# GRB 221009A, its precursor and two afterglows in the Fermi data

Boris E. Stern,1 I.I. Tkachev,1,2
1Institute for Nuclear Research of the Russian Academy of Sciences, Moscow
117312, Russia
2Physics Department and Laboratory of Cosmology and Elementary Particle
Physics, Novosibirsk State University, Novosibirsk 630090, Russia
E-mail<EMAIL_ADDRESS>
###### Abstract

We study GRB 221009A, the brightest gamma-ray burst in the history of
observations, using Fermi data. To calibrate them for large inclination
angles, we use the Vela X gamma-ray source. Light curves in different spectral
ranges demonstrate a 300 s overlap of afterglow and delayed episodes of soft
prompt emission. We demonstrate that a relatively weak burst precursor that
occurs 3 minutes before the main episode has its own afterglow, i.e.,
presumably, its own external shock. The main afterglow is the brightest one,
includes a photon with an energy of 400 GeV 9 hours after the burst, and is
visible in the LAT data for up to two days.

###### keywords:

gamma-ray burst: individual, methods: data analysis

††pubyear: 2023††pagerange: GRB 221009A, its precursor and two afterglows in
the Fermi data–GRB 221009A, its precursor and two afterglows in the Fermi data

##  1 Introduction

The recent and brightest gamma-ray burst, GRB 221009A, has been detected by
many space X-ray – $\gamma$-ray observatories, including Fermi  (Veres et al.,
2022; Lesage et al., 2022), Swift (for more detailed analysis see Williams et
al. (2023)), SRG/ART-XC (Lapshov et al., 2022), Konus-Wind (Frederiks et al.,
2022) and others. The burst and its afterglow were also registered by LHAASO
on Earth in the range of hundreds GeV to several TeV (Huang et al., 2022).
Carpet-2 on Baksan Neutruno Observatory has detected an atmospheric shower 250
TeV photon from the location of GRB 221009A (Dzhappuev et al., 2022). On the
other hand HAWC collaboration reports no detection of photons from the
afterglow in TeV range beyond 8 hours after the trigger (Ayala et al., 2022)
and claim the upper limit on the energy flux $4.16\cdot 10^{-12}$ TeV cm-2
s-1.

The burst was intrinsically strong and relatively nearby, z = 0.151. The
apparent brightness of GRB 221009A is exceptional. In Fermi GBM burst catalog
it exceeds the next brightest by factor 15 in energy fluence. However the
strongest impact of this event is due to claims of two photons 18 GeV (LHAASO)
and 250 TeV (Carpet 2) which cannot come from z = 0.15 because of the
absorption on extragalactic background light. There already have appeared
numerous e-prints suggesting new physics to explain these photons.

Taking advantage of the brightness of GRB 221009A we try to find something new
about GRBs themselves in publicly available Fermi data. Namely:

– Is there anything interesting between the precursor of the burst and its
main emission three minutes later.

– What does the transition from the prompt phase of gamma-ray bursts to
afterglow look like?

– How bright is the afterglow and how long can it be traced in the GeV range.

##  2 Data and their calibration

Some raw Fermi data for GRB 221009A are shown in Fig. 1. Large Area Telescope
(LAT) and NaI detectors of Gamma Burst Monitor (GBM) were oversaturated while
Bismuth Germanate (BGO) scintillation detectors satisfactory reproduce the
peak flux in energy channels above $\sim$ 1 MeV. There are no LAT data in the
most interesting intervals 220 – 240 and 260 – 270 seconds (photon detections
lack not only from the GRB direction but from the whole sky).

Figure 1: Time evolution of GRB 221009A in raw data. Prompt phase and early
afterglow. (a) Count rates in various energy ranges on a logarithmic scale,
upper curve GBM NaI 295 - 540 keV, middle curve GBM BGO 22 - 38 MeV, lower
blue curve LAT, $8^{\circ}$ circle around the location of GRB 221009A, yellow
curve - all LAT photons. Dips at 210 and 260 s result from detector
saturation. (b) Individual LAT photons in the $8^{\circ}$ circle around the
position of GRB 221009A (left logarithmic energy scale) and the angle between
the LAT axis and the burst direction (right scale). Note the emission in the
10-200 MeV range 50 s after the precursor.

The next circumstance that makes the direct interpretation of LAT data
problematic is a large angle $\theta$ between the direction of LAT z-axis and
the burst location. The burst has occurred at the very edge of the telescope
field of view where the detection efficiency is low. Fig 1b shows time
dependence of $\theta$ on time during the event. The inclination angle varied
from $75^{\circ}$ to more than $80^{\circ}$ at $t\sim 500$ s when the source
leaved the field of view for an hour.

The angular dependence of LAT effective area is given in Ajello et al. (2021),
see also Fermi LAT Performance. However these data have insufficient
resolution for this specific problem, therefore we have performed a detailed
calibration of LAT detection efficiency for large incident angles using the
brightest GeV source Vela X (both the pulsar and the nebula). The same object
was used by Fermi team for calibration of the point spread function (Ajello et
al., 2021) .

We use photons in $8^{\circ}$ circle around Vela X location as the calibration
sample. The size of this circle is a result of a trade-off between sufficient
containment for $\sim 100$ MeV photons $(>68\%)$ and the contamination of the
sample with background photons. The result of our calibration is shown in Fig.
2.

Figure 2: Calibration data for LAT performance at large incident angles using
Vela-X gamma-pulsar. The ratio of effective area to on-axis effective area as
a function of angle between the photon source and LAT z-axis. Curves from top
to bottom correspond to energy intervals : $>1$ GeV, 1 GeV – 562 MeV, 562 MeV
– 316 MeV, 316 MeV – 178 MeV, 178 MeV – 100 MeV.

The number of photons in our calibration sample is $9.4\cdot 10^{6}$ while the
number of background photons is $~{}2\cdot 10^{6}$ as we have estimated from a
neighbouring site of Milky Way. So, the background is considerable, but its
effect should be moderate because the angular dispersion of background photons
contributes the detection efficiency with different signs: the number of
photons incident at $\theta+\Delta$ is comparable with those arrived at
$\theta-\Delta$. Neverphtheless wide angular cut in calibration sample mimics
some extension of the field of view. We have checked the effect of the wide
angular cut with the same calibration cutting the sample at $4^{\circ}$. The
difference at $\theta=78^{\circ}$ for photons with $E\gtrsim 1$ GeV is
$~{}40\%$ and factor 2.5 for $100-178$ MeV interval. The latter large value is
certainly the effect of widening of point spread function at edge of the field
of view. Therefore we prefer to use the calibration with $8^{\circ}$
calibration sample as it better reproduces the soft end of the spectrum and an
extra background just slightly affects its hard range.

Except angular dependence of detection efficiency one should take into account
the energy dependence. We use that described by Ajello et al. (2021). Note
that for lowest energy bin 100 – 178 MeV that we use the efficiency is 0.45 of
the maximal efficiency. Therefore a 100 MeV photon detected by LAT at 400 s
after the trigger represents $\sim$ 300 photons of the same energy crossing
the detector area (see Fig. 1b and Fig. 2). In our analysis and calibration we
do not distinguish the conversion type and the quality class of photons, using
total effective area for the "source" event class.

##  3 Main episode and onset of the afterglow

Normalized Fermi data for the first 600 seconds are shown in Fig. 3. LAT data
were normalized using our Vela X calibration results (Fig. 2) and energy
dependent effective area from Ajello et al. (2021). Total effective area of
two BGO detectors was set to 200 cm2 independently on energy and incident
angle as such assumption is sufficient for qualitative demonstration. For the
energy and angular performance of BGO detectors see Meegan et al. (2009).

BGO 22 - 38 MeV energy flux in 300 - 600 s interval is very sensitive to the
background model. We use three parametric description: constant plus one
sinusoidal half period with fitting intervals 300 - 150 s and 600 - 1300 s.
Resulting $\chi^{2}$ is good, however the error in energy fluence in BGO us
large therefore we do not take BGO data into consideration when reconstructing
photon spectra. We see a striking transition in time behaviour at $\sim$ 300
s: a sharp decline of main pulse changes to a flat smooth slope. The exception
is another soft episode of prompt emission at 400 - 600 s which we discuss
below. It would be reasonable to suggest then the high energy emission after
300 s can be considered as the main afterglow.

