# Using Semantic Similarity and Text Embedding to Measure the Social Media
Echo of Strategic Communications

Tristan J.B. Cann, Ben Dennes, Travis Coan, Saffron O’Neill and Hywel T.P.
Williams

###### Abstract

Online discourse covers a wide range of topics and many actors tailor their
content to impact online discussions through carefully crafted messages and
targeted campaigns. Yet the scale and diversity of online media content make
it difficult to evaluate the impact of a particular message. In this paper, we
present a new technique that leverages semantic similarity to quantify the
change in the discussion after a particular message has been published. We use
a set of press releases from environmental organisations and tweets from the
climate change debate to show that our novel approach reveals a heavy-tailed
distribution of response in online discourse to strategic communications.

## Introduction

Social media have exerted profound changes on established hierarchies of
communication (Pearce et al. 2019). Diverse actors such as advertisers,
charities, celebrities and governments seek to exert influence on the content
and flow of information to promote their perspectives, products and agendas.
In recent years, these attempts have seen increased awareness given their
impact on democratic processes (Haenschen and Jennings 2019) but precise
measurement of the impact of these strategic communications campaigns on
public discourse remains a challenge given the difficulty of defining suitable
metrics and the volume of media content now available.

Typically, attempts to understand the response to media interventions make use
of survey methods. (Zhang et al. 2018) surveyed individuals for their
responsiveness to climate change information and found that exposure had a
greater effect in more conservative areas. (Feezell 2018) explored the impact
of political content on social media, and found that not only did exposure
increase awareness, it also reduced the rate of attention decay. (Goldberg et
al. 2021) tested the response to banner ads on attitudes to climate change in
the US, and found that the response was stronger among Republicans. (Nyhan,
Porter, and Wood 2022) enhanced these findings by comparing the effects of
factual, partisan and opinion pieces on climate change issue awareness. While
individual attitudes were affected by each type of content, the short-lived
gains from factual content were able to be quickly eroded by sceptical pieces.
Survey-based experiments do present advantages by controlling the level of
exposure that participants receive, but they are necessarily incapable of
controlling all aspects of content exposure through longitudinal studies.
Despite this, the impact of survey effects (and in particular social
desirability bias in polarised contexts (Nederhof 1985)) mean that scholarly
efforts would be enhanced by tools to study message effectiveness in real-
world settings.

The rapid growth of digital media presents challenges for the scalability of
analysis of media dynamics, including the measurement of influence. The volume
of online news and social media content necessitates computational methods,
but these have typically lacked the qualitative skill of survey methods. One
potential avenue for analysing strategic messaging campaigns is using machine
learning techniques to identify and measure influence in social media
campaigns. (Alizadeh et al. 2020) used labelled data from Chinese, Russian and
Venezuelan accounts to train a classifier for accounts linked to influence
operations over both Twitter and Reddit posts using a range of content and
metadata features. Their efforts were successful in specific contexts, but
they note that the evolution of tactics by malicious actors hampers efforts to
find a general set of features common to members of influence operations.
Other attempts have investigated whether semantically coherent topics can be
developed using content types on Twitter, such as that of (JafariAsbagh et al.
2014) who developed an efficient process capable of identifying organic trends
in real time since they may indicate influence. Complementary to attempts to
understand popular topics is the identification of individual user accounts
that drive these trends. Previous efforts have considered user metadata and
posting habits online (Azcorra et al. 2018) or social network position (Hu et
al. 2018) to predict user influence. Most such efforts are hampered, however,
by weak definitions (that use a partial substitute or proxy for influence) or
incomplete data (since most data samples do not capture all interactions
between individuals or content exposure due to practicalities and privacy
constraints). Thus the challenge of measuring influence remains unsolved and
new efforts are required to offer a more complete picture of interpersonal and
organisational influence in online spaces.

Attempts to quantify information contagion or influence campaigns on contact
networks have seen success. Some expand on the ideas of influence prediction
discussed previously to give an estimate of the level of activity a particular
topic will receive, rather than a binary indicator for whether the campaign
was successful (e.g. (Weng, Menczer, and Ahn 2014)). Many studies of
information diffusion online rely on the use of specific parameters such as
users, keywords, hashtags and URLs (see e.g. (Gabielkov et al. 2016; Alperin,
Gomez, and Haustein 2019; Alizadeh et al. 2020; Cruickshank and Carley 2020))
but these necessarily miss relevant discussions that do not use the
predetermined search parameters.

Work by (Bollenbacher, Loynes, and Bryden 2022) is of particular relevance to
the aims of this manuscript. Their research presented a means of quantifying
the flow of attention between different conversational formats, using averaged
word co-occurrence of texts to measure the change in agreement after the
publication of a reference text. Regression analysis revealed that a small,
but significant, flow of words emerged from news articles shared on Facebook
to speeches made by MPs in the UK Houses of Parliament. Moreover, there was
variation in the responses of different political parties with Labour
politicians being more likely to use languange from news articles and social
media interactions. This work has two main shortcomings, however. Using word
co-occurrence will underestimate similarity when the two texts express the
same message but with a different vocabulary. Additionally, although the
authors note the short-lived engagement with online content (Gabielkov et al.
2016), they still choose to consider a two week post-event period, which may
lead to any shorter term signals becoming lost.

In this paper we present a new framework for quantifying the uptake of
strategic organisational messaging on social media. Our semantic similarity-
based approach improves upon previous competitor techniques by adding
robustness under changes to distinct but contextually similar vocabularies. We
also present an alternative method that accounts for variation in background
activity rates for effective comparison of messages released at different
stages of the baseline activity cycle. To demonstrate the effectiveness of our
metrics, we collect press releases from ten environmentally-active
organisations and analyse their “echo” within the Twitter discussion of
climate change. We find that our new metric allows for differentiation between
messages and provides valuable context for understanding their impact.

The rest of this paper proceeds as follows. In the next section we present the
conceptual framework we use to investigate influence. Later, we detail the
development of our novel metrics, alongside the datasets used to test it.
Finally, we detail the findings of our experiments before summarising our
recommendations for future users.

## Conceptual model

From an engineering perspective, exerting influence on public debate can be
conceptualised as the problem of transmitting information through a ‘noisy
channel’. A source seeks to use communication to change some characteristic of
a receiver; if successful, some change in the receiver state will be
detectable and correlated with the communicated information. One possible
example of this is a lobbying group using messaging to change public opinion.
If their campaign is successful, then the target audience will, in some form,
adopt aspects of the message. While the true desired effect might be an
internal characteristic of the receiver, such as belief, opinion or feeling,
these typically remain hidden. So to measure an effect we must focus on
external characteristics, such as utterances or messages that reveal internal
state.

A number of confounding factors add challenges to the types of inference that
can be made about source influence. The primary complication is incomplete
information about the transmission channel (we do not know the content to
which the receiver was exposed, only the source message and receiver outputs)
and the diversity of other potential influences (communications do not happen
in isolation and we must assume that all receivers are exposed to many
competing messages). As highlighted in Figure 1, this leaves the problem as
one of correlation, not causation; we cannot attribute causes, but can still
seek to measure a correlated change in a receiver characteristic that occurs
following a source communication. One final confounding factor in any analysis
of communication channels is the impact of background media trends: any
measure of a change in receiver behaviour should account for their ‘normal’
behaviour from comparable periods.

As discussed in the previous section, much of the previous work in this area
has used approaches that consider co-occurrence of keywords and tokens and
other measures that are unaware of the context within which these words
appear. Moreover, these methods are often reduced to considerations of
frequency with limited consideration of the background trends. We propose
taking advantage of the advances in the availability and scalability of text
embedding techniques and semantic models to encode this context in our
comparison of messages before and after source communications. By accounting
for this change appropriately we can calculate receiver changes relative to
the baseline media trends, and begin to reduce the impact of this confounding
factor.

Figure 1: Illustrative schematic of our conceptual model. Strategic
communications are published by groups and individuals who seek to shift the
narrative in legacy or social media. How these messages filter through the
media ecosystem and interpersonal influence is partially observable at best.
The utterances of end users however can be observed and changes in message
content can indicate they were potentially influenced by the strategic
communication.

## Operationalising the model

In order to test the application of our conceptual model, we chose the context
of strategic communications from environmental advocacy organisations sharing
messages around climate change and seek to understand how they impact the
online discussion of climate change on Twitter. This study makes use of two
forms of text data: press releases from environmentally-focused organisations
with a global reach and tweets about climate change, both for a two year
period between 2019-11-01 and 2021-10-31.

The organisations were selected from a list of environmental organisations
used in previous work (Weaver et al. 2023). From this list, we selected the 10
groups with the largest following on their main Twitter accounts. Table 1
gives details of the included organisations and their reach in terms of
Twitter followers and press releases published. For each organisation, a web
crawler was constructed to parse the web page listing statements published as
press releases by the organisations and extracting the text content within. In
total 4,334 press releases were collected across the 10 organisations, and our
analysis focuses on the 4,221 press releases with a full week of tweets before
and after their release date (i.e. those between 8/11/19 and 24/10/21).

Tweets about climate change were collected via the Twitter streaming API by
searching for the keywords _climate change_ and _global warming_. The majority
of the collection was completed in real-time, but short periods of
interruption were recovered using historic searches through the V2 Academic
API. While there are necessarily inconsistencies between the collection
methods (e.g. the removal of posts in violation of the terms of service before
historic collection), the volume of posts in our dataset within a given
interval is comparable between the two collection methods. In total our
dataset includes 67,672,904 tweets.

Organisation | Twitter handle | Followers | Press releases
---|---|---|---
World Wildlife Fund | @wwf | 3,936,636 | 135
Greenpeace | @greenpeace | 1,908,122 | 135
Oxfam | @oxfam | 838,981 | 335
Avaaz | @avaaz | 728,685 | 17
Climate Reality Project | @climatereality | 611,028 | 73
National Wildlife Federation | @nwf | 602,290 | 503
Public Citizen | @public_citizen | 526,268 | 775
350 | @350 | 391,928 | 335
Sierra Club | @sierraclub | 384,747 | 1,363
Natural Resources Defense Council | @nrdc | 345,191 | 550
Table 1: Details for the 10 environmental organisations defining our sample of
press releases. Followership reflects the values recorded during collection in
May 2022.

### Measuring text similarity

To identify the semantic similarity between two texts, the gold standard
method is manual analysis and labelling by a subject matter expert. However,
due to the large volume of texts involved in our study – in addition to the
desire for a generalisable method that can be applied to new texts at any time
– such methodology is impractical. Instead we utilised a pre-trained deep
learning model to perform text embedding, representing a sequence of text as a
vector and thereby encoding the meaning of the text such that “semantically
similar sentences are close in vector space” (Reimers and Gurevych 2019).

