chapter0in0.5in section0in0.5in subsection0in0.5in Interactions between
topological defects in (1+1) dimensions João Guilherme Ferreira Campos Recife
2022 Profa. Dra. Azadeh Mohammadi Universidade Federal de Pernambuco Centro de
Ciências Exatas e da Natureza Departamento de Física Tese de Doutorado Tese
apresentada ao Programa de Pós-Graduação em Física da Universidade Federal de
Pernambuco, como requisito parcial para obtenção do título de Doutor em
Física. english

See capa See contra See catalogo See assinatura

Aos meus pais, Itaciana Maria de Souza Ferreira e Hélio Araújo Cavalcanti
Campos Filho e à minha irmã Mariana Lídia Ferreira Campos.

AGRADECIMENTOS

Gostaria de agradecer ao departamento de física da UFPE, por ter proporcionado
o ambiente para a realização desse trabalho e à professora Azadeh Mohammadi,
pela leveza em sua condução durante o árduo processo que é o doutorado. Além
disso, gostaria de agradecer aos meus colegas do grupo de física de altas
energia pelas discussões e ao professor Mauro Copelli por permitir o uso do
cluster de seu laboratório, sem o qual essa pesquisa não seria possível.
Gostaria de agradecer também ao CNPq, à CAPES, e à UFPE pelo apoio financeiro.

###### Abstract

In this thesis, we study interactions between topological defects in two-
dimensional spacetimes. These defects are called kinks. They are solutions of
scalar field theories with localized energy which propagate without losing its
shape. In order to understand the resonance phenomenon exhibited by those
models, we built a toy model where the kink’s vibrational mode turns into a
quasinormal mode. This causes the suppression of resonance windows and,
consequently, its fractal structure is lost. Considering a higher order
polynomial as the scalar field potential, we find kinks with long-range tails,
which decay as a power law. We developed a numerical method to correctly
initialize this systems and applied it to a scalar field model containing
kinks with long-range tails in both sides. After the collision, the kink-
antikink pair is annihilated for velocities below an ultra-relativistic
critical velocity without bion formation. We also investigated a collision
between wobbling kinks of the double sine-Gordon model. When the kinks are
already wobbling before colliding, there appears resonance windows with
separation after a single bounce. On the second half of the thesis, we focused
on fermion-kink interactions. We studied what happens when a fermion binds to
a wobbling kink. The result is that the fermion escapes from the kink as
radiation and at a constant rate. This occurs if the energy gap between the
initial state and the continuum threshold is not too large. Lastly, we
investigated the interaction of a fermion with a background scalar field with
an impurity that preserves half of the Bogomol’nyi–Prasad–Sommerfield (BPS)
property. We found an adiabatic evolution near the BPS regime, which means
that the system is at a static BPS solution at every moment.

Keywords: topological defect; field theory; kink; collision; normal and
quasinormal modes; fermion.

RESUMO

Nesta tese, estudamos interações entre defeitos topológicos em uma dimensão
espacial e uma temporal. Esses defeitos são chamados de kinks. Eles são
soluções de teorias de campos escalares que possuem energia localizada e se
propagam sem perder sua forma. Para entender melhor o fenômeno de ressonância
exibido nesses modelos, construímos um modelo simplificado onde o modo de
vibração do kink se torna um modo quasinormal. Isso acarreta na supressão das
janelas de ressonância e, consequentemente, na perda da estrutura fractal
formada pelas mesmas. Já no caso em que o pontencial do campo escalar é um
polinômio de ordem alta, a cauda do kink pode ser de longo alcance, por decair
como uma lei de potência. Nós desenvolvemos um método numérico para
inicializar esses sistemas corretamente e aplicamos a um modelo de kinks com
caudas longas em ambos os lados. Após a colisão, o sistema se aniquila para
velocidades abaixo de uma velocidade crítica ultra-relativística e não forma
bions. Também investigamos uma colisão entre kinks vibrantes dentro do modelo
sine-Gordon duplo. Um dos efeitos de excitar o modo de vibração antes da
colisão é a formação de janelas de ressonância onde ocorre apenas um contato
entre os kinks. Já na segunda metade da tese, focamos em interações entre
kinks e férmions. Estudamos o que acontece com o férmion quando ele se liga a
um kink que está vibrando. O férmion escapa do kink na forma de radiação a uma
taxa contínua se a diferença de energia entre o estado inicial e o contínuo
não for muito grande. Por último, nós investigamos a interação de um férmion
na presença de um campo escalar com uma impureza que preserva metade da
propriedade Bogomol’nyi–Prasad–Sommerfield (BPS) do sistema. O resultado desse
processo é que perto do regime BPS a evolução do sistema é adiabática, pois
sempre corresponde a uma solução BPS estática.

Palavras-chave: defeito topológico; teoria de campos; kink; colisão; modos
normais e quasinormais; férmion.

###### CONTENTS

  1. 0 INTRODUCTION
    1. 1 Solitons, solitary waves and topological defects
    2. 2 A little bit of history
      1. 1 Integrable models
      2. 2 Interactions between non-integrable kinks
    3. 3 Some relevant works
    4. 4 Applications of kink models
  2. 1 INTRODUCTION TO KINK-ANTIKINK COLLISIONS
    1. 1 The $\phi^{4}$ model
    2. 2 Sine-Gordon model
    3. 3 kink-antikink collision in the $\phi^{4}$ model
  3. 2 KINK INTERACTIONS WITH QUASINORMAL MODES
    1. 1 Overview
    2. 2 Model
    3. 3 Stability equation
    4. 4 Collision
    5. 5 Conclusion
    6. 6 Appendix: Numerical method
    7. 7 Appendix: Transmission and reflection coefficients
    8. 8 Appendix: Bound states of the square well potential
  4. 3 INTERACTIONS OF KINKS WITH DOUBLE LONG-RANGE TAILS
    1. 1 Overview
    2. 2 Model
    3. 3 Inter-kink force
    4. 4 Numerical method
    5. 5 Collisions
    6. 6 Conclusion
  5. 4 WOBBLING DOUBLE SINE-GORDON KINKS
    1. 1 Overview
    2. 2 Model
    3. 3 Collision simulations
    4. 4 Conclusion
    5. 5 Appendix: Numerical method
  6. 5 FERMION-KINK SYSTEMS
    1. 1 Definition and motivation
    2. 2 Literature revision
    3. 3 Some theoretical aspects
    4. 4 Appendix: Fermion bound states of the $\phi^{4}$ model
  7. 6 FERMIONS ON WOBBLING KINKS
    1. 1 Overview
    2. 2 Model
    3. 3 $\phi^{4}$ model
    4. 4 Toy model
      1. 1 Normal mode
      2. 2 Quasinormal mode
    5. 5 Conclusion
    6. 6 Appendix: Numerical method
  8. 7 FERMION TRANSFER IN THE $\phi^{4}$ MODEL WITH AN IMPURITY
    1. 1 Overview
    2. 2 Model
      1. 1 Lagrangian and field equations
      2. 2 Scalar field solutions
      3. 3 Fermion bound states
    3. 3 Results
      1. 1 Scalar field collisions
      2. 2 Bogoliubov coefficients
      3. 3 Adiabatic evolution
      4. 4 Relativistic evolution
    4. 4 Conclusion
    5. 5 Appendix: Numerical method
  9. 8 FINAL REMARKS

*

## 0 INTRODUCTION

### 1 Solitons, solitary waves and topological defects

We will start this thesis by defining our objects of study. These are
solitons, solitary waves and topological defects. The definition of solitons
can be found, for instance, in the excellent book by Rajaraman [Rajaraman
1982]. A soliton is a nontrivial solution of a wave field theory with three
properties

  * •

It has localized and finite energy;

  * •

It propagates without losing the shape;

  * •

When two or more solitons interact, they emerge with the same velocities as
the initial ones. The only effect of the interaction is a displacement from
their original trajectories.

The first two properties define a solitary wave. When the third property is
included, the solution is called a soliton.

In many cases, these solutions are called topological defects. We use this
term when the wave properties and stability come from topological reasons. To
understand that, let us first recall what topology is. Topology is the study
of properties that are preserved when a geometric object is continuously
deformed. A topological defect can be defined as a solution that cannot be
continuously deformed to the trivial vacuum without reaching an infinite
energy configuration. Therefore, the topology of the solution guarantees its
stability. We will give a concrete example of how this works when we analyze
the $\phi^{4}$ model.

### 2 A little bit of history

This section will discuss a few historical discoveries regarding solitons and
solitary waves. We will give special attention to solutions in (1+1)
dimensions, one time and one spatial dimension. These types of solutions are
the subject of this thesis.

#### 1 Integrable models

A model is called integrable when it possesses an infinite number of conserved
quantities. In general, models that exhibit solitons are integrable because
the conserved quantities allows the initial configuration to be recovered
after a collision, making it ellastic. The study of those systems was very
important for the understanding of topological defects in field theories. For
this reason, let us review a few key results. The first documented
experimental observation of solitons dates back to the observation of John
Scott Russell in 1834. He observed a mass of water that propagated in a narrow
channel. He followed the mass for one or two miles and observed that it
retained its shape during the whole path. In 1895, Kortweg and de Vries (KdV)
found an equation describing waves’ propagation in a narrow channel with
shallow water. This equation was able to describe the phenomenon observed by
John Scott Russel. An important method used to solve the KdV equation
analytically is the inverse scattering method. It was discovered 1967 [Gardner
et al. 1967]. The history of the inverse scattering method is summarized, for
instance in [Novikov et al. 1984]. This method is very powerful and allows one
to find all the exact solutions of the KdV equation. With this formalism, it
was proven that the KdV equation is integrable and exhibits solitons. A few
years later, other equations such as the nonlinear Schrödinger equations and
the sine-Gordon model could also be solved by the inverse scattering method.
This also means that they are integrable and exhibit solitons.

#### 2 Interactions between non-integrable kinks

In integrable theories, we obtain soliton solutions. Their interaction is
elastic and can be computed analytically. On the other hand, interactions
between topological defects in non-integrable models are much more complex.
Their investigation started later because it had to wait for the development
of modern computers. These defects do not collide elastically due to the non-
integrable behavior and are actually solitary waves.

In (1+1) dimensions, topological defects are usually called kinks. In general,
the kink solution possesses a reverse solution, the antikink. The sine-Gordon
and $\phi^{4}$ models are examples of models containing kinks. The former is
integrable, while the latter is non-integrable. One of the earliest
investigations of interaction in the $\phi^{4}$ model is the work by Sugiyama
[Sugiyama 1979]. There, the author analyzed the collision between a kink and
an antikink. He showed that after a critical velocity, the two reflect and,
below that velocity, they form a long-lived bound state. Nowadays, this bound
state is called a bion. Moreover, Sugiyama also proposed an effective model
for the kink-antikink system. This effective model is called the collective
coordinates method and consists of reducing the infinite degrees of freedom of
a kink-antikink system to only two: the relative position and the vibration
amplitude. However, the equations had a typo that was only corrected many
years later [Kevrekidis e Goodman 2019, Takyi e Weigel 2016].

A few years after the work of Sugiyama, Campbell et al. published a triplet of
seminal papers [Campbell, Schonfeld e Wingate 1983, Peyrard e Campbell 1983,
Campbell, Peyrard e Sodano 1986]. There, the authors also observed that, in a
collision, the kink and the antikink reflect after a critical velocity.
However, below the critical velocity, there are also regions where the
kinks111Sometimes, we will abbreviate kink and antikink by the term kinks.
reflect, but only after multiple bounces. These regions alternate between
regions where the kinks annihilate and are called resonance windows. They
argued, using simplified models, that the resonances occur due to a resonant
energy exchange mechanism between the translational and vibrational modes of
the kink. This mechanism states that, in the first bounce, the translational
energy of the kink is transformed into vibrational energy. If the
translational energy is not large enough afterward, the kinks will not be able
to separate and will collide again due to the mutual attraction. However, the
vibrational energy can be converted back into translational energy at
subsequent bounces if the timing is right. This would allow the kinks to
separate. Moreover, the authors found that at the border of two-bounce
resonance windows, there is a nested structure of three-bounce windows, while,
at the border of three-bounce resonance windows, there is a nested structure
of four-bounce resonance windows, and so on.

A few years later, Anninos et al. showed numerically that the resonance
windows exhibit a fractal structure [Anninos, Oliveira e Matzner 1991]. This
result was backed up by an analysis of the reduced Sugiyama model. At the
time, the authors were not aware of the typo, and surprisingly, the reduced
model worked exceptionally well. This analysis was considered as a
quantitative explanation of the resonant energy exchange mechanism. Many years
later, the typo was corrected [Takyi e Weigel 2016], and the resemblance of
the reduced model with the full one was lost. Even worse, the reduced model
was singular for vanishing separation. More recently, Manton et al. showed
that the collective coordinates model had a poor choice of coordinates [Manton
et al. 2021, Manton et al. 2021]. After finding a more suitable set of
coordinates, they were able to reproduce the behavior of the full system very
well. Thus, the resonant energy exchange mechanism finally acquired a correct
quantitative explanation.

### 3 Some relevant works

There are many excellent books about solitons, where the essential theoretical
aspects are discussed. To cite a few, see [Vilenkin e Shellard 2000, Manton e
Sutcliffe 2004, Vachaspati 2006, Rajaraman 1982].

The area of interactions between kinks is a very active field of research with
many theoretical developments. Let us briefly mention some relevant works in
the field. The reduced Sugiyama model was discussed in detail in [Goodman e
Haberman 2005, Goodman e Haberman 2007]. The interaction of kinks with
impurities was discussed in [Kivshar, Fei e Vázquez 1991, Fei, Kivshar e
Vázquez 1992, Fei, Kivshar e Vázquez 1992, Malomed 1992, Goodman e Haberman
2004]. Interestingly, when kinks interact with impurities, resonance windows
also appear due to a vibrational mode of the impurity.

The vibrational mode of a single kink was studied in [Manton e Merabet 1997,
Barashenkov e Oxtoby 2009, Oxtoby e Barashenkov 2009]. The important property
is that the vibration amplitude decays slowly in time due to the coupling with
the radiation modes. This property was also observed in oscillons
[Romańczukiewicz e Shnir 2018].

The double sine-Gordon model, which is a modification of the integrable sine-
Gordon, was studied perturbatively in [Malomed 1989, Kivshar e Malomed 1989,
Kivshar e Malomed 1987] using the inverse scattering method. This is an
interesting modification of the sine-Gordon model, which is not integrable. A
significant result is that there is still no energy and momentum transfer
between the kinks in a collision in first-order perturbation theory. Then,
this model was studied numerically in [Campbell, Peyrard e Sodano 1986, Gani e
Kudryavtsev 1999, Gani et al. 2018, Gani, Marjaneh e Saadatmand 2019, Simas et
al. 2020]. It contains kink solutions that are composed of two sine-Gordon
subkinks. This means that the kinks have an inner structure. This property is
also found in other models such as [Zhong et al. 2020, Dorey et al. 2021]. The
double sine-Gordon model possesses resonance windows due to the presence of a
vibrational mode. However, we will see in section 4 that near the integrable
regimes, the resonance windows are hidden. The work we developed on collisions
between wobbling kinks in this model was published in [Campos e Mohammadi
2021], and will be discussed in detail in the aforementioned section.

