# Prediction of the Realisation of an Information Need: An EEG Study

Niall McGuire<EMAIL_ADDRESS>0009-0005-9738-047X University of
StrathclydeGlasgowUnited Kingdom and  Yashar Moshfeghi
<EMAIL_ADDRESS>0000-0003-4186-1088 University of
StrathclydeGlasgowUnited Kingdom

(2024)

###### Abstract.

One of the foundational goals of Information Retrieval (IR) is to satisfy
searchers’ Information Needs (IN). Understanding how INs physically manifest
has long been a complex and elusive process. However, recent studies utilising
Electroencephalography (EEG) data have provided real-time insights into the
neural processes associated with INs. Unfortunately, they have yet to
demonstrate how this insight can practically benefit the search experience. As
such, within this study, we explore the ability to predict the realisation of
IN within EEG data across 14 subjects whilst partaking in a Question-Answering
(Q/A) task. Furthermore, we investigate the combinations of EEG features that
yield optimal predictive performance, as well as identify regions within the
Q/A queries where a subject’s realisation of IN is more pronounced. The
findings from this work demonstrate that EEG data is sufficient for the real-
time prediction of the realisation of an IN across all subjects with an
accuracy of 73.5% (SD 2.6%) and on a per-subject basis with an accuracy of
90.1% (SD 22.1%). This work helps to close the gap by bridging theoretical
neuroscientific advancements with tangible improvements in information
retrieval practices, paving the way for real-time prediction of the
realisation of IN.

EEG, Information Retrieval, Information Need, Machine Learning, NeuraSearch

††journalyear: 2024††copyright: rightsretained††conference: Proceedings of the
47th International ACM SIGIR Conference on Research and Development in
Information Retrieval; July 14–18, 2024; Washington, DC, USA††booktitle:
Proceedings of the 47th International ACM SIGIR Conference on Research and
Development in Information Retrieval (SIGIR ’24), July 14–18, 2024,
Washington, DC, USA††doi: 10.1145/3626772.3657981††isbn:
979-8-4007-0431-4/24/07††ccs: Information systems Search interfaces††ccs:
Information systems††ccs: Information systems Information retrieval††ccs:
Information systems Users and interactive retrieval

##  1\. Introduction

The primary objective of any Information Retrieval (IR) system is to fulfil a
searcher’s Information Need (IN) (Cooper, 1971; Michalkova et al., 2022;
Moshfeghi et al., 2016). In the realm of IR, numerous endeavours have been
dedicated to unravelling and defining the intricate concept of IN. Pioneering
works, such as Taylor’s Question Negotiation Process (Taylor, 1968), Anomalous
State of Knowledge Model (Belkin et al., 1982), and Wilson’s Information
Seeking Behavior (Wilson, 1981), have significantly contributed to this
pursuit. These works explore the essence of IN by examining user behaviour
through techniques like user-system interactions (Bendersky and Croft, 2009),
self-reflective notes (Kuhlthau, 1991), and interviews/questionnaires (Markey,
1981). While these methods offer valuable insights, reporting IN by subjects
is often challenging due to its intricate and elusive nature, thereby
constraining the efficacy of user-based studies (Moshfeghi et al., 2016).

Consequently, over the last decade, a new line of research has endeavoured to
address the inherent limitations by directly examining the neurological
activity in the searcher’s brain through the utilisation of neuroimaging
technologies(Moshfeghi et al., 2016; Moshfeghi and Pollick, 2019; Moshfeghi et
al., 2019; Michalkova et al., 2022). Research conducted at the crossroads of
neuroscience and information retrieval is often referred to as NeuraSearch
(Moshfeghi, 2021). This interdisciplinary field has yielded numerous findings
focused on the tangible representation of information needs (INs) within
specific brain regions. For instance, Functional Magnetic Resonance Imaging
(fMRI) was employed to observe subjects’ brain activity in a Q/A task
(Moshfeghi et al., 2016, 2019). The results revealed a distributed network of
brain regions commonly associated with IN, with varying activity levels in
these regions based on whether the subject knew the answer to a question or
needed to search for it. Further exploration (Moshfeghi et al., 2019) utilised
fMRI data from a similar Q/A task to train a support vector machine (SVM)
capable of distinguishing instances when a searcher possesses an IN. These
investigations have presented compelling evidence regarding the existence and
expression of INs within the minds of searchers. However, employing fMRI for
such analyses presents several limitations. Firstly, the physical hardware of
an fMRI machine is both sizable and costly, necessitating the subject to lie
supine within the central bore of the apparatus while maintaining stillness,
as outlined by Moshfeghi et al. (Moshfeghi et al., 2019). Secondly, despite
its fine spatial resolution, fMRI exhibits suboptimal temporal resolution,
with each measurement taking a duration of 2 seconds (Moshfeghi et al., 2019).
This limitation is further exacerbated by the Blood Oxygenation Level
Dependent (BOLD) signal’s inherent delays (Moshfeghi et al., 2019, 2016).
Despite the valuable insights offered by fMRI, the cumbersome nature and high
cost of the equipment, coupled with its temporal constraints, hinder its
seamless integration into current IR systems.

Acknowledging the limitations inherent within fMRI data, researchers sought
alternative neuroimaging methods to better depict the dynamics of INs with
higher temporal resolution. One such approach involves the utilisation of
Electroencephalography (EEG) data, a cheaper and more practical method, where
electrical activity from the brain is recorded at a millisecond scale through
electrodes placed on the subject’s scalp (Teplan et al., 2002). In the
research presented by (Michalkova et al., 2022), EEG data is employed to
observe subjects’ brain activity during a Q/A session. This investigation aims
to understand the temporal dynamics of IN formation, detecting the presence of
INs even before searchers consciously acknowledge them. This exploration opens
avenues for a proactive search process, offering insights into the early
stages of information needs. Although previous works (Michalkova et al., 2022)
provided an excellent analysis of the physical manifestations of the
realisation of an IN within real-time through the use of EEG data, the
question of “Can the realisation of an IN be predicted in real-time?” is still
unanswered. From this hypothesis, we formulate these four research questions:
RQ1: ”Is it possible to predict the realisation of an IN in real-time from EEG
data?”, RQ2: ”Can prediction of the realisation of IN be generalised across
subjects, or is it subject-specific?”, RQ3: ”During the Q/A session, where are
the strongest indicators of the realisation of an IN?”, RQ4: ”What combination
of features is optimal for the realisation of an IN prediction?”.

In order to address our first research question, within this study we
incorporated the EEG data gathered from 14 subjects whilst they took part in a
Q/A task which involved the subjects observing queries word-by-word and
determining if they could correctly answer the question or had a need to
search (IN). This data is then provided to machine learning models to predict
the subject’s realisation of an IN. Additionally, we investigate the inter and
intra-variability of EEG data across a variety of subjects by exploring how
the prediction of the realisation of an IN is affected when the models are
trained to generalise across subjects compared to when they are trained on a
single subject at a time. Moreover, whilst subjects examine the queries from
the Q/A tasks word-by-word, we determine which segments within the given
sentences are the strongest indicator of the realisation of an IN. As well as
this, we perform an ablation analysis to discern the combination of commonly
extracted EEG features that enables the models to best distinguish between
different search states.

##  2\. Methodology

subjects. The subjects were recruited by the University of Strathclyde. They
received no monetary payments but were eligible for academic credits. The
subjects consisted of 13 females (93%) and 1 male (7%) within an age range
between 18 and 39 years and a mean age of 23 years (SD 6.5).

Recording. The EEG data was captured using a 40-electrode NeuroScan Ltd.
system with a 10/20 cap, sampled at a frequency of 500Hz. The Q/A task was
made of general knowledge questions taken from TREC-8 and TREC-2001 and
B-KNorms Database2.

Q/A Dataset. Two independent assessors separately evaluated the question
difficulty (Cohen’s Kappa: 0.61). A subset of 120 questions was then selected,
and both annotators agreed upon their difficulties. The difficulty of the
questions was equally distributed between easy and difficult for the overall
dataset.

Figure 1. Task Procedure.

Experimental Procedure. Ethical permission to conduct the study was approved
by the Universities Ethics Committee, with the tasks being conducted in a
laboratory setting and all subjects meeting the inclusion criteria, i.e.
healthy subjects of ages 18 - 55 years, fluent English ability, and no
prior/current neurological disorders that may influence the task. Before any
trials began, consent was obtained from the subjects. To ensure the subjects
had a solid grasp of the procedure, before the main trial, they were supplied
with a practice example, which consisted of five questions not included in the
main trial. For the practice session, there was no time limit, and subjects
were allowed to repeat if required, until comfortable to proceed. The
following task procedure was repeated for each trial. The trial began by
viewing a fixation cross in the middle of the screen for a duration of 2000ms,
indicating the location of the next stimuli on the screen, which was a way to
minimise eye movements on the screen. The subjects then viewed a sequential
presentation of a question randomly selected from the dataset. Each word
within the question was displayed for 800ms on the screen one at a time.
Within this step, the subject processed the information as it was being
presented word-by-word. Following the presentation of all the words within the
question, the subjects were presented with a now fully-displayed question and
three on-screen answer choices associated with the question. They were
requested to select the correct answer or the option ”I do not know”, see
Figure 1. If the subjects correctly or incorrectly answered the question, the
answer was displayed onscreen (NoNeedToSearch), where the trial terminated and
moved on to the next question. However, if the subject selected the ”I do not
know”, they were presented with two options: whether they wanted to search
(NeedToSearch) for the correct answer or not (NoNeedToSearch). For this task,
there was no search process as the overall goal was to analyse the presence of
an information need based on the decision to search by the subject. After
selecting one of the two options, the trial would terminate, and the next
question would be presented. This was repeated for all 120 questions. Upon
task completion, analysis of the 14 subjects revealed that 85% of the
responses were classed as NoNeedToSearch, and the remaining 15% were
NeedToSearch, in order to balance the dataset, the number of NoNeedToSearch
classes was made equal to the number of NeedToSearch classes. The subjects
completed the task (without breaks) on average in 44 min (sd=4.62, med=43.40).

Pre-processing. During EEG recording, the individual’s actions often introduce
electrical activities that can affect measurements and distort results. To
address this issue, it is essential to eliminate these artefacts as
effectively as possible. Initially, we utilised a bandpass filter (Daud and
Sudirman, 2015; Kingphai and Moshfeghi, 2021) with a range of 0.5 to 50Hz.
This range is commonly used because research indicates that the brain’s
recorded electrical activity falls within this spectrum. Additionally, we
implemented average re-referencing (Lepage et al., 2014), a technique that
establishes a reference point by aggregating the activity measured across all
electrodes. The objective is to capture any noise or interference impacting
all electrodes within this reference. Subsequently, we subtract this reference
from each electrode’s signal, effectively eliminating the noise from each
electrode’s signal.