The deep learning model used was a sentence transformer, a type of model
trained on large collections of text (such as Wikipedia and collections of
literature (Bao et al. 2020)) and fine-tuned with semantic similarity data
(Reimers and Gurevych 2019) to improve the quality of text representations for
the purpose of pairwise comparison. The sentence transformer used was all-
MiniLM-L6-v2 (Reimers 2021), chosen for its performance and efficiency. As
outlined in the documentation of this model, the cosine similarity of vectors
is an appropriate measure for similarity (Reimers 2022, 2021).

The all-MiniLM-L6-v2 model is designed to take short texts (less than 256 word
pieces (Reimers 2021), “subword units [which] retain linguistic meaning” (Song
et al. 2021)) as input. As press releases are typically longer than this
limit, the embeddings of all constituent sentences of a press release were
averaged (after sentencisation with spaCy (Honnibal et al. 2021) following
removal of extra white space characters) to result in an overall document
embedding. The only pre-processing applied to the tweet text was the removal
of all _t.co_ (Twitter-shortened) URLs due to their ubiquity in the texts and
lack of linguistic information at face value.

The choice of similarity threshold at which a human reader would consider two
texts to be similar is not an obvious one. Moreover in the context of
strategic messaging by NGOs it is important to distinguish between tweets that
cover the same topic and those that discuss the same aspects within a topic
(such as references to a particular event or advocating a particular course of
action).

To test this, and validate our use of an embedding model over competing
techniques, we prepared a sample of pairs of tweets and press releases for
labelling by two human coders under a binary scheme where texts are labelled
as similar under the stricter condition of requiring more specificity than the
general topic. Under this scheme consider the following (fictional) press
release:

_Unprecedented warming in the Arctic, warmest February on record_

Temperature records from this year show that the average monthly temperature
in the Arctic circle set a new record for the month of February. As global
heating intensifies, such extremes will continue and new records set with
increasing frequency. Global action is required to prevent a dangerous tipping
point from being reached - contact your local representatives using the
templates below. With reference to this press release, the tweet “ _Sunny
first day on my expedition in the Arctic_ ” would be rated as dissimilar since
it misses the key factor of extreme temperatures as a consequence of climate
change. The tweet “ _OMG, can’t believe how extreme global heating is! Warmest
Feb on record in Arctic, poor polar bears_ ” would be rated as similar as it
references both the extreme heat and geographic location.

For each of the press releases in our dataset we found their temporally
relevant tweets (those falling within a week before or after the publication
date of the press release) and calculated the cosine similarity between each
pair of embeddings and from this set found press releases with which a range
of similarity values was observed in the set of temporally relevant tweets. A
number of press releases had very few (or even no) tweets with a similarity of
greater than 0.7. However, as preliminary experiments had suggested that a
suitable similarity threshold was at least this high, including such press
releases would not give any useful information for calculating this threshold.
Therefore, in order to filter such press releases out, the number of tweets
for each press release with embedding similarity falling in 0.05-width bins
between 0.5 and 0.8 were counted. Any press release without at least 4 unique
tweet texts in each bin was excluded from sampling.

From the remaining press releases, four were sampled from each publishing
organisation, and for each, four tweets were sampled from each similarity bin
for a total of 960 pairs. These were shuffled and provided to two human coders
without the similarity information, who discussed any disagreement in their
labels to reach an agreement.

This labelled dataset allowed for comparison against two common methods for
text similarity to assess their utility for the purposes of evaluating
strategic communication campaigns. Jaccard similarity is the simplest of
these, measuring the proportion of tokens appearing in either of two reference
texts that appear in both. TF-IDF similarity is instead based on the frequency
of tokens relative to their appearance across all texts in the corpus. Terms
common in many of the texts have their weights reduced and consequently TF-IDF
can be thought of as highlighting the characteristic terms for each text in
turn.

One major area in which the TF-IDF and Jaccard similarity methods differ from
semantic similarity is in their need for careful text pre-processing. We used
sklearn (Pedregosa et al. 2011) to handle tokenisation and English stop word
removal, alongside Porter stemming from NLTK (Bird, Klein, and Loper 2009) to
allow optimal matching with variation in phrasing. Although the TF-IDF and
Jaccard similarity metrics have no limitation in the lengths of text they are
applied to, we use the same average sentence similarity as in the text
embedding case for consistency. Investigation of alternative approaches using
TF-IDF and Jaccard similarity measures saw qualitatively similar trends in
similarity across the labelled testing set.

To identify the preferred similarity threshold required to distinguish similar
and disssimilar pairs of texts we use the similarity scores found by text
embedding, TF-IDF and Jaccard methods for pairs in our labelled set to define
a binary classification problem. We used a range of values in the interval
[0,1] as candidates and for each calculate the accuracy, F1 score and adjusted
Rand index with respect to our human annotations for similarity.

The results from our analysis are shown in Figure 2. We note that given our
sampling procedure for the pairs of tweets and press releases, the range of
similarity values under text embedding is [0.5,0.8]. In regions outside of
this interval, each of the evaluation statistics is unchanged for the
embedding similarity as the threshold varies. This behaviour is repeated in
the competitor metrics beyond their own ranges of similarity values. The first
pattern revealed in Figure 2 is the comparatively low similarity values
observed by the TF-IDF and Jaccard metrics, which reach maxima of 0.495 and
0.264 respectively. Despite this shift in the similarity values observed, we
can see that each of the metrics was able to distinguish between similar and
dissimilar pairs more accurately than labelling by the most-common class
(dissimilar) as evidenced by Figure 2(a). Moreover, running pairwise
comparisons of the labels assigned using threshold values in their respective
preferred regions showed good agreement across approaches: each pairwise
accuracy was greater than 0.7 using a threshold of 0.7 for embedding
similarity, 0.2 for TF-IDF similarity and 0.08 for Jaccard similarity. Figs.
2(b) and 2(c) suggest that the Jaccard similarity metric performs the worst
when accounting for the imbalanced classes in the labelled data (approximately
25% were rated as similar by our human coders). We expect that these
shortcomings in the TF-IDF and Jaccard similarities come from the challenges
of our particular test case. Tweets are shorter pieces of text given their
character limits, and therefore can only include a limited number of words.
Since the TF-IDF and Jaccard metrics rely only on presence of tokens and not
the context within which those tokens appear, they are unable to capture the
maximum amount of information from the limited text available in tweets. As
such, our choice of using text embedding as a preferred metric is validated in
our context.

(a) Accuracy

(b) F1 score

(c) Adjusted Rand index

Figure 2: Effectiveness of different similarity thresholds at discriminating
between similar and dissimilar texts.

From Figure 2 we can see that there is consistently peaked behaviour under
each of the similarity metrics and each of the reported benchmarking
statistics. Focusing now on the results for the text embedding case, we can
see that the maximum value observed for accuracy, F1 score and adjusted Rand
index is found close to a similarity threshold of 0.7 (accuracy at 0.736, F1
score at 0.696, adjusted Rand index at 0.712). In the context of this
consistency, we choose 0.7 as an appropriate threshold to recognise texts as
‘sufficiently similar’ for our purposes.

### Measuring the echo

We now detail the method for calculating the echo of a given press release,
for which we require the press release text and its release date. Given the
release date, we collect all tweets in our dataset within a predetermined
window around the release date (including the release date itself).

The press release and tweet texts are each embedded into a 384-dimensional
vector space wherein we can calculate the semantic alignment between texts
using cosine similarity. Using the preferred threshold of 0.7, identified in
the previous section, we can then label tweets as “sufficiently similar” if
the cosine similarity of their embedding and the press release embedding is
greater than this threshold.

Our first proposed echo metric considers the absolute change in sufficiently
similar tweets. We establish the baseline topic visibility by calculating the
average number of sufficiently similar tweets on the days preceding press
release publication and similarly measure the publication impact by
calculating the average number of sufficiently similar tweets on the days
after publication (typically including the release date in this window). The
metric, which we refer to as $\Delta_{raw}$, is finally calculated as the
difference between the post-day calculation and the pre-day calculation. In
other words,

$\Delta_{raw}=\frac{1}{|\mathrm{Post}|}\sum_{i\in\mathrm{Post}}|S_{i}|-\frac{1}{|\mathrm{Pre}|}\sum_{j\in\mathrm{Pre}}|S_{j}|,$
(1)

where $S_{i}$ is the set of sufficiently similar tweets on day $i$, and Pre
and Post are the sets of pre- and post-release days considered respectively.
Positive values of $\Delta_{raw}$ indicate more sufficiently similar tweets
after the publication date, negative values indicate fewer sufficiently
similar tweets after publication and values around zero indicate little change
overall.

Social media activity is inherently variable on a day-to-day basis, and as
such we present a complementary metric called $\Delta_{prop}$ to normalise for
this variation. Where $\Delta_{raw}$ considers the absolute number of tweets,
$\Delta_{prop}$ considers the rate of sufficiently similar tweets relative to
daily changes in background activity, that is

$\Delta_{prop}=\frac{1}{|\mathrm{Post}|}\sum_{i\in\mathrm{Post}}\frac{|S_{i}|}{|T_{i}|}-\frac{1}{|\mathrm{Pre}|}\sum_{j\in\mathrm{Pre}}\frac{|S_{j}|}{|T_{j}|},$
(2)

where $T_{i}$ is the total number of tweets on day $i$.

The size of the pre- and post-release windows poses a question of which
combination is preferable. We tested each combination of one, three and seven
days before and after the release date on our sample of press releases from
environmental organisations to determine the impact of these choices. In
Figure 3 we show how our metrics perform under these conditions, and also
highlight two of the most successful examples under $\Delta_{raw}$. We
highlight a comment from 350 about Jeff Bezos’ $10 billion climate pledge in
February 2020 in Figs. 3(a) and 3(b). We see here that in most cases there is
some variation in the echo scores returned for a given piece of text, but a
successful message will still be considered as such in most combinations of
pre- and post-release window size. The case of one day preceding release and
seven days after release stands out as anomalous for this press release where
it sees a large, negative echo in contrast to the other cases. This situation
occurs due to a combination of a large amount of activity on the day before
release coupled with short-lived activity after release resulting in the post-
release window average being comparatively small. This example highlights an
extreme case and illustrates the potential pitfalls with failing to establish
a suitable baseline comparison for the post-release window and in observing
beyond the lifetime of attention to the topic.

