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# Scattering of kinks in coreless potentials

Ondřej Nicolas Karpíšek  karponius(at)gmail.com Institute of Physics, Silesian
University in Opava, Bezručovo nám. 1150/13, 746 01 Opava, Czech Republic.
Lukáš Rafaj  lukasrafaj(at)gmail.com Institute of Physics, Silesian University
in Opava, Bezručovo nám. 1150/13, 746 01 Opava, Czech Republic.  Filip
Blaschke  filip.blaschke(at)physics.slu.cz Research Centre for Theoretical
Physics and Astrophysics, Institute of Physics, Silesian University in Opava,
Bezručovo nám. 1150/13, 746 01 Opava, Czech Republic.

###### Abstract

We explore the relevance of the central hill for a symmetric double-well
potential and its impact on the scattering of kinks in a scalar field theory
in (1+1)-dimensions. This region controls the inner core structure of the
kink. We study how the disappearance of analyticity in this region of the
potential affects the resonant features in $K\bar{K}$ scattering.

##  I Introduction

Classical field theory with non-linear field interactions often leads to the
presence of solitons. These are particle-like, extended objects that are
remarkably stable under the effects of perturbation or soliton-soliton
interactions and lead to a richness of dynamical phenomena. When the
underlying model has in addition non-trivial topology, there exist topological
solitons, which are absolutely stable Manton:2004tk ; Vilenkin:2000jqa ;
Shnir:2018yzp .

Among the conceptually and mathematically simplest topological solitons are
the so-called _kinks_. These particle-like objects in one spatial dimension
(manifesting as strings and domain walls in higher dimensions) are described
by a single, real, self-interacting scalar field, say $\phi$. A local, Lorentz
invariant description is afforded via the Lagrangian density

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

where the potential $V(\phi)$ encodes the self-interaction. The kinks are
present as static solutions for any potential $V(\phi)$ that has multiple
vacua, i.e. field values $\phi=v_{a}$ for which
$V(v_{a})=V^{\prime}(v_{a})=0$.

The mathematical framework embodied in Eq. (1) is so simple that one would not
expect the dynamics of kinks to be particularly complicated. Since kinks have
been the object of scientific interest for many decades now, one could be
tempted to guess that the overall dynamic picture of how kinks interact with
themselves or with the environment is fully mapped out. Therefore, it is more
surprising to learn that the actual state of affairs is still far from ideal.

Indeed, while the interactions of kinks with other kinks and/or anti-kinks
have been numerically investigated since the late 70ties Sugiyama:1979mi ;
Campbell:1983xu ; Moshir:1981ja ; Anninos:1991un ; Belova:1985fg , the
quantitative and qualitative dynamical picture of the associated phenomena,
such as bouncing, bion formation, the role of radiation, spectral wall
phenomenon, dynamical generation of delocalized modes, etc., has been achieved
in both classical and quantum settings (with various degrees of completeness)
only recently Manton:2020onl ; Manton:2021ipk ; Adam:2021gat ; Dorey:2011yw ;
Adam:2022mmm ; Weigel:2013kwa ; Dorey:2023izf ; Adam:2023kel ; Adam:2023qgx ;
Dorey:2023izf ; Adam:2023kel ; Adam:2022bus ; Adam:2019xuc ; Adam:2023qgx ;
Dorey:2017dsn ; Evslin:2022xmp ; Evslin:2023oub ; Evslin:2023egm ;
Evslin:2024czr  (one can also read about the somewhat intricate history of
investigations of kink-anti-kink collisions in 2019arXiv190903128K ).

The source for all this complexity must be somehow intricately encoded into
the potential $V(\phi)$.

Indeed, the choice of the potential dictates the underlying dynamics with
precarious exactness. For instance, the so-called sine-Gordon (sG) model
$V(\phi)=2\sin^{2}(\phi/2)$ is completely integrable, and the kink-anti-kink
($K\bar{K}$) and kink-kink $(KK)$ solutions (and many others) are known in a
closed form. The collisions of kinks are completely elastic, exemplifying
perhaps the simplest behavior across the spectrum of all models.

On the other hand, $K\bar{K}$ collisions in the $\phi^{4}$ (or double-well)
model with the potential $V(\phi)=\bigl{(}1-\phi^{2}\bigr{)}^{2}/2$, which is
regarded as a canonical representative, are very rich in dynamical aspects,
such as the fractal structure of bouncing windows, bion chimneys, etc.
Moreover, these features are universal, in that they are present for generic
choices of potentials unless special circumstances prevent them from
occurring, such as integrability or absence of resonant modes that facilitate
energy transfer mechanism.

This is exemplified in the so-called $\phi^{6}$ model with
$V(\phi)=\phi^{2}\bigl{(}1-\phi^{2}\bigr{)}^{2}/2$ in which the $K\bar{K}$
collisions are devoid of this fractal structure, while the collisions of anti-
kinks with kinks ($\bar{K}K$) produce it Adam:2022mmm . Multiple other
potentials have been explored, such as $\phi^{8}$ potential
$V(\phi)=\phi^{4}\bigl{(}1-\phi^{2}\bigr{)}^{2}/2$ in which kinks have long,
polynomial tails Christov:2018wsa ; Manton:2018deu ; Gani:2015cda ;
Gani:2021ttg  and the parametrically-dependent Christ-Lee potential, i.e.
$V(\phi)=\bigl{(}\varepsilon^{2}+\phi^{2}\bigr{)}\bigl{(}1-\phi^{2}\bigr{)}^{2}/(2+2\varepsilon^{2})$
that smoothly interpolates between $\phi^{4}$ and $\phi^{6}$ potentials
Dorey:2023izf , to name just a few.

###  I.1 The shape of a kink

The shape of a static kink is in one-to-one correspondence with the shape of
the potential between the minima that the kink interpolates. This is most
easily seen from the static equation of motion

$\phi^{\prime\prime}(x)=V^{\prime}(\phi)\,,$ (2)

which is equivalent to an equation of motion for a particle under the
influence of the upside-down potential $-V(\phi)$. The “time” is the
$x$-coordinate and the particle starts at one of the maxima, corresponding to
the “left” vacuum, $v_{\rm L}$, at $x=-\infty$ and rolls down towards the
other maximum, $v_{\rm R}$, that is reached at $x=+\infty$.

For a generic potential with two vacua, and no other local minima, such as the
one depicted in Fig. 1, we can talk about three distinct regions with
qualitatively different impacts on the shape of the kink.

Figure 1: A pictograph of how the shape of a kink, interpolating between
$v_{\rm L}$ and $v_{\rm R}$ vacua is affected by different parts of the
potential.

The regions near vacua encode the _tails_ of the kink. Indeed, for non-zero
curvature, i.e. $V^{\prime\prime}(v_{\rm L,R})\equiv m_{\rm L,R}^{2}>0$, the
kink approaches vacua exponentially fast as $|\phi-v_{\rm
L,R}|\sim\exp\bigl{(}\pm m_{\rm L,R}x\bigr{)}$. On the other hand, if $m=0$,
the tails are polynomial Christov:2018ecz .