Polynomial potentials are very important due to their simplicity. They are
usually called $\phi^{n}$, where $n$ is the polynomial order. Modification of
the $\phi^{4}$ model have been discussed in [Bazeia, Belendryasova e Gani
2018, Gomes et al. 2018, Simas et al. 2016, Yan et al. 2020], the $\phi^{6}$
model has been studied in [Dorey et al. 2011, Gani, Kudryavtsev e Lizunova
2014]. Modification of the $\phi^{6}$ model has been studied in [Demirkaya et
al. 2017] and a hybrid model between $\phi^{4}$ and $\phi^{6}$ has been
studied in [Bazeia et al. 2019]. The $\phi^{6}$ model is an interesting
exception to the resonant energy exchange mechanism because it exhibits
resonance windows even though the kink does not have a vibrational mode. In
[Dorey et al. 2011], the authors showed that the resonance appears due to the
exchange of the translational energy with a vibrational mode of the kink-
antikink pair.

Scalar field theories $\phi^{n}$ containing higher-order polynomials with
$n\geq 8$ were studied in [Gomes, Menezes e Oliveira 2012, Khare, Christov e
Saxena 2014, Gani, Lensky e Lizunova 2015, Bazeia, Menezes e Moreira 2018,
Belendryasova e Gani 2019, Manton 2019, Khare e Saxena 2019, Christov et al.
2019, Christov et al. 2019, Christov et al. 2021]. Interestingly, in these
theories, the kinks may contain one or two long-range tails, which decay as a
power-law. The simulation of these systems requires specialized methods, such
as the method we developed in [Campos e Mohammadi 2021]. It will be discussed
in section 3. The opposing limit, where the kink possesses short-range tails,
was discussed in [Bazeia, Gomes e Simas 2021].

An interesting question is whether the resonance phenomenon still exists when
the vibrational mode of the kink is turned into a quasinormal mode. The reason
is that it tests other regimes where the resonant energy exchange mechanism
may be valid. A quasinormal mode is a solution of the kink stability equation
that obeys purely outgoing boundary conditions. In some cases, including the
one we are interested, it signals that the kink had a normal mode that became
unstable and can slowly decay. The aforementioned question was studied in
[Dorey e Romańczukiewicz 2018]. The authors found that the resonance structure
is gradually lost as the decay rate of the quasinormal mode increases. We also
studied this effect for a toy model in [Campos e Mohammadi 2020]. It will be
discussed in section 2. In other models, the potential approaches a vacuumless
configuration, which has a local maximum but no local minimum. In this limit,
the vibrational mode of the kink ceases to exist. Thus, the resonance windows
also disappear [Bazeia e Moreira 2017, Simas, Gomes e Nobrega 2017].

One can also study a scalar field in the half-line subject to a boundary
condition at the origin [Arthur, Dorey e Parini 2016, Dorey et al. 2017, Lima
et al. 2019]. In this case, a great variety of processes can occur, and in
many cases, the system exhibits resonance. Another type of study with a great
variety of outcomes is multi-kink collisions [Marjaneh et al. 2017, Marjaneh
et al. 2017, Marjaneh et al. 2018]. In those works, the authors studied the
interaction of three and four kinks colliding at a single point. One can also
study scalar fields with two components [Halavanau, Romanczukiewicz e Shnir
2012, Alonso-Izquierdo 2018, Alonso-Izquierdo 2020, Alonso-Izquierdo et al.
2021]. This system also exhibits a wide variety of behaviors due to the
increasing complexity.

In scalar field models, the field is also interacting with radiation.
Interestingly, this interaction may create kink-antikink pairs
[Romańczukiewicz 2006, Dutta, Steer e Vachaspati 2008, Romanczukiewicz e Shnir
2010]. Also radiation incident on a kink generates negative radiation pressure
[Forgács, Lukács e Romańczukiewicz 2008]. This could explain the absence of
topological defects in a cosmological scale [Romańczukiewicz 2017].

Another interesting question is how the kink-antikink collision is modified if
the vibrational mode is excited beforehand. This was studied in [Izquierdo,
Queiroga-Nunes e Nieto 2021]. The authors argued that this investigation is
relevant to understanding the resonance phenomenon because the multiple
bounces in resonance can be viewed as an iteration of wobbling kink
collisions.

Finally, we would like to mention a series of papers about scalar field models
with an impurity that preserves half of the Bogomol’nyi–Prasad–Sommerfield
(BPS) property [Manton, Oleś e Wereszczyński 2019, Adam et al. 2019, Adam et
al. 2019, Adam, Romanczukiewicz e Wereszczynski 2019, Adam, Queiruga e
Wereszczynski 2019, Adam et al. 2020]. Solutions with BPS property are static
and stable, meaning that they have vanishing force. Interestingly, the BPS
sector in these models has nontrivial dynamics. The authors were able to show
a phenomenon called a spectral wall. A spectral wall appears when a
vibrational mode of the BPS solution enters a continuum. An excited defect
passing through a spectral wall experiences a force that may cause it to be
reflected by the wall. This phenomenon can be isolated in those models because
the inter-defect force vanishes. Interestingly, this phenomenon was also
observed in models with two scalar fields without the need for an impurity
[Adam et al. 2021]. Moreover, the authors found a model which displays another
exception to the resonant energy exchange mechanism. They found that the
resonant structure appears due to the exchange between the translational
energy with the vibrational mode of a sphaleron, an unstable static solution
[Adam et al. 2021].

### 4 Applications of kink models

Many systems in physics are described by fields. Moreover, if they are
effectively one-dimensional in space and have degenerate lowest energy states,
they probably will exhibit kinks. Examples of such models are deformation in
polyacetylene [Su, Schrieffer e Heeger 1979] and in graphene [Yamaletdinov,
Slipko e Pershin 2017], the potential in Josephson junctions [Ustinov 1998],
vortices in superfluid Helium-3 [Volovik 2003], properties of DNA [Yakushevich
2006], signals in optical fibers [Mollenauer e Gordon 2006] and in surface
displacement in a long channel of shallow liquid [Denardo et al. 1990].
Moreover, kinks can be used to describe domain walls in the direction
perpendicular to the wall. Domain walls appear in magnetic materials [Kardar
2007, Koyama et al. 2011, Parkin, Hayashi e Thomas 2008]. Moreover, in some
cosmological theories, the universe is described by a domain wall [Rubakov e
Shaposhnikov 1983] and the Big Bang by domain wall collisions [Khoury et al.
2001].

Polyethylene is another effectively one-dimensional system that exhibit
topological defects. These are called twistons. They are related to 180∘
degrees twists in the polymeric chain and can be described by coupled scalar
fields [Bazeia e Ventura 1999]. We should also mention that one-dimensional
solitons also appear in optics [Radhakrishnan e Lakshmanan 1995, Abdullaev et
al. 2014], but these solitons are not kinks. The same is true for systems
described by the KdV equation, such as the soliton observed by John Scott
Russell and deformations in plasma [Chen 2016].

As one of the simplest topological defects available, kinks serve as a guide
to study more complicated defects in higher dimensions. It is more difficult
to study these structures in higher dimensions because they are much harder to
simulate numerically. Hence, a better understanding of kinks can elucidate the
behavior of higher dimensional topological defects. Examples of such defects
are vortices in superconductors [Abrikosov 2004, Auslaender et al. 2009,
Fetter e Hohenberg 2018], knots in fluids [Kleckner e Irvine 2013], instantons
in quantum field theory [Polyakov e Son 2018, Schaefer 2002, Schneider,
Torgrimsson e Schützhold 2018], magnetic skyrmions [Fert, Reyren e Cros 2017]
and skyrmions in nuclear physics [Skyrme 1962]. Other topological defects that
are relevant theoretically are cosmic strings [Hindmarsh, Rummukainen e Weir
2016] and monopoles [Hooft 1974, Csáki et al. 2018]. All these structures
appear in either in real physical systems or effective models. There are some
contexts where they have not been observed yet, but it is possible that they
will be observed in the future.

## 1 INTRODUCTION TO KINK-ANTIKINK COLLISIONS

In this main section, we will discuss two prototypical models that exhibit
kink solutions. The models are the sine-Gordon, which is integrable, and the
$\phi^{4}$, non-integrable. Then we will discuss collisions in these models
and discuss how they differ. None of the results in this main section is
original. They can be found in [Rajaraman 1982, Manton e Sutcliffe 2004,
Vachaspati 2006] for instance. However, the section serves as an introduction
to the more complicated cases discussed in the following main sections.

### 1 The $\phi^{4}$ model

One of the simplest examples that exhibits kinks is the $\phi^{4}$. It
consists of a single scalar field theory in $(1+1)$ dimensions described by
the following Lagrangian

$\mathcal{L}=\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-V(\phi).$ (1)

We are considering the metric $g_{00}=-g_{11}=1$ and $g_{01}=g_{10}=0$. This
Lagrangian describes relativistic theories in units where $c=\hbar=1$ as well
as non-relativistic wave equations in units where the wave velocity has been
set to unit. It leads to the following equation of motion

$\partial_{t}^{2}\phi-\partial_{x}^{2}\phi+V^{\prime}(\phi)=0.$ (2)

So far, this is very general. However, for the $\phi^{4}$ model we set
$V(\phi)=\frac{\lambda}{2}(\phi^{2}-\eta^{2})^{2}$. The constant $\lambda$ has
units of mass squared, and the constant $\eta$ is dimensionless. They will
depend on the system that we are modeling. However, we can remove these
constants by working with adimensional units. For the $\phi^{4}$, this is done
by setting $\phi\to\eta\phi$ and $x^{\mu}\to x^{\mu}/(\sqrt{\lambda}\eta)$.
The rescaled system possesses the following potential
$V(\phi)=\frac{1}{2}(\phi^{2}-1)^{2}$. In the following sections, we will
always work with adimensional units. The rescaled potential contains two
symmetric vacua at $\phi=\pm 1$ as shown in Fig. 1. In this case the $Z_{2}$,
$\phi\to-\phi$, symmetry is spontaneously broken.

Figure 1: (left) Potential and (right) kink solution for the $\phi^{4}$ model.
Source: The author (2022).

The equation of motion for the $\phi^{4}$ model becomes

$\partial_{t}^{2}\phi-\partial_{x}^{2}\phi+2\phi(\phi^{2}-1)=0.$ (3)

We see that the trivial vacua $\phi=\pm 1$ are solutions to this equation. We
are looking for static solutions which are nontrivial with finite energy.
First, let us take a look at the system’s total energy. It can be easily found
from the Lagrangian. The result is

$E=\int
dx\left[\frac{1}{2}(\partial_{t}\phi)^{2}+\frac{1}{2}(\partial_{x}\phi)^{2}+V(\phi)\right].$
(4)

From this expression, we see that a static solution with finite energy must
reach a vacuum configuration for the boundaries $x\to\pm\infty$, that is,
$V(\phi(x\to\pm\infty))=0$. This is necessary for the integral to converge.
For the $\phi^{4}$ model, the vacuum is degenerate. It can be either $\phi=\pm
1$. If the vacua at the limits $x\to\pm\infty$ are identical, the static
solution is the trivial vacuum, which has zero total energy. If the vacuum at
the limits $x\to\pm\infty$ is different, we may have nontrivial, static, and
finite energy solutions111A solution is nontrivial when it depends on the
position $x$. These three properties only appear if the topology is also
nontrivial, which means that the value of the field at $\pm\infty$ are
different. It is also possible to have nontrivial solution (depending on $x$)
with trivial topology, but these are not static. This can be proven using the
static equation of motion, which will be derived below..

Now, we will derive these solutions with nontrivial boundary conditions. The
static kink solutions can be found from eq. (2) after setting the time
derivatives to vanish. We find

$\partial_{x}^{2}\phi=V^{\prime}(\phi).$ (5)

Multiplying this equation by $\partial_{x}\phi$ on both sides and integrating,
we obtain

$\frac{1}{2}(\partial_{x}\phi)^{2}=V(\phi).$ (6)

The constant of integration is zero due to the boundary conditions obtained by
the finite energy condition. Substituting the $\phi^{4}$ potential, this
equation is easily integrated and has two nontrivial solutions. The first one
is the kink

$\phi_{K}(x)=\tanh(x-x_{0}).$ (7)

It interpolates between $-1$ and $1$. The integration constant $x_{0}$ is the
center of the kink. The freedom to set the center at any point is related to
the translation symmetry of the system. There is also the antikink solution,
which interpolates between $1$ and $-1$

$\phi_{\bar{K}}(x)=-\tanh(x-x_{0}).$ (8)

These solutions are shown in Fig. 1. It is easy to see that the energy density
is $\text{sech}^{4}(x-x_{0})$ and total energy is $E=4/3$. As the model obeys
Lorentz symmetry, to find the kink solutions in a boosted frame, one only
needs to Lorentz transform the coordinates. The boosted kink solution is

$\phi_{K}(x,t)=\tanh(\gamma(x-x_{0}\pm vt)),$ (9)

where $v$ is the kink’s velocity and $\gamma\equiv 1/\sqrt{1-v^{2}}$.

Due to the boundary conditions, the kink and antikink solutions are stable and
cannot decay into the trivial vacuum. To prove this property, let us define a
topological current

$j^{\mu}=\frac{1}{2}\epsilon^{\mu\nu}\partial_{\nu}\phi.$ (10)

This current is conserved by construction due to the presence of the
antisymmetric tensor $\epsilon^{\mu\nu}$. The conserved topological charge of
this current, or winding number, is

$Q=\int dx\,j^{0}=\frac{1}{2}[\phi(x=+\infty)-\phi(x=-\infty)].$ (11)

Therefore, the kink cannot decay to the trivial vacuum because they have
different winding numbers. Of course, if we consider finite boundaries, the
winding number is not conserved anymore.

Now, we would like to study what happens to the field when the kink is
perturbed. The stability equation for fluctuations around the kink solution
can be obtained by considering the behavior of solutions close to the kink
solution. We write $\phi=\phi_{K}+\psi\cos(\omega t)$, where $\psi$ is a small
perturbation. Substituting in the equations of motion, it yields

$H\psi\equiv-\partial_{x}^{2}\psi+V^{\prime\prime}(\phi_{K})\psi=\omega^{2}\psi.$
(12)

It can be shown that the eigenvalues of this equation are non-negative and,
therefore the kink is stable. For the $\phi^{4}$ model, it reads

$-\partial_{x}^{2}\psi+(6\tanh^{2}(x)-2)\psi=\omega^{2}\psi,$ (13)

which is a Pöschl-Teller potential with discrete solutions $\omega^{2}=0,3$.
The existence of a mode with $\omega=0$, called zero mode, is expected due to
the translation invariance of the system. The mode with $\omega=\sqrt{3}$ is
the vibrational mode of the kink, also known as shape mode. Its solution is
given by $\psi_{S}=\text{sech}(x)\tanh(x)$.

Another way to obtain the equation for a static kink is to complete squares in
the energy functional. This is done as follows

$\displaystyle E$ $\displaystyle=\int
dx\left[\frac{1}{2}(\partial_{t}\phi)^{2}+\frac{1}{2}(\partial_{x}\phi)^{2}+V(\phi)\right]$
(14) $\displaystyle=\int
dx\left[\frac{1}{2}\left(\partial_{t}\phi\right)^{2}+\frac{1}{2}\left(\partial_{x}\phi\mp\sqrt{2V(\phi)}\right)^{2}\pm\sqrt{2V(\phi)}\partial_{x}\phi\right]$
(15)
$\displaystyle\geq\int^{\phi(\infty)}_{\phi(-\infty)}d\phi^{\prime}\sqrt{2V(\phi^{\prime})}$
(16)

Therefore, the energy is minimized, for fixed boundary conditions, when the
equality is satisfied. Consequently, the action will be extremized. The
equality occurs for static fields obeying

$\partial_{x}\phi=\pm\sqrt{2V(\phi)}.$ (17)

This equation is know as the Bogomol’nyi–Prasad–Sommerfield (BPS) [Prasad e
Sommerfield 1975, Bogomol’Nyi 1976]. It is equivalent to eq. (6). This
construction from the energy functional is important because it is also used
to simplify the equations of motion in more complicated models.

