# Multikink scattering in the $\phi^{6}$ model revisited

C. Adam<EMAIL_ADDRESS>Departamento de Física de Partículas, Universidad
de Santiago de Compostela and
Instituto Galego de Física de Altas Enerxias (IGFAE), E-15782 Santiago de
Compostela, Spain  P. Dorey<EMAIL_ADDRESS>Department of Mathematical
Sciences, Durham University, UK  A. García Martín-Caro
<EMAIL_ADDRESS>Departamento de Física de Partículas, Universidad de
Santiago de Compostela and
Instituto Galego de Física de Altas Enerxias (IGFAE), E-15782 Santiago de
Compostela, Spain  M. Huidobro<EMAIL_ADDRESS>Departamento de
Física de Partículas, Universidad de Santiago de Compostela and
Instituto Galego de Física de Altas Enerxias (IGFAE), E-15782 Santiago de
Compostela, Spain  K. Oles<EMAIL_ADDRESS>Institute of
Theoretical Physics, Jagiellonian University, Lojasiewicza 11, Kraków, Poland
T. Romanczukiewicz<EMAIL_ADDRESS>Institute of Theoretical
Physics, Jagiellonian University, Lojasiewicza 11, Kraków, Poland  Y. Shnir
<EMAIL_ADDRESS>Institute of Physics, University of Oldenburg, Oldenburg
D-26111, Germany  A. Wereszczynski<EMAIL_ADDRESS>Institute
of Theoretical Physics, Jagiellonian University, Lojasiewicza 11, Kraków,
Poland

###### Abstract

Antikink-kink ($\bar{\rm K}$${\rm K}$) collisions in the $\phi^{6}$ model
exhibit resonant scattering although the $\phi^{6}$ kinks do not support any
bound states to which energy could be transferred. In Phys. Rev. Lett. 107
(2011) 091602 it was conjectured that, instead, energy is transferred to a
collective bound mode of the full $\bar{\rm K}$${\rm K}$ configuration. Here
we present further strong evidence for this conjecture.

Further, we construct a collective coordinate model (CCM) for $\bar{\rm
K}$${\rm K}$ scattering based on this collective bound mode trapped between
the $\bar{\rm K}$${\rm K}$ pair which allows us to reproduce the full dynamics
of $\bar{\rm K}$${\rm K}$ scattering with striking accuracy. We also study
kink-antikink (${\rm K}$$\bar{\rm K}$) scattering and its description by a
CCM. In this case a significant role of radiation is discovered.

##  I Introduction

Topological solitons are spatially localized, stable solutions of nonlinear
field equations which carry a nonzero topological charge [1, 2, 3]. The
quantitative and sometimes even the qualitative understanding of their
interactions is far from satisfactory in many cases. Since solitons exist both
in fundamental theories (e.g., sphalerons or monopoles in the electroweak
theory) as well as in numerous effective models, a comprehensive understanding
of their interactions is vital not only for a deeper insight into the
mathematical structure of the theories but also for applications.

There are three main contributions to solitonic interactions.

Firstly, two solitons at a finite distance can act on each other with a static
force. This force can be attractive (as typically happens for a kink-antikink
pair) or repulsive. In some particular cases, the so-called Bogomonly-Prasad-
Sommerfeld (BPS) models [4], there is no static force between the constituent
solitons in a multi-soliton state [5], [6]. There are famous examples of this
in higher dimensions such as the Abelian Higgs model at critical coupling or
BPS monopoles [7, 8, 9], but they also exist in impurity-deformed [10, 11] or
multi-field models in (1+1) dimensions [12, 13, 14, 15].

Secondly, the dynamics may be significantly affected by the excitation of
internal degrees of freedom (DoF). These are often massive normal or
quasinormal modes supported by solitons, found in linear perturbation theory.
Later on we will see, however, that other possibilities also play a very
important role. A well-known example for the impact of internal DoF on soliton
dynamics is the so-called resonance phenomenon which is responsible for the
fractal structure observed in the final state production in kink-antikink
($\bar{\rm K}$ ${\rm K}$) collisions in various (1+1) dimensional solitonic
models [16, 17], [18]. Here, during the collision, the initial kinetic energy
of the incoming kink and antikink is transferred to internal DoF. Then, it can
be transferred back to the kinetic DoF, allowing for the re-appearance of the
solitons in the final state. However, it is also possible that a significant
fraction can be kept in the internal DoF, not allowing the solitons to escape,
which eventually leads to their complete annihilation. Both scenarios occur in
a chaotic manner, resulting in the well-known fractal pattern. This mechanism
has been recently confirmed in ${\rm K}$ $\bar{\rm K}$ scattering in the
$\phi^{4}$ theory by a derivation of a collective coordinate model (CCM) based
on the zero mode (kinetic DoF) and a massive normal mode, the so-called shape
mode [19]. In fact, the resulting CCM reproduces the fractal structure with
quite good agreement. This has been further improved by considering, instead,
a tower of Derrick modes, that is, scaling deformations. This framework, the
so-called perturbative relativistic CCM not only allows to incorporate
relativistic corrections in a systematic, well defined perturbative fashion,
but it also provides an arbitrary number of collective coordinates, which
results in a more accurate description of solitonic processes. E.g., the
inclusion of the two lowest Derrick modes already led to a CCM which describes
the ${\rm K}$ $\bar{\rm K}$ process in the $\phi^{4}$ model with a few percent
precision [20].

Finally, in multi-soliton processes the constituent solitons interact with
radiation which is easily produced during the scattering. Radiation is an
excitation of the continuous spectrum (scattering modes) which is not confined
to the solitons but propagates in the full space. The interaction between
solitons and radiation is a very important but, at the same time, a very
complicated topic where the famous soliton resolution conjecture still waits
to be proven [21, 22]. Due to nonlinearities, radiation may have a rather
surprising effect on soliton motion like, e.g., the negative radiation
pressure, e.g., [23]. Note that, in contrast to internal DoF, radiation
provides a channel in which the energy can escape from the solitons.

All these three types of interactions are obviously coupled to each other,
rendering the rigorous analysis extremely difficult even in the simplest case
of ${\rm K}$ $\bar{\rm K}$ collisions in (1+1) dimensions. Note that the use
of Derrick modes may probably take into account both the massive normal modes
as well as the continuous spectrum, at least partially. This is because higher
Derrick modes have frequencies above the mass threshold [20]. On the other
hand, Derrick modes, of course, do not represent proper radiation, since they
are confined to solitons, although higher Derrick modes are more and more
widely spread out.

Although the recent progress in the explanation of the ${\rm K}$ $\bar{\rm K}$
collisions in the $\phi^{4}$ model [19, 20] is a significant step forward in
the understanding of the dynamics of solitons, it is just a beginning. There
are many kink models whose dynamics are still understood to a rather
unsatisfactory extent, see, e.g., deformations of the $\phi^{6}$ model [24,
25, 26], higher order potentials [28, 29, 30, 31], double sine-Gordon [32, 33,
34, 35] and other models [36, 37, 38, 39, 40, 41, 42].

In the present paper we revisit the antikink-kink ($\bar{\rm K}$${\rm K}$) and
kink-antikink (${\rm K}$$\bar{\rm K}$) collisions in the $\phi^{6}$ model [43,
44, 45, 46]. This is the simplest theory which goes beyond $\phi^{4}$ model
and which, for the $\bar{\rm K}$ ${\rm K}$ case, exhibits a fractal structure
in the final state formation despite the nonexistence of shape modes hosted by
a single (anti)kink. However, as originally proposed in [43], the resonant
phenomenon can be triggered by dynamically created modes trapped between the
colliding solitons. The fact that solitonic dynamics can be significantly
affected by temporal variations of the mode structure has been recently
appreciated, see, for instance, the spectral wall phenomenon [47] or the role
played by unstable solutions, i.e., sphalerons, occurring during a multi-kink
evolution [48].

On the other hand, there is no corresponding fractal structure in the ${\rm
K}$ $\bar{\rm K}$ case as no trapped modes exist.

Here, we provide a detailed exploration of $\bar{\rm K}$${\rm K}$ and ${\rm
K}$$\bar{\rm K}$ collisions based on full numerical simulations. Further, we
shall propose strategies for the construction of CCMs for kink collisions. In
particular, we will present a CCM which reproduces the full numerical
$\bar{\rm K}$ ${\rm K}$ collisions with a high precision. We will also
discover some interesting difficulties concerning the construction of a CCM
for the ${\rm K}$ $\bar{\rm K}$ collisions. Here the inclusion of radiation
effects seems to be inevitable.

##  II Road map to restricted set of configurations

Let us consider a scalar field theory in (1+1) dimensions

$L[\phi]=\int_{-\infty}^{\infty}dx\left(\frac{1}{2}\phi_{t}^{2}-\frac{1}{2}\phi_{x}^{2}-U(\phi)\right)\,,$
(1)

where the field theoretical potential $U$ has at least two vacua,
$\phi_{v}^{(1)}>\phi_{v}^{(2)}$. This guarantees the existence of a
topological (anti)kink $\Phi_{K(\bar{K})}$ which interpolates between the
vacua. In principle, we do not have to assume that the vacua are approached
quadratically. This means that the mass of small perturbations around a
vacuum, i.e., the meson mass, can take any value $0\leq m_{1,2}\leq\infty$. A
finite, non-zero mass means a quadratic approach, as for example in the
$\phi^{4}$ and $\phi^{6}$ models

$U_{\phi^{4}}=\frac{1}{2}(1-\phi^{2})^{2},\;\;\;U_{\phi^{6}}=\frac{1}{2}\phi^{2}(1-\phi^{2})^{2}.$
(2)

On the other hand, the $\phi^{8}$ potential

$U_{\phi^{8}}=\frac{1}{2}\phi^{4}(1-\phi^{2})^{2}$ (3)

has a vacuum at $\phi_{v}=0$ which is approached with a higher than quadratic
power. Therefore, the mass of mesons vanishes and there is no mass gap in this
vacuum. On the other hand a formally infinite mass of vacuum excitations is
characteristic for compactons, which are solitons with finite support [49].