The region near the maximum controls the shape of the very center of the kink,
which we shall call its _core_. Denoting
$\alpha^{2}\equiv-V^{\prime\prime}(\phi_{\rm max})$, the kink’s profile at its
core (where also the most energy is concentrated and is identified with the
kink’s position) will be approximately given by a combination of $\sin(\alpha
x)$ and $\cos(\alpha x)$. On the other hand, if $\alpha=0$, such is the case
for $\phi^{8}$ potential, the core’s profile can be approximated by Jacobi
functions.

Lastly, the regions near inflection points dictate what we subsume into the
notion of the kink’s _skin_. Around the skin, the kink behaves as a quadratic
function of $x$.111This is also true around a generic point where the first
derivative is non-vanishing.

We can be more precise and place the above notions onto a firm ground by
defining the tails, core, and skin of a kink in a perturbative sense.

Expanding the field around the vacuum, i.e. $\phi=v+\delta\phi$ and putting it
into the equation of motion, we get

$\delta\phi^{\prime\prime}=m^{2}\delta\phi+\frac{1}{2}V^{\prime\prime\prime}(v)\delta\phi^{2}+\ldots$
(3)

At the leading order, the above equation is solved as $\delta\phi\sim
A\exp\bigl{(}\pm mx\bigr{)}$, where the sign depends on the boundary
conditions. We can estimate at which value of $\delta\phi$ the leading order
approximation breaks down by comparing the size of the next term on the right-
hand side. Indeed, the value of $\delta\phi$ for which both terms are equal
gives us the rough extent of the tail region, i.e.

$L_{\rm
tail}\equiv\frac{2m^{2}}{\bigl{|}V^{\prime\prime\prime}(v)\bigr{|}}\,.$ (4)

The $L_{\rm tail}$ corresponds to the horizontal extend of either of the red
regions on Fig. 1. Note, that the above formula is only valid if
$V^{\prime\prime\prime}(v)\not=0$. If the third derivative vanishes, comparing
the leading term with the third order term gives us instead $L_{\rm
tail}=\sqrt{6m^{2}/\bigl{|}V^{(4)}(v)\bigr{|}}$. Similar consideration must be
taken when dealing with potentials with vanishing $m$.

The size of the core and skin regions of the potential can be estimated using
the same reasoning through expansion around the maximum or an inflection
point, respectively. Denoting $\alpha^{2}\equiv-V^{\prime\prime}(\phi_{\rm
max})$ and $\eta\equiv V^{\prime}(\phi_{\rm inf})$ the corresponding sizes of
these regions read (assuming non-vanishing third derivatives)

$L_{\rm core}\equiv\frac{2\alpha^{2}}{\bigl{|}V^{\prime\prime\prime}(\phi_{\rm
max})\bigr{|}}\,,\quad L_{\rm
skin}\equiv\sqrt{\frac{2|\eta|}{\bigl{|}V^{\prime\prime\prime}(\phi_{\rm
inf})\bigr{|}}}\,.$ (5)

The numbers $L_{\rm tail}$, $L_{\rm core}$ and $L_{\rm skin}$ should be used
as rough gauges of how much the tails, core, and skin dominate the kink’s
shape. Of course, these regions can have overlaps and typically do not add up
to the total value of a field span for a static (anti-)kink, that is $|v_{\rm
R}-v_{\rm L}|$.

For illustration, the corresponding values for a $\phi^{4}$ potential are
$L_{\rm tail}^{\phi^{4}}=2/3$, $L_{\rm core}^{\phi^{4}}=1$ and $L_{\rm
skin}^{\phi^{4}}=2/9$, while the same for the sG model reads $L_{\rm
tail}^{\rm sG}=\sqrt{6}$, $L_{\rm core}^{\rm sG}=\sqrt{6}$ and $L_{\rm
skin}^{\rm sG}=\sqrt{2}$. The $\phi^{6}$ potential, having three minima with
the middle one of different curvature, thus has two different tails and skin
regions, depending on which vacuum and which inflection point is considered.

Let us also note that there can be additional local minima in between the true
vacua that the kink is interpolating. In that case, it is reasonable to dub
the corresponding region of the kink near a local minimum as its _pseudocore_.
The reason for this nomenclature – as opposed to calling it a pseudotail – is
simply that a pseudocore region affects the kink’s center and it would be
confusing to associate it with the semi-infinitely extended tails. In the
double sine-Gordon model (dsG), for instance, the kink solution is
approximately equal to two sG kinks separated by a fixed distance. In our
picture, each of these sG kinks will have its core and skin regions, while the
region between them corresponds to a pseudocore.

###  I.2 The Frankensteinian potentials

So far, we have discussed the structural aspects of a static kink. The issue
now is, whether these notions have any relevance for the dynamics and if so,
whether their contributions are approximately independent.

Intuitively, the answer should be cautious “yes”. It is well known, for
instance, that kink tails are responsible for attractive interaction between
well-separated kink-anti-kink ($K\bar{K}$) pairs and are, therefore key
ingredients in the initial (and final) stages of the $K\bar{K}$ collisions
Manton:2004tk . On the other hand, it is also intuitively clear that they are
of little consequence when the kinks are on top of each other where their
cores play a key role.222Assuming, of course, that kinks retain their
individuality during the collisions and it is hence still reasonable to talk
about their cores. In this regard, the successes of collective coordinate
models based on separated kink ansatzes in reproducing qualitative features of
$K\bar{K}$ scattering indicate that this is not an unreasonable assumption.

The role of skin, however, if any, is hard to appreciate intuitively. However,
the skin region of the potential (i.e. its inflection point) is deeply
connected with the longevity or even existence of oscillons Gleiser:2008ty ,
which are important, if not crucial, for the understanding of the $K\bar{K}$
scattering.

These observations may help us to appreciate the underlying reason for the
complexity of $K\bar{K}$ scattering – the fact that it may be composed of
multiple structural pieces that contribute differently and simultaneously to
its dynamics. In other words, it could be the case that $K\bar{K}$ scattering
is a tapestry of interwoven but otherwise only loosely dependent contributing
phenomena.

To facilitate this intuition, it should be worthwhile to investigate the
dynamics of kinks in potentials lacking some or most of these structural
pieces. To that end, let us briefly discuss a particular class of potentials
that – for lack of a better term – we call the _Frankensteinian_ potentials.
As the name suggests, they are composed of pieces of functions of the field
that are continuously and/or differentiably sewn together at chosen field
values. For our purposes, we see their primary utility in the fact that inside
each patch, a given Frankensteinian potential could be deficient in one or two
structural regions. This makes them an ideal tool for exploring the relevance
of these regions on the dynamics.

In particular, we will focus on a subset of Frankensteinian potentials that
are constructed out of linear or quadratic functions. Within these pieces, the
leading order approximations presented above for either a tail, core, or skin
hold exactly. Thusly constructed potentials, therefore, possess the simplest
kink solutions as far as their static characteristics are concerned.