### 2 Sine-Gordon model

The most familiar example of a soliton is the sine-Gordon model. It is a
soliton because the model is integrable with infinite conserved quantities.
Hence, the solitons emerge after a collision with the same velocity as before
and the only effect of the interaction is a time delay in the original
trajectory. The Lagrangian for this model is given by

$\mathcal{L}=\frac{1}{2}(\partial_{\mu}\phi)^{2}+(\cos\phi-1)$ (18)

The BPS equation can be easily solved. It yields the following kink solution

$\phi_{K}(x)=4\tan^{-1}\left(e^{x-x_{0}}\right).$ (19)

The solution and potential are shown in Fig. 2. Due to the periodicity of the
potential, one can obtain a different kink solution for any integer $n$ by
adding a constant $2\pi n$ to the field. Moreover, one can obtain an antikink
solution reversing the field. It is simple to show that these solutions have
energy density $4\text{sech}^{2}(x-x_{0})$ and total energy $E=8$.

Figure 2: (left) Potential and (right) kink solutions for the sine-Gordon
model. Source: The author (2022).

Due to the integrability of the model, there are exact solutions for soliton
collision and oscillating bound states, which are called breathers. They can
be obtained using specialized methods like the Bäcklund transformation or the
inverse scattering method. We will not derive these solutions but merely state
them. A breather is described by

$\phi_{v}(x,t)=4\tan^{-1}\left[\frac{\sin(\gamma vt)}{v\cosh(\gamma
x)}\right],$ (20)

where $v$ is a parameter related to the frequency of the breather and
$\gamma=1/\sqrt{1-v^{2}}$. The evolution of this field configuration in
spacetime is shown in Fig. 3.

Another example is the kink-antikink collision

$\phi_{KA}(x,t)=4\tan^{-1}\left(\frac{\sinh(\gamma vt)}{v\cosh(\gamma
x)}\right).$ (21)

One can show that the asymptotic behavior of this solution for $t\to-\infty$
describes widely separated kink and antikink approaching each other.
Similarly, for $t\to\infty$, it consists of widely separated kink and antikink
moving away from each other with the same velocity but with a time delay. This
is what one expects from an interaction between solitons. The evolution of
this field configuration in spacetime is shown in Fig. 3.

Figure 3: Evolution in spacetime of (left) a breather solution and (right) a
kink-antikink collision. The model considered is the sine-Gordon and we set
$v=0.2$. Source: The author (2022).

### 3 kink-antikink collision in the $\phi^{4}$ model

Now, let us consider a kink-antikink configuration. In this case, there is no
analytical expression, but we can find good approximations. One widely used
approximation is the additive ansatz

$\phi(x,t)=\phi_{K}(x+x_{0})+\phi_{\bar{K}}(x-x_{0})-1.$ (22)

If $x_{0}$ is large, the kinks are far away from each other and can be
linearly superposed. This field configuration approximates a kink and an
antikink located at $-x_{0}$ and $+x_{0}$, respectively. To understand the
need for the constant $-1$, we can examine the ansatz for $x>0$. In this
region, we are far from the kink and, thus, the first term is approximately
$1$. Therefore, it is canceled by the third term, and only the second term is
left, as desired.

The kink-antikink configuration does not solve the BPS equation (17). In this
model, the BPS equation is equivalent to the static equation of motion (5) for
configurations with finite energy. Therefore, the kink-antikink configuration
is not static. In fact, there is an attractive force between the two. One can
show that, for a large separation $R=2x_{0}$, the force is given by
$F=32e^{-2R}$.

The additive ansatz can be used to study the interaction of boosted kink-
antikink configurations. In this case, the initial condition for the field
equations is given by the following expression

$\phi(x,t)=\phi_{K}(\gamma(x+x_{0}-vt))+\phi_{\bar{K}}(\gamma(x-x_{0}+vt))-1.$
(23)

The evolution of this solution can be obtained by integrating the equations of
motion numerically. By virtue of the non-integrability of the model, there are
more possible outcomes to this interaction. If the velocity is small, the
kinks annihilate, as shown in the left plot of Fig. 4. Before decaying into
the trivial vacuum, they form an oscillating bound state known as bion. This
bound state is long-lived but slowly decays by emitting radiation at each
bounce. Another possibility is shown in the right plot of Fig. 4. In this
case, the kinks bounce, reflect and separate to infinity. Surprisingly, there
is a third possibility called resonance. In this scenario, the kinks also
separate, but only after multiple bounces. A two-bounce scenario is shown in
the center plot of Fig. 4.

Figure 4: Evolution in spacetime of kink-antikink collisions in the $\phi^{4}$
model. The different scenarios are (left) annihilation, (center) resonance and
(right) reflection. Source: The author (2022).

The regions of initial velocity $v_{i}$ where each behavior occurs are
summarized in Fig. 5. In this figure, we show the final velocity $v_{f}$ as a
function of the initial one. The final velocity is computed as follows. While
we integrate the equations of motions, we keep track of the kink’s position.
In order to do so, we look for the position where $\phi=\phi_{K}(x=0)$. Then,
we wait until the distance between the kink and the origin is larger than
$\Delta x\in[10.0,15.0]$. If this separation is never reached, we set $v_{f}$
to zero, meaning that there is annihilation for the corresponding value of
$v_{i}$. If the kinks separate, we start using the position measurements to
compute the final velocity. We perform a linear regression using 80 position
values separated by $\Delta t=0.5$. Therefore, we obtain a nonzero $v_{f}$
when there is separation after one or multiple bounces. The region where there
is only a single bounce is drawn in blue. The other colors correspond to
multiple bounces. The regions where multiple bounces occur are called
resonance windows. The term $n$-bounce window can be used to describe
separation after $n$ bounces. Interestingly, there is a sequence of
$(n+1)$-bounce windows at the edge of $n$-bounce windows, forming a fractal
structure.

Figure 5: Final velocity $v_{f}$ as a function of the initial velocity $v_{i}$
for the $\phi^{4}$ model. The color code indicates the number of bounces
before separation. Source: Reproduction made by the author of results obtained
in [Izquierdo, Queiroga-Nunes e Nieto 2021].

The resonance phenomenon was explained in [Campbell, Schonfeld e Wingate
1983]. The mechanism behind its appearance is called the resonant energy
exchange mechanism. According to this mechanism, there is an exchange between
the translational and vibrational energy of the kink at each bounce. At the
first bounce, part of the translational energy is lost and is stored as
vibrational energy. Due to the mutual attraction, if the translational energy
is too small afterward, the kinks will be forced to bounce one more time. The
vibrational energy can be converted back into the translational energy at the
subsequent bounces, and the kinks may separate. Whether it happens or not
should be sensitive to the state of vibrational mode at the moment of the
subsequent bounces.

Figure 6: Evolution of the field at the origin for the first six two-bounce
resonance windows. We are considering the $\phi^{4}$ model. Source:
Reproduction made by the author of results obtained in [Campbell, Schonfeld e
Wingate 1983].

The authors plotted the field’s evolution at the origin of several two-bounce
resonance windows to prove this conjecture. The result for the first six two-
bounce windows is shown in Fig. 6. The figure shows that the field starts at
$1$ because the kinks are separated. Then, there is a sharp inverted peak
corresponding to the first bounce. After that, there is an important
oscillation of the field at the center and, after a second bounce, the kinks
separate. If it is hard to interpret the figure, try following the field at
the origin in the center plot of Fig. 4. The crucial point is that, as we move
from one resonance window to the next, the number of oscillations between
bounces increases by one. If $\omega$ is the angular frequency of oscillation
and $T_{n}$ is the time between bounces for the $n$-th two-bounce resonance
window, this can be written mathematically as

$\omega T_{n}=\delta+2\pi n,$ (24)

where $\delta$ is a constant. This means that whether the system separates
depends on the state of the vibration appearing between the sharp inverted
peaks in Fig. 6. To confirm the resonant energy exchange mechanism, the
authors had to show that this oscillation at the center corresponds to the
kink’s vibration measured at the origin. Comparing the numerical value of
$\omega$ with the theoretical value of the vibrational mode’s frequency
$\omega=\sqrt{3}$, they found an excellent agreement, confirming the exchange
mechanism.

One important consequence of the resonant energy exchange mechanism is that
resonance windows are absent when the kink does not have a vibrational mode.
As explained in the introduction, there are exceptions to this rule, but it is
generally valid. In the three subsequent main sections, we will study kink-
antikink collisions in three different models. These consist of a toy model
with quasinormal modes, kinks with double long-range tails, and double sine-
Gordon, respectively. The second half of the thesis will be dedicated to
studying fermion-kink systems.

## 2 KINK INTERACTIONS WITH QUASINORMAL MODES

### 1 Overview

The work described in this main section resulted in the following publication
[Campos e Mohammadi 2020]. We were interested in the effect of turning the
normal mode of the kink into a quasinormal mode (QNM). QNMs may appear in
models where the stability equation has a potential with a “volcano" shape. A
few examples are described in [Gomes, Menezes e Oliveira 2012, Bazeia e
Moreira 2017, Campos e Mohammadi 2021] and in section 3. This modification is
important, taking into consideration the role of the normal mode of the kink
on the resonant energy exchange mechanism. This mechanism states that the
existence of resonance windows in kink-antikink collisions is caused by the
energy exchange between the translational and vibrational modes of the kink.
The vibrational mode is excited at the first bounce and, if it is turned into
a QNM, its energy will start to leak, and only the remaining part will be
converted back to translational energy at the successive collisions.
Therefore, it is expected that in this case, the resonance structure will be
gradually lost.

The first work to investigate this effect was [Dorey e Romańczukiewicz 2018].
There, the authors constructed a modified $\phi^{4}$ model exhibiting a QNM.
They showed, numerically, that when the decay rate of the QNM is increased,
the resonance windows gradually disappear. This result was corroborated by a
numerical analysis of the reduced Sugiyama model. As discussed before, this
model has a typo, but the authors argued that it would be considered as a
phenomenological model.

Our contribution to this problem was to build a model that is similar to
[Dorey e Romańczukiewicz 2018] but simple enough to allow an analytical
solution of the kink profile and the QNM properties, such as shape, decay
rate, and frequency. Hence, we were able to corroborate the results in [Dorey
e Romańczukiewicz 2018], explicitly showing how the QNM mode appears in an
analytical model. In the next section, we describe the construction of the
model and its kink solution.

It is important to mention that most models possess an infinite discrete set
of QNMs. For the Schrödinger and Klein-Gordon equation the set of QNMs can be
found analytically for a few cases, as discussed in [Boonserm e Visser 2011].
However, according to the resonant energy exchange mechanism, the most
important excitation mode in kink-antikink collisions is the shape mode.
Therefore, we are actually interested in turning the shape mode into a QNM and
this is the only QNM that we will focus on.

### 2 Model

We start with the following scalar field Lagragian

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

where we assume that the system has $Z_{2}$ symmetry with $V(-\phi)=V(\phi)$
and has two symmetric vacua at $\pm\phi_{0}$. Therefore, the system possesses
a kink solution, $\phi_{K}$, which interpolates between the vacua. Then,
considering perturbations around the kink solutions
$\phi=\phi_{K}+\eta(x)e^{i\omega t}$, one arrives at the stability equation

$H\eta\equiv\left[-\frac{d^{2}}{dx^{2}}+\left.\frac{d^{2}V}{d\phi^{2}}\right|_{\phi=\phi_{K}}\right]\eta=\omega^{2}\eta.$
(2)

This is a Schrödinger-like equation with a linearized potential defined as
$U(x)\equiv V^{\prime\prime}(\phi_{K}(x))$. The desired form of the linearized
potential is in the form of a square well, which is one of the simplest
quantum mechanics potentials with the correct qualitative behavior and
analytical solutions. It reads

$U(x)=\begin{cases}-\lambda,&0\leq x<L,\\\ \gamma,&x>L,\end{cases}$ (3)

for positive $x$ and has even symmetry. In the definition above, $\lambda$,
$\gamma$, and $L$ are positive constants. A typical profile of the linearized
potential is shown in Fig. 1(a). The linearized potential can be easily
modified to exhibit QNMs as follows

$U(x)=\begin{cases}-\lambda,&0\leq x<L,\\\ \gamma,&L<x<L+\Delta L,\\\
\gamma-\Delta\gamma,&x>L+\Delta L,\end{cases}$ (4)

with new constants $\Delta L$ and $\Delta\gamma$. Again, we only write the
potential for positive $x$ and consider even symmetry. This modified potential
consists of a square well with two barriers and is shown in Fig. 1(b).
Clearly, if the square well potential has bound states with sufficiently high
energy, it will become a QNM as $\Delta\gamma$ increases.

Figure 1: Profile of the linearized potential in (a) the normal mode case and
(b) the QNM case. Parameters are: (a) $\gamma=3.0$, $\lambda=1.0$ and
$L=1.047$. (b) $\gamma=3.0$, $\Delta\gamma=2.0$, $\lambda=1.0$, $L=1.045$ and
$\Delta L=1.365$. Source: Results obtained by the author in [Campos e
Mohammadi 2020].

The task now is to construct a potential $V(\phi)$ that leads to the desired
linearized one. This is accomplished by a piecewise-defined function with
maximum powers of two in $\phi$, namely

$V(\phi)=\begin{cases}\frac{\lambda}{2}(-\phi^{2}+A^{2}),&0\leq\phi<\phi_{1},\\\
\frac{\gamma}{2}(\phi-\phi_{0})^{2},&\phi>\phi_{1},\end{cases}$ (5)

and

$V(\phi)=\begin{cases}\frac{\lambda}{2}(-\phi^{2}+A^{2}),&0\leq\phi<\phi_{1},\\\
\frac{\gamma}{2}(\phi-\phi_{0}+\delta)^{2}+B^{2},&\phi_{1}<\phi<\phi_{0}-\epsilon\\\
\frac{\gamma-\Delta\gamma}{2}(\phi-\phi_{0})^{2},&\phi_{0}-\epsilon<\phi<\phi_{0}+\epsilon\\\
\frac{\gamma}{2}(\phi-\phi_{0}-\delta)^{2}+B^{2},&\phi>\phi_{0}+\epsilon\end{cases}$
(6)

for the normal modes and QNM cases, respectively. In the above definitions,
all quantities are positive constants, except $\phi$. The expression for the
potential above is written only for positive $\phi$, and we assume even
symmetry. In total, there are five free parameters in the normal mode case and
nine in the QNM case. To ensure that the system is well behaved, we choose the
constants such that $V(\phi)$ and its derivative are continuous. This results
in the following set of equations

$\displaystyle\lambda(-\phi_{1}^{2}+A^{2})$
$\displaystyle=\gamma(\phi_{1}-\phi_{0})^{2},$ (7)
$\displaystyle-\lambda\phi_{1}$ $\displaystyle=\gamma(\phi_{1}-\phi_{0}),$ (8)

for the normal mode case and

$\displaystyle\lambda(-\phi_{1}^{2}+A^{2})$
$\displaystyle=\gamma(\phi_{1}-\phi_{0}+\delta)^{2}+B^{2},$ (9)
$\displaystyle-\lambda\phi_{1}$
$\displaystyle=\gamma(\phi_{1}-\phi_{0}+\delta),$ (10)
$\displaystyle\gamma(\delta-\epsilon)^{2}+B^{2}$
$\displaystyle=(\gamma-\Delta\gamma)\epsilon^{2}$ (11)
$\displaystyle\gamma(\delta-\epsilon)$
$\displaystyle=(\Delta\gamma-\gamma)\epsilon,$ (12)

for the QNM one. These equations reduce the number of free parameters in the
normal mode case to three and the QNM one to five. We can also rescale the
variables according to $x\to x/\sqrt{\lambda}$ and $\phi\to\phi_{0}\phi$. This
is equivalent to set $\phi_{0}=1$ and $\lambda=1$, after redefining all
constants. Therefore, in the end, the number of free parameters in eq. (5) and
(6) are one and three, respectively. After fixing some constant in eq. (5),
the others can be found analytically using the continuity equations. After
fixing three constants in eq. (6), the other can be found either numerically
or analytically, depending on which constants one chooses to fix. Typical
profiles of the potential are shown in Fig. 2.