The collective coordinate model (CCM) approach is a standard approach which
allows for a semi-analytical treatment of the dynamics of topological solitons
and, therefore, offers a tool for the explanation of a variety of phenomena
occurring in multi-soliton collisions, see [2] for a review. In this
framework, the infinitely many DoF of the original field $\phi$ whose dynamics
is governed by the Lagrangian $L[\phi]$ are replaced by a finite number of DoF
arising from the truncation of the space of fields to a restricted set of
static configurations

$\mathcal{M}=\\{\Phi(x;\,X^{i}),\,i=1..N\\}\,.$ (4)

$\mathcal{M}$ is called the moduli space and it should contain configurations
which capture the main features of a given solitonic process. Its dimension,
$N$, depends of the number of parameters, i.e., moduli, $X^{i}$, which are
then promoted to time dependent variables $X^{i}(t)$. The configurations
$\Phi$ are inserted into the original Lagrangian ${L}[\phi]$ and after
performing spatial integration we obtain an effective CCM theory

$L({\bf X})=\int_{-\infty}^{\infty}\mathcal{L}[\Phi(x;{\bf
X})]dx=\frac{1}{2}g_{ij}({\bf X})\dot{X}^{i}\dot{X}^{j}-V({\bf X})\,,$ (5)

where

$g_{ij}({\bf X})=\int_{-\infty}^{\infty}\frac{\partial\Phi}{\partial
X^{i}}\frac{\partial\Phi}{\partial X^{j}}\,dx$ (6)

is the metric on $\mathcal{M}$ while

$V({\bf
X})=\int_{-\infty}^{\infty}\left(\frac{1}{2}\left(\frac{\partial\Phi}{\partial
x}\right)^{2}+U(\Phi)\right)\,dx$ (7)

is the positive-definite potential.

The essential ingredient is, therefore, the correct choice of the restricted
set of configurations. Unfortunately, except for the BPS models, there is no
canonical way to construct this set. An obvious criterion is the following:

  * (i)

The restricted configurations should reproduce, as much as possible, the
actual field profiles occurring during a given process.

We begin with the simplest single kink sector. As a kink is a solution of a
Poincaré invariant theory, it enjoys translational symmetry and, therefore,
can be located at any spatial point $a$. This leads to a whole family of
energetically equivalent kinks $\Phi_{K}(x;a)=\Phi_{K}(x-a)$. This is also
related to the fact that a kink is a static solution of a first order
equation, the so-called Bogomolny equation,

$\phi_{x}=\pm\sqrt{2U},$ (8)

which implies the appearance of one integration constant, the modulus $a$. Of
course, the transition between energetically equivalent solutions costs an
arbitrarily small amount of energy. Therefore, the modulus $a$ can be
identified with the presence of a zero mode and its change describes a kinetic
DoF, i.e., the rigid motion of the kink.

In the next step, one may also include massive excitations. They can be
vibrational modes obtained in the linear perturbation analysis (normal modes
or quasinormal modes). Here, we perturb the kink solution by a small
deformation, $\phi(x,t)=\Phi_{K}(x;a)+\eta^{K}(x,t;a)$. If inserted into the
equation of motion, this leads to the linear Schroedinger-type equation

$\left.\left(\frac{d^{2}}{dx^{2}}-\frac{d^{2}U}{d\phi^{2}}\right|_{\Phi_{K}(x;a)}\right)\eta^{K}(x;a)=-\omega^{2}\eta^{K}(x;a),$
(9)

where we assumed periodic time dependence of the perturbation,
$\eta^{K}(x,t;a)=\eta^{K}(x;a)e^{i\omega t}.$ Normal modes require
$\omega\in\mathbb{R}$ while for quasi-normal modes $\omega$ has a nontrivial
imaginary part.

However, this is not the only possible choice. For example, it has recently
been proposed to use Derrick modes arising in a scaling perturbation [20].
This not only introduces an arbitrary number of modes (hence, collective
coordinates) but also recovers, in a perturbative fashion, the Lorentz
contraction of solitons. In addition, higher Derrick modes have frequencies
above the mass threshold (the mass of the meson squared), which may take into
account some features of radiation.

Both choices simply represent internal DoF of a kink. As a result, we arrive
at the following restricted set of configurations

$\mathcal{M}_{K}=\\{\Phi_{K}(x;a)+\sum_{i=1}^{N}X^{i}\eta^{K}_{i}(x,a)\\}\,,$
(10)

where for concreteness we choose the normal modes $\eta^{K}_{i}(x;a)$ hosted
by a kink. $X^{i}$ are the amplitudes of the modes and serve as new moduli.
Note that due to the Poincare invariance $\eta^{K}_{i}(x,a)=\eta^{K}_{i}(x-a)$
and the internal DoF are confined to the kink. The construction for the
antikink is identical.

Now, we can move to a multi soliton processes like antikink-kink or kink-
antikink collisions. The starting point is the simplest, one-parameter
restricted set of configurations, $\\{\Phi(x;a)\\}$, which should provide the
crudest description of the considered process. This one-dimensional moduli
space is not meant to give a good approximation but rather a correct arena in
which the multi-kink scattering happens. This means the following.

  * (ii)

The one-parameter set $\\{\Phi(x;a)\\}$ should include the initial and final
states occurring in the process.

In other words, $\lim_{a\to a_{\pm}}\Phi(x;a)=\Phi_{in(out)}$, where $a_{\pm}$
are some of the values of the modulus $a$. It is important to fulfill this
condition, which is more restrictive than the condition of the correct
topology. In many cases, it is enough to take a simple sum of the scattered
(initial) solitons (modulo an additive constant setting the proper value of
the vacuum). This happens, e.g., for ${\rm K}$$\bar{\rm K}$ collisions in the
$\phi^{4}$ model, where $\Phi(x;a)=\Phi_{K}(x+a)+\Phi_{\bar{K}}(x-a)$. This
will also be a valid construction for antikink-kink collisions in the
$\phi^{6}$ model. Nonetheless, we will present an example where such a naive
sum leads to configurations which do not interpolate between the initial and
final states, see the kink-antikink scattering in $\phi^{6}$ theory.

Let us assume that we have found the correct one-dimensional moduli space
obeying condition (ii) above. For the sake of simplicity we also assume that
the simple sum of the initial states is a good choice. Now we want to include
further DoF. The most straightforward choice for the restricted set of
configurations is simply the naive sum of single soliton sets. For example,
for an antikink-kink collision this results in the following moduli space

$\displaystyle\mathcal{M}_{\bar{K}K}=\mathcal{M}_{\bar{K}}\cup\mathcal{M}_{K}=$
(11)
$\displaystyle\\{\Phi_{\bar{K}}(x;-a)+\Phi_{K}(x;a)+\sum_{i=1}^{N}X^{i}\left(\eta^{\bar{K}}_{i}(x,-a)+\eta^{K}_{i}(x,a)\right)\\},$

where we restrict our consideration to a symmetric scattering. Here the
antikink and kink are located at $-a$ and $+a$, respectively. In general, this
choice may lead to the appearance of apparent singularities on the moduli
space. Usually, they can be removed by an appropriate redefinition (or
extension) of the collective coordinates [50]. Another important observation
is that, by construction, all internal DoF are confined to the constituent
solitons.

Figure 1: Schematic plot of the single kink eigen-problem potential
$U_{\phi\phi}\left|\Phi_{K}\right.$. Left: asymmetric case, e.g., in
$\phi^{6}$ model. Right: symmetric case, e.g., in $\phi^{4}$ model.

Figure 2: Schematic plot of the two-kink eigen-problem potential for the
asymmetric solitons. Left: KAK case, $U_{\phi\phi}\left|\Phi_{KAK}\right.$.
Right: AKK case, $U_{\phi\phi}\left|\Phi_{AKK}\right.$.

Although this framework very successfully reproduced, e.g., kink-antikink
collisions in the $\phi^{4}$ model, especially in the case where Derrick modes
are used, in general it may miss an important detail, namely the possible
existence of modes which are not confined to individual solitons.

To see this possibility, we consider again a kink and its linear perturbation.
In general, the effective potential in the eigen-problem can tend to two
different values at $x\to\pm\infty$. This corresponds to two different meson
masses in the two vacua, $m_{1}<m_{2}$. This is schematically plotted in Fig.
1, left panel. This happens, for example, in $\phi^{6}$ theory. The mass at
the vacuum $\phi=0$ is $m_{2}=1$ while at the vacuum $\phi=1$ it takes the
bigger value $m_{1}=2$. For the $\phi^{4}$ model the masses are identical and
we have a symmetric situation, see Fig. 1, right panel.

Now, if we consider a two-soliton configuration built as a naive sum of single
soliton states, we find two qualitatively distinct possibilities for well-
separated constituent solitons. The single soliton effective potentials can be
joined by a plateau with the bigger mass threshold $m_{2}$, see Fig. 2, left
panel. This corresponds to a kink-antikink state, e.g., in the $\phi^{6}$
model. Obviously, the resulting modes are simply a sum of modes of each
soliton. Furthermore, they are still confined to the solitons. However, if the
effective potentials are connected by a plateau with smaller mass $m_{1}$,
some new modes may show up, see Fig. 2, right panel. These modes are not
confined to the colliding solitons but are delocalized in the space between
them. These are new, two-soliton, trapped or delocalized modes and they play
an important role in a scattering process. We call them delocalized modes
because, in contrast to the usual normal modes, they are not bounded to
individual solitons but, instead, extend to the whole space between the
colliding kinks. Exactly this situation occurs in antikink-kink collisions in
the $\phi^{6}$ model where, in addition, single solitons do not host any
normal modes. Therefore, the observed fractal structure in the final state
formation was attributed to the resonant phenomenon between the kinetic DoF
and the two-soliton modes [43]. A similar mechanism was later considered in
the case of the $\phi^{8}$ model [27].

This leads to an improved proposal for the restricted set of configurations.

  * (iii)

The multi-parameter restricted set of configurations,
$\\{\Phi(x;a,X^{1},..,X^{N})\\}$, should include the internal DoF (internal
modes) arising in the perturbation theory of the corresponding multi-soliton
configuration $\Phi(x;a)$.

Concretely, for antikink-kink collisions this leads to the following moduli
space

$\displaystyle\tilde{\mathcal{M}}_{\bar{K}K}=\\{\Phi(x;a,X^{1},..,X^{N})\\}$
(12)
$\displaystyle\\{\Phi_{\bar{K}}(x;-a)+\Phi_{K}(x;a)+\sum_{i=1}^{N}X^{i}\eta^{\bar{K}K}_{i}(x,a)\\},$

where $\eta^{\bar{K}K}_{i}(x,a)$ contains both the localized and the
delocalized modes obtained for the effective potential

$\left.\frac{d^{2}U}{d\phi^{2}}\right|_{\Phi_{\bar{K}}(x;-a)+\Phi_{K}(x;a)}.$
(13)

Here we use our assumption that the one-dimensional moduli space is spanned by
a simple sum of the antikink and kink,
$\Phi(x;a)=\Phi_{\bar{K}}(x;-a)+\Phi_{K}(x;a)$. In general, also the localized
soliton modes should be added as in (11). Indeed, if there are no delocalized
modes, such a naive superposition works quite well. Therefore, this improved
set contains the set obtained by naive superposition,
$\tilde{\mathcal{M}}\supset\mathcal{M}$.

Several comments are in order. Firstly, while the localized soliton modes are
assumed to have the same form at any intersoliton distance $2a$, this is not
the case for the delocalized modes. On the contrary, by construction their
form as well as their number changes as the solitons approach each other. In
particular, at a certain distance between the solitons a delocalized mode hits
the mass threshold and becomes a non-normalizable mode. This means that it
ceases to be a valid DoF. Such a change of the structure of the modes may
potentially have a very nontrivial impact on the kink dynamics, which has to
be carefully taken into account in the restricted set of configurations. This
fact also is behind the spectral wall phenomenon [47, 52].