###  I.3 Single-component kinks

To provide some examples of Frankensteinian potentials, let us consider those
that support static solutions of the greatest structural simplicity, namely
kinks made of a single type of component. In Fig. 2, we present basic
characteristics for these potentials and their kinks. We use the labeling
$TT$, $C$, and $SS$ that simply tally the structural pieces of the kinks as
seen from the left to right.

Figure 2: The “simplest” kinks and their potentials. The vacua are placed at
$\pm 1$ and kinks are centered at $x=0$ for simplicity. The last row shows the
effective potential that enters the Schrödinger-like equation for the
determination of normal modes of a given kink.

The potential labeled by $TT$ (“tail-tail”) consists of two quadratic wells
sewn together at the center $\phi=0$. For simplicity, both wells have the same
curvature $m^{2}$, and the respective vacua are placed at $\pm 1$, which is
always possible to enforce via rescaling of the field. In fact, it is also
possible to set $m=1$ by rescaling the coordinates.

Consequently, the $TT$ kink is made of two exponential tails sewn
differentiably together at the center of the kink where $\phi=0$. With the
energy density given as

$\mathcal{E}_{TT}=m^{2}\mathrm{e}^{-2m|x|}\,,$ (6)

the mass of the $TT$ kink works out to be equal to the perturbative mass, i.e.
$M_{TT}\equiv\int\limits_{-\infty}^{\infty}\mathcal{E}_{TT}\,\mathrm{d}x=m$.
In turn, the second moment of energy density reads
$\int\limits_{-\infty}^{\infty}x^{2}\mathcal{E}_{TT}\,\mathrm{d}x=1/(2m)$,
which provides a simple measure of how the energy is concentrated around the
kink’s center.

This can be equivalently expressed using the so-called Derrick’s frequency,
which is defined as the ratio of the mass and the second moment and for a $TT$
kink reads $\omega_{TT}^{2}=2m^{2}$.

Derrick’s frequency is associated with Derrick’s mode of the kink, which is
derived by observing how the energy of a static kink changes under an
infinitesimal scaling of the spatial coordinate. Derrick’s mode, however, is
not a normal mode, but it has been recognized to be important for restoring
Lorentz invariance of collective coordinate models Adam:2021gat . That being
said, the $TT$ kink has Derrick’s frequency well above the mass threshold and
hence should have no impact on the dynamics.

The normal modes of $TT$ kink can be also easily established by solving the
Schrödinger-like equation with the effective potential given by $U(x)\equiv
V^{\prime\prime}(\phi_{TT})$. As we see in the last row of Fig. 2, the
effective potential for $TT$ kink consists of a single $\delta$ well with
strength $-m^{2}$ plus a constant $m^{2}$. As is well known, a Dirac
$\delta$-well potential supports only a single bound mode, which is the zero
mode associated with uniform translations. Hence, no massive bound modes exist
for $TT$ kink. Note, that the effective potential $U(x)$ is the only quantity
in Fig. 2 that is sensitive to the regions of the potential outside the vacua,
i.e. $|\phi|>1$. The kink solution itself and its static characteristics given
in the fourth row of Fig. 2 would remain unaltered if the potential is
modified in these outer regions.

Let us now briefly turn our attention to the remaining two kinks, that are
made of a core ($C$) and two skins $(SS)$. These represent the simplest
examples of _compact_ kinks whose spatial extents are finite. This is due to
the non-analytic minima, which effectively means an infinite perturbative
mass, rendering the tails non-existent. We will not comment on the dynamics of
these kinks in this paper outside what is given in Fig. 2. The reason is that
these compact kinks have been proposed and investigated before. In fact, the
origin of compact solitons can be traced back to the 90ties, when they were
introduced in the context of modified KdV equation Rosenau:1993zz ,
investigation of which continues to this day Francisco .

The compact solitons in the relativistic scalar field theory were first
explored in the early 2000s in Arodz:2002yt , later in the context of the so-
called $V$-shaped potentials Arodz:2005gz , the most famous of which is the
so-called signum-Gordon potential Arodz:2007ek . The signum-Gordon model is
probably the simplest Frankensteinian model consisting only of two linear
functions sewn at the middle forming a $V$-shaped potential well, which
supports exactly soluble compact oscillons Arodz:2007jh ; Hahne:2019ela .
Furthermore, the compact kinks were introduced as a limit of certain
mechanical linear systems with chained pendula in Arodz ; Arodz:2003mx .

The Frankensteinian potential that was proposed for studying the interaction
between compact kinks and oscillons Hahne:2022wyl  and the scattering of
compact kinks themselves Hahne:2023dic  was a periodic version of the
potential $C$ in Fig. 2. As shown in Hahne:2023dic , collisions of compact
$K\bar{K}$ pairs lead to long-living oscillating bound states or the
$K\bar{K}$ pair reemerges with accompanying “shockwave” that disintegrates
into a cascade of compact oscillons. Curiously, the characteristic bouncing of
kink anti-kink has not been observed. This is, perhaps, because the potential
used in Hahne:2023dic  was a periodic piece-wise quadratic function that
lacked any other non-linearities besides the non-analytic sewing at minima.

As far as we are aware, no dedicated study has been published on compact kinks
made of skins.

A different kind of Frankensteinian potential was also presented in
campostoymodel , which is a piece-wise quadratic potential resembling
$\phi^{4}$ potential with the sewing points around the region controlling the
skin of the kink. As a result, the kink solution possesses both exponentially
dampened tails and a core region, but no skin. In our notation, such a
solution is a symmetric $TCT$ kink. These kinks feature all characteristics of
generic $K\bar{K}$ scattering: bouncing, bion formation, and radiation
production.

Let us also mention that there have been few works that investigate potentials
near Frankensteinian (or otherwise singular) limits that support compact kinks
Bazeia:2014hja ; Bazeia:2019tgt .

The central task of this paper is to both continue the investigation of the
Frankensteinian potentials and to address a specific question: namely how does
the core region affect the outcomes of $K\bar{K}$ scattering? In particular,
we will focus on the presence or absence of characteristic dynamical feature
due to the resonant-energy transfer mechanism, namely the bouncing.

In Sec. II, we begin addressing this question by briefly discussing $K\bar{K}$
scattering in models of increasing structural complexity. We will first show
that scattering of $TT$ kinks is particularly simple: the only outcome is a
total annihilation into massive waves. Then we show that this result is
unaltered by adding non-linearities inside the tail region, without
introducing inflection points. It is only in the third example, with a
potential possessing inflection points and therefore skin region, where we see
oscillons together with critical velocity below one.

In Sec. III we change tactics and present two parametric families of
potentials interpolating between $\phi^{4}$ and particular coreless
potentials. Here we study the disappearance of resonant structures in
$K\bar{K}$ collisions as the potential becomes more and more singular at its
center.