Figure 3: Prompt GRB and early afrerglow in different energy ranges.
Histogram: energy flux of LAT photons normalized to angular-dependent
effective area calibrated with Vela-X source.Green line: energy flux in 0.93 -
2.1 MeV range from Gamma-Burst Monitor BGO detectors. Blue circles: the same
in 22 - 38 MeV energy band.

Phenomenologically, prompt emission usually undergoes a very fast variation of
intensity and spectrum while afterglow has a smooth long decline with a stable
wide spectral energy distribution. Prompt emission sometimes consists of
multiple pulses of different duration and spectra, this pulses can overlap in
time producing in some cases complex structures with a wide temporal Fourier
power spectrum (Beloborodov et al., 2000). Their time behaviour is very
diverse. On the contrary, all afterglows have typical time behaviour: a power
law decline slightly faster than $t^{-1}$.

Theoretically, there exists a paradigm that the prompt emission arises from
internal shocks (or magnetic reconnection or both) in the jet while the
afterglow from external shock form collision of the jet with ambient medium
(see e.g. Piran (2005)), dominated by the stellar wind of the GRB progenitor
(Chevalier & Li, 1999). Physically, the prompt emission in many cases can be
described as a radiation of an optically thick medium due to multiple
Comptonization (e.g. Ito et al. (2018)), while afterglow better corresponds to
synchrotron/Compton radiation of electrons accelerated in an optically thin
environment, see Nava (2018) for a review.

In the case of GRB 221009A we can describe as the prompt emission the
precursor, a soft pulse at $t\sim 180\,-\,200$ s, two main hard pulses and a
soft long structure in $400\,-\,600$ s interval. Photon flux detected by LAT
since 250 s with its spectrum (see Fig. 4) and light curve resembles an
afterglow rather than a prompt emission. The structure at 400 - 600 s is
probably an independent prompt emission episode overlapping in time with the
early afterglow.

Probably the afterglow mechanism (presumably the external shock) has turned on
slightly earlier, e.g at 230 s. Thereafter we use this time as a reference
point for a power law decline of the afterglow.

Figure 4: Spectral energy distribution of photons detected by LAT at time
intervals 230 - 500 s and 4000 - 6000 s reconstructed with calibrated
effective area (see Fig. 2).

Our estimate of the energy fluence represented by 229 photons detected by LAT
in the time interval 220 - 500 s is $1.55\cdot 10^{-3}$ erg/cm2. Actual
fluence could be several times higher since we don not know how many photons
are lost in over-saturation gaps. The low energy fluence preliminary estimated
by Lesage et al. (2022) is $2.9\cdot 10^{-2}$ erg/cm2, the total energy
fluence can be much higher, see Frederiks et al. (2023). The spectral energy
distribution in 100 MeV - 100 GeV range is shown in Fig. 4. We made a forward-
folding power law fit to the numbers of detected photons in energy bins. The
resulting photon index for LAT photons in background-free 8∘ field is
-1.92$\pm 0.04$ which is consistent with estimate of Pillera et al. (2022)

##  4 The main afterglow

After 500 seconds since the trigger the GRB came out of LAT field. The next
time window was open from $\sim 4100$ to $\sim 5700$ seconds. Then the
location of GRB221009 periodically appeared in the field of view with duty
cycle $\sim 20$ %.

Figure 5: Events in the $1^{\circ}$ circle around GRB 221009A detected by LAT.
(a) Photons in the time interval $10^{3}$ \- $3\cdot 10^{5}$ s since the
trigger (green squares) and background photons in the interval of the same
length but before trigger (red circles). (b) The same but as a histogram. The
average background normalized to the same logarithmic bins is shown by the
green line. The estimate of the background has been done with $8^{\circ}$
field around the GRB location for 300000 seconds before the burst.

Fig. 5 shows counts of photons detected by LAT versus logarithm of time.
Unlike the main episode which is essentially background free, the background
during the late afterglow is considerable and is too large in $8^{\circ}$
field of view which we accepted for the description of the main episode. For
this reason we analyse the afterglow using $1^{\circ}$ circle. The background
estimated with photons detected from the same direction during 300 000 seconds
prior the trigger is shown in Fig. 5. With this window the afterglow is
significant up to two days (see Fig. 5b). Further contraction of the field of
view does not improve the significance.

Fig. 5a shows LAT photons in log-log scale for $1^{\circ}$ field. It looks
like the afterglow is getting softer with time, however note a 400 GeV photon
at $\log(t)\sim 4.5$ (33554 s). The angular deviation of this photon from the
GRB location is $0.06^{\circ}$, the probability of chance coincidence with
such angular deviation during a day is $\sim 10^{-6}$. This photon is missing
in the telegram of Fermi team, but was noticed by Xia et al. (2022). The
impression of softening can result from the lack of soft photons in the main
episode and their excess at $t\sim 10^{5}$ s. The former can be explained by a
stronger suppression of soft photons at large incident angles (see Fig. 2a)
and the latter by the contribution of a softer background. The statistics is
insufficient to reveal an evolution of the afterglow spectrum.

Figure 6: The afterglow of several $\gamma$-ray bursts. Black: GRB221009A,
squares with error bars – this work, dot with an upper limit – HAWC
Collaboration. Blue: GRB 190114C, dots – MAGIC (0.3 - 1 TeV), stars with error
bars – Fermi LAT. Green: GRB 190829A – H.E.S.S. (0.2 - 4 TeV). Violet: GRB
180720B – H.E.S.S. (100 - 440 GeV). Red: GRB 130427A – Fermi LAT.

The spectral energy distributions for the beginning of the afterglow and for
the second window are shown in Fig. 4. They are consistent with each other and
just slightly differ from the "canonical" spectrum with flat SED (photon index
$\alpha=-2$). We suggest that it would be reasonable to set the spectral index
to -2 when fitting the afterglow energy flux. Fig. 6 shows the photon flux of
the afterglow versus time. The flux is normalized to $3^{\circ}$ field of view
using energy dependent relative containment in $3^{\circ}$ and $1^{\circ}$
circles averaged over spectrum with $\alpha=-2$. We add for comparison some
data points for high energy fluxes of other bright GRBs: GRB 130427SA,
(Ackermann et al., 2014), GRB 180720B (Abdalla et al., 2019), GRB 190114C
(Acciari et al., 2019) and GRB 190829 (Abdalla et al., 2021). These data are
summarised and discussed by Miceli & Nava (2022). The afterglow of GRB is
slightly brighter than afterglow of GRB 130427A and almost an order of
magnitude brighter than GRB 18072B , 190114C and 190829 which were detected by
Cherenkov telescopes. This is the first case when the afterglow is still
visible in the Fermi range two days after the GRB. Intriguingly, it was not
detected by the HAWC Collaboration (Ayala et al., 2022), see Fig. 6.

The afterglow is still $3\sigma$ significant in 100 000 – 200 000 s interval:
the number of photons is 40 versus 23 photons in the "mirror" -100 000 –
-200000 s interval. The afterglow light curve in 0.1 - 10 GeV range can be
described as a power law $F\sim t^{-\alpha_{\tau}}$, where
$\alpha_{\tau}=1.32\pm 0.05$.

##  5 The afterglow of the precursor

Fermi LAT detected 6 photons in the range 20 - 200 MeV in time interval 7 - 50
seconds after the trigger (Fig. 1b) when the angle $\theta$ varied from
$65^{\circ}$ to $70^{\circ}$ and LAT effective area was several times less
than for on-axis photons (see Fig. 2). Note that the prompt precursor emission
already has relaxed when LAT has detected first of these photons. We estimate
the energy fluence represented by these 6 photons as $~{}1.5\cdot 10^{-6}$ erg
cm-2.

How significant is this "preafterglow"? We have counted photons from
$8^{\circ}$ circle when the orientation of LAT z-axis to the GRB was
$63^{\circ}<\theta<71^{\circ}$ during 30 000 seconds before the burst. The
result is 302 photons for 23400 seconds which gives the expectation 0.64
photons for 50 s interval after the precursor. The probability to sample 6
photons by chance is $0.5\cdot 10^{-4}$ ($4\sigma$). Note that there is no
"look elsewhere" effect in this case: the photons appear in a proper place
with no sample manipulation. Therefore this significance is quite sufficient
to claim that Fermi has detected an afterglow of a precursor. This is the
first case of such detection. The idea that a GRB precursor could produce its
own afterglow was suggested by Nappo et al. (2014).

The spectral fit to 6 photons is, of course, quite loose, the resulting
spectral index is $\alpha=-2.5\pm 0.5$ which is consistent with the spectrum
of the main afterglow with $\alpha\sim-2$.