Figure 2: Profile of (a) the potential and (b) kink for the normal mode case
($\Delta\gamma=0.0$). (c), (d) the same for the QNM case with
$\Delta\gamma=2.0$. The dashed lines mark the limiting points in the
definition $V(\phi)$ and $\phi_{K}(x)$. Other parameters are: (a), (b)
$\gamma=3.0$. (c), (d) $\gamma=3.0$ and $\epsilon=0.05$. Source: Results
obtained by the author in [Campos e Mohammadi 2020].

From the definition of the potential it is easy to compute the kink profile
using the Bogomol’nyi–Prasad–Sommerfield (BPS) equation. First, for the normal
mode case, we find

$\phi_{K}(x)=\begin{cases}A\sin(x),&0\leq x<L,\\\
1-(1-\phi_{1})e^{-\sqrt{\gamma}(x-L)},&x>L,\end{cases}$ (13)

for positive $x$ with odd symmetry. The constants $\phi_{1}$ and $L$ are
related by the expression $\phi_{1}=A\sin(L)$. For the QNM case, we find

$\phi_{K}(x)=\begin{cases}A\sin(x),&0\leq x<L,\\\
1-\delta+B\sinh(C+\sqrt{\gamma}(x-L)),&L<x<L+\Delta L,\\\ 1-\epsilon
e^{-\sqrt{\gamma-\Delta\gamma}[x-(L+\Delta L)]},&x>L+\Delta L,\end{cases}$
(14)

for positive $x$ and also with odd symmetry. In this expression, we defined
$C\equiv\ln(B/D)$, where

$D\equiv\sqrt{B^{2}+(1-\delta-\phi_{1})^{2}}+1-\delta-\phi_{1}.$ (15)

The constants $L$ and $L+\Delta L$ are the points where $\phi_{K}=\phi_{1}$
and $\phi_{K}=1-\epsilon$, respectively, and, thus, are not independent from
the other constants. The first relation is the same obtained in the normal
mode case and the second one reads

$\displaystyle\epsilon=\delta-B\text{sinh}(C+\sqrt{\gamma}\Delta L).$ (16)

The kinks’ profiles are shown Fig. 2 in the right panels.

This model is somewhat unusual, because it is piecewise defined. However, we
found that it is very well-behaved with all the expected properties of a non-
integrable model containing kinks. In the next section we will show how to
solve the stability equation for the kink configuration.

### 3 Stability equation

The stability equation for the normal mode case is the same as the Schrödinger
equation with the square well potential, which has well-known analytical
solutions [Griffiths e Schroeter 2018]. The eigenvalues of the even and odd
eigenfunctions are the solutions of the following transcendental equations

$\displaystyle k=$ $\displaystyle p\tan(pL),$ (17) $\displaystyle-k=$
$\displaystyle p\cot(pL),$ (18)

respectively, where $k\equiv\sqrt{\gamma-\omega^{2}}$ and
$p\equiv\sqrt{\omega^{2}+1}$. The analytical expressions of the eigenfunctions
are listed in section 8. It can be shown that, due to the continuity relation
of the potential, this system always possesses a zero-mode solution, where
$\omega=0$. As $\gamma$ increases, a tower of alternating odd and even
solutions starts to appear. In all simulations performed here, we choose the
parameters such that the stability equation admits only the zero-mode and one
normal mode. In this case, the system exhibits resonance windows, as expected
by the resonant energy exchange mechanism (see below).

Now, let us focus on the QNM case. We are interested in comparing the
stability potential in (6) as we vary the parameters. In particular, we want
to isolate the effect of $\Delta\gamma$ and $\Delta L$ variations while
keeping the other parameters of the linearized potential fixed. Fixing the
values of $L$ and $\gamma$, there remains only one degree of freedom in the
potential due to the continuity conditions. In this case, $\Delta\gamma$ and
$\Delta L$ are related as shown in Fig. 3. Notice that the point
$\Delta\gamma=\Delta L=0$ is a solution to the continuity equations for the
potential (6) only if $L$ is precisely the solution of the continuity
equations for the potential (5) with the same $\gamma$.

Figure 3: Relation between the parameters $\Delta L$ and $\Delta\gamma$ for
several values of $L$. We fix $\gamma=4.0$. Source: Results obtained by the
author in [Campos e Mohammadi 2020].

For the QNM case, we are interested in the scattering states and, naturally,
in the QNMs. The scattering states of eq. (2) with the linearized potential
described by eq. (6) can be written as

$\eta(x)=\begin{cases}F_{1}e^{imx}+F_{2}e^{-imx},&x<-L-\Delta L,\\\
G_{1}e^{-kx}+G_{2}e^{kx},&-L-\Delta L<x<-L,\\\
H_{1}\sin(px)+H_{2}\cos(px),&-L<x<L,\\\ I_{1}e^{-kx}+I_{2}e^{kx},&L<x<L+\Delta
L,\\\ Je^{imx},&x>L+\Delta L,\end{cases}$ (19)

where $p$ and $k$ are defined as before and
$m\equiv\sqrt{\omega^{2}+\Delta\gamma-\gamma}$. Imposing continuity of
$\eta(x)$ and its derivative, we find a linear system, which can be solved
straightforwardly. The solution of the linear system is given in section 7.
Using these expressions, we can obtain the transmission and reflection
coefficients. They are defined, respectively, as $T=|\frac{J}{F_{1}}|^{2}$ and
$R=|\frac{F_{2}}{F_{1}}|^{2}$ and are given by

$\displaystyle\begin{split}T=&4\bigg{|}e^{2k\Delta
L}(1+i\alpha_{km}^{-})[\cos(2pL)+\alpha_{kp}^{-}\sin(2pL)]-2i\alpha_{km}^{+}\alpha_{kp}^{+}\sin(2pL)\\\
&+e^{-2k\Delta
L}(1-i\alpha_{km}^{-})[\cos(2pL)-\alpha_{kp}^{-}\sin(2pL)]\bigg{|}^{-2},\end{split}$
(20) $\displaystyle\begin{split}R=&\frac{T}{4}\bigg{\\{}e^{2k\Delta
L}\alpha_{km}^{+}[\cos(2pL)+\alpha_{kp}^{-}\sin(2pL)]-2\alpha_{km}^{-}\alpha_{kp}^{+}\sin(2pL)\\\
&-e^{-2k\Delta
L}\alpha_{km}^{+}[\cos(2pL)-\alpha_{kp}^{-}\sin(2pL)]\bigg{\\}}^{2},\end{split}$
(21)

where

$\alpha_{km}^{\pm}\equiv\frac{k^{2}\pm
m^{2}}{2mk},\quad\alpha_{kp}^{\pm}\equiv\frac{k^{2}\pm p^{2}}{2pk}.$ (22)

From these analytical expressions, it is now easy to find the QNMs. They are
defined as solutions with purely outgoing boundary conditions. This means that
$F_{1}$ is set to zero in eq. (19). In turn, this causes $T$ to diverge
because the denominator vanishes. Similarly, it is possible to find the bound
states by setting $im\to-q$ in eq. (19), where
$q\equiv\sqrt{\gamma-\Delta\gamma-\omega^{2}}$, and, again, setting $F_{1}$ to
zero.

Figure 4: Transmission and reflection coefficients as a function of
$\omega^{2}$ for several values of $L$ and $\Delta\gamma$. We fix
$\gamma=4.0$. The grey vertical lines are the QNM or bound states frequencies.
Source: Results obtained by the author in [Campos e Mohammadi 2020].

In Fig. 4 we plot $T$ and $R$ as a function of the eigenvalue $\omega^{2}$. We
clearly see that if $\Delta\gamma$ is large enough, the system exhibits
transmission resonances, where the coefficient reaches the maximum value, one.
These are linked to the presence of the QNMs and occur near the QNM frequency.

The transcendental equation obtained by setting the denominator of eq. (20)
equal to zero can be solved numerically for the normal mode and QNM. Writing
the real part of $\omega$ as $\Omega$ and the imaginary part as $\Gamma$, we
find the curves shown in Fig. 5. For small $\Delta\gamma$ only the bound state
equation has a solution and, of course, this solution possesses the decay rate
$\Gamma=0$. At a critical value of $\Delta\gamma$, the bound state disappears,
and the QNM appears, which is indicated by a nonzero decay rate $\Gamma$.
These two regimes are separated in Fig. 5(b) by the red dotted curve. At this
curve, both bound state solution and quasinormal mode solutions are identical.
This can only happen if $im=-q$, which implies that $m=q=0$ or
$\omega=\sqrt{\gamma-\Delta\gamma}$. In the next section, we will see how the
appearance of the QNM affects the kink-antikink interaction.

Figure 5: Analytical values of the frequency and decay rate of the QNMs and
bound states as a function of $\Delta\gamma$. We consider several values of
$L$ and fix $\gamma=4.0$. Source: Results obtained by the author in [Campos e
Mohammadi 2020].

### 4 Collision

In this section, we will consider kink-antikink collision for our toy model.
The collision is defined by the following initial conditions

$\displaystyle\phi(x,0)=$
$\displaystyle\phi_{K}(\gamma(x+x_{0}))-\phi_{K}(\gamma(x-x_{0}))-1,$ (23)
$\displaystyle\dot{\phi}(x,0)=$ $\displaystyle-\gamma
v_{i}\left[\phi_{K}^{\prime}(\gamma(x+x_{0}))+\phi_{K}^{\prime}(\gamma(x-x_{0}))\right],$
(24)

where $v_{i}$ is the initial velocity of both kinks and $\gamma\equiv
1/\sqrt{1-v_{i}^{2}}$. The kinks are initially separated symmetrically with
respect to the origin by a distance equal to $2x_{0}$. After setting the
initial conditions, the Euler-Lagrange equations of motion are integrated
according to the numerical method described in section 6.

Considering first the case with $\epsilon=0$ or, equivalently,
$\Delta\gamma=0$, we choose the free parameter $\gamma=2.667$. In this case,
the system possesses only one vibrational mode in addition to the zero-mode.
In agreement with the resonant energy exchange mechanism, the system exhibits
a nested structure of resonance windows.

Typical spacetime evolutions of the field during collisions are shown in Fig.
6. In all cases, the kinks start approaching each other, and in (a) they
annihilate and form a bion that slowly radiates away the energy, in (b) the
kinks bounce twice before separating, and in (c) the kinks separate after the
first bounce. These are all typical scenarios in usual nonintegrable systems
with kink solutions, such as the $\phi^{4}$ model.

Figure 6: Spacetime evolution of the scalar field for several initial
velocities showing: (a) annihilation at $v=0.200$, (b) two-bounce resonance at
$v=0.228$ and (c) reflection at $v=0.340$. Other parameters are $\gamma=2.667$
and $\epsilon=0$.

(a)

(b)

(c)

Source: Results obtained by the author in [Campos e Mohammadi 2020].

We can simulate collisions in a range of initial velocities and measure the
final velocity of the kinks. The result is shown in Fig. 7(a). There are many
isolated peaks that correspond to resonance windows. After a critical
velocity, the final velocity is always nonzero and monotonically increasing.
This corresponds to the reflection curve. Interestingly, we see that the
resonance windows accumulate at the border of the reflection curve. Moreover,
measuring the field at the center of the collision at the resonance windows,
we obtain Fig. 7(c). The sharp peaks in the figure show the instants where the
bounce occurs, and we find that the field has a small oscillation between
bounces. We indexed the resonance windows in order with the integer $n$. The
number of small oscillations increases by one as we move from one resonance
window to the next, meaning that this oscillation has to be in phase for the
separation to occur. According to [Campbell, Schonfeld e Wingate 1983] this
implies in the following relation between the frequency of small oscillations
$\omega_{1}$ and the time between collisions $T$

$\omega_{1}T=\delta+2\pi n.$ (25)

The time between bounces as a function of $n$ is shown in Fig. 7(b). To
confirm the conjecture that the frequency of the small oscillations is due to
the shape mode, we fit the curve according to eq. (25) and obtain
$\omega_{1}=1.510$. This result should be compared to the theoretical value
$\omega=1.533$ obtained from eq. (18). We chose the values of $n$ such that
the constant $\delta$ is between $0$ and $2\pi$. The value of $\delta$ should
be approximately equal to $\pi$, and the fit gives $\delta=3.719$, which is a
reasonable result.

Figure 7: (a) Final velocity as a function of the initial velocity. (b) Time
between first and second bounce as a function of the two-bounce resonance
windows order. (c) Field at the center of the collision as a function of time
for the first eight two-bounce resonance windows. Parameters are
$\gamma=2.667$ and $\epsilon=0$. Source: Results obtained by the author in
[Campos e Mohammadi 2020].

Let us turn now to the QNM case. In this case, there are three free parameters
in the potential. We fix $\gamma=2.667$ and $\Delta\gamma=2.0$. This value of
$\Delta\gamma$ is enough to guarantee that the normal mode has turned into a
QNM. The last free parameter that we will vary is $\epsilon$ because we find
it the most intuitive way to measure how far we are from the normal mode
regime.

In Fig. 8, we repeat the plot of the final velocity versus the initial one for
several values of $\epsilon$. It is clear from the figure that, as $\epsilon$
increases, the resonance windows gradually start to disappear. This effect is
linked to the following qualitative argument. The resonance windows occur
because when translational energy is converted into vibrational energy at the
bounces, it is stored and can be recovered. However, if the vibrational mode
becomes a QNM, this energy will leak, and now only part of the initial energy
can be recovered, making it increasingly harder for the kinks to separate.
Notice that, accordingly, the critical velocity increases with $\epsilon$.

Figure 8: Final velocity as a function of the initial velocity for several
values of $\epsilon$. Parameters are $\gamma=2.667$, $\Delta\gamma=2.0$ and
(a) $\epsilon=0.0$, (b) $\epsilon=0.02$, (c) $\epsilon=0.04$, (d)
$\epsilon=0.06$. Source: Results obtained by the author in [Campos e Mohammadi
2020].

The set of resonance intervals is known to exhibit fractal structure. To
verify whether this is the case in our model, we depict the intervals where
the kinks escape after the collision in Fig. 9. More specifically, the
intervals are exhibited for three cases. The first one corresponds to the
region near the border of the reflection curve. Then, we zoom in near the edge
of an arbitrarily chosen two-bounce resonance window and, again, near the edge
of an arbitrarily chosen three-bounce resonance window, as indicated in the
figure. Accordingly, we vary $v_{i}$ between simulations in increasingly
smaller steps as we zoom in. The resonance structure consists of a nested
structure of three-bounce windows at the border of two-bounce windows, four-
bounce windows at the border of the three-bounce windows, and so on. This is
shown for $\epsilon=0.0$ in the upper panels.