In Sec. IV we discuss our results.

##  II Coreless kinks: three examples

In this section, we shall consider three examples of coreless potentials. As
we will see, for the first two, the absence of both core and skin regions
seems to render the $K\bar{K}$ scattering entirely trivial: the only result of
the collision is annihilation into massive waves. Only in the third example,
where we consider a potential with inflection point = non-zero skin region, do
we obtain oscillons, however, the bouncing is still absent.

###  II.1 Scattering of $TT$ kinks

Let us comment on $K\bar{K}$ scattering of the kinks made from only
exponential tails. As already described, the potential is constructed by
gluing together two quadratic wells of the same curvature at the origin. We
can express such potential as

$\displaystyle V_{TT}(\phi)=$
$\displaystyle\frac{1}{2}m^{2}\bigl{(}1-|\phi|\bigr{)}^{2}\,.$ (7)

The kink solution can be obtained by sewing together two exponentials. Up to
arbitrary shift along the $x$-axis, the solution reads:

$\displaystyle\phi_{TT}={\rm sign}(x)\Bigl{(}1-\mathrm{e}^{-m|x|}\Bigr{)}\,,$
(8)

where ${\rm sign}(x)$ is the sign function. Both the potential (7) and the
kink solution are illustrated in Fig. 3.

Figure 3: An example of piece-wise quadratic potential with the $\phi_{TT}$
kink solution compared with $\phi^{4}$ model.

It is easy to verify that $\phi_{TT}$ solves the second-order equations of
motion in a weak sense. There are no $\delta$-function contributions since not
only the field itself but also its first derivative is continuous. However,
there is a jump singularity for the second derivative.

As discussed in the previous section, there are no massive normal modes and
Derrick’s frequency is above the mass threshold, i.e.
$\omega_{D}^{2}=2m^{2}>m^{2}$. It is thus not surprising that collisions of a
TT kink with an anti-kink are quite boring. However, this featureless-ness is
of a different kind than, say in sG model, where $K\bar{K}$ pairs collide
elastically and do not annihilate each other due to the underlying
integrability. In contrast, $TT$ kinks _always annihilates_ each other.

It is easy to understand why. When the $K\bar{K}$ pair is sufficiently close
to each other, the whole field becomes localized entirely within the left
quadratic well. From that point on, the dynamics is equivalent to a time-
evolution of some initial data via the Klein-Gordon equation with the mass
$m$. As is well known, this results in the disintegration of the initial shape
into a train of massive waves.

As a check of the above intuition, we plot a “map” of $K\bar{K}$ scattering in
Fig. 4, where we show the dependence of the central value of the field on
time, i.e. $\phi(0,t)$, for the whole range of initial velocities of the
$K\bar{K}$ pair.

Figure 4: A plot of values of $\phi(0,t)$ as dependent on initial velocity $v$
of the $K\bar{K}$ pair. We see that for any $v$, the initially separated pair
(blue color indicating $\phi\sim+1$ vacuum at the center before the collision)
always disintegrates into massive waves around the $\phi\sim-1$ vacuum (the
yellow colors). Here, $m=2$.

###  II.2 Scattering of kinks in piece-wise quartic wells.

The triviality of scattering of $TT$ kinks can be blamed on the linearity of
the respective equation of motion in each half-plane of the target space.
Therefore, it is unclear whether this dull result has anything to do with the
absence of core and skin region in the $V_{TT}(\phi)$ potential or, rather,
whether it is an artifact of its piece-wise integrability. For this reason, in
this subsection, we will investigate a potential that is non-linear in each
potential well. This is achieved by adding a quartic term, i.e.

$V_{TT\lambda}(\phi)\equiv\frac{1}{2}m^{2}\bigl{(}1-|\phi|\bigr{)}^{2}+\frac{\lambda}{12}\bigl{(}1-|\phi|\bigr{)}^{4}\,,$
(9)

where $\lambda>0$ is an arbitrary positive constant. This potential leads to
the equation of motion that is neither integrable nor linear in the respective
halves of the target space. However, it still lacks inflection points, and
thus the kink solution, i.e.

$\phi_{TT\lambda}={\rm
sign}(x)\biggl{(}1-\frac{m\sqrt{6/\lambda}}{\sinh\bigl{(}m|x|+\sinh^{-1}(m\sqrt{6/\lambda})\bigr{)}}\biggr{)}\,,$
(10)

has neither a core nor a skin, but possesses non-linear tails.

Notice that in the limit $\lambda\to 0$, the solution becomes ${\rm
sign}(x)\bigl{(}1-\mathrm{e}^{-m|x|}\bigr{)}$, which is nothing but a $TT$
kink. We display both the potential and its kink solution in Fig. 5.

Figure 5: The potential (9) and its kink solution compared with $\lambda=0$
case. Here $m=2$ and $\lambda=6$.

The Bogomol’nyi-Prasad-Sommerfield (BPS) mass Manton:1981mp ; Bogomolny:1975de
of the kink reads

$M_{TT\lambda}=\frac{4m^{3}}{\lambda}\biggl{(}\Bigl{(}1+\frac{\lambda}{6m^{2}}\Bigr{)}^{3/2}-1\biggr{)}\,,$
(11)

which is an increasing function of $\lambda$. Derrick’s frequency can be also
shown to be an increasing function of $\lambda$. Hence, it always stays above
the perturbative threshold for all $\lambda>0$. We also did not find any
massive modes. Indeed, a quick glance at the effective potential
$V^{\prime\prime}(\phi_{TT\lambda})$ reveals that the extra term acts as a
potential barrier rather than a well, hence there is no hope for any bound
mode besides the zero mode that is there due to the delta-peak potential well
at $x=0$.

Given that there are no modes that could facilitate resonant energy transfer,
we expect that the dynamics of $K\bar{K}$ scattering is equally trivial as for
$TT$ kinks. Indeed, this is what we found. We show our result in Fig. 6.

Figure 6: The plot of values of $\phi(0,t)$ as dependent on the initial
velocity $v$ of the $K\bar{K}$ pair in the model with additional quartic term.

###  II.3 Scattering of kinks in log-corrected Klein-Gordon model

Lastly, let us consider a coreless potential, which does have inflection
points, namely the Klein-Gordon model with a logarithmic “correction”, i.e.

$V_{\rm
KG+log}(\phi)=\frac{1}{4}m^{2}\Bigl{(}1-\phi^{2}+\phi^{2}\log(\phi^{2})\Bigr{)}\,.$
(12)

We display it in Fig. 7.

Figure 7: The potential (12) and its kink solution compared with $\phi^{4}$
model. Here $m=2$.

This potential has a degenerate minima at $\phi=\pm 1$ with
$V^{\prime\prime}(\pm 1)=m^{2}$. Note that we can fix $m^{2}$ to any value by
rescaling the coordinates. As was the case for the previous two potentials,
the maximum at $\phi=0$ is non-analytic since the second derivative at
$\phi=0$ is undefined (it goes to $-\infty$). This makes the Taylor expansion
around the origin impossible, rendering $V_{\rm KG+log}(\phi)$ coreless.
However, unlike for both $V_{TT}(\phi)$ and $V_{TT\lambda}(\phi)$, there are
inflection points located at $\phi_{\rm inf}=\pm 1/{\rm e}$, hence it makes
sense to talk about the skin.