##  6 Discussion and conclusions

Probably the most interesting fact that we see in Fermi data is the afterglow
of the precursor. This is the first direct evidence of such phenomenon at
least if we treat a precursor as a relatively weak event separated from the
main episode by a long time interval.

A weak precursor that occurs long before (up to several minutes) the main GRB
is a feature of many GRBs. The estimates of their occurrence varies from 3%
GRBs (Koshut et al., 1995) up to 10% (Troja et al., 2010) or even 20%
(Lazzati, 2005). A useful review of the phenomenon is given by Zhu (2015). In
fact, this fraction may be even higher, since the precursor can be easily
lost. The precursor of GRB 221009A has a few hundred times lower peak count
rate and several thousand times lower energy fluence than the main emission
episode. Such relatively weak precursor can by detected only in rare cases of
very strong GRBs and one can not exclude that this phenomenon is common for
the majority of GRBs and we observe just strongest precursors.

However this weak precursor has a bright afterglow. The energy fluence of the
precursor and its afterglow is: $\sim 1.7\cdot 10^{-5}$ erg cm-2 (0.1 - 4.8
MeV, BGO) and $\sim 1.5\cdot 10^{-5}$ erg cm-2 correspondingly. The estimate
by Minaev et al. (2022) of the precursor fluence is $2.38\pm 0.04\cdot
10^{-5}$ erg cm-2. Therefore, they are comparable, while the main afterglow is
much weaker than the main prompt emission episode: $4.7\cdot 10^{-4}$ erg cm-2
versus $3.3\cdot 10^{-2}$ erg cm-2 (our estimate using BGO counts in 0.1 - 4.8
MeV range), or even 0.21 erg cm-2 in 0.02 - 10 MeV band according to estimate
of Frederiks et al. (2023). The difference in this ratio is two orders of
magnitude. This is a hint that the precursor could differ from the main GRB in
its nature. If we follow the paradigm that the prompt emission originates from
internal shocks in the jet and the afterglow comes from external shock in the
ambient medium (see Piran (2005)), then we have to conclude that internal
shock is pathologically weak in the case of the precursor. In principle, one
cannot exclude that a precursor with its afterglow is an essentially different
phenomenon preceding the main emission. Unfortunately, there is a very small
chance to observe such afterglow directly in the nearest future, nevertheless
it probably could be sensed statistically using existing data bases.

As for the main afterglow, this is the brightest one as well as the GRB 221009
itself. In other respects (relative energy flux, spectrum, decay law) it looks
typical. The afterglow due to its brightness is visible in Fermi data during
two days and could be visible even longer by Cherenkov telescopes unless bad
observational conditions including bright moon.

HAWC collaboration has reported no detections beyond 8 hours after the event
and set the upper limit $4.16\cdot 10^{-12}$ erg cm-2 s-1 (Ayala et al.,
2022). This upper limit is in apparent tension with Fermi data unless one
implies a sharp spectral decline between $\sim 10$ GeV and TeV energy ranges.
However other GRB do not have such decline in their afterglows which is
supported by data of Cherenkov telescopes for other GRBs (cf. Fermi and MAGIC
data in Fig. 6). Moreover, the 400 GeV photon detected at 9 hours is an
argument against a spectral cutoff. The publication of LHAASO results may
clarify this issue. Note that the second brightest afterglow of GRB 130427A
also has not been detected above 100 GeV, the upper limit set by VERITAS (Aliu
et al., 2014) $\sim 10^{-11}$ erg cm-2 s-1 at the next day could be in some
tension with Fermi data too.

## Acknowledgements

We thank Fermi team for the excellent data base and NASA for the open data
policy.

## Data Availability

This work is based on publicly available Fermi database and the quantitative
description of the Fermi Large Area Telescope performance. The work of I.T.
was supported by the Russian Science Foundation grant 23-42-00066.

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# WikiGUM: Exhaustive Entity Linking for Wikification in 12 Genres

Jessica Lin
Department of Linguistics
Georgetown University
<EMAIL_ADDRESS>
&Amir Zeldes
Department of Linguistics
Georgetown University
<EMAIL_ADDRESS>

###### Abstract

Previous work on Entity Linking has focused on resources targeting non-nested
proper named entity mentions, often in data from Wikipedia, i.e. Wikification.
In this paper, we present and evaluate WikiGUM, a fully wikified dataset,
covering all mentions of named entities, including their non-named and
pronominal mentions, as well as mentions nested within other mentions. The
dataset covers a broad range of 12 written and spoken genres, most of which
have not been included in Entity Linking efforts to date, leading to poor
performance by a pretrained SOTA system in our evaluation. The availability of
a variety of other annotations for the same data also enables further research
on entities in context.

##  1 Introduction

Entity linking (EL) involves identifying entities within a text and
subsequently linking their mentions to a knowledge base or table of
authorities. The former step is often referred to as Named Entity Recognition
(NER) and the latter may also be referred to as entity disambiguation. In this
study, we will focus on the latter task by following the popular approach of
mapping named entities to Wikipedia entities Milne and Witten (2008);
Shnayderman et al. (2019), i.e. Wikification.

Wikification is the task of adding links to Wikipedia pages to mentions of
named entities in a written or spoken text. This task supports Natural
Language Understanding in downstream tasks such as question answering,
summarization, and relation extraction. However the scope and structure of EL
depends heavily on datasets which are either automatically derived from
hyperlinked text and thus suffer some limitations, or are created via human
annotation, a time-consuming and expensive task. Despite numerous existing EL
datasets Cucerzan (2007); Ji et al. (2015); Kulkarni et al. (2009); Milne and
Witten (2008); Ratinov et al. (2011), few have attempted to capture nested
entity structure, as in Figure 1, which never occurs in hyperlinks, which
cannot be nested.

Figure 1: Nested entity linking.

Instead, annotations have focused on flat mention structure from popular
online sources, leaving out important information in nested entities that can
be useful for downstream tasks. Closest to the resource presented here is the
Nested Named Entities (NNE) dataset Ringland et al. (2019), which is a large,
manually-annotated, nested named entity dataset over English newswire,
however, it does not include entity linking. Although NNE includes fine-
grained semantic information in nested entity types, it is not linked to any
identifiers (e.g. a Wikipedia page). Furthermore, even if used for mention
recognition, the data is not ideal for testing on diverse genres, as NNE
solely covers news text. We also note other datasets capturing some nested
entity structure, such as the Abstract Meaning Representation (AMR) corpus
(Banarescu et al., 2013), which includes compositional nesting e.g. in
possessives such as Toronto’s international airport, composed of a city and an
airport. However, since AMR is not word-aligned to text, even those nested
entities that are covered are not aligned to their textual position.

In this paper, we present and evaluate a gold standard wikified dataset,
called WikiGUM, in which named and non-named entities have been annotated
manually. WikiGUM is based on the existing GUM dataset (Georgetown University
Multilayer corpus, Zeldes 2017), and goes beyond other EL corpora, in covering
all mentions of named entities (NEs), including non-named and pronominal
mentions, as well as nested mentions, for 12 genres of English text. WikiGUM
also enables assessment of EL annotations by highlighting challenges that are
common in our dataset, and reveals the relatively poor coverage of state-of-
the-art NLP systems for EL in diverse genres (Section 4). Taken together, we
aim to facilitate new research on nested NER and EL, to promote recognition of
all NE mentions and a deeper understanding of the hierarchical structure of
entities in text.

##  2 WikiGUM

The underying GUM Zeldes (2017) corpus is a manually annotated dataset with
multiple layers, including POS tagging (Penn tags, CLAWS5, Universal POS),
sentence types (e.g. declarative, imperative, yes/no question), UD dependency
trees Nivre et al. (2016), coreference resolution (including bridging anaphora
and split antecedents), and RST discourse parses Mann and Thompson (1988).
Data covers 12 genres: academic, biographies, conversation, fiction, forums,
how-to, interviews, news, speeches, textbooks, travel and vlogs.