Figure 9: Escape intervals of the initial velocity $v_{i}$. We zoom in the
regions marked by the black box. (a) The blue region corresponds to
$\epsilon=0.0$ and (b) the red region to $\epsilon=0.005$ and
$\Delta\gamma=2.0$. We fixed $\gamma=2.667$ in both cases. Source: Results
obtained by the author in [Campos e Mohammadi 2020].

Moving on to the QNM case, we set a small value $\epsilon=0.005$ to
investigate the resonance structure’s fate when the QNM mode is still close to
the non-decaying normal mode. Again, we draw escape intervals for three cases.
The first one is near the edge of the reflection curve, and we zoom twice near
the edge of arbitrarily chosen resonance windows. We also vary $v_{i}$ with
increasingly smaller steps as we zoom in. Clearly, a resonance structure is
still visible near the critical velocity and even when we zoom in near the
two-bounce window indicated in the figure. However, when we zoom in the three-
bounce window indicated in the figure, we find a very poor structure of four-
bounce windows. This effect is already visible despite the value of $\epsilon$
being small. On the other hand, for $\epsilon=0.0$, the structure goes to
higher orders. This implies that the resonance structure is gradually
disappearing, as expected by the leaking effect of the QNM.

We quantify the results described above by measuring the box fractal dimension
of the intervals drawn in Fig. 9. We divide the interval into boxes and
measure the number of boxes $n_{b}$ needed to cover the whole escape interval
for several box lengths $l$. Then, the box fractal dimension is given by
$D=-\lim_{l\to 0}\log n_{b}/\log l$. The plot of $n_{b}$ versus $1/l$ is shown
in Fig. 10 and the box fractal dimension is estimated by the slope of a linear
fit to the data. In the figure, $l$ is normalized by the smallest box size
$l_{min}$. The values of the slopes are shown in Table 1. For the case with
$\epsilon=0.0$, we obtain the box fractal dimension $D$ in the range
$0.84-0.86$. On the other hand, for $\epsilon=0.005$, we obtain $D$ in the
interval $0.88-0.885$ for the first two cases. However, a much larger
dimension of $D=0.956$ is obtained for the last one, which is not very
different from the non-fractal value $D=1$. This suggests that in the third
interval, the fractal structure disappeared, as can be inferred from Fig.
9(b).

Figure 10: Number of boxes to cover the escape regions in Fig. 9 as a function
of the box size. The slope is the box fractal dimension. Parameters are
$\gamma=2.667$ and, in (b), $\Delta\gamma=2.0$.

(a) $\epsilon=0.0$

(b) $\epsilon=0.005$

Source: Results obtained by the author in [Campos e Mohammadi 2020]. Table 1: Values of the slopes in Fig. 10 as well as the range of $v_{i}$ and smallest box size used to compute the box fractal dimension. We consider $\gamma=2.667$ for both values of $\epsilon$ and $\Delta\gamma=2.0$ when $\epsilon=0.005$. Range number | $\epsilon$ | Range in $v_{i}$ | Smallest box size $l_{min}$ | Slope
---|---|---|---|---
1 | $0.0$ | $0.20-0.32$ | $1\times 10^{-4}$ | $0.860\pm 0.006$
2 | $0.0$ | $0.2586-0.2610$ | $2\times 10^{-6}$ | $0.849\pm 0.006$
3 | $0.0$ | $0.25980-0.25986$ | $5\times 10^{-8}$ | $0.84\pm 0.01$
1 | $0.005$ | $0.23-0.35$ | $1\times 10^{-4}$ | $0.885\pm 0.008$
2 | $0.005$ | $0.2878-0.2902$ | $2\times 10^{-6}$ | $0.88\pm 0.02$
3 | $0.005$ | $0.28894-0.28900$ | $5\times 10^{-8}$ | $0.956\pm 0.009$
Source: Results obtained by the author in [Campos e Mohammadi 2020].

To investigate further the disappearance of the fractal structure due to the
change from normal mode to QNM, we measure the widths of the resonance windows
$\overline{\Delta v}$. In a system exhibiting a true fractal structure, such
as the $\phi^{4}$ model, the widths of the two-bounce windows should obey the
same relation as the width of any other set of consecutive higher-bounce
windows. In [Campbell, Schonfeld e Wingate 1983], they found that the widths
of resonance windows follow an approximate power-law behavior as a function of
$n$. Repeating the analysis for our system, we also find approximate power-law
decay for the widths. This can be seen in Fig. 11, where we plot the window’s
width as a function of the order of the windows for three different sets of
resonance windows. These sets are the two-bounce windows and two chosen sets
of three-bounce windows, which are the three-bounce windows near the first and
second two-bounce ones. We plot the widths starting from the index $n=3$, and
we normalize the value of all widths by the initial one. When $\epsilon=0.0$,
the exponent of $n$ is approximately the same for the three sets. Repeating
the calculation for the $\epsilon=0.005$ case, we did not find the same
scaling behavior for different sets of resonance windows, meaning that the
fractal structure is already lost to some degree for this value of $\epsilon$.
Actually, we find that higher-order resonance windows decay faster with $n$,
and this occurs because higher-bounce resonance windows take longer to occur
and, therefore, there is more energy leak.

Figure 11: Normalized width of resonance windows as a function of the window
number for two bounce windows and some sets of three bounce windows.
Parameters are $\gamma=2.667$, $\Delta\gamma=2.0$ and $\epsilon=0.0$, $0.005$.
Source: Results obtained by the author in [Campos e Mohammadi 2020].

In the next section we will discuss and summarize the main findings of our
work.

### 5 Conclusion

We started our work by considering the desired shape of the linearized
potential in the stability equation. From the desired shape, we were able to
construct the scalar field potential $V(\phi)$ systematically and find the
kink profile. Our approach is novel and could be repeated to construct
different scalar potential as was done later in [Basak, Roy e Kar 2021], for
instance, where the authors considered a linearized potential with a simple
harmonic well in the center and constant elsewhere.

Another interesting feature of our work was that we were able to find the QNMs
and normal modes analytically. Thus, we found explicitly that, at a critical
value of $\Delta\gamma$, the bound state becomes a QNM. Moreover, the presence
of the QNM mode is linked to resonances in the transmission coefficient.

We simulated kink-antikink collisions for our model, and we found
annihilation, resonance, and reflection depending on the initial kink-antikink
velocity, as expected. Furthermore, the resonances were shown to obey the
resonant energy exchange mechanism. On the other hand, the resonant effect
gradually disappears as the normal mode is gradually turned into a QNM due to
the energy leakage of this mode. Hence, we were able to confirm the results in
[Dorey e Romańczukiewicz 2018]. The disappearance of the self-similar
structure for small values of $\epsilon$ was quantified by measuring the
fractal structure via the box fractal dimension and comparing the scaling
relation of the resonance windows width. We concluded that, even though
resonance windows were still present in this case, higher-order windows are
suppressed, and the structure loses self-similarity.

### 6 Appendix: Numerical method

This work employs the simplest method for integrating partial differential
equations, which is a second-order finite difference method. We divide
spacetime in a grid with spacings $\tau=0.002$ and $h=0.01$. We define the
field at the gridpoints $(x_{i},t_{j})$ as $\phi_{i,j}$. Therefore, the
partial derivatives are approximated by

$\displaystyle\frac{\partial\phi_{i,j}}{\partial
t^{2}}=\frac{\phi_{i,j+1}-2\phi_{i,j}+\phi_{i,j-1}}{\tau^{2}}$ (26)
$\displaystyle\frac{\partial\phi_{i,j}}{\partial
x^{2}}=\frac{\phi_{i+1,j}-2\phi_{i,j}+\phi_{i-1,j}}{h^{2}}.$ (27)

After substituting these expressions in the equations of motion, one can solve
for $\phi_{i,j+1}$ in terms of $\phi_{i,j}$ and $\phi_{i,j-1}$

$\phi_{i,j+1}=2(1-\lambda^{2})\phi_{i,j}+\lambda^{2}(\phi_{i+1,j}+\phi_{i-1,j})-\phi_{i,j-1}-\tau^{2}V^{\prime}(\phi_{i,j}),$
(28)

where we defined $\lambda=\tau/h$. To perform the first iteration the method
is modified and we need only the $\phi$ and $\phi_{t}$ at $t=0$ [Burden,
Faires e Burden 2015]

$\phi_{i,1}=(1-\lambda^{2})\phi_{i,0}+\frac{\lambda^{2}}{2}(\phi_{i+1,0}+\phi_{i-1,0})-\frac{\tau^{2}}{2}V^{\prime}(\phi_{i,j})+\tau\phi_{t}(x_{i},0).$
(29)

The final time in the simulations is fixed at $t=400.0$ and the boundaries are
set at $x=\pm 100.0$ with periodic boundary conditions. The box is large
enough to guarantee that the boundary condition do not interfere with the bulk
evolution. We tested the method’s precision and observed that the energy is
conserved up to a relative error of $10^{-4}$ during the full evolution of the
system.

### 7 Appendix: Transmission and reflection coefficients

The coefficients of eq. (19) are given by

$\displaystyle I_{1}=$ $\displaystyle\frac{J}{2}e^{k(L+\Delta
L)}e^{im(L+\Delta L)}\beta^{-},$ $\displaystyle I_{2}=$
$\displaystyle\frac{J}{2}e^{-k(L+\Delta L)}e^{im(L+\Delta L)}\beta^{+},$
$\displaystyle H_{1}=$ $\displaystyle\frac{Je^{im(L+\Delta
L)}}{2}\big{\\{}e^{k\Delta
L}\beta^{-}[\sin(pL)-\frac{k}{p}\cos(pL)]+e^{-k\Delta
L}\beta^{+}[\sin(pL)+\frac{k}{p}\cos(pL)]\big{\\}},$ $\displaystyle H_{2}=$
$\displaystyle\frac{Je^{im(L+\Delta L)}}{2}\big{\\{}e^{k\Delta
L}\beta^{-}[\cos(pL)+\frac{k}{p}\sin(pL)]+e^{-k\Delta
L}\beta^{+}[\cos(pL)-\frac{k}{p}\sin(pL)]\big{\\}},$ $\displaystyle G_{1}=$
$\displaystyle\frac{Je^{im(L+\Delta L)}}{2}e^{-kL}\big{\\{}e^{k\Delta
L}\beta^{-}[\cos(2pL)+\alpha_{kp}^{-}\sin(2pL)]-e^{-k\Delta
L}\beta^{+}\alpha_{kp}^{+}\sin(2pL)\big{\\}},$ $\displaystyle G_{2}=$
$\displaystyle\frac{Je^{im(L+\Delta L)}}{2}e^{kL}\big{\\{}e^{k\Delta
L}\beta^{-}\alpha_{kp}^{+}\sin(2pL)+e^{-k\Delta
L}\beta^{+}[\cos(2pL)-\alpha_{kp}^{-}\sin(2pL)]\big{\\}},$
$\displaystyle\begin{split}F_{1}=&\frac{Je^{2im(L+\Delta
L)}}{2}\big{\\{}e^{2k\Delta
L}[1+i\alpha_{km}^{-}][\cos(2pa)+\alpha_{kp}^{-}\sin(2pa)]-2i\alpha_{km}^{+}\alpha_{kp}^{+}\sin(2pL)\\\
&+e^{-2k\Delta
L}[1-i\alpha_{km}^{-}][\cos(2pL)-\alpha_{kp}^{-}\sin(2pL)]\big{\\}},\end{split}$
$\displaystyle\begin{split}F_{2}=&\frac{J}{2}\big{\\{}-ie^{2k\Delta
L}\alpha_{km}^{+}[\cos(2pa)+\alpha_{kp}^{-}\sin(2pa)]+2i\alpha_{km}^{-}\alpha_{kp}^{+}\sin(2pL)\\\
&+ie^{-2k\Delta
L}\alpha_{km}^{+}[\cos(2pL)-\alpha_{kp}^{-}\sin(2pL)]\big{\\}},\end{split}$
(30)

where used the definition in eq. (22) and defined

$\beta^{\pm}=1\pm\frac{im}{k}.$ (31)

The transmission coefficient is given by $T=|\frac{J}{F_{1}}|^{2}$ and the
reflection coefficient by $R=|\frac{F_{2}}{F_{1}}|^{2}$.

### 8 Appendix: Bound states of the square well potential

The even bound states of the square well potential have eigenvalues
$\omega_{2n}$, which are solutions of the transcendental equation 17. Their
respective eigenfunctions are given by

$\eta_{2n}(x)=\begin{cases}C_{2n}e^{k_{2n}x},&x<-L,\\\
C_{2n}e^{-k_{2n}L}\cos(p_{2n}x)/\cos(p_{2n}L),&-L<x<L,\\\
C_{2n}e^{-k_{2n}x},&x>L,\end{cases}$ (32)

where $k_{2n}\equiv\sqrt{\gamma-\omega^{2}_{2n}}$,
$p_{2n}\equiv\sqrt{\omega^{2}_{2n}+\lambda}$ and $C_{2n}$ are normalization
constants.

The odd bound states of the square well potential have eigenvalues
$\omega_{2n+1}$, which are solutions of the transcendental equation 18. Their
respective eigenfunctions are given by

$\eta_{2n+1}(x)=\begin{cases}-C_{2n+1}e^{k_{2n+1}x},&x<-L,\\\
C_{2n+1}e^{-k_{2n+1}L}\sin(p_{2n+1}x)/\sin(p_{2n+1}L),&-L<x<L,\\\
C_{2n+1}e^{-k_{2n+1}x},&x>L,\end{cases}$ (33)

where $k_{2n+1}\equiv\sqrt{\gamma-\omega^{2}_{2n+1}}$,
$p_{2n+1}\equiv\sqrt{\omega^{2}_{2n+1}+\lambda}$ and $C_{2n+1}$ are
normalization constants.

## 3 INTERACTIONS OF KINKS WITH DOUBLE LONG-RANGE TAILS

### 1 Overview

Recently, some attention has been given to kinks with long-range tails, which
decay as a power-law. These kinks appear in scalar field models when the
potentials are polynomials of higher-order [Khare, Christov e Saxena 2014,
Khare e Saxena 2019]. They can be used to describe several physical systems,
such as Rydberg atoms [Saffman, Walker e Mølmer 2010], and quantum gases
[Lahaye et al. 2009]. Other applications appear in cosmology [Valle e Mielke
2016, Greenwood et al. 2009], statistical mechanics [Campa, Dauxois e Ruffo
2009], and supersymmetric quantum mechanics [Bazeia e Bemfica 2017].

In [Christov et al. 2019], the authors showed that usual initialization
methods lead to wrong results in simulations of kinks with long-range tails,
such as kink-antikink repulsion. To fix this issue, they developed a method
that consists of a nonlinear least-square minimization of the initial
condition such that the configuration of the system is as close as possible to
a solution of the static field equation. This specialized method was proven to
be very reliable and generated a very smooth scalar field evolution. However,
the method could not generate initial conditions where the kink and the
antikink are boosted.

Another difficulty due to the long-range tail of kinks is the computation of
inter-kink force. The standard method used to estimate this force was
developed by Manton and consists of integrating the momentum density around
the kink position and computing the time derivative of the resulting
expression [Manton e Sutcliffe 2004]. However, this method fails to estimate
the force between long-range kinks due to the large overlap between them. In
[Manton 2019], the same author developed a more suitable method for long-range
tails which consists of inserting an accelerating kink ansatz in the equations
of motion and making some further approximations, which will be described
below. This new method was shown to give very accurate results for a class of
models with long-range tails [Christov et al. 2019].