The kink solution cannot be obtained analytically and we showcase a numerical
solution in Fig. 7. The BPS mass reads

$M_{\rm
KG+log}=\frac{m}{\sqrt{2}}\int\limits_{-1}^{1}\sqrt{\eta^{2}\log\eta^{2}+1-\eta^{2}}\,\mathrm{d}\eta\approx
0.84\times m\,.$ (13)

Derrick’s frequency is numerically determined to be $\omega_{D}^{2}\approx
1.34m^{2}$, so it is above the perturbative threshold. Similarly, we have not
found any massive mode. Given that there are no bound modes, we could expect
the scattering of $K\bar{K}$ pairs to be still completely sterile as in the
previous two models, but this expectation would be incorrect.

In fact, compared with $V_{TT}(\phi)$ and $V_{TT\lambda}(\phi)$, the
$K\bar{K}$ scattering in $V_{\rm KG+log}$ potential exhibits two new features.
The first one is the appearance of a critical velocity, $v_{\rm crit}\approx
0.79$, above which the scattering is quasi-elastic, and second, there are
certain velocities below $v_{\rm crit}$ for which oscillons are produced. This
can be seen in Fig. 8.

Figure 8: The evolution of the central value of the field $\phi(0,t)$ as a
function of initial velocity of the $K\bar{K}$ pair. In the insets, we see two
particular choices of $v$ that showcase the formation of an oscillon and a
quasi-elastic collision. Here $m=2$.

The results of this section show that if a potential lacks both core and skin
regions, it implies a lack of any interesting dynamics in $K\bar{K}$
collisions. This was true for both $V_{TT}(\phi)$ and $V_{TT\lambda}(\phi)$.
When we considered coreless potential with inflection points, i.e. $V_{\rm
KG+log}(\phi)$, we immediately got more interesting dynamics. Of course, we
cannot prove that this must be the case in general, but we take these
observations as a confirmation that the core and skin regions greatly affect
the outcomes of $K\bar{K}$ collisions.

##  III Disappearance of bouncing windows and non-analyticity of the maximum

In this section, we investigate $K\bar{K}$ scattering in two parametric
families of models that start from $\phi^{4}$ potential and develop a cusp
singularity around the origin ($V(\phi)\sim|\phi|$) as $\varepsilon$ goes from
1 to 0. In this way, we explore how the rate of divergence of
$V^{\prime\prime}(\phi)$ at $\phi\to 0$ affects the bouncing windows. As we
shall see, the bouncing windows disappear in an almost linear fashion well
before the point $\varepsilon=0$ is reached.

###  III.1 Two families of coreless potentials

We define the first family in such a way that it begins as $\phi^{4}$
potential at $\varepsilon=1$ and deforms into $V_{TT}$ potential (7) as
$\varepsilon\to 0$, i.e.

$V_{\varepsilon}^{(1)}(\phi)=\frac{2}{(1+\varepsilon)^{2}}\Bigl{(}1-|\phi|^{1+\varepsilon}\Bigr{)}^{2}\,.$
(14)

We display $V_{\varepsilon}$ in Fig. 9 (top).

The second family aims to deform the $\phi^{4}$ model only around the maximum
and keep the quartic term for all $\varepsilon\leq 1$.

$V_{\varepsilon}^{(2)}(\phi)=\frac{1}{1+\varepsilon}-\frac{4|\phi|^{1+\varepsilon}}{(3-\varepsilon)(1+\varepsilon)}+\frac{\phi^{4}}{3-\varepsilon}\,.$
(15)

We display $V_{\varepsilon}$ in Fig. 9 (bottom).

Figure 9: Two families of potentials that start as $\phi^{4}$ model (thick
black line) and gradually develop a cusp singularity at the central hill as
$\varepsilon\to 0$.

Note that both families are coreless in the entire range $\varepsilon\in[0,1)$
as $V^{\prime\prime}(0)$ is undefined.

As $\varepsilon\to 0$, the potential $V_{\varepsilon}^{(1)}$ looses its skin
regions. Indeed, the positions of inflection points, given as $|\phi_{\rm
inf}^{(1)}|=(\varepsilon/(1+2\varepsilon))^{1/(1+\varepsilon)}$, gradually
decreases from $\phi^{4}$ values $\pm 1/\sqrt{3}$ to zero as $\varepsilon\to
0$, so that they merge with the maximum. However, in $V_{\varepsilon}^{(2)}$
family, the inflection points stay separate from the central maximum for all
$\varepsilon$. Indeed, $|\phi_{\rm
inf}^{(2)}|=\bigl{(}1/3\bigr{)}^{1/(3-\varepsilon)}$ and the $\varepsilon=0$
positions are $\pm 1/\sqrt[3]{3}$.

For both $V_{\varepsilon}^{(1)}$ and $V_{\varepsilon}^{(2)}$ the curvatures of
the vacua remain constant, i.e. $m^{2}=4$.

For the first family, the kink solution can be given implicitly as

$\phi\ _{2}F_{1}\biggl{(}\begin{matrix}1\quad 1/(1+\varepsilon)\\\
(2+\varepsilon)/(1+\varepsilon)\end{matrix}\biggr{|}\
|\phi|^{1+\varepsilon}\biggr{)}=\frac{2x}{1+\varepsilon}\,,$ (16)

where ${}_{2}F_{1}$ is the hypergeometric function. For the second family, the
BPS equation can be solved only numerically. However, for both families, the
shapes of kinks remain quite close to the $\phi^{4}$ kink, i.e. $\tanh(x)$, as
seen from Fig. 10. As expected, the absolute deviation is smaller for
$V_{\varepsilon}^{(2)}$.

Figure 10: Absolute differences between shapes of deformed kinks and
$\phi^{4}$ kink for various values of $\varepsilon$ for both families given in
Eqs. (14)-(15).

The BPS mass of the first family can be obtained analytically as
$M_{K}^{(1)}=4/(2+\varepsilon)$, while for the second family it is known only
implicitly through the integral
$2\int\limits_{0}^{1}\mathrm{d}\phi\,\sqrt{2V_{\varepsilon}^{(2)}}$. The
frequencies of Derrick’s modes must be obtained numerically for both families.
We display both these quantities in Fig. 11.

Figure 11: The BPS masses and Derrick’s frequencies for families as functions
of $\varepsilon$. Figure 12: Dependence of normal modes on $\varepsilon$ for
families $V_{\varepsilon}^{(1,2)}$. For $V_{\varepsilon}^{(1)}$, the massive
mode crosses the continuum threshold around $\varepsilon_{\rm
threshold}=0.365$.

