# A Methodology to Support Automatic Cyber Risk Assessment Review

Marco Angelini  Silvia Bonomi  Alessandro Palma

###### Abstract

Cyber risk assessment is a fundamental activity for enhancing the protection
of an organization, identifying and evaluating the exposure to cyber threats.
Currently, this activity is carried out mainly manually and the identification
and correct quantification of risks deeply depend on the experience and
confidence of the human assessor. As a consequence, the process is not
completely objective and two parallel assessments of the same situation may
lead to different results. This paper takes a step in the direction of
reducing the degree of subjectivity by proposing a methodology to support risk
assessors with an automatic review of the produced assessment. Our methodology
starts from a controls-based assessment performed using well-known
cybersecurity frameworks (e.g., ISO 27001, NIST) and maps security controls
over infrastructural aspects that can be assessed automatically (e.g., ICT
devices, organization policies). Exploiting this mapping, the methodology
suggests how to identify controls needing a revision. The approach has been
validated through a case study from the healthcare domain and a set of
statistical analyses.

Keywords: Risk Assessment. Security Governance. Cybersecurity frameworks.

##  1 Introduction

The growing number of cyber threats puts more and more organizations at risk
of being the victim of a cyber attack. In this picture, it is thus paramount
to design and implement efficient and effective processes to timely identify
and manage potentially dangerous situations. Cyber risk management is one of
those processes, maybe the most important one. It is challenging due to the
peculiarities of cyberspace (e.g., wide attack surface and a high degree of
uncertainty) and its main activities are identified by several standards and
best practices ([12],[11],[20]). Evaluating cyber risk raises several issues:
human errors are always possible in each step of the assessment, as well as
different sensibilities from different assessors to similar situations that
may influence the evaluation. Those factors could negatively affect the
reliability of the final outcome. Moreover, the bias during the assessment
could compromise its accuracy. This paper proposes a methodology to review the
results of a risk assessment depending on the actual organization’s
infrastructure exposure. We introduce the concept of _comprehensive score_ to
quantify the security of the different parts of the infrastructure at
different levels (i.e., technical, human-based) and take into account
governance aspects. To perform this comprehensive analysis, we exploit the
multi-layer attack graph (MLAG), over which the results of an assessment based
on well-known frameworks (e.g., ISO 27001, NIST) are mapped. Through this
approach, the assessor has more fitting quantification of the cyber risk and
the critical exposures in the infrastructure.

##  2 Related work

In the current literature, there exist several methodologies (and sometimes
related tools) developed in compliance (sometimes partially) with the
standards, to support the risk identification, analysis, and treatment. For
example, MEHARI risk management methodology [5] and EBIOS [2] detail the main
steps that every organization should implement to deal with cyber risk.
However, they leave to the assessor the duty to decide how to quantify the
risk by leveraging on security frameworks (e.g., ISO/IEC 27002 [10] and NIST
CSF [20]). Such frameworks can be used in combination with different models of
risk quantification. They range from the simple _two-factor_ risk model that
considers the risk as the product of the _likelihood_ that an asset is
compromised times the _impact_ of the compromising, to the _multi-factor_ risk
model where both likelihood and impact can be further decomposed into more
detailed factors quantified independently and then aggregated. One of these
latter models is the OWASP risk rating methodology [14], in which the
likelihood is estimated by considering 4 factors: threat agent,
vulnerabilities, technical impact, and business impact. This rating method is
quite simple to adopt and OWASP provides guidelines about how to score each
factor. However, the scoring is done by the human who needs to spend a big
effort in the analysis and in remaining objective and consistent during it.
More objective evidences, coming from the infrastructure setup, should be
considered to address such issues. A suitable approach to do it is leveraging
an attack graph. An attack graph allows representing possible ways via which
an attacker can intrude into a target network by exploiting a series of
vulnerabilities on various hosts. Attack graphs can be computed automatically
starting from the data gathered in the system and thus they represent an
objective model over which the risk may be computed [15]. A large body of
literature exists about using attack graphs to compute network security
evaluation metrics ([21],[24]), perform security risk analysis ([3],[8]) and
compute near-optimal proactive defense measures ([25],[8]). In this scenario,
Gonzalez et al. [8] propose an approach to perform data-driven cyber risk
analysis through an attack graph, with a quantitative risk-aware system to
dynamically manage risk response by considering likelihood, the induced
impact, the cost of the possible responses, and the negative side-effects of a
response. However, the limitation of using the classical approaches to attack
graphs is that almost all of them can support risk estimation by considering
only technical aspects, without considering a comprehensive view of the
organization environment (e.g., human factors, policies, and governance
compliance). There exist few proposals trying to address this limitation.
Bonomi et al.[4] introduce a novel approach to attack graphs that is the
multi-layer attack graph (MLAG). In the context of the PANACEA project [17]
the architecture introduced by Gonzalez is enriched with comprehensive data
through the MLAG. Although these advancements represent a good starting point
for comprehensive risk analysis, information about security governance and
compliance is still not considered due to the difficulty to integrate them
with technical data. We faced such a challenge in [1] by proposing a
methodology towards a comprehensive risk analysis, including governance
aspects. It consists of the review of a checklist-based cyber risk assessment,
that we include in our methodology as detailed in Section 3.

