# Inclusion of radiation in the CCM approach of the $\phi^{4}$ model

S. Navarro-Obregón<EMAIL_ADDRESS>L.M. Nieto
<EMAIL_ADDRESS>Departamento de Física Teórica, Atómica y
Óptica, Universidad de Valladolid, 47011 Valladolid, Spain  J.M. Queiruga
<EMAIL_ADDRESS>Department of Applied Mathematics, University of
Salamanca, Casas del Parque 2, 37008 - Salamanca, Spain  Institute of
Fundamental Physics and Mathematics, University of Salamanca, Plaza de la
Merced 1, 37008 - Salamanca, Spain

(August 28, 2024)

###### Abstract

We present an effective Lagrangian for the $\phi^{4}$ model that includes
radiation modes as collective coordinates. The coupling between these modes to
the discrete part of the spectrum, i.e., the zero mode and the shape mode,
gives rise to different phenomena which can be understood in a simple way in
our approach. In particular, the energy transfer between radiation,
translation and shape modes is carefully investigated in the single-kink
sector. Finally, we also discuss the inclusion of radiation modes in the study
of oscillons. This leads to relevant phenomena such as the oscillon decay and
the kink-antikink creation.

##  1 Introduction

Topological solitons are non-linear field theory solutions that appear in many
branches of physics, from condensed matter to cosmology Rajaraman ; Manton ;
Shellard ; Shnir , and that have gained interest over the last decades. The
stability of these objects is guaranteed by their topological charge, which is
a topological invariant conserved during time evolution. They have particle-
like behaviour, so they can interact with each other, with external fields or
with radiation, as well as being annihilated and even created in pairs. Among
all of them, the $\phi^{4}$ model is particularly interesting: it can be
formulated in $1+1$ dimensions, which makes it simpler from a computational
point of view. In addition, the static solitons (called kinks) as well as the
spectrum of perturbations can be determined analytically. The study of low-
energy perturbations around the kinks shows that, in the linearised theory,
there exists a discrete spectrum formed by two states localised around the
core of the topological soliton - the zero mode and the shape mode - as well
as a continuum of states representing radiation moving away from the kink.

However, the analysis of the dynamics in the full non-linear theory is
extremely involved, which often means that the equations have to be solved
numerically. This complexity comes, in part, from the multiple interaction
channels, namely: attractive or repulsive static forces, excitation of the
internal degrees of freedom, and interaction with radiation. One method to
reduce the complexity of the topological soliton dynamics is through the
Collective Coordinate Method (CCM). Within this approach, the field theory
Lagrangian is reduced to a mechanical one with a finite number of degrees of
freedom. In Bogomolnyi-Prasad-Sommerfield (BPS) models, this approach gives
rise to the so-called canonical moduli space. Here, the relevant degrees of
freedom are the positions of the solitons, and the dynamics can be described
effectively as a geodesic motion in the manifold given by the BPS solitons.
When we deal with non-BPS sectors this study requires the introduction of a
potential, which accounts for the static interactions between solitons, but
the formalism is basically the same as in the BPS sector.

This effective point of view can be improved by introducing new coordinates
which take into account internal degrees of freedom. But, even in the
apparently simple case of $\phi^{4}$, there is a complicated pattern of final
states in scattering processes related to the non-integrability of the model,
which cannot be explained satisfactorily by a simple choice of translational
and internal oscillatory degrees of freedom Sugiyama ; Campbell . Recently, an
important improvement has been made by means of the introduction of the
relativistic moduli space Relativistic ; Relativistic1 . This approach, unlike
the standard CCM, can also accommodate some relativistic degrees of freedom.
As shown in Relativistic , this quantitatively improves the agreement between
the effective model and the field theory. However, neither of these approaches
considers dissipative degrees of freedom as generalised coordinates, i.e.,
they cannot describe radiation. The effect of radiation in soliton dynamics
can be very relevant in certain violent processes, such as the kink-antikink
annihilation, but it is also determinant in long-time dynamics and may
contribute to the fractal structure of the kink scattering. One of the
purposes of this work is to introduce the radiation modes as generalised
coordinates and study their role in certain dynamical processes.

Apart from the topologically non-trivial solutions of the $\phi^{4}$ model,
there are other time dependent soliton-like structures that deserve special
attention, the oscillons. They are topologically trivial solutions (they are
in the topological sector of the vacuum) that are long-lived, and in contrast
to other time-dependent solitons such as Q-balls, they are not associated to
any conserved charge. They are ubiquitous in a wide range of models from one
to three dimensions Segur ; Fodor1 ; Fodor3 ; Hindmarsh2 ; Hindmarsh3 ;
Manton3 , and they have found applications in many scenarios in theoretical
physics, from dark matter Olle ; Kawasaki ; Arvanitaki  to cosmology
Hindmarsh5 ; Gorghetto ; Blanco . Some of their characteristics such as
profiles and life-time have been studied, mostly numerically, in the
literature. Much less is known about their internal structure (see dissel  for
a recent publication). We will introduce radiation degrees of freedom in an
effective model for the $\phi^{4}$ oscillon. They are able to provide a decay
channel for oscillons below the critical amplitude. In addition, the
scattering modes, or more precisely, an effective version of them, are able to
describe qualitatively some features of the internal modes hosted by the
oscillon, including the decay into kink-antikink pairs.

This paper is organised as follows. In Sec. 2 we briefly review the $\phi^{4}$
model and its spectrum of perturbations. In Sec. 3 we introduce the radiation
modes as collective coordinates and analyse the radiation emitted by a
wobbling kink. After that, in Sec. 4 we explore some analytic solutions to the
lowest order in perturbation theory involving radiation. In Sec. 5 we
introduce the zero mode as a collective coordinate and study its interaction
with the rest of the modes. In Sec. 6 we extend our approach to describe
oscillons. Finally, Sec. 7 contains our conclusions and further comments. We
also add two appendices with some computational details.

##  2 The model and the linearised spectrum

The field theory model that we will discuss is the so-called $\phi^{4}$ model,
which is described by the following $(1+1)$ dimensional Lagrangian for a real
scalar field $\phi(x,t)$

$\mathcal{L}=\dfrac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-\dfrac{1}{2}(\phi^{2}-1)^{2}\,.$
(2.1)

The field equation for this model reads

$\square\phi+2\phi(\phi^{2}-1)=0\,.$ (2.2)

It can be shown by using a Bogomolnyi rearrangement that the static field
configurations also satisfy the following first order BPS equations

$\phi^{\prime}(x)\pm(\phi(x)^{2}-1)=0\,.$ (2.3)

In addition to vacuum solutions ($\phi(x)=\pm 1$), there are non-trivial
solutions interpolating the vacua which can be computed analytically

$\phi_{K(\bar{K})}(x)=\pm\tanh(x-x_{0})\,.$ (2.4)

The solution with positive sign is called kink ($K$), and the one with
negative sign is called antikink ($\bar{K}$). They depend on a free parameter,
$x_{0}$, which is interpreted as the position of the kink (antikink). There is
a topological charge associated to these solutions, $Q=\pm 1$, which is
conserved during time evolution. The perturbations around the kink (antikink)
do not depend on the position or the topological charge of the solution,
therefore we will consider a kink centred at the origin perturbed as follows

$\phi(x,t)=\phi_{K}(x)+\eta(x,t),$ (2.5)

with $\eta(x,t)=\eta(x)e^{i\omega t}$. Substituting (2.5) into (2.2), at
linear order, the field equation looks like

$-\eta^{\prime\prime}(x)+\left(6\phi_{K}(x)^{2}-2\right)\eta(x)=\omega^{2}\eta(x).$
(2.6)

The equation (2.6) can be considered a Sturm–Liouville differential equation,
where we can identify $U(x)=6\phi_{K}(x)^{2}-2$ with a Pöschl-Teller
potential. The spectrum of eigenstates and eigenvalues associated to this
Schrödinger-like equation is

$\displaystyle\eta_{0}(x)$ $\displaystyle=$
$\displaystyle\frac{\sqrt{3}}{2}\operatorname{sech}^{2}x,\qquad\omega_{0}=0,$
(2.7) $\displaystyle\eta_{s}(x)$ $\displaystyle=$
$\displaystyle\sqrt{\dfrac{3}{2}}\sinh
x\operatorname{sech}^{2}x,\qquad\omega_{s}=\sqrt{3},$ (2.8)
$\displaystyle\eta_{q}(x)$ $\displaystyle=$
$\displaystyle\dfrac{3\tanh^{2}x-q^{2}-1-3iq\tanh
x}{\sqrt{(q^{2}+1)(q^{2}+4)}}e^{iqx},\qquad\omega_{q}=\sqrt{q^{2}+4},$ (2.9)

with $q\in\mathbb{R}$. Altogether, the following relations are satisfied:

$\displaystyle\left\langle\eta_{0}(x),\eta_{s}(x)\right\rangle$
$\displaystyle=$
$\displaystyle\left\langle\eta_{0}(x),\eta_{q}(x)\right\rangle=\left\langle\eta_{s}(x),\eta_{q}(x)\right\rangle=0,$
(2.10)
$\displaystyle\left\langle\eta_{q}(x),\eta_{q^{\prime}}(x)\right\rangle$
$\displaystyle=$ $\displaystyle 2\pi\,\delta(q-q^{\prime}).$ (2.11)

We have chosen the normalization of the scattering modes such that
asymptotically they are plane waves of amplitude one. Moreover, the general
theory of the Sturm–Liouville systems ensures that the eigenstates of (2.6)
form a basis, so
$\mathcal{B}=\left\\{\eta_{0}(x),\eta_{s}(x),\eta_{q}(x)\right\\}$ may be used
to build a general configuration belonging to the linearised field
configuration space. As a consequence, a general field configuration close to
the kink solution can be expanded as follows

$\phi(x,t)=\phi_{K}(x)+c_{0}(t)\eta_{0}(x)+c_{s}(t)\eta_{s}(x)+\int_{\mathbb{R}}dq\,c_{q}(t)\eta_{q}(x).$
(2.12)

This natural assumption contains all possible degrees of freedom of the kink:
the zero mode $\eta_{0}(x)$ is responsible for the infinitesimal rigid
translation of the kink, the shape mode $\eta_{s}(x)$ is responsible for the
modification of the width of the kink, and the radiation modes (or scattering
states) $\eta_{q}(x)$ are related to the continuum of perturbative
fluctuations around the vacuum, that propagate freely to infinity. Such a
general ansatz will be used as the basis for the field configurations that we
will study throughout this work.

##  3 Leading radiation from the wobbling kink

In this section we study in detail the radiation emitted by a kink whose shape
mode is excited with a small amplitude. A similar analysis was performed in
Manton2 , albeit with a different approach, so we will use those results as a
first check of the validity of our guess for the dissipative modes. In order
to do that, let us assume that the kink is at rest at the origin so that we
can disregard the translational degree of freedom, $\eta_{0}(x)$. Hence, we
consider the following simplified ansatz

$\phi(x,t)=\phi_{K}(x)+c_{s}(t)\eta_{s}(x)+\int_{\mathbb{R}}dq\,c_{q}(t)\eta_{q}(x)\,.$
(3.1)

Since the shape mode solution is exact at linear order in perturbation theory,
we will assume that $c_{q}(t)\sim\mathcal{O}\left(c_{s}^{2}(t)\right)$.
Substituting (3.1) in (2.2) we obtain, at linear order in $c_{s}(t)$,

$\eta_{s}(x)\bigg{(}\ddot{c}_{s}(t)+\omega_{s}^{2}c_{s}(t)\bigg{)}=0\,.$ (3.2)

Consequently, the shape mode oscillates with frequency $\omega_{s}$, i.e.,

$\ddot{c}_{s}(t)+\omega_{s}^{2}c_{s}(t)=0\Rightarrow
c_{s}(t)=A_{0}\cos(\omega_{s}t)\,.$ (3.3)

Since this solution solves exactly the equation at this order, we may conclude
that there is no source for radiation at linear order in the shape mode
amplitude. At second order in $c_{s}(t)$ we get

$\eta_{s}(x)\bigg{(}\ddot{c}_{s}(t)+\omega_{s}^{2}c_{s}(t)\bigg{)}+\int_{\mathbb{R}}dq\,\eta_{q}(x)\bigg{(}\ddot{c}_{q}(t)+\omega_{q}^{2}c_{q}(t)\bigg{)}+6c_{s}^{2}(t)\phi_{K}(x)\eta_{s}^{2}(x)=0\,.$
(3.4)

Projecting onto $\eta_{s}(x)$ and assuming the orthogonality relations (2.10),
equation (3.4) reduces to