Feature Extraction. For this study, we extract a commonly adopted core
(Kingphai and Moshfeghi, 2021; Zhang et al., 2017) set of features from the
EEG signals to determine which combination is optimal for IN classification,
with each feature being extracted per-electrode (channel) signals across four
specific frequency bands: Delta (1–4 Hz), theta (4–8 Hz), alpha (8–12 Hz),
beta (12–30 Hz) and gamma (30–40 Hz). The features are extracted across every
800ms block where the question words are presented to the subject and when
they respond to the question (NoNeedToSerach and NeedToSearch). The list of
the features is as follows: Mean: Calculates the average amplitude of the EEG
signal within the specified frequency band over the 800ms block. It indicates
the central tendency of the signal, helping to characterise the overall
activity level. (Zhang et al., 2017). Standard Deviation: Measures the
variability or spread of the EEG signal within the frequency band (Zhang et
al., 2017). Skewness: Quantifies the asymmetry of the EEG signal’s
distribution within the frequency band (Kingphai and Moshfeghi, 2021).
Kurtosis: Measures the ”tailedness” of the EEG signal’s distribution within
the frequency band (Kingphai and Moshfeghi, 2021). Curve Length: Calculates
the cumulative Euclidean distance between consecutive data points in the EEG
signal within the frequency band (Zhang et al., 2017). Number of Peaks: Counts
the number of local maxima or peaks in the EEG signal within the frequency
band (Zhang et al., 2017). Average Non-Linear Energy: Quantifies the non-
linear dynamics of the EEG signal within the frequency band (Zhang et al.,
2017).

Experiment Conditions. As the overall goal of this study is to explore the
best methods for predicting IN from searchers RQ1, we found it key to explore
several experimental parameters outlined by our research questions within
Section 1.

Generalised & Personalised: To address the research question RQ2 during
training, two methods are devised. For the first approach, the samples
relating to IN and non-IN from every subject were combined into a single
dataset that would then be passed onto the model, this being the generalised
training strategy to asses how well our classifier can discern IN across all
subjects as EEG has been noted to be heavily subject dependant. The second
approach maintained the IN and non-IN EEG data at a subject level, allowing us
to assess the variability of subject performance for IN prediction.

Window Size: To address RQ3, we adjusted the size of question segments (words)
utilised by our classifier. This modification involved implementing an
expanding window, starting from the onset of the subject’s search decisions:
NoNeedToSearch and NeedToSearch. On average, each question comprised seven
segments, encompassing both words and responses. The minimum segment count was
4, while the maximum reached 16 segments. In this investigation, we explored
the expanding window with four distinct sizes: 2, 4, 8, and 16. These sizes
represented the range from the moment of question response to the full length
of the question, including the response, see Figure 2. The objective was to
ascertain the segments that the classifier favoured for distinguishing between
IN and non-IN instances, potentially revealing where the realisation of IN was
most pronounced during the question review process.

Feature Combination: In accordance with RQ4, one of the primary aims of this
study is to determine the optimal combinations of features commonly employed
in EEG classification for effectively predicting the realisation of IN. As
elaborated in Section 1, we identified and extracted seven key features for
this investigation. Generating an exhaustive list of all possible combinations
from these features resulted in 127 combinations. Each of these combinations
was then input into the classifier, with the model’s performance serving as
the metric to assess the effectiveness of the various feature combinations.

Figure 2. Expanding Window Size.

Predictive Models. For this study, we incorporated each of the aforementioned
experiment conditions into a training loop, where Generalised & Personalised
were separated where each would iterate through every possible Feature
Combination and Window Size. The classifiers selected for this task were the
Support Vector Machine (SVM) (Hearst et al., 1998), Random Forest Classifier
(Breiman, 2001), and AdaBoost (Freund et al., 1996) models, as they have seen
substantial success within the realm of EEG classification (Hu, 2017; Li et
al., 2013) and are well suited to the limited quantity of data available for
this task. Prior to this investigate several Deep and Recurrent Neural
Networks were trained on the collected EEG data, however, their performance
was sub-optimal as they were limited by the number of samples within the
dataset. Each dataset provided to the model through each of the possible
combinations of experiment conditions was cross-validated with a k-fold size =
5. Each fold returned the following metrics: Accuracy, Precision, and Recall,
where their average across each fold was calculated along with their standard
deviation. Baseline. Since there are no prior works to compare to, we
introduce a baseline that represents an untrained model where all its
predictions are based on a random choice, i.e. where its accuracy is set to
50%.

##  3\. Results & Conclusion

The results produced for the Generalised and Personalised conditions are
detailed in Table 1 and 2 respectively. Each of these tables denotes the
Model, the selected window size (W-Size), and the best-performing feature
combination at the given window size with its subsequent Accuracy, Precision,
and Recall scores. We also performed a paired Wilcoxon test between the
predictions obtained for each model to check the significance of the
difference with the baseline. All of the results obtained from our models
trained on a set of features were different from that of the baseline with a
confidence level of (p ¡ 0.01).

Table 1. This table shows the prediction accuracy of our Generalised approach.
The standard deviation is presented in parentheses. The best-performing model
is highlighted in bold.

Model | W-Size | Features | Accuracy (SD) | Precision (SD) | Recall (SD)
---|---|---|---|---|---
Baseline (Random) | - | - | 50% | 50% | 50%
RandomForest | 2 | Mean-SD-Curve | 73.5% (2.6%) | 73.6% (2.5%) | 73.5% (2.6%)
| 4 | Mean-SD-Curve | 71.9% (2.3%) | 72.0% (2.2%) | 71.9% (2.3%)
| 8 | Mean-SD-Curve-AvEn | 71.8% (0.9%) | 71.8% (0.9%) | 71.8% (0.9%)
| 16 | Mean-Curve-AvEn | 71.5% (1.1%) | 71.6% (1.1%) | 71.5% (1.1%)
SVM | 2 | Mean-AvEn | 69.7% (2.1%) | 69.8% (2.2%) | 69.7% (2.1%)
| 4 | Mean-AvEn | 69.1% (2.4%) | 69.2% (2.4%) | 69.1% (2.4%)
| 8 | Mean-Skew-AvEn | 69.0% (1.0%) | 69.0% (1.0%) | 69.0% (1.0%)
| 16 | Mean-Peaks | 69.0% (1.1%) | 69.1% (1.1%) | 69.0% (1.1%)
AdaBoost | 2 | Mean-SD | 70.6% (0.2%) | 70.7% (0.2%) | 70.6% (0.2%)
| 4 | Mean-Curve | 70.4% (0.2%) | 70.0% (0.1%) | 69.9% (0.2%)
| 8 | Mean-Curve | 70.3% (0.1%) | 70.5% (0.1%) | 70.3% (0.1%)
| 16 | Mean-Curve-Peaks | 70.0% (0.1%) | 70.2% (0.1%) | 70.0% (0.1%)

Table 2. This table shows the prediction accuracy of our Personalised
approach. The standard deviation is presented in parentheses. The best-
performing model is highlighted in bold.

Model | W-Size | Features | Accuracy (SD) | Precision (SD) | Recall (SD)
---|---|---|---|---|---
Baseline (Random) | - | - | 50% | 50% | 50%
RandomForest | 2 | Mean-SD-AvEn | 68.9% (19.7%) | 71.4% (20.3%) | 68.9% (19.7%)
| 4 | Mean | 73.0% (21.5%) | 75.7% (21.9%) | 73.0% (21.5%)
| 8 | Mean | 75.9% (21.5%) | 77.9% (21.5%) | 75.9% (21.5%)
| 16 | Mean | 76.6% (21.3%) | 77.5% (21.4%) | 76.6% (21.3%)
SVM | 2 | Mean-Curver-AvEn | 74.2% (21.5%) | 75.4% (21.3%) | 74.2% (21.5%)
| 4 | Mean-Kur-Curve | 74.3% (21.9%) | 76.5% (22.1%) | 74.3 (21.9%)
| 8 | Mean-AvEn | 73.5% (21.8%) | 75.7% (22.3%) | 73.5% (21.8%)
| 16 | Mean-Kur-Curve-AvEn | 71.7% (22.1%) | 74.6% (22.1%) | 71.7% (22.1%)
AdaBoost | 2 | Mean-Peaks | 73.9% (23.4%) | 74.5% (23.9%) | 73.9% (23.4%)
| 4 | Mean | 80.5% (22.3%) | 81.3% (22.4%) | 80.5% (22.3%)
| 8 | Mean-Peaks | 88.9% (22.5%) | 89.0% (22.2%) | 88.9% (22.5%)
| 16 | Mean | 90.1% (22.1%) | 90.3% (22.1%) | 90.1% (22.1%)

We first address RQ1 by reviewing the results produced in both the Generalised
and Personalised conditions. As we can see the prediction of the realisation
of IN is possible, as every model in Table 1 and Table 2 was able to achieve
an accuracy score above that of random classification (50%), with the lowest
reported accuracy score being the RandomForest classifier with an accuracy of
68.9% (SD 19.7%) and the highest being the AdaBoost model with a score of
90.1% (22.1%) as seen in Table 2. These results demonstrate that EEG data is
capable of achieving greater Generalised and Perosnalised accuracy performance
for the prediction of the realisation of IN than that of alternative
neuroimaging techniques such as fMRI (Moshfeghi et al., 2019).

By comparing the performance of the Genralised and Personalised models, it can
be observed that the Personalised approach achieves the highest overall
prediction accuracy, evidenced by the AdaBoost model that obtained 90.1% (SD
22.1%) in Table 2. When trained using the Personalised method, the
RandomForest, SVM, and AdaBoost model’s accuracy on average across window
sizes increases over its Generalised counterparts by 1.4%, 4.2%, and 13%
respectively. However, this increased accuracy also comes with an increased
Standard Deviation, with the Personalsied RandomForest, SVM, and AdaBoost
models on average across window sizes having a higher Standard Deviation of
19.3%, 21.8%, and 22.4% respectively than the Generalised models. These
findings help to address RQ2 as they suggest, on average, creating a model
tailored to each subject is the best approach for predicting the realisation
of IN as evidenced by the performance of the AdaBoost model in Table 2.
However, the variation in Standard deviation indicates that the Generalised
models offer a more robust and reliable prediction accuracy. This difference
follows the trend observed in prior works (Moshfeghi et al., 2019) and is
likely due to the natural variability in EEG data collected across subjects.
As such, for future systems aiming to predict the realisation of an IN from
EEG, the best approach may be to assess the model performance on individual
subjects and determine if the trade-off between accuracy and standard
deviation is acceptable or if a generalised model with a lower accuracy but
more reliable standard deviation is more suitable for their specific research
purposes.

Regarding RQ3, the results presented in Table 1 and Table 2 are in contrast to
each other. In the Generalised condition Table 1, we observe that all models
achieve their peak performance when the window size is set to 2, with the
RandomForest, SVM and AdaBoost models achieving an accuracy of 73.5%, 69.7%,
and 70.6% respectively. As the window size is increased from 2 up to 16, the
performance of the RandomForest, SVM, and AdaBoost models decreases by 2%,
0.7%, and 0.6% respectively. Conversely, in the Personalised results, Table 2
we observe that at the window size of 16, the RandomForest and AdaBoost models
achieve their highest performance of 76.6% and 90.1% respectively. As the
window size is increased from 2 up to 16 the RandomForest and AdaBoost models
accuracy increases by 7.7% and 16.2% respectively, except the SVM model, which
follows the same trend as the Generalised Models.