##  3 Problem Statement and Background Notions

Before describing our methodology, we introduce two models that are the inputs
for our solution: a _controls-based assessment_ representing the governance
perspective and a _multi-layer attack graph_ (MLAG) handling the technical
perspective. We define as _controls-based assessment_ a list of security
controls and their individual evaluation, as for example the outcome of an
assessment performed using ISO 27001 [9]. A security control is everything
that may require the definition and deployment of a specific process, policy,
device, practice, or other actions which may contribute to the definition and
quantification of risk. The evaluation for us is performed by assessing the
coverage level of each control. Without loss of generality, we will assume a
qualitative evaluation based on three levels i.e., completely covered (C),
partially covered (PC), or not covered at all (NC) and it can be done by the
assessor using data collected through interviews, workshops or questionnaires.
Let us note that compiling a controls-based assessment is a complex activity
and prone to human bias because of the interpretability of several controls.
Thus, our methodology proposes to reduce such bias. Concerning the MLAG, we
will consider the one presented in [4]. This model allows to create an
objective representation of the possible paths that an attacker may follow to
compromise an asset and their likelihood can be quantified in an unbiased way
depending on the severity of the vulnerabilities involved. The MLAG is based
on three interconnected layers: (i) the _human layer_ modeling how an attacker
can compromise individual identities by exploiting human vulnerabilities of
the personnel and their relationships; (ii) the _network layer_ modeling the
ICT part of the company used by individuals represented in the human layer;
(iii) the _access layer_ modeling the credentials that humans may use to
access devices (residing in the network layer). Each layer is a multi-graph,
meaning that multiple edges between the same pair of nodes are characterized
by different vulnerabilities.

Using these models, we initially made a step towards their integration in [1],
in which the MLAG is used to validate the controls-based assessment performed
by a security expert, allowing to detect errors due to human bias. Beyond the
fact that the approach was completely manual, it just informs about the
trustworthiness of a controls-based assessment. Instead, the methodology we
propose in this paper aims to quantify and re-asses the cyber exposure using
comprehensive data and indicating the specific parts of the infrastructure
they impact on, as well as to automate the computation steps.

##  4 Methodology

In this section we present the methodology by firstly introducing the general
description and then detailing the application of each activity.

###  4.1 Methodology description

The proposed methodology (see Figure 1) starts taking as input (i) the
reference frameworks, (ii) a controls-based assessment, (iii) domain models
(either extracted from assessment and attack graph or modeled from scratch),
and (iv) a technical assessment (through a MLAG).

Figure 1: Scored assessment review methodology.

_Controls classification_ (Fig 1a) is the first activity we must complete,
which is labeling each security control and categorizing it according to a
taxonomy that will allow us to map it to specific elements in the environment.
A classified set of controls will be the result of this step. This task is
needed as security controls describe sets of activities to be implemented to
secure the organization and are expressed using different levels of
granularity.