Figure 3 reveals another trend in the responses of our metrics to different
window sizes when looking at the broader distribution trends. Primarily we can
see that the size of the post-release window has a clear effect on the range
of values observed, whereas the size of the pre-release window has much less
impact. As the post-release window expands to include days further from the
release, we see that the range of values observed across all press releases
decreases, as does the echo score observed for the specific examples
highlighted. These observations make intuitive sense in the context of media
ecosystems and our conceptual model. In the case of a truly novel message
transmitted in the press release, the number of sufficiently similar messages
produced before this content is shared will be low and stable over time,
therefore removing any dependence on pre-window size from our echo metrics.
This is not always the case however since some messages can be retrospective
(referencing events that occur in the past) or periodic (referencing events
that occur with some frequency). Such cases would particularly benefit from a
choice of pre-release window size that covers the range of relevant
behavioural rhythms among individuals such as daily or weekly activity habits.
The suitability of a choice for post-release window size has similar
considerations around the lasting relevancy of a message. On social media,
narratives in the media ecosystem evolves rapidly and quickly shift from issue
to issue (Castillo et al. 2014). Typically however, most awareness of media
issues is short-lived which means that shorter post-release windows should be
preferred.

Given these observations we present the following recommendations for the
choice of pre- and post-release window sizes. In all cases, we recommend
avoiding the use of a single day in the pre- and post-release windows to limit
the impact of anomalous days. Our context of politically relevant messages
observed through the lens of social media gives a rapid turnover of events
that can have relevance of several days and in this case recommend a choice of
three days for the post window. Seven days is our preferred choice for the
size of the pre-release window to fully capture weekly baseline trends and
allow for large echo scores for reactive messages such as the example in
Figure 3(a).

(a) $\Delta_{raw}$ for a press release from 350 on 2020-02-18 discussing Jeff
Bezos’ $10 billion pledge to create a fund fighting climate change.

(b) $\Delta_{prop}$ for a press release from 350 on 2020-02-18 discussing Jeff
Bezos’ $10 billion pledge to create a fund fighting climate change.

(c) $\Delta_{raw}$ for a press release from Greenpeace on 2021-08-09
highlighting the need for urgent climate action following the publication of
the latest IPCC WG1 report.

(d) $\Delta_{prop}$ for a press release from Greenpeace on 2021-08-09
highlighting the need for urgent climate action following the publication of
the latest IPCC WG1 report

Figure 3: Selected examples from the press releases that score most highly by
$\Delta_{raw}$. The black points in each panel show the position of the
highlighted press release relative to all other press releases in our sample
under the tested combinations of pre- and post-release window sizes.

## Interpreting the echo

We now explore the general behaviours of both $\Delta_{raw}$ and
$\Delta_{prop}$ over the press releases in our dataset using the preferred
parameter choices of a similarity threshold of 0.7, pre-release window size of
seven days and post-release window size of three days, as shown in Figure 4.
Here we can see that the majority of press releases have very little impact on
the Twitter conversation of climate change (of those with sufficiently similar
tweets, 95th percentiles: $\Delta_{raw}$ 351.67, $\Delta_{prop}$ 0.0041)
compared to the maximum values observed ($\Delta_{raw}$ 21,655.52,
$\Delta_{prop}$ 0.157) and suggest a strongly positively skewed distributions.
Some press releases have negative $\Delta$ values, which indicate that
interest in the topic was greater before the publication date, but this is
most apparent in the case of $\Delta_{raw}$ when a press release is referring
to a recent natural disaster (the Australian bush fires in January 2020).

Across the set of all press releases studied a few patterns emerge. Firstly,
more than 50% of all texts (2,364/4,221) had no sufficiently similar tweets
within the period defined by our pre- and post-release windows, that is their
messages were not repeated on Twitter close to their release date and we omit
such cases from all figures. We can also see that $\Delta$ values are not
driven solely by baseline activity in the Twitter collection but do see some
evidence of temporal clusters of influential communications. These typically
arise from notable events such as John Kerry being named as US International
Climate Envoy after the victory of President Biden in the 2020 US presidential
election (2020-11-23), the release of the IPCC Sixth Assessment Report
(2021-08-09) or the Leaders’ Climate Summit on Earth Day (2021-04-22).

Figure 4: Scores for the echo metrics for all press releases in the dataset
that had sufficiently similar tweets and data available for the seven days
before and after the release date. The bottom panel plots the number of tweets
collected each day for context.

When comparing trends observed for $\Delta$ values in Figure 4, the successful
press releases under one metric tend to be similarly successful under the
other. $\Delta_{prop}$ however is more successful at highlighting important
communications in periods of lower baseline Twitter activity, given its
inherent normalisation of background activity trends. Figure 5 complements
this by showing a clear correlation between $\Delta_{raw}$ and $\Delta_{prop}$
(Pearson’s $R=0.808,p<0.01$). The two press releases with the largest
$\Delta_{raw}$ (as shown in Table 2) are outliers from this trend, as is the
press release with the smallest $\Delta_{raw}$ (in which 350 discussed the
catastrophic Australian bushfires in January 2020 and called on the Australian
government to take action). These outliers highlight considerations to make
when comparing our $\Delta$ metrics. Firstly, $\Delta_{raw}$ is affected by
changes in the background rate of Twitter posts. The Australian bushfires
example shows this clearly as a reactive message to a major event. Calculating
$\Delta_{prop}$ for the same message we instead find a small positive effect,
which shows that the large negative value of $\Delta_{raw}$ observed was
linked to the decline in climate change activity on Twitter. A similar reason
explains the outliers with high $\Delta_{raw}$. The IPCC Sixth Assessment WG1
report release is a major event in the climate change discussion on Twitter,
producing a more than five-fold increase in daily posts compared to the
previous week. Relevant press releases published towards the beginning of this
spike therefore received large $\Delta_{raw}$ values.

Figure 5: Scatter plot of $\Delta_{raw}$ against $\Delta_{prop}$. Correlation
between the two metrics is strong (Pearson’s $R=0.808,p<0.01$).

We focus now on the most influential press releases as highlighted by our new
metrics. Table 2 lists the five most successful by $\Delta_{raw}$ and
$\Delta_{prop}$. A few patterns are revealed from these highly successful
communications. Firstly, 350 and Climate Reality Project are well-represented
and each contributes a message that is highly rated on both of our metric
formulations: discussion of a philanthropic pledge by Amazon founder Jeff
Bezos and the selection of John Kerry to a position of climate leadership by
President-elect Biden respectively. More broadly, the most influential
communications are tied into major events such as the incoming Biden
administration, the IPCC Sixth Assessment Report publication and investigative
reporting into the lobbying efforts of ExxonMobil.

Rank | $\Delta_{raw}$ |  URL | $\Delta_{prop}$ |  URL
---|---|---|---|---
1 | 21,655.5 |  350.org/press-release/bezos-climate-pledge/ | 0.157 |  climaterealityproject.org/press/statement-ken-berlin-secretary-john-kerry
2 | 21,537.4 |  greenpeace.org/international/press-release/49125/ipcc-report-signals-decisive-moment-for-humanity-urgent-climate-action-needed/ | 0.119 |  350.org/press-release/bezos-climate-pledge/
3 | 13,038.9 |  climaterealityproject.org/press/statement-ken-berlin-secretary-john-kerry | 0.105 |  350.org/press-release/kerry-international-climate-envoy/
4 | 10,479.7 |  350.org/press-release/350-org-reacts-to-ipcc-report-on-the-state-of-our-climate/ | 0.105 |  wwf.panda.org/wwf_news/press_releases/ ?1174191/Climate-Ambition-Summit-2020
5 | 9,340.3 |  350.org/press-release/exxon-tapes-climate-crimes/ | 0.100 |  climaterealityproject.org/press/statement-former-vice-president-al-gore-fifth-anniversary-paris-climate-agreement
Table 2: The five most influential press releases by each $\Delta$ metric.

The interpretations of $\Delta_{raw}$ and $\Delta_{prop}$ in an experimental
context vary. $\Delta_{raw}$ measures the average change in sufficiently
similar pieces of content and therefore gives an indication of the size of the
following gained by a particular message. Analysis of this figure should
always consider the context of changing background activity rates, and
comparisons between contexts should also be aware of any differences between
their respective audience sizes. $\Delta_{prop}$ on the other hand gives a
measure of change relative to the population of interest and therefore should
be thought of as the change in the proportion of the whole reference
conversation that is sufficiently similar to the chosen message. More
specifically, a $\Delta_{prop}$ of 0.1 means that on average 10% more of the
conversation is similar to the chosen message after its release.

Given our example dataset and the interpretation of each metric we can now
discuss how these metrics can be used to determine whether a particular
message is successful. We observed highly-skewed distributions for both
$\Delta_{raw}$ and $\Delta_{prop}$ where most of observed values are small
relative to the maximum. In eight of the nine combinations of pre- and post-
release window sizes for $\Delta_{prop}$, the 97th percentile is less than
0.01 which indicates that such a value is a consistent threshold for the most
impactful communications. Identifying such a threshold for $\Delta_{raw}$ is
more challenging given the inherent variability of the measure in different
contexts. With reference to the Twitter conversation surrounding climate
change the 97th percentile ranges from approximately 400 to approximately 1200
(which is not surprising given the patterns shown in Figs. 3(b) and 3(d)). The
lower end of this range is observed when using seven days post-release which
is likely to include too much ‘business as usual’ activity after attention to
the message has faded. Excluding these cases, the lower range limit increases
to approximately 850 and shows a similar approximately one percentage point
change relative to average daily tweets in our sample (approximately 93,000).

We can summarise the recommendations for our echo metrics as follows.
$\Delta_{raw}$ and $\Delta_{prop}$ should be used as complementary metrics
that capture absolute and relative change respectively. We recommend using a
value of 0.7 as the threshold for semantic similarity for the best balance of
relevance and coverage when comparing reference texts. The choice of pre-
release window size was less impactful on the trends observed so we therefore
recommend using seven days to account for daily and weekly activity rhythms.
In the context of fast-moving social media discussions it is important to
choose a shorter post-release window size that avoids including many days
after attention has waned. As such, three days is a suitable choice for post-
release window size. Finally, the distribution of $\Delta_{raw}$ and
$\Delta_{prop}$ scores we observed suggest that echo scores equivalent to an
average increase in coverage of one percentage point should be considered
indicative of messages with the largest social media discussion.

## Discussion

In this paper we introduced a new methodology for evaluating the impact of
strategic communications. Our new metrics use text embedding to measure the
change in semantic similarity between the text of the reference strategic
message and a set of texts from an audience of interest close to its release
date. Given the variability in posting behaviours over time and among groups,
we presented two variants to measure the impact or ‘echo’ of a press release:
$\Delta_{raw}$ which measures the change in absolute activity and
$\Delta_{prop}$, which measures the change in activity relative to the
baseline activity rate. To test this approach, we focused on ten organisations
promoting action on climate change and tweets to capture the conversational
echo of organisational messaging over a two year period.