Text Types | Source | Documents | # of NE Mentions | # of Nested Wikified Mentions | Total Mentions | Tokens
---|---|---|---|---|---|---
Interviews | Wikinews | 19 | 1,146 | 107 | 5,204 | 18,037
News stories | Wikinews | 21 | 1,221 | 217 | 4,130 | 14,094
Travel guides | Wikivoyage | 17 | 1,327 | 174 | 4,087 | 14,955
How-to guides | WikiHow | 19 | 94 | 8 | 4,469 | 16,920
Academic writing | various | 16 | 329 | 48 | 4,486 | 15,110
Biographies | Wikipedia | 20 | 2,450 | 413 | 5,763 | 17,951
Fiction | various | 18 | 195 | 9 | 4,737 | 16,307
Forum discussions | Reddit | 18 | 196 | 2 | 4,530 | 16,286
Conversations | UCSB corpus | 5 | 31 | 0 | 1,477 | 5,698
Political speeches | various | 5 | 316 | 32 | 1,423 | 4,831
CC Vlogs | YouTube | 5 | 39 | 1 | 1,355 | 5,180
Textbooks | OpenStax | 5 | 188 | 21 | 1,507 | 5,376
Total | 168 | 7,352 | 1,032 | 43,168 | 150,745

Table 1: Statistics on WikiGUM

WikiGUM adds a layer of Wikipedia identifiers to all NEs in GUM, which are
identified automatically by having the gold PTB POS tag NNP(S) for their
syntactic head, based on gold syntax trees (for some resulting issues, see
below), as well as non-named mentions coreferring to them based on coreference
annotations. Since GUM is expanded by students in classroom annotation every
year, and we plan to continue adding Wikification in the future, no closed or
pre-prepared ontology is applied to the Wiki identifiers, making the task
simpler for student annotators who only need to find a corresponding Wikipedia
article.

That said, the existing 10 entity types in GUM111Types: person, place, org,
animal, plant, event, time, substance, abstract and inanimate object. mean
that our EL benefits from the same categorization scheme as a rough ontology,
and the availability of semantic information from WikiData means that many
relationships between entities can be explored. All referential NPs, including
pronouns and even clauses (if they co-refer with a named entity based on GUM’s
coreference annotations, for example movie titles), were selected as markables
for annotation. Note that nested markables are always included, for example:

. [the airport in [Cuba]place]place

Our general guideline for entity linking is that NEs, including pronominal and
non-named mentions, were manually linked to the corresponding Wikipedia
article whenever one exists, using the version controlled online editor GitDox
Zhang and Zeldes (2017). For example:

. Kim likes [The Terminator]abstract. [This movie]abstract is her favorite.

In this example, the span This movie should also be linked to the Wikipedia
page that refers to The Terminator (the movie). Statistics on WikiGUM, which
is freely available under the same Creative Commons license as GUM, are shown
in Table 1.

Although the basic Wikification task is fairly straightforward, some
ambiguous/tricky cases during annotation included:

  * •

Generic terms: some capitalized common nouns that have Wikipedia links appear
within NEs, and are tagged NNP(S), but do not correspond to named entities.
For example, Oil is incorrectly proposed as a NE due to capitalization within
the NE the Oil Capital of the World (referring to Tulsa, OK) and due to the
POS tag NNP. It can be tempting to link ‘oil’ as a NE candidate to the
Wikipedia article ‘Petroleum’. However in context, ‘oil’ is a generic, non-
named modifier to ‘Capital’, and should not be linked as a NE. Annotators
should be mindful of context of terms tagged NNP(S) within NEs, rather than
linking any NNP span.

  * •

Subset of entity with the same type: a common type of ambiguity for place
entities arises when names are reused in different countries, regions, cities
or villages. For example, terms like ‘North’, ‘South’, ‘East’, and ‘West’ as a
subset of a region are hard to disambiguate, and they are common in street
names in North America. In this case, annotators must look at the broader
context and carefully check whether the entity refers to a subset or not, for
example cities and their metropolitan areas, streets with and without cardinal
directions, or other parts of cities which sometimes have separate Wikipedia
entries.

  * •

Distinct links for identical mentions: It is sometimes hard for annotators to
realize that an entity string has several EL variants. This happens often in
abbreviations, which may be labelled with the wrong entity type. For example,
‘JFK’ can be a person’s name (the 35th US President) or a place name (JFK
Airport in New York), depending on context. We instructed annotators to
prioritize the existing entity type annotation: if JFK is tagged as a place,
it is linked to the article about the airport. Another common issue affects
ancient place names which do not exist nowadays, resulting in difficulty for
EL. For instance, Jorvik is the viking name of York, and was therefore linked
to the closest equivalent article, ‘Scandinavian York’ rather than ‘York’. In
other cases, we relied on the coreference annotations to establish
equivalence: for example England’s City of Festivals was labeled as
coreferring with York, and was therefore considered equivalent to York for EL
purposes.

  * •

Lack of background knowledge: In some cases context alone cannot help
annotators decide on the right sense of an entity, especially in academic
texts, but also in discussion forums. Academic articles often assume readers
have detailed knowledge of the topic and thus provide little context for the
target entity. For example, ‘Su’ in ‘Su et al. 2016’ is a named entity, but it
may be difficult to know whether there is a corresponding Wikipedia article
based solely on the author’s name.

##  3 Related work

Table 2 compares WikiGUM to other EL corpora. Most current EL datasets are
based on newswire text, overlooking the impact of genre on EL – for example,
Dai (2018) notes that the biomedical domain involves complex and unique entity
mentions. As EL datasets are developed for evaluation of EL systems, out-of-
domain data could create challenges for conventional tools. Furthermore, most
previous work Cucerzan (2007); Ji et al. (2015); Kulkarni et al. (2009); Milne
and Witten (2008); Ratinov et al. (2011) has focused on identifying and
classifying atomic, flat mention structures, leaving out the semantic
information available in nested mentions.

As shown in Table 2, most datasets do not contain nested entity linking
annotations, with ACE2004 being the exception Ratinov et al. (2011).
Unfortunately, no dataset covers all mentions of named entities, including
their non-named mentions (‘the same airport’, or ‘it’). As mentioned above, we
do see some research on nested entity structure Glavaš and Šnajder (2014);
Hong et al. (2016), for example the NNE corpus Ringland et al. (2019) contains
fine-grained semantic information including e.g. the category city nesting a
state, which could easily be used for EL. However they are not disambiguated
or linked to a table of authorities, in addition to excluding non-named
mentions of the same entities. WikiGUM thus differs from previous EL datasets
and is rich in terms of both genre and entity structure, as well as being
among the larger available datasets as shown in the Table.

Dataset | Paper | # of documents | # of NEs | # of genres | NE/N | Pronouns | Nested Entities
---|---|---|---|---|---|---|---
WikiGUM | – | 168 | 7,352 | 12 | NE&N |  |
ACE2004 | Ratinov et al. (2011) | 36 | 256 | 1 | NE |  |
AIDA-A | Hoffart et al. (2011) | 216 | 5,917 | 1 | NE |  |
AIDA-B | Hoffart et al. (2011) | 231 | 5,616 | 1 | NE |  |
AQUAINT | Milne and Witten (2008) | 50 | 727 | 1 | NE&N |  |
Derczynski | Derczynski et al. (2015) | 182 | 210 | 1 | NE |  |
IITB | Kulkarni et al. (2009) | 107 | 17,200 | 1 | NE |  |
KORE50 | Hoffart et al. (2012) | 50 | 148 | 1 | NE |  |
MSNBC | Cucerzan (2007) | 20 | 656 | 1 | NE |  |
n3-RSS-500 | Röder et al. (2014) | 500 | 1,000 | 1 | NE |  |
n3-Reuters-128 | Röder et al. (2014) | 128 | 880 | 1 | NE |  |
OKE2015 | Nuzzolese et al. (2015) | not specified | 718 | 1 | NE&Roles |  |
OKE2016 | Nuzzolese et al. (2016) | not specified | 940 | 1 | NE&Roles |  |

Table 2: English EL datasets. NE/N indicates whether only named entities (NE)
or also common nouns (N) are included. Note that ACE 2004 is a subset of the
documents used in the ACE 2004 coreference documents.

##  4 Evaluation

In this section we evaluate inter-annotator agreement, as well as the extent
to which existing Wikification technology already captures the information in
WikiGUM.