The results produced by the Generalised approach suggest that the distinctive
EEG patterns associated with the realisation of an IN may be more strongly
concentrated immediately after the subject concludes their review of the
question. This might be indicative of a universal or commonly shared cognitive
response that occurs promptly after the comprehension of a question,
highlighting a quick and standardised recognition process for INs across
subjects. In Contrast to this, the performance of the Personalsied models
indicates that for individual subjects, the discernible EEG patterns unfold
over a more extended period. This could be influenced by varying cognitive
styles, attention spans, or information processing speeds unique to each
subject. subjects might take more time to process and formulate their
information needs, leading to a prolonged period of activity associated with
information-seeking. Lastly, addressing RQ4, we can observe that the best-
performing feature is the Mean value taken from the EEG segments, as it
appeared in every single best-performing combination at each window size for
both generalised and personalised training Table 1 and 2 respectively.
However, a large subset of key features see substantial use across both
training conditions, for Generalised condition the following features are
listed in order of occurrence: with Curve Length appearing in 7 optimal
combinations, Average Energy in 5, Standard Deviation in 4, Number of Peaks in
2, and Skewness in 1. Similarly for Personalised: Average Energy appears in 4,
Curve Length in 3, Kurtosis in 2, and Standard Deviation appears in one. Our
results indicate that these features are strong performers in predicting the
realisation of IN.

In conclusion, the findings of this study demonstrate that through the use of
Electroencephalography (EEG) data, we were able to predict the realisation of
IN substantially above the random baseline classification accuracy of 50%,
with models achieving up to 90.1% accuracy. This work is the first to ever
demonstrate the prediction of the realisation of IN through the use of EEG
data, and at an accuracy higher than any other previously utilised
neuroimaging techniques, paving the way to real-time realisation of IN
prediction. Furthermore, the encouraging results obtained from the Generalised
and Personalised conditions will help to inform future research and
Information Retrieval (IR) systems that seek to incorporate the realisation of
IN prediction, by taking into consideration the inter and intra-variability of
EEG data cross subjects and examine the trade-off between a potentially more
accurate subject-specific models and a more reliable generalised model.
Moreover, we also highlighted optimal ranges within queries that should be
examined to provide the strongest indicators of the realisation of IN, as well
as the optimal combinations of features that should be considered for the
prediction of the realisation of IN within subjects.

###### Acknowledgements.

This work was supported by the Engineering and Physical Sciences Research
Council [grant number EP/W522260/1].

## References

  * (1)
  * Belkin et al. (1982) Nicholas J Belkin, Robert N Oddy, and Helen M Brooks. 1982.  ASK for information retrieval: Part I. Background and theory.  _Journal of documentation_ 38, 2 (1982), 61–71.
  * Bendersky and Croft (2009) Michael Bendersky and W Bruce Croft. 2009.  Analysis of long queries in a large scale search log. In _Proceedings of the 2009 workshop on Web Search Click Data_. 8–14.
  * Breiman (2001) Leo Breiman. 2001.  Random forests.  _Machine learning_ 45 (2001), 5–32.
  * Cooper (1971) William S Cooper. 1971.  A definition of relevance for information retrieval.  _Information storage and retrieval_ 7, 1 (1971), 19–37.
  * Daud and Sudirman (2015) SS Daud and R Sudirman. 2015.  Butterworth bandpass and stationary wavelet transform filter comparison for electroencephalography signal. In _2015 6th international conference on intelligent systems, modelling and simulation_. IEEE, 123–126.
  * Freund et al. (1996) Yoav Freund, Robert E Schapire, et al. 1996\.  Experiments with a new boosting algorithm. In _icml_ , Vol. 96. Citeseer, 148–156.
  * Hearst et al. (1998) M.A. Hearst, S.T. Dumais, E. Osuna, J. Platt, and B. Scholkopf. 1998.  Support vector machines.  _IEEE Intelligent Systems and their Applications_ 13, 4 (1998), 18–28.  https://doi.org/10.1109/5254.708428
  * Hu (2017) Jianfeng Hu. 2017.  Automated detection of driver fatigue based on AdaBoost classifier with EEG signals.  _Frontiers in computational neuroscience_ 11 (2017), 72.
  * Kingphai and Moshfeghi (2021) Kunjira Kingphai and Yashar Moshfeghi. 2021.  On EEG preprocessing role in deep learning effectiveness for mental workload classification. In _Human Mental Workload: Models and Applications: 5th International Symposium, H-WORKLOAD 2021, Virtual Event, November 24–26, 2021, Proceedings 5_. Springer, 81–98.
  * Kuhlthau (1991) Carol C Kuhlthau. 1991.  Inside the search process: Information seeking from the user’s perspective.  _Journal of the American society for information science_ 42, 5 (1991), 361–371.
  * Lepage et al. (2014) Kyle Q Lepage, Mark A Kramer, and Catherine J Chu. 2014.  A statistically robust EEG re-referencing procedure to mitigate reference effect.  _Journal of neuroscience methods_ 235 (2014), 101–116.
  * Li et al. (2013) Shufang Li, Weidong Zhou, Qi Yuan, Shujuan Geng, and Dongmei Cai. 2013.  Feature extraction and recognition of ictal EEG using EMD and SVM.  _Computers in biology and medicine_ 43, 7 (2013), 807–816.
  * Markey (1981) Karen Markey. 1981.  Levels of question formulation in negotiation of information need during the online presearch interview: A proposed model.  _Information Processing & Management_ 17, 5 (1981), 215–225.
  * Michalkova et al. (2022) Dominika Michalkova, Mario Parra-Rodriguez, and Yashar Moshfeghi. 2022.  Information Need Awareness: an EEG study. In _Proceedings of the 45th International ACM SIGIR Conference on Research and Development in Information Retrieval_. 610–621.
  * Moshfeghi (2021) Yashar Moshfeghi. 2021.  Neurasearch: Neuroscience and information retrieval. In _CEUR Workshop Proceedings_ , Vol. 2950. 193–194.
  * Moshfeghi and Pollick (2019) Yashar Moshfeghi and Frank E Pollick. 2019.  Neuropsychological model of the realization of information need.  _Journal of the Association for Information Science and Technology_ 70, 9 (2019), 954–967.
  * Moshfeghi et al. (2019) Yashar Moshfeghi, Peter Triantafillou, and Frank Pollick. 2019.  Towards predicting a realisation of an information need based on brain signals. In _The world wide web conference_. 1300–1309.
  * Moshfeghi et al. (2016) Yashar Moshfeghi, Peter Triantafillou, and Frank E Pollick. 2016.  Understanding information need: An fMRI study. In _Proceedings of the 39th International ACM SIGIR conference on Research and Development in Information Retrieval_. 335–344.
  * Taylor (1968) Robert S Taylor. 1968.  Question-negotiation and information seeking in libraries.  _College & research libraries_ 29, 3 (1968), 178–194.
  * Teplan et al. (2002) Michal Teplan et al. 2002\.  Fundamentals of EEG measurement.  _Measurement science review_ 2, 2 (2002), 1–11.
  * Wilson (1981) Tom D Wilson. 1981.  On user studies and information needs.  _Journal of documentation_ 37, 1 (1981), 3–15.
  * Zhang et al. (2017) Yong Zhang, Bo Liu, Xiaomin Ji, and Dan Huang. 2017.  Classification of EEG signals based on autoregressive model and wavelet packet decomposition.  _Neural Processing Letters_ 45 (2017), 365–378.


Truncated Solutions of Painlevé Equation ${\rm P}_{\rm V}$

Truncated Solutions of Painlevé Equation $\boldsymbol{{\rm P}_{\rm V}}$††This
paper is a contribution to the Special Issue on Painlevé Equations and
Applications in Memory of Andrei Kapaev. The full collection is available at
https://www.emis.de/journals/SIGMA/Kapaev.html

Rodica D. COSTIN

R.D. Costin

The Ohio State University, 231 W 18th Ave, Columbus, OH 43210, USA
<EMAIL_ADDRESS>https://people.math.osu.edu/costin.10/

Received May 01, 2018, in final form October 25, 2018; Published online
October 31, 2018

We obtain convergent representations (as Borel summed transseries) for the
five one-parameter families of truncated solutions of the fifth Painlevé
equation with nonzero parameters, valid in half planes, for large independent
variable. We also find the position of the first array of poles, bordering the
region of analyticity. For a special value of this parameter they represent
tri-truncated solutions, analytic in almost the full complex plane, for large
independent variable. A brief historical note, and references on truncated
solutions of the other Painlevé equations are also included.

Painlevé trascendents; the fifth Painlevé equation; truncated solutions; poles
of truncated solutions

33E17; 34M30; 34M25

##  1 Introduction

###  1.1 Historical notes

By the middle of the 19th century it became apparent that solutions of many
linear differential equations should be considered new, “special” functions.
The natural question then arose: can nonlinear differential equations have
solutions that could be thought of being special functions? Fuch’s intuition
was that the answer is affirmative if “solutions have only fixed branch points
none of which depend on the initial conditions” [15]. This is now called the
Painlevé property, the fact that solutions of an equation are meromorphic on a
common Riemann surface. Fuchs studied first order equations having this
property, concluding in 1884 that all such equations can be solved in terms of
previously known functions; his results were later extended by Poincaré [15].

The idea that absence of movable branch points would mean integrability was
then used by Sofie Kowalevski in the study of the rotation of a solid about a
fixed point, for which she discovered a third integrable case [29] (previous
two cases being discovered by Euler and Lagrange), a discovery for which she
was awarded the Prix Bordin of the French Academy of Science in 1888. Around
the turn of the 20th century, Painlevé, Picard and Gambier studied nonlinear
second order differential equations, rational in $y$ and $y^{\prime}$,
discovering that those possessing the Painlevé property can be brought to
fifty canonical forms. Of these fifty equations, all but six could be solved
in terms of earlier known functions. Painlevé went on to show that generic
solutions of the remaining six equations cannot be expressed in terms of
earlier known functions, or in terms of each other [32]. These equations are
now known as the Painlevé equations, denoted ${\rm P}_{\rm I}$ up to ${\rm
P}_{\rm VI}$, and their solutions as the Painlevé transcendents.

Having been discovered as a result of a purely theoretical inquiry, the
Painlevé transcendents have appeared later in modern geometry, integrable
systems [1], statistical mechanics [21, 36, 37], and recently in quantum field
theory. Their practical importance makes it necessary to study their
properties in detail and to develop good numerical methods for their
calculation [4].

###  1.2 Truncated solutions of Painlevé equations

It is known that generic Painlevé transcendents have poles in any sector
towards infinity. But there are special solutions, called tronquée (or
truncated) which are free of poles in some sectors, at least for large values
of the independent variable. It was later conjectured that truncated solutions
have no poles whatsoever in these sectors, the Dubrovin–Novokshenov conjecture
[12, 31].

For ${\rm P}_{\rm I}$, Boutroux showed that there are five special sectors in
the complex plane, each of opening $2\pi/5$, where solutions may lack poles.
There are truncated solutions free of poles in two adjacent such sectors, for
large $x$. Among these, there are solutions free of poles in four sectors –
tri-truncated solutions [3].

In each sector, there is a one parameter family of truncated solutions, all
asymptotic to the same power series; these solutions can be distinguished by a
parameter multiplying a small exponential; a complete formal solution can be
obtained as a transseries, mixing negative powers of the variable and
exponentially small terms. The form of the exponentially small terms as well
as the location of the first array of poles of truncated solutions were found
in [9]. The Dubrovin–Novokshenov conjecture was proved for ${\rm P}_{\rm I}$
[11].