$\ddot{c}_{s}(t)+\omega_{s}^{2}c_{s}(t)+\frac{9\pi\sqrt{6}}{32}\,c_{s}^{2}(t)=0\,.$
(3.5)

This is the equation of an anharmonic oscillator corrected by a quadratic
term. If we now project onto $\eta^{\ast}_{q^{\prime}}(x)$ and assume again
the orthogonality relations (2.10)-(2.11), the equation (3.4) reads as

$\ddot{c}_{q}(t)+\omega_{q}^{2}c_{q}(t)+\dfrac{3}{\pi}c_{s}^{2}(t)\int_{\mathbb{R}}dx\,\eta^{\ast}_{q}(x)\phi_{K}(x)\eta_{s}^{2}(x)=0\,,$
(3.6)

where the last term can be interpreted as the overlap between the scattering
state of frequency $\omega_{q}$ with the combination
$\phi_{k}(x)\eta_{s}^{2}$. Such a term can be computed exactly:

$\mathcal{F}(q)=\int_{\mathbb{R}}dx\,\eta^{\ast}_{q}(x)\phi_{K}(x)\eta_{s}^{2}(x)=-\dfrac{i\pi}{32}\sqrt{\dfrac{q^{2}+4}{q^{2}+1}}\dfrac{q^{2}(q^{2}-2)}{\sinh\left(\pi
q/2\right)}\,.$ (3.7)

Therefore, (3.6) looks like

$\ddot{c}_{q}(t)+\omega_{q}^{2}c_{q}(t)+\dfrac{3}{\pi}c_{s}^{2}(t)\mathcal{F}(q)=0\,.$
(3.8)

Let us assume that the amplitude of the shape mode $c_{s}(t)$ is given by its
linear approximation, i.e., $c_{s}(t)=A_{0}\cos(\omega_{s}t)$. We are
considering that the shape mode is the only source of radiation, so in the
absence of the shape mode there is no radiation. We will take this into
account imposing the initial conditions $c_{q}(0)=0$ and $\dot{c}_{q}(0)=0$.
With this choice, the general solution of (3.8) takes the form

$c_{q}(t)=-\dfrac{3}{2\pi}\dfrac{(4\omega_{s}^{2}-\omega_{q}^{2})-\omega_{q}^{2}\cos(2\omega_{s}t)-(4\omega_{s}^{2}-2\omega_{q}^{2})\cos(\omega_{q}t)}{\omega_{q}^{2}(4\omega_{s}^{2}-\omega_{q}^{2})}\mathcal{F}(q)\,.$
(3.9)

This expression provides time-dependent amplitudes to radiation modes. As a
consequence, the radiation emitted by an oscillating kink is

$R(x,t)=\int_{\mathbb{R}}\,dq\,c_{q}(t)\eta_{q}(x),$ (3.10)

with $c_{q}(t)$ given by (3.9). This is the exact form of the radiation at
leading order for a static wobbling kink valid at all distances and times.
Notice that the structure of $\mathcal{F}(q)$ indicates that some frequencies
are suppressed in the radiation. It has a maximum at $q\approx 2\sqrt{2}$,
which is consistent with the fact that the shape mode is the quadratic source
for radiation. Moreover, it can be shown analytically that at large distances
all frequencies but $q=2\sqrt{2}$ are suppressed. A non-trivial calculation
(see Appendix A for details) leads to the following expression for radiation
at infinity

$R_{\infty}(x,t)=\dfrac{3\,\pi
A_{0}^{2}}{2\sinh(\sqrt{2}\pi)}\sqrt{\frac{3}{8}}\cos\big{(}2\sqrt{3}t-2\sqrt{2}x-\delta\big{)}.$
(3.11)

This expression agrees with the one obtained in Manton2 . Following Manton2 ,
the decay of the amplitude of the shape mode into radiation can be determined
directly from (3.11). Taking into account that the average energy flux carried
by the wave (3.11) has to be equal to the rate of change of the energy of the
excited kink (for details see Manton2 ), we arrive at

$\displaystyle A(t)=\dfrac{1}{\sqrt{A_{0}^{-2}+0.03\,t}}\,,$ (3.12)

where $A_{0}$ is the initial amplitude of the shape mode. Notice that in this
approximation there is no back-reaction of the radiation into the shape mode.
Therefore, strictly speaking, the shape mode oscillates harmonically with
frequency $\omega_{s}$ and constant amplitude. In addition, this approximation
suggests that it is possible to excite resonantly the shape mode with
radiation of the appropriate frequency. In order to obtain these results, we
have to consider the shape mode amplitude as a free collective coordinate
interacting with the radiation coordinates. We will study these issues in
detail in the next section.

##  4 Interaction of radiation and shape mode

In this section we will derive, within the collective coordinate approach, the
effective Lagrangian involving the shape mode coupled to radiation, i.e.,
$c_{s}(t)$ and $c_{q}(t)$ will be consider genuine collective coordinates. In
order to achieve that, let us suppose a perturbation of the kink solution of
the form (3.1), where $c_{s}(t)\sim\mathcal{O}(A_{0})$ and
$c_{q}(t)\sim\mathcal{O}(A_{0}^{2})$ with $\mathinner{\\!\left\lvert
A_{0}\right\rvert}\ll 1$. Substituting (3.1) into (2.1) we obtain

$\displaystyle\mathcal{L}_{s,q}$ $\displaystyle=$
$\displaystyle-\dfrac{4}{3}+\dfrac{1}{2}\left(\dot{c}_{s}^{2}(t)-\omega_{s}^{2}c_{s}^{2}(t)\right)+\pi\int_{\mathbb{R}}dq\,\left(\dot{c}_{q}(t)\dot{c}_{-q}(t)-\omega_{q}^{2}c_{q}(t)c_{-q}(t)\right)$
(4.1)
$\displaystyle-\int_{\mathbb{R}}dx\,\bigg{(}2\phi_{K}(x)c_{s}^{3}(t)\eta_{s}^{3}(x)+6\phi_{K}(x)c_{s}^{2}(t)\eta_{s}^{2}(x)\int_{\mathbb{R}}dq\,c_{q}(t)\eta_{q}(x)\bigg{)}$
$\displaystyle-\int_{\mathbb{R}}dx\bigg{(}6\phi_{K}(x)c_{s}(t)\eta_{s}(x)\int_{\mathbb{R}^{2}}dqdq^{\prime}c_{q}(t)c_{q^{\prime}}(t)\eta_{q}(x)\eta_{q^{\prime}}(x)\bigg{)},$

where the normalization of the shape mode and the delta-normalization relation
(2.11) were taken into account. Integrating in the $x-$variable we get

$\displaystyle\int_{\mathbb{R}}dx\,\phi_{K}(x)\,\eta_{s}(x)^{3}=\dfrac{3}{32}\sqrt{\dfrac{3}{2}}\pi\,,$
(4.2)
$\displaystyle\int_{\mathbb{R}}dx\,\phi_{K}(x)\,\eta_{s}^{2}(x)\,\eta_{q}(x)=\dfrac{i\pi}{32}\sqrt{\dfrac{q^{2}+4}{q^{2}+1}}\dfrac{q^{2}(q^{2}-2)}{\sinh\left(\pi
q/2\right)}\,,$ (4.3)
$\displaystyle\int_{\mathbb{R}}dx\,\phi_{K}(x)\eta_{s}(x)\eta_{q}(x)\eta_{q^{\prime}}(x)=$
$\displaystyle\frac{\pi}{16}\sqrt{\frac{3}{2}}\dfrac{17+17q^{2}+17q^{\prime
2}+10q^{2}q^{\prime 2}-q^{4}-q^{\prime 4}+q^{2}q^{\prime 4}+q^{4}q^{\prime
2}-q^{6}-q^{\prime 6}}{\sqrt{(q^{2}+1)(q^{2}+4)}\sqrt{(q^{\prime
2}+1)(q^{\prime 2}+4)}\cosh(\frac{\pi}{2}(q+q^{\prime}))}\,.$ (4.4)

One finally obtains the following expression

$\displaystyle\mathcal{L}_{s,q}$ $\displaystyle=$
$\displaystyle\dfrac{1}{2}\bigg{(}\dot{c}_{s}^{2}(t)-\omega_{s}^{2}c_{s}^{2}(t)\bigg{)}+\pi\int_{\mathbb{R}}dq\,\left(\dot{c}_{q}(t)\dot{c}_{-q}(t)-\omega_{q}^{2}c_{q}(t)c_{-q}(t)\right)-\dfrac{3\pi}{16}\sqrt{\dfrac{3}{2}}c_{s}^{3}(t)$
(4.5) $\displaystyle-\dfrac{3\pi
i}{16}\int_{\mathbb{R}}\,dq\,\sqrt{\dfrac{q^{2}+4}{q^{2}+1}}\dfrac{q^{2}(q^{2}-2)}{\sinh(\pi
q/2)}c_{s}^{2}(t)c_{q}(t)+c_{s}(t)\int_{\mathbb{R}^{2}}dqdq^{\prime}\,f_{sq}(q,q^{\prime})c_{q}(t)c_{q^{\prime}}(t)\,,$

where we have removed from (4.1) a constant term (kink rest energy with
opposite sign) since it does not contribute to the field equations, and
$f_{sq}(q,q^{\prime})$ can be read from (4.4). The Lagrangian (4.5) describes
a system of harmonic oscillators coupled by the last two terms. The equations
of motion governing the evolution of $c_{q}(t)$ and $c_{s}(t)$ are yielded by

$\displaystyle\ddot{c}_{-q}(t)$ $\displaystyle+$
$\displaystyle\omega_{q}^{2}\,c_{-q}(t)+\dfrac{3i}{32}\sqrt{\dfrac{q^{2}+4}{q^{2}+1}}\dfrac{q^{2}(q^{2}-2)}{\sinh\left(\pi
q/2\right)}c_{s}^{2}(t)-\dfrac{1}{\pi}c_{s}(t)\int
dq^{\prime}f_{sq}(q,q^{\prime})c_{q^{\prime}}(t)=0,$ (4.6)
$\displaystyle\ddot{c}_{s}(t)$ $\displaystyle+$
$\displaystyle\omega_{s}^{2}c_{s}(t)+\dfrac{9\pi}{16}\sqrt{\dfrac{3}{2}}c_{s}^{2}(t)+\dfrac{3\pi}{8}i\int_{\mathbb{R}}dq\,\sqrt{\dfrac{q^{2}+4}{q^{2}+1}}\dfrac{q^{2}(q^{2}-2)}{\sinh(\pi
q/2)}c_{s}(t)c_{q}(t)$ (4.7) $\displaystyle-$
$\displaystyle\int_{\mathbb{R}^{2}}dqdq^{\prime}\,f_{sq}(q,q^{\prime})c_{q}(t)c_{q^{\prime}}(t)=0.$

This coupled system has a straightforward interpretation: Eq. (4.6) is a
forced harmonic oscillator of fundamental frequency $\omega_{q}$ with external
force proportional to $c_{s}^{2}(t)$ (notice that this equation was derived in
the previous section, but now we allow for back-reaction of radiation in the
shape mode). Regarding Eq. (4.7), its structure is more involved. It describes
an anharmonic oscillator of fundamental frequency $\omega_{s}$ coupled
linearly to the $c_{q}(t)$. Note that, for sufficiently small amplitudes of
$c_{s}(t)$, this expression reduces to the well-known Hill’s equation provided
that $c_{q}(t)$ is periodic.

In order to solve the system (4.6)-(4.7) numerically, we have to choose a
discretization in $q$, i.e., we have to select $N$ scattering modes labelled
by $q_{i}$ and solve the coupled system of $N+1$ ordinary differential
equations. Note that the discretization of the integrals in $q$ gives rise a
Hamiltonian system that conserves energy. Then, the discretization fixes a
time cut-off of order $t_{c}=1/\Delta q$ (where $\Delta q=q_{i}-q_{i-1}$),
which implies that for times bigger than $t_{c}$ our computations are no
longer trustable. We will see in Sec. 6 that there are other effective ways to
mimic the effect of dissipative degrees of freedom.

In our first numerical experiment we will show that the Lagrangian (4.5)
describes accurately the decay of the shape mode for $t\lesssim t_{c}$. We
will choose the following initial conditions (IC)

$c_{s}(0)=A_{0}\,,\quad c_{s}^{\prime}(0)=0\,,\quad
c_{q}(0)=0\,,\quad\text{and}\quad c_{q}^{\prime}(0)=0\,.$ (4.8)

These IC describe a kink with the shape mode excited with an initial amplitude
of $A_{0}$ and no radiation. Due to the addition of the dissipative degrees of
freedom (radiation modes), the effective system (4.6)-(4.7) naturally allows
the decay of the shape mode. In Fig. 1 we compare the decay of the shape mode
obtained from field theory with the solution of the effective system
(4.6)-(4.7). Notice that, the bigger the number of scattering modes added to
the effective model, the better the fit to field theory. We see that, for
times $t_{c}<1/\Delta q$ and number of scattering modes $N>5$, both solutions
agree with great accuracy.