Later, in [Christov et al. 2021], the authors were able to develop a
specialized method to initialize kink-antikink as well as kink-kink
configurations with a nonzero initial velocity. The method consists of
starting the system at an initial guess and integrating the equations of
motion for a short period. Then, the initial guess is modified in order to
minimize the two-norm of the equations of motion in that period. However, this
method was very computationally costly.

Here, we will consider a class of models where the kinks possess double long-
range tails. This means that not only the tail that is facing the opposite
kink is long-range, but also the one in the opposite direction. We will show
that this property drastically alters the behavior of the system. Moreover, we
will develop a method to initialize kinks with a nonzero velocity that is
physically intuitive and more computationally efficient. This work resulted in
the following publication [Campos e Mohammadi 2021].

The following section will describe the family of models containing kinks with
double long-range tails.

### 2 Model

Figure 1: (a) Potential, (b) Kink and (c) Linearized potential of the
stability equation for the model containing kinks with double long-range
tails. We consider the first few values of $n$. Source: Results obtained by
the author in [Campos e Mohammadi 2021].

We start with a scalar field Lagrangian in $(1+1)$ dimensions

$\mathcal{L}=\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-V_{4n}(\phi).$
(1)

The equations of motion are obviously

$\frac{\partial^{2}\phi}{\partial t^{2}}-\frac{\partial^{2}\phi}{\partial
x^{2}}+\frac{dV_{4n}}{d\phi}=0.$ (2)

Moreover, the family of potentials we consider are of the following form

$V_{4n}(\phi)=\frac{1}{2^{2n+1}}(1-\phi^{2})^{2n}.$ (3)

The constant $1/2^{2n+1}$ was chosen for later convenience without loss of
generality. The first few potentials are shown in Fig. 1(a). It exhibits
spontaneous symmetry breaking and $Z_{2}$ symmetry. The $n=1$ case corresponds
to the well-known $\phi^{4}$ model, while for $n\geq 2$ the perturbations
around the vacua are massless, which leads to the appearance of long-range
tail.

We can write the potentials in terms of a superpotential $W_{4n}$. The two are
related as follows

$V_{4n}(\phi)=\frac{1}{2}\left(\frac{dW_{4n}}{d\phi}\right)^{2}.$ (4)

Computing the superpotential explicitly for the first two relevant cases, we
find

$W_{8}=\frac{\phi}{4}-\frac{\phi^{3}}{6}+\frac{\phi^{5}}{20}+c_{1},\quad
W_{12}=\frac{\phi}{8}-\frac{\phi^{3}}{8}+\frac{3\phi^{5}}{40}-\frac{\phi^{7}}{56}+c_{2}.$
(5)

We can also set to zero the two integration constants $c_{1}$ and $c_{2}$
without loss of generality.

The kink solutions can be found using the BPS equation

$\frac{d\phi}{dx}=\frac{dW_{4n}}{d\phi}.$ (6)

Substituting the superpotentials in the expression above, we obtain
respectively for the $\phi^{8}$ and $\phi^{12}$ models [Khare, Christov e
Saxena 2014]

$\frac{2\phi}{1-\phi^{2}}+\log\left(\frac{1+\phi}{1-\phi}\right)=x-A,\quad\frac{10\phi-6\phi^{3}}{(1-\phi^{2})^{2}}+3\log\left(\frac{1+\phi}{1-\phi}\right)=2(x-A).$
(7)

These equations can be solved numerically for $\phi$, and the resulting kink
profiles are shown in Fig. 1(b). Notice that both tails decay much more slowly
than the solid curve, corresponding to the $\phi^{4}$ kink. The asymptotic
form can be easily found as $1/x$ for $\phi^{8}$ and $1/\sqrt{2x}$ for the
$\phi^{12}$. The masses of the kinks are given by the expression
$M=W(1)-W(-1)$ leading to $4/15$ for $\phi^{8}$ and $4/35$ for $\phi^{12}$.

We can also study the behavior of perturbations around the kink solutions
numerically. The linearized potential of the resulting stability equation is
shown in Fig. 1(c). For $n\geq 2$, the potentials have a volcano shape and
tend to zero on both sides. This means that the only possible discrete
solution is the zero mode, which is guaranteed to exist by translational
symmetry. Therefore, the absence of a vibrational mode means we do not expect
this system to exhibit resonance windows like other non-integrable models.

The next section will discuss how to compute the force between kinks with
long-range tails.

### 3 Inter-kink force

In this section, we want to compute the inter-kink force for a kink-antikink
configuration starting from rest. The following analysis was developed in
[Manton 2019] and generalized in [Christov et al. 2019]. We will start with a
review of the previous works. The analysis starts by considering a symmetric
configuration with an antikink at position $-A$ and a kink at position $A$.
The solution for the kink at the right of the origin will be written as
$\phi(x,t)=\eta(x-A(t))$. We denote the derivative with respect to $X\equiv
x-A$ by a prime. Substituting this ansatz in the equation of motion, we find

$\eta^{\prime\prime}-a\eta^{\prime}-\frac{dV(\eta)}{d\eta}=0,$ (8)

where we defined the acceleration $a\equiv\ddot{A}$. We have neglected the
term proportional to $\dot{A}^{2}$ because we are considering that the kink
starts from rest and, therefore, has a small velocity. Then, assuming that the
accelerating kink solution approximately solves the BPS equation, we can
substitute $\eta^{\prime}=dW/d\eta$. Then, integrating the resulting equation
once this yields

$\eta^{\prime}=\sqrt{2[V_{eff}(\eta)-aW(1)]},$ (9)

where $V_{eff}(\eta)\equiv V(\eta)+aW(\eta)$.

So far, the approximations we made are nearly exact as long as the kinks’
velocity is still small. The next simplification made in [Manton 2019] was to
expand $V_{eff}(\eta)$ until first order around $\eta=-1$, which is the
approximate value of the scalar field in the overlapping region. We will see
that it is necessary to include higher-order terms in some models. Ignoring
this issue, for now, we obtain

$\eta^{\prime}=\sqrt{(1+\eta)^{2n}-2Ma}.$ (10)

Now that we found an equation for an accelerating kink near the overlapping
region, we can ask the value of $a$ such that the solution best fits the left
tail of the static kink solution centered at $A$. This can be achieved by
integrating eq. (10) with the appropriate boundary conditions. At $x=0$, or
$X=-A$, $\eta^{\prime}=0$ due to the even symmetry of the kink-antikink
configuration. According to eq. (10), we have that $\eta^{\prime}=0$ when
$(1+\eta)^{2n}=2Ma$. The second boundary condition at $x=A$, or $X=0$, is the
point where the extrapolation of the static kink tail diverges, and we should
have $\eta\to\infty$. Thus, we get

$\int_{(2Ma)^{\frac{1}{2n}}}^{\infty}\frac{d\xi}{\sqrt{\xi^{2n}-2Ma}}=A,$ (11)

where $\xi=1+\eta$. After a change of variables, we find

$A(2Ma)^{(n-1)/2n}=\int_{1}^{\infty}\frac{d\lambda}{\sqrt{\lambda^{2n}-1}}=\frac{-\sqrt{\pi}\Gamma\left(\frac{n-1}{2n}\right)}{\Gamma\left(-\frac{1}{2n}\right)},$
(12)

or equivalently

$a=\left[\frac{-\sqrt{\pi}\Gamma\left(\frac{n-1}{2n}\right)}{\Gamma\left(-\frac{1}{2n}\right)}\right]^{2n/(n-1)}\frac{1}{2M}A^{2n/(1-n)}.$
(13)

This result matches the expression found in [Christov et al. 2019] for a
similar model. In that case, the analytical estimate was shown to have a very
good agreement with numerical simulations. Observe that in this method, we
match the tail of the accelerating kink facing the antikink to the same tail
of the static kink. This means that the tail not facing the antikink, which we
will call backtail, is not relevant for the interaction. This is a correct
assumption. However, we will see that the behavior after the collision can be
very different depending on the character of the backtail.

We compared these analytical results to numerical simulations in our model,
and we did not find a good agreement between the two. The reason for this
discrepancy was that, while the second-order term in the tail expansion of the
kink is absent in [Christov et al. 2019], it is present in our model.
Therefore, the approximation of $V_{eff}$ by the first-order term is much
worse in our case. Here, the analytical result improves as the separation
between the kink and the antikink increases because the second-order term
becomes less relevant relative to the first-order one. However, the
convergence of the numerical acceleration, and the analytical expression is
very slow due to the long-range character of the tail.

In order to estimate the effect of the second-order term in the tail
expansion, we make the following approximations. First, we write the expansion
of the kink tail for the $\phi^{8}$ model up to second-order

$\phi=-1+\frac{1}{A-x}+\frac{\log[2(A-x)]-\frac{1}{2}}{(A-x)^{2}}+\mathcal{O}\left((A-x)^{-3}\right).$
(14)

Then, we approximate $\log[2(A-x)]\simeq\log(2A)$ because the interaction
between the kinks occurs in the overlapping region, where $x\ll A$. With this
approximation, the second-order term can be interpreted as a shift in the
point where the extrapolation diverges from $A\to A-\Delta A$. From the above
expression, we find $\Delta A=\log(2A)-\frac{1}{2}$. Using this prescription
in eq. (13), we find better results for the $\phi^{8}$ model. For the
$\phi^{12}$ model, however, the second-order term cannot be interpreted as a
shift in the point where the extrapolated tail diverges. If we wish to include
this term in the estimation of the force between the kinks, we need to
integrate eq. (9) numerically. In this case, we chose to integrate this
equation numerically, including all terms in $V_{eff}$ because the inclusion
of more terms does not increase the difficulty in numerical integration.

In the next section, we will discuss the numerical method we used to simulate
the kink-antikink collisions for the class of models where the kinks exhibit
double long-range tails.

### 4 Numerical method

The first step in the simulation of the model is to construct a suitable
initial condition for both the field and its time derivative, which we call
the velocity field. We need both initial conditions because the equations of
motion are second order in time. In order to understand the algorithm, it is
important to investigate a simpler case of a single traveling kink. We will
follow [Christov et al. 2021] in this discussion. We start by writing the
field as $\phi(x,t)=u(x-vt)$, which is the functional form of the field of a
traveling kink. Then, substituting in the equations of motion, we find that
$u$ obeys the following equation

$(1-v^{2})u^{\prime\prime}(\xi)-V^{\prime}(u(\xi))=0,$ (15)

where $\xi\equiv x-vt$ and prime denotes derivative with respect to the
argument. Moreover, the velocity field of a traveling kink is given by
$\phi_{t}(x,t)\equiv f(\xi)=-vu^{\prime}(\xi)$. The derivative of the kink
solution corresponds to the zero mode, which is the mode related to the
translation invariance of the system. For a traveling kink, the velocity field
is proportional to the zero mode of the kink. This is an important property
that the authors did not give enough attention in [Christov et al. 2021]. This
property means that the velocity field obeys the following equation

$(1-v^{2})\frac{\partial^{2}f}{\partial\xi^{2}}-V^{\prime\prime}(u(\xi))f=0.$
(16)

In the above expression, the term $(1-v^{2})$ is related to the Lorentz
contraction of the traveling kink similar to the same factor in eq. (15).
After this digression, we can now turn to the numerical method.

First, to initialize the field, we start with a guess, which will be given by
the split-domain ansatz

$\phi(x,t)=(1-\Theta(x))\phi_{K}(\gamma(x+A-vt))+\Theta(x)\phi_{\bar{K}}(\gamma(x-A+vt)).$
(17)

In words, this ansatz considers a kink to the left of the origin and an
antikink to the right. Moreover, the kink is boosted to the right with
velocity $v$, and the antikink is boosted to the left with velocity $-v$.
Thus, this initial guess has a discontinuity at the center. This initial guess
was suggested in [Christov et al. 2019] for the static case and in [Christov
et al. 2021] for the boosted one. There, the authors optimized this field
configuration to a solution that obeys eq. (15) as close as possible. This is
done via a nonlinear least-square minimization, which we also perform here.
This minimization is supplemented by the constraint that the centers of the
kinks should be fixed at $x=\pm A$, which in our case means that the field
should vanish in those positions. After discretizing the field and using a
Fourier spectral method to compute the derivatives, we perform the
minimization via the least_square function in the SciPy library in python. The
program minimizes the following functional

$\mathcal{I}[u]=\lVert(1-v^{2})D_{2}u-V^{\prime}(u)\rVert_{2}^{2}+C|u(A)|^{2}+C|u(-A)|^{2}.$
(18)

The first term is a euclidean norm. The argument of the norm becomes a vector
after discretizing the field, and the euclidean norm can be easily calculated.
The second derivative is obtained by the Fourier Spectral matrix $D_{2}$
[Trefethen 2000]. The two last terms enforce the constraints for suitable
values of the constant $C$. We found good results for $C=5.0$.

The next step is to find an initial condition for the velocity. We start with
a guess given by

$\phi_{t}(x,t=0)=v\text{sgn}(x)u^{\prime}(x).$ (19)

This guess is the analog of the split-domain ansatz to the velocity field
because we are using the relation of the velocity field of a traveling kink
with positive (negative) velocity to the left (right) of the origin. This
guess is discontinuous at the center as well. In [Christov et al. 2021], the
authors suggested this initial guess and optimized it according to the
equations of motion. They had to integrate the initial condition for a short
time. This required a large computational time. Our approach is much more
computationally efficient. We minimized the initial guess according to eq.
(16) via the same nonlinear least-square minimization. This means that we find
the velocity field that obeys as close as possible the zero-mode equation for
the field configuration $u(x)$. This was supplemented by the constraint that,
near the centers of the kinks, the solution should be as close as possible to
the initial guess. More specifically, the constraint was set in the intervals
$-A-2<x<-A+2$ and $A-2<x<A+2$. The initial guesses and the minimized ones are
shown in Fig. 2. Observe that the minimization procedure smooths out the
configuration at the center and preserves the configuration near the kinks.

Figure 2: Initial guess (solid) and minimized one (dashed) for the scalar
field (left) and the velocity field (right). Source: Results obtained by the
author in [Campos e Mohammadi 2021].

The next step is to compute the actual evolution of the field. This is
accomplished by discretizing the space in the interval $-200.0<x<200.0$ with
spacings $h=0.2$ for $\phi^{8}$ and in the interval $-400.0<x<400.0$ with
spacings $h=0.4$ for $\phi^{12}$. Then, we set periodic boundary conditions,
but the integration is short enough to guarantee that the boundaries will not
interfere with the bulk evolution. We approximate the second-order derivative
with respect to $x$ by a Fourier spectral matrix. This consists in taking the
Fourier transform of the field, performing the derivative in Fourier space,
which is a trivial multiplication, and then transforming back to real space
[Trefethen 2000]. After discretizing in space, the equations of motion become
a set of first order ordinary diferential equations for a vector containing
both the values field $\phi$ and its time derivative $\phi_{t}$ at the grid
points. This procedure is known as the method of lines. The resulting ordinary
differential equations are integrated in time using the solve_ivp method from
the SciPy library in python. This method implements an eighth order Runge-
Kutta method with adaptive step size and error control.

Figure 3: Evolution of the velocity field in spacetime for (a) the reference
method and (c) for an initial condition using our minimization methods. (b)
Evolution of the velocity and (d) initial value of the velocity field. Dashed
lines correspond to the reference method and solid ones for an initial
condition using our minimization methods. Source: Results obtained by the
author in [Campos e Mohammadi 2021].