The Stokes constant (which can be viewed as a complex number, attached to the
equation, controlling the relation between the exponentially small terms
associated to different sectors) was calculated for the tritronquée solution
for ${\rm P}_{\rm I}$ for the first time by Kapaev [24], see also [28], using
the isomonodromy method. Kapaev then obtained the complete description of the
global asymptotic behavior of the tronquée solutions of ${\rm P}_{\rm I}$ and
connection formulae [26]. The Stokes constant was re-calculated later using
WKB methods [35], and again later by continuation of a tri-truncated solution
through the sector with poles [10].

More recently, Kapaev considered the study of the tronquée solution of the
${\rm P}_{\rm I2}$ equation (the second member of the ${\rm P}_{\rm I}$
hierarchy); a detailed global asymptotic analysis of the trintronqués
solutions is found in [14].

For ${\rm P}_{\rm II}$ Boutroux showed that there are six special sectors,
with truncated solutions free of poles in two adjacent sectors, and tri-
truncated solutions free of poles in four sectors. Using the Riemann–Hilbert
approach, Kapaev gave a complete description of the global asymptotic behavior
of the tronquée solutions of ${\rm P}_{\rm II}$, together with all relevant
connection formulae [20, 27], see also Chapter 11 of the monograph [13].
Existence of tri-truncated solutions of the ${\rm P}_{\rm II}$ hierarchy was
shown in [23]. The Dubrovin–Novokshenov conjecture was established for the
Hastings–McLeod solution (tri-truncated in pairs of non-adjacent sectors)
[18].

For ${\rm P}_{\rm IV}$, a quite detailed global asymptotic analysis of its
tronquée solutions was obtained in Kapaev’s work [25]. The connection formulae
were found by Its and Kapaev [19], based on the Riemann–Hilbert isomonodromy
method. Existence of truncated solution fo ${\rm P}_{\rm III}$ and ${\rm
P}_{\rm IV}$ was re-established in [30] following methods in [22], and using a
different method in [38], where the location of the first array of poles was
also found. An overview of ${\rm P}_{\rm VI}$ is contained in [17]; see also
[8]. The truncated solutions of the fifth Painlevé equation are the subject of
the present article. It is to be noted that fixed singularities of ${\rm
P}_{\rm V}$ can only be located at $0$ and $\infty$.

###  1.3 Truncated solutions of Painlevé equation $\boldsymbol{{\rm P}_{\rm
V}}$

The fifth Painlevé equation

$\displaystyle{\rm P}_{\rm V}(\alpha,\beta,\gamma,\delta)\colon$
$\displaystyle
w^{\prime\prime}=\left(\frac{1}{2w}+\frac{1}{w-1}\right){w^{\prime}}^{2}-{\frac{w^{\prime}}{x}}+{\frac{(w-1)^{2}}{{x}^{2}}\left(\alpha
w+{\frac{\beta}{w}}\right)}+{\frac{\gamma w}{x}}+{\frac{\delta w(w+1)}{w-1}}$
(1.1)

is known to be reducible to ${\rm P}_{\rm III}$ if $\delta=0$ [16].

We assume $\alpha\beta\delta\neq 0$. The following algebraic behaviors towards
infinity are then possible for solutions of (1.1) [33]:

$\displaystyle{\rm I.}\quad$ $\displaystyle
w=\pm\sqrt{\frac{\beta}{\delta}}x^{-1}+O\big{(}x^{-2}\big{)},\qquad$
$\displaystyle x\to\infty,$ $\displaystyle{\rm II.}\quad$ $\displaystyle
w=\pm\sqrt{\frac{\delta}{-\alpha}}x+O(1),\qquad$ $\displaystyle x\to\infty,$
$\displaystyle{\rm III.}\quad$ $\displaystyle
w=-1+O\big{(}x^{-1}\big{)},\qquad$ $\displaystyle x\to\infty.$

It is known that these five families represent asymptotic behaviors of
truncated solutions, analytic in (almost) a half plane for large $|x|$; the
position of the half plane is determined by the exponentially small terms [2,
34].

In the present paper we express the exponentially small terms using a full
formal solution (transseries), Borel summable to actual solutions. This yields
one-parameter families of truncated solutions as series which converge in
appropriate (almost) half-planes and large $|x|$. For special values of the
parameter we obtain tri-truncated transcendents. Moreover, we find the
location of the first array of poles beyond the sector of analyticity.

We use techniques used before in [9]. For completeness, the statements of the
theorems used here are included in Appendix A.

###  1.4 Relations between different truncated solutions

######  Remark 1.1 ([16]).

  1. (i)

If $w(x)$ satisfies ${\rm P}_{\rm V}(\alpha,\beta,\gamma,\delta)$, then
$1/w(x)$ satisfies ${\rm P}_{\rm V}(-\beta,-\alpha,-\gamma,\delta)$.

  2. (ii)

If $w(x)$ solves ${\rm P}_{\rm V}(\alpha,\beta,\gamma,\delta)$ then
$w(x/\lambda)$ solves ${\rm P}_{\rm
V}\big{(}\alpha,\beta,\gamma\lambda,\delta\lambda^{2}\big{)}$, for any
$\lambda\neq 0$.

By Remark 1.1(i), truncated solutions in the family II. are obtained as
reciprocals of truncated of the family I.

By Remark 1.1(ii), if $w(x)$ solves ${\rm P}_{\rm
V}(\alpha,\beta,\gamma,\delta)$ with
$w(x)\sim\sqrt{\frac{\beta}{\delta}}x^{-1}$, $x\to\infty$, then $w(-x)$ solves
${\rm P}_{\rm V}(\alpha,\beta,-\gamma,\delta)$ and satisfies
$w(-x)\sim-\sqrt{\frac{\beta}{\delta}}x^{-1}$, $x\to\infty$.

We can assume any nonzero value for $\delta$, by rescaling $x$ and using
Remark 1.1(ii).

It is interesting to note that there are other Bäcklund transformations as
explained in [16, Theorem 39.2]. These transform truncated solutions of the
first two families into truncated solutions of the first two families, and
truncated solutions in the family III into solutions in the same family.

Due to these relations, it suffices to obtain results for the truncated
Painlevé transcendents satisfying

$\displaystyle{\rm I_{0}}\quad
w=\sqrt{{\beta}/{\delta}}x^{-1}+O\big{(}x^{-2}\big{)},\qquad
x\to\infty,\qquad\text{for}\quad\delta=-1/2$

and for

$\displaystyle{\rm III_{0}}\quad w=-1+O\big{(}x^{-1}\big{)},\qquad
x\to\infty\qquad\text{for}\quad\delta=2.$

##  2 Truncated solutions in the family $\boldsymbol{{\rm I}_{0}}$

In this section we state the main results; their proofs are found in Section
4.

######  Theorem 2.1.

Assume $\alpha\beta\gamma\neq 0$. Let $w(x)$ be a solution of ${\rm P}_{\rm
V}\big{(}\alpha,\beta,\gamma,-\tfrac{1}{2}\big{)}$ such that there exists some
$\phi\in\big{(}{-}\tfrac{\pi}{2},\tfrac{\pi}{2}\big{)}$ so that

$\displaystyle w(x)=\sqrt{-2\beta}x^{-1}(1+o(1))\qquad\text{as}\quad
x\to\infty\quad\text{along}\quad\arg x=-\phi.$ (2.1)

Then (2.1) holds along any
$\phi\in\big{(}{-}\tfrac{\pi}{2},\tfrac{\pi}{2}\big{)}$.

  1. $(i)$

Furthermore, $w(x)$ is asymptotic to a unique power series solution:

$\displaystyle
w(x)\sim\tilde{w}_{0}(x)=\sum_{n=1}^{\infty}{w}_{0;n}x^{-n}\qquad\text{as}\quad
x\to\infty\quad\text{along any}\quad\arg x=-\phi,$

where $|\phi|<\tfrac{\pi}{2}$, ${w}_{0;1}=\sqrt{-2\beta}$.

  2. $(ii)$

The complete formal solution $($transseries$)$ along any half-line $\arg
x=-\phi$ with $0<|\phi|<\tfrac{\pi}{2}$ is

$\displaystyle\tilde{w}(x)=\tilde{w}_{0}(x)+\sum_{k=1}^{\infty}C^{k}{\rm
e}^{-kx}x^{-qk}\tilde{w}_{k}(x)\qquad\text{where}\quad\tilde{w}_{k}(x)=\sum_{n=0}^{\infty}w_{n;k}x^{-n},$
(2.2)

$C$ is an arbitrary constant and

$\displaystyle q=\gamma+2\sqrt{-2\beta}.$

  3. $(iii)$

$w(x)$ has a Borel summed transseries representation: there exist $C_{\pm}$,
where $C_{+}-C_{-}$ is a multiple of the Stokes constant, such that for any
$\epsilon>0$ there is $R_{\epsilon}>0$

$\displaystyle w(x)=\begin{cases}\displaystyle
w_{0}(x)+\sum_{k=1}^{\infty}C_{+}^{k}{\rm
e}^{-kx}x^{-qk}w_{+;k}(x)&\text{for}\ \arg
x\in\big{(}0,\frac{\pi}{2}-\epsilon\big{)},\ |x|>R_{\epsilon},\\\
\displaystyle w_{0}(x)+\sum_{k=1}^{\infty}C_{-}^{k}{\rm
e}^{-kx}x^{-qk}w_{-;k}(x)&\text{for}\ \arg
x\in\big{(}{-}\frac{\pi}{2}+\epsilon,0\big{)},\
|x|>R_{\epsilon},\end{cases}\\!\\!\\!\\!$ (2.3)

where $w_{0}(x)$, $w_{\pm;k}(x)$ are the Borel sums of $\tilde{w}_{0}(x)$,
$\tilde{w}_{k}(x)$ along half-lines ${\rm e}^{{\rm i}\phi}\mathbb{R}_{+}$
$($where $\phi=-\arg x)$. For $\arg x=0$ the transseries is Écalle–Borel
summable.111It is obtained by special averages of Laplace transforms in the
upper and the lower half plane [5]. The function series (2.3) converge for $x$
satisfying $|x|>R$ and $|C_{\pm}{\rm e}^{-x}x^{-q}|<\mu$ for suitable $\mu>0$
and $R>0$. In particular, if $\operatorname{Re}q>0$ then $w(x)$ is analytic in
the right half plane for $|x|>R$ $($for some $R>0)$.

  4. $(iv)$

If $\operatorname{Re}q>0$ the constants $C_{\pm}$ can be determined from

$\displaystyle
w(x)\sim\sum_{n=1}^{\lfloor\operatorname{Re}q\rfloor}{w}_{0;n}x^{-n}+C_{+}x^{-q}{\rm
e}^{-x}\qquad\text{as}\quad x\to+{\rm i}\infty$

and

$\displaystyle
w(x)\sim\sum_{n=1}^{\lfloor\operatorname{Re}q\rfloor}{w}_{0;n}x^{-n}+C_{-}x^{-q}{\rm
e}^{-x}\qquad\text{as}\quad x\to-{\rm i}\infty.$

######  Corollary 2.2 (existence of tri-truncated solutions).

Consider the unique truncated solution $w(x)$ as in Theorem 2.1 with
$C_{+}=0$. Then $w(x)$ is analytic for large $|x|$ in the left-half plane, for
$\arg x\in\big{(}{-}\tfrac{\pi}{2}+\epsilon,\tfrac{3\pi}{2}-\epsilon\big{)}$,
$|x|>R_{\epsilon}$ $($for any $\epsilon>0)$.