Figure 1: Decay of the shape mode in field theory (solid line) and in the
effective model (4.5) (dashed line) compared to the analytical decay law given
by (3.12) (dotted-dashed line). For the computation we have chosen $n=20$
equidistant scattering modes in the interval $q\in[-3,3]$.

As we have learnt from the previous section, the most relevant scattering
modes to describe the decay should have frequencies close to
$\omega=2\omega_{s}$, since these modes carry the energy to infinity. In terms
of the system (4.6)-(4.7) this has a simple explanation: if only modes with
frequency $\omega$ far from $2\omega_{s}$ are allowed, the forced harmonic
oscillator equation (4.6) never enters in the resonant region
$\omega=2\omega_{s}$, and as a result the scattering modes excited are not
able to carry enough energy from the shape mode. For small shape mode
amplitudes and radiation frequencies close to $2\omega_{s}$ the equation (4.7)
enters in the unstable region, and the amplitude of the shape mode grows
exponentially. Assuming a monochromatic wave of the form (4.16), the equation
(4.7) reduces to

$\displaystyle\ddot{c}_{s}(t)+\left(\omega_{s}^{2}+f(q_{0})\sin(\omega_{q_{0}}t)\right)c_{s}(t)=0\,,$
(4.9)

with

$\displaystyle f(q_{0})=-\frac{3\pi
A_{q_{0}}}{4}\dfrac{q_{0}^{2}(q_{0}^{2}-2)}{\sinh\left(\pi
q_{0}/2\right)}\sqrt{\dfrac{q_{0}^{2}+4}{q_{0}^{2}+1}}\,.$ (4.10)

The equation (4.9) constitutes a Mathieu equation. By means of the general
theory of Mathieu equations we can compute the bands of instability. For small
$A_{q_{0}}$ they satisfy the relation $\omega_{s}/\sqrt{q_{0}^{2}+4}=k/2$,
where $k\in\mathbb{Z}$. For $k=1$ the instability appears when the frequency
of the radiation is twice the frequency of the shape mode. Hence, the
radiation triggers resonantly the shape mode and one should expect an
exponential amplification of its amplitude. Due to energy conservation, as
$c_{s}(t)$ grows, the third term in (4.6) transfers energy to the radiation
modes and the exponential growth stops. Notice that this resonant transfer
mechanism between internal modes has been also observed in two and three-
dimensional solitons q1 ; q2 .

Figure 2: Frequency spectrum of the radiation emitted by a wobbling kink
obtained from (4.6)-(4.7). We have taken $n=40$ equidistant scattering modes
in the interval $q\in[-4,4]$, time of simulation $t=t_{c}$, and the initial
values of the shape mode $c_{s}(0)=0.03,\,0.05$ and $0.07$ (solid, dashed and
dotted-dashed line respec.). It can be appreciated that the maxima take place
approximately at $\omega_{q}=2\omega_{s}$ for small shape mode amplitudes.

In our second numerical experiment we will explore the excitation of the shape
mode when the kink is illuminated with radiation of frequency $\omega_{q}$. We
choose the following IC

$\displaystyle c_{s}(0)=0\,,\quad c_{s}^{\prime}(0)=0\,,$ (4.11)
$\displaystyle c_{q}(0)=0\,,\quad\dot{c}_{q}(0)=0\,,\,\,\text{for}\,\,q\neq
q_{0},-q_{0}\,,$ (4.12) $\displaystyle
c_{q}(0)=A_{q}\,,\quad\dot{c}_{q}(0)=iA_{q}\omega_{q},\,\,\text{for}\,\,q=q_{0},$
(4.13) $\displaystyle
c_{q}(0)=A_{q}\,,\quad\dot{c}_{q}(0)=-iA_{q}\omega_{q},\,\,\text{for}\,\,q=-q_{0}\,.$
(4.14)

In linear theory, the IC (4.13) and (4.14) correspond to the solution

$c_{q}(t)=A_{q}e^{i\omega_{q}t}\delta(q-q_{0})+A_{q}e^{-i\omega_{q}t}\delta(q+q_{0})\,.$
(4.15)

This choice describes a superposition of a kink with a combination of
scattering modes of frequency $\omega_{q_{0}}$. Initially, the radiation has
the form

$\displaystyle\int_{\mathbb{R}}dq\,c_{q}(t)\eta_{q}(x)$ $\displaystyle=$
$\displaystyle\frac{A_{q_{0}}}{\sqrt{(q_{0}^{2}+1)(q_{0}^{2}+4)}}\left[6\tanh(x)(\cos(\omega_{0}t+q_{0}x)\tanh(x)+q_{0}\sin(\omega_{0}t+q_{0}x))\right.$
(4.16) $\displaystyle\left.-2(1+q_{0}^{2})\cos(\omega_{0}t+q_{0}x)\right].$

Notice that, even though asymptotically (4.16) is a plane wave of frequency
$\omega_{0}$, close to the origin it gets distorted by the presence of the
kink. In Fig. 3 we compare the excitation of the shape mode in field theory
with the Eqs. (4.6)-(4.7) with IC (4.11)-(4.14).

(a) $q=2.0$ (b) $q=3.0$

Figure 3: Shape mode excitation by external radiation of different frequencies
and amplitude $A_{q}=0.02$ in field theory (solid line) and in the effective
theory (dashed line) (4.5). We have taken into account $n=30$ equidistant
scattering modes in the interval $q\in[-3,3]$.

Quite remarkably, we can obtain an approximate analytical solution for the
shape mode in the background of radiation for frequencies different from the
resonance frequency. For small radiation amplitudes the relevant terms in Eq.
(4.7) are the harmonic oscillator part and the last term (radiation source).
This equation has the form of a forced harmonic oscillator. Regarding the
initial conditions given by (4.11)-(4.14) we get

$c_{s}(t)=A_{q_{0}}^{2}\Omega(q_{0})\left(\dfrac{1}{\omega_{s}^{2}}+\dfrac{\left(4\omega_{q_{0}}^{2}-(\operatorname{sech}(\pi
q_{0})+1)\omega_{s}^{2}\right)\cos\left(t\omega_{s}\right)}{\omega_{s}^{2}\left(\omega_{s}^{2}-4\omega_{q_{0}}^{2}\right)}+\dfrac{\operatorname{sech}(\pi
q_{0})\cos\left(2t\omega_{q_{0}}\right)}{\omega_{s}^{2}-4\omega_{q_{0}}^{2}}\right),$
(4.17)

where

$\Omega(q_{0})=-\dfrac{3\sqrt{\dfrac{3}{2}}\pi\left(8q_{0}^{4}+34q_{0}^{2}+17\right)}{4\left(q_{0}^{4}+5q_{0}^{2}+4\right)}\,.$
(4.18)

For larger radiation amplitudes this expression is not valid anymore, and new
phenomena may appear (see for example Tom2  for $K\bar{K}$ creation). A
comparison between the approximate analytical solution and the field theory
results can be found in Fig. 4. There we can appreciate the accuracy between
both results. Indeed, this analytic expression allows us to explain the
negative amplitude excitation of the shape mode.

(a) $q=2.0$ (b) $q=3.0$

Figure 4: Shape mode excitation by external radiation of different frequencies
and amplitude $A_{q}=0.02$ in field theory (solid line) and with the
approximate solution (dashed line) (4.17). We have taken into account $n=30$
equidistant scattering modes in the interval $q\in[-3,3]$.

So far we have not taken into account the translational degree of freedom in
our model. However, the translational mode can interact with radiation and
produce unexpected phenomena like the negative radiation pressure Roman . We
will explore some features of the interaction of the translational mode in our
approach in the following section.

##  5 Adding the translational mode

In this section we aim to generalise the previous approach allowing for
translations of the kink. This new case will give rise to a richer structure
and will allow for new couplings among the different collective coordinates.
Let us now consider a configuration of the form

$\phi(x,t)=\phi_{K}\big{(}x-a(t)\big{)}+c_{s}(t)\eta_{s}\big{(}x-a(t)\big{)}+\int_{\mathbb{R}}dq\,c_{q}(t)\eta_{q}\big{(}x-a(t)\big{)},$
(5.1)

where $a(t)$ is the collective coordinate that describes the translation of
the kink (or the kink zero mode). Once more, we have to substitute the field
configuration ansatz into the Lagrangian density of the full theory (2.1).
Notice that (5.1) only adds new contributions to the kinetic part with respect
to (3.1), whereas the potential terms remain the same. The additional
contributions to the effective Lagrangian due to the presence of the zero mode
are given by

$\displaystyle\mathcal{L}_{t}$ $\displaystyle=$
$\displaystyle\dfrac{1}{2}\int_{\mathbb{R}}dx\,\bigg{(}\dot{a}^{2}(t)\bigg{(}\int_{\mathbb{R}}dq\,dq^{\prime}\,c_{q}(t)c_{q^{\prime}}(t)\eta^{\prime}_{q}(x)\eta^{\prime}_{q^{\prime}}(x)+\phi^{\prime}\hskip
0.28436pt_{K}{}^{2}(x)+c_{s}^{2}(t)\eta^{\prime}\hskip 0.28436pt_{s}{}^{2}(x)$
(5.2)
$\displaystyle\\!\\!+2\phi^{\prime}_{K}(x)c_{s}(t)\eta^{\prime}_{s}(x)+2\phi^{\prime}_{K}(x)\int_{\mathbb{R}}dq\,c_{q}(t)\eta^{\prime}_{q}(x)+2c_{s}(t)\eta^{\prime}_{s}(x)\int_{\mathbb{R}}dq\,c_{q}(t)\eta^{\prime}_{q}(x)\bigg{)}$
$\displaystyle\\!\\!-\dot{a}(t)\bigg{(}2\dot{c}_{s}(t)\eta_{s}(x)\int_{\mathbb{R}}dq\,c_{q}(t)\eta^{\prime}_{q}(x)+2c_{s}(t)\eta^{\prime}_{s}(x)\int_{\mathbb{R}}dq\,\dot{c}_{q}(t)\eta_{q}(x)$
$\displaystyle\\!\\!+2\int_{\mathbb{R}}\,dq\,dq^{\prime}\,c_{q^{\prime}}(t)\dot{c}_{q}(t)\eta_{q}(x)\eta^{\prime}_{q^{\prime}}(x)\bigg{)}\bigg{)}.$

Integrating in the $x-$coordinate, adding $\mathcal{L}_{s,q}$, and assuming
$\mathinner{\\!\left\lvert\dot{a}(t)\right\rvert}\ll 1$, we finally get the
following effective Lagrangian

$\displaystyle\mathcal{L}_{s,q,t}$ $\displaystyle=$
$\displaystyle\dfrac{1}{2}\left(\dot{c}^{2}_{s}(t)-\omega_{s}^{2}c_{s}^{2}(t)\right)+\pi\int
dq\left(\dot{c}_{q}(t)\dot{c}_{-q}(t)-\omega_{q}^{2}c_{q}(t)c_{-q}(t)\right)+c_{s}^{2}(t)\int
dqf_{s}(q)c_{q}(t)$ (5.3) $\displaystyle+$ $\displaystyle c_{s}(t)\int
dqdq^{\prime}f_{sq}(q,q^{\prime})c_{q}(t)c_{q^{\prime}}(t)+\dfrac{2}{3}\dot{a}^{2}(t)+\dfrac{\pi}{4}\sqrt{\dfrac{3}{2}}\dot{a}^{2}(t)c_{s}(t)+\dot{a}^{2}(t)\int
dqf_{aa}(q)c_{q}(t)$ $\displaystyle+$ $\displaystyle\dot{a}(t)\int
dqf_{as}(q)\left(\dot{c}_{s}(t)c_{q}(t)-c_{s}(t)\dot{c}_{q}(t)\right)+\dot{a}(t)\int
dqdq^{\prime}f_{a}(q,q^{\prime})\dot{c}_{q}c_{q^{\prime}}(t).$