Our analysis of this new technique focused on understanding the responsiveness
of Twitter users to messaging within the online debate around climate change.
We found a highly skewed distribution of echo scores, with the majority of the
press releases collected seeing very little change in the level of
semantically similar posts in the discourse. However, we did observe several
highly successful strategic communications with an increase of the order of
ten thousand semantically similar posts after their release date. As a result,
we can be confident that our approach is suitable for differentiating between
content with varying levels of uptake in the public discourse on social media.

We also find some evidence of temporal co-occurrence in successful messaging.
In Figure 4 we can see vertical bands of increased responsiveness to strategic
communications. Many of these arise around events in the US political
discourse such as the 2020 US presidential election and the initial actions by
newly-elected President Biden. In some cases these are driven by major events
promoting wider awareness of and participation in the climate change debate
(e.g. the 2021 Earth Day Summit, the 2021 IPCC WG1 report release) but in
others, the same events do not precipitate an increase in activity. This
suggests that timing and consistency in messaging may be a key component in
promoting wider success.

One important consideration arising from our $\Delta$ metrics is in the use of
either the raw or proportional variant. The results presented here suggest
that in many cases, they should be considered as complementary metrics.
$\Delta_{raw}$ communicates the absolute size of the conversational change and
gives a better indication of volume of relevant messages being posted, whereas
$\Delta_{prop}$ is designed to account for the variable activity rates common
in social media and allow for effective comparisons between messages released
at different times and in different contexts. Each of these variants has
shortcomings at the extremes of the activity distribution, however:
$\Delta_{raw}$ can be misleading when the baseline activity changes rapidly or
press releases follow major events (as highlighted in Figure 3); whereas
$\Delta_{prop}$ is susceptible to periods of low baseline activity, during
which time any increase is proportionally larger. While the level of activity
we see in the English language Twitter discourse of climate change means we do
not observe this latter challenge in this instance, future applications of
$\Delta_{prop}$ to smaller topics or audiences should be aware of this
behaviour. Considering both $\Delta_{raw}$ and $\Delta_{prop}$ together
provides valuable context and clarification.

In this work, we have presented a new tool with value to communications
scholars and particularly those with an interest in agenda setting. Our
$\Delta$ metrics help to improve understanding of the influence of specific
messages on the public discourse and presents a quantification to aid in the
identification of successful strategies. Communications professionals are also
likely to benefit from better evaluation of the ‘echo’ of their communications
and robust quantification of reception. It should be stressed here however
that our metrics should not be used to make causal inference. In the context
of our conceptual framework we can consider a null result of small $\Delta$ as
indicative of no influence, whereas a large $\Delta$ could arise from a number
of (typically impossible to measure and distinguish) confounding factors. When
such characterisations are important, we recommend using the $\Delta$ values
to identify promising candidates for subsequent qualitative analysis.

This work suggests several promising avenues for future research. The
calculation of messaging echo is agnostic to the text domains compared, and
could be readily applied to study influence on political conversations or news
media. Varying the reference texts may help identify sources of particular
messaging and illustrate the path of influence from source to wider usage.
Comparing the echo of the same texts across a range of such sources would
provide another valuable dimension to this analysis. Being able to
differentiate the uptake of strategic communications across target groups will
enable more granular analyses and answers to questions such as the extent to
which messages reach outside of already engaged individuals.

While our experiments show that sentence representations are of good quality
for our purposes (as discussed in the context of our set of labelled text
pairs), there may still be bias encoded in the text representations of the
model. For example, language containing gender and racial bias has been shown
to exist in text corpora commonly used to train large language models (Ferrer
et al. 2021; Field et al. 2021). While “numerous debiasing techniques [have
been] proposed” (Joniak and Aizawa 2022), evaluating and adapting such a
method to suitably fine-tune our model was beyond the scope of our study.
However, the modular nature of our method means future work could include the
use of a debiased language model or those that are specifically tuned for
particular corpora.

Overall, the results presented here demonstrate the value of considering
semantic similarity as a means of tracking content alignment between
communication mediums. Further investigation of the methods we introduce here
presents exciting opportunities for communications scholars and practitioners
for richer analysis of agenda setting at much larger scales and will inform
future successful campaigning strategies.

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# On spatial Fourier spectrum
of rogue wave breathers

Natanael Karjanto 🖂<EMAIL_ADDRESS>Department of Mathematics, University
College, Natural Science Campus
Sungkyunkwan University, Suwon 16419, Republic of Korea

( Updated )

###### Abstract

In this article, we derive exact analytical expressions for the spatial
Fourier spectrum of the soliton family on a constant background. Also known as
breathers, these solitons are exact solutions of the nonlinear Schrödinger
equation and are considered as prototypes for rogue wave models. Depending on
the periodicity in the spatial-temporal domain, the characteristics in the
wavenumber-temporal domain may feature either a continuous or discrete
spectrum.

Keywords: rogue waves, nonlinear Schrödinger equation, breathers, solitons on
a constant background, spatial Fourier spectrum.

##  1 Introduction

The study of rogue waves has been of great interest ever since the discovery
of soliton solutions of some nonlinear evolution equations. In particular, the
focusing nonlinear Schrödinger (NLS) equation and its family of exact
analytical solutions of solitons on a non-vanishing background, also known as
breathers, has been proposed as prototypes for rogue wave modeling. The
literature is not exhaustive in disseminating rogue waves study based on the
NLS model and its corresponding breather solutions, for which the phenomena
are observed and encounter applications in hydrodynamics, optical fibers, and
photonic crystals among others [1, 2].

Consider the focusing NLS equation in $(1+1)$-dimension written in a canonical
form [3, 4]:

$iq_{t}+q_{xx}+2|q|^{2}q=0,\qquad(x,t)\in\mathbb{R}^{2},\qquad
q(x,t)\in\mathbb{C}.$ (1)

In surface gravity waves, the variables $(x,t)$ represent spatial and temporal
quantities, respectively, while in nonlinear optics, $t$ denotes the
transversal pulse propagation in space and $x$ designates the time variable.
Unless otherwise indicated, we adopt the interpretation for water waves for
the rest of the article. The complex-valued amplitude $q(x,t)$ describes an
envelope of the corresponding wave packet profile $\eta$, where usually given
by the relationship
$\eta(\bar{x},\bar{t})=\text{Re}\left\\{q(x,t)e^{i(k\bar{x}-\omega\bar{t})}\right\\}$,
where $x=\varepsilon(\bar{x}-c_{g}\bar{t})$, $t=\varepsilon^{2}\bar{t}$,
$\varepsilon\ll 1$, $c_{g}$ is the group velocity, and the wavenumber $k$ and
wave frequency $\omega$ are related by the linear dispersion relationship for
the corresponding medium.

The simplest nontrivial solution is space-independent,
$q(x,t)=q_{0}(t)=e^{2it}$, known as the continuous or plane wave solution. The
simplest solution with dependence on both variables is
$q(x,t)=q_{0}(t)\,\mathrm{sech}\,(x)$ [5]. In particular, the NLS equation
admits the following family of solitons on a constant background, and this is
where our interest lies:

$\displaystyle q_{K}(x,t)$ $\displaystyle=q_{0}(t)\left(\frac{\mu^{3}\cos(\rho
t)+i\mu\rho\sin(\rho t)}{\rho\cosh(\mu x)-2\mu\cos(\rho
t)}-1\right),\qquad\rho=\mu\sqrt{4+\mu^{2}}$ (2) $\displaystyle q_{P}(x,t)$
$\displaystyle=q_{0}(t)\left(\frac{4(1+4it)}{1+(4t)^{2}+(2x)^{2}}-1\right)$
(3) $\displaystyle q_{A}(x,t)$
$\displaystyle=q_{0}(t)\left(\frac{\kappa^{3}\cosh(\sigma
t)+i\kappa\sigma\sinh(\sigma t)}{2\kappa\cosh(\sigma t)-\sigma\cos(\kappa
x)}-1\right),\qquad\sigma=\kappa\sqrt{4-\kappa^{2}}.$ (4)

In our context, $q_{K}$, $q_{P}$, and $q_{A}$ denote the Kuznetsov-Ma (KM),
Peregrine/rational, and Akhmediev breathers, respectively [6, 7, 8, 9, 10,
11]. The chosen order is more historical rather than its fame, simplicity, or
fame, even though $q_{P}$ is the limiting case for both $q_{K}$ and $q_{A}$.
See [12] for an extensive discussion and visual exploration of the limiting
behavior of these NLS breathers. While the discussion of these breathers in
the spatio-temporal domain has reached maturity, this article fills a gap in
covering their features in the wavenumber-temporal domain by observing their
spectra. Note that this should not be confused with spectrum in the context of
the inverse scattering transform (IST), but rather the physical spectrum that
can be calculated by switching from the space domain into the wavenumber
domain by means of the spatial Fourier transform.

Recently, _Frontiers of Physics_ has published a sequence of topical research
articles describing the latest progress of NLS breathers while commemorating
the tenth anniversary of the observation of the Peregrine soliton in nonlinear
media, including optical fibers, water waves, and plasma physics [13, 14, 15].
In particular, the stability properties in energy spaces of these three
important NLS breathers have been reviewed in [16]. By characterizing the
spectral properties of each of these breathers, the authors demonstrated that
they are unstable in the Lyapunov sense.

We begin with the following well-known definitions on Fourier transforms,
Fourier series, and spectral decomposition [17].

######  Definition 1 (Spatial Fourier transform).

Let $f(x,t)$ be a square-integrable function on the spatial real line, then it
can be represented in a dual spatial-wavenumber $(x,k)$ space by fixing the
time variable $t$ as integral transforms

$f(x,t)=\frac{1}{2\pi}\int_{-\infty}^{\infty}\widehat{f}(k,t)e^{ikx}\,dk,$ (5)

where $\widehat{f}(k,t)$ is the spatial non-unitary Fourier transform written
in the terms of angular wavenumber $k$, and is defined by

$\widehat{f}(k,t)=\int_{-\infty}^{\infty}q(x,t)e^{-ikx}\,dx.$ (6)

######  Definition 2 (Temporal Fourier transform).

Alternatively, for a square-integrable function $f(x,t)$ on the temporal real
line, it can also be expressed at a fixed location $x$ in space as integral
transforms between dual temporal-frequency $(t,\omega)$ domain

$f(x,t)=\frac{1}{2\pi}\int_{-\infty}^{\infty}\widehat{f}(x,\omega)e^{i\omega
t}\,d\omega,$ (7)

where $\widehat{f}(x,\omega)$ is the temporal non-unitary Fourier transform
written in the terms of angular frequency $\omega$, and is defined by

$\widehat{f}(x,\omega)=\int_{-\infty}^{\infty}f(x,t)e^{-i\omega t}\,dt.$ (8)

In this case, the relationship between (5) and (6) is between space and
wavenumber domains, where the spatial and wavenumber variables become the
transform variables for a fixed time. On the other hand, for (7) and (8), they
relate between the time and frequency domains at a fixed location in space by
letting time and frequency as the transform variables [18].