Secondly, it may be necessary to include radiation, which in particular cases
can strongly affect the dynamics. Interestingly, the delocalized modes can be
identified with localized radiation with a frequency below the higher mass
threshold. Indeed, for sufficiently separated solitons, an initial
perturbation localized at the origin will decay into radiation identically as
if being created in the vacuum with lower meson mass. The fraction of
radiation with $\omega<m_{1}$ will be trapped and eventually can excite
delocalized modes. The number of these modes goes to infinity and the energy
level separation decreases as the intersoliton distance grows, and effectively
we flow into the continuous spectrum. This is a rather remarkable mode-
radiation duality.

In the next section we will present further arguments that delocalized,
trapped modes are indeed responsible for the fractal structure in $\bar{\rm
K}{\rm K}$ collisions in the $\phi^{6}$ model, as originally proposed in [43].

Later on, we will consider ${\rm K}\bar{\rm K}$ collisions where no
delocalized modes exist. As there are also no single soliton massive normal
modes, the dynamics seems to be governed by radiation, which quite efficiently
transfers the energy from the kinetic DoF leading to a fast annihilation. This
will lead to a new challenge for a description within a CCM.

##  III $\bar{\rm K}{\rm K}$ collisions in the $\phi^{6}$ model revisited

The $\phi^{6}$ model is defined by the following Lagrangian

$L_{\phi^{6}}[\phi]=\int_{-\infty}^{\infty}\left(\frac{1}{2}\phi_{t}^{2}-\frac{1}{2}\phi_{x}^{2}-\frac{1}{2}\phi^{2}\left(1-\phi^{2}\right)^{2}\right)dx$
(14)

and has a static kink

$\Phi_{K}(x;a)\equiv\phi_{(0,1)}(x;a)=\sqrt{\frac{1+\tanh(x-a)}{2}}$ (15)

which interpolates between two vacua: $\Phi_{K}(x=-\infty)=0$ and
$\Phi_{K}(x=+\infty)=1$. The antikink joins the same vacua but in opposite
order and reads

$\Phi_{\bar{K}}(x;a)\equiv\phi_{(1,0)}(x;a)=\sqrt{\frac{1-\tanh(x-a)}{2}}$
(16)

In contrast to $\phi^{4}$ solitons, kink and antikink are not related by a
simple multiplication by $(-1)$. There are also mirror kinks $\Phi^{*}_{K}$
and antikinks $\Phi^{*}_{\bar{K}}$ interpolating between the vacua at $-1$ and
$0$. They are $\Phi^{*}_{K}(x;a)\equiv\phi_{(-1,0)}(x;a)=-\phi_{(0,1)}(-x;-a)$
and $\Phi^{*}_{\bar{K}}(x;a)\equiv\phi_{(0,-1)}(x;a)=-\phi_{(0,1)}(x;a)$.

The solitons host only a zero mode reflecting the existence of the free
parameter $a$. There are no massive normal modes. In addition, as we have
already mentioned, the mass of small perturbations around the $\phi_{v}=0$
vacuum is smaller than for the $\phi_{v}=\pm 1$ vacua. Namely, $m_{0}=1$ while
$m_{\pm 1}=2$. This leads to an asymmetric effective potential in the linear
perturbation problem.

Figure 3: Time dependence of the value of the field at the origin,
$\phi(x=0,t)$ for various initial velocities $v_{in}$ in the $\bar{\rm K}{\rm
K}$ collision in the $\phi^{6}$ model. Figure 4: Dependence of the trapped
(delocalized) 2-soliton even mode structure on $a$ obtained for the effective
potential 13 in $\phi^{6}$ model.

It is known that $\bar{\rm K}{\rm K}$ collisions have a fractal structure in
the final state formation [43]. Indeed, the incoming antikink and kink may be
back scattered, via a sequence of bounces, or may annihilate to the
$\phi_{v}=1$ vacuum, which occurs by the formation of a quasi-periodic kink-
antikink bound state, the bion, which decays to the vacuum by the emission of
radiation. The actual behaviour depends on the initial velocity of the
colliding solitons and reveals a fractal-like pattern of bounce windows and
bion chimneys, see Fig. 3. This is very similar to the famous ${\rm K}\bar{\rm
K}$ (and $\bar{\rm K}{\rm K}$) scattering in the $\phi^{4}$ model. There is,
however, one important difference. While solitons in $\phi^{4}$ theory have a
well defined soliton confined DoF, i.e. a massive normal mode, the so-called
shape mode or the very similar Derrick mode, the situation in the $\phi^{6}$
model is more subtle.

Figure 5: Upper left: an example of a two bounce $\bar{\rm K}{\rm K}$
collision in the $\phi^{6}$ model with initial velocity of the solitons
$v_{in}=0.04542$. Upper right: plot of several profiles for $t\in[422,427]$.
Lower left: an example of two bounce ${\rm K}\bar{\rm K}$ collision in
$\phi^{4}$ model with initial velocity of the solitons $v_{in}=0.2599$. Lower
right: plot of several profiles for $t\in[197,202]$.

Figure 6: Field profile with subtracted static configuration (17) for
$\bar{\rm K}{\rm K}$ solution in $\phi^{6}$ model with $v_{in}=0.04542$ (solid
line) and the delocalized mode fit (dashed line).

Figure 7: The lowest four delocalized, massive normal modes arising from (17)
with $a=6(?)$. Figure 8: Time dependence of the amplitudes of the first four
delocalized modes found in the decomposition of the $\bar{\rm K}{\rm K}$ field
profiles for $v_{in}=0.04542$.

As we have already noted, a single (anti)kink in $\phi^{6}$ theory does not
possess any massive bound modes which could enter into the resonant energy
transfer mechanism and explain the fractal structure. On the other hand, there
are soliton confined Derrick modes playing the role of internal DoF. In
addition, there are delocalized, two-soliton modes. It was observed in [43],
that the naive $\bar{\rm K}{\rm K}$ sum

$\Phi_{\bar{K}K}(x;a)=\Phi_{\bar{K}}(x;-a)+\Phi_{K}(x;a),$ (17)

gives rise to a two-soliton effective potential which hosts trapped or
delocalized modes $\eta_{i}^{\bar{K}K}(x;a)$. Their form, frequencies and even
number depends on the distance between the colliding solitons, see Fig. 4.
Indeed, there is a zero mode $\eta^{\bar{K}K}_{0}$ (which corresponds to a
simultaneous translation of the $\bar{\rm K}{\rm K}$ pair), an unstable mode
$\eta^{\bar{K}K}_{-1}$ (which reflects the fact that the naive superposition
is not a static solution of the model) and, finally, there are massive modes
$\eta^{\bar{K}K}_{i},i\geq 1$, whose number grows with the intersoliton
distance. These two-soliton, trapped modes were used to explain the resonant
phenomena in the $\bar{\rm K}{\rm K}$ collision in the $\phi^{6}$ model [43].

Here we present further convincing arguments that this is indeed the case -
the fractal structure in the final state formation is triggered by the
delocalized modes. In addition, we find evidence that the Derrick modes have
some importance.

Let us consider a $\bar{\rm K}{\rm K}$ pair colliding with initial velocity
$v_{in}=0.04542$. This is an example of a two bounce solution with a rather
long lasting bounce window, see Fig. 5, upper left panel. There are two
important distinct regimes: (i) the formation of a quasi-stationary state
where the positions (centers of mass) of the kink and antikink do not change,
and (ii) the transition through the vacuum, i.e., the annihilation moment.

As far as the quasi-stationary state is considered, the first observation is
that the change of the field profiles occurs mainly in the region between the
solitons. This is shown in Fig. 5, upper right panel, where we plot various
field profiles for $t\in[422,427]$. As said, the antikink and kink practically
do not change their positions, staying all the time at $a=a_{\rm qs}\approx\mp
6$, while the field between them fluctuates. This can be contrasted with a two
bounce ${\rm K}\bar{\rm K}$ solution in the $\phi^{4}$ model, see Fig. 5,
lower panels, where the field fluctuates mainly around the colliding solitons,
due to excitation of the shape (or Derrick) mode. Undoubtedly, in the
$\phi^{6}$ model the delocalized excitations seem to govern the dynamics.

This is particularly visible if we consider the field profiles where we
subtract the $\bar{\rm K}{\rm K}$ pair configuration, (17) with $a=a_{\rm
st}=6$ see Fig. 7, solid line. Here, the profiles are taken at $t=400,401,402$
and $403$. Indeed, the deformation extends over the whole intersoliton region
and concentrates in the center. This deformation is fully described by a
superposition of the first few two-soliton modes with positive frequency,
$\eta^{\bar{K}K}_{i}$ where $i\geq 1$. As the soliton separation remains
basically constant at $a=6$, we use this value to obtain the lowest four
normal modes $\eta^{\bar{K}K}_{i}$, $i=\\{1,2,3,4\\}$, see Fig. 7. We found
that the actual field profiles can be decomposed into these first four
positive two-soliton modes with a high precision, see, Fig. 7, dashed line.
For example, for $t=400$ the corresponding amplitudes are:
$A_{1}=-0.07519,A_{2}=0.00881,A_{3}=0.00292,A_{4}=-0.00083$. Note that the
amplitude of the first delocalized mode $\eta^{\bar{K}K}_{1}$ is approximately
an order of magnitude bigger than the amplitude of the next mode.

The same analysis can be repeated for any $t\in[400,425]$, which still
corresponds to the regime when the antikink and kink positions do not change.
The resulting time dependence of the amplitudes of the modes is shown in Fig.
8. Again, the amplitudes of the modes decreases quickly with the mode number.
After fitting the periodic functions $A_{i}^{0}\cos(\omega_{i}t+\delta)$, we
found that the fitted frequencies very well correspond to the frequencies of
the modes emerging from the small perturbation around the $\bar{\rm K}{\rm K}$
superposition (17) with $a=6$. The fitted values are, respectively,
$\omega_{1}=1.0377,\omega_{2}=1.2756,\omega_{3}=1.5999,\omega_{4}=1.9046$
while the frequencies derived in the linear problem are
$\omega_{1}=1.0376,\omega_{2}=1.2795,\omega_{3}=1.6083,\omega_{4}=1.9191$. The
agreement is spectacular.

This fully confirms that the two-soliton (delocalized) modes trigger the
resonant mechanism and, therefore, are responsible for the appearance of the
fractal structure. It also firmly shows that these two-soliton modes should be
included in any restricted set of configurations, which would be considered as
the starting point for a CCM.

Figure 9: A $\bar{\rm K}{\rm K}$ collision with $v_{in}=0.04542$ at a time
close to passing the vacuum. Comparison of field profiles (solid line), naive
superposition (17) (dotted line) and naive superposition with Lorentz factor
(dashed line).