The couplings between modes are collected in the following functions

$\displaystyle f_{s}(q)$ $\displaystyle=$
$\displaystyle-\frac{3i\pi}{16}\sqrt{\frac{q^{2}+4}{q^{2}+1}}\dfrac{q^{2}(q^{2}-2)}{\sinh\left(\pi
q/2\right)},$ (5.4) $\displaystyle f_{as}(q)$ $\displaystyle=$
$\displaystyle-\dfrac{\pi}{4}\sqrt{\frac{3}{2}}\sqrt{\frac{q^{2}+1}{q^{2}+4}}\dfrac{q^{2}+3}{\cosh\left(\frac{\pi
q}{2}\right)}\,,$ (5.5) $\displaystyle f_{aa}(q)$ $\displaystyle=$
$\displaystyle-i\dfrac{\pi}{4}\sqrt{\dfrac{q^{2}+4}{q^{2}+1}}\dfrac{q^{2}}{\sinh\left(\pi
q/2\right)}\,,$ (5.6) $\displaystyle f_{a}(q,q^{\prime})$ $\displaystyle=$
$\displaystyle\dfrac{3i\pi}{4}\frac{\left(4+q^{2}+q^{\prime
2}\right)}{\sqrt{(q^{2}+1)(q^{2}+4)}\sqrt{(q^{\prime 2}+1)(q^{\prime
2}+4)}}\dfrac{q^{2}-q^{\prime 2}}{\sinh(\dfrac{\pi}{2}(q+q^{\prime}))}$ (5.7)
$\displaystyle-\dfrac{2i\pi q^{\prime}\left(4-9qq^{\prime}-2q^{2}-2q^{\prime
2}+q^{2}q^{\prime 2}\right)}{\sqrt{(q^{2}+1)(q^{2}+4)}\sqrt{(q^{\prime
2}+1)(q^{\prime 2}+4)}}\delta(q+q^{\prime})\,,$ $\displaystyle
f_{sq}(q,q^{\prime})$ $\displaystyle=$
$\displaystyle-\frac{3\pi}{8}\sqrt{\frac{3}{2}}\frac{17+17q^{2}+17q^{\prime
2}+10q^{2}q^{\prime 2}-q^{4}-q^{\prime 4}+q^{2}q^{\prime 4}+q^{4}q^{\prime
2}-q^{6}-q^{\prime 6}}{\sqrt{(q^{2}+1)(q^{2}+4)}\sqrt{(q^{\prime
2}+1)(q^{\prime 2}+4)}\cosh(\frac{\pi}{2}(q+q^{\prime}))}.$ (5.8)

The quadratic terms in (5.3) are just a collection of harmonic oscillators
plus the kinetic term of the kink. On the other hand, the cubic terms describe
the interaction between the modes. The field equations associated to (5.3) for
the radiation modes are yielded by

$\displaystyle\ddot{c}_{-q}(t)$ $\displaystyle+$
$\displaystyle\omega_{q}^{2}c_{-q}(t)-\dfrac{1}{2\pi}c^{2}_{s}(t)f_{s}(q)-\dfrac{1}{\pi}c_{s}(t)\int
dq^{\prime}f_{sq}(q,q^{\prime})c_{q^{\prime}}(t)-\dfrac{1}{2\pi}\dot{a}^{2}(t)f_{aa}(q)$
(5.9) $\displaystyle-$
$\displaystyle\dfrac{1}{2\pi}\ddot{a}(t)f_{as}(q)c_{s}(t)-\dfrac{1}{\pi}\dot{a}(t)f_{as}(q)\dot{c}_{s}(t)+\dfrac{1}{2\pi}\dot{a}(t)\int
dq^{\prime}\dot{c}_{q^{\prime}}(t)\left(f_{a}(q,q^{\prime})-f_{a}(q^{\prime},q)\right)$
$\displaystyle+$ $\displaystyle\dfrac{1}{2\pi}\ddot{a}(t)\int
dq^{\prime}f_{a}(q,q^{\prime})c_{q^{\prime}}(t)=0,$

whilst the equations of motion determining the evolution of $c_{s}(t)$ and
$a(t)$ take the form

$\displaystyle\ddot{c}_{s}(t)$ $\displaystyle+$
$\displaystyle\omega_{s}^{2}c_{s}(t)-2c_{s}(t)\int dqf_{s}(q)c_{q}(t)-\int
dqdq^{\prime}f_{sq}(q,q^{\prime})c_{q}(t)c_{q^{\prime}}(t)-\frac{\pi}{4}\sqrt{\frac{3}{2}}\dot{a}^{2}(t)$
(5.10) $\displaystyle+$ $\displaystyle 2\dot{a}(t)\int
dqf_{as}(q)\dot{c}_{q}(t)+\ddot{a}(t)\int dqf_{as}(q)c_{q}(t)=0\,,$

and

$\displaystyle\frac{4}{3}\ddot{a}(t)$ $\displaystyle+$
$\displaystyle\dfrac{\pi}{2}\sqrt{\dfrac{3}{2}}\left(\ddot{a}(t)c_{s}(t)+\dot{a}(t)\dot{c}_{s}(t)\right)+2\int
dqf_{aa}(q)\left(\ddot{a}(t)c_{q}(t)+\dot{a}(t)\dot{c}_{q}(t)\right)$ (5.11)
$\displaystyle+$ $\displaystyle\int
dqf_{as}(q)\left(\ddot{c}_{s}(t)c_{q}(t)-c_{s}(t)\ddot{c}_{q}(t)\right)+\frac{d}{dt}\int
dqdq^{\prime}f_{a}(q,q^{\prime})\dot{c}_{q}c_{q^{\prime}}(t)=0\,.$

The first configuration we will discuss is the translating kink without any
excitation. In the non-relativistic approach this corresponds to
$a(t)=x_{0}+vt$, while the rest of the modes vanish. However, a simple
inspection of the system (5.9)-(5.11) reveals something surprising at first
glance: if $\dot{a}(t)\neq 0$, the terms proportional to $\dot{a}(t)^{2}$ in
(5.9) and (5.10) act as sources for the shape mode and radiation. This seems
contradictory since, due to the Lorentz invariance of the model, the boosted
kink is an exact solution. The Lorentz boosted version of the kink has the
following form

$\phi(x,t)=\tanh\left(\frac{x-vt}{\sqrt{1-v^{2}}}\right).$ (5.12)

Of course, (5.12) does not contain any radiation. This apparent pathology of
the effective model can be easily resolved. The standard CCM approach is not
Lorentz invariant at first order. This is obvious from the asymmetry between
time and spatial coordinates in the ansatz (5.1). However, once we consider
higher order terms, approximate Lorentz invariant solutions exist. Actually,
the mentioned source terms for $c_{s}(t)$ and $c_{q}(t)$ are responsible for
the Lorentz contraction of a moving kink. Let us consider a moving kink with
velocity $v<\\!\\!<1$ located at the origin, then initially $\dot{a}(t)=v$.
Assuming that $c_{s}(t)$ and $c_{q}(t)$ do not depend on time, an approximate
solution of (5.10) and (5.9) is given by

$c_{s}(t)=\frac{\pi}{4\sqrt{6}}v^{2},\quad
c_{q}(t)=-\frac{iq^{2}\operatorname{csch}(\pi
q/2)}{8\sqrt{(q^{2}+1)(q^{2}+4)}}v^{2},$ (5.13)

where we disregard corrections of order $\mathcal{O}(v^{4})$. On the other
hand, if we expand (5.12) with respect to $v$, at $t=0$ we get

$\phi(0,x)=\tanh(x)+\frac{1}{2}\left(x-x\tanh^{2}(x)\right)v^{2}+\mathcal{O}(v^{4}).$
(5.14)

We denote the first correction to the Lorentz contraction by
$\phi^{(1)}(x)=\frac{1}{2}\left(x-x\tanh^{2}(x)\right)v^{2}$. The projection
of $\phi^{(1)}(x)$ onto the spectral modes gives

$\displaystyle\langle\phi^{(1)}(x),\eta_{s}(x)\rangle$ $\displaystyle=$
$\displaystyle\frac{\pi}{4\sqrt{6}}v^{2}\,,$ (5.15)
$\displaystyle\langle\phi^{(1)}(x),\eta_{q}(x)\rangle$ $\displaystyle=$
$\displaystyle-\frac{i\pi q^{2}\operatorname{csch}(\pi
q/2)}{4\sqrt{(q^{2}+1)(q^{2}+4)}}v^{2}\,.$ (5.16)

Note that there is an extra $2\pi$ factor due to the normalization of the
scattering modes. This is easily interpreted: this effective model already
describes relativistic effects at this order. The solution (5.13) can be
interpreted as a first order Lorentz boost, i.e., in terms of the coordinates
of our model the Lorentz boosted kink takes the form

$\displaystyle\phi_{B}(x,t)$ $\displaystyle=$
$\displaystyle\tanh(x-vt)+\phi^{(1)}(x-vt)+\mathcal{O}(v^{4})=$ (5.17)
$\displaystyle=$
$\displaystyle\tanh(x-vt)+v^{2}\frac{\pi}{4\sqrt{6}}\eta_{s}(x-vt)-v^{2}\int
dq\frac{iq^{2}\operatorname{csch}(\pi
q/2)}{8\sqrt{(q^{2}+1)(q^{2}+4)}}\eta_{q}(x-vt),$

for $x_{0}=0$. If one is interested in the description of relativistic
processes, this approach does not seem appropriate since, even to describe a
simple boosted solution, one needs to excite a large number of modes. Still,
we would like to emphasise that, already at quadratic order, the “non-
relativistic” CCM approach describes relativistic effects, although these
effects require in general, the excitation of scattering modes. There is a
simple ansatz that describes the exact Lorentz boost with only two degrees of
freedom given by the following expression

$\phi(x,t)=\phi_{K}\left(\frac{x-a(t)}{\delta(t)}\right)\,.$ (5.18)

This type of ansatz was first considered in Rice  (for recent discussions see
Relativistic ). However, after the introduction of the scattering modes, it
seems not possible to obtain analytical results, thus we will not pursue this
line in this section. However, as we will see in Sec. 6, some small
generalizations of (5.18) may be used to describe effectively dissipative
degrees of freedom.

In our second experiment we will study the excitation of the translational
mode by radiation. Notice that the first and last term of (5.11) resemble the
Newton equation, where we can identify $m_{K}=4/3$. However, for the IC
representing a kink at rest illuminated by linear radiation (4.16), the rate
of change of the total momentum of radiation given by the last term in (5.11)
vanishes, so the kink remains at rest. As a matter of fact, it is well-known
that the $\phi^{4}$ kink is transparent to linear radiation as it was
emphasised in Roman , and that the negative radiation pressure appears at
fourth order in perturbation theory. Nevertheless, the shape mode may act as
an intermediary between the zero mode and radiation. Namely, a non-trivial
interplay between the modes could be as follows: the radiation triggers the
shape mode due to the term proportional $f_{sq}(q,q^{\prime})$ in (5.10). Once
$c_{s}(t)$ is excited, there is a source for $a(t)$ in (5.11) given by the
term proportional to $f_{as}(q)$. The excitation of $a(t)$ is roughly of order
$A_{0}A_{q}$, where $A_{0}$ is the amplitude of the shape mode and $A_{q}$ is
the amplitude of the $q-$scattering mode. Since from (4.17) $A_{0}$ is of
order $A_{q}^{2}$, the excitation of the zero mode due to this mechanism
appears at order $\mathcal{O}(A_{q}^{3})$. However, contrary to the negative
radiation pressure effect, there is no net momentum transfer at this order and
the kink simply oscillates about its rest position. This can be seen easily
from (5.11) disregarding the last term and keeping terms of
$\mathcal{O}(A_{q}^{3})$ order. In Fig. 5 we show the maximum amplitude of the
zero mode as a function of the incident radiation frequency. This shows that
for small $q$ the term proportional to $f_{a}$ is indeed subleading.

Figure 5: Maximum displacement of the kink due to radiation of different
frequencies and amplitude $A_{q}=0.05$. The solid line represents the complete
effective model, the dashed line represents the simulation when only the
$f_{as}$ source is considered, and the dotted dashed represents the simulation
when only the source $f_{a}$ is taken into account. We have taken into account
$n=30$ equidistant scattering modes in the interval $q\in[-5,5]$. The time of
simulation is $t=t_{c}$.

For larger times, higher order corrections to radiation start to play a role
and the last term in (5.11) cannot be neglected anymore. This term transfers a
net momentum to the kink Roman  and it is pulled back in the opposite
direction of the incoming radiation.

So far, we have discussed only the single-kink sector. Of course, one expects
that the combination of translational and scattering modes should play an
important role in the $K\bar{K}$ sector, allowing for energy dissipation in
scattering processes or even describing $K\bar{K}$ annihilation. This is a
more ambitious goal that we leave for future research. However, during a kink-
antikink scattering process, when the solitons completely overlap, an
intermediate state that resembles the profile of an oscillon is formed. This
suggest that the study of oscillons, with appropriate initial conditions, may
provide useful information about these violent processes. This is the point of
view we adopt in the following section.