An interesting question is what happens to the structure of normal modes as
$\varepsilon\to 0$? In particular, we are interested in the change of the sole
bound mode of $\phi^{4}$ kink with the frequency $\omega=\sqrt{3}$. In Fig. 12
we see that with decreasing $\varepsilon$, the frequency of this mode
gradually increases and merges with the continuum around the value
$\varepsilon_{\rm threshold}\approx 0.365$. On the other hand, for the second
family, the massive mode never merges with the continuum.

###  III.2 Bouncing windows

Given how we set up our models, it is expected that all dynamical features
connected with resonant energy transfer disappear for the first family
$V_{\varepsilon}^{(1)}$ as $\varepsilon\to 0$. The only question is how fast.

We explore this issue by performing numerical analysis of $K\bar{K}$
collisions (see details of our numerical method in the Appendix) for various
values of $\varepsilon$. The main result is displayed in Fig. 16 (left
column). Surprisingly, the bouncing windows disappear well before
$\varepsilon_{\rm threshold}$ where the massive mode crosses into the
continuum. In fact, the actual value $\varepsilon\approx 0.7$ coincides almost
perfectly with the merging of Derrick’s frequency into the continuum (see Fig.
11). This is surprising, as it is usually the massive mode and not Derrick’s
mode that is linked with the presence of the bouncing windows.

Figure 13: Closing of two-bounce windows for both families. The numbers
represent approximate values of $\varepsilon$ at which that particular window
closes, given our numerical accuracy.  Figure 14: Detail of closing of windows
for two-bounce, three-bounce, and four-bounce for both families.

This observation is reinforced by very similar results seen for the second
family (Fig. 16 right column). There we see a more gradual closing of the
bouncing windows, but they too disappear in spite of the fact that the massive
mode is present for all $\varepsilon<1$. Again, the threshold below which no
bouncing windows are observed, i.e. $\varepsilon\approx 0.4$ is quite close to
the point of merging of Derrick’s mode into the continuum $\varepsilon\approx
0.37$ (see Fig. 11).

In Fig. 13, we display the range of velocities at which two-bounce windows are
found for both families of potentials. There we see that – perhaps contrary to
intuition – the windows close starting with the largest ones from left to
right. This means, for example, that there exist ranges of $\varepsilon$ for
which the two-bounce windows only contain no less than $n$ internal
oscillations of the field, with $n=2,3,\ldots$.

In Fig. 14, we display a similar picture with added three-bounce and four-
bounce windows. The $\varepsilon$ dependence of these windows closely follows
the trend set by the two-bounce windows.

Figure 15: Dependence of critical velocities on $\varepsilon$ for families
$V_{\varepsilon}^{(1,2)}$. For $V_{\varepsilon}^{(1)}$, the critical velocity
crosses the speed of light around $\varepsilon\approx 0.215$.

Let us also comment on the critical velocity, whose dependence on
$\varepsilon$ we display in Fig. 15. There we see a nearly linear dependence
on $\varepsilon$ that is very similar to the dependence of Derrick’s frequency
on Fig. 11 (bottom).

##  IV Summary and outlook

In this paper, we have introduced the notions of a core, tails, and skin of a
kink – approximate regions of a generic two-vacuum potential, where the
solutions to the leading-order Taylor-expanded equation of motion around the
central maximum, minima, and inflection points, respectively, holds. Further,
we have introduced a class of potentials, that we call Frankensteinian, as
ideal laboratories for exploring the significance of these structural pieces
of a kink on the dynamics. In particular, in this paper, we were concerned
with the presence or absence of bouncing in $K\bar{K}$ scattering.

In Sec. II, we have shown three concrete examples of coreless potentials or
increasing complexity and we demonstrated a lack of bouncing in all of them.
We pointed out that the absence of both core and skin regions results in
trivial $K\bar{K}$ scattering outcomes: namely complete annihilation of the
pair. Only when the potential had skin regions, do we see oscillons and the
appearance of critical velocity.

In Sec. III, we explored the impact of non-analyticity at the maximum for the
bouncing phenomenon by considering two parametric families of potentials that
are increasingly more singular at $\phi=0$ as $\varepsilon\to 0$.

The first family, $V_{\varepsilon}^{(1)}$, approaches the harmonic double-well
TT potential (7) as $\varepsilon\to 0$ and, therefore, departs the most from
the canonical $\phi^{4}$ model. Indeed, as $\varepsilon\to 0$, the scattering
windows rapidly close, the critical velocity increases almost linearly and all
resonant features disappear well before $\varepsilon=0$.

Perhaps surprisingly, the second family which was designed to stay quite close
to the $\phi^{4}$ model for the entire range of $\varepsilon$, showed very
similar results. The most visible difference compared with the first family is
that the critical velocity remains below one and, as Fig. 16 indicates, some
oscillons are formed as the outcome of the collision even for very small
$\varepsilon$. In that regard, the $\varepsilon\to 0$ limit is qualitatively
similar to the $V_{\rm KG+log}$ potential discussed in Sec. II.

Perhaps most surprisingly, in both families, we have seen that the instance of
$\varepsilon$ when Derrick’s frequency merges with the continuum (see Fig. 11)
is a far superior predictor of when the bouncing windows disappear compared
with the massive mode. This is certainly unexpected, but let us speculate on
why this could be the case.

As $\varepsilon\to 0$, the range of initial velocities containing bouncing
windows gets quickly shifted towards higher, relativistic values. It is known
that Derrick’s mode plays a key role in relativistic dynamics, hence it is
also important here. To test this hypothesis, it would be of interest to find
a family of models for which critical velocity remains small for the entire
range of the deformation parameter.

Our results suggest that the core region of the potential plays a central (pun
intended) role in the bouncing phenomenon in $K\bar{K}$ scattering. Of course,
it is hard to quantify this statement, and, at this point, we do not have
sufficient knowledge to turn the observations of this paper into a predictive
heuristic. Nevertheless, we feel that the presented results give credence to
the notion that core, tails, and skins, as defined above, are of some
importance to the dynamical phenomena and that they merit further
investigation.

In particular, we plan to present a detailed investigation of lifetimes and
other characteristics of oscillons in a separate publication. We also plan to
investigate further examples of Frankensteinian potentials and, in particular,
determine the role of the skin on the dynamics of kinks.

Figure 16: Gradual disappearance of resonant features in $K\bar{K}$ collisions
for $V_{\varepsilon}^{(1)}$ and $V_{\varepsilon}^{(2)}$ as $\varepsilon$ is
lowered from $\phi^{4}$ model ($\varepsilon=1.0$).

###### Acknowledgements.

We would like to thank T. Romańczukiewicz, K. Sławińska, and A. Wereszczyński
for many fruitful discussions and hospitality at Jagiellonian University in
Krakow. This work benefited greatly from the feedback provided by the
participants of the SIG XII workshop held at Jagiellonian University in June
2024. We also want to say a huge thanks to Martin Urbanec who helped us
immensely with running the numerical code. We also acknowledge the
institutional support of the Research Centre for Theoretical Physics and
Astrophysics, Institute of Physics, Silesian University in Opava. This work
has been supported by the grant no. SGS/24/2024 Astrophysical processes in
strong gravitational and electromagnetic fields of compact object.