Similarly, the unique truncated solution with $C_{-}=0$ is analytic for
$|x|>R_{\epsilon}$ with $\arg
x\in\big{(}{-}\tfrac{3\pi}{2}+\epsilon,\tfrac{\pi}{2}-\epsilon\big{)}$. If
$\operatorname{Re}q>0$ then we can take $\epsilon=0$.

######  Remark 2.3.

Truncated solutions in the left half-plane, satisfying

$\displaystyle w(x)=-\sqrt{-2\beta}x^{-1}(1+o(1))\qquad\big{(}x\to\infty\text{
along }\arg x=-\phi,\ |\phi+\pi|<\tfrac{\pi}{2}\big{)}$

form a one-parameter family, which, by Remark 1.1 are obtained by replacing
$(x,\gamma)$ with $(-x,-\gamma)$ in the representation given by Theorem 2.1.

Truncated solutions $w(x)$ as in Theorem 2.1 with $C_{+}\neq 0$ develop arrays
of poles. The following result shows the position of the array of poles
closest to ${\rm i}\mathbb{R}^{+}$.

######  Theorem 2.4.

Assume $\alpha\beta\gamma\neq 0$ and

$\displaystyle 2\alpha\neq\big{(}\sqrt{-2\beta}-q-1\big{)}^{2}.$

Let $w(x)$ be as in Theorem 2.1 with nonzero constant $C_{+}$ in (2.3). Them
$w(x)$ has two arrays of poles located at

$\displaystyle x_{n;1,2}=2n\pi{\rm
i}+\big{(}\gamma+2\sqrt{-2\beta}+2\big{)}\ln(2n\pi{\rm i})+\ln
C_{+}-\ln\zeta_{1,2}+o(1),\qquad n\to+\infty,$

where

$\displaystyle\zeta_{1,2}=\frac{2}{\sqrt{-2\beta}}\frac{1}{\sqrt{-2\beta}-q-1\pm\sqrt{2\alpha}}.$
(2.4)

##  3 Truncated solutions in the family $\boldsymbol{{\rm III}_{0}}$

In this section we state the main results; their proofs are found in Section
5.

######  Theorem 3.1.

Assume $\alpha\beta\gamma\neq 0$. Let $w(x)$ be a solution of ${\rm P}_{\rm
V}(\alpha,\beta,\gamma,2)$ such that there exists some
$\phi\in\big{(}{-}\tfrac{\pi}{2},\tfrac{\pi}{2}\big{)}$ so that

$\displaystyle w(x)=-1+o(1)\qquad\text{as}\quad
x\to\infty\quad\text{along}\quad\arg x=-\phi.$ (3.1)

Then (3.1) holds along any
$\phi\in\big{(}{-}\tfrac{\pi}{2},\tfrac{\pi}{2}\big{)}$.

  1. $(i)$

Furthermore, $w(x)$ is asymptotic to a unique power series solution:

$\displaystyle
w(x)\sim-1+\tilde{w}_{0}(x)=-1+\sum_{n=1}^{\infty}{w}_{0;n}x^{-n}\qquad
x\to\infty\quad\text{along any}\quad\arg x=-\phi,$

where $|\phi|<\tfrac{\pi}{2}$.

  2. $(ii)$

The complete formal series solution $($transseries$)$ along any half-line in
the right half-plane has the form

$\displaystyle\tilde{w}(x)=\tilde{w}_{0}(x)+\sum_{k=1}^{\infty}C^{k}{\rm
e}^{-kx}x^{-k/2}\tilde{w}_{k}(x),$

where $C$ is an arbitrary constant, $\tilde{w}_{k}(x)$ are series in $x^{-n}$,
$n\in\mathbb{N}$.

  3. $(iii)$

There are unique constants $C_{\pm}$ so that

$\displaystyle w(x)\sim-1+C_{+}x^{-1/2}{\rm e}^{-x}\qquad\text{as}\quad
x\to+{\rm i}\infty$ (3.2)

and

$\displaystyle w(x)\sim-1+C_{-}x^{-1/2}{\rm e}^{-x}\qquad\text{as}\quad
x\to-{\rm i}\infty$

and $C_{+}-C_{-}$ is a constant which does not depend on the particular
solution $($and it is a multiple of the Stokes constant of the equation$)$.

  4. $(iv)$

$w(x)$ is analytic in the right half plane for $|x|>R$ $($for some $R>0)$ and
has the Borel summed transseries representation

$\displaystyle w(x)=\begin{cases}\displaystyle
w_{0}(x)+\sum_{k=1}^{\infty}C_{+}^{k}{\rm
e}^{-kx}x^{-k/2}w_{+;k}(x)&\text{for}\ \arg
x\in\big{(}0,\frac{\pi}{2}\big{]},\\\ \displaystyle
w_{0}(x)+\sum_{k=1}^{\infty}C_{-}^{k}{\rm
e}^{-kx}x^{-k/2}w_{-;k}(x)&\text{for}\ \arg
x\in\big{[}{-}\frac{\pi}{2},0\big{)},\end{cases}$ (3.3)

where $w_{0}(x)$, $w_{\pm;k}(x)$ are Laplace transforms on the half-lines
${\rm e}^{{\rm i}\phi}\mathbb{R}_{+}$ $($where $\phi=-\arg x$ is between $0$
and $\pm\tfrac{\pi}{2})$ of the Borel transforms of the series
$\tilde{w}_{0}(x)$, $\tilde{w}_{k}(x)$. The domain of convergence of the
series in (3.3) is the set of all $x$ with $|x|>R$ and $\big{|}C_{\pm}{\rm
e}^{-x}x^{-1/2}\big{|}<\mu$ for some suitable $R,\mu>0$.

######  Corollary 3.2 (existence of tri-truncated solutions).

Consider the unique truncated solution $w(x)$ as in Theorem 3.1 with
$C_{+}=0$. Then $w(x)$ is analytic for large $|x|$ also in the left-half
plane, for $\arg
x\in\big{(}{-}\tfrac{\pi}{2}+\epsilon,\tfrac{3\pi}{2}-\epsilon\big{)}$.

Similarly, the unique truncated solution with $C_{-}=0$ is analytic for large
$|x|$ with $\arg
x\in\big{(}{-}\tfrac{3\pi}{2}+\epsilon,\tfrac{\pi}{2}-\epsilon\big{)}$.

######  Remark 3.3.

Truncated solutions in the left half-plane, satisfying

$\displaystyle w(x)=-1+o(1)\qquad\big{(}x\to\infty\text{ along }\arg
x=-\phi\text{ with }0<|\phi+\pi|<\tfrac{\pi}{2}\big{)}$

form a one-parameter family, which, by Remark 1.1 are obtained by replacing
$(x,\gamma)$ with $(-x,-\gamma)$ in the representation given by Theorem 3.1.

Truncated solutions $w(x)$ as in Theorem 3.1 with $C_{+}\neq 0$ have arrays of
poles near ${\rm i}\mathbb{R}_{+}$. The following result shows the position of
the closest array of poles.

######  Theorem 3.4.

Assume $\alpha\beta\gamma\neq 0$. Let $w(x)$ be as in Theorem 3.1 with nonzero
constant $C_{+}$ in (3.2) and (3.3). Them $w(x)$ has an array of poles located
at

$\displaystyle x_{n}=2n\pi{\rm i}-\frac{1}{2}\ln(2n\pi{\rm i})-\frac{{\rm
i}\pi}{2}-\ln(-C_{+}/2)+o(1),\qquad n\to\infty.$

We conjecture that there are, in fact, two poles near each $x_{n}$.

##  4 Proofs for family $\boldsymbol{{\rm I}_{0}}$

###  4.1 Proof of Theorem 2.1

Denote

$\displaystyle m=\sqrt{-2\beta},\qquad q=\gamma+2m.$

Then $w(x)$ satisfies

$\displaystyle w^{\prime\prime}={\frac{(3w-1)w^{\prime
2}}{2w(w-1)}}-{\frac{w^{\prime}}{x}}+{\frac{\alpha
w(w-1)^{2}}{{x}^{2}}}-\frac{{m}^{2}}{2}{\frac{(w-1)^{2}}{{x}^{2}w}}+{\frac{(q-2m)w}{x}}-\frac{1}{2}{\frac{w(w+1)}{w-1}}$
(4.1)

for which we study solutions satisfying $w=mx^{-1}+O\big{(}x^{-2}\big{)}$ as
$x\to\infty$.

To normalize the equation, substitute

$\displaystyle w(x)=\frac{m}{x}\left(1-\frac{q}{x}+u(x)\right),$ (4.2)

which transforms (4.1) into

$\displaystyle
u^{\prime\prime}=\left(1+\frac{2q}{x}\right)u+f\big{(}x^{-1},u,u^{\prime}\big{)},$
(4.3)

where $f$ is analytic at $(0,0,0)$ and
$f=O\big{(}x^{-2}\big{)}+O\big{(}u^{2}\big{)}+O\big{(}u^{\prime
2}\big{)}+O(uu^{\prime})$. To show that this a normal form for second order
equations, we turn it into a first order system by substituting

$\displaystyle u(x)=y_{1}(x)+y_{2}(x),\qquad
u^{\prime}(x)=\left(-1-{\frac{q}{x}}\right)y_{1}(x)+\left(1+{\frac{q}{x}}\right)y_{2}(x),$
(4.4)

upon which (4.3) is turned into a first order system in normal form (A.1):

$\displaystyle\frac{{\rm d}}{{\rm d}x}\left[\begin{matrix}y_{1}\\\
y_{2}\end{matrix}\right]=\left[\begin{matrix}-1-{\dfrac{q}{x}}&0\\\
0&1+{\dfrac{q}{x}}\end{matrix}\right]\left[\begin{matrix}y_{1}\\\
y_{2}\end{matrix}\right]+g\big{(}x^{-1},y_{1},y_{2}\big{)}\left[\begin{matrix}1\\\
-1\end{matrix}\right],$ (4.5)

where

$\displaystyle
g\big{(}x^{-1},y_{1},y_{2}\big{)}=\frac{1}{2}\frac{x}{x+q}\left[-f\left(x^{-1},y_{1}+y_{2},\left(1+\frac{q}{x}\right)(y_{2}-y_{1})\right)\right.$
$\displaystyle\left.\hphantom{g\big{(}x^{-1},y_{1},y_{2}\big{)}=}{}+\frac{q(q+1)}{x^{2}}y_{1}+\frac{q(q-1)}{x^{2}}y_{2}\right].$
(4.6)

There are general theorems that can be applied for differential equations in
normal form; these are presented, for convenience, in the Appendix A: applying
Theorem A.3 to the system (4.5), (4.6) and then reverting the substitutions
(4.4), (4.2), Theorem 2.1 follows.