Now we move to the second regime, that is the annihilation moment. Although
the velocities relevant for two and higher bounce windows as well as for bion
chimneys in the $\bar{\rm K}{\rm K}$ scattering in $\phi^{6}$ model are
definitely non-relativistic, with the critical velocity being approximately
$v_{c}\approx 0.048$, passing through the vacuum is a rather violent,
relativistic process. To see this, we again consider the $\bar{\rm K}{\rm K}$
solution with $v_{in}=0.04542$ and plot the field profiles at $t\approx 144$,
Fig. 9 solid line. These profiles can be approximated by the naive
superposition expression (17) with a moderate success, Fig. 9 dotted line. The
fit is significantly improved if we take into account the Lorentz contraction,
Fig. 9 dashed line. Here the Lorentz $\gamma$ factor grows from the initial
non-relativistic value $\gamma_{in}\approx 1.001$ up to $\gamma=1.5$, which
corresponds to the highly relativistic velocity $v\approx 0.75$. This suggests
a relativistic nature of the field when it crosses the vacuum, even though the
solitons start their evolution with totally non-relativistic velocities.

The highly relativistic property of the field in the moment of annihilation
can be important for the construction of the correct restricted set of
configurations capturing the $\bar{\rm K}{\rm K}$ (or ${\rm K}\bar{\rm K}$)
dynamics. The Lorentz contraction of a soliton can be taken into account in
terms of the (perturbative) relativistic moduli space by including the Derrick
modes [51, 20]. Therefore, their amplitudes may play an important role in the
construction of the correct CCM, in addition to the amplitudes of the
delocalized modes.

It is instructive to compare this annihilation process with the ${\rm
K}\bar{\rm K}$ solution in the sine-Gordon theory, which is an example of a
the very special integrable model, for which

$U_{sG}=1-\cos\phi.$ (18)

Here, the ${\rm K}\bar{\rm K}$ scattering solution is exactly known and reads

$\Phi_{KAK}=4\arctan\left(\frac{\sinh(\gamma vt)}{v\cosh(\gamma x)}\right).$
(19)

Importantly, at any time of the evolution, the ${\rm K}\bar{\rm K}$ solution
can be written as a sum of a single kink and antikink at certain positions
$\mp a$

$\displaystyle\phi_{{\rm K}\bar{\rm K}}=4\arctan e^{b(x+a)}-4\arctan
e^{b(x-a)}$ (20)

where

$b=\gamma,\;\;\;a=\frac{1}{\gamma}\mbox{arsinh}\frac{\sinh(\gamma vt)}{v}.$
(21)

As $a$ is the position of the colliding solitons, its time derivative,
$\dot{a}$, can be viewed as a velocity of the center of the constituent kink
and antikink. Here,

$\dot{a}=\frac{v\cosh(\gamma vt)}{\sqrt{v^{2}+\sinh^{2}(\gamma vt)}}$ (22)

At $t\to-\infty$, it tends to $v$ which is simply the initial velocity of the
infinitely separated, free kink and antikink. However, as $t$ grows, $\dot{a}$
also grows and at $t=0$ it reaches its maximal value $\dot{a}(t=0)=1$. This is
the instant where the solitons annihilate completely, temporarily forming the
vacuum. Even though the velocities of the soliton centers tend to 1, one
cannot say that the process experiences here a highly relativistic phase. The
Lorentz contraction factor $b$ is all the time equal to its initial value
$b=\gamma(v)$. Hence, if $v\ll 1$ then the non-relativistic approximation,
$\gamma\approx 1$ is a valid approximation at any time of the evolution.

Similar conclusions can be drawn for the sine-Gordon breather. In this case
the solution is

$b=\sqrt{1-\omega^{2}},\;\;\;a=\frac{1}{\sqrt{1-\omega^{2}}}\mbox{arsinh}\frac{\sqrt{1-\omega^{2}}\sin(\omega
t)}{\omega}$ (23)

where $\omega$ is the frequency of the oscillations. As before, $|\dot{a}|\leq
1$ with the equality for $t=n\pi$, while the Derrick factor, $b$, does not
change. In the non-relativistic approximation $\omega\ll 1$ and once again
$b\approx 1$, see e.g., [20].

Thus, for the sine-Gordon model we can conclude that an initially non-
relativistic kink and antikink remain non-relativistic during the whole
evolution.

From the above analysis of the field profiles the following qualitative
picture of the $\bar{\rm K}{\rm K}$ scattering in the $\phi^{6}$ model (in the
interesting fractal regime) emerges. Initially, we have an antikink and a kink
approaching each other with a small velocity. The profiles are just the naive
superposition of the single soliton solutions (17) and no internal DoF are
excited. As the solitons come closer, the profiles require a growing Lorentz
contraction factor while simultaneously some additional deformation on the
antikink and kink shows up. This deformation spreads quickly over the whole
intersoliton region as the solitons approach each other, which eventually
leads to the excitation of the lowest delocalized two-soliton modes. Now, the
process is very rapid with a highly relativistic $\gamma$. This corresponds to
a significant excitation of Derrick modes. A further effect of this is that
for even smaller antiki-kink separation, i.e., smaller $a$, for which the
positive frequency delocalized modes formally ceases to exist, the profiles
still carry such modes. An identical effect occurs in the ${\rm K}\bar{\rm K}$
collision in the $\phi^{4}$ model, where the shape mode is not an excitation
of the vacuum and, therefore, formally disappears during the annihilation.
Apparently, because of the rapidity of the process, the modes are frozen while
the field passes through the vacuum. After crossing the vacuum, the
delocalized modes are more and more excited, while the excitation of Derick
modes decreases. Thus, the $\gamma$ factor approaches again $1$ and finally,
if $v_{in}$ is suitably chosen, the quasi-stationary state is formed.

In the next section we will analyze several CCM models based on restricted
sets of configurations which follow from this general picture.

##  IV Collective coordinate models

###  IV.1 Relativistic CCM with soliton confined modes

First of all, the traditional way of constructing a CCM, which takes into
account only the zero and shape modes of single solitons, fails completely. As
we said, (anti)kinks in $\phi^{6}$ theory do not host any massive bound modes
and, therefore, one is left only with the naive antikink-kink superposition
(17), that is, with the modulus $a$. This is too simple a set of
configurations to describe bounces. All solutions of the corresponding CCM
with energy bigger than twice the soliton mass are simply one bounce
solutions.

To improve the description, we apply the (perturbative) relativistic moduli
space approach for the construction of a (perturbative) relativistic CCM
((p)RCCM). This approach in its perturbative version provides an arbitrary
large number of coordinates [20]. However, in the simplest set-up, the moduli
correspond to single localized soliton excitations and, therefore, cannot be
fully suitable for the $\bar{\rm K}{\rm K}$ collisions in the $\phi^{6}$
model. As a result, the predictions of this minimal version of the (p)RCCM,
that is, a version without any addition of the two-soliton delocalized modes,
do not agree with the full field theory computations. This is, of course, not
surprising in the view of the previous section. We underline that this is not
a failure of the perturbative relativistic moduli space framework but rather
it reflects a problem of the construction of the two-soliton restricted set of
configurations as a naive sum of two single soliton sets, as we explained
above.

Let us start with the relativistic moduli space approach applied to the single
kink sector, where the BPS kink static solutions, parametrized by the position
$a$, are extended to include the scaling deformation $b$. This results in the
following restricted set of configurations

$\Phi_{K}(x;a,b)=\sqrt{\frac{1+\tanh b(x-a)}{2}}$ (24)

which give rise to a RCCM possessing a stationary solution equal to a Lorentz
contracted kink

$\dot{a}=v=\mbox{const},\;\;\;b=\gamma=\frac{1}{\sqrt{1-v^{2}}}$ (25)

where $v$ is the velocity of a free kink, while $\gamma$ is the Lorentz factor
[51]. Note that $b\in\mathbb{R}_{+}$. The same holds for the antikink as well
as the mirror solitons. Hence, the inclusion of the new modulus, $b$, leads to
a relativistic motion of the kink. In other words, the Lorentz invariance of
the original field theory is realized at the level of the CCM.

Now, we construct the restricted set of configuration for the AKK scattering
as a naive sum of two single soliton sets

$\displaystyle\mathcal{M}_{\bar{K}K}[a,b]=\\{\Phi_{\bar{K}K}(x;a,b)\\}=$
$\displaystyle\left\\{\sqrt{\frac{1-\tanh b(x+a)}{2}}+\sqrt{\frac{1+\tanh
b(x-a)}{2}}\right\\}$ (26)

For $a\to\infty$ we obtain the antikink and kink located at $x=-\infty$ and
$x=\infty$ respectively. As $a$ decreases, the solitons come closer and then
pass through each other for $a<0$ forming a positive bump whose maximum can
never cross $\phi=2$. These configurations provide a two dimensional CCM
(5)-(7).

Let us now focus on the moduli space metric resulting from the restricted set
of configurations (IV.1). The main observation is that, contrary to the kink-
antikink collision in $\phi^{4}$ theory, the relativistic moduli space metric
has no singularity for any finite $a$ as there is no null vector issue. This
follows from the fact that the field configurations (IV.1) never equal the
vacuum solution $\phi\equiv 1$. Indeed, for any $a$, $\Phi_{\bar{K}K}$, never
is a constant field configuration. Specifically, at $a=0$, which is a singular
point in the $\phi^{4}$ case (corresponding to the situation when colliding
solitons are on top of each other), the restricted fields have the following
expansions

$\displaystyle\Phi_{\bar{K}K}(x;a,b)=\sqrt{\frac{1-\tanh
bx}{2}}+\sqrt{\frac{1+\tanh bx}{2}}$
$\displaystyle\left(\frac{1-\tanh^{2}bx}{\sqrt{1-\tanh
bx}}+\frac{1-\tanh^{2}bx}{\sqrt{1+\tanh bx}}\right)\frac{ab}{2\sqrt{2}}+o(a)$
(27)

Hence, $\partial_{a}\Phi_{\bar{K}K}$ is not identically zero for $a=0$,
leading to non-zero metric components. There is still a singularity at $b=0$.
However, as we will discuss below, this point cannot be attained by a finite
energy configuration and, therefore, is excluded from the dynamics. Hence, the
RCCM is globally well defined for the scattering case.

Finally, we consider the effective potential $V$ (7). In the limit $b\to 0$,
where the metric has the singularity, the restricted configurations tend to a
constant which is not a vacuum. Namely, $\Phi_{\bar{K}K}(x,a,b=0)=\sqrt{2}$.
Obviously, this corresponds to infinite potential energy. This means that the
singularity of the metric at $b=0$ is never accessible for a finite energy
configuration.

To conclude, the relativistic moduli space modeling the antikink-kink
collisions in $\phi^{6}$ theory is a smooth two-dimensional manifold without
any singularities. Therefore, it leads to a globally well-defined dynamical
RCCM. Owing to this property we can study the collisions already in the RCCM.
This is a nice feature which is not shared by the RCCM for scatterings of
symmetric kinks, as in the $\phi^{4}$ model, where a regularizing scheme had
to be introduced.