_Contextualized assessment mapping_ (Fig 1b) is the second phase, in which we
map security controls over the technical assessment result to identify the
layers that each control affects. Controls are mapped in one or more layers of
the attack graph to accomplish this. The result is a classification of
security controls based on their context (that could be human, access, and/or
network perspective). We denote such classification as contextualized
controls.

_Assessment review_ (Fig 1c) is performed once we have the classified and
contextualized controls, and it assigns a reliability evaluation to the
assessment carried out for each control. This task will result in a flagged
assessment i.e., a controls-based assessment with additional information
indicating the subset of controls that should be revised, further
investigated, or could be implemented involving different aspects. This first
part of the methodology represents the evaluation of governance and compliance
aspects.

_Vulnerability exposure_ (Fig 1d) is concurrently assessed by using the MLAG
and analyzing attack paths from human to network layer, using an approach
inspired by Gonzalez [8]. Finally, we map controls over the edges of the
attack graph to obtain, for each edge, information about technical
vulnerabilities, security controls, and assessed risk impacting specific parts
of the organization. After putting this information together, a _scored risk
assessment_ (Fig 1e) is produced, computed with comprehensive and more
objective information.

###  4.2 Methodology application

####  4.2.1 Controls classification.

Given the generality of the security controls, the first step of the
methodology consists of classifying controls based on their features to assess
how much they are clear to be interpreted for implementation. Some controls
are more focused because they refer to specific aspects of the organization
(e.g., login and logout of devices), while others are more general and impact
on different parts of the infrastructure (e.g., design of access control
policies impact both the administrative Board and the ICT department). This
information will affect the evaluation of compliance with governance
frameworks. General controls are more difficult to fully comply with due to
the large number of concepts they include. We consider two criteria to assess
these aspects. The first is the _lifetime_ , which is the point of the risk
management process in which the controls must be put in place. It specifies
whether the control is implemented at _design time_ or _run time_. We assume
that design-time controls have a greater impact on the risk management process
because decisions made at design time impact also their implementation at run
time as appearing in controls-based frameworks ([9],[20]). The second
criterion is the _management level_ as it is introduced by Von Solms [22]: it
indicates whether the nature of the security controls is _operational_ ,
meaning that they include practical activities, or _compliance_ , related to
administrative responsibilities.

We use two ontologies to classify lifetime and management levels of each
control: one containing the semantics of security controls inspired by UCO
[23] and the other containing relations between lifetime operations (e.g.,
process execution monitoring, communication plans policies) and management
levels. Then, we use ontology alignment between them using the automated tool
AML [7]. Ontology alignment is the process of determining correspondences
between concepts in different ontologies [18]: for each couple of concepts,
such a process outputs a value, which describes the similarity of two terms of
the input ontologies. Such a value ranges from 0 (no similarity) to 1 (full
similarity). We compare the concept of each control to the concept of each
feature (run time, design time, operational, and compliance), obtaining an
output as shown in table 1.

ID | Run time | Design time | Operational | Compliance
---|---|---|---|---
A.9.4.3 | 0 | 0.923 | 0.926 | 0
Table 1: Example of alignment values for specificity degree.

The classification of the controls labels the lifetime class with the label
associated to the highest value between run time and design time alignments,
while the management level class with the label of the highest alignment
between operational and compliance. If the alignment produces equal values for
both run time and design time, or for both operational and compliance, the
related class is labeled“Not defined”, indicating that lifetime or management
level cannot be distinguished unambiguously. Considering such classification,
we define the _specificity degree_ of a control ($spec(c)$) as a function
mapping the classification into numerical values. The specificity degree is
higher for run time and operational controls, while it decreases for design
time and compliance ones. This is according to the assumption that run time
and operational controls are easier to comply with because they are more
focused on practical activities and more clear to interpret, while compliance
and design tasks are harder to meet due to their generality. Thus, we set a
maximum value $\alpha$ for run time and operational controls, then we halve
such a value each time that the feature is design time or compliance,
representing that more general is the control, more effort is required to
comply with it. In the “Not defined” case we do not decrease $\alpha$ because
the related control has both operational/run-time and compliance/design-time
nature. The specificity degree values based on controls classification are
reported in Table 2.