###  4.2 Proof of Theorem 2.4

Searching for asymptotic expansions of the form (A.8) setting $\xi=C{\rm
e}^{-x}x^{-q}$, plugging in an asymptotic series $u(x)\sim
F_{0}(\xi)+\tfrac{1}{x}F_{1}(\xi)+\tfrac{1}{x^{2}}F_{2}(\xi)+\cdots$ in (4.3)
and expanding under the assumption that $\xi\gg x^{-k}$ for all $k$ we obtain
that all $F_{n}$ are polynomials:

$\displaystyle F_{0}(\xi)=\xi,\qquad F_{1}(\xi)=c_{1}\xi,$ $\displaystyle
F_{2}(\xi)=m(m-q-1){\xi}^{2}+c_{2}\xi-\tfrac{1}{2}{m}^{2}+\tfrac{3}{2}{q}^{2}-a+\tfrac{1}{2},\qquad\ldots,$

where $c_{1}$, $c_{2}$ are uniquely determined in terms of the parameters. By
Theorem A.4 we have

$\displaystyle
u(x)\sim\sum_{m=0}^{\infty}x^{-m}{F}_{m}(\xi(x))\qquad\text{for}\ x\to\infty\
\text{with}\ \arg
x\in\big{[}{-}\tfrac{\pi}{2}+\delta,\tfrac{\pi}{2}-\delta\big{]},\
|\xi(x)|<\delta_{1}$ (4.7)

for some $\delta_{1}>0$.

On the other hand, $F_{0}$ has no singularities, so Theorem A.6 yields no
additional information on the position of the first array of poles, beyond the
right half plane.

A direct calculation of the first few terms of the transseries (2.2) unveils a
non-generic structure, namely that $w_{n,k}=0$ for all $k=0,1,\ldots,n-1$ (at
least for $n=1,\dots,4$). This peculiar structure suggests (by formal re-
arrangement of the transseries) to use instead the second scale

$\displaystyle\zeta=C{\rm e}^{-x}x^{-q-2}$

and look for an expansion of the form

$\displaystyle u(x)\sim
x^{2}\Phi(\zeta)+\sum_{n=-1}^{\infty}\frac{1}{x^{n}}\Phi_{n}(\zeta),\qquad\zeta(x)\gg
x^{-k}\ \text{for all}\ k,$ (4.8)

where $\Phi(\zeta)=\zeta+O\big{(}\zeta^{2}\big{)}$ when $\zeta\to 0$ and
$\Phi_{n}$ are analytic at $\zeta=0$, and for $x$ so that
$|\zeta|=\big{|}C{\rm e}^{-x}x^{-q-2}\big{|}<\mu$, where $\mu$ is small enough
(to be determined).

Introducing the formal expansion (4.8) in (4.3) and expanding in powers of
$x^{-1}$, we obtain that $\Phi(\zeta)$ must satisfy

$\displaystyle{\zeta}^{2}\Phi^{\prime\prime}+\zeta\Phi^{\prime}+\frac{1}{2}\Phi=\alpha{m}^{2}\Phi^{3}+\frac{3}{2}\frac{\zeta^{2}\Phi^{\prime
2}}{\Phi}$

having the general solution

$\displaystyle\Phi(\zeta)=\frac{2\zeta/C_{1}}{8\alpha m^{2}/C_{1}^{2}-(\zeta-
C_{2})^{2}},$ (4.9)

where $C_{1,2}$ are determined from the condition that
$\Phi(\zeta)=\zeta+O\big{(}\zeta^{2}\big{)}$ and that $\Phi_{-1}(\zeta)$ be
analytic at $\zeta=0$, yielding

$\displaystyle C_{1}=-2\beta\big{[}2\alpha-(m-q-1)^{2}\big{]},\qquad
C_{2}=-\frac{2}{m}\frac{m-q-1}{2\alpha-(m-q-1)^{2}}.$ (4.10)

With the notation (2.4) (recall that $m=\sqrt{-2\beta}$) and using (4.9), then
(4.10) becomes

$\displaystyle\Phi(\zeta)=\frac{\zeta\zeta_{1}\zeta_{2}}{(\zeta-\zeta_{1})(\zeta-\zeta_{2})}.$

Letting $\mu<\min|\zeta_{1,2}|$, the next term in the expansion is

$\displaystyle\Phi_{-1}(\zeta)={\frac{\zeta(2\zeta-\zeta_{{1}}-\zeta_{{2}})C_{{3}}}{(\zeta-\zeta_{{2}})^{2}(\zeta-\zeta_{{1}})^{2}}}+{\frac{\zeta\big{(}{\zeta}^{2}-\zeta_{{1}}\zeta_{{2}}\big{)}C_{{4}}}{(\zeta-\zeta_{{2}})^{2}(\zeta-\zeta_{{1}})^{2}}}$
(4.11)
$\displaystyle\hphantom{\Phi_{-1}(\zeta)=}{}-\frac{1}{2a}{\frac{\zeta\big{(}\zeta_{{1}}\zeta_{{2}}\sqrt{2}(\zeta_{{1}}+\zeta_{{2}}){a}^{3/2}+(\zeta_{{1}}-\zeta_{{2}})\big{(}{-}\frac{1}{\sqrt{2}}\left(\zeta_{{1}}-\zeta_{{2}}\right)\sqrt{a}+a\zeta_{{2}}\zeta_{{1}}\big{)}\big{)}(\zeta_{{1}}-\zeta_{{2}})}{(\zeta-\zeta_{{2}})^{2}(\zeta-\zeta_{{1}})^{2}}},$

where $C_{3,4}$ are determined from the condition that the next term,
$\Phi_{0}$, be analytic at $\zeta=0$.

To justify that the actual truncated solution $w(x)$ is also singular near
$\zeta_{1}$ and near $\zeta_{2}$, we first note that, since $\zeta=\xi x^{-2}$
and $u(x)\sim x^{2}\Phi$ then

$\displaystyle
x^{2}\Phi(\zeta)=\frac{\xi}{\big{(}1-x^{-2}\xi\zeta_{1}^{-1}\big{)}\big{(}1-x^{-2}\xi\zeta_{2}^{-1}\big{)}}=\xi+\frac{1}{x^{2}}\xi\big{(}\zeta_{1}^{-1}+\zeta_{2}^{-1}\big{)}+\cdots,$

which is a convergent series in powers of $x^{-2}$. Also $x\Phi_{-1}(\zeta)$
has a similar convergent expansion.

Therefore $x^{2}\Phi(\zeta)=F_{0}(\xi)+O\big{(}x^{-2}\big{)}$ and, in view of
(4.11), $x\Phi_{-1}(\zeta)=O\big{(}x^{-1}\big{)}$ therefore (4.7) implies that

$\displaystyle u(x)\sim x^{2}\Phi(\zeta)+O\big{(}x^{-1}\big{)}$

for $x\to\infty$ in the same region where (4.7) holds (and $|x|>1$,
$|\zeta|<\mu$). The same argument as in [7, Section 4.6] implies that $u(x)$
has singularities within $o(1)$ distance of the singularities of
$\Phi(\zeta)$.

##  5 Proofs for the family $\boldsymbol{{\rm III}_{0}}$

The existence of a unique power series formal solution
$\tilde{w_{0}}(x)=-1+o(1)$ is established by standard techniques.

It is also relatively algorithmic to obtain the form of the transseries. Since
the procedure may not be well known some details are provided here, also
illustrating why $\delta=2$ is a natural choice.

Plugging in a formal solution of the type
$\tilde{w}(x)=\tilde{w_{0}}(x)+\epsilon g(x)+O\big{(}\epsilon^{2}\big{)}$ in
the equation (1.1) and expanding in power series in $\epsilon$, the
coefficient of $\epsilon^{0}$ vanishes, since $\tilde{w_{0}}(x)$ is already a
formal solution. Next, the coefficient of $\epsilon$ is a linear second order
differential equation for $g(x)$, with coefficients given in terms of
$\tilde{w_{0}}(x)$ and its derivatives. It has the form

$\displaystyle
g^{\prime\prime}(x)+\frac{1}{x}g^{\prime}(x)-\frac{\delta}{2}g(x)=\frac{1}{x^{2}}R\big{(}x^{-1},g,g^{\prime}\big{)}$

with two independent solutions

$\displaystyle g_{\pm}(x)=x^{-1/2}\exp\big{(}\pm
x\sqrt{\delta/2}\big{)}\big{(}1+O\big{(}x^{-1}\big{)}\big{)},$

where we see that it is convenient to take $\delta=2$.

Let $w(x)$ satisfy the assumptions of Theorem 3.1. To normalize the equation,
substitute

$\displaystyle w(x)=-1+\frac{\gamma}{x}+u(x),$

so that $u(x)=O\big{(}x^{-2}\big{)}$ and satisfies an equation of the form

$\displaystyle
u^{\prime\prime}=\left(1-\frac{1}{x}\right)u+f\big{(}x^{-1},u,u^{\prime}\big{)},$

where $f$ is analytic at $(0,0,0)$ and
$f=O\big{(}x^{-2}\big{)}+O\big{(}u^{2}\big{)}+O\big{(}u^{\prime
2}\big{)}+O(uu^{\prime})$. This is a normal form for a second order equation,
by the argument in Section 4.1: the normalizing transformation is (4.4) with
$q=-1/2$ and the normal form as a first order system is of the type (4.5) with
$q=-1/2$. Theorem A.3 applies, and reverting the substitutions, Theorem 3.1
follows.

###  5.1 The first array of poles: proof of Theorem 3.4

Setting $\xi={C{{\rm e}^{-x}}}{x}^{-1/2}$, plugging in
$\mathcal{P}(\alpha,\beta,\gamma,2)$ an expansion (A.8) and expanding, it
follows that $F_{0}(\xi)$ must satisfy

$\displaystyle{\xi}^{2}F_{0}^{\prime\prime}+\xi
F_{0}^{\prime}={\frac{{\xi}^{2}(3F_{0}-4)F_{0}^{\prime
2}}{2(F_{0}-2)(F_{0}-1)}}+2{\frac{F_{0}(F_{0}-1)}{F_{0}-2}},$

whose unique solution analytic at $\xi=0$ with
$F_{0}(\xi)=\xi+O\big{(}\xi^{2}\big{)}$, $\xi\to 0$, is

$\displaystyle F_{0}(\xi)={\frac{\xi}{(\xi/4+1)^{2}}}.$

Let $x_{n}$ be solutions of $\xi(x)/4+1=0$. By Theorem A.6, if $x_{n}$ solve
$C{\rm e}^{-x}x^{-1/2}=-4$, then solutions $u(x)$ have singularities located
at $x_{n}+o(1)$ for large $n$ and Theorem 3.4 follows.

######  Remark 5.1.

While $F_{0}(\xi)$ has a pole of order two, it is known that the poles of
$P_{V}$ have order one if $\alpha\neq 0$. This suggests that there are two
array of poles within $O\big{(}n^{-1}\big{)}$ distance from each other, as the
expansion in $x^{-1}$ would collapse them into a double pole. The author has
no rigorous argument at this time that this is indeed the case, and it is
formulated here as a conjecture.

##  Appendix A Results used

###  A.1 Summation of transseries formal solutions

######  Theorem A.1 ([5]).

Consider the (nonlinear) system of first order differential equations:

$\displaystyle\mathbf{y}^{\prime}+\left(\Lambda-\frac{1}{x}A\right)\mathbf{y}=\mathbf{g}(x^{-1},\mathbf{y}),\qquad\mathbf{y}\in\mathbb{R}^{d},$
(A.1)

with $\mathbf{g}=O\big{(}x^{-2}\big{)}+O\big{(}|\mathbf{y}|^{2}\big{)}$ for
$x\to\infty$, $|\mathbf{y}|\to 0$ and $\Lambda={\rm
diag}(\lambda_{1},\ldots,\lambda_{d})$, $A={\rm
diag}(\alpha_{1},\ldots,\alpha_{d})$. Assume the following non-resonance
condition: any collection among $\lambda_{1},\ldots,\lambda_{d}$ which are in
the same open half-plane are linearly independent over $\mathbb{Z}$.