Figure 10: Time dependence of the value of the field at the origin,
$\phi(x=0,t)$ for various initial velocities $v_{in}$ in CCMs for the
$\bar{\rm K}{\rm K}$ collision in $\phi^{6}$ model. Upper: RCCM with moduli
$(a,b)$. Lower: pRCCM with moduli $(a,c)$.

In Fig. 10, upper panel, we show the scan of the evolution obtained in the
corresponding RCCM (5)-(7). The initial conditions are

$a(0)=a_{0},\;\;\;\dot{a}(0)=v_{in},\;\;\;b(0)=\gamma_{in},\;\;\;\dot{b}(0)=0$
(28)

where the initial separation $2a_{0}=24$. We show the value of the field as a
function of time for different values of the initial velocities.
Qualitatively, the effective model possesses desirable properties. It predicts
two, three and higher bounce windows immersed between bion chimneys. However,
the critical velocity, which separates the fractal regime and one bounce
scattering is much higher than in the original theory. Instead of
$v_{c}\approx 0.048$ it is $v_{c}\approx 0.37$. There are two related reasons
for this failure. On the one hand, the restricted set of configurations does
not approximate the true profiles too well. For example,
$\Phi_{\bar{K}K}(x;a,b)$ is always bounded between $0<\phi(x,t)<2$. On the
other hand, it does not have any wiggles in the central region. This
corresponds to the fact that the scale deformation is confined to the
solitons. Hence, our RCCM contains only soliton confined DoF.

This restricted set of configurations can be improved if we consider a variant
of the upper construction, known as the perturbative relativistic moduli
space. We present it starting with the single soliton sector,
$\Phi_{K}(x;a,b)$. Now we expand the scaling deformation of the (anti)kink,
$b=1+c$, assuming that $c$ is a small parameter. Here we consider only the
first term in the expansion

$\displaystyle\Phi_{K}(x;a,c)$
$\displaystyle\approx\sqrt{\frac{1+\tanh(x-a)}{2}}$ (29)
$\displaystyle+\frac{(x-a)c}{2\sqrt{2}\cosh^{2}(x-a)\sqrt{1+\tanh(x-a)}}$

In principle, we can keep an arbitrary number of the terms in the expansion,
which is equivalent to the fact that we take into account further Derrick
modes. The crucial idea is to replace each power of the expansion parameter,
$c^{n}$, by a new, independent collective coordinate (amplitude) $C_{n}$, of
the pertinent Derrick mode. Inserting the perturbative expansion of the
restricted configurations (29), we obtain a pRCCM with the following metric
functions

$\displaystyle g_{aa}^{K}$ $\displaystyle=$
$\displaystyle\frac{1}{96}(24+24c+(6+\pi^{2})c^{2})$ (30) $\displaystyle
g_{cc}^{K}$ $\displaystyle=$
$\displaystyle\frac{\pi^{2}}{48},\;\;\;g_{ac}^{K}=\frac{1}{8}$ (31)

and the effective potential

$\displaystyle V_{K}$ $\displaystyle=$
$\displaystyle\frac{1}{4}+\frac{c^{2}}{8}+\frac{c^{3}}{96}(\pi^{2}-6)+\frac{c^{4}}{19200}(570-275\pi^{2}+21\pi^{4})$
(32) $\displaystyle+$
$\displaystyle\frac{c^{5}}{11520}(-90+75\pi^{2}-7\pi^{4})$ $\displaystyle+$
$\displaystyle\frac{c^{6}}{1024}\left(\frac{6}{7}-\frac{7\pi^{4}}{20}+\frac{31\pi^{6}}{882}\right).$

In the case of the antikink sector, the moduli space metric differs only in
the off-diagonal term, $g^{\bar{K}}_{ac}=-1/8$, while the effective potential
is exactly the same.

The pRCCM in the kink or antikink sector supports a stationary solution with
$\dot{a}=v=\mbox{const}$, $c=\tilde{c}=\mbox{const}$ following an algebraic
equation

$\frac{v^{2}}{2}\frac{dg_{aa}}{dc}=\frac{dV}{dc}$ (33)

This solution is an approximation of a Lorentz boosted (anti)kink.

The construction of the pRCCM describing antikink-kink collisions goes along
standard lines. The corresponding restricted set of configurations is built as
a simple sum of the single soliton sectors. Hence,

$\Phi_{\bar{K}K}=\Phi_{\bar{K}}(x;-a,c)+\Phi_{K}(x;a,c),$ (34)

which leads to a pertinent collective model. Again, one can easily verify that
the restricted configurations do not produce any null vector problem and the
moduli space metric is well defined everywhere.

As the initial states in the soliton collisions are a free, boosted kink or
antikink, we have to specify the corresponding initial conditions. These are
the stationary solutions found in the single soliton pRCCM. Thus,

$a(0)=a_{0},\;\;\;\dot{a}=v_{in},\;\;\;c(0)=\tilde{c},\;\;\;\dot{c}=0$ (35)

Here $a_{0}=12$ is half of the initial separation between the colliding kink
and antikink and $v_{in}$ their initial velocity.

The results are better than in the RCCM but the critical velocity is still
approximately four times bigger than in reality, see Fig, 10, lower panel. The
improvement is probably related to a slightly better set of field
configurations whose value is now not bounded to the segment $(0,2)$. Of
course, the first Derrick mode is confined to the solitons and therefore
cannot alone correctly model the true dynamics.

We expect that a further improvement can be achieved by considering higher
Derrick modes. They are more and more widespread modes which potentially may
better approximate the true delocalized two-soliton modes.

Summarizing the results of this section, we can draw the conclusion that a CCM
based entirely on soliton confined DoF cannot successfully describe the
$\bar{\rm K}{\rm K}$ dynamics in the $\phi^{6}$ model. The inclusion of the
delocalized, two-soliton modes seems to be inevitable.

###  IV.2 CCM with one delocalized mode

As we have clearly shown above, during the scattering one observes the
excitation of the delocalized modes $\eta^{i}_{\bar{K}K}(x;a)$, which arise in
the linear perturbation theory of the kink-antikink state. Therefore, we
propose the following restricted set of configurations

$\displaystyle\tilde{\mathcal{M}}_{\bar{K}K}[a;X^{1}...X^{N}]=$
$\displaystyle\\{\Phi_{\bar{K}}(x;-a)+\Phi_{K}(x;a)+\sum_{i=1}^{N}X^{i}\eta^{\bar{K}K}_{i}(x,a)\\}.$
(36)

We also observed that the first delocalized mode $\eta^{\bar{K}K}_{1}$ plays
the most significant role, as it stores the biggest amount of energy. Thus, it
is reasonable to start with as simple a CCM as possible and consider only the
first delocalized mode

$\tilde{\mathcal{M}}_{\bar{K}K}[a;X^{1}]=\\{\Phi_{\bar{K}}(x;-a)+\Phi_{K}(x;a)+X^{1}\eta^{\bar{K}K}_{1}(x,a)\\}.$
(37)

The mode still has quite an involved dependence on the position $a$ of the
colliding solitons. Furthermore, for sufficiently small $a$ it disappears from
the spectrum of linear normal modes. To simplify our considerations, however,
we will not take into account this variation with $a$. Instead, in our
simplest CCM we assume that the mode does not change and is equal to the first
delocalized mode at a given $a=a_{*}$. This is a standard assumption for the
construction of a CCM for kink collisions referred to as the frozen
vibrational moduli space approximation. For example, it has been successfully
applied to ${\rm K}\bar{\rm K}$ scattering in the $\phi^{4}$ model [19]. This
defines the final form of the restricted configurations

$\tilde{\mathcal{M}}_{\bar{K}K}[a;X^{1}]=\\{\Phi_{\bar{K}}(x;-a)+\Phi_{K}(x;a)+X^{1}\eta^{\bar{K}K}_{1}(x,a=a_{*})\\}.$
(38)

The reason why this is an admissible assumption is related to the fact that
the process is always rapid (relativistic) when the kinks are very close to
each other. This explains why the disappearance of the internal modes
(strictly speaking their crossing through the mass threshold) in many models
does not seem to affect the ${\rm K}\bar{\rm K}$ scattering. The reason is
that the mode ceases to exist approximately in the moment when the field is
close to the vacuum value, i.e., when the solitons are almost on top of each
other. However, precisely in the moment of temporary annihilation, the
scattering solitons behave very relativistic which translates into a rapid
evolution of the field. If the typical time scale of the evolution is smaller
than the time scale of the internal mode oscillation, then the mode freezes,
preserving its form despite the change of the underlying field. This is very
much alike the non-adiabatic evolution of a quantum-mechanical wave function
if the quantum Hamiltonian changes very rapidly (instantaneously) from
$H(t_{0})$ to $H(t_{1})$. After the temporary annihilation, when the solitons
reappear and the system returns to the adiabatic evolution, the mode is again
able to evolve with its soliton(s).

Figure 11: $\bar{\rm K}{\rm K}$ collisions in the $\phi^{6}$ model in the CCM
based on $\tilde{\mathcal{M}}[a,X^{1}]$: time dependence of the field at the
origin, $\phi(x=0,t)$, and final velocity of the backscattered antikink,
$v_{out}$, for various initial velocities $v_{in}$.

All this results in the following final form of the restricted set of
configurations

$\displaystyle\tilde{\mathcal{M}}_{\bar{K}K}[a;X^{1}]=$ (39)
$\displaystyle\left\\{\sqrt{\frac{1-\tanh(x+a)}{2}}+\sqrt{\frac{1+\tanh(x-a)}{2}}+X^{1}\eta^{\bar{K}K}_{1}(x)\right\\},$

where

$\displaystyle\eta^{\bar{K}K}_{1}(x)$ $\displaystyle=$
$\displaystyle\frac{\cos(x\pi/7)}{2}\left(\tanh(x+3)-\tanh(x-3)\right)$ (40)
$\displaystyle+$
$\displaystyle\frac{1}{10}\left(-2+\tanh(x+3)-\tanh(x-3)\right)\times$
$\displaystyle\times$
$\displaystyle\left(\frac{1}{\cosh(x+3)}+\frac{1}{\cosh(x+3)}\right)$

is an approximate analytical form of the first delocalized mode for the
particular value $a_{*}=3$.

This restricted set of configurations defines a two dimensional CCM with a
non-singular moduli space and potential for any finite
$(a,X^{1})\in\mathbb{R}^{2}$. Hence we get a well defined dynamical system
$L(a,X^{1})$. We do not present the explicit formulas for $g_{ij}$ and
$V(a,X^{1})$ because they are very complicated and because in our numerical
scheme we do not use analytical expressions but compute these functions
numerically.

Finally, the initial conditions for the evolution in the CCM are as follows

$a(0)=a_{0},\;\;\dot{a}(0)=v_{in},\;\;\;A(0)=0,\;\;\dot{A}(0)=0$ (41)

These are natural initial conditions corresponding to non-relativistic kinks
with no excited modes.