We compare our initial conditions for a boosted kink-antikink configuration
with a reference solution to test our minimization method. We obtain the
reference solution by the following method. We start with a kink-antikink
configuration where the elements are at rest with $A=30.0$. When the system
starts at rest, the initialization method was shown to produce good results
[Christov et al. 2019]. If we let the kinks interact, the attraction makes
them accelerate towards each other. After some time, the configuration becomes
a boosted kink-antikink configuration. When the elements reach the position
$A=10.0$, we measure the velocity $v\simeq 0.0832$. Then, we initialize
another simulation according to our minimization methods with the same $A$ and
$v$ and let it evolve in time. Thus, we can compare the evolution of the
simulation using our method and the continuation of the evolution of the
reference method. The result is shown in Fig. 3.

In Fig. 3(a) and (c), we plot the evolution of the velocity field in both
simulations. There are no visible differences between the two in the scale of
the figure. This shows that our minimization procedure matches the natural
evolution from a static configuration to a boosted one. In particular, we see
in Fig. 3(d) that the initial condition of the velocity field matches
perfectly the velocity field of the reference solution at $A=10.0$. Moreover,
we can measure the position of the kinks as the point where the field crosses
the value zero. Then, computing the derivative of the position with respect to
time, we find the velocity of the kinks. The result is shown in Fig. 3(b).
Both methods agree perfectly in the scale of the figure.

In the next section, we will summarize all the possible behaviors of kink-
antikink collisions varying the initial velocity of the system.

### 5 Collisions

Figure 4: Evolution of the field in spacetime for kink-antikink interactions
starting from a statitc configuration. We consider the $\phi^{8}$ model (left)
and the $\phi^{12}$ model (right). Lines correspond to the evolution of a
particle with acceleration given by (solid) eq. (9) without further
approximations and (dashed) eq. (13) after setting $A\to A-\Delta A$. Source:
Results obtained by the author in [Campos e Mohammadi 2021].

We start this analysis by the simulation of kink-antikink collisions that are
initially static. The result is shown in Fig. 4. First, we compare the
evolution of the system before the collision to the analytical expression of
the acceleration given by eq. (13). The two do not match very well due to the
presence of the second term in the tail expansion, as argued before. In the
figure, we show for the $\phi^{8}$ the evolution of a particle accelerating
according to that expression after prescribing $A\to A-\Delta A$. This
solution has a reasonable agreement with simulations. On the other hand, we do
not have a simple prescription for improving the acceleration estimate for the
$\phi^{12}$ model. As a consistency check, we can compare the full evolution
of the system with the acceleration obtained by integrating eq. (9)
numerically without further approximations. More specifically, we find the
acceleration numerically starting from an initial guess for $a$, and we
integrate the equation starting with the value of $\eta$ such that
$\eta^{\prime}=0$ from $x=0$ to $x=A$. Then, it becomes a root-finding
problem. We search for a value of $a$ such that $\eta=0$ at $x=A$. This
condition means that the kink is centered at $x=A$. As expected, the two
results match very well, as shown in the solid lines in the figure.

The comparison between the different methods to estimate the acceleration is
shown in fig. 5. We consider both the $\phi^{8}$ and $\phi^{12}$ models. To
obtain the initial acceleration in the simulations, we evolve the system for a
short time for several values of $A$ and obtain the acceleration using a
stencil approximation for each simulation. The integration of eq. (9) without
further approximations as described above is shown as the numerical method. It
agrees perfectly with simulations and serves as a consistency check. For the
$\phi^{8}$ model, we also show the result in eq. (13) after including the
prescription $A\to A-\Delta A$. In the figure, we call it the corrected
asymptotic method. It has a reasonable agreement with the previous ones.

Figure 5: Comparison of approximate calculations of the initial acceleration
of an interacting kink to numerical simulations. Left plot corresponds to
$\phi^{8}$ model and the right one, to the $\phi^{12}$ model. Source: The
author (2022).

In Fig. 4, it should be noticed that the system annihilates directly into
radiation. This is in stark contrast to the behavior of kink-antikink
collisions in most models, where a bion is formed. We argue that this occurs
because the backtail is also long-range. The qualitative argument for this
phenomenon will be described below.

First, let us discuss the behavior in the $\phi^{4}$ model or any similar
models where both tails of the kinks decay exponentially. The interaction
starts with the kink on the left and the antikink on the right. Then, the two
elements temporarily annihilate, and the system reaches a state where the
field is near the vacuum all over space. Due to the system’s momentum, the
field keeps decreasing, bounces back, and the kink-antikink pair reappears.
The pair always separate above the critical velocity, while, for smaller
velocities, it can either separate after multiple bounces or form a bion,
which slowly decays to the trivial vacuum. During this process, the
perturbations around the vacuum are massive, and it takes a long time to
dissipate the energy. Therefore, the kink-antikink configuration can be easily
recovered.

If the tail facing the opposing kink in a kink-antikink collision is long-
range, the same picture applies. The kink and the antikink approach each
other, temporarily annihilate and reach a vacuum value all over space. As the
perturbations around this vacuum are also massive, the behavior remains the
same. However, if the backtail is also long-range, perturbations around that
vacuum are massless. Therefore, kink-antikink energy will dissipate much
faster, meaning the kink-antikink configuration cannot be recovered, and the
bion is not formed.

Figure 6: Field configuration in spacetime for kink-antikink collisions. We
consider the $\phi^{8}$ model with several initial velocities. Source: Results
obtained by the author in [Campos e Mohammadi 2021].

The next step is to consider kink-antikink collisions with a nonzero initial
velocity. In this case, we need to use the full minimization method for the
initial configuration of the field and the velocity field. The result is shown
in Fig. 6 for the $\phi^{8}$ model. We see that the system annihilates
directly into radiation until very large velocities. This behavior is quite
similar to the static case. However, we find that the system can separate
after a critical velocity that is ultra-relativistic. The critical velocity
$v_{c}$ depends on the initial separation of the simulations. For $A=10.0$ we
find $v_{c}\simeq 0.94$. We also observe that, in the reflection regime, the
system still produces a large amount of radiation. Interestingly, at
$v=0.935$, we observe that the system tries to separate, but the final
velocity is not large enough to overcome the attraction between the kink and
the antikink. Thus, they collide one more time and annihilate completely. This
is the closest configuration we have to a bion, but they are not long-lived at
all. This is because the velocity at the second collision is not large enough,
and the system will always annihilate into radiation at the second bounce.

Figure 7: Field configuration in spacetime for kink-antikink collisions. We
consider the $\phi^{12}$ model with several initial velocities. Source:
Results obtained by the author in [Campos e Mohammadi 2021].

Finally, we repeat the process for the $\phi^{12}$ model. The result is shown
in Fig. 7. Again we find annihilation directly into radiation. Moreover, we
expect that the critical velocity, in this case, is even larger than the
previous one because the long-range character is even more pronounced.
However, the minimization method does not converge to the correct solution if
the velocity is too large. Hence, we were not able to find the critical
velocity in this case.

It is clear from the results above that this model does not exhibit resonance
windows. It is important to emphasize that this result was already expected
due to the absence of a vibrational mode. Moreover, the long-range character
of the backtail also suppresses bion formation and causes the system to
annihilate directly into radiation. The next section will summarize our
findings and discuss ideas for future works.

### 6 Conclusion

In this main section, we have discussed a class of scalar field models
exhibiting kink solutions with long-range tails at both ends. We called them
kinks with double long-range tails. We found the kink solutions explicitly and
showed that the stability equation does not allow vibrational modes. Moreover,
we showed that the method developed by Manton [Manton 2019] can be applied in
our case, but the approximations are considerably worse here because there is
a second-order term in the tail expansion of the kinks.

In order to simulate kink-antikink collision, we developed a minimization
method for the velocity field, building upon previous works on kinks with
long-range tails [Christov et al. 2019, Christov et al. 2019, Christov et al.
2021]. The idea behind the method is to find a configuration of the velocity
field that obeys as close as possible the equation of the zero-mode of the
field configuration. The field configuration, in turn, is obtained by the
minimization of the equations of motion for a traveling wave. We checked the
accuracy of our model against a reference configuration obtained from the
evolution of a nearly static kink-antikink configuration.

We simulated kink-antikink collisions for several initial velocities, and we
found that the system annihilates directly into radiation, except for ultra-
relativistic velocities. We argued that this is related to the long-range
character of the kink’s backtail. After the ultra-relativistic critical
velocity, the kinks separate, and near the critical velocity, a short-lived
bound state appears. Furthermore, the system does not exhibit resonance
windows as expected from the absence of a vibrational mode.

Ideas for future works include applying our initialization procedures for
other classes of models as can be found, for instance, in [Khare, Christov e
Saxena 2014, Khare e Saxena 2019]. Moreover, one could study interactions of
long-range kinks with boundaries, which should be an exciting but challenging
problem.

## 4 WOBBLING DOUBLE SINE-GORDON KINKS

### 1 Overview

Notably, the vibrational mode has an essential influence on the kink’s
behavior. First of all, in non-integrable theories, the vibrational mode is
usually excited when the kink interacts with other elements. This mode can
store energy and, therefore, if the translational energy is converted into
vibrational during an interaction, it can be recovered in subsequent events.
This property leads to the well-known resonance phenomenon. Furthermore, a
vibrating kink will likely come up after the interaction. Therefore, it is
indispensable to investigate the properties of vibrating kinks. Some of these
properties are described in [Manton e Merabet 1997, Barashenkov e Oxtoby 2009,
Oxtoby e Barashenkov 2009]. For instance, it is now well known that the kink’s
vibration decays in time via the coupling between the first harmonic and
radiation.

These ideas lead to the question of how kinks interact if they are vibrating
beforehand. This problem was investigated for the first time recently in
[Izquierdo, Queiroga-Nunes e Nieto 2021] for the $\phi^{4}$ model. The authors
argued that this analysis is important because the subsequent bounces in
resonance windows can be interpreted as an iteration of wobbling kink
collisions. Therefore, these collisions are important to understand the
resonance phenomenon.

In [Izquierdo, Queiroga-Nunes e Nieto 2021] the authors found many exciting
results, such as the appearance of one-bounce resonance windows. Moreover,
they analyzed the final amplitude and frequency of the kinks after they
separate and found that amplitude is usually large and that the frequency is
Lorentz contracted.

Building upon the work mentioned above, we study collisions between wobbling
kinks in the double sine-Gordon model. This model differs form the $\phi^{4}$
because the potential is periodic and has a free parameter $R$. Moreover, the
model becomes integrable in both limits where $R$ is very small and very
large. Interestingly, it gives an effective description of a wide range of
physical systems, such as gold dislocations [El-Batanouny et al. 1987],
optical pulses and spin waves [Bullough, Caudrey e Gibbs 1980], pseudo 1-D
ferromagnets [Rettori 1986] and Josephson structures [Alfimov, Malishevskii e
Medvedeva 2014].

There are quite a few works already on the double sine-Gordon model. Examples
are [Kivshar e Malomed 1987, Malomed 1989, Kivshar e Malomed 1989], where it
is studied perturbatively using the inverse scattering method. Examples
including analyses with a numerical focus are [Campbell, Peyrard e Sodano
1986, Gani e Kudryavtsev 1999, Gani et al. 2018, Gani, Marjaneh e Saadatmand
2019, Simas et al. 2020]. However, the model is still not fully comprehended.
For instance, the dependence of the critical velocity on the parameter $R$ was
only found recently [Gani et al. 2018]. We will see that the study of
colliding wobbling kinks in the double sine-Gordon leads to some novel results
such as the appearance of spines in the region of one-bounce windows as well
as of a structure of resonance windows near the integrable limits which is
hidden in collisions between kinks that are not wobbling. The work described
below resulted in the following publication [Campos e Mohammadi 2021].

In the next section we describe the double sine-Gordon and its main
properties.

### 2 Model

Figure 1: Double sine-Gordon (a) potential, (b) kink and (c) linearized
potential for several values of the paramter $R$. Source: Results obtained by
the author in [Campos e Mohammadi 2021].

We will investigate the double sine-Gordon model described by a scalar
Lagrangian

$\mathcal{L}=\frac{1}{2}(\partial_{\mu}\phi)^{2}-V_{\eta}(\phi),$ (1)

with the following potential

$V_{\eta}(\phi)=\frac{4}{1+4|\eta|}\left(\eta(1-\cos\phi)+1+\cos\frac{\phi}{2}\right).$
(2)

The case where $\eta<0$ was investigated in detail in [Simas et al. 2020]. We
are interested, however, in the regime with $\eta>0$. In this case, the
constant $\eta$ can be written as $\eta=\frac{1}{4}\sinh^{2}R$ and the
potential becomes

$V_{R}(\phi)=\tanh^{2}R(1-\cos\phi)+\frac{4}{\cosh^{2}R}\left(1+\cos\frac{\phi}{2}\right).$
(3)

This potential is periodic with degenerate vacua at $\phi=2\pi(2n+1)$, with
$n$ integer. The potential as a function of the field is shown in Fig. 1 for
some values of the parameter $R$. For $R=0$, the potential corresponds to the
sine-Gordon one, which is integrable. At $R>\text{arcsinh}(1)\simeq 0.881$,
the potential possesses a local minima at $\phi=4\pi n$, for $n$ integer.
These minima are responsible for the inner structure of the kink. For large
values of $R$, the local minima of the potential gradually become global
minima and the potential approaches again the sine-Gordon one, but with half
the period.

Using the BPS equation (17), finding the kink solutions is a trivial task.
They are given by

$\phi_{k(\bar{k})}=4\pi n\pm 4\tan^{-1}\left(\frac{\sinh(x)}{\cosh R}\right).$
(4)

This expression can be written in terms of the sine-Gordon kink expression,
$\phi_{SG}=4\tan^{-1}\exp(x)$. One can express the double sine-Gordon kink as
a linear combination of two sine-Gordon subkinks

$\phi_{k(\bar{k})}=4\pi n\pm[\phi_{SG}(x+R)-\phi_{SG}(R-x)].$ (5)

When a kink is composed of more than one subkink, it possesses an inner
structure. This general property occurs when the potential possesses a local
minimum between its vacua. The kinks’ profiles are shown in Fig. 1 for some
values of the parameter $R$. The inner structure can be clearly seen for
$R=3.0$.

Figure 2: Frequency of the shape mode as a function of the parameter $R$. The
zero mode is indicated by a dashed line. Source: Results obtained by the
author in [Campos e Mohammadi 2021].

The resonance behavior depends on the spectrum of perturbations around the
kink solution. Writing the field as $\phi=\phi_{k}+\psi\cos(\omega t)$ and
substituting in the equations of motion we arrive at the following stability
equation

$-\psi^{\prime\prime}+U_{R}\psi=\omega^{2}\psi,$ (6)

where $U_{R}$ is called the linearized potential and is defined as
$U_{R}\equiv V_{R}^{\prime\prime}(\phi_{k})$. More explicitly, one can arrive
at [Gani et al. 2018]

$U_{R}=\frac{8\tanh^{2}R}{(1+\text{sech}^{2}R\sinh^{2}x)^{2}}+\frac{2(3-4\cosh^{2}R)}{\cosh^{2}R(1+\text{sech}^{2}R\sinh^{2}x)}+1.$
(7)

This stability equation possesses two discrete solutions. The first one is the
zero mode, which is related to the translational invariance of the model, and
the second one is the vibrational mode, also known as the shape mode. Finally,
we have the continuum solutions for $\omega>1$. We denote the shape mode’s
frequency by $\omega_{D}$ and the normalized profile by $\psi_{D}$. Using the
expression for the energy of the scalar field $\phi$, it is straightforward to
show that the energy associated with the perturbation
$A\psi_{D}\cos(\omega_{D}t)$ is $E_{D}=\frac{1}{2}\omega_{D}^{2}A^{2}$.