|  | Lifetime
---|---|---
|  | Runtime | DesingTime | Not defined
Management | Operational | $\alpha$ | $\alpha/2$ | $\alpha/2$
Compliance | $\alpha/2$ | $\alpha/4$ | $\alpha/4$
Not defined | $\alpha/2$ | $\alpha/4$ | $\alpha/4$
Table 2: Values of specificity degree.

The parameter $\alpha$ is the percentage of maximum simplicity of
interpretation of the most focused controls and it is assignable by the
assessor depending on the overall generality of the controls. For example, a
possible value we can assign to $\alpha$ is 0.5, indicating that at least half
of the controls’ activities can be completely complied with. Considering such
a value for the example in Table 1, the specificity degree for control A.9.4.3
is $0.25$ since it is classified as design time and operational.

####  4.2.2 Contextualized assessment mapping.

This step consists of mapping the controls-based assessment onto the layers of
the attack graph to contextualize each security control to the infrastructure.
This is achieved with automatic ontology alignment [7] between the ontology of
security controls previously designed for controls classification and the
ontological domain of the attack graph obtained by associating the nodes of
the attack graph to ontology classes and the edges to ontology relations. In
this way, each control has an alignment value between 0 (not fitting) and 1
(full fitting) for each layer (human, access, network), as reported in the
example of Table 3.

ID | Human | Access | Network
---|---|---|---
A.9.4.3 | 0.7 | 0.371 | 0
Table 3: Example of alignment between controls and layers.

The highest mapping value of each control represents the layer in which it is
mainly contextualized. It is possible for a security control to be mapped on
none of the layers, on one specific layer, or on multiple layers. Controls
that are mapped in a single layer are focused within the infrastructure of the
organization (i.e., easier to comply due to homogeneity of the impacted
resources), while the ones mapped in multiple layers impact different parts of
the infrastructure (i.e., more effort is required to comply).

Considering this rationale, we define the _fitting degree_ of a control
($fit(c)$) to express how much it fits the infrastructure. Let $h_{i}$,
$a_{i}$, $n_{i}$ be the alignment values for the control $c_{i}$ in the human,
access and network layers respectively. Let $l_{M}=max(h_{i},a_{i},n_{i})$ be
the highest between these values (most fitting layer) and
$l_{m}=min(h_{i},a_{i},n_{i})$ the smallest one (less fitting layer). We
define the fitting degree of a control as $l_{M}-l_{m}$, that is the range of
the alignment values representing the spread of the control in the
infrastructure. Indeed, the fitting degree decreases as the control is more
spread between layers. Considering the example in Table 3, the most fitting
layer for control A.9.4.3 is human (with value $0.7$), while the layer in
which it fits less is network (with value 0).

####  4.2.3 Assessment review.

Assessment review is performed following the approach we introduced in [1],
which expresses the reliability of a controls-based assessment performed by a
cybersecurity expert, in which each control can be assessed totally
(C-Covered), partially (PC-Partially Covered), or nothing (NC-Not Covered). In
such an approach the security controls are associated with the layers of the
attack graph and to management levels. Then the retrieved information is
averaged to output a percentage of how much the performed assessment is
reliable for each security control. In particular, it is computed the average
of the following four data: lifetime, management level, mapping value of a
control in the layers, and the coverage factor of the assessment. While the
calculation of these data is based on manual quantification, we extend it by
automatizing the computation of these values through ontology alignment using
the ontologies introduced in the previous sections. We define the output we
obtain by such a computation as _reliability degree_ of a control ($rel(c)$),
which represents the trustworthiness of the assessment of each control.