In Fig. 11 we present our results. In particular, we plot the time dependence
of the field at the origin for initial velocity $v_{in}\in[0.01,0.05]$. We
found a spectacular quantitative agreement with the full field theory
computation.

First of all, exactly as in the full field theory, the fractal structure shows
up in the non-relativistic range of initial velocities. The critical velocity,
above which only one-bounce scattering occurs, is $v_{crit}^{CCM}=0.0457$
which is exactly equal to the true value $v_{crit}=0.0457$ within our
numerical precision.

Secondly, almost all positions of the two-bounce windows, as well as their
widths, coincide with the full theory computation. Only the two-bounce windows
located around $v_{in}\approx 0.015$ and $v_{in}\approx 0.0275$ correspond to
false windows in the full model. This suggests that the existence of false
windows may be related with higher delocalized modes or with radiation.
Undoubtedly, our collective model gives a tool in which this question can be
analyzed.

It is important to remark that the results of the CCM depend on the value of
the modulus $a=a_{*}$ at which we freeze the first delocalized mode. The
critical velocity grows from $v_{cr}=0.0320$ for $a_{*}=2.6$ to
$v_{cr}=0.0598$ for $a_{*}=3.5$, see Table 1. Details of the fractal structure
also vary with the freezing point.

$a_{*}$ |  $v_{cr}$
---|---
2.6 | 0.0320
2.7 | 0.0355
2.8 | 0.0391
2.9 | 0.0424
3.0 | 0.0457
3.1 | 0.0487
3.2 | 0.0517
3.3 | 0.0545
3.4 | 0.0572
3.5 | 0.0598
Table 1: Dependence of the value of the critical velocity $v_{cr}$ obtained in
the CCM based on (38) on the freezing point $a_{*}$.

Of course, in a more complete CCM based on the dynamical restricted set of
configurations (37) one should take into account the fact that the delocalized
mode changes during the collision and does not freeze at a given $a$. Here,
however, we follow the simplest strategy. Namely, we just select a particular
shape of the first delocalized mode, i.e., freeze it at a certain $a_{*}$.
Therefore, this freezing point $a_{*}$ may be viewed as a free parameter of
the simplest CCM. For $a=a_{*}=3$ we found the most striking agreement. This
value is ”natural” in the sense that quasi-stationary states with a $\bar{\rm
K}{\rm K}$ distance of $2a_{\rm qs}\sim 2a_{*}\sim 6$ form in many bounce
windows. As a general rule, the value of $2a_{\rm qs}$ grows with the lifetime
of the quasi-stationary state. In section III, e.g., we analyzed in detail the
case of a particularly long-lived quasi-stationary state with $a_{\rm qs}=6$.
Probably, the correct separation in each bounce window can be reproduced in a
more complete CCM provided by (37).

In any case, we must admit that at the moment we do not fully understand why
our simple CCM for $a_{*}=3$ agrees so spectacularly well with the full field
theory for all initial velocities. Naively, one would probably expect a
certain variation of the optimal value for $a_{*}$ with the initial velocity.
A more complete understanding of this issue most likely requires a dynamical
treatment of the delocalized modes where $a$ is not frozen but allowed to vary
with time. It must be stressed, however, that such a fully dynamical treatment
within a CCM framework is a very challenging problem. This reflects the
existence of essential singularities of any moduli space based on delocalized
normal modes. Indeed, any of these modes ceases to exist for a sufficiently
small antikink-kink distance. One by one they hit the mass threshold and
become non-normalizable threshold modes which further change into anti-normal
or quasinormal modes. As a consequence, some of the metric components diverge
at these points which results in a breakdown of a CCM based only on normal
modes [47].

##  V ${\rm K}\bar{\rm K}$ collisions

In this section we turn to the kink-antikink collisions. It is known that in
this case there is no fractal structure in the final state formation. Instead,
there are two regimes which meet at $v_{cr}\approx 0.289$. For smaller initial
velocities, the solitons undergo annihilation via the formation of a quickly
oscillating bion/oscillon, while for $v>v_{cr}$ they simply pass through each
other and change into their mirror kinks which escape to infinity, see Fig.
12. Note that the critical velocity is quite big if compared with the
$\bar{\rm K}{\rm K}$ case.

Despite the relatively simple outcome of this scattering, it is surprisingly
nontrivial to model it within the collective coordinate framework. This
originates in the fact that now radiative modes play a more significant role
than in the previous $\bar{\rm K}{\rm K}$ case or in ${\rm K}\bar{\rm K}$
collisions in the $\phi^{4}$ model. Indeed, there are no single-kink or
delocalized kink-antikink normal modes involved here. Therefore, the main
channel in which energy escapes from the kinetic DoF of the solitons is
radiation. Ideally we would like to have a CCM which includes such radiative
DoF. This is a complicated and still not solved issue within the CCM
framework. However, as we have noticed a few times, the Derrick modes may
capture some properties of radiation, at least at short time scales. The
description in terms of these modes should become better as we increase their
number. In the subsequent analysis we apply this approach, again in the
simplest version, i.e., with only one Derrick mode taken into account.
Obviously, this is a very crude approximation which cannot lead to
satisfactory results. Nonetheless, some findings are encouraging and give new
insights into the understanding of this process.

To see that there is significant radiation in a ${\rm K}\bar{\rm K}$ collision
in the $\phi^{6}$ model we consider a kink and antikink boosted towards each
other with initial velocity $v_{in}=0.288$, which is just below the regime
where the simple passing-through scenario occurs. We compare it with a
$\bar{\rm K}{\rm K}$ collision with $v_{in}=0.042$. In Fig. 13 we plot the
time evolution of the field in the origin (upper panels) together with the
energy stored in the space region $-15<x<15$ (lower panels). Thus it shows how
much of the energy is radiated away from the solitons.

Figure 12: Time dependence of the value of the field at the origin,
$\phi(x=0,t)$, for various initial velocities $v_{in}$ in ${\rm K}\bar{\rm K}$
collisions in the $\phi^{6}$ model.  Figure 13: Value of the field at the
origin (upper panels) and energy stored in the region $|x|<15$ (lower panels)
for a $\bar{\rm K}{\rm K}$ collision with $v_{in}=0.042$ (left panels) and a
${\rm K}\bar{\rm K}$ collision with $v_{in}=0.288$ (right panels).

In the left panels we show the $\bar{\rm K}{\rm K}$ collision. We see that
after each bounce, where the field passes $+1$ vacuum, the part of the energy
is emitted from the solitons. The amount of emitted energy is relatively
small, $\Delta E\approx 0.00004$, and it requires many bounces to lower the
energy below the annihilation threshold which is twice of the soliton mass. In
the right panels the ${\rm K}\bar{\rm K}$ collision is plotted. Now, after the
first bounce, the radiation takes a significant amount of energy, $\Delta
E\approx 0.022$, which brings the solitons very close to the annihilation
threshold. This confirms that the energy is radiated out in a very efficient
way and, consequently, is an important factor in the dynamics.

The first step in the construction of a correct CCM is the right choice of the
one-parameter restricted set of configurations which interpolates between the
initial and final states. In this case, the initial state, i.e., the
infinitely separated pair of kink $\Phi_{K}$ and antikink $\Phi_{\bar{K}}$
located at $\mp a$ respectively, can produce a final state consisting of a
pair of mirror antikink $\Phi_{\bar{K}}^{*}$ and mirror kink $\Phi_{K}^{*}$
located at $\mp a$. Schematically it can be represented as the following
process $(0,1)+(1,0)\to(0,-1)+(-1,0)$. Surprisingly, the usual naive
superposition

$\mathcal{M}^{naive}_{K\bar{K}}[a]=\\{\Phi_{K}(x;-a)+\Phi_{\bar{K}}(x;a)-1\\}$
(42)

fails. Topologically, changing the modulus from $a=\infty$ to $a=-\infty$
gives a correct transition. In the initial state $(a\to\infty)$ we do have an
infinitely separated kink-antikink pair, with a profile starting from 0 at
$x=-\infty$, tending to 1 at $x=0$ and then decreasing to 0 for $x=\infty$. In
the final state ($a\to-\infty$) we get a configuration which again tends to 0
for $x=\mp\infty$ and takes $-1$ at $x=0$. Although this is a correct
topological behaviour, the final state is not a pair of mirror solitons,

$\displaystyle\lim_{a\to-\infty}\left(\Phi_{K}(x;-a)+\Phi_{\bar{K}}(x;a)-1\right)$
$\displaystyle\neq$ (43) $\displaystyle\Phi^{*}_{\bar{K}}(x;-\infty)$
$\displaystyle+$ $\displaystyle\Phi^{*}_{K}(x;\infty)+1$

Hence, the assumed naive sum does not provide the correct final state.

Specifically, the large $|x|$ asymptotic of the naive sum (42) for
$a\to-\infty$ does not agree with the asymptotic of a largely separated mirror
antikink and mirror kink. This leads to slightly different asymptotical values
of the effective potentials in the corresponding CCM. Specifically,
$V_{-}>V_{+}$, where $V_{\pm}=\lim_{a\to\pm\infty}V(a)$, see Fig. 14 dashed
curve. This unphysical jump in the CCM potential can have an unwanted impact
on the resulting dynamics.

Figure 14: The effective potential in the one-dimensional CCM for ${\rm
K}\bar{\rm K}$ collisions: dashed curve - the naive sum (42); orange, green
and purple curves - the improved choice with $D=0.5,1,2,200$.

A restricted one-parameter set of configurations which provides the correct
initial and final states can have the following form

$\displaystyle\Phi_{K\bar{K}}(x;a)=$ (44)
$\displaystyle\tanh(Da)\left(\Phi_{K}(x;-a\tanh(Da))+\Phi_{\bar{K}}(x;a\tanh(Da))-1\right)$

or explicitly

$\displaystyle\Phi_{K\bar{K}}(x;a)$ $\displaystyle=$
$\displaystyle\tanh(Da)\left(\sqrt{\frac{1+\tanh(x+a\tanh(Da))}{2}}+\right.$
(45) $\displaystyle\left.\sqrt{\frac{1-\tanh(x-a\tanh(Da))}{2}}-1\right),$

where, for the moment, $D$ is a free constant. It is easy to verify that this
set interpolates between the right initial and final states. Indeed,

$\lim_{a\to\infty}\Phi_{K\bar{K}}(x;a)=\lim_{a\to\infty}\left(\Phi_{K}(x;-a)+\Phi_{\bar{K}}(x;a)-1\right)$
(46)

which is an infinitely separated pair of kink and antikink. Similarly,

$\displaystyle\lim_{a\to-\infty}\Phi_{K\bar{K}}(x;a)$ $\displaystyle=$
$\displaystyle\lim_{a\to-\infty}\left(-\Phi_{K}(x;a)-\Phi_{\bar{K}}(x;-a)+1\right)$
(47) $\displaystyle=$
$\displaystyle\lim_{a\to\infty}\left(\Phi_{\bar{K}}^{*}(x;-a)+\Phi^{*}_{K}(x;a)+1\right),$

which is an analogous pair of mirror antikink and mirror kink. This choice has
the additional advantage that these states are approached exponentially fast.
Thus, the naive superposition, both in the normal and mirror sector, is
deformed only for small $a$.