##  6 Effective model for the radiating oscillon

As we have shown, the scattering modes play an important role in the dynamics
of the single-kink sector. In this section we will illustrate the importance
of such modes for a non-topological soliton, the oscillon of the 1-dimensional
$\phi^{4}$ model. In particular, we will describe the decay of oscillons below
the critical amplitude, the possibility of internal modes and the $K\bar{K}$
formation. Since oscillons have been observed starting from rather generic
initial data, we decide to follow first Fodor1 ; Fodor2 , and take the
following ansatz

$\Phi_{\text{o}}(x;a)=-1+a\operatorname{sech}(x/R),$ (6.1)

where $a$ is identified with the amplitude of the oscillon and $R$ accounts
for its size. Later, in this section, we will change to a Gaussian profile
used in Hindmarsh . The effective Lagrangian associated to (6.1) assuming that
$a$ depends on time takes the following form

$\mathcal{L}^{o}=R\left(\dot{a}^{2}(t)-\dfrac{1}{3}(12+\dfrac{1}{R^{2}})a^{2}(t)+\pi
a^{3}(t)-\dfrac{2}{3}a^{4}(t)\right),$ (6.2)

This is the Lagrangian of an anharmonic oscillator of frequency
$\omega_{o}=\sqrt{\frac{1}{3}(12-\frac{1}{R^{2}})}$. For small amplitudes, the
frequency of $a(t)$ is above the threshold frequency $\omega_{t}=2$. As a
result, $a(t)$ couples directly to the continuum and collapses into radiation.
As a consequence, for $a$ small enough, the initial data (6.1) does not evolve
into an oscillon. However, for $a(t)$ large enough, the non-linearities in
(6.2) decrease the oscillator frequency avoiding the direct coupling to
radiation. Therefore, the oscillon weakly couples to radiation and its
amplitude decreases very slowly. The details of this coupling are, of course,
not described in (6.2), and one needs to add dissipative degrees of freedom.
This can be done as in the previous sections considering an ansatz of the form

$\Phi_{\text{o,
rad}}(x;a,c_{q})=-1+a(t)\operatorname{sech}(x/R)+\int_{\mathbb{R}}dqc_{q}(t)\eta_{q}(x/R).$
(6.3)

It is enough to consider the truncated Lagrangian at second order in
$c_{q}(t)$ since we are interested in the description of the oscillon coupled
to the slow-amplitude emitted radiation. However, as we have mentioned, it is
fundamental to consider the action at all orders in $a(t)$, since the
existence of the oscillon is linked to the non-linear structure of the
Lagrangian. Proceeding as in the previous sections we arrive at

$\displaystyle\mathcal{L}^{o}_{r}$ $\displaystyle=$
$\displaystyle\pi\left(\int_{\mathbb{R}}dq\,\dot{c}_{q}(t)\dot{c}_{-q}(t)-\int_{\mathbb{R}}dq\,w_{q,R}^{2}\,c_{q}(t)c_{-q}(t)\right)+\dot{a}^{2}(t)-\omega_{o}^{2}a^{2}(t)+\pi
a^{3}(t)-\dfrac{2}{3}a^{4}(t)$ (6.4)
$\displaystyle+\int_{\mathbb{R}^{2}}dqdq^{\prime}\,f_{1}(q,q^{\prime},R)\,c_{q}(t)c_{q^{\prime}}(t)+\dot{a}(t)\int_{\mathbb{R}}dq\,f_{2}(q)\dot{c}_{q}(t)+a(t)\int_{\mathbb{R}}dq\,f_{3}(q)c_{q}(t)$
$\displaystyle+a(t)\int_{\mathbb{R}^{2}}dqdq^{\prime}\,f_{4}(q,q^{\prime})c_{q}(t)c_{q^{\prime}}(t)+a^{2}(t)\int_{\mathbb{R}}dqdq^{\prime}\,f_{5}(q,q^{\prime})c_{q}(t)c_{q^{\prime}}(t)$
$\displaystyle+a^{3}(t)\int_{\mathbb{R}}dq\,f_{6}(q)c_{q}(t),$

with $w_{q,R}=\sqrt{\left(q/R\right)^{2}+4}$ and where

$\displaystyle f_{1}(q,q^{\prime},R)$ $\displaystyle=$
$\displaystyle-\dfrac{3\pi}{5R^{2}}\dfrac{4q+4q^{\prime}+5q^{3}+5q^{\prime
3}+q^{5}+q^{\prime
5}}{\sqrt{\left(q^{2}+1\right)\left(q^{2}+4\right)}\sqrt{\left(q^{\prime
2}+1\right)\left(q^{\prime
2}+4\right)}}\operatorname{csch}\left(\dfrac{\pi}{2}(q+q^{\prime})\right),$
(6.5) $\displaystyle f_{2}(q)$ $\displaystyle=$
$\displaystyle\dfrac{\pi}{2}\dfrac{\sqrt{q^{2}+1}}{\sqrt{q^{2}+4}}\operatorname{sech}\left(\dfrac{\pi
q}{2}\right),$ (6.6) $\displaystyle f_{3}(q,R)$ $\displaystyle=$
$\displaystyle\pi\dfrac{\sqrt{q^{2}+1}}{\sqrt{q^{2}+4}}\left(\dfrac{q^{2}+3}{4R^{2}}-2\right)\operatorname{sech}\left(\dfrac{\pi
q}{2}\right),$ (6.7) $\displaystyle f_{4}(q,q^{\prime})$ $\displaystyle=$
$\displaystyle\dfrac{3\pi}{4}\dfrac{11+14q^{2}+14q^{\prime 2}+2q^{2}q^{\prime
2}+3q^{4}+3q^{\prime
4}}{\sqrt{\left(q^{2}+1\right)\left(q^{2}+4\right)}\sqrt{\left(q^{\prime
2}+1\right)\left(q^{\prime
2}+4\right)}}\operatorname{sech}\left(\dfrac{\pi}{2}(q+q^{\prime})\right),$
(6.8) $\displaystyle f_{5}(q,q^{\prime})$ $\displaystyle=$
$\displaystyle-\dfrac{3\pi}{5}\dfrac{4q+4q^{\prime}+5q^{3}+5q^{\prime
3}+q^{5}+q^{\prime
5}}{\sqrt{\left(q^{2}+1\right)\left(q^{2}+4\right)}\sqrt{\left(q^{\prime
2}+1\right)\left(q^{\prime
2}+4\right)}}\operatorname{csch}\left(\dfrac{\pi}{2}(q+q^{\prime})\right),$
(6.9) $\displaystyle f_{6}(q)$ $\displaystyle=$
$\displaystyle\dfrac{\pi}{4}\dfrac{(q^{2}+1)^{3/2}}{\sqrt{q^{2}+4}}\operatorname{sech}\left(\dfrac{\pi
q}{2}\right).$ (6.10)

Notice that we have omitted a global factor $R$ in (6.4) since it does not
have any effect on the equations of motion. In Fig. 6 we illustrate the
evolution of the oscillon profile at $x=0$ for different initial amplitudes
and sizes.

(a) $n=20$, $a_{0}=0.3$, $R=1.5.$ (b) $n=30$, $a_{0}=0.2$, $R=1.5.$

(c) $n=10$, $a_{0}=0.4$, $R=2.$ (d) $n=10$, $a_{0}=0.5$, $R=2.$

Figure 6: Comparison between the effective model (dashed line) (6.4) and field
theory (solid line). The scattering modes have been taken in the interval
$q\in[-5,5]$.

In the upper panel we show two configurations with small initial amplitudes.
This results into a fast decay of the initial configuration into radiation. It
is worth mentioning even with a large number of radiation modes ($n>20$) the
amplitude in the effective model decays slower than in the field theoretical
simulation. This suggests that the dissipative mechanism provided by the set
of scattering modes is actually not very efficient. In the lower panel we show
a genuine oscillon. The numerical simulation reveals that the oscillon hosts
an internal non-dissipative mode, due to the amplitude modulation during the
time evolution. This mode can be related to the variation of the oscillon
size. Hence, let us consider a small variation of the oscillon width as
follows

$\Phi_{\text{o}}(x;a,\delta)=-1+a\operatorname{sech}\left(\frac{x}{R+\delta}\right).$
(6.11)

If $\delta\ll 1$ we may expand about $\delta=0$ up to first order

$\Phi_{\text{o}}(x;a,\delta)=-1+a\operatorname{sech}(x/R)+\frac{a\delta}{R^{2}}\,x\,\operatorname{sech}(x/R)\tanh(x/R)\,.$
(6.12)

The additional term to the unperturbed oscillon corresponds to the so-called
Derrick mode. This correction should codify the possible changes of the
oscillon size, and luckily represent the behaviour expected from the full
numerical result. This simple choice has a problem. At $a=0$, the ansatz
(6.11) does not depend on $\delta$. This implies that the moduli metric
associated to $(a,\delta)$ is not well-defined at this point (see for example
Manton1  for a discussion about the null vector problem). In order to cure
this issue we may perform a simple change of coordinates
$\delta\rightarrow\delta/a$. Finally, we get

$\Phi_{\text{o,
s}}(x;a,\delta)=-1+a\operatorname{sech}(x/R)+\frac{\delta}{R^{2}}\,x\,\operatorname{sech}(x/R)\tanh(x/R)\,.$
(6.13)

Treating $a$ and $\delta$ as collective coordinates (recall that $R$ remains
fixed) we get the below effective Lagrangian

$\displaystyle\mathcal{L}^{o}_{s}$ $\displaystyle=$ $\displaystyle
R\dot{a}(t)^{2}-\omega_{o}^{2}a(t)^{2}+\dfrac{1}{R}\left(\dfrac{\pi^{2}}{36}+\dfrac{1}{3}\right)\dot{\delta}(t)^{2}-\dfrac{1}{R}\left(\pi^{2}\left(\dfrac{1}{9}+\dfrac{7}{180R^{2}}\right)+\dfrac{4}{3}\right)\delta(t)^{2}+\pi
Ra(t)^{3}$ (6.14)
$\displaystyle-\dfrac{2}{3}Ra(t)^{4}+\dfrac{\pi}{R^{2}}\left(\dfrac{11\pi^{2}}{80}-1\right)\delta(t)^{3}-\dfrac{1}{R^{3}}\left(\dfrac{\pi^{4}}{600}+\dfrac{\pi^{2}}{90}-\dfrac{2}{15}\right)\delta(t)^{4}+\dot{a}(t)\dot{\delta}(t)$
$\displaystyle+\left(\dfrac{1}{3R^{2}}-4\right)a(t)\delta(t)+\dfrac{\pi}{R}\left(\dfrac{3\pi^{2}}{16}-1\right)a(t)\delta(t)^{2}-\dfrac{1}{R^{2}}\left(\dfrac{7\pi^{2}}{90}-\dfrac{1}{3}\right)a(t)\delta(t)^{3}$
$\displaystyle+\pi
a(t)^{2}\delta(t)-\dfrac{2}{3}a(t)^{3}\delta(t)-\dfrac{\pi^{2}}{15R}a(t)^{2}\delta(t)^{2}.$

The associated equations of motion are collected below

$\displaystyle\ddot{a}(t)$ $\displaystyle+$
$\displaystyle\dfrac{1}{3}\left(\dfrac{1}{R^{2}}+12\right)a(t)-\dfrac{3}{2}\pi
a(t)^{2}+\dfrac{4}{3}a(t)^{3}+\dfrac{1}{2R}\ddot{\delta}(t)+\dfrac{\left(12R^{2}-1\right)}{6R^{3}}\delta(t)$
(6.15) $\displaystyle-$
$\displaystyle\dfrac{\pi\left(3\pi^{2}-16\right)}{32R^{2}}\delta(t)^{2}+\dfrac{\left(7\pi^{2}-30\right)}{180R^{3}}\delta(t)^{3}-\dfrac{\pi}{R}a(t)\delta(t)+\dfrac{\pi^{2}}{15R^{2}}a(t)\delta(t)^{2}$
$\displaystyle+$ $\displaystyle\dfrac{1}{R}a(t)^{2}\delta(t)=0\,,$

and

$\displaystyle\ddot{\delta}(t)$ $\displaystyle+$
$\displaystyle\dfrac{\left(\pi^{2}\left(20R^{2}+7\right)+240R^{2}\right)}{5\left(\pi^{2}+12\right)R^{2}}\delta(t)-\dfrac{27\pi\left(11\pi^{2}-80\right)}{40\left(\pi^{2}+12\right)R}\delta(t)^{2}+\dfrac{\left(3\pi^{4}+20\pi^{2}-240\right)}{25\left(\pi^{2}+12\right)R^{2}}\delta(t)^{3}$
(6.16) $\displaystyle+$
$\displaystyle\dfrac{18R}{\pi^{2}+12}\ddot{a}(t)+\dfrac{\left(72R^{2}-6\right)}{\left(\pi^{2}+12\right)R}a(t)-\dfrac{18\pi
R}{\pi^{2}+12}a(t)^{2}+\dfrac{12R}{\pi^{2}+12}a(t)^{3}-\dfrac{9\pi\left(3\pi^{2}-16\right)}{4\left(\pi^{2}+12\right)}a(t)\delta(t)$
$\displaystyle+$
$\displaystyle\dfrac{3\left(7\pi^{2}-30\right)}{5\left(\pi^{2}+12\right)R}a(t)\delta(t)^{2}+\dfrac{12\pi^{2}}{5\pi^{2}+60}a(t)^{2}\delta(t)=0\,.$

Notice that the frequency of the width perturbation $\delta(t)$ is above the
threshold. At first glance it may seem that any excitation of $\delta(t)$
should be dissipated very fast through the coupling with radiation. However,
as it happens with the oscillon itself, if $\delta(t)$ is excited at
sufficiently high amplitudes, the non-linear terms may decrease its frequency
below $\omega_{t}$, becoming a confined mode. Let us now investigate the
accuracy of our new effective model with field theory for the initial
conditions of Fig. 6 (d). The corresponding comparison is depicted in Fig. 7.