In [Campbell, Peyrard e Sodano 1986], the authors were able to derive
analytical approximations for the shape mode frequency both near $R=0$ and for
large $R$. We will give a brief discussion of their analysis. For $R=0$, the
model corresponds to the sine-Gordon, which does not possess a vibrational
mode. However, the authors showed that as $R$ increases, it could be
considered a good approximation that the threshold state becomes a vibrational
mode. The threshold state is the lowest energy state in the continuum modes.
For large $R$, the kink is composed of two widely separated sine-Gordon
subkinks. As can be seen in Fig. 1, in this limit, the linearized potential
$U_{R}$ becomes two widely separated sine-Gordon wells. The separate sine-
Gordon potentials contains a zero mode each. Accordingly, the zero mode and
shape mode of the double sine-Gordon kink correspond, respectively, to the
symmetric and antisymmetric combinations of these two modes. The authors
showed that this is a good approximation for the exact shape mode, even for
intermediate values of the parameter $R$.

Figure 3: (a) Amplitude and (b) frequency of the kink’s vibration as a
function of time for several values of $R$. (c)-(f) Power spectrum of the
vibration near and far from the kink for large times with same values of $R$.
The initial amplitude is $A=1.0$. Source: Results obtained by the author in
[Campos e Mohammadi 2021].

It is known that the amplitude of the kink’s vibration slowly decays with
time. This occurs due to the coupling of the first harmonic of the wobbling
frequency with the radiating modes [Manton e Merabet 1997]. This coupling
appears when one considers discrete and continuous modes as a perturbation to
the kink solution and expands the equations of motion up to higher orders.
Moreover, if the shape mode’s frequency is lower than half the threshold, the
first harmonic is not large enough to couple with radiation, and the decay
occurs via harmonics higher than the first one. It can be shown that this
suppresses the decay [Romańczukiewicz e Shnir 2018].

The effects described above are also observed in the double sine-Gordon model
as shown in Fig. 3. We simulate a vibrating kink at the origin. In the left
panels, we plot the amplitude and frequency of the vibration at a point
shifted by $\Delta x=1.465$ from the origin. This point is near the position
where the amplitude of the shape mode is maximum. The vibration is measured by
the difference $\phi(x)-\phi_{k}(x)$. From the figure, a few things can be
noticed. First, as expected, for $R=0.5$ and $R=1.0$, we observe the expected
decay for a frequency above half the threshold. However, for $R=0.5$, the
decay is less pronounced. This effect becomes even more noticeable as we get
closer to the integrable limit at $R=0$. Now, turning to large values of $R$,
we see that the decay is suppressed for $R=1.8$ and $R=2.1$ because the
frequency is below half the threshold. To corroborate these results, we plot
the power spectrum of the time series of the vibration. Near the kink at
$x=\Delta x$, we find that the largest peak is at the shape mode’s frequency.
Far from the kink at $x=25.0$, the largest peak is at harmonics higher than
the threshold. These results are in agreement with the picture described
above.

In the next section, we will discuss the behavior of kink-antikink collisions
in the double sine-Gordon model. We will investigate two different scenarios,
the usual collision, and collisions between wobbling kinks.

### 3 Collision simulations

Figure 4: (a) Final velocity as a function of initial velocity for usual kink-
antikink collisions with $R=1.0$. (b) Bounce regions as a function of $R$ and
$v_{i}$ for the same type of collisions. Source: Results obtained by the
author in [Campos e Mohammadi 2021].

Now, let us turn to collisions between wobbling double sine-Gordon kink and
antikink. This corresponds to the initial conditions given by the following
expression

$\phi=\phi_{k}(\xi_{+})-\phi_{k}(\xi_{-})-2\pi+A\sin(\omega_{D}\tau_{+})\psi_{D}(\xi_{+})-A\sin(\omega_{D}\tau_{-})\psi_{D}(\xi_{-}),$
(8)

evaluated at $t=0$. We defined the Lorentz transformation of the coordinates
as $\xi_{\pm}=\gamma(x\pm x_{0}\mp v_{i}t)$ and $\tau_{\pm}=\gamma(t\mp
v_{i}x)$, where $\gamma=1/\sqrt{1-v_{i}^{2}}$. These initial conditions
correspond to a kink on the left vibrating with frequency $A$ and an antikink
on the right vibrating with the same frequency. The two are approaching each
other with velocities $\pm v_{i}$, respectively. The details of the
simulations are described in section 5. It is clear from eq. (8) that the
wobbling frequency is Lorentz contracted in the center of mass frame. To see
that, we measure the field at a position moving with the kink
$x=v_{i}t+\alpha$. In this case, we find that the wobbling term becomes

$A\sin\left(\frac{\omega_{D}t}{\gamma}-\gamma\omega_{D}v_{i}\alpha\right)\psi_{D}(\gamma(x_{0}+\alpha)),$
(9)

which oscillates with the Lorentz contracted frequency $\omega_{D}/\gamma$. It
should be clear that, to measure this effect, we need to choose $\alpha\neq-
x_{0}$, so that contribution from $\psi_{D}$ does not vanish at the point that
we are measuring.

Figure 5: Value of the field at the center of the collision as a function of
$t$ and $v_{i}$. We consider usual kink-antikink collisions with (a) a small
value of $R=0.3$ and (b) a large value of $R=3.0$. Source: Results obtained by
the author in [Campos e Mohammadi 2021].

We start the analysis by considering usual kink-antikink collisions with
$A=0$. It is important to mention an important difference between the double
sine-Gordon and the $\phi^{4}$ model. In the former, the potential is
periodic, and a bounce occurs every time the kink and the antikink cross,
while, in the latter, a bounce corresponds to a reflection between the two.
The result considering $R=1.0$ is shown in Fig. 4 showing the usual behavior.
For $v_{i}$ above a critical value, there is crossing after one bounce, while
there are either annihilation or resonance windows for smaller velocities. We
depict the number of bounces in the resonance window by a color code explained
in the figure. In the right panel of the same figure, we summarize the regions
where resonances occur using this color code. The blue region corresponds to
crossing after one bounce and the white region to annihilation. The interface
between the two corresponds to the critical velocity, which is in agreement
with the non-monotonic behavior found in [Gani et al. 2018]. Moreover, the
fractal structure appears in the red region. From the figure, we can estimate
that fractal behavior occurs for $0.52<R<1.78$, which corresponds to wobbling
frequencies $0.437<\omega_{D}<0.963$. Of course, the actual region should be
slightly larger because the resonance windows get increasingly narrower and
become harder to localize. These results indicate no resonance windows near-
integrable regimes, presumably because the degrees of freedom decouple. This
is in agreement with first-order perturbation theory, which predicts that
there is no energy and momentum transfer between the kinks in this limit
[Kivshar e Malomed 1989]. However, a more detailed analytical explanation for
this phenomenon is still lacking.

We analyzed the system’s behavior for large and small $R$ to explore the near-
integrable limits and find out whether resonance windows exist or not. In Fig.
5(a), we plot the value of the field at the center of the collision as a
function of time and the initial velocity for $R=0.3$. This figure should be
read as follows. Every vertical line corresponds to the field at the center of
the collision for the corresponding initial velocity. Every time the field
changes color in a vertical line, a bounce occurs. For large $v_{i}$, the
field at the center starts yellow and then becomes purple. This means that the
kink and the antikink bounced once and then separated. Below the critical
velocity, the field oscillates between yellow and purple and never reaches a
constant value. This means that a bion is formed and, therefore, annihilation
occurs. Thus, we confirm that there are no resonance windows and, instead, we
find false resonance windows. The false resonance windows are small peaks when
the third bounce occurs. This means that the system is slightly closer to
separating at a few points but returns and bounces again. The resonant energy
exchange mechanism can also describe the position of false resonance windows.
Hence, one can say that they are a weaker type of resonance. Likewise, in Fig.
5(b), we do the same analysis for $A=3.0$, and we only observe false resonance
windows. The absence of resonance windows in nearly integrable models has also
been reported in [Simas et al. 2020, Dorey et al. 2021].

Figure 6: (a) Final velocity as a function of the initial velocity for $R=1.0$
and $A=0.1$. (b) Number of bounces as a function of $R$ and $v_{i}$ for
$A=0.1$. Source: Results obtained by the author in [Campos e Mohammadi 2021].

The next step is to turn on the wobbling. We initialize the system according
to eq. (8) with $A=0.1$. The result is shown in Fig. 6. We plot the final
velocity as a function of the initial velocity on the left panel for $R=1.0$.
The blue curves correspond to the regions where crossing occurs after one
bounce. Due to the wobbling of the kinks, this curve oscillates. More
specifically, this effect originates from the fact that the wobbling phase at
the moment of a collision varies with $v_{i}$. The wobbling amplitude evolves
approximately as

$S(t)=A\exp\left[i\left(\frac{\omega_{D}t}{\gamma}-\theta_{0}\right)\right],$
(10)

where $\theta_{0}$ is an initial constant. We consider it complex for
simplicity, but the real part should be taken in the end. As the collision
takes a time $t=x_{0}/v_{i}$ to happen, the phase at the moment of the
collision is given by

$\theta=\sqrt{1-v_{i}^{2}}\frac{\omega_{D}x_{0}}{v_{i}}+\theta_{0}.$ (11)

The variation of the phase with $v_{i}$ is the origin of the oscillation in
the blue curve.

Another intriguing effect is that the blue curve splits, meaning that there
are two separate blue regions. The small one is called a one-bounce resonance
window, while the large is called the crossing curve. Interestingly, near the
edges of one-bounce resonance windows, there appears a nested structure of
higher-bounce resonance windows. As discussed in [Izquierdo, Queiroga-Nunes e
Nieto 2021], the appearance of one-bounce resonance windows can occur in two
ways. The first one occurs as the oscillation of the crossing curve increases
and eventually touches the $v_{i}$-axis. In this case, we say that the
crossing curve splits. This occurs because the interaction between the
vibrational and translational modes depends on the oscillation phase at the
collision. Thus, if the oscillation is large and out of phase, it can lead to
the annihilation of the kinks at some point in the crossing curve and split
it. The second one occurs when a one-bounce window appears where previously it
did not exist. This occurs when the wobbling is significant and in phase at
the moment of the collision. In this case, the energy of the wobbling is
transferred to the translational mode and allows the separation of the kinks.

In Fig. 6(b), we show the number of bounces before separation as a function of
$R$ and $v_{i}$. The color code is the same as before, and the white regions
correspond to annihilation. One can see that the effect of wobbling is to
create an intricate oscillating structure. These branches in the structure are
called spines in [Dorey et al. 2021]. In this work, the authors described how
these structures appear in detail. In short, this structure can be understood
if we follow how the resonance windows appear, move, and merge with the
crossing curve as we vary the parameter $R$.

Figure 7: (a) Final velocity as a function of the initial velocity for $R=1.0$
and $A=0.8$. (b) Number of bounces as a function of $R$ and $v_{i}$ for
$A=0.8$. Source: Results obtained by the author in [Campos e Mohammadi 2021].

Increasing the wobbling amplitude to $A=0.8$, we arrive at Fig. 7. We plot the
final velocity versus the initial velocity on the left panel. In this case,
the wobbling energy is much larger, and its effect is much more noticeable.
Accordingly, the oscillation in the crossing curve is more significant, and
there is a larger number of one-bounce resonance windows. The one-bounce
resonance windows appear in the two ways discussed above. Moreover, the final
velocity can be above the identity curve due to the extra vibrational energy
available. As the initial velocity decreases, we find many two-bounce
resonance windows as well as higher bounce ones. They also appear due to the
extra wobbling energy. This structure should continue all the way to
$v_{i}=0$, but the curve is truncated because the computational time for the
kinks to separate becomes increasingly larger.

The number of bounces as a function of $v_{i}$ and $R$ is shown in Fig. 7(b).
One can see that the spine structure is even more intricate in this case
because there are many more one-bounce windows. Interestingly, it is possible
to find annihilation for values of $v_{i}>0.35$. This occurs because if the
wobbling is out of phase at the collision, it favors annihilation.

Figure 8: (a) Final velocity versus initial velocity for collisions between
wobbling kinks with (a) $R=0.3$ and (b) $R=3.0$. (c) and (d) Value of the
field at the center of the collision as a function of $t$ and $v_{i}$ for the
same values of $R$. The amplitude is $A=0.8$.

Source: Results obtained by the author in [Campos e Mohammadi 2021].

If we take a look at Fig. 6(b) and Fig. 7(b), we will notice that there are
two-bounce resonance windows, denoted as red dots, for small and large values
of $R$. Previously, we found that these regions would not have resonance
windows if the kinks were not wobbling initially. Therefore, we conclude that
the resonant structure was hidden before turning the wobbling on. The plots in
Fig. 8 can confirm it. First, we consider wobbling kink collisions with
$A=0.8$ and compute the kinks’ final velocity as a function of the initial one
for both $R=0.3$ and $R=3.0$. In the former case, there are many higher-bounce
windows at the border of the one-bounce ones, while, in the latter, there are
only a few two-bounce windows. This confirms the hidden resonant structure.
There are more higher-bounce resonance windows for $R=0.3$ than for $R=3.0$.
This occurs because the wobbling energy $E_{D}$ is larger for larger wobbling
frequencies, and $\omega_{D}$ is much larger for $R=0.3$ than for $R=3.0$ as
shown in Fig. 2. Second, as a complement, we compute the field of the center
of the collision as a function of time and $v_{i}$ for the same values of the
parameters. In the scale of the figure, it is not possible to see higher-
bounce resonance windows, but the structure of one-bounce resonance windows
can be easily seen. In particular, for $R=0.3$, the one-bounce windows appear
much more frequently. This is expected from eq. (11) because, in this case,
the wobbling frequency $\omega_{D}$ is much larger than the $R=3.0$ case.

Figure 9: (a) Final velocity as a function of the initial velocity for several
values of $A$ and $R=1.0$. (b) Bounce regions as a function of $A$ and $v_{i}$
for $R=1.0$. Source: Results obtained by the author in [Campos e Mohammadi
2021].

In Fig. 9(a), we plot the final velocity as a function of the initial one but
only for one-bounce windows and the crossing curve. We fix $R=1.0$ and show
the result for several values of the wobbling amplitude. This is the best way
to visualize the appearance of one-bounce resonance windows. As the amplitude
is increased, the crossing curve splits, giving rise to some one-bounce
windows, whereas others appear as well in regions outside the crossing curve.
The same behavior can be seen from a different angle in Fig. 9(b). There, we
show in blue the regions in the $v_{i}A$-plane where separation after one
bounce occurs. The other colors indicate higher-bounce windows. Again, we are
fixing $R=1.0$. As we move from the bottom, with $A=0$, to the top, we find
that many one-bounce windows appear. The number of windows becomes larger as
$A$ increases because the wobbling becomes more energetic. Again, the two ways
that one-bounce windows appear can be distinguished in the figure. In the
first one, the blue region is split as $A$ becomes larger due to the
appearance of a white region in between. A blue region appears in a white
region as $A$ increases in the second one.

Figure 10: Velocity at the center of a one-bounce resonance window as a
function of an index. We fix $R=1.0$. Source: Results obtained by the author
in [Campos e Mohammadi 2021].

In Fig. 9(a), one can notice that the position of the center of one-bounce