Note also that our configurations are antisymmetric in $a$,
$\Phi_{K\bar{K}}(x;-a)=-\Phi_{K\bar{K}}(x;a)$.

Since (44) provides the correct initial and final states, the corresponding
effective potential has the same asymptotical values. Hence, there is no
unwanted jump. However, the effective potential can still have a local maximum
which again may lead to unwanted one-bounce windows. This in fact happens for
$D\lesssim 1.27$, see Fig. 14. For bigger values the potential has only the
global minimum for $a=0$, $V(a=0)=0$, where we pass through the vacuum
$\Phi_{v}=0$.

Of course, a CCM based on one collective DoF cannot explain the dynamics even
in the crudest way. Here, however, there are no soliton-confined or
delocalized normal modes whose amplitudes could provide new collective
coordinates. Fortunately, the perturbative relativistic moduli space framework
supplies us with the required additional collective excitations, which are
Derrick modes. Thus, the simplest two dimensional moduli space with only one
Derrick mode reads

$\displaystyle\mathcal{M}_{K\bar{K}}[a,c]=\\{\Phi_{K\bar{K}}(x;a)+c\tanh(Da)\times$
(48)
$\displaystyle\left(\frac{a\tanh(Da)-x}{\left(e^{-2(x-a\tanh(Da))}+1\right)\sqrt{e^{2(x-a\tanh(Da))}+1}}\right.$
$\displaystyle\left.\left.+\frac{a\tanh(Da)+x}{\sqrt{e^{-2(a\tanh(Da)+x)}+1}\left(e^{2(a\tanh(Da)+x)}+1\right)}\right)\right\\}$

where $\Phi_{K\bar{K}}(x;a)$ is the improved one-parameter configurations (44)
and $c$ is the amplitude of the corresponding superposition of the modified
first Derrick mode obtained by a scaling perturbation of (44).

It is easy to notice that this restricted set of configurations suffers from a
null vector problem. Indeed, $\partial_{c}\Phi(x;a=0,c)=0$. Therefore,
$g_{ca}=g_{cc}=0$ at $a=0$. Fortunately, this singularity can be removed by a
redefinition of the collective coordinate $c$ [19], [20]. Namely,

$c\to\frac{c}{\tanh(Da)}.$ (49)

Thus, finally, the regular two-dimensional moduli space is defined as follows

$\displaystyle\mathcal{M}_{K\bar{K}}[a,c]=\\{\Phi_{K\bar{K}}(x;a)+$ (50)
$\displaystyle
c\left(\frac{a\tanh(Da)-x}{\left(e^{-2(x-a\tanh(Da))}+1\right)\sqrt{e^{2(x-a\tanh(Da))}+1}}\right.$
$\displaystyle\left.\left.+\frac{a\tanh(Da)+x}{\sqrt{e^{-2(a\tanh(Da)+x)}+1}\left(e^{2(a\tanh(Da)+x)}+1\right)}\right)\right\\}.$

The optimal value of the parameter $D$ can be determined by comparison of the
configurations (50) with the actual profiles observed in the scattering
process. We found that $D=2$ is a good choice. We also remark that all
$D\in(1,5)$ work similarly well and no big changes are observed.

Figure 15: Comparison of true field profiles in a ${\rm K}\bar{\rm K}$
collision with $v_{in}=0.288$ with the best fit in terms of the two
dimensional moduli space (50). Figure 16: Left: Time dependence of the best
fit of the moduli $a,c$ for the ${\rm K}\bar{\rm K}$ collision with
$v_{in}=0.288$. Right: $a(t),c(t)$ computed from the CCM based on (50).

In Fig. 15 we show the actual field profiles (solid line) together with the
best fit for the two-dimensional moduli space configurations (dashed line) for
several time moments. Here again $v_{in}=0.288$ and we demonstrate the
situation close to the first collision. It is clearly visible that the
configurations (50) very well approximate the field profiles except the
inability to model the radiation far away from the center of the solitons.
This is, of course, expected. Furthermore, once again we see how big is the
radiation emitted during the collision.

In Fig. 16, left panels, we show how the fitted values of the moduli $a$ and
$c$ change in time. This is not a CCM result but a fit of the configurations
(50) to the true profiles obtained in the full partial differential evolution.
We see a relatively long first window and fast decay of the excited first
Derrick mode after the first collision. This agrees with the emission of
radiation which quickly releases energy from the soliton-confined DoF. The
late time chaotic changes of the amplitude of the Derrick mode probably imply
that the moduli (50) does not work too well and some other DoF must be
included.

Now we show the results obtained in the CCM based on the moduli (50). The
initial conditions are again

$a(0)=a_{0},\;\;\dot{a}=v_{in},\;\;c(0)=\tilde{c},\;\;\dot{c}=0$ (51)

where $\tilde{c}$ is a stationary solution of the single soliton case, exactly
as for the $\bar{\rm K}{\rm K}$ collision in pRCCM. In our analysis this is
computed numerical for each $v_{in}$.

In Fig. 17 we show the time dependence of the value of the field at the origin
for initial velocities $v_{in}\in[0.1,0.35]$. An encouraging result is that
this simple CCM reproduces the critical velocity quite well. Namely, we found
$v_{cr}^{CCM}\approx 0.298$ which is quite close to the true value
$v_{cr}\approx 0.289$. On the other hand, the whole structure of the formation
of the final state is not accurately predicted by the CCM. For example,
besides the nice bion chimneys, we also see quite well pronounced, unwanted
multi-bounce windows. The origin of the appearance of these unwanted features
is quite clear. The CCM has only soliton confined DoF and, therefore, there is
no way to dissipate energy located on a kink or antikink. As a consequence,
during the CCM evolution the energy stored in the first Derrick mode can at
later times be back-transferred to the kinetic motion, which eventually leads
to back-scattering of the solitons. This, of course, does not happen in the
full field theoretical collision, where the radiation quickly reduces the
energy of the kinks, leading to a complete annihilation.

The correct prediction of the critical velocity may suggest that the first
Derrick mode is good enough to describe the early phase of the collision
properly, where it is decided which amount of the energy is transferred from
kinetic motion to other DoF and, therefore, it is decided whether we have a
simple passing-through collision, or we enter in a more complicated scenario.
This is in fact visible if we plot $a(t)$ and $c(t)$ obtained from the CCM and
compare it with the best fit, see Fig. 16, right panels. We found a reasonable
good agreement for the first collision. Namely, we see a wider first bounce
window. Also the amplitude of the excited Derrick mode agrees. However, as
there is no dissipation, this amplitude remains basically constant. This has a
significant impact on the late time behaviour and leads to the observed
disagreement with the full theory.

Figure 17: ${\rm K}\bar{\rm K}$ collisions in the $\phi^{6}$ model in a CCM
based on the moduli (50). We plot the time dependence of the field at the
origin as a function of the initial velocity $v_{in}$.

##  VI Conclusions

In this work, we fully confirmed that the fractal structure observed in the
final state formation in antikink-kink collision in the $\phi^{6}$ model is
caused by delocalized, two-soliton modes. These modes arise in the linear
perturbation theory of the antikink-kink static configuration and, contrary to
the usual shape modes, are not hosted by a single soliton solution. So, they
are non-perturbatively related to the antikink-kink sector. As a consequence,
the properties of these delocalized modes (their shapes and frequencies) as
well as their number depends on the inter-soliton distance.

This was confirmed by a close analysis of the actual field profiles as well as
by a construction of a simple two-dimensional collective coordinate model
(CCM) which amazingly well reproduces the full theory computations. We would
like to underline that the results obtained in the CCM agree with the full
field theory with a precision which goes significantly beyond previously
presented collective approximations such as, for example, kink-antikink
scattering in the $\phi^{4}$ model [19].

Based on the success of the CCM description of the $\bar{\rm K}{\rm K}$
collisions in the $\phi^{6}$ model (and the ${\rm K}\bar{\rm K}$ collision in
$\phi^{4}$ theory) we propose a general, robust way for the construction of
CCM for a wide class of multikink processes. The main ingredient is the
restricted set of configurations (parametrized by a finite set of collective
coordinates) which includes: (1) a one-parameter subset configurations
interpolating between the initial and final states of the process. This is
typically, but not always, given by the naive superposition of static
solutions of solitons which participate in the process; (2) the linear modes
arising in the multi-soliton linear perturbation theory; and finally (3)
Derrick modes which take into account the Lorentz contraction of the
constituent solitons and also may serve to model some effects of radiation.
These components should provide a CCM which may be treated as a precision tool
allowing for a quantitative understanding of many kink collisions. This
approach should lead to the explanation of fractal structures found in
multikink collisions in various models as, for example, $\phi^{8}$ theory or
other models having solitons with so-called fat tails. We expect that this
should work especially well whenever there is a (single- or multi-soliton)
normal mode involved in the process.

If no such modes are present, the situation is surprisingly more subtle,
because the main factor governing the dynamics is the interaction of kinks
with radiation. Unfortunately, the inclusion of radiative (not soliton
confined) DoF into a collective model framework is still not fully developed.
However, the Derrick modes may provide a simple and useful approximation. This
follows from the observation that for sufficiently high Derrick modes the
frequencies begin to be higher than the mass threshold. Furthermore, higher
modes are more widespread and therefore give a better approximation to more
distant regions. Of course, the main limitation is that these modes are also
soliton confined and cannot transfer energy to infinity.

Thus, although the proposed general scheme for the construction of CCMs is
based on discrete DoF ((quasi)normal modes, Derrick modes), it does take into
account some feature of radiation, which is also an important step forward.
Interestingly, delocalized modes can also be viewed as localized radiation,
i.e., a kind of standing wave trapped between the colliding solitons. This
includes only radiation with frequency lower than the mass threshold of the
outer vacuum. In addition, Derrick modes also enjoy some features of
radiation, especially at a short time scale. This we saw in a quite accurate
value of the critical velocity for the ${\rm K}\bar{\rm K}$ collision.

There are many directions in which the current work should be continued
besides the straightforward application to multikink processes in other single
scalar field theories.

Firstly, we need a CCM where the delocalized, two-soliton modes are treated
dynamically, i.e., without any freezing procedure. As we already mentioned,
this a rather nontrivial task which, in its full shape, might require to
incorporate threshold modes, anti-normal modes and quasi-normal modes, i.e.,
the modes into which the delocalized normal modes transmute as the solitons
approach each other. This would allow us to get rid of the free parameter
$a_{*}$ which enters the current version of the CCM for the $\bar{\rm K}{\rm
K}$ collision.