Figure 7: Comparison between field theory (solid line) and the effective model
with $a_{0}=0.5,\,\delta_{0}=0$, and $R=2$ (dashed line).

Note that the prediction of the effective model (6.14) and field theory agree
with great precision. Therefore, we can confirm that the previous quasi-
periodic behaviour is related to the existence of an internal state bounded to
the oscillon. Despite the accuracy of the model we can go beyond and include
the radiation modes as well, since these degrees of freedom may be significant
for certain field configurations. However, as shown in Fig. 6, the addition of
genuine radiation modes to the effective model does not seem to dissipate
efficiently the energy. If one is interested in the dynamics close to the
oscillon core and for not very large times, one could add instead modes that
resemble the scattering modes, i.e., with spatial frequencies above the mass
threshold, but confined to the oscillon. Following this strategy we consider
the following ansatz

$\Phi_{\text{o,
rad}}(x;a,\delta)=-1+a(t)e^{-\left(\frac{x}{R}\right)^{2}}+\sum_{k=1}^{n}\delta_{k}(t)h_{k}(x/r)\,.$
(6.17)

For simplicity, we have changed the initial oscillon profile to a Gaussian
profile. Although (6.1) seems to model better the oscillon tails, the profile
(6.17) greatly simplifies the calculations. We choose the modes as follows

$h_{k}(x/R)=\dfrac{1}{k!}\frac{d^{k}}{dr^{k}}e^{-\left(\frac{x}{r}\right)^{2}}.$
(6.18)

The first one is, in fact, the Derrick mode associated to the new profile. The
precise choice of the rest of the modes is not very relevant as long as they
have increasing spatial frequencies in the oscillon core. The effective
Lagrangian can be written symbolically in a very simple way

$\mathcal{L}^{o}_{r}=\sum_{k,l=0}^{n}m_{k,l}\dot{\xi}_{k}(t)\dot{\xi}_{l}(t)-\sum_{k,l=0}^{n}\omega^{2}_{k,l}\xi_{k}(t)\xi_{l}(t)-V(\xi_{k}(t))\,,$
(6.19)

where $\xi_{0}(t)=a(t)$ and $\xi_{k}(t)=\delta_{k}(t)$ for $k=1,...n$, the
matrices $m_{k,l}$ and $\omega^{2}_{k,l}$ are constant, and $V(\xi_{k}(t))$ is
a potential that couples non-linearly all the modes. Using standard techniques
we can diagonalize simultaneously $m_{k,l}$ and $\omega_{k,l}$ and rewrite
(6.19) in normal coordinates $\eta_{k}(t)$ as follows

$\mathcal{L}^{\text{o}}_{\text{r}}=\sum_{k=0}^{n}m_{k}\dot{\eta}_{k}^{2}(t)-\sum_{k=0}^{n}\omega_{k}^{2}\eta_{k}^{2}(t)-V(\eta_{k}(t))\,.$
(6.20)

Therefore, in our approach, the oscillon is described by $n+1$ anharmonic
oscillators of proper frequencies $\omega_{k}$ coupled non-linearly by the
potential $V(\eta_{k}(t))$. The system (6.20) is conservative, therefore it
cannot dissipate energy. However, the modes $\delta_{k}(t)$ act effectively as
dissipative degrees of freedom, storing energy from the mode amplitude $a(t)$.
The energy transfer mechanism works actually very efficiently as we will show.
As a consequence, for not very large times, the model is able to describe how
the initial data decays into radiation.

In our following experiment we study the decay of an initial configuration of
the form

$\Phi_{\text{o}}(x;a_{0})=-1+a_{0}e^{-\left(\frac{x}{R}\right)^{2}}\,.$ (6.21)

Below a critical value of $a_{0}$, the initial configuration given by (6.21)
decays into radiation. This value can be taken as the minimal $a_{0}$ such
that the proper frequency of a(t) coincides with the mass threshold frequency.
By expanding the frequency and imposing that the secular terms vanish, we can
compute the frequency correction due to higher order terms. This gives

$\omega=\omega_{0}+\frac{a_{0}^{2}\left(3\sqrt{2}\omega_{0}^{2}-80\right)}{8\omega_{0}^{3}}+\mathcal{O}(a_{0}^{4}),\,\,\omega_{0}=\sqrt{4+1/R^{2}}\,.$
(6.22)

From this expression we can compute the critical amplitude value for the
oscillon formation

$a_{0}^{\text{critical}}(R)=\frac{\sqrt{\frac{2}{191}\left(20+3\sqrt{2}\right)}}{R}\,.$
(6.23)

One situation with $a_{0}<a_{0}^{\text{critical}}(R)$ is illustrated in Fig.
8a. The effective model mimics the decay accurately for $t\lesssim 60$. For
$t>60$ the energy stored in the internal modes is transferred back to the
amplitude. In Fig. 8b we show a genuine oscillon with an internal mode
excited.

(a) $a_{0}=0.1$, $R=4$, $r=4$, $n=7$ (b) $a_{0}=0.5$, $R=4$, $r=4$, $n=7$

Figure 8: $\phi(0,t)$ for the initial data (6.21) for different values of the
initial amplitude in full numerics (solid line) and in the effective model
from (6.20) (dashed line).

Interestingly, the effective model (6.20) also describes the creation of a
$K\bar{K}$ pair from the oscillon profile. In order to understand this
phenomenon it is enough to analyse the effective action for $a(t)$. Similarly
to (6.2), the effective Lagrangian for $a(t)$ with the profile (6.17) is given
by

$\mathcal{L}^{o}=\sqrt{\frac{\pi}{2}}R\left(\frac{1}{2}\dot{a}(t)^{2}-\frac{\left(4R^{2}+1\right)a(t)^{2}}{2R^{2}}-\frac{a(t)^{4}}{2\sqrt{2}}+2\sqrt{\frac{2}{3}}a(t)^{3}\right).$
(6.24)

The potential for $a(t)$ is depicted in Fig. 9. For $R\gtrsim 2.6$ the
potential develops a new local minimum around $a\approx 2$. For large enough
initial amplitudes, $a(t)$ is able to climb the potential barrier and sit on
the upper minimum. If internal modes are absent and $a(t)$ possesses
sufficient energy to overcome the potential barrier, it must descend to the
minimum at zero due to the conservation of energy. But when the internal modes
are included, they are able to store the energy excess allowing $a(t)$ to
oscillate in the upper minimum. This situation is identified with the
formation of a $K\bar{K}$ pair which leaves the positive vacuum $\phi=1$ at
$x=0$.

Figure 9: Effective potential for $a(t)$ at $R=4$.

In Fig. 10 we show the value of the field at the origin $\phi(t,0)$ for
different initial amplitudes $a_{0}$ and $R=4$. The regions where the field
takes a value close to $1$ correspond to the $K\bar{K}$ creation. In order to
produce a pair the system needs to have an energy $E>2M_{k}$. But above this
value the pair is not always produced, leading to oscillon regions whose
energy is dissipated into radiation. This of course resembles the
characteristic fractal pattern in the $K\bar{K}$ scattering processes. In the
intermediate stages right after the scattering, the field profile looks like a
bump above the $-1$ vacuum, resembling the profile of an oscillon. This is the
connection between oscillons and $K\bar{K}$ scattering we have mentioned
before.

(a) Full numerics. $R=4$. (b) Effective model. $R=4$, $r=2$ and $n=8$.

Figure 10: Comparison between the effective model (6.20) and field theory for
the initial data (6.21). The color palette indicates the value of the field
$\phi$ at the origin, $\phi(0,t)$.

In order to reproduce the fractal pattern visible at $a_{0}<2$ one needs to
add modes of higher frequencies. This suggests that the scattering modes are
essential to explain this structure. Similar results are obtained for values
for $2\lesssim R\lesssim 6$. For $R<2.6$, the effective potential has only a
local minima at $a=0$, therefore the field cannot sit on the upper vacuum
describing the creation of the $K\bar{K}$ pair. On the other hand, for very
large values of $R$, our modes decrease their spatial frequency, and the model
as given by the ansatz (6.17) does only reproduce qualitatively the pair
creation. This can be solved easily by adding higher frequency modes.

We would like to emphasise that in our approach, the moduli space metric is
trivial (i.e. constant), and the relevant dynamics is completely encoded in
the specific coupling between modes given by the potential.

To close this section we are going to illustrate the rich structure given by
the internal structure of the oscillons. Apart from the internal mode
described by the Derrick mode illustrated in Figs. 7 and 8 (b), it is possible
to have oscillations with more than one internal mode excited. This gives rise
to complicated long-live oscillatory patterns which are very sensitive to the
initial conditions.

Figure 11: $\phi(0,t)$ with initial data given by (6.1) with $R=10$ and
$a_{0}=0.23$.

In Fig. 11 we show an interesting pattern. This behaviour is characterized by
the excitation of two internal modes. From the point of view of the effective
model, the amplitudes of these modes are big enough to decrease their
frequencies below the mass threshold. As a consequence, they remain excited
for very long times (numerically, at $t\approx 7000$ in our time units, the
oscillon ceases to exist).

The results of this section show the relevance of the radiative modes in the
oscillon dynamics. Although for certain initial data the oscillon evolution is
well-approximated by the amplitude degree of freedom plus one internal mode,
in general, one needs higher frequency modes (which in our approach are
confined to the oscillon core) to describe correctly the dynamics.
Interestingly, a reasonably simple model (a set of coupled anharmonic
oscillators) is able to describe the main features of the oscillon dynamics,
including the decay into radiation and the pair $K\bar{K}$ creation.

##  7 Summary and Conclusions

In this paper, we have introduced dissipative degrees of freedom in the moduli
space approximation of the $\phi^{4}$ model. We have computed, within this
approach, the radiation emitted at infinity by a wobbling kink at the lowest
order in the shape mode amplitude. Our results are in complete agreement with
the well-know calculations from Manton2 .

We begin our investigation by studying in detail the interaction of radiation
with the vibrational mode. In terms of the effective model, there are two
leading mechanisms that explain the energy transfer between them: a Mathieu
instability and a resonance. These mechanisms show that the strongest coupling
between radiation and the shape mode occurs for $\omega_{q}=2\omega_{s}$. We
contemplate two different experiments: in the first one we analyse a kink with
its shape mode initially excited and discuss the main frequency of the
radiation emitted. In the second one, we irradiate a kink with linear
radiation and study how the shape mode is triggered. Notably, we found an
analytic expression for the excitation of the shape mode for frequencies away
from the unstable region.

We have also studied the role of the translational mode. Despite the fact that
the standard CCM approach is non-relativistic, when considered up to second
order in $\dot{a}^{2}(t)$, it is able to reproduce at second order the Lorentz
contraction of the kink. The vibrational degree of freedom is not enough to
reproduce correctly this relativistic effect, however, the inclusion of
scattering modes allows for an exact Lorentz contraction at second order. This
suggest that an effective model able to describe dissipative effects should be
a nice candidate to describe detailed features of non-linear processes such as
kink scattering. Of course, the model looses its usefulness if all radiation
modes are included, since basically one recovers field theory. However, a
judicious choice of modes describing effectively the scattering modes could
shed more light on the understanding of many non-linear processes.