####  4.2.4 Vulnerability exposure.

We use the approach of Gonzalez et al. [8] to evaluate the probability that an
attack step over the MLAG is performed, indicating the difficulty to exploit
the related vulnerability. They model each attack path as a Markov chain and
then evaluate the exit rates ($\lambda$) of each node as the sojourn time in
each state: such a rate is the likelihood that the vulnerability in the
related edge of the attack graph is exploited. Such exit rates are evaluated
based on the attributes of each vulnerability, therefore we need to
distinguish the network layer from the human/access ones, because in the
former such information is available through CVSS [6], while in the latter we
retrieve human vulnerabilities from ISO 27005 [13]. Then the qualitative
values of each attribute are normalized in the range [0,1] such that the
higher the values, the higher the severity.

In the network layer we can retrieve the following metrics valued according to
CVSS: Attack Complexity (AC), Attack Vector (AV), Privilege Required (PR),
Exploit Code Maturity (CM) and Report Confidence (RC). The attacker ability
must be considered for each attribute and it can be {Naive, Advanced,
Professional}. Thus, given a network vulnerability $v_{k}$ of the k-th step of
the attack path, assuming attacker with abilities
$A=<t_{AC},t_{AV},t_{PR},t_{CM},t_{RC}>$ and considering the attributes of
network vulnerability as $X(v_{k})$, the difficulty of exploitation of the
edge $e_{k}$ in the network layer is:

$\lambda_{k}=\prod_{X\in{AC,AV,PR,CM,RC}}H(X(v_{k})-t_{X})*X(v_{k})$

where $H(z)$ is the Heaviside step function equal to 0 if z$<$0 and 1
otherwise.

In human and access layers, we refer to vulnerabilities and their evaluation
guidelines contained in ISO 27005, as “No logout”, “Sharing credentials”,
“E-mail misuse”. We consider the attack complexity (AC) and the access vector
(AV) attributes of each vulnerability, such that AC can assume values {Low,
High} and AV can assume values {Proximity, Knowledge}. We do not consider the
attacker ability because the vulnerability exposures in these layers do not
depend from the attacker. Following this approach, given a vulnerability
$v_{k}$ related to the k-th step of an attack path, with attributes
$X(v_{k})$, then:

$\lambda_{k}=\prod_{X\in{AC,AV}}X(v_{k})$

####  4.2.5 Scored assessment.

At this point, we have the necessary data to compute a _comprehensive score_
for each edge of the attack graph including both governance compliance and
technical aspects. On one side we have information about controls (i.e.,
specificity degree ($spec(c)$), fitting degree ($fit(c)$), reliability degree
($rel(c)$)) and their classification in the attack graph layers based on the
information computed in the contextualized assessment mapping phase of the
methodology. We use such information to introduce a _governance factor_ of a
layer expressing the degree of compliance of the layer with respect to
security governance frameworks (i.e., security controls). To do so, we group
the controls according to the layers they belong to. Let $C_{i}$ be the set of
controls mapped into layer $l_{i}$, then the governance factor is:

$governanceFactor(l_{i})=gov(l_{i})=avg\\{\forall c\in
C_{i}[f(spec(c),fit(c),rel(c))]\\}$

Function _f_ is a generic aggregation function that maps the relation between
specificity, fitting, and reliability degrees into a value between 0 and 1.
Suitable solutions for such a function could be the average to assign to the
three values the same importance, as well as the minimum (worst-case scenario
usually considered by cyber-risk assessment and management activities). Once
aggregated, the average between the compliance degree of all the controls
belonging to the same layer is computed: this allows to consider the mean
compliance of the governance within each layer.