### Inter-annotator agreement

Measuring agreement for Wikification involves two main complementary aspects:
span detection and Wikification (including the decision whether to link an
entity and to what). Since GUM already contains mention boundaries and
named/non-named status, we focus on the latter task, measuring linking
agreement. To calculate agreement, we carried out an inter-annotator agreement
experiment by double annotating 3,103 tokens of corpus data containing 237
entities after adjudication, about 3% of the data. We compute both Cohen’s
Kappa and simple percent agreement (percentage of exact match), shown in Table
3. Note that computing Cohen’s Kappa here is somewhat artificial, as in the
real world there is an (almost) unlimited space of possible Wikipedia
identifiers. For simplicity, we define the space of possible links as the
union of any values annotators used in this subset, meaning that any link
chosen at any point by any annotator is considered a possible value for the
annotation, and any disagreement is penalized by the metric.222An anonymous
reviewer has asked whether this means that search ambiguity and an
overwhelming number of options impacted our process: this is certainly true,
and somewhat inevitable given that annotators were unrestricted in the
identifiers they could choose from Wikipedia. However the high level of
absolute agreement suggests that in practice annotators were surprisingly
internally consistent.

Metric | Score
---|---
Agreement | 0.8903
Cohen’s $\kappa$ | 0.8782
Table 3: Results of the inter-annotator agreement experiment

Results in Table 3 show that agreement is far beyond chance, with Kappa=.87
and simple agreement of .89. While this indicates very good agreement, raters
did disagree on ambiguous cases, which is worth discussing. A major source of
disagreement involves linking the same entity string to distinct but related
identifiers, i.e. the name variants issue highlighted in Section 2. This is
often due to lack of context information: for example, in the sentence “CC
makes things more complex”, the mention CC could be linked to Creative Commons
license (public copyright license) or Creative Commons (organization that
produced the Creative Commons license). In this case, broader context and
reasoning are required to make consistent decisions. Another example is the
sentence “According to the Arts and Humanities Citation Index Professor
Chomsky is the eighth most cited scholar of all time.”, the mention Arts is
not a NE and should not be linked, but was linked by one annotator to “The
arts”, which has a linkable article and can easily be confused with a named
concept of sorts.

### NLP coverage

To evaluate the usefulness of WikiGUM beyond existing resources, we test a
recent SOTA pretrained end-to-end neural system (e2e, Kolitsas et al. 2018) on
the test set and compare it to a baseline strategy. Our baseline system uses a
neural constituent parser Mrini et al. (2020) to identify predicted noun
phrases and simply checks the exact string of every phrase headed by a proper
noun to see if it has a Wikipedia article (using the Python library
wikipedia). Since the SOTA system cannot identify nested mentions, but we do
not know which one of two nested mentions it might identify (the bigger or
smaller one), we evaluate in multiple scenarios: counting all mentions, only
unnested mentions, and, since the system cannot identify pronouns, with and
without them.

| data | P | R | F1 | links
---|---|---|---|---|---
e2e | all | 0.398 | 0.192 | 0.259 | 827
| -pron | 0.441 | 0.259 | 0.327 | 677
| -nest | 0.363 | 0.203 | 0.260 | 713
| -both | 0.363 | 0.253 | 0.298 | 573
baseline | all | 0.480 | 0.182 | 0.264 | 827
| -pron | 0.480 | 0.223 | 0.304 | 677
| -nest | 0.363 | 0.203 | 0.260 | 713
| -both | 0.391 | 0.214 | 0.277 | 573
Table 4: Baseline and e2e results on WikiGUM test set.

The results in 4 show that e2e, even when trained on the largest available
Wikification dataset (AIDA, $\approx$1,000 documents, 18K links) does not
generalize well to the domains found in our corpus, barely outperforming a
naive lookup baseline. Comparing the scenarios, we see that best performance
for both systems is achieved when removing pronouns, which is unsurprising
since neither strategy can be expected to link them. However removing nested
mentions does not result in higher scores: this is because some common
targets, such as places, are often nested in larger names (organizations,
office-holders), and removing them disrupts score gains from their correct
identification. Nevertheless, the most lenient possible evaluations are in the
low 30s, as opposed to a score of 82.6 on AIDA (Kolitsas et al., 2018, 524).
This suggests that, unsurprisingly, the corpus covers a range of entities and
contexts that are under- or unrepresented in previous benchmarks.

##  5 Conclusion

This paper presented WikiGUM, the first exhaustive Wikification dataset for
named entity linking, including nested and pronominal mentions in 12 genres of
English text. Our evaluation suggests a high level of agreement, as well as
coverage for a significant amount of entities not retrieved by a SOTA neural
linking system. We hope that this dataset will enable further research on
entity linking and increase coverage for all types of linkable named entities
across a broad spectrum of genres, both spoken and written.

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# Kink-antikink scattering-induced breathing bound states and oscillons in a
parametrized $\phi^{4}$ model

F. Naha Nzoupe  Laboratory of Mechanics, Department of Physics, Faculty of
Science, University of Yaoundé I P.O. Box 812 Yaoundé, Cameroon
Alain M. Dikandé  Laboratory of Research on Advanced Materials and Nonlinear
Science (LaRAMaNS), Department of Physics, Faculty of Science, University of
Buea P.O. Box 63 Buea, Cameroon.
<EMAIL_ADDRESS> C. Tchawoua  Laboratory of Mechanics, Department of
Physics, Faculty of Science, University of Yaoundé I P.O. Box 812 Yaoundé,
Cameroon

###### Abstract

Recent studies have emphasized the important role that a shape deformability
of scalar-field models pertaining to the same class with the standard
$\phi^{4}$ field, can play in controlling the production of a specific type of
breathing bound states so-called oscillons. In the context of cosmology, the
built-in mechanism of oscillons suggests that they can affect the standard
picture of scalar ultra-light dark matter. In the present work kink
scatterings are investigated in a parametrized model of bistable system
admitting the classical $\phi^{4}$ field as an asymptotic limit, with focus on
the formation of long-lived low-amplitude almost harmonic oscillations of the
scalar field around a vacuum. The parametrized model is characterized by a
double-well potential with a shape-deformation parameter that changes only the
steepness of the potential walls, and hence the flatness of the hump of the
potential barrier, leaving unaffected the two degenerate minima and the
barrier height. It is found that the variation of the deformability parameter
promotes several additional vibrational modes in the kink-phonon scattering
potential, leading to suppression of the two-bounce windows in kink-antikink
scatterings and the production of oscillons. Numerical results suggest that
the anharmonicity of the potential barrier, characterized by a flat barrier
hump, is the main determinant factor for the production of oscillons in
double-well systems.

###### keywords:

scalar field; parametrized $\phi^{4}$ model; instantons; kink-antikink
collision; oscillons.

Received (Day Month Year)Revised (Day Month Year)

PACS Nos.: 03.50.-z; 05.45.Yv; 11.10.St; 03.65.Nk.

##  1 Introduction

The generation and interactions of solitary waves and solitons have attracted
a great deal of interest over the past years, due to the fact that they can
control many features related to the dynamics of natural systems ranging from
biology and organic polymers, to classical and quantized fields in condensed-
matter and high-energy physics [1, 2, 3, 4, 5, 6]. The simplest localized
solutions known in field theory are kink and antikink solitons, they display
topological profiles in ($1+1$) space-time dimensions and can be generated in
classical as well as quantum scalar field systems.

In non-integrable scalar field theories such as the $\phi^{4}$ field [4, 7],
scatterings of a kink-antikink pair usually give rise to a competition between
a bion state and a two-soliton solution characterized by a fractal structure
in the parameter space of scattering velocity [8]. For some impact velocities
the kink-antikink collision will give birth to a breather-like bound-state
(bion) solution, that radiates progressively until a total annihilation of the
pair. For other ranges of velocities, the pair performs an inelastic
scattering with the solitons colliding once and separating thereafter. There
also exist particular regions in velocity ($n-$bounce windows), where the
scalar field at the center of mass can bounce several times ($n$ times) before
the final separation of the pair. The later $n-$bounce windows have been
explained as the consequence of a resonance mechanism for the exchange of
energy between the vibrational and translational modes, resulting from
discrete eigenstates of the Schrödinger-like equation inherent to the
stability analysis of the $\phi^{4}$ kink [8, 9].