Assume, for simplicity, that the independent variable, $x$, is scaled so that
$\lambda_{1}=1$.

  1. $(i)$

Then (A.1) has formal solutions

$\displaystyle\tilde{\mathbf{y}}=\tilde{\mathbf{y}}(x;\mathbf{C})=\tilde{\mathbf{y}}_{\mathbf{0}}(x)+\sum_{\mathbf{k}\in\mathbb{N}^{d}\setminus\mathbf{0}}\mathbf{C}^{\mathbf{k}}{\rm{e}}^{-\boldsymbol{\lambda}\cdot\mathbf{k}x}\tilde{\mathbf{y}}_{\mathbf{k}}(x)\qquad\text{with}\quad\tilde{\mathbf{y}}_{\mathbf{k}}(x)=x^{\boldsymbol{\alpha}\cdot\mathbf{k}}\tilde{\mathbf{s}}_{\mathbf{k}}(x),$
(A.2)

where $\tilde{\mathbf{s}}_{\mathbf{k}}(x)$ are integer power series in
$x^{-1}$. A formal solution (A.2) is a transseries for $x\to\infty$ along any
direction along which all the exponentials present are decaying, i.e., along
any direction in the sector

$\displaystyle S_{\rm
trans}=\\{x\in\mathbb{C}\,|\,\operatorname{Re}(\lambda_{j}x)>0\ \text{for
all}\ j\ \text{with}\ C_{j}\neq 0\\}.$ (A.3)

  2. $(ii)$

A transseries is Borel summable for large $x$ along any direction of argument
$\phi$, where the sector $a_{1}<\arg x<a_{2}$ contains only one Stokes line,
$\arg x=0$. The Borel sum is an actual solution: for any $\epsilon>0$ there is
$R_{\epsilon}>0$ so that

$\displaystyle\mathbf{y}(x)=\begin{cases}\displaystyle\mathcal{L}_{\phi}\mathbf{Y}_{\mathbf{0}}(x)+\\!\sum_{\mathbf{k}\in\mathbb{N}^{d}\setminus\mathbf{0}}\\!\mathbf{C_{+}}^{\mathbf{k}}{\rm{e}}^{-\boldsymbol{\lambda}\cdot\mathbf{k}x}\mathcal{L}_{\phi}\mathbf{Y}_{\mathbf{k}}(x),\
-\phi=\arg x\in(0,a_{2}\\!-\\!\epsilon),\ |x|>R_{\epsilon},\\\
\displaystyle\mathcal{L}_{\phi}\mathbf{Y}_{\mathbf{0}}(x)+\\!\sum_{\mathbf{k}\in\mathbb{N}^{d}\setminus\mathbf{0}}\\!\mathbf{C_{-}}^{\mathbf{k}}{\rm{e}}^{-\boldsymbol{\lambda}\cdot\mathbf{k}x}\mathcal{L}_{\phi}\mathbf{Y}_{\mathbf{k}}(x),\
-\phi=\arg x\in(a_{1}\\!+\\!\epsilon,0),\ |x|>R_{\epsilon},\end{cases}$ (A.4)

where
$\mathbf{Y}_{\mathbf{k}}=\mathcal{B}_{\phi}\tilde{\mathbf{y}}_{\mathbf{k}}$
$($the analytic continuation of the Borel transform of
$\tilde{\mathbf{y}}_{\mathbf{k}}$ along the direction of argument $\phi)$.
Along $\arg x=0$ balanced averages of
$\mathcal{L}_{\phi}\mathbf{Y}_{\mathbf{k}}$ sum to the solution
$\mathbf{y}(x)$ $($Écalle–Borel summation$)$. The solution $\mathbf{y}(x)$ is
analytic for $|x|>R_{\epsilon}$ with $\arg
x\in(a_{1}+\epsilon,a_{2}-\epsilon)$. Conversely, any solution asymptotic to
$\tilde{\mathbf{y}}_{0}$ for $x\to+\infty$ has a representation (A.4).

  3. $(iii)$

Only the first component $C_{1}$ of the constant beyond all orders in (A.4)
changes when $\arg x$ crosses the Stokes line $\arg x=0$, corresponding to
$\lambda_{1}=1$. The change of this constant depends only on the equation: it
is a multiple of the first Stokes constant.

######  Note A.2.

More recent results showed that the region of convergence of (A.4) is in fact
given by conditions of the form $|x|>R$ and $\big{|}C_{i}{\rm
e}^{-\lambda_{i}x}x^{\alpha_{i}}\big{|}<\mu_{i}$, $i=1,\ldots,d$ for suitable
constants $\mu_{i}$ and $R$ ($C_{i}=0$ if $\operatorname{Re}\lambda_{i}>0$)
[7], see also [6].

We use Theorem A.1 in the particular case when $d=2$, $\lambda_{1}=1$,
$\lambda_{2}=-1$, in which case the sectors (A.3) become the right
(respectively left) half plane and the constants have the form
$\mathbf{C}=(C,0)$ (respectively $\mathbf{C}=(0,C)$); we also assume that
solutions have only poles as moving singularities, as it is the case for the
Painlevé equations (otherwise domains of analyticity would have to extend on
Riemann sheets). Theorem A.1 takes the following simpler formulation:

######  Theorem A.3.

Consider the two-dimensional system of first order differential equations
(A.1) with $\Lambda={\rm diag}(1,-1)$, $A={\rm diag}(\alpha_{1},\alpha_{2})$.
Then:

  1. $(i)$

The system has a one parameter family of transseries solutions

$\displaystyle\tilde{\mathbf{y}}=\tilde{\mathbf{y}}(x;C)=\tilde{\mathbf{y}}_{0}(x)+\sum_{k\geq
1}{C}^{k}{\rm{e}}^{-{k}x}\tilde{\mathbf{y}}_{k}(x),\ C\in\mathbb{C},\
\text{along}\ \arg x=-\phi,\ |\phi|<\tfrac{\pi}{2},$ (A.5)

where
$\tilde{\mathbf{y}}_{\mathbf{k}}(x)=x^{{k}\alpha_{1}}\tilde{\mathbf{s}}_{\mathbf{k}}(x)=x^{{k}\alpha_{1}}\sum\limits_{n=0}^{\infty}\tilde{\mathbf{s}}_{\mathbf{k,n}}x^{-n}\text{
and }\tilde{\mathbf{y}}_{0}(x)=O\big{(}x^{-2}\big{)}$.

  2. $(ii)$

For any $C$ the formal solution (A.5) is Borel summable for large $x$ along
any direction of argument $\phi=-\arg x$ if $0<|\phi|<\tfrac{\pi}{2}$.
$($Along $\arg x=0$ it is Écalle–Borel summable.$)$ The Borel sum is an actual
solution: for any constant $C_{+}$, and any $\epsilon>0$ there exists $C_{-}$
$(C_{+}-C_{-}$ is a multiple of the Stokes constant, [9]) and $R_{\epsilon}>0$
such that

$\displaystyle\mathbf{y}(x)=\begin{cases}\displaystyle\mathcal{L}_{\phi}\mathbf{Y}_{\mathbf{0}}(x)+\sum_{k\geq
1}C_{+}^{k}{\rm{e}}^{-{k}x}\mathcal{L}_{\phi}\mathbf{Y}_{k}(x),&-\phi=\arg
x\in\big{(}0,\frac{\pi}{2}-\epsilon\big{)},\ |x|>R_{\epsilon},\\\
\displaystyle\mathcal{L}_{\phi}\mathbf{Y}_{\mathbf{0}}(x)+\sum_{k\geq
1}C_{-}^{k}{\rm{e}}^{-{k}x}\mathcal{L}_{\phi}\mathbf{Y}_{k}(x),&-\phi=\arg
x\in\big{(}{-}\frac{\pi}{2}+\epsilon,0\big{)},\ |x|>R_{\epsilon}.\end{cases}$
(A.6)

The domain of convergence of the function series (A.6) given by

$\displaystyle\big{\\{}x\,|\,|x|>R,\,\big{|}C_{\pm}{\rm
e}^{-x}x^{\alpha_{1}}\big{|}<\mu\big{\\}}$

for $\mu>0$ small enough and $R>0$ large enough. Conversely, any solution
asymptotic to $\tilde{\mathbf{y}}_{0}$ for $x\to+\infty$ has a representation
(A.6) for some $C_{\pm}$.

  3. $(ii)$

Similarly, the system has a transseries solution

$\displaystyle\tilde{\mathbf{y}}=\tilde{\mathbf{y}}(x;C)=\tilde{\mathbf{y}}_{0}(x)+\sum_{k\geq
1}{C}^{k}{\rm{e}}^{{k}x}\tilde{\mathbf{y}}_{k}(x)\qquad\text{for}\ \arg
x=-\phi,\ 0<|\phi+\pi|<\frac{\pi}{2},$ (A.7)

where
$\tilde{\mathbf{y}}_{\mathbf{k}}(x)=x^{{k}\alpha_{2}}\tilde{\mathbf{s}}_{\mathbf{k}}(x)$
with $\tilde{\mathbf{s}}_{\mathbf{k}}(x)$ are integer power series.

Statements similar to those in $(i)$ hold regarding Borel summability (A.7)
along any direction of argument $\arg x\neq\pi$ in the left half plane, and
Écalle–Borel summability along $\arg x=\pi$, the sum being an actual solution,
analytic in a domain $|x|>R$ and $\big{|}C_{\pm}{\rm
e}^{x}x^{\alpha_{2}}\big{|}<\mu$ for some $\mu>0$ and $R>0$. Conversely, any
solutions asymptotic to $\tilde{\mathbf{y}}_{0}$ for $x\to-\infty$ is the
Borel sum of such a transseries.

###  A.2 Arrays of singularities bordering the sector of analyticity

Theorem A.1 establishes existence of solutions, in one-to-one correspondence
with formal transs̄eries solutions, and which are analytic for large $x$ in
the sector (A.3) where these formal solutions are defined (i.e., they are
well-ordered with respect to $\gg$). In [7] it is further shown that on the
boundary of the sector of analyticity, these solutions develop arrays of
singularities.

In the particular case when $d=2$, $\lambda_{1}=1$, $\lambda_{2}=-1$ of this
paper the results in [7] are as follows. Denote

$\displaystyle\xi=\xi(x)=C{\rm e}^{-x}x^{\alpha_{1}}.$

For $x$ near ${\rm i}\mathbb{R}_{+}$, i.e., when $\xi\gg x^{-k}$ for all
$k>0$, the transseries (A.5) can be formally reordered as

$\displaystyle\sum_{k\geq
0}\xi^{k}\tilde{\mathbf{s}}_{k,0}(x)+\frac{1}{x}\sum_{k\geq
0}\xi^{k}\tilde{\mathbf{s}}_{k,1}+\frac{1}{x^{2}}\sum_{k\geq
0}\xi^{k}\tilde{\mathbf{s}}_{k,2}+\cdots.$

It turns out that the series in $\xi$ are convergent, and the resulting
expansion is asymptotic to the solution the original transseries summed to:

######  Theorem A.4 ([7]).

Let $\delta,c>0$. There exists $\delta_{1}>0$ so that for $|\xi|<\delta_{1}$
the power series

$\displaystyle\mathbf{F}_{m}(\xi)=\sum_{k=0}^{\infty}\xi^{k}\tilde{\mathbf{s}}_{k,m},\qquad
m=0,1,2,\ldots$

converge. Furthermore

$\displaystyle\mathbf{y}(x)\sim\sum_{m=0}^{\infty}x^{-m}\mathbf{F}_{m}(\xi(x))\qquad\text{for}\
x\to\infty\ \text{with}\ \arg
x\in\big{[}{-}\tfrac{\pi}{2}+\delta,\tfrac{\pi}{2}-\delta\big{]},\
|\xi(x)|<\delta_{1}.\\!\\!\\!\\!\\!$ (A.8)

The asymptotic series is uniform, it is differentiable and satisfies Gevery-
like estimates.