On the other hand we know the difficulty of technical exploitability of each
vulnerability through the computation of $\lambda$ rates associated with the
edges of the attack graph. The combination of the governance factor with
$\lambda$ values represents the association of governance and technical
knowledge. We associate the governance factor of a layer to all the edges
belonging to that layer: this indicates an equal impact of that factor on each
edge. Considering this assumption, we define the _comprehensive score_ in the
edge $e_{k}$ in layer $l_{i}$ of the attack graph as:

$comprehensiveScore(e_{k})=f(gov(l_{i}),\lambda_{k})*cv$

where $cv$ is a constant set by the security expert that takes into account
how rigorous is the assessment evaluation, and it weights the comprehensive
score. It is defined as:

$cv=\frac{\sum_{x\in X}\alpha_{x}*|x|}{|X|}$

where X is the set of possible values used to assess the security controls and
$|X|$ is the number of such different values (i.e., 3 if we consider C, PC,
and NC); $\alpha_{x}$ is the percentage of coverage related to each assessment
value (e.g., $\alpha_{PC}=0.5$ means that the assessor evaluates ”PC” all the
controls in which 50% of the activities have been implemented); $|x|$ is the
number of controls that have been assessed with value “x”.

The aggregation function _f_ in the _comprehensive score_ calculation can be
chosen depending on the relation between the compliance governance and the
technical exposure (e.g., average, min, max). The comprehensive scores
projected onto the technical assessment represent the cyber exposure of the
infrastructure in a more fitting way.

##  5 Validation

We illustrate a case study to evaluate a realistic implementation of the
methodology and its behavior first in a controlled fitting assessment and then
in two controlled biased assessments. Then, we relax the hypothesis of
controlled bias by simulating possible scenarios of random human errors.
Finally, we test limit cases to study the bounds of the methodology.

###  5.1 Case study: controlled assessment ISO/IEC 27001:2013

We consider a healthcare organization (hospital) whose network is reproduced
in the PANACEA emulation environment [17], while the assessment is simulated
to capture a realistic scenario. The infrastructure is represented in Fig. 2
through the MLAG.

Figure 2: Output of scored assessment review methodology.

The human layer is reported in green in the left part of the figure, composed
of the IT department, a resident, two doctors, and an employee simulating a
possible internal attacker. The yellow nodes in the middle part of the graph
represent the access layer in which everyone accesses the systems through
username and password and _doctor1_ has to use a badge to access the device
_n9_. Blue nodes in the right part of the graph compose the network layer
which is a dense network composed of physical devices and several routers.
The MLAG contains a set of vulnerabilities from CVE [19], namely: 2010-1883,
2019-2018, 2009-2412, 2018-4846, 2018-3110. The vulnerabilities in human and
access layers are the following: (i) sharing credential (i.e., the person
shares the personal credentials with others); (ii) no log-out when leaving the
workstation (i.e., the person leaves unattended device(s) logged in with
personal credential); (iii) poor password management (i.e., the person uses
weak passwords and do not update them). The assessment has been conducted
using ISO/IEC 27001:2013 controls because it is common for the organization to
be certified by ISO and it is such that higher assessment scores correspond to
a lower cyber risk. We label the fittest controls-based assessment as _ground
truth_ and in Fig. 2 the assessment values before the review (i.e.,
considering only technical perspective) are reported in red, and the
comprehensive scores after the review are shown in green. The comprehensive
scores are lower than the technical assessment in 94% of the edges with a
range of variation between 10-30% (mean 25%). This highlights that while the
technical assessment would indicate a certain level of cyber risk due to the
presence of specific vulnerabilities, this is not correct. In fact the
comprehensive scores (that consider governance factor across multiple layers)
indicate that the organization is overall not compliant with the standards,
resulting in higher cyber exposure. Considering just the technical perspective
would give a distorted overview of the cyber risk.
Then we introduce two controlled biased assessments to study the behavior for
the following cases: (i) the _conservative assessment_ represents the
assessment performed by an expert that gives the certification in a meticulous
way. The bias is that if something is only partially covered, the related
controls are flagged as “not covered”; (ii) the _not rigorous assessment_
represents the opposite case: it simulates the assessment biased by an auditor
more oriented to flag partially covered controls as “covered”. We perform the
methodology for the assessments using the average as aggregation function for
_governance factor_ and _comprehensive score_. We obtain as result the
distribution of the scores reported in Fig. 3: the ground truth has a mean
score equals to 0.115 and a standard deviation of 0.0746; the conservative
assessment has a mean of 0.0314 and a standard deviation of 0.036; the not
rigorous assessment has a mean of 0.206 and a standard deviation of 0.125. The
trend is the same in all the cases; however, they differ for the values the
scores assume: the box plots in Fig. 3b make clear the difference with the
ground truth. The conservative assessment has some outliers due to the high
number of “NC” that appears with respect to the rest of the assessment. In the
not rigorous assessment, the great amplitude of the box underlines the
underestimation of cyber exposures. A big difference with respect to the
ground truth can be dangerous: it either means that the risk coverage is not
sufficient, therefore several outliers appear (conservative), or that the risk
is underestimated, therefore the distribution of scores is amplified (non-
rigorous). This information (i.e., presence of outliers and amplified
distribution) can be used to detect which parts of the infrastructure have
overestimated or underestimated exposures by considering the line-chart of
Fig. 3a which reports the edges IDs in the x-axis and the comprehensive scores
in the y-axis.