The last decade has witnessed a regain of interest in kink scatterings in non-
integrable models, marked by intensive studies for instance of multi-kink
collisions[10, 11, 12, 13, 14, 15], the interactions of a kink or an anti-kink
with a boundary or a defect[16, 17], the scattering processes in models with
generalized dynamics [18], nonpolynomial models [19, 20, 21, 22], polynomial
models with one [23, 24, 16, 25, 26, 27, 28, 29, 30, 31, 32] and two [33, 34,
35, 36, 37] scalar fields and so on. However all these studies involve mostly
two universal models which are the sine-Gordon model [1], assumed to describe
systems with periodic one-site potentials, and the $\phi^{4}$ model intended
for physical systems with double-well (DW) potentials. Although the $\phi^{4}$
kink for example has very recently been linked with topological excitations
observed in buckled graphene nanoribbon [38], real physical systems to which
the two universal models address are actually rather quite diverse, and most
often unique in some aspects of their physical features. Indeed the $\phi^{4}$
and sine-Gordon model have fixed extrema while their shape profiles, including
their their potential barriers, are rigid which confine their applicability to
a very narrow class of physical systems. To lift the shortcomings related to
the rigidity of shape profiles, these two universal models have been
parametrized leading to two hierarchies of deformable-shape one-site
potentials i.e. the Remoissenet-Peyard periodic potential [39, 40, 41], and
the family of Dikandé-Kofané (DK) DW potentials [41, 42, 43].

In two recent studies [22, 44], Bazeia et al addressed the issue of the
influence of shape deformability of DW potentials, on kink-antikink
scatterings with production of oscillon bound states [14, 22, 44]. Thus they
first applied the shape deformability procedure to the standard $\phi^{4}$ by
introducing a bistable model with non-polynomial potential, which they called
sinh-deformed $\phi^{4}$ potential [22]. Despite this new model showing
similar features with the $\phi^{4}$ model a new phenomenon was observed,
indeed under certain conditions the kink-antikink pair in the new model was
found to convert, after collision, into long-lived low amplitude and almost
harmonic oscillations of the scalar field around one vacuum. They interpreted
these almost harmonic oscillations as a bound state of individual oscillons
[45]. Later on the authors investigated [44] kink-antikink collisions with
production of oscillons, considering two members of the family of DK DW
potentials [42, 43, 46, 47]. One member was a DW potential with variable
separation between the two degenerate minima but with fixed barrier height
[42], and the other member was the DW potential with variable barrier height
but fixed positions of the two degenerate minima. The oscillons production in
these two members of the family of DK DW potentials was established, and shown
to occur when the distance between the minima gets smaller for the first
member, and when the barrier height becomes lower for the second member. Based
on their results with the two DK DW models, the authors concluded that the
lowering of kink energy with increase of the shape deformability parameter was
the determinant factor favoring the production of oscillons in the two models.

Stimulated by the studies of Bazeia et al [22, 44], in the present work we
investigate kink-antikink collisions and the possible production of oscillons
in a DW model with fixed barrier height and fixed separation between the two
degenerate minima, but a variable curvature of the barrier hump. With this we
wish to establish that parametrized DW models with increasing kink energy as a
function of a deformability parameter, are also quite prone to production of
oscillons upon kink-antikink collisions. In fact we will show that the
anharmonicity of the potential at its maximum, characterized by a flat barrier
hump with increasing deformability parameter, is more likely to represent the
unifying factor favoring the production of oscillons in the DK DW hierarchy.
Proceeding with we shall introduce a new member to the family of DK DW
potentials [43], characterized by a parametrization that leaves unaffected the
barrier height and positions of the two potential minima, but allows tuning
the steepness (or the curvatures) of the potential walls causing the barrier
hump to flatten out.

In Section 2 we introduce the member of parametrized DK DW potential with
variable steepness, and formulate its field-theoretical dynamics. This enables
us determine some associate characteristic quantities such as its kink and
antikink solutions and the kink creation energy. In section 3 we examine the
kink-antikink scatterings, with emphasis on the production of oscillons as the
deformability paramater is varied. Section 4 is devoted to a summary of
results and to conclusion.

##  2 The model, kink solution and kink-phonon scattering spectrum

Consider a field-theoretical model in ($1+1$) dimensional space-time, the
dynamics of which is described by the Lagrangian:

$L=\frac{1}{2}\left(\frac{\partial\varphi}{\partial
t}\right)^{2}-\frac{1}{2}\left(\frac{\partial\varphi}{\partial
x}\right)^{2}-V(\varphi,\mu),$ (1)

where $\varphi(x,t)$ is a real scalar field in one space ($x$) and temporal
($t$) dimensions. $V(\varphi,\mu)$ is a one-body scalar potential which can be
expressed more generally [43]:

$V(\varphi,\mu)=\frac{1}{8}\left(\frac{\sinh^{2}(\alpha(\mu)\varphi)}{\mu^{2}}-1\right)^{2},\hskip
8.5359pt\mu>0.$ (2)

In the present study we pick:

$\alpha(\mu)=asinh(\mu),$ (3)

which is a function of a real parameter $\mu$ assumed to control shape profile
of the DW potential. For arbitrary values of the shape defomability parameter
$\mu$, the scalar potential $V(\varphi,\mu)$ is a bistable function symmetric
around a potential barrier located at the equilibrium state $\varphi=0$. The
potential possesses two degenerate vacuum states at $\varphi=\pm 1$. Thus,
unlike the two members of the DK DW potential discussed by Bazeia et al in
ref. [44], the barrier height and minima positions of the parametrized DW
potential (2) are always fix. However, on fig. 1, where $V(\varphi,\mu)$ is
sketched for some values of $\mu$, one sees that the variation of $\mu$
influences the steepness of the potential walls.

Figure 1: (Color online) Plot of the double-well potential $V(\varphi,\mu)$,
for some values of $\mu$: $\mu=0$ (Solid line), $\mu=2.0$ (Dashed line),
$\mu=4.0$ (Dot-dashed line), $\mu=8.0$ (Dotted line).

Quite interesting fig. 1 suggests that the change in steepness of the
potential walls, caused by a variation of the deformability parameter $\mu$,
has the consequence of rendering the top of the potential barrier either flat
or sharp. Indeed, when $\mu$ tends to zero the parametrized DW potential (2)
reduces exactly to the standard $\phi^{4}$ potential [48, 49]:

$V(u)=\frac{1}{8}\left(u^{2}-1\right)^{2}.$ (4)

As $\mu$ increases the minima positions and the barrier height remain
unchanged, but the slope of the potential walls gets steeper: the narrowest
part of the potential barrier broadens while the flatness (i.e. the
anharmonicity) of the barrier hump (or top) becomes more pronounced, resulting
in an enhancement of the confinement of the two potential wells.

The Lagrangian in formula (1) leads to the following equation of motion for
the field $\varphi$:

$\frac{\partial^{2}\varphi}{\partial
t^{2}}-\frac{\partial^{2}\varphi}{\partial
x^{2}}+\frac{d}{d\varphi}V(\varphi,\mu)=0.$ (5)

In the static regime, the solitary-wave solution to this equation is given by:

$\varphi_{K,\bar{K}}(x)=\pm\frac{1}{\alpha(\mu)}\tanh^{-1}\left[\frac{\mu}{\sqrt{1+\mu^{2}}}\tanh\frac{\sqrt{2}x}{d(\mu)}\right],$
(6)

where:

$d(\mu)=\frac{2\mu}{\alpha(\mu)\sqrt{(1+\mu^{2})}}.$ (7)

The solution with ”+” sign stands for a kink $\varphi_{K}(x)$, while the
solution with ”-” sign stands for an antikink $\varphi_{\bar{K}}(x)$ of width
$d(\mu)$. The characteristic energy (or rest mass) associated with the static
kink and static antikink solution eq. (6), is obtained by using the general
expression:

$E_{K}=\int^{+\infty}_{-\infty}\rho_{\mu}(x)dx,$ (8)

with:

$\rho_{\mu}(x)=\frac{1}{2}\left(\frac{\partial\varphi}{\partial
x}\right)^{2}+V(\varphi,\mu)$ (9)

the kink energy density. Substituting the solitary-wave solution obtained in
formula (6) this yields:

$E_{K}=\frac{1}{4\alpha(\mu)\mu^{2}}\left[2\alpha(\mu)(1+\mu^{2})-sinh(2\alpha(\mu))\right].$
(10)

In fig. 2, shape profiles of the static kink solution $\varphi_{K}(x)$ (a) and
of the kink energy density $\rho_{\mu}(x)$ (b), are plotted versus the spatial
coordinate $x$ for some values of the deformability parameter $\mu$. The
bottom graph in the figure, i.e. graph (c), represents the variation of the
kink rest energy as a function of the deformability parameter $\mu$.

Figure 2: (Color online) (a) Shape of the kink $\varphi_{K}(x)$ and (b) of the
energy density $\rho_{\mu}(x)$ as a function of $x$, for: $\mu=0$ (Solid
line), $\mu=2.0$ (Dashed line), $\mu=4.0$ (Dot-dashed line) and $\mu=8.0$
(Dotted line). (c) Variation of the kink creation energy $E_{K}$, as a
function of $\mu$.