######  Definition 3 (Spectral decomposition and Complex spatial Fourier
series).

For a spatially periodic function $f(x,t)$ with a period $L\neq 0$, i.e.,
$f(x,t)=f(x+L,t)$, for all $x\in\mathbb{R}$, the spectral decomposition of $f$
is given as a discrete summation of the harmonic oscillation $e^{ik_{n}x}$

$f(x,t)=\sum_{-\infty}^{\infty}\widehat{f}_{n}(t)e^{ik_{n}x}$ (9)

where $\widehat{f}_{n}(t)=\widehat{f}(k_{n},t)$ are Fourier coefficients of
the complex spatial Fourier series for $f(x,t)$.

$\widehat{f}_{n}(t)=\frac{1}{2L}\int_{-L}^{L}f(x,t)e^{-ik_{n}x}\,dx.$ (10)

######  Definition 4 (Complex temporal Fourier series).

For a temporally periodic function $f(x,t)$ with a period $T\neq 0$, i.e.,
$f(x,t)=f(x,t+T)$, $t>0$, the spectral decomposition of $f$ is given as a
discrete summation of the harmonic oscillation $e^{i\omega_{n}x}$

$f(x,t)=\sum_{-\infty}^{\infty}\hat{f}_{\omega_{n}}(x,\omega_{n})e^{i\omega_{n}t}$
(11)

where $\hat{f}_{\omega_{n}}(x)=\hat{f}(x,\omega_{n})$ are Fourier coefficients
of the complex temporal Fourier series for $f(x,t)$.

$\hat{f}_{\omega_{n}}(x)=\frac{1}{2T}\int_{-T}^{T}f(x,t)e^{-i\omega_{n}t}\,dt.$
(12)

Throughout the rest of this article, unless otherwise mentioned, we focus our
discussion on the spatial Fourier transform and (complex) Fourier series for
the family of breather soliton solutions of the NLS equation (1), i.e., (5),
(6), (9), and (10). Before moving on to the next section, we end this
introduction by stating the spatial Fourier transform of a unitary constant
function as the following lemma.

######  Lemma 1.

The Fourier transform of the constant function $f(x)=1$ is given by the Dirac
delta function $\delta(k)$, generally in the sense of distributions rather
than measures, i.e.,

$\widehat{f}(k)=\int_{-\infty}^{\infty}1\,e^{-ikx}\,dx=2\pi\delta(k).$

The article is organized as follows. Section 2 covers the spatial Fourier
transform for the breather solitons. It covers the three breathers mentioned
earlier and provides analytical expressions for the corresponding spectra. We
also illustrate the spectra by plotting them. Section 3 concludes our
discussion.

##  2 Spatial Fourier spectrum

In this section, we derive the spatial Fourier spectrum for the breather
solitons of the NLS equation. For the KM breather and Peregrine soliton, the
spectra are continuous in wavenumber $k$ while for the Akhmediev breather, the
spectrum is discrete due to its spatial periodicity.

###  2.1 Kuznetsov-Ma breather

Before proving the theorem for the spectrum of the KM breather, we need to
verify the following lemma.

######  Lemma 2.

For $k\neq 0$, $\mu>0$, and $b>|c|>0$, we have the following definite integral
(Formula 3.983.1(b) in [19]):

$\int_{-\infty}^{\infty}\frac{\cos(kx)\,dx}{b\cosh(\mu
x)+c}=2\frac{\pi}{\mu}\frac{\sinh\left(\frac{k}{\mu}\arccos\frac{c}{b}\right)}{\sqrt{b^{2}-c^{2}}\sinh\frac{k}{\mu}\pi}.$

###### Proof.

We consider the complex-valued function

$f(z)=\frac{e^{ikz}}{b\cosh(\mu z)+c}$

and the rectangular contour $\gamma_{\text{\tiny$K$}}$ with corners at $\pm R$
and $\pm R+2\pi i/\mu$, for $R>0$. See Figure 1. To find the poles and
residues of $f$, we need to solve $b\cosh(\mu z)+c=0$, $z\in\mathbb{C}$. This
gives

$\displaystyle\frac{b}{2}\left(e^{2\mu z}+1\right)+ce^{\mu z}$
$\displaystyle=0$ $\displaystyle e^{\mu z}$
$\displaystyle=\frac{-c\pm\sqrt{c^{2}-b^{2}}}{b}=-\frac{c}{b}\pm
i\sqrt{1-\frac{c^{2}}{b^{2}}}=\cos(\pi\mp\theta)\pm i\sin(\pi\mp\theta)$
$\displaystyle z$ $\displaystyle=\ln
1+\frac{i}{\mu}\left(\pi\mp\theta+2n\pi\right),\qquad n\in\mathbb{Z}$

where for $0\leq\theta<\pi/2$, we take

$\cos\theta=\frac{c}{b},\qquad\sin\theta=\sqrt{1-\frac{c^{2}}{b^{2}}},\qquad\tan\theta=\sqrt{\frac{b^{2}}{c^{2}}-1}>0.$

Re($z$)Im($z$)O$R$$-R$$R+2\pi i/\mu$$-R+2\pi
i/\mu$$z_{3}$$z_{2}$$\gamma_{\text{\tiny$K$}}$ Figure 1: A rectangular contour
$\gamma_{\text{\tiny$K$}}$ with corners $\pm R$ and $\pm R+2\pi i/\mu$
determined by a closed, simple, piecewise smooth curve traversed in positive
orientation. The function $f$ is holomorphic in an open connected domain
containing the interior of $\gamma_{\text{\tiny$K$}}$ and its closure, except
at the poles $z=z_{2}$ and $z=z_{3}$.

Inside the rectangular contour, the function has two simple poles at the
following points:

$\displaystyle
z_{2}=\frac{i}{\mu}(\pi-\theta),\qquad\qquad\text{and}\qquad\qquad
z_{3}=\frac{i}{\mu}(\pi+\theta)$

that lie on the imaginary axis. The corresponding residue for each of these
poles is given as follows:

$\displaystyle\text{res}(f,z_{2})$
$\displaystyle=\frac{e^{-\frac{k}{\mu}(\pi-\theta)}}{b\mu\sinh
i(\pi-\theta)}=\frac{e^{-\frac{k}{\mu}(\pi-\theta)}}{ib\mu\sin(\pi-\theta)}=\frac{e^{-\frac{k}{\mu}(\pi-\theta)}}{i\mu\sqrt{b^{2}-c^{2}}}$
$\displaystyle\text{res}(f,z_{3})$
$\displaystyle=\frac{e^{-\frac{k}{\mu}(\pi+\theta)}}{b\mu\sinh
i(\pi+\theta)}=\frac{e^{-\frac{k}{\mu}(\pi+\theta)}}{ib\mu\sin(\pi+\theta)}=\frac{-e^{-\frac{k}{\mu}(\pi+\theta)}}{i\mu\sqrt{b^{2}-c^{2}}}.$

Using the Residue Theorem, we know that

$\displaystyle\int_{\gamma_{\text{\tiny$K$}}}f(z)\,dz$ $\displaystyle=2\pi
i\sum_{n=2}^{3}\text{res}(f,z_{n})=\frac{2\pi}{\mu\sqrt{b^{2}-c^{2}}}\left[e^{-\frac{k}{\mu}(\pi-\theta)}-e^{-\frac{k}{\mu}(\pi+\theta)}\right]$
$\displaystyle=\frac{2\pi
e^{-\frac{k}{\mu}\pi}}{\mu\sqrt{b^{2}-c^{2}}}\left[e^{\frac{k}{\mu}\theta}-e^{-\frac{k}{\mu}\theta}\right]=\frac{2\pi\cdot\left(2e^{-\frac{k}{\mu}\pi}\right)}{\mu\sqrt{b^{2}-c^{2}}}\sinh\left(\frac{k}{\mu}\theta\right).$
(13)

On the other hand, expressing the integral on the left-hand side above as a
combination of line integrals yields

$\displaystyle\int_{\gamma_{\text{\tiny$K$}}}f(z)\,dz$
$\displaystyle=\int_{-R}^{R}\frac{e^{ikx}\,dx}{b\cosh(\mu
x)+c}+\int_{0}^{2\pi}\frac{e^{ik(R+iy)}\,i\,dy}{b\cosh\mu(R+iy)+c}$
$\displaystyle\quad-\int_{-R}^{R}\frac{e^{ik(x+2\pi i/\mu)}\,dx}{b\cosh(\mu
x+2\pi i)+c}-\int_{0}^{2\pi}\frac{e^{ik(-R+iy)}\,i\,dy}{b\cosh\mu(-R+iy)+c}$
$\displaystyle=\left(1-e^{-2\frac{k}{\mu}\pi}\right)\int_{-R}^{R}\frac{e^{ikx}\,dx}{b\cosh(\mu
x)+c}$ $\displaystyle\quad+i\int_{0}^{2\pi}\frac{e^{ikR}e^{-\omega
y}\,dy}{b\cosh\mu(R+iy)+c}-i\int_{0}^{2\pi}\frac{e^{-ikR}e^{-\omega
y}\,dy}{b\cosh\mu(R-iy)+c}$ (14)

since $\cosh(\mu x+2\pi i)=\cosh(\mu x)\cos(2\pi)+i\sinh(\mu
x)\sin(2\pi)=\cosh(\mu x)$. Furthermore, $\left|\cosh\mu(\pm
R+iy)\right|^{2}=\cosh^{2}(\mu R)-\sin^{2}(\mu y)\geq\cosh^{2}(\mu
R)-1\geq\cosh^{2}(\mu R)$, $|b\cosh\mu(\pm R+iy)|\geq b\cosh(\mu R)$, and for
sufficiently large $R$, $|b\cosh\mu(\pm R+iy)+c|\geq b\cosh(\mu R)-c$. Hence

