^{4}$
model. Since there are now two different discrete eigenmodes instead of just
one, there is less kinetic energy available for the translational mode, which
explains why the escape velocity of the kink-antikink pair after the last
collision $v_{f}$ is never greater than the initial kink velocity collision
$v_{0}$. This is in contrasts to what was found in the $\phi^{4}$ model, since
in the latter, when the amplitude of the only shape mode was triggered with
enough amplitude, $v_{f}$ could be larger than $v_{0}$.

Another notable phenomenon found in this study is the changing of the velocity
diagrams as $\widehat{a}_{0}$ and $\sigma$ change their value. For low values
of $\sigma$, when $\widehat{a}_{0}$ increases, one-bounce windows were created
in the resonant part of the velocity diagram. However, for high values of
$\sigma$ and a small value for $\widehat{a}_{0}$ the velocity diagram
resembles the behavior found for an unexcited kink in the $\phi^{4}$ field
theory. This phenomenon does not manifest itself for higher values of the
initial orthogonal amplitude, since in the latter case the resonant part of
the diagram becomes chaotic and the amplitude of the oscillations of the one-
bounce regime decreases.

The system response changes dramatically when the value of $\sigma$ is set to
$\sqrt{7}/2$. Under these conditions, the orthogonal mode discharges most of
its energy into the longitudinal mode and radiation modes. This implies that
the longitudinal mode is activated with a certain amplitude before the
scattering process. Such circumstance clarifies the similarity between the
velocity diagrams for the resonance value $\sigma_{1}=\sqrt{7}/2$ and the
velocity diagrams coming from the wobbler scattering in the $\phi^{4}$ model
[21, 22].

In addition to what has already been explained, it is worth highlighting the
correspondence between the maxima and minima of the final wobbling amplitudes
explained in detail in Section 4.2, which has shed light on the energy
transfer mechanism between modes which, as has being shown, strongly depends
on the value of $\sigma$ since the orthogonal mode can only be triggered again
after the scattering process for values of $\sigma\approx 1$.

As a future line of research that serves as a natural continuation of this
work, we consider extending the numerical techniques presented here in order
to study the scattering of excited topological defects defined in higher
dimensions. Among the innumerable possibilities that are contemplated in this
regard, one of them is generalize the results found for the excited vortex
scattering process in the Abelian-Higgs model recently presented in [39, 38]
studying the same situation in generalized models that support the existence
of vortices. Another potential avenue of exploration considered involves the
study of kink-antikink collisions in two-component scalar field models whose
kink solutions possess more than three internal modes. In this regard, a
suitable candidate for this research would be the two-component coupled
$\phi^{4}$ model [27, 59], in which the internal structure and the possible
number of eigenmodes depend on the value of the coupling constant that the
potential defining the model depends on.

## Acknowledgments

This research was supported by the European Union-Next Generation
UE/MICIU/Plan de Recuperacion, Transformacion y Resiliencia/Junta de Castilla
y Leon (PRTRC17.11), and also by RED2022-134301-T and PID2020-113406GB-I00,
both financed by MICIU/AEI/10.13039/501100011033. DMC acknowledges financial
support from the European Social Fund, the Operational Program of Junta de
Castilla y Leon and the regional Ministry of Education. This research has made
use of the high-performance computing resources of the Castilla y León
Supercomputing Center (SCAYLE), financed by the European Regional Development
Fund (ERDF).

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