One sees that as the deformability parameter $\mu$ increases, the asymptotic
values of $\varphi_{K}(x)$ as $|x|\rightarrow\infty$ remains the same but a
decrease in the kink width is noticeable. On the other hand, an increase of
$\mu$ leaves the maximum of the energy density unaffected but affects the
width of the energy density in the region covered by the barrier and the
potential wells. Remarkably the energy density seems to decrease with $\mu$ as
we go far in the region covered by the repulsive walls of the potential, such
that in this region the kink is expected to become more localized.

Fig. 2c depicts the kink rest energy as a monotonically increasing function of
the shape deformability parameter. In other words, an increase in $\mu$ will
enhance the kink stability and hence the sharpness of the kink profile.

Most of the processes from the kink-antikink collisions arise as a consequence
of vibrational modes inherent to the kink scattering excitation spectrum.
Usually a perturbation theory is utilized to derive the spectrum of localized
excitations around a kink [47, 48]. To this last point, perturbing linearly
the scalar field $\varphi(x,t)$ around the one kink solution $\varphi_{K}(x)$
i.e. $\varphi(x,t)=\varphi_{K}(x)+\eta(x)\exp(-i\omega t)$, yields the
following Schrödinger-like eigenvalue problem [47, 48]:

$\left[-\frac{\partial^{2}}{\partial
x^{2}}+V_{sch}(x,\mu)\right]\eta=\omega^{2}\eta.$ (11)

In this eigenvalue equation the quantity
$V_{sch}(x,\mu)=\frac{d^{2}V}{d\varphi^{2}}|_{\phi_{K}}$ is the scattering
potential, which in the present case is given by:

$\displaystyle V_{sch}(x,\mu)$ $\displaystyle=$ $\displaystyle
a_{0}\frac{\left[\mu^{2}\tanh^{4}\left(\frac{x}{d(\mu)}\right)+3\tanh^{2}\left(\frac{x}{d(\mu)}\right)-c(\mu)\right]}{\left[d(\mu)\left(\mu^{2}\tanh^{2}\left(\frac{x}{d(\mu)}\right)-c(\mu)\right)\right]^{2}},$
$\displaystyle c(\mu)$ $\displaystyle=$ $\displaystyle 1+\mu^{2}.$ (12)

Note that this scattering potential determines the kink stability upon
scattering with phonons [47, 48]. Instructively an identical expression for
$V_{sch}(x,\mu)$ is obtained by taking a linear perturbation around an
antikink solution.

Distinct profiles of the scattering potential $V_{sch}(x,\mu)$, for different
values of the deformability parameter $\mu$, are represented in fig. 3.

Figure 3: (Color online) Plot of the scattering potential $V_{sch}(x,\mu)$ as
a function of $x$, for $\mu=0$ (Solid line), $\mu=1.0$ (Dashed line),
$\mu=2.0$ (Dot-dashed line) and $\mu=3.0$ (Dotted line).

One sees that as $\mu$ increases, the scattering potential has its width that
gradually decreases and its asymptotic limit growing drastically higher. In
the range $0<\mu\lesssim 1.2$ the potential possesses a global minimum located
at $x=0$, which transforms into a local maximum together with the appearance
of two degenerate minima in the potential as the value of $\mu$ rises larger
than $1.2$.

The same way as the scattering potential, the occurrence of bound states holds
a key importance in grasping some relevant features of the scattering
structure of the system [22, 44]. In particular a resonance mechanism for the
exchange of energy between the translational mode and a vibrational mode, may
result in rich consequences in the spectral features of the system [44]. To
gain insight onto this last featurte, we solved the eigenvalue equation (11)
for $\mu$ and results emphasizing the influence of the parametrization on the
appearance of bound states, are shown in fig. 4.

Figure 4: (Color online) Plot of the squared frequencies $\omega^{2}$ of the
vibrational states, as a function of $\mu$.

We note the presence of a zero-mode for all the values of the shape
deformability parameter, moreover the appearance of new bound states is
observed as $\mu$ rises. For instance, as $\mu$ lies in the range
$0.55\lesssim\mu\lesssim 1.8$ we notice the presence of two vibrational
states, and in the ranges $1.8\lesssim\mu\lesssim 4.0$ and $\mu\gtrsim 4$ a
third and a fourth bound state emerge respectively. Furthermore the lower
vibrational has its frequency increasing as $\mu$ grows to a specific value,
then decreasing while the frequencies of higher vibrational states are
monotonically increasing functions of the deformability parameter.

##  3 Analysis of kink-antikink collisions

The dynamical equation pertaining to colliding kink-antikink pairs will be
solved numerically in this section, with the aim to explore some
characteristic spectral features of kink-antikink scatterings and identify
vibrational models associated with the collisions. To this end, eq. (5) is
discretized on a spacial grid with periodic boundary conditions. The grid is
divided into $N$ nodes such that zone widths $\delta x$ in the simulations
have fixed size, with the location of the $n^{th}$ point on the grid given by
$x_{n}=n\Delta x$. The scalar field is then defined by
$\varphi_{n}(t)=\varphi(x_{n},t)$ for $n=1,2,...,N$. The second-order spatial
derivative is approximated using a fourth-order central-difference scheme
[50], which leads to a set of $N$ coupled second-order ordinary differential
equations in $\varphi_{n}$ i.e.:

$\displaystyle\frac{\partial^{2}\varphi_{n}}{\partial
t^{2}}=\frac{1}{12(\Delta
x)^{2}}(-\varphi_{n-2}+16\varphi_{n-1}-30\varphi_{n}$
$\displaystyle+16\varphi_{n+1}-\varphi_{n+2})-\frac{dV(\varphi_{n},\mu)}{d\varphi_{n}},$
(13)

which is solved numerically using a fourth-order runge-kutta scheme with fixed
step. The accuracy of our algorithm stands with errors that scale as $(\Delta
x)^{2}$ and $(\Delta t)^{4}$.

The initial data used in our simulations represent a kink and antikink
centered at the points $x=-x_{0}$ and $x=x_{0}$ respectively, and moving
forward each other with initial velocities $\upsilon$ in the laboratory frame.
The definition of the starting function can therefore be expresssed:

$\displaystyle\varphi(x,0)=\varphi_{K}(x+x_{0},v,0)-\varphi_{K}(x-x_{0},-v,0)-\varphi_{m},$
(14)

where $\varphi_{m}=\pm 1$ for the kink-antikink and the antikink-kink initial
configurations, respectively. We set the grid to be sufficiently large, with
left and right boundaries respectively at $x_{l}=-400$ and $x_{r}=+400$, and
the separation distance to be $2x_{0}=24$. The choice of a large grid together
with periodic boundary conditions, was to avoid the reflected kink forms to
travel to the boundary, and also to prevent any radiation emitted during the
collision process to eventually find itself back to interact with the kinks.
The grid is discretized with $N=10^{5}$ nodes and all simulations were run
with a temporal step size $\Delta t=0.7(\Delta x)$, found to be a good
consensus between the costly computational time and the production of results
form high-resolution runs.

Several outputs obtained from numerical simulations at some different initial
velocities are now doscussed. For weak velocities, the kink and antikink are
expected to bound upon collision and have enough time to radiate sufficient
energy forming a bion state. This is shown in figs. 5(a) and 5(e), where we
plotted the evolution of the center of mass $\varphi(x=0,t)$ of the kink-
antikink pair, for some values of the shape deformability parameter for which
the system has just one vibrational mode. Irrespective of the barrier
deformation, the kink-antikink pair moving with an initial velocity lower than
a critical velocity $\upsilon_{c}$ settles to an erratically oscillating bion
state. But for large velocities $\upsilon>\upsilon_{c}$, the kink-antikink
pair does not have enough time to radiate sufficient energy to form a bion
state. Thus the kink and antikink will once collide and permanently reflect
each other during the scattering process. This is represented in figs. 5(b)
and 5(f).

Figure 5: (Color online) Possible results for a kink-antikink collision at
several initial velocities, considering two values of the shape deformability
parameter $\mu$. $\mu=0$: (a), (b),(c) and (d). $\mu=0.5$: (e), (f),(g) and
(h). Values of $\mu$ were choosen such that only one vibrational mode sppears
in the excitation spectrum.