## Appendix: Comments on the numerical method

All numerical calculations were performed in the programming language Julia
1.10.3 using the library DifferentialEquations.jl. We evolved the second-order
equations of motion via the Bogacki-Shampine 5/4 Runge-Kutta method (BS5()) in
the time domain, while the spatial domain was discretized (typically into
$\sim$6000 segments), approximating the second derivative using the central
finite difference of the third-order.

We also exploited the reflection symmetry through the center of collision at
$x=0$, so that the calculations were performed only in the half interval
$x\in[0,x_{end}]$. For the last 10% of the length of the interval, we
introduced an absorbing layer in which an additional damping term is included
in the equation of motion,i.e.

$\partial_{t}^{2}\phi=\partial_{x}^{2}\phi-V^{\prime}(\phi)-\alpha_{D}\partial_{t}\phi\,,$

where the damping value $\alpha_{D}$ was chosen in the range of values
$\alpha_{D}\in[0.2,0.4]$ depending on the complexity of the numerical
calculation. This approach allows us to use a smaller spatial interval
$x\in[0,25]$ without fear of reflections from the end of the interval.

## References

  * (1) N. S. Manton and P. Sutcliffe, “Topological solitons,” Cambridge University Press, 2004, ISBN 978-0-521-04096-9, 978-0-521-83836-8, 978-0-511-20783-9 doi:10.1017/CBO9780511617034
  * (2) A. Vilenkin and E. P. S. Shellard, “Cosmic Strings and Other Topological Defects,” Cambridge University Press, 2000, ISBN 978-0-521-65476-0
  * (3) Y. M. Shnir, “Topological and Non-Topological Solitons in Scalar Field Theories,” Cambridge University Press, 2018, ISBN 978-1-108-63625-4
  * (4) T. Sugiyama, “Kink-antikink collisions in the two-dimensional $\phi^{4}$ model”, Prog. Theor. Phys. 61 (1979), 1550-1563 doi:10.1143/PTP.61.1550
  * (5) D. K. Campbell, J. F. Schonfeld and C. A. Wingate, “Resonance Structure in Kink - Antikink Interactions in $\phi^{4}$ Theory,” Physica D 9 (1983), 1 FERMILAB-PUB-82-051-THY.
  * (6) M. Moshir, “Soliton - Anti-soliton Scattering and Capture in $\lambda\phi^{4}$ Theory,” Nucl. Phys. B 185 (1981), 318-332 doi:10.1016/0550-3213(81)90320-5
  * (7) T. I. Belova and A. E. Kudryavtsev, “QUASIPERIODICAL ORBITS IN THE SCALAR CLASSICAL lambda phi**4 FIELD THEORY,” Physica D 32 (1988), 18 ITEP-94-1985.
  * (8) P. Anninos, S. Oliveira and R. A. Matzner, “Fractal structure in the scalar lambda (phi**2-1)**2 theory,” Phys. Rev. D 44 (1991), 1147-1160 doi:10.1103/PhysRevD.44.1147
  * (9) N. S. Manton, K. Oleś, T. Romańczukiewicz and A. Wereszczyński, “Kink moduli spaces: Collective coordinates reconsidered,” Phys. Rev. D 103 (2021) no.2, 025024 doi:10.1103/PhysRevD.103.025024 [arXiv:2008.01026 [hep-th]].
  * (10) N. S. Manton, K. Oles, T. Romanczukiewicz and A. Wereszczynski, “Collective Coordinate Model of Kink-Antikink Collisions in $\phi^{4}$ Theory,” Phys. Rev. Lett. 127 (2021) no.7, 071601 doi:10.1103/PhysRevLett.127.071601 [arXiv:2106.05153 [hep-th]].
  * (11) C. Adam, N. S. Manton, K. Oles, T. Romanczukiewicz and A. Wereszczynski, “Relativistic Moduli Space for Kink Collisions,” [arXiv:2111.06790 [hep-th]].
  * (12) P. Dorey, K. Mersh, T. Romanczukiewicz and Y. Shnir, “Kink-antikink collisions in the $\phi^{6}$ model,” Phys. Rev. Lett. 107 (2011), 091602 doi:10.1103/PhysRevLett.107.091602 [arXiv:1101.5951 [hep-th]].
  * (13) C. Adam, P. Dorey, A. Garcia Martin-Caro, M. Huidobro, K. Oles, T. Romanczukiewicz, Y. Shnir and A. Wereszczynski, “Multikink scattering in the $\phi^{6}$ model revisited,” Phys. Rev. D 106 (2022) no.12, 125003 doi:10.1103/PhysRevD.106.125003 [arXiv:2209.08849 [hep-th]].
  * (14) H. Weigel, “Kink-Antikink Scattering in $\varphi^{4}$ and $\phi^{6}$ Models,” J. Phys. Conf. Ser. 482 (2014), 012045 doi:10.1088/1742-6596/482/1/012045 [arXiv:1309.6607 [nlin.PS]].
  * (15) P. Dorey, A. Gorina, T. Romańczukiewicz and Y. Shnir, “Collisions of weakly-bound kinks in the Christ-Lee model,” [arXiv:2304.11710 [hep-th]].
  * (16) C. Adam, C. Halcrow, K. Oles, T. Romanczukiewicz and A. Wereszczynski, “Moduli space for kink collisions with moving center of mass,” [arXiv:2304.07895 [hep-th]].
  * (17) C. Adam, D. Ciurla, K. Oles, T. Romanczukiewicz and A. Wereszczynski, “Relativistic Moduli Space and critical velocity in kink collisions,” [arXiv:2304.14076 [hep-th]].
  * (18) P. Dorey and T. Romańczukiewicz, “Resonant kink-antikink scattering through quasinormal modes,” Phys. Lett. B 779 (2018), 117-123 doi:10.1016/j.physletb.2018.02.003 [arXiv:1712.10235 [hep-th]].
  * (19) C. Adam, K. Oles, T. Romanczukiewicz and A. Wereszczynski, “Spectral Walls in Soliton Collisions,” Phys. Rev. Lett. 122 (2019) no.24, 241601 doi:10.1103/PhysRevLett.122.241601 [arXiv:1903.12100 [hep-th]].
  * (20) C. Adam, K. Oles, T. Romanczukiewicz and A. Wereszczynski, “Spectral walls in antikink-kink scattering in the $\phi$6 model,” Phys. Rev. D 106 (2022) no.10, 105027 doi:10.1103/PhysRevD.106.105027 [arXiv:2209.11479 [hep-th]].