(a)

(b)

Figure 3: Comparative charts of biased assessments with ground truth.

###  5.2 Assessment review sensitivity analysis.

To test the methodology sensitivity we relax the hypothesis of controlled bias
by perturbing the ground truth assessment of a certain percentage of error and
studying how this perturbation affects the methodology scores. We tested 28
cases: seven cases for each percentage of error equal to $15\%$, $45\%$,
$65\%$, and $90\%$. These values are chosen by considering experimental
results that help in differentiating the different cases of errors.

Figure 4: Distribution of the 28 error cases and ground truth.

Fig. 4 represents the trend of the different cases. We notice that the scores
affected by errors are more variable for human and access layers (the first
ones in the x-axis): this is because in such layers we have fewer edges,
therefore the information about controls is more compacted in each edge and
this increases the variability. Instead in the network layer it is more
distributed among the edges, therefore the assessment scores are more uniform.
Also, we can see how the variation of the errors (in green) differs from the
ground truth (in red) in the four cases. In the human and access layers the
errors are detectable because they tend to amplify the value of the scores,
while in the network layer (last edges in the x-axis) the errors are
detectable because they tend to decrease the values of the scores. This means
that it is possible to detect the presence of error with respect to the ground
truth in the vast majority of cases by analyzing the peaks. Thus the
methodology is able to distinguish the degree of bias in the perturbed
assessments from the unbiased one (ground truth).

###  5.3 Scores distribution validation.

We finally remove the hypothesis of perturbed biased assessment by studying
the scores the methodology produces independently from the context. It is
useful to ensure adaptability to a generic context and it allows to understand
the bounds of the scores variability. The data are obtained firstly by
considering all the possible combinations of assessment (Fig. 5.3) for 114
security controls. Then, we considered the three borderline cases, where the
controls are all covered, partially covered, or all not covered (Fig. 5.3).

figureScores trend.

figureBox plots of borderline cases.

In Fig. 5.3 the Gaussian distribution (5.3a) and the box plot (5.3b) of the
scores are reported. They have a mean of 0.117 and a standard deviation of
0.120, with distribution concentrated in the part of the chart with lower
values. The outliers are all located in correspondence with high values: they
can be considered the more secure points of the infrastructure. The presence
of a few of those points indicates that the methodology is oriented to
identify the vulnerable parts of the organization, and this is in line with
expectations due to the applicability of this approach to critical
infrastructures. In Fig. 5.3 the three borderline cases of the possible bias
are reported. For the assessment with all controls covered (Fig. 5.3a) the
distribution is more amplified than the other cases because several security
operations are implemented, therefore the organization is compliant from both
governance and technical perspectives and the scores assume higher values. For
some elements they are still low because some controls deal with critical
parts of the organization or they are too general and difficult to be
compliant with (e.g., managing changes to supplier services is a control that
depends also on third parties). Also, we set the parameter at the minimum
(e.g., $\alpha=0.5$) because we are considering a critical infrastructure,
therefore we tend to be conservative. In the other cases (Fig. 5.3b,5.3c) the
distribution is similar to the case of full coverage, but the mean is halved.
This is a good result because it indicates that the methodology takes the
assessment variation into account. Also, the standard deviation is 0.1 in case
(a), 0.078 in case (b), and 0.016 in case (c), indicating good stability of
the computed scores independently from the context.