For all the considered values of $\mu$ one can note the appearance of a spike
illustrating the collision, followed by a leveling off at $\varphi=+1$
implying that the kink and antikink have reflected and traveled far from each
other. Still, the transition between the bion state and the reflection state
is not smooth as the initial velocities increase. There are regions of values
of $\upsilon$ for which these two states alternate. This regions were reported
in several works as ”windows” [8, 9, 24, 25, 44]. For instance, in figs. 5(c)
and 5(g), we note two spikes in the evolution of the center of mass implying
that the kink and antikink collide and reflect, then return to collide again
before receding to a permanent reflection. Referring to the number of
collisions before the last and permanent reflection, the sets of contiguous
initial velocities leading to this state can be identified as forming a two-
bounce window. The presence of a three-bounce window is evidenced in figs.
5(d) and 5(h), the velocities lying in the three-bounce windows are found on
the edge of the two-bounce regions. When values of the deformability parameter
$\mu$ are located in the range where there exists only one vibrational mode,
the system presents a similar fractal structure as the one observed from the
scattering process in the $\phi^{4}$ model. For example we can note the
existence of a four-bounce window in fig. 6.

Figure 6: (Color online) A four-bounce window taking $\mu=0.5$ is evidenced by
plotting $\varphi(x=0,t)$ for $\upsilon=0.1445$. Note the presence of four
large spikes illustrating collision after which the kink and antikink reflect
and recede from each other forming a two-soliton state.

The appearance of n-bounce windows is also expected to be observed for this
range of shape deformability parameter values. To further understand the
structure of scattering in the system, in fig. 7 we plotted the time of the
three bounces as a function of the initial velocity.

Figure 7: (Color online) Kink-antikink collision times to first (blue), second
(red) and third (dotted-black) bounces, as a function of initial velocity for
(a) $\mu=0.5$ and (b) $\mu=5.0$.

The intervals in which the time for the third collisions diverges, range in
the two-bounce windows. A case with only one vibrational mode is illustrated
in Fig. 7(a), which shows plot of the collision times for $\mu=0.5$ (compare
with figs. 5(e), 5(f), 5(g) and 5(h)). We can note that bion states are formed
for $\upsilon<\upsilon_{c}\sim 0.183$ while for $\upsilon>\upsilon_{c}$, the
collision results in an inelastic scattering between the pair corresponding to
one-bounce around one vacuum. Moreover, the figure captures the complete set
of two-bounce windows of which the width continuously decreases and
accumulates around $\upsilon_{c}$. The scattering times in the presence of
several vibrational modes are plotted in fig. 7(b) where we considered the
shape of the potential for $\mu=5$. We first see that $\upsilon_{c}$ grows
larger with $\mu$. From our simulations we observed that the variation of the
critical velocities is not a monotonic function of the shape deformability
parameter, we first observed a decrease of $\upsilon_{c}$ as $\mu$ increases
till $\mu\sim 1$ then recedes to an increasing behavior (this result is not
presented here). This reflects that the attraction between the kink and
antikink lessens as the system is deformed departing from the $\phi^{4}$
model, but the attractive interaction gets stronger and stronger as
$\mu\gtrsim 1$. Suppression of two-bounce windows is also observed in fig.
7(b). This is justified by the presence of several vibrational modes
complicating the energy transfer from the translational mode to just one
vibrational mode to achieve resonance conditions.

One of our main point of focus in this work is the possible production of
oscillons. For relatively small values of the deformability parameter $\mu$,
for which only one vibrational mode is generated in the excitation spectrum,
oscillon structures cannot appear, the kink-antikink collisions with initial
velocities lower than the critical velocity can only result in bion and
n-bounce states. The $\phi^{4}$ model being an asymptotic limit of the
parametrized DW model in this range of values of $\mu$, this agrees with the
$\phi^{4}$ model showing no evidence for the formation of oscillons as a
result of the scattering process. We see from fig. 8 that for larger values of
$\mu$, the presence of more than one vibrational mode favors the production of
oscillons.

Figure 8: (Color online) Oscillons resulting from kink-antikink collisions:
(a) $\mu=3.0$ and $\upsilon=0.23$ $(\upsilon_{c}=0.245)$ showing one bion and
two oscillons, (b) $\mu=5.0$ and $\upsilon=0.236$ $(\upsilon_{c}=0.27)$
showing one bion and four oscillons.

At some velocities of the colliding kink, lower than the critical velocities,
the bion is formed and travels with oscillons which can oscillate around each
other, or escape to infinities. We can compare the appearance of one bion and
two oscillons travelling together with a large flux of emitted radiation in
fig. 8(a), and the appearance of one bion and four oscillons in fig. 8(b),
travelling with quite no radiation and a larger degree of harmonicity. Note
that the larger the shape deformability parameter the greater the number of
created oscillons.

##  4 Conclusion

Oscillons are breather-like bound states generated by self-interactions of
kink-antikink pairs that exist in some scalar-field models [51, 52, 53, 54,
55], in the context of cosmology their built-in mechanism suggests that they
can affect the standard picture of scalar ultra-light dark matter. In two
recent studies [22, 44] the generation of oscillons in bistable systems,
characterized by a parametrized double-well potential, was discussed with
emphasis on the influence of the shape deformability on the oscillon
production. First [22] the authors considered a deformable $\phi^{4}$
potential represented by an hyperbolic double-well potential, and established
that the deformability favors the emergence of oscillon modes from kink-ankink
collisions and for well selected intial velocities of the colliding kinks.
Later on [44] they extended the study to two members of the family of Dikandé-
Kofané DW potentials. One of these members has its double-well minima fixed
but a variable height of the potential barrier, whereas the other member has
fixed barrier height but a variable separation between the two potential
minima.

To determine more exactly which of the characteristic features introduced by
the potential deformability-i.e. the variable positions of potential minima,
or the variable height of the potential barrier- effectively controls the
oscillon production, in this work we revisited the study by considering a
parametrized DW potential with fixed potential minima and fixed barrier height
fixed. However the steepness of the potential walls, and hence the flatness of
the barrier top, can be tuned by varying a deformability parameter. The
parametrized DW potential has the particularity to reduce to a $\phi^{4}$
potential, just the same as with the already known family of DW potentials
proposed in refs. [42, 43, 47] and referred to as Dikandé-Kofané potential.
Examining the kink-antikink scattering processes, we found that the
parametrized bistable model inherits some of the general features of the
$\phi^{4}$ model that is the possibility of formation of bion states,
reflected states and also $n-$bounce windows. However the appearance of
additional modes in the scattering spectrum, as the DW potential deformation
becomes predominant in our model, suggests the possibility of suppression of
the two-bounce windows due to a kind of interference, as was already detailed
in some other works [25, 44].

Long-lived, quasi-harmonic and low-amplitude structures called oscillons were
shown to form after kink-antikink collisions with some initial velocities less
than a critical velocity. This is not observed for low values of $\mu$, where
the model has only one vibrational state and is more close to the $\phi^{4}$
model. The rising number of vibrational states as $\mu$ increases yields to an
intricate situation where the realization of the mechanism of resonant energy
exchange between the translational and one vibrational more becomes more
difficult. The appearance of oscillons is thus favored by the deformation in
our model.

In the works of Bazeia et al, reporting the appearance of oscillons in
hyperbolic models [44] for two deformable double-well potentials, they showed
that the production of oscillons is boosted by applying the conformational
changes from those potentials’ deformability such as reducing the distance
between the minima keeping the barrier height fixed, or decreasing the barrier
height while keeping the minima fixed. They pointed out that the factor
unifying the two contexts is the lowering of the kink energy by the
deformability in the two models. The scattering dynamics at the center of mass
in our present model are roughly the same as the one in the work of ref. [22,
44], however the increase of the kink energy in our model disagrees with a
tentative idea to extend the consideration of kink energy being a determinant
unifying factor to a more general case. The deformation in our model is
manifest through an increase of the steepness of the potential walls, with the
barrier top becoming flattened hence imposing an anharmonic shape to the
potential barrier. This trend can also be observed as an implicit result of
the deformation in the two models considered by Bazeia et al [44], and also in
the sinh-deformed $\phi^{4}$ model [22] shown to also allow the creation of
oscillons. Bistable systems modeled by potentials with anharmonic barrier are
thus suggested to be good candidates to observe the formation of oscillons in
kink-scattering processes.

## acknowledgements

the work of A. M. Dikandé is supported by the Alexander von Humboldt (AvH)
Foundation.

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