######  Note A.5.

$\mathbf{F}_{0}(0)=0$ and $\mathbf{F}^{\prime}_{0}(0)=1$.

In fact (A.8) is valid in a larger domain, up to distance $o(1)$ of the
singularities of $\mathbf{F}_{0}$ ($\mathbf{F}_{m}$ with $m>0$ can have no
other singularities). In general $\mathbf{F}_{0}$ has branch point
singularities, and a Riemann surface needs to be considered. But Painlevé
equations have no movable branch points, so the singularities of
$\mathbf{F}_{0}$ can only be poles. Then for simplicity we state here the
general result of [7] in this case only.

Let $\rho_{1,2}$ so that the small term $\mathbf{g}$ in (A.1) is analytic in
the polydisk $\big{|}x^{-1}\big{|}<\rho_{1}$, $|\mathbf{y}|<\rho_{2}$.

Let $\Xi$ be a finite set (possibly empty) of poles of $\mathbf{F}_{0}$. Let
$\mathcal{D}\subset\mathbb{C}\setminus\Xi$ be open, connected, relatively
compact, containing $|\xi|<\delta_{1}$, so that $\mathbf{F}_{0}$ is analytic
in an $\epsilon$-neighborhood of $\mathcal{D}$ ($\epsilon>0$), so that
$\sup\limits_{\mathcal{D}}|\mathbf{F}_{0}(\xi)|=\rho_{3}<\rho_{2}$.

We need the counterpart of $\mathcal{D}$ in the $x$-plane. Let

$\displaystyle X=\xi^{-1}(\mathcal{D})\cap\big{\\{}|x|>R,\,\arg
x\in\big{[}{-}\tfrac{\pi}{2}+\delta,\tfrac{\pi}{2}-\delta\big{]}\big{\\}}.$

######  Theorem A.6 ([7]).

All $\mathbf{F}_{m}$ with $m\geq 1$ are analytic on $\mathcal{D}$ and for $R$
large enough the asymptotic expansion (A.8) holds for $x\to\infty$ with $x\in
X$.

Furthermore, assume that $\mathbf{F}_{0}$ is singular at
$\xi_{s}\in\mathcal{D}$. Then $\mathbf{y}(x)$ is singular near points $x$ with
$\xi(x)=\xi_{s}$, more precisely at

$\displaystyle x_{n}=2n\pi{\rm i}+\alpha_{1}\ln(2n\pi{\rm i})+\ln
C-\ln\xi_{s}+o(1)\qquad\text{as}\quad n\to\infty.$

### Acknowledgements

The author is grateful to the editors for valuable references and information,
and to the referees careful reading of the manuscript and for their helpful
comments.

## References

  * [1] Ablowitz M.J., Clarkson P.A., Solitons, nonlinear evolution equations and inverse scattering, London Mathematical Society Lecture Note Series, Vol. 149, Cambridge University Press, Cambridge, 1991.
  * [2] Andreev F.V., Kitaev A.V., Exponentially small corrections to divergent asymptotic expansions of solutions of the fifth Painlevé equation, Math. Res. Lett. 4 (1997), 741–759.
  * [3] Boutroux P., Recherches sur les transcendantes de M. Painlevé et l’étude asymptotique des équations différentielles du second ordre (suite), Ann. Sci. École Norm. Sup. (3) 31 (1914), 99–159.
  * [4] Clarkson P.A., Painlevé equations – nonlinear special functions, in Orthogonal Polynomials and Special Functions, Lecture Notes in Math., Vol. 1883, Springer, Berlin, 2006, 331–411.
  * [5] Costin O., On Borel summation and Stokes phenomena for rank-$1$ nonlinear systems of ordinary differential equations, Duke Math. J. 93 (1998), 289–344, math.CA/0608408.
  * [6] Costin O., Asymptotics and Borel summability, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, Vol. 141, CRC Press, Boca Raton, FL, 2009.
  * [7] Costin O., Costin R.D., On the formation of singularities of solutions of nonlinear differential systems in antistokes directions, Invent. Math. 145 (2001), 425–485, math.CA/0202234.
  * [8] Costin O., Costin R.D., Asymptotic properties of a family of solutions of the Painlevé equation $\rm P_{VI}$, Int. Math. Res. Not. 2002 (2002), 1167–1182, math.CA/0202235.
  * [9] Costin O., Costin R.D., Huang M., Tronquée solutions of the Painlevé equation PI, Constr. Approx. 41 (2015), 467–494, arXiv:1310.5330.
  * [10] Costin O., Costin R.D., Huang M., A direct method to find Stokes multipliers in closed form for ${\rm P}_{1}$ and more general integrable systems, Trans. Amer. Math. Soc. 368 (2016), 7579–7621, arXiv:1205.0775.
  * [11] Costin O., Huang M., Tanveer S., Proof of the Dubrovin conjecture and analysis of the tritronquée solutions of ${\rm P}_{\rm I}$, Duke Math. J. 163 (2014), 665–704, arXiv:1209.1009.
  * [12] Dubrovin B., Grava T., Klein C., On universality of critical behavior in the focusing nonlinear Schrödinger equation, elliptic umbilic catastrophe and the tritronquée solution to the Painlevé-I equation, J. Nonlinear Sci. 19 (2009), 57–94, arXiv:0704.0501.
  * [13] Fokas A.S., Its A.R., Kapaev A.A., Novokshenov V.Yu., Painlevé transcendents: the Riemann–Hilbert approach, Mathematical Surveys and Monographs, Vol. 128, Amer. Math. Soc., Providence, RI, 2006.
  * [14] Grava T., Kapaev A., Klein C., On the tritronquée solutions of ${\rm P}_{\rm I}^{2}$, Constr. Approx. 41 (2015), 425–466, arXiv:1306.6161.
  * [15] Gray J.J., Fuchs and the theory of differential equations, Bull. Amer. Math. Soc. (N.S.) 10 (1984), 1–26.
  * [16] Gromak V.I., Laine I., Shimomura S., Painlevé differential equations in the complex plane, De Gruyter Studies in Mathematics, Vol. 28, Walter de Gruyter & Co., Berlin, 2002.
  * [17] Guzzetti D., A review of the sixth Painlevé equation, Constr. Approx. 41 (2015), 495–527, arXiv:1210.0311.
  * [18] Huang M., Xu S.-X., Zhang L., Location of poles for the Hastings–McLeod solution to the second Painlevé equation, Constr. Approx. 43 (2016), 463–494, arXiv:1410.3338.
  * [19] Its A.R., Kapaev A.A., Connection formulae for the fourth Painlevé transcendent; Clarkson–McLeod solution, J. Phys. A: Math. Gen. 31 (1998), 4073–4113.
  * [20] Its A.R., Kapaev A.A., Quasi-linear Stokes phenomenon for the second Painlevé transcendent, Nonlinearity 16 (2003), 363–386, nlin.SI/0108010.
  * [21] Jimbo M., Miwa T., Môri Y., Sato M., Density matrix of an impenetrable Bose gas and the fifth Painlevé transcendent, Phys. D 1 (1980), 80–158.
  * [22] Joshi N., Kitaev A.V., On Boutroux’s tritronquée solutions of the first Painlevé equation, Stud. Appl. Math. 107 (2001), 253–291.
  * [23] Joshi N., Mazzocco M., Existence and uniqueness of tri-tronquée solutions of the second Painlevé hierarchy, Nonlinearity 16 (2003), 427–439, math.CA/0212117.
  * [24] Kapaev A.A., Asymptotic behavior of the solutions of the Painlevé equation of the first kind, Differential Equations 24 (1988), 1107–1115.
  * [25] Kapaev A.A., Global asymptotics of the fourth Painlevé transcendent, Steklov Math. Inst. and IUPUI, Preprint # 96-06, 1996, available at http://www.pdmi.ras.ru/preprint/1996/index.html.
  * [26] Kapaev A.A., Quasi-linear Stokes phenomenon for the Painlevé first equation, J. Phys. A: Math. Gen. 37 (2004), 11149–11167, nlin.SI/0404026.
  * [27] Kapaev A.A., Quasi-linear Stokes phenomenon for the Hastings–McLeod solution of the second Painlevé equation, nlin.SI/0411009.
  * [28] Kapaev A.A., Kitaev A.V., Connection formulae for the first Painlevé transcendent in the complex domain, Lett. Math. Phys. 27 (1993), 243–252.
  * [29] Kowalevski S., Sur le problème de la rotation d’un corps solide autour d’un point fixe, in The Kowalevski Property (Leeds, 2000), CRM Proc. Lecture Notes, Vol. 32, Amer. Math. Soc., Providence, RI, 2002, 315–372, Reprinted from Acta Math. 12 (1889), 177–232.
  * [30] Lin Y., Dai D., Tibboel P., Existence and uniqueness of tronquée solutions of the third and fourth Painlevé equations, Nonlinearity 27 (2014), 171–186, arXiv:1306.1317.
  * [31] Novokshenov V.Yu., Distributions of poles to Painlevé transcendents via Padé approximations, Constr. Approx. 39 (2014), 85–99.
  * [32] Painlevé P., Oeuvres de Paul Painlevé, Tome III, Équations différentielles du second ordre, Mécanique, Quelques documents, Éditions du Centre National de la Recherche Scientifique, Paris, 1975.
  * [33] Parusnikova A., Asymptotic expansions of solutions to the fifth Painlevé equation in neighbourhoods of singular and nonsingular points of the equation, in Formal and Analytic Solutions of Differential and Difference Equations, Banach Center Publ., Vol. 97, Polish Acad. Sci. Inst. Math., Warsaw, 2012, 113–124.
  * [34] Shimomura S., Truncated solutions of the fifth Painlevé equation, Funkcial. Ekvac. 54 (2011), 451–471.
  * [35] Takei Y., On the connection formula for the first Painlevé equation – from the viewpoint of the exact WKB analysis, Sūrikaisekikenkyūsho Kōkyūroku 931 (1995), 70–99.
  * [36] Tracy C.A., Widom H., On exact solutions to the cylindrical Poisson–Boltzmann equation with applications to polyelectrolytes, Phys. A 244 (1997), 402–413, cond-mat/9701067.
  * [37] Wu T.T., McCoy B.M., Tracy C.A., Barouch E., Spin-spin correlation functions for the two-dimensional Ising model: exact theory in the scaling region, Phys. Rev. B 13 (1976), 316–374.
  * [38] Xia X., Tronquée solutions of the third and fourth Painlevé equations, SIGMA 14 (2018), 095, 28 pages, arXiv:1803.11230.