We devoted the last section to the derivation of an effective model for the
$\phi^{4}$ oscillon. Although its natural frequency is above the mass
threshold limit, the non-linear terms decrease the frequency below such a
limit, avoiding the direct coupling with radiation. The numerical simulations
indicate that the oscillon can host a discrete mode responsible for
modifications of the width. We implemented this behaviour through the
inclusion of the Derrick mode associated to the change of size of the
oscillon. This new proposal gives a good agreement with the full numerical
simulations. We have added higher frequency modes confined to the oscillon
core. They represent scattering modes which may store energy for certain time,
acting effectively as dissipative degrees of freedom. The effective equations
are a system of coupled anharmonic oscillators with a trivial moduli space
metric. Interestingly, once these degrees of freedom are added, this simple
effective model is able to describe the KAK creation from initial oscillon
data.

Our results suggest that the radiation modes play a crucial role in the study
of solitons dynamics. They are of course necessary to explain the decay of
non-topological solution such as the oscillon, or long-lived internal modes
such as the shape mode. But they also seem to be fundamental to disentangle
the complicated patterns in soliton scattering processes. In addition, the
results presented here can be easily generalised to other models. The study of
the internal structure of oscillons within this approach in different models
deserve further research, therefore is left for a future investigation.

## Acknowledgments

J.Q. thanks J.J. Blanco-Pillado and A. Wereszczynski for useful discussions.
This research was supported by Spanish MCIN with funding from European Union
NextGenerationEU (PRTRC17.I1) and Consejeria de Educacion from JCyL through
QCAYLE project, as well as MCIN project PID2020-113406GB-I00. SNO’s research
is carried out thanks to a pre-doctoral contract financed by the Junta de
Castilla y León through the European Social Fund Plus (ESF+).

##  Appendix A Radiation from the shape mode

In this appendix we provide some details of the calculation of the radiation
emitted by a wobbling kink with its shape mode excited. We start with (3.10)
with the scattering amplitudes given by (3.9) and split it as follows

$R(x,t)=R_{1}(x)+R_{2}(x,t),$ (A.1)

where

$\displaystyle R_{1}(x)$ $\displaystyle=$ $\displaystyle
i\int_{\mathbb{R}}\,dq\,R_{0}(q)\,\left(4\omega_{s}^{2}-\omega_{q}^{2}\right)\eta_{q}(x)\,,$
(A.2) $\displaystyle R_{2}(x,t)$ $\displaystyle=$
$\displaystyle-i\int_{\mathbb{R}}\,dq\,R_{0}(q)\,\left(\omega_{q}^{2}\cos(2\omega_{s}t)+(4\omega_{s}^{2}-2\omega_{q}^{2})\cos(\omega_{q}t)\right)\eta_{q}(x)\,,$
(A.3)

and

$R_{0}(q)=\dfrac{3A_{0}^{2}}{64}\dfrac{q^{2}\left(q^{2}-2\right)}{\sqrt{q^{2}+1}\,\omega_{q}(4\omega_{s}^{2}-\omega_{q}^{2})\sinh(\pi
q/2)}\,.$ (A.4)

###  A.1 Evaluation of the function $R_{1}(x)$

Let us begin with (A.2). In the first place, notice that we can divide the
integrand into an odd and even contribution. Due to the symmetric interval of
integration, only the even contribution is not null, which results in

$R_{1}(x)=-2\int_{0}^{\infty}\,dq\,R_{0}(q)\,(4\omega_{s}^{2}-\omega_{q}^{2})\operatorname{Im}(\eta_{q}(x)),$
(A.5)

where

$\operatorname{Im}(\eta_{q}(x))=-(q^{2}+1)\sin qx-3\,q\tanh x\cos
qx+3\tanh^{2}x\sin qx\,.$ (A.6)

Now we will split the integral as

$\displaystyle R_{1}(x)$ $\displaystyle=$
$\displaystyle\frac{3A_{0}^{2}}{32}\bigg{(}\int_{0}^{\infty}dq\,\dfrac{q^{2}\,(q^{2}-2)\sin
qx}{(q^{2}+4)\sinh(\pi q/2)}+3\tanh
x\int_{0}^{\infty}dq\,\dfrac{q^{3}\,(q^{2}-2)\cos
qx}{(q^{2}+4)(q^{2}+1)\sinh(\pi q/2)}$ (A.7)
$\displaystyle-3\tanh^{2}x\int_{0}^{\infty}dq\,\dfrac{q^{2}\,(q^{2}-2)\sin
qx}{(q^{2}+4)(q^{2}+1)\sinh(\pi q/2)}\bigg{)}\,,$

and we will solve each term separately by transforming the problem in the
calculation of the solution of an inhomogeneous differential equation. To
simplify the notation, let us write

$\displaystyle R_{1}(x)$ $\displaystyle=$
$\displaystyle\frac{3A_{0}^{2}}{32}\bigg{(}\alpha(x)+3\tanh
x\,\beta(x)-3\tanh^{2}x\,\gamma(x)\bigg{)}\,.$ (A.8)

The inhomogeneous differential equation satisfied by $\alpha(x)$ is

$\alpha^{\prime\prime}(x)-4\alpha(x)=-\int_{0}^{\infty}dq\,\dfrac{q^{2}(q^{2}-2)\sin
qx}{\sinh(\pi q/2)}=-6\,(3-\cosh 2x)\tanh x\operatorname{sech}^{4}x.$ (A.9)

Given the obvious condition $\alpha(x)=0$, and to avoid divergences as
$\mathinner{\\!\left\lvert x\right\rvert}\rightarrow\infty$, the solution to
this differential equation turns out to be

$\alpha(x)=6e^{2x}\log\left(1+e^{-2x}\right)-6e^{-2x}\log\left(1+e^{2x}\right)-2\tanh
x\left(3-\operatorname{sech}^{2}x\right)\,.$ (A.10)

Regarding the function $\beta(x)$, it must be solution of the forth order
differential equation

$\beta^{(4)}(x)-5\beta^{\prime\prime}(x)+4\beta(x)=\int_{0}^{\infty}dq\,\dfrac{q^{3}(q^{2}-2)\cos
qx}{\sinh(\pi q/2)}=3\,(21-18\cosh 2x+\cosh 4x)\operatorname{sech}^{6}x,$
(A.11)

with conditions $\beta^{\prime}(0)=0$ and $\beta^{(3)}(0)=0$. In order to
avoid divergences at $\mathinner{\\!\left\lvert
x\right\rvert}\rightarrow\infty$, the solution must be

$\displaystyle\beta(x)$ $\displaystyle=$
$\displaystyle-4e^{2x}\log(1+e^{-2x})-4e^{-2x}\log(1+e^{2x})+5-\dfrac{4}{(1+e^{2x})^{2}}+\frac{\pi
e^{x}}{2}-2\tanh x$ (A.12) $\displaystyle-2\sinh x\arctan e^{x}.$

Finally, the $\gamma(x)$ term of $R_{1}(x)$ will be the solution of the
differential equation

$\gamma^{(4)}(x)-5\gamma^{\prime\prime}(x)+4\gamma(x)=\int_{0}^{\infty}dq\,\frac{q^{2}(q^{2}-2)}{\sinh(\pi
q/2)}\sin qx=6\,(3-\cosh 2x)\tanh x\operatorname{sech}^{4}x,$ (A.13)

with conditions $\gamma(0)=0$ and $\gamma^{\prime\prime}(0)=0$. The well
behaved solution at $\mathinner{\\!\left\lvert
x\right\rvert}\rightarrow\infty$ is

$\displaystyle\gamma(x)$ $\displaystyle=$ $\displaystyle
2e^{-2x}\log(1+e^{2x})-2e^{2x}\log(1+e^{-2x})-e^{-2x}(1+\tanh x)+1+\frac{\pi
e^{x}}{2}$ (A.14) $\displaystyle-2\cosh x\arctan e^{x}.$

Substituting (A.10), (A.12) and (A.14) into (A.8), we get the complete
expression of $R_{1}(x)$

$\displaystyle R_{1}(x)=\dfrac{3A_{0}^{2}}{64\cosh^{2}x}\left(3\pi\sinh
x+16\tanh x-24x\right)\,.$ (A.15)

A plot of this function $R_{1}(x)$ can be seen in Fig. 12.

Figure 12: Representation of $R_{1}(x)$ given by (A.15) for $A_{0}=1$.

###  A.2 Evaluation of the function $R_{2}(x,t)$

The evaluation of the contribution of $R_{2}(x,t)$ seems to be more
challenging. Although an analytical calculation seems impossible, some
properties of the radiation can be obtained by certain approximations. We will
focus now on the asymptotic radiation. As we have seen, the $R_{1}(x)$
contribution is exponentially suppressed for large $x$, therefore the
asymptotic radiation is entirely given by the $R_{2}(x,t)$ contribution.

Through some elemental algebra is straightforward to verify that the integrand
of $R_{2}(x,t)$ can be written as

$(4\omega_{s}^{2}-\omega_{q}^{2})\bigg{(}\cos(\omega_{q}t)-\cos(2\omega_{s}t)+\cos(\omega_{q}t)-4\omega_{s}^{2}\dfrac{\cos(\omega_{q}t)-\cos(2\omega_{s}t)}{4\omega_{s}^{2}-\omega_{q}^{2}}\bigg{)}\,,$
(A.16)

where we have removed temporarily a $-iR_{0}(q)\hskip 0.28436pt\eta_{q}(x)$
factor. This expression can be arranged as

$(4\omega_{s}^{2}-\omega_{q}^{2})\bigg{(}-2\sin(\dfrac{\omega_{q}+2\omega_{s}}{2}t)\sin(\dfrac{\omega_{q}-2\omega_{s}}{2}t)+\cos(\omega_{q}t)-\dfrac{8\omega_{s}^{2}\sin(\dfrac{\omega_{q}+2\omega_{s}}{2}t)\sin(\dfrac{\omega_{q}-2\omega_{s}}{2}t)}{\omega_{q}^{2}-4\omega_{s}^{2}}\bigg{)}\,.$
(A.17)

Let us define the following variables

$\Omega_{q}^{+}=\dfrac{\omega_{q}+2\omega_{s}}{2},\qquad\Omega_{q}^{-}=\dfrac{\omega_{q}-2\omega_{s}}{2}\,.$
(A.18)

As it is well known, there are some sequence of functions that converge weakly
to the Dirac delta. One of them is

$f_{n}(x)=\dfrac{\sin(nx)}{\pi
x}\overset{n\rightarrow\infty}{\longrightarrow}\delta(x).$ (A.19)

In our case, we could identify

$\dfrac{\sin(\Omega_{q}^{-}t)}{\pi\Omega_{q}^{-}}\overset{t\rightarrow\infty}{\longrightarrow}\delta(\Omega_{q}^{-})=\sqrt{6}\bigg{(}\delta(q-2\sqrt{2})+\delta(q+2\sqrt{2})\bigg{)}\,,$
(A.20)

where it was used a property of the Dirac delta. In order to achieve this, let
us rewrite (A.17) as

$(4\omega_{s}^{2}-\omega_{q}^{2})\bigg{(}-2\pi\Omega_{q}^{-}\sin(\Omega_{q}^{+}t)\dfrac{\sin(\Omega_{q}^{-}t)}{\pi\Omega_{q}^{-}}+\cos(\omega_{q}t)-2\pi
t\,\omega_{s}^{2}\operatorname{sinc}(\Omega_{q}^{+}t)\dfrac{\sin(\Omega_{q}^{-}t)}{\pi\Omega_{q}^{-}}\bigg{)}\,.$
(A.21)

Substituting (A.20) into (A.21) and integrating, we can realise that the
contribution from the first term in (A.21) vanishes. This fact is supported by
the numerical analysis, which shows that the contribution is suppressed in
time. The second term may be computed as follows: let us call such
contribution as

$\Psi(x,t)=i\int_{\mathbb{R}}\,dq\,R_{0}(q)\,(4\omega_{s}^{2}-\omega_{q}^{2})\cos(\omega_{q}t)\eta_{q}(x)\,.$
(A.22)

To facilitate the calculation, let’s divide $\Psi(x,t)$ in the same way as we
did for $R_{1}(x)$