Secondly, especially in the cases where there is no normal mode involved in
the process (as in the ${\rm K}\bar{\rm K}$ collision in $\phi^{6}$ theory)
one should couple a bigger number of Derrick modes. This should allow for a
better modeling of the effects of radiation, which in this case is the crucial
factor. We expect that the addition of higher Derrick modes will improve the
medium time CCM dynamics by introducing a channel in with the energy can be
transferred in quite an efficient way. This should lead to the disappearance
of unwanted bounce windows which exist in our CCM, leading to a much better
accordance of the CCM with the full theory.

Another issue is related to the construction of the improved one-dimensional
moduli space which replaces the invalid usual naive sum. It would be desirable
to find a unique and general way for such a construction.

## Acknowledgements

The authors acknowledge financial support from the Ministry of Education,
Culture, and Sports, Spain (Grant No. PID2020-119632GB-I00), the Xunta de
Galicia (Grant No. INCITE09.296.035PR and Centro singular de investigación de
Galicia accreditation 2019-2022), the Spanish Consolider-Ingenio 2010
Programme CPAN (CSD2007-00042), and the European Union ERDF. AGMC is grateful
to the Spanish Ministry of Science, Innovation and Universities, and the
European Social Fund for the funding of his predoctoral research activity
(_Ayuda para contratos predoctorales para la formación de doctores_ 2019). MHG
thanks the Xunta de Galicia (Consellería de Cultura, Educación y Universidad)
for the funding of his predoctoral activity through _Programa de ayudas a la
etapa predoctoral_ 2021\. KO was supported by the Polish National Science
Centre (Grant NCN 2021/43/D/ST2/01122). The research of PED was supported in
part by the National Science Foundation under Grant No. NSF PHY-1748958, and
in part from the STFC under consolidated grant ST/T000708/1.

## References

  * [1] R. Rajaraman, Solitons and Instantons, Elsevier Science, Amsterdam, 1982.
  * [2] N. Manton and P. Sutcliffe, Topological Solitons, Cambridge University Press, Cambridge U.K., 2004.
  * [3] Y. M. Shnir, Topological and Non-Topological Solitons in Scalar Field Theories, Cambridge University Press, Cambridge U.K., 2018.
  * [4] E. B. Bogomolny, The stability of classical solutions, Sov. J. Nucl. Phys. 24 (1976) 449.
  * [5] A. Jaffe and C. Taubes, Vortices and Monopoles, Boston, Birkhäuser, 1980.
  * [6] N. S. Manton, The force between ’t Hooft-Polyakov monopoles, Nucl. Phys. B126 (1977) 525.
  * [7] M. F. Atiyah and N. J. Hitchin, The Geometry and Dynamics of Magnetic Monopoles, Princeton University Press, Princeton NJ, 1988.,
  * [8] T. M. Samols, Vortex scattering, Commun. Math. Phys. 145 (1992) 149.
  * [9] N. S. Manton, A remark on the scattering of BPS monopoles, Phys. Lett. B110 (1982) 54.
  * [10] C. Adam, T. Romanczukiewicz and A. Wereszczynski, The $\phi^{4}$ model with the BPS preserving impurity, JHEP 03 (2019), 131, [arXiv:1812.04007].
  * [11] C. Adam, K. Oles, J. Queiruga, T. Romanczukiewicz, and A. Wereszczynski, Solvable self-dual impurity models, JHEP 07 (2019) 150.
  * [12] D. Bazeia, J.R.S. Nascimento, R.F. Ribeiro and D. Toledo, Soliton stability in systems of two real scalar fields, J. Phys. A30 (1997) 8157.
  * [13] A.A. Izquierdo, M.A.G. Leon and J.M. Guilarte, The kink variety in systems of two coupled scalar fields in two space-time dimensions, Phys. Rev. D65 (2002) 085012.
  * [14] A.A. Izquierdo, M.A. Gonzalez Leon, J.M. Guilarte and M. de la Torre Mayado, Adiabatic motion of two component BPS kinks, Phys. Rev. D66 (2002) 105022.
  * [15] C. Adam, K. Oles, T. Romanczukiewicz, A. Wereszczynski, and W. Zakrzewski, Spectral walls in multifield kink dynamics, JHEP 08 (2021) 147.
  * [16] T. Sugiyama, Kink-antikink collisions in the two-dimensional $\phi^{4}$ model, Prog. Theor. Phys. 61 (1979) 1550.
  * [17] D. K. Campbell, J. F. Schonfeld, and C. A. Wingate, Resonance structure in kink-antikink interactions in $\phi^{4}$ theory, Physica D9 (1983) 1.
  * [18] R. H. Goodman and R. Haberman, Kink-antikink collisions in the $\phi^{4}$ equation: The n-bounce resonance and the separatrix map, SIAM J. Appl. Dyn. Syst. 4, 1195 (2005).
  * [19] 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) 071601.
  * [20] C. Adam, N. S. Manton, K. Oles, T. Romanczukiewicz, and A. Wereszczynski, Relativistic Moduli Space for Kink Collisions, Phys. Rev. D105 (2022) 065012.
  * [21] A. Soffer, Soliton dynamics and scattering, ICM-2006 Conference Book, 2007.
  * [22] T. Tao, Why are solitons stable?, Bull. Amer. Math. Soc. 46 (2009) 1.
  * [23] P. Forgacs, A. Lukacs, T. Romanczukiewicz, Negative radiation pressure exerted on kinks, Phys. Rev. D77 (2008) 125012.
  * [24] A. Demirkaya, R. Decker, P. G. Kevrekidis, I. C. Christov, and A. Saxena, Kink dynamics in a parametric $\phi^{6}$ system: A model with controllably many internal modes, JHEP 12 (2017) 071.
  * [25] A. M. Marjaneh, F. C. Simas, D. Bazeia, Collisions of kinks in deformed $\phi^{4}$ and $\phi^{6}$ models, arXiv:2207.00835.
  * [26] I. Takyi, S. Gyampoh, B. Barnes, J. Ackora-Prah, G.A. Okyere, Kink Collision in the Noncanonical $\phi^{6}$ Model: A Model with Localized Inner Structures, 2209.05902.
  * [27] E. Belendryasova, V. A. Gani, Scattering of the $\phi^{8}$ kinks with power-law asymptotics, Commun. Nonlinear Sci. Numer. Simulat. 67 (2019) 414
  * [28] N. Manton, Forces between Kinks and Antikinks with Long-range Tails, J. Phys. A52 (2019) 065401.
  * [29] 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, 171601 (2019)
  * [30] I. C. Christov, R. J. Decker, A. Demirkaya, V. A. Gani, P.G. Kevrekidis, et al., Kink-Antikink Collisions and Multi-Bounce Resonance Windows in Higher-Order Field Theories, Commun.Nonlinear Sci.Numer.Simul. 97 (2021) 105748.
  * [31] J. G. F. Campos and A. Mohammadi, Interaction between kinks and antikinks with double long-range tails, Phys. Lett. B818 (2021) 136361.
  * [32] D. Campbell, M. Peyrard, and P. Sodano, Kink-antikink interactions in the double sine-Gordon equation, Physica D19 (1986) 165.
  * [33] V. A. Gani and A. E. Kudryavtsev, Kink-antikink interactions in the double sine- Gordon equation and the problem of resonance frequencies, Phys. Rev. E60, 3305 (1999)
  * [34] F. C. Simas, F. C. Lima, K.Z. Nobrega, A. R. Gomes, Solitary oscillations and multiple antikink-kink pairs in the double sine-Gordon model, JHEP 12 (2020) 143.
  * [35] J. G. F. Campos and A. Mohammadi, Wobbling double sine-Gordon kinks, JHEP 09 (2021) 067.
  * [36] F.C. Simas, A.R. Gomes, K.Z. Nobrega and J.C.R.E. Oliveira, Suppression of two-bounce windows in kink-antikink collisions, JHEP 09, (2016) 104.
  * [37] A. Alonso-Izquierdo, Kink dynamics in a system of two coupled scalar fields in two space-time dimensions, Physica D: Nonlinear Phenomena 365, (2018) 12.
  * [38] A. Alonso-Izquierdo, Asymmetric kink scattering in a two-component scalar field theory model, Commun. Nonlinear Sci. Numer. Simulat. 75, (2019) 200.
  * [39] D. Bazeia, A. R. Gomes, K. Z. Nobrega, and F. C. Simas, Kink scattering in hyperbolic models, Int. J. Mod. Phys. A34, 1950200 (2019)
  * [40] D. Bazeia, Adalto R. Gomes, K. Z. Nobrega, F. C. Simas, Oscillons in hyperbolic models, Phys. Lett. B803 (2020) 135291.
  * [41] D. Bazeia, A. R. Gomes, F. C. Simas, Semi-compactness and multiple oscillating pulses in kink scattering, Eur. Phys. J. C81 (2021) 532.
  * [42] F. C. Simas, K.Z. Nobrega, D. Bazeia, A. R. Gomes, Degeneracy and kink scattering in a two coupled scalar field model in (1,1) dimensions, arXiv:2201.03372.
  * [43] P. Dorey, K. Mersh, T. Romanczukiewicz, Y. Shnir, Kink-antikink collisions in the $\phi^{6}$ model, Phys. Rev. Lett. 107 (2011) 091602.
  * [44] H. Weigel, Kink-Antikink Scattering in $\varphi^{4}$ and $\varphi^{4}$ Models, J. Phys. Conf. Ser. 482 (2014) 012045.
  * [45] V. A. Gani, A. E. Kudryavtsev, and M. A. Lizunova, Kink interactions in the (1+1)-dimensional $\phi^{6}$ model, Phys. Rev. D89 (2014) 125009.
  * [46] I. Takyi and H. Weigel, Collective coordinates in one- dimensional soliton model revisited, Phys. Rev. D94 (2016) 085008.
  * [47] C. Adam, K. Oles, T. Romanczukiewicz, and A. Wereszczynski, Spectral walls in soliton collisions, Phys. Rev. Lett. 122 (2019) 241601.
  * [48] C. Adam, D. Ciurla, K. Oles, T. Romanczukiewicz, and A. Wereszczynski, Sphalerons and resonance phenomenon in kink-antikink collisions, Phys. Rev. D104, 105022 (2021).
  * [49] H. Arodz, Topological compactons, Acta Phys. Pol. B33, 1241 (2002).
  * [50] N. S. Manton, K. Oles, T. Romanczukiewicz, and A. Wereszczynski, Kink moduli spaces: Collective coordinates reconsidered, Phys. Rev. D103 (2021) 025024.
  * [51] M. J. Rice, Physical dynamics of solitons, Phys. Rev. B28 (1983) 3587.
  * [52] C. Adam, K. Oles, T. Romanczukiewicz, and A. Wereszczynski, Kink-antikink scattering in the $\phi^{4}$ model without static intersoliton forces, Phys. Rev. D101, 105021 (2020).