$\displaystyle\left|i\int_{0}^{2\pi}\frac{e^{ikR}e^{-ky}\,dy}{b\cosh\mu(R+iy)+c}\right|$
$\displaystyle\leq\int_{0}^{2\pi}\left|\frac{e^{ikR}e^{-ky}}{b\cosh\mu(R+iy)+c}\right|\,dy\leq\int_{0}^{2\pi}\frac{e^{-ky}}{b\cosh\mu
R-c}\,dy$ $\displaystyle\leq\frac{1-e^{-2k\pi}}{k(b\cosh\mu R-c)}$

which tends to $0$ as $R\to\infty$. Similarly, for sufficiently large $R$

$\displaystyle\left|i\int_{0}^{2\pi}\frac{e^{-ikR}e^{-ky}\,dy}{b\cosh\mu(-R+iy)+c}\right|$
$\displaystyle\leq\int_{0}^{2\pi}\left|\frac{e^{-ikR}e^{-ky}}{b\cosh\mu(-R+iy)+c}\right|\,dy\leq\int_{0}^{2\pi}\frac{e^{-ky}}{b\cosh\mu
R-c}\,dy$ $\displaystyle\leq\frac{1-e^{-2k\pi}}{k(b\cosh\mu R-c)}$

which again tends to $0$ as $R\to\infty$. Hence, by letting $R\to\infty$, we
observe from (14) and (13) that

$\left(1-e^{-2\frac{k}{\mu}\pi}\right)\int_{-\infty}^{\infty}\frac{e^{ikx}\,dx}{b\cosh(\mu
x)+c}=\frac{2\pi\cdot\left(2e^{-\frac{k}{\mu}\pi}\right)}{\mu\sqrt{b^{2}-c^{2}}}\sinh\left(\frac{k}{\mu}\theta\right).$

By taking the real-part of the integral on the left-hand side and pulling out
the factor $2e^{-\frac{k}{\mu}\pi}$ from the product in front of it, we obtain

$\displaystyle\int_{-\infty}^{\infty}\frac{\cos\left(kx\right)\,dx}{b\cosh(\mu
x)+c}$
$\displaystyle=\frac{2\pi}{\mu\sqrt{b^{2}-c^{2}}}\frac{\sinh\left(\frac{k}{\mu}\theta\right)}{\sinh\left(\frac{k}{\mu}\pi\right)}=\frac{2\pi}{\mu}\frac{\sinh\left[\frac{k}{\mu}\arccos\left(\frac{c}{b}\right)\right]}{\sqrt{b^{2}-c^{2}}\sinh\left(\frac{k}{\mu}\pi\right)}.$

This completes the proof. ∎

Figure 2: The spatial Fourier spectrum evolution for the KM breather
$\widehat{q}_{K}(k_{0},t)$ as a function of time for $\mu=1$, $k_{0}=0.1$ (top
left), $k_{0}=0.5$ (top right), $k_{0}=1$ (bottom left), and $k_{0}=2$ (bottom
right). The real part (dashed blue), the imaginary part (dotted red), and the
modulus (solid black) are depicted separately.

######  Theorem 1.

The spatial Fourier spectrum of the KM breather is given by

$\displaystyle\widehat{q}_{K}(k,t)$
$\displaystyle=\left\\{\begin{array}[]{ll}2\pi
e^{2it}\left\\{\frac{\left[\mu^{3}\cos(\rho t)+i\mu\rho\sin(\rho
t)\right])}{\mu^{2}\sqrt{\mu^{2}+\sin^{2}(\rho
t)}}\cdot\frac{\sinh\left[\frac{k}{\mu}\arccos\left(-2\frac{\mu}{\rho}\cos(\rho
t)\right)\right]}{\sinh\left(\frac{k}{\mu}\pi\right)}-\delta(k)\right\\},&\text{for}\;k\neq
0\\\ 2e^{2it}\left\\{\frac{\left[\mu^{3}\cos(\rho t)+i\mu\rho\sin(\rho
t)\right]}{\mu^{2}\sqrt{\mu^{2}+\sin^{2}(\rho
t)}}\cdot\arccos\left(-2\frac{\mu}{\rho}\cos(\rho
t)\right)-\pi\delta(0)\right\\},&\text{for}\;k=0.\end{array}\right.$

###### Proof.

Using Definition 1 and the fact that $q_{K}(x,t)$ is an even function with
respect to $x$, the spatial Fourier transform of the KM breather $q_{K}$ can
be written as follows:

$\displaystyle\widehat{q}_{K}(k,t)$
$\displaystyle=\int_{-\infty}^{\infty}e^{2it}\left(\frac{\mu^{3}\cos(\rho
t)+i\mu\rho\sin(\rho t)}{\rho\cosh(\mu x)-2\mu\cos(\rho
t)}\right)e^{-ikx}\,dx-\int_{-\infty}^{\infty}1\,e^{-ikx}\,dx$
$\displaystyle=e^{2it}\left[\mu^{3}\cos(\rho t)+i\mu\rho\sin(\rho
t)\right]\int_{-\infty}^{\infty}\left(\frac{\cos(kx)}{\rho\cosh(\mu
x)-2\mu\cos(\rho t)}\right)\,dx-\int_{-\infty}^{\infty}e^{-ikx}\,dx.$

Using Lemma 1, taking $b=\rho$ and $c=-2\mu\cos(\rho t)$ in Lemma 2, and since
$\sqrt{b^{2}-c^{2}}=\mu\sqrt{\mu^{2}+\sin^{2}(\rho t)}$ for $\mu>0$, we obtain

$\widehat{q}_{K}(k,t)=2\pi e^{2it}\left\\{\frac{\left[\mu^{3}\cos(\rho
t)+i\mu\rho\sin(\rho t)\right]}{\mu^{2}\sqrt{\mu^{2}+\sin^{2}(\rho
t)}}\cdot\frac{\sinh\left[\frac{k}{\mu}\arccos\left(-2\frac{\mu}{\rho}\cos(\rho
t)\right)\right]}{\sinh\left(\frac{k}{\mu}\pi\right)}-\delta(k)\right\\},\qquad\text{for}\;k\neq
0.$

By taking the limit of the expressions in Lemma 2 as $k\to 0$ yields

$\int_{-\infty}^{\infty}\frac{dx}{\rho\cosh(\mu x)-2\mu\cos(\rho
t)}=\frac{2}{\mu^{2}\sqrt{\mu^{2}+\sin^{2}(\rho
t)}}\arccos\left(-2\frac{\mu}{\rho}\cos(\rho t)\right).$

Hence

$\widehat{q}_{K}(0,t)=2e^{2it}\left\\{\frac{\left[\mu^{3}\cos(\rho
t)+i\mu\rho\sin(\rho t)\right]}{\mu^{2}\sqrt{\mu^{2}+\sin^{2}(\rho
t)}}\cdot\arccos\left(-2\frac{\mu}{\rho}\cos(\rho
t)\right)-\pi\delta(0)\right\\}.$

This completes the proof. ∎

Figure 3: The spatial Fourier spectrum of the KM breather
$\widehat{q}_{K}(k,t_{0})$ as a function of the wavenumber $k$ for $\mu=1$
when $t_{0}=0$ (top left), $t_{0}=\pi/(8\rho)$ (top right),
$t_{0}=\pi/(6\rho)$ (bottom left), and $t_{0}=\pi/(4\rho)$ (bottom right). The
real and imaginary parts are given by dashed blue and dotted red plots,
respectively.

Figure 2 displays the spatial Fourier spectrum of the KM breather as a
function of time for several values of wavenumber $k=k_{0}$. All panels
correspond to the parameter value $\mu=1$. The real part, imaginary part, and
modulus of the spectrum are depicted as dashed blue, dotted red, and solid
black curves, respectively. We observe that all spectrum components are
periodic with respect to time. For small $k$, the spectrum modulus reaches
local maximum values of larger than six while the local minimum values are
also positive and larger than one. As the values of $k$ increase, both local
maximum and minimum values of the spectrum decrease, where the latter could
vanish for particular temporal values. Furthermore, the spectrum moduli
feature a local minimum that is sandwiched between local maxima for $k\leq 1$
in the neighborhood when $t=0$. This feature starts to disappear as the
wavenumber increases and eventually, only local maxima are visible.

Figure 3 illustrates the spatial Fourier spectrum of the KM breather as a
function of the wavenumber for several values of time $t=t_{0}$. We also take
the breather parameter $\mu=1$ in all cases. The real and imaginary parts of
the spectrum are given by dashed blue and dotted red curves, respectively. For
the former, the presence of the Dirac delta function is noticeable at $k=0$,
where the curves go to infinity negatively. The spectrum reduces to a real-
valued function when $t=0$ and its corresponding periodic time-scale. Both
spectrum components vanish as $k\to\pm\infty$. While the imaginary part always
reaches a local maximum at $k=0$ for $t\neq 0$, the real-part does not always
behave like that. As shown at the bottom right panel, the real part may reach
a local minimum in a limiting sense when $k\to 0$ at $k=\pi/(4\rho)$.

###  2.2 Peregrine soliton

Before deriving the spectrum for the Peregrine soliton, we state and verify
the following lemma.

######  Lemma 3.

For $a>0$ and $k\in\mathbb{R}$, we have the following integral:

$\int_{-\infty}^{\infty}\frac{\cos(kx)}{a+(2x)^{2}}\,dx=\frac{\pi}{2\sqrt{a}}e^{-\frac{|k|}{2}\sqrt{a}}.$
(15)

###### Proof.

We show the proof using the semi-circular contour theorem from complex
analysis, e.g., Theorem 9.1 in [20] (on page 154–155). Let

$f(z)=\frac{e^{i|k|z}}{a+(2z)^{2}}$

be a complex-valued function and meromorphic in the upper half-plane. We
observe that $f$ has no poles on the real axis and it has only two poles at
$z=i\sqrt{a}/2$ and $z=-i\sqrt{a}/2$. Only the first of these is in the upper
half-plane. We calculate that

$\text{res}\left(f,i\frac{\sqrt{a}}{2}\right)=\frac{e^{-\frac{|k|}{2}\sqrt{a}}}{4i\sqrt{a}}.$

In the upper half-plane, for all sufficiently large $|z|$, we have

$\displaystyle\left|zf(z)\right|=\left|\frac{z\,e^{i|k|z}}{a+(2z)^{2}}\right|=\left|\frac{z\,e^{i|k|x}\,e^{-|k|y}}{4z^{2}+a}\right|\leq\frac{|z|}{4|z|^{2}-a}.$

Re($z$)Im($z$)O$R$$-R$$i\frac{\sqrt{a}}{2}$$\gamma_{\text{\tiny$P$}}$ Figure
4: A semicircle contour
$\gamma_{\text{\tiny$P$}}=\left[-R,R\right]\cup\left\\{z:|z|=R,\;\text{Im}\;z\geq
0\right\\}$, traversed in the positive direction with $R>\sqrt{a}/2$.