$\displaystyle\Psi(x,t)$ $\displaystyle=$
$\displaystyle\frac{3iA_{0}^{2}}{64}\bigg{(}\int_{\mathbb{R}}dq\,\dfrac{q^{2}\,(q^{2}-2)\cos(\omega_{q}t)}{(q^{2}+4)\sinh(\pi
q/2)}e^{iqx}+3i\tanh
x\int_{\mathbb{R}}dq\,\dfrac{q^{3}\,(q^{2}-2)\cos(\omega_{q}t)}{(q^{2}+1)(q^{2}+4)\sinh(\pi
q/2)}e^{iqx}$ (A.23)
$\displaystyle-3\tanh^{2}x\int_{\mathbb{R}}dq\,\dfrac{q^{2}\,(q^{2}-2)\cos(\omega_{q}t)}{(q^{2}+1)(q^{2}+4)\sinh(\pi
q/2)}e^{iqx}\bigg{)}\,.$

Each of the terms from $\Psi(x,t)$ can be conceived as wave packets where the
dispersion relation is non-linear and where the constituent amplitudes are
exponentially suppressed in $q$

$\displaystyle\Psi(x,t)$ $\displaystyle=$
$\displaystyle\frac{3iA_{0}^{2}}{138}\bigg{(}\int_{\mathbb{R}}dq\,\dfrac{q^{2}\,(q^{2}-2)}{(q^{2}+4)\sinh(\pi
q/2)}\left(e^{i(qx+\omega_{q}t)}+e^{i(qx-\omega_{q}t)}\right)$ (A.24)
$\displaystyle+3i\tanh
x\int_{\mathbb{R}}dq\,\dfrac{q^{3}\,(q^{2}-2)}{(q^{2}+1)(q^{2}+4)\sinh(\pi
q/2)}\left(e^{i(qx+\omega_{q}t)}+e^{i(qx-\omega_{q}t)}\right)$
$\displaystyle-3\tanh^{2}x\int_{\mathbb{R}}dq\,\dfrac{q^{2}\,(q^{2}-2)}{(q^{2}+1)(q^{2}+4)\sinh(\pi
q/2)}\left(e^{i(qx+\omega_{q}t)}+e^{i(qx-\omega_{q}t)}\right)\bigg{)}\,.$

In the literature, the problem of the calculation of wave packets is well
known when the amplitude is a Gaussian function, that is,
$A(q)=e^{-\alpha^{2}(q-q_{c})^{2}}$. For that case, as the amplitude is
exponentially suppressed away from the maximum, it is a good approximation to
Taylor expand the dispersion relation around $q_{c}$. Then, $\omega(q)\approx
q_{c}v_{p}+(q-q_{c})v_{g}+\dfrac{1}{2}\Gamma(q-q_{c})^{2}+\dots$ where $v_{p}$
is the phase velocity, $v_{g}$ is the group velocity, and $\Gamma$ is the
dispersion parameter. The final result of this approximation is collected by
the following expression

$\int_{\mathbb{R}}dq\,A(q)e^{i(qx\mp\omega(q)t)}\approx\dfrac{\sqrt{2\pi}}{\sqrt{2\alpha^{2}\pm
i\Gamma t}}\exp\left(-\dfrac{1}{2}\left(\dfrac{x\mp
v_{g}t}{\sqrt{2\alpha^{2}\pm i\Gamma
t}}\right)^{2}\right)\exp\left(iq_{c}(x\mp v_{p}t)\right).$ (A.25)

Notice that this integral is also suppressed in time. Finally, using the
approximation (A.20) in the last term of (A.21), we obtain the only non-
vanishing contribution, that looks like

$\displaystyle R_{2}(x,t)=$ $\displaystyle\dfrac{9\pi
A_{0}^{2}}{2\sqrt{8}\sinh(\sqrt{2}\pi)}\sin(2\sqrt{3}\,t)\sin(2\sqrt{2}\,x)+\dfrac{3\pi
A_{0}^{2}}{2\sinh(\sqrt{2}\pi)}\sin(2\sqrt{3}\,t)\cos(2\sqrt{2}\,x)\tanh x$
(A.26) $\displaystyle-\dfrac{3\,\pi
A_{0}^{2}}{2\sqrt{8}\sinh(\sqrt{2}\pi)}\sin(2\sqrt{3}\,t)\sin(2\sqrt{2}\,x)\tanh^{2}x\,.$

In the asymptotic spatial regime, the previous expression reduces to

$\displaystyle R_{2}(x,t)=\dfrac{3\,\pi
A_{0}^{2}}{2\sinh(\sqrt{2}\pi)}\sqrt{\dfrac{3}{8}}\bigg{(}\cos\big{(}2\sqrt{3}\,t\mp
2\sqrt{2}\,\mathinner{\\!\left\lvert
x\right\rvert}\mp\delta\big{)}-\cos\big{(}2\sqrt{3}\,t\pm
2\sqrt{2}\,\mathinner{\\!\left\lvert
x\right\rvert}\pm\delta\big{)}\bigg{)}\,,$ (A.27)

with

$\delta=\arctan\sqrt{2}\,.$ (A.28)

Note that the Dirac delta approximation picks a single frequency in the
$q-$integral, that is why we have obtained a superposition of two travelling
waves of the same frequencies and opposite directions, i.e., a standing wave.
Choosing the outgoing wave in the positive direction we get

$R_{\infty}(x,t)=\dfrac{3\,\pi
A_{0}^{2}}{2\sinh(\sqrt{2}\pi)}\sqrt{\frac{3}{8}}\cos\big{(}2\sqrt{3}\,t-2\sqrt{2}\,x-\delta\big{)}.$
(A.29)

An alternative derivation of the same result can be achieved as follows. Let
us split the integral of the last term in (A.16) as follows

$\displaystyle I(x,t)=$
$\displaystyle-4i\omega_{s}^{2}\int_{-\infty}^{0}dq\tilde{R}_{0}(q)\tilde{\eta}_{q}(x)\frac{\cos\left(\omega_{q}t\right)-\cos\left(2\omega_{s}t\right)}{4\omega_{s}^{2}-\omega_{q}^{2}}e^{iqx}$
(A.30)
$\displaystyle-4i\omega_{s}^{2}\int_{0}^{\infty}dq\tilde{R}_{0}(q)\tilde{\eta}_{q}(x)\frac{\cos\left(\omega_{q}t\right)-\cos\left(2\omega_{s}t\right)}{4\omega_{s}^{2}-\omega_{q}^{2}}e^{iqx}$
$\displaystyle\equiv I_{1}+I_{2},$

where

$\displaystyle\tilde{R}_{0}(q)$ $\displaystyle=$
$\displaystyle\dfrac{3A_{0}^{2}}{64}\sqrt{\dfrac{q^{2}+4}{q^{2}+1}}\dfrac{q^{2}\left(q^{2}-2\right)}{\omega_{q}^{2}\sinh(\pi
q/2)},$ (A.31) $\displaystyle\tilde{\eta}_{q}(x)$ $\displaystyle=$
$\displaystyle\dfrac{3\tanh^{2}x-q^{2}-1-3iq\tanh
x}{\sqrt{(q^{2}+1)(q^{2}+4)}}.$ (A.32)

Note that values of $q$ close to $0$ as well as large values are suppressed by
the function $R_{0}(q)$. Since the main contribution to the integrals (A.30)
at large $t$ comes from a neighbourhood of $q_{\pm}=\pm 2\sqrt{2}$ (notice
that the modes corresponding to $q=q_{\pm}$ grow linearly with time), the
frequency $\omega_{q}$ can be linearly approximated by

$\omega_{q}\approx 2\sqrt{3}\pm\sqrt{\frac{2}{3}}\left(q\mp
2\sqrt{2}\right)+\mathcal{O}\left(q\mp 2\sqrt{2}\right)^{2}.$ (A.33)

Then, using the linear approximation for $\omega_{q}$ we may approximate
$I_{2}$ by the following expression

$I_{2}(x,t)\approx-4i\tilde{R}_{0}(q_{+})\tilde{\eta}_{q_{+}}(x)\omega_{s}^{2}\int_{0}^{\infty}dq\frac{\cos\left(\widetilde{\omega}_{q}t\right)-\cos\left(2\omega_{s}t\right)}{4\omega_{s}^{2}-\omega_{q}^{2}}e^{iqx}\,,$
(A.34)

where
$\widetilde{\omega_{q}}=2\sqrt{3}+\sqrt{\frac{2}{3}}\left(q-2\sqrt{2}\right)$.
We are assuming implicitly that in a neighbourhood of $q=2\sqrt{2}$, both
$\tilde{R}_{0}(q)$ and $\tilde{\eta}_{q}(x)$ are approximately constant in
$q$. Similarly, a straightforward manipulation leads to the following
expression for $I_{1}$

$I_{1}(x,t)\approx
4i\tilde{R}_{0}(q_{+})\tilde{\eta}^{\ast}_{q_{+}}(x)\omega_{s}^{2}\int_{0}^{\infty}dq\frac{\cos\left(\widetilde{\omega}_{q}t\right)-\cos\left(2\omega_{s}t\right)}{4\omega_{s}^{2}-\omega_{q}^{2}}e^{-iqx}\,.$
(A.35)

We have finally

$I(x,t)=8\omega_{s}^{2}\tilde{R}_{0}(q_{+})\left(\text{Re}\left(\tilde{\eta}_{q_{+}}(x)\right)\mathcal{F}_{s}(t)+\text{Im}\left(\tilde{\eta}_{q_{+}}(x)\mathcal{F}_{c}(t)\right)\right)\,,$
(A.36)

where

$\displaystyle\mathcal{F}_{s}(t)$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}dq\frac{\cos\left(\widetilde{\omega}_{q}t\right)-\cos\left(2\omega_{s}t\right)}{4\omega_{s}^{2}-\omega_{q}^{2}}\sin\left(qx\right),$
(A.37) $\displaystyle\mathcal{F}_{c}(t)$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}dq\frac{\cos\left(\widetilde{\omega}_{q}t\right)-\cos\left(2\omega_{s}t\right)}{4\omega_{s}^{2}-\omega_{q}^{2}}\cos\left(qx\right)\,.$
(A.38)

The integrals (A.37) and (A.38) can be computed analytically for all $x$, but
the expressions are not particularly illuminating. For $x>>0$ and large $t$ we
have

$\displaystyle\mathcal{F}_{s}(t)$ $\displaystyle=$
$\displaystyle-\dfrac{\pi\cos\left(2\sqrt{3}\,t-2\sqrt{2}\,x\right)}{4\sqrt{2}}\,,$
(A.39) $\displaystyle\mathcal{F}_{c}(t)$ $\displaystyle=$
$\displaystyle-\dfrac{\pi\sin\left(2\sqrt{3}\,t-2\sqrt{2}\,x\right)}{4\sqrt{2}}\,.$
(A.40)

Substituting (A.39) and (A.40) into (A.36) we obtain (for $x>\\!\\!>0$)

$I(x,t)=\dfrac{3\,\pi
A_{0}^{2}}{2\sinh(\sqrt{2}\pi)}\sqrt{\dfrac{3}{8}}\cos\big{(}2\sqrt{3}\,t-2\sqrt{2}\,x-\operatorname{arctanh}\sqrt{2}\big{)}\,.$
(A.41)

##  Appendix B Details of the numerical simulations

In this appendix we discuss the numerical coding scheme that we have used to
solve numerically the equation of motion

$\ddot{\phi}-\phi^{\prime\prime}+2\phi(\phi^{2}-1)=0\,,$ (B.1)

where dots and primes denote derivatives with respect to time and space
respectively. In order to solve (B.1) we have discretized the second order
spatial derivative as

$\phi^{\prime\prime}(x,t)=\dfrac{\phi(x+\Delta x,t)-2\phi(x,t)+\phi(x-\Delta
x,t)}{(\Delta x)^{2}}\,,$ (B.2)

and performed the temporal evolution through the leapfrog method. In addition,
we have employed absorbing boundary conditions to avoid that the radiation is
scattered-back and interferes with the system. These conditions read as

$\displaystyle(\partial_{t}+\partial_{x}\phi)\Big{|}_{x=\frac{L}{2},t}$
$\displaystyle=$ $\displaystyle 0\,,$ (B.3)
$\displaystyle(\partial_{t}-\partial_{x}\phi)\Big{|}_{x=-\frac{L}{2},t}$
$\displaystyle=$ $\displaystyle 0\,.$ (B.4)

The simulations have been run in a lattice of length $L=30$ with $\Delta
x=0.0375$, so the number of spatial points is $N_{space}=1600$. Finally, to
guarantee the stability of the method we have chosen $\Delta
t=\dfrac{L}{10(N_{space}+2)}$ , which satisfies the Courant condition with
$\Delta t/\Delta x<0.5$.

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