  * (21) J. Evslin, C. Halcrow, T. Romanczukiewicz and A. Wereszczynski, “Spectral walls at one loop,” Phys. Rev. D 105 (2022) no.12, 125002 doi:10.1103/PhysRevD.105.125002 [arXiv:2202.08249 [hep-th]].
  * (22) J. Evslin, “Asymptotic states for kink–meson scattering,” Eur. Phys. J. C 83 (2023) no.8, 743 doi:10.1140/epjc/s10052-023-11919-8 [arXiv:2304.01564 [hep-th]].
  * (23) J. Evslin and H. Liu, “Elastic Kink-Meson scattering,” JHEP 04 (2024), 072 doi:10.1007/JHEP04(2024)072 [arXiv:2311.14369 [hep-th]].
  * (24) J. Evslin and H. Liu, “Reflection coefficient of a reflectionless kink,” Phys. Rev. D 109 (2024) no.8, 085019 doi:10.1103/PhysRevD.109.085019 [arXiv:2402.17968 [hep-th]].
  * (25) Kevrekidis, P. G. & Goodman, R. H. “Four Decades of Kink Interactions in Nonlinear Klein-Gordon Models: A Crucial Typo, Recent Developments and the Challenges Ahead,” 2019, arXiv:1909.03128
  * (26) I. C. Christov, R. J. Decker, A. Demirkaya, V. A. Gani, P. G. Kevrekidis and R. V. Radomskiy, “Long-range interactions of kinks,” Phys. Rev. D 99 (2019) no.1, 016010 doi:10.1103/PhysRevD.99.016010 [arXiv:1810.03590 [hep-th]].
  * (27) N. S. Manton, “Forces between Kinks and Antikinks with Long-range Tails,” J. Phys. A 52 (2019) no.6, 065401 doi:10.1088/1751-8121/aaf9d1 [arXiv:1810.03557 [hep-th]].
  * (28) V. A. Gani, A. M. Marjaneh and K. Javidan, “Exotic final states in the $\varphi^{8}$ multi-kink collisions,” Eur. Phys. J. C 81 (2021) no.12, 1124 doi:10.1140/epjc/s10052-021-09935-7 [arXiv:2106.06399 [hep-th]].
  * (29) V. A. Gani, V. Lensky and M. A. Lizunova, “Kink excitation spectra in the (1+1)-dimensional $\varphi^{8}$ model,” JHEP 08 (2015), 147 doi:10.1007/JHEP08(2015)147 [arXiv:1506.02313 [hep-th]].
  * (30) I. C. Christov, R. J. Decker, A. Demirkaya, V. A. Gani, P. G. Kevrekidis, A. Khare and A. Saxena, “Kink-kink and kink-antikink interactions with long-range tails,” Phys. Rev. Lett. 122 (2019) no.17, 171601 doi:10.1103/PhysRevLett.122.171601 [arXiv:1811.07872 [hep-th]].
  * (31) M. Gleiser and D. Sicilia, “Analytical Characterization of Oscillon Energy and Lifetime,” Phys. Rev. Lett. 101 (2008), 011602 doi:10.1103/PhysRevLett.101.011602 [arXiv:0804.0791 [hep-th]].
  * (32) P. Rosenau and J. M. Hyman, “Compactons: Solitons with finite wavelength,” Phys. Rev. Lett. 70 (1993), 564-567 doi:10.1103/PhysRevLett.70.564
  * (33) R. Garralon-LĂłpez, F. Rus and F. R. Villatoro, “Robustness of the absolute Rosenauâ€“Hyman $|K|(p,p)$ equation with non-integer $p$,” Chaos, Solitons & Fractals, Volume 169, 2023, 113216, ISSN 0960-0779, https://doi.org/10.1016/j.chaos.2023.113216.
  * (34) H. Arodz, “Topological compactons,” Acta Phys. Polon. B 33 (2002), 1241-1252 [arXiv:nlin/0201001 [nlin.PS]].
  * (35) H. Arodz, P. Klimas and T. Tyranowski, “Field-theoretic models with V-shaped potentials,” Acta Phys. Polon. B 36 (2005), 3861-3876 [arXiv:hep-th/0510204 [hep-th]].
  * (36) H. Arodz, P. Klimas and T. Tyranowski, “Signum-Gordon wave equation and its self-similar solutions,” Acta Phys. Polon. B 38 (2007), 3099-3118 [arXiv:hep-th/0701148 [hep-th]].
  * (37) H. Arodz, P. Klimas and T. Tyranowski, “Compact oscillons in the signum-Gordon model,” Phys. Rev. D 77 (2008), 047701 doi:10.1103/PhysRevD.77.047701 [arXiv:0710.2244 [hep-th]].
  * (38) F. M. Hahne, P. Klimas, J. S. Streibel and W. J. Zakrzewski, “Scattering of compact oscillons,” JHEP 01 (2020), 006 doi:10.1007/JHEP01(2020)006 [arXiv:1909.01992 [hep-th]].
  * (39) H. Arodz, “Symmetry breaking transition and appearance of compactons in a mechanical system,” Acta Phys. Polon. B 35 (2004), 625-638 [arXiv:hep-th/0312036 [hep-th]].
  * (40) Arodz, H. and Klimas, P., “Chain of Impacting Pendulums as Non-Analytically Perturbed Sine-Gordon System”, Acta Physica Polonica B, vol. 36, no. 3, p. 787, 2005. doi:10.48550/arXiv.cond-mat/0501112.
  * (41) F. M. Hahne and P. Klimas, “Compact kink and its interaction with compact oscillons,” JHEP 09 (2022), 100 doi:10.1007/JHEP09(2022)100 [arXiv:2207.07064 [hep-th]].
  * (42) F. M. Hahne and P. Klimas, “Scattering of compact kinks,” JHEP 01 (2024), 067 doi:10.1007/JHEP01(2024)067 [arXiv:2311.09494 [hep-th]].
  * (43) D. Bazeia, L. Losano, M. A. Marques and R. Menezes, “From Kinks to Compactons,” Phys. Lett. B 736 (2014), 515-521 doi:10.1016/j.physletb.2014.08.015 [arXiv:1407.3478 [hep-th]].
  * (44) D. Bazeia, T. S. Mendonça, R. Menezes and H. P. de Oliveira, “Scattering of compactlike structures,” Eur. Phys. J. C 79 (2019) no.12, 1000 doi:10.1140/epjc/s10052-019-7519-4 [arXiv:1910.05458 [hep-th]].
  * (45) Campos, J.G.F., Mohammadi, “A. Quasinormal modes in kink excitations and kinkâ€“antikink interactions: a toy model,” Eur. Phys. J. C 80, 352 (2020). https://doi.org/10.1140/epjc/s10052-020-7856-3
  * (46) N. S. Manton, “A Remark on the Scattering of BPS Monopoles,” Phys. Lett. B 110 (1982), 54-56 doi:10.1016/0370-2693(82)90950-9
  * (47) E. B. Bogomolny, “Stability of Classical Solutions,” Sov. J. Nucl. Phys. 24 (1976), 449 PRINT-76-0543 (LANDAU-INST.).