We have adopted the relationship Re$(z)=x$ and Im$(z)=y$ and the fact that
$\left|e^{i|k|x}\right|=1$ and $\left|e^{-|k|y}\right|\leq 1$ for $y\geq 0$.
It follows that $\left|zf(z)\right|$ tends uniformly to $0$ as $|z|\to\infty$
in the upper half-plane. Let $\gamma_{\text{\tiny$P$}}$ be a semicircle
contour given by
$\gamma_{\text{\tiny$P$}}:\left[-R,R\right]\cup\left\\{z:|z|=R,\;\text{Im}\;z\geq
0\right\\}$, traversed in the positive direction with $R>\sqrt{a}/2$, as shown
in Figure 4, then

$\displaystyle\text{(PV)}\int_{-\infty}^{\infty}f(x)\,dx$
$\displaystyle=\lim\limits_{R\to\infty}\int_{-R}^{R}f(x)\,dx=\int_{\gamma_{\text{\tiny$P$}}}f(z)\,dz-\lim\limits_{R\to\infty}\int_{0}^{\pi}f(Re^{i\theta})\,iRe^{i\theta}\,d\theta$

where PV stands for the principal value. Indeed, from the condition of
$\left|zf(z)\right|$ uniformly in the upper half-plane as $|z|\to\infty$, it
implies that for all $\varepsilon>0$, there exists $S>0$ such that
$\left|zf(z)\right|<\varepsilon$ for all $z$ in the upper half-plane such that
$|z|>S$. Consequently, for all $R>S$

$\left|\int_{0}^{\pi}f(Re^{i\theta})\,iRe^{i\theta}\,d\theta\right|\leq\int_{0}^{\pi}\left|\frac{e^{i|k|Re^{i\theta}}iRe^{i\theta}}{a+4R^{2}e^{2i\theta}}\right|\,d\theta\leq\frac{\pi
R}{4R^{2}-a}<\pi\varepsilon$

and thus

$\int_{0}^{\pi}f(Re^{i\theta})\,iRe^{i\theta}\,d\theta\to
0\qquad\text{as}\quad R\to\infty.$

Since there exists a constant $\widehat{S}>0$ such that
$\left|f(x)\right|\leq\widehat{S}/x^{2}$ for large $|x|$, then
${\displaystyle\int_{\infty}^{\infty}f(x)\,dx}$ exists and equals to its
Cauchy principal value. Hence, we can dispense with the principal value and by
applying the Residue Theorem, we deduce that

$\int_{-\infty}^{\infty}\frac{e^{i|k|x}}{a+(2x)^{2}}\,dx=\int_{-\infty}^{\infty}\frac{\cos(kx)+i\sin\left(|k|x\right)}{a+(2x)^{2}}\,dx=2\pi
i\;\text{res}\left(f,i\frac{\sqrt{a}}{2}\right)=\frac{\pi}{2\sqrt{a}}e^{-\frac{|k|}{2}\sqrt{a}}.$

Equating the real parts gives the desired result and the proof is completed. ∎

Figure 5: The spatial Fourier spectrum of the Peregrine breather
$\widehat{q}_{P}(k_{0},t)$ as a function of time $t$ for $k_{0}=0.1$ (top
left), $k_{0}=0.2$ (top right), $k_{0}=0.5$ (bottom left), and $k_{0}=1$
(bottom right). The real and imaginary parts are given by dashed blue and
solid red plots, respectively. The solid black curve is the modulus spectrum
amplitude.

######  Theorem 2.

The spatial Fourier spectrum of the Peregrine soliton $q_{P}(x,t)$ is given by

$\widehat{q}_{P}(k,t)=2\pi
e^{2it}\left(\frac{1+4it}{\sqrt{1+(4t)^{2}}}e^{-\frac{k}{2}\sqrt{1+(4t)^{2}}}-\delta(k)\right).$
(16)

###### Proof.

Using Definition (1) and the fact that $q_{P}(x,t)$ is an even function with
respect to $x$, we can write the spatial Fourier transform
$\widehat{q}_{P}(k,t)$ as follows:

$\displaystyle\widehat{q}_{P}(k,t)$
$\displaystyle=\int_{-\infty}^{\infty}e^{2it}\left(\frac{4(1+4it)}{1+(4t)^{2}+(2x)^{2}}\right)e^{-ikx}\,dx-\int_{-\infty}^{\infty}1\,e^{-ikx}\,dx$
(17)
$\displaystyle=4e^{2it}(1+4it)\int_{-\infty}^{\infty}\frac{\cos(kx)}{1+(4t)^{2}+(2x)^{2}}\,dx-\int_{-\infty}^{\infty}e^{-ikx}\,dx.$
(18)

Using Lemmas 1 and 3, we obtain the desired result:

$\displaystyle\widehat{q}_{P}(k,t)$
$\displaystyle=4e^{2it}(1+4it)\left(\frac{\pi}{2\sqrt{1+(4t)^{2}}}e^{-\frac{k}{2}\sqrt{1+(4t)^{2}}}\right)-2\pi\delta(k)$
(19) $\displaystyle=2\pi
e^{2it}\left(\frac{1+4it}{\sqrt{1+(4t)^{2}}}e^{-\frac{k}{2}\sqrt{1+(4t)^{2}}}-\delta(k)\right).$
(20)

This completes the proof of the theorem. ∎

Figure 6: The spatial Fourier spectrum of the Peregrine breather
$\widehat{q}_{P}(k,t_{0})$ as a function of the wavenumber $k$ when $t_{0}=0$
(top left), $t_{0}=1/4$ (top right), $t_{0}=1/2$ (bottom left), and $t_{0}=1$
(bottom right). The real and imaginary parts are given by dashed blue and
dotted red plots, respectively.

Figure 5 shows the spatial Fourier spectrum of the Peregrine breather as a
function of time for various values of wavenumber $k=k_{0}$. The real-part,
imaginary part, and modulus of the spectrum are depicted in dashed blue,
dotted red, and solid black curves, respectively. All spectrum components tend
to vanish as $t\to\pm\infty$. Both the real-part and modulus of the spectrum
reach local maxima at $t=0$ while the imaginary part takes negative and
positive values for $t<0$ and $t>0$, respectively. It reaches local minimum
and maximum for some values $t$ not too far away from $t=0$. The value of
local maxima for the spectrum modulus decreases as the value of the wavenumber
increases. For small wavenumber, the local maximum reaches the value of six as
$k\to 0$.

Figure 6 features the spatial Fourier spectrum of the Peregrine breather as a
function of the wavenumber for several values of time $t=t_{0}$. The real and
imaginary parts of the spectrum are plotted as dashed blue and dotted red
curves, respectively. Similar to the previous case, the appearance of the
Dirac delta function is visible at $k=0$ as the curve extends infinitely
negative. The imaginary part vanishes when $t_{0}=0$ and thus the spectrum
reduces to a real-valued function. For $t>0$, both real and imaginary parts
tend to vanish as $k\to\pm\infty$. They also reach local maxima at $k=0$, and
depending on the values of $k$, the local maxima for the real part can be
larger, equal, or smaller than the ones for the imaginary part. The largest of
these quantities takes the value of slightly larger than six at $k=0$.

###  2.3 Akhmediev breather

######  Lemma 4.

For $0<|a|<b$, the following trigonometric integral holds, c.f. Formula
3.613.1 in [19]:

$\int_{0}^{2\pi}\frac{\cos(n\xi)\,d\xi}{b-a\,\cos\xi}=\frac{2\pi}{\sqrt{b^{2}-a^{2}}}\left(\frac{b-\sqrt{b^{2}-a^{2}}}{a}\right)^{n},\qquad\;\textmd{for}\;\;n\in\mathbb{Z}.$
(21)

###### Proof.

By performing the integration in the complex plane, we can write the integral
as the real part of another integral expressed on the right-hand side of the
following equation:

$\int_{0}^{2\pi}\frac{\cos(n\xi)\,d\xi}{b-a\,\cos\xi}=\text{Re}\left\\{\int_{0}^{2\pi}\frac{e^{in\xi}\,d\xi}{b-a\,\cos\xi}\right\\}.$

Let $C:z=e^{i\xi}$, $0\leq\xi\leq 2\pi$ be the unit circle in the complex
plane, then $e^{in\xi}=z^{n}$, $\cos\xi=\frac{1}{2}(z+z^{-1})$, and
$d\xi=-idz/z$. See Figure 7. Substituting these into the integral above, we
can express it as

$\int_{0}^{2\pi}\frac{\cos(n\xi)\,d\xi}{b-a\,\cos\xi}=2i\oint_{C}\frac{z^{n}\,dz}{az^{2}-2bz+a}=2i\oint_{C}\frac{z^{n}\,dz}{a(z-z_{1})(z-z_{2})}$

where

$z_{1}=\frac{b+\sqrt{b^{2}-a^{2}}}{a}\qquad\text{and}\qquad
z_{2}=\frac{b-\sqrt{b^{2}-a^{2}}}{a}$

are the roots of the quadratic equation $az^{2}-2bz+a=0$. The integrand has
simple poles at $z=z_{1}$ and $z=z_{2}$. Observe that

$|z_{1}|=\frac{b+\sqrt{b^{2}-a^{2}}}{a}>\frac{b}{a}>1$

and thus $z_{1}$ does not lie within the contour of $C$. Furthermore, since
$z_{1}z_{2}=a/a=1$, we deduce that $|z_{2}|<1$. The residue at $z=z_{2}$ is
given by

$\frac{z_{2}^{n}}{a(z_{2}-z_{1})}=\frac{1}{-2\sqrt{b^{2}-a^{2}}}\left(\frac{b+\sqrt{b^{2}-a^{2}}}{a}\right)^{n}.$

Using Cauchy’s Residue Theorem, we obtain the desired trigonometric integral:

$\int_{0}^{2\pi}\frac{\cos(n\xi)\,d\xi}{b-a\,\cos\xi}=\frac{(2\pi
i)\,(2i)}{-2\sqrt{b^{2}-a^{2}}}\left(\frac{b+\sqrt{b^{2}-a^{2}}}{a}\right)^{n}=\frac{2\pi}{\sqrt{b^{2}-a^{2}}}\left(\frac{b-\sqrt{b^{2}-a^{2}}}{a}\right)^{n}.$

This completes the proof. ∎

Re($z$)Im($z$)O$1$$-1$$z_{2}$$z_{1}$$\gamma_{\text{\tiny$A$}}$ Figure 7: A
unit circle contour $\gamma_{\text{\tiny$A$}}=\left\\{z:|z|=1\right\\}$
traversed in the positive direction with $z_{2}<1<z_{1}$.

######  Corollary 1.

For $0\leq|a|<1$, we have the following integral:

$\int_{0}^{2\pi}\frac{d\xi}{1-a\,\cos\xi}=\frac{2\pi}{\sqrt{1-a^{2}}}.$ (22)

###### Proof.

By taking $b=1$ and $n=0$ in Lemma 4, i.e., Equation (21), we obtain the
result immediately. ∎

In what follows, we state and prove the spatial Fourier spectrum for the
Akhmediev breather. A similar expression of this spectrum is available in
[21], but the authors did not provide any proof. On the other hand, a
derivation proof using trigonometric integrals and series has been attempted
by the author in an appendix of his PhD thesis [22].

######  Theorem 3.

The spatial Fourier amplitude spectrum for the Akhmediev breather $q_{A}(x,t)$
is given by

$\widehat{q}_{A}^{(n)}(k,t)=\widehat{q}_{A}^{(n)}(\kappa,t)\left\\{\begin{array}[]{ll}e^{2it}\left(\frac{\kappa^{3}\cosh(\sigma
t)+i\kappa\sigma\sinh(\sigma t)}{\sqrt{4\kappa^{2}\cosh^{2}(\sigma
t)-\sigma^{2}}}-1\right),&\quad\text{for}\;n=0,\\\
e^{2it}\left(\frac{\kappa^{3}\cosh(\sigma t)+i\kappa\sigma\sinh(\sigma
t)}{\sqrt{4\kappa^{2}\cosh^{2}(\sigma
t)-\sigma^{2}}}\right)\left(\frac{2\kappa\cosh(\sigma
t)-\sqrt{4\kappa^{2}\cosh^{2}(\sigma
t)-\sigma^{2}}}{\sigma}\right)^{n},&\quad\text{for}\;n\in\mathbb{N}.\end{array}\right.$
(23)

###### Proof.

The Akhmediev breather is periodic in space with the period $L=2\pi/\kappa$.
Furthermore, we consider the wavenumber values $k_{n}$ at a discrete multiple
of the modulation wavenumber $\kappa$, i.e., $k_{n}=n\kappa$,
$n\in\mathbb{Z}$. Hence,

$\widehat{q}_{A}^{(n)}(\kappa,t)=\frac{\kappa}{4\pi}\int_{-2\pi/\kappa}^{2\pi/\kappa}q_{A}(x,t)e^{-in\kappa
x}\,dx.$

Employing the variable substitution $\xi=\kappa x$, and using the fact that
$q_{A}$ is an even function with respect to the spatial variable $x$, it
reduces to

$\displaystyle\widehat{q}_{A}^{(n)}(\kappa,t)$
$\displaystyle=\frac{1}{4\pi}\int_{-2\pi}^{2\pi}e^{2it}\left(\frac{\kappa^{3}\cosh(\sigma
t)+i\kappa\sigma\sinh(\sigma t)}{2\kappa\cosh(\sigma
t)-\sigma\cos\xi}-1\right)\cos(n\xi)\,d\xi$
$\displaystyle=\frac{1}{2\pi}\int_{0}^{2\pi}e^{2it}\left(\frac{\kappa^{3}\cosh(\sigma
t)+i\kappa\sigma\sinh(\sigma t)}{2\kappa\cosh(\sigma
t)-\sigma\cos\xi}-1\right)\cos(n\xi)\,d\xi$

For $n=0$, applying Lemma 4 or Corollary 1, we obtain

$\widehat{q}_{A}^{(0)}(\kappa,t)=e^{2it}\left(\frac{\kappa^{3}\cosh(\sigma
t)+i\kappa\sigma\sinh(\sigma t)}{\sqrt{4\kappa^{2}\cosh^{2}(\sigma
t)-\sigma^{2}}}-1\right).$

For $n\neq 0$, taking $a=\sigma$ and $b=2\kappa\cosh(\sigma t)$, we obtain

$\widehat{q}_{A}^{(n)}(\kappa,t)=e^{2it}\left(\frac{\kappa^{3}\cosh(\sigma
t)+i\kappa\sigma\sinh(\sigma t)}{\sqrt{4\kappa^{2}\cosh^{2}(\sigma
t)-\sigma^{2}}}\right)\left(\frac{2\kappa\cosh(\sigma
t)-\sqrt{4\kappa^{2}\cosh^{2}(\sigma t)-\sigma^{2}}}{\sigma}\right)^{n}.$

This completes the proof. ∎

Figure 8: The spatial Fourier spectrum of the Akhmediev breather
$\widehat{q}_{A}^{(0)}(\kappa_{0},t)$ as a function of time $t$ for
$\kappa_{0}=0.5$ (top left), $\kappa_{0}=1$ (top right), $\kappa_{0}=\sqrt{2}$
(bottom left), and $\kappa_{0}=\sqrt{3}$ (bottom right). The real and
imaginary parts are given by dashed blue and solid red plots, respectively.
The solid black curve is the modulus spectrum amplitude.

Figure 8 displays the spatial Fourier spectrum of the Akhmediev breather for
$n=0$ as a function of time $t$ for various values of the modulation
wavenumber $\kappa=\kappa_{0}$, i.e., $\widehat{q}_{A}^{(0)}(\kappa_{0},t)$.
The real and imaginary parts of $\widehat{q}_{A}^{(0)}$ are depicted by the
dashed blue and dotted red curves, respectively, while the solid black curve
represents the modulus of the amplitude spectrum evolution. For all values of
$\kappa$, the spectrum modulus reaches a minimum value during $t=0$, and for
an increasing value of $\kappa$, these minimum values of the modulus are
decreasing until they vanish at $\kappa_{0}=1$ and then increase again towards
one as $\kappa_{0}\to 2$. For $t\to\pm\infty$, the amplitude modulus also goes
to one.

Figure 9: The modulus of spatial Fourier amplitude spectrum of the Akhmediev
breather $\left|\widehat{q}_{A}^{(n)}(\kappa_{0},t)\right|$ as a function of
time $t$ for $\kappa_{0}=0.5$ (top left), $\kappa_{0}=1$ (top right),
$\kappa_{0}=\sqrt{2}$ (bottom left), and $\kappa_{0}=\sqrt{3}$ (bottom right)
for $n=0$ (solid black), $n=1$ (dashed blue), $n=2$ (dotted red), and $n=3$
(dash-dotted green).

Similar to Figure 8, Figure 9 displays the moduli of amplitude spectra for
several values of $n$ and $\kappa$. The solid black curves correspond to the
moduli for the main/central wavenumber when $n=0$, the dashed blue curves
correspond to the first pair of sidebands ($n=1$), the dotted red curves
correspond to the second pair of sidebands ($n=2$), and the dash-dotted green
curves correspond to the third pair of sidebands ($n=3$). While the moduli for
the central wavenumber reach a local minimum when $t=0$, all the sidebands
moduli reach a local maximum when $t=0$. The further the sideband pairs from
the main wavenumber, the smaller they attain the maximum value. For
$t\to\pm\infty$, all sideband moduli tend to zero while the central sideband
goes to one.

Figure 10: The modulus of spatial Fourier amplitude spectrum of the Akhmediev
breather $\left|\widehat{q}_{A}^{(n)}(\kappa,t)\right|$ when $t=0$ as a
function of the wavenumber $k$ for $\kappa=0.5$ (top left), $\kappa=1$ (top
right), $\kappa=\sqrt{2}$ (bottom left), and $\kappa=\sqrt{3}$ (bottom right)
for $n=0$ (black), $n=1$ (blue), $n=2$ (red), $n=3$ (green), $n=4$ (cyan), and
$n=5$ (magenta).

Figure 10 displays the moduli of the spatial Fourier amplitude spectrum of the
Akhmediev breather when $t=0$ as a function of wavenumber for several values
of $\kappa$. Each panel shows only up to five pairs of sideband wavenumber as
indicated by different colors, the first pair is blue, the second one is red,
the third one is green, the fourth and fifth pairs are cyan and magenta,
respectively. While the moduli of the sidebands are getting smaller as they
progress toward higher-order pairs, the modulus corresponds to the main
wavenumber does not always occupy the highest position. Depending on the
sideband wavenumber $\kappa$, it might be taller than the first pairs of
sidebands (for $\kappa_{0}=\frac{1}{2}$ and $\kappa_{0}=\sqrt{3}$) or shorter
than the first pairs of sidebands (for $\kappa_{0}=1$ and
$\kappa_{0}=\sqrt{2}$).

Different from the previous two cases where the spectra are continuous in $k$
(the KM breather and Peregrine soliton), the spectrum for this breather is
discrete due to its periodic wave profile. While the initial modulated waves
(when $t<0$ but not $t\to-\infty$) exhibit the spectrum of the main/central
frequency accompanied by a pair of sidebands, more pairs of sidebands were
generated as the waves evolves in time. As time increases (when $t>0$),
higher-order sidebands in the spectrum disappear and return to the state of
the initial condition when $t<0$ with the central frequency with a pair of
sidebands. This behavior suggests that the wave energy is distributed from the
main wavenumber to the sideband wavenumbers and is recollected back during the
evolution. Interestingly, for a particular value of $\kappa$, i.e.,
$\kappa_{0}=1$, all energy from the central wavenumber is transferred
completely to its sidebands while for other values of $\kappa$, only partial
energy is distributed from the central wavenumber to its sidebands.

##  3 Conclusion

In this article, we have considered the Fourier spectrum for rogue wave
prototypes from the NLS equation. Also known as the soliton waves on a non-
vanishing background, all three breathers are related with a complex-valued
parameter. While the Peregrine soliton was discovered theoretically after the
discovery of the KM breather and before the Akhmediev breather, it serves as
the limiting case for both the KM and Akhmediev breathers. We need to perform
integration in the complex plane when deriving the analytical expressions for
the spatial Fourier spectrum of the breathers. Although the computations
performed in this article are relatively straightforward, it turns out that
the derivation is technically challenging nonetheless.

Since both the KM and Peregrine breathers have infinite periodicity in the
spatial domain, their spectra are continuous functions in both wavenumber and
temporal domains. The spectrum modulus for the former in the temporal domain
shows a pattern of periodicity while for the latter, the period is infinity.
In the wavenumber domain, the corresponding spectrum moduli for both breathers
exhibit infinite periodicity. For the Akhmediev breather, since it is periodic
in space, its spatial spectrum remains continuous in the time domain but
transforms into a discrete-type in the wavenumber domain. Remarkably, higher-
order sidebands were generated as the initial sideband pairs evolve in time
and then returned to an initial state.

### Dedication

The author would like to dedicate this article to his late father Zakaria
Karjanto (Khouw Kim Soey, 許金瑞) who introduced and taught him the alphabet,
numbers, and the calendar in his early childhood. Karjanto senior was born in
Tasikmalaya, West Java, Japanese-occupied Dutch East Indies on 1 January 1944
(Saturday Pahing) and died in Bandung, West Java, Indonesia on 18 April 2021
(Sunday Wage).

### Acknowledgment

The author gratefully acknowledges E. (Brenny) van Groesen for the long-
lasting guidance and concerted cultivation in thinking and growing up
mathematically.

### Conflict of Interest

The author declares no conflict of interest.

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