##  6 Conclusion

In this paper we proposed a methodology for obtaining a more fit risk
assessment of a generic organization exploiting a multi-layer attack graph.
The methodology has been validated through a case study and statistical tests,
proving its capability to correct cyber risk underestimation due to human
assessor errors or biases. Some limitations can be identified, that will be
the object of further research: the aggregation functions between governance
and technical factors could be more complex than the average we used for the
validation, as well as the ontology models for security controls and
management levels could be not exhaustive in terms of modeled information.
Indeed, we plan as future work (i) to provide an aggregation function able to
capture with suitable weights the governance and technical aspects and the
refinement of the proposed ontologies; (ii) to develop a Visual Analytics [16]
tool for the proposed approach, to better support the application of this
methodology.

## References

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11footnotetext:  School of Mathematical Sciences, Fudan University, Shanghai
200433, P.R. China.

#  The Riemann-Hilbert approach to focusing Kundu-Eckhaus equation with
nonzero boundary conditions

Li-Li Wen1, En-Gui Fan1 Corresponding author and email address:
<EMAIL_ADDRESS>
###### Abstract

In this article, we focus on investigating the focusing Kundu-Eckhaus equation
with nonzero boundary condition. A appropriate two-sheeted Riemann surface is
introduced to map the spectral parameter $k$ into a single-valued parameter
$z$. Starting from the Lax pair of Kundu-Eckhaus equation, two kind of Jost
solutions are construed. Further their asymptotic, analyticity, symmetries as
well as spectral matrix are detailed analyzed. It is shown that the solution
of Kundu-Eckhaus equation with nonzero boundary condition can characterized
with a matrix Riemann-Hilbert problem. Then a formula of $N$-soliton solutions
is derived by solving Riemann-Hilbert problem. As applications, the first-
order explicit soliton solution is obtained.

Keywors: the focusing Kundu-Eckhaus equation; nonzero boundary conditions;
Riemann-Hilbert problem; soliton solution.

##  1 Introduction

It is well-known that the nonlinear Schr$\ddot{o}$dinger equation plays an
important role and achieved great success in physical fields such as nonlinear
optics, nonlinear water waves, plasma physics and so on. But
Schr$\ddot{o}$dinger equation with higher-order nonlinear terms, such as the
self-steepening and self-frequency shift, has an significant effect in
specific physical background, for example, optic fiber communication and Bose-
Einstein condensates [1, 2].

The modified Gross-Pitaevskii equation [3]

$iu_{t}+u_{xx}+2f(t)|u|^{2}u+\frac{1}{2}\delta^{2}x^{2}u+\kappa_{1}|u|^{4}u+i\kappa_{2}(|u|^{2})_{x}u=0$
(1)

was firstly proposed by Wadati for describing the interactions between two-
body and three-body on the condensates. The variable parameters $\kappa_{1}$
and $\kappa_{2}$ are constants. The two-body scattering lengths $f(t)$ can be
adjusted by Feshbach resonance. Neither $\kappa_{1}=0$ nor $\kappa_{2}=0$,
equation (1) is suitable for Bose-Einstein condensates with higher densities.
Reversely, if $\kappa_{1}=0$ and $\kappa_{2}=0$, equation (2) only for lower
densities in Bose-Einstein condensates. This situation can be described by
Gross-Pitaevskii equation [4]

$iu_{t}+u_{xx}+2f(t)|u|^{2}u+\frac{1}{2}\delta^{2}x^{2}u=0.$ (2)

If $f(t)\in\mathbb{R}$ and independent of $t$, the equation (1) reduce to the
Kundu-Eckhaus (KE) equation [5]

$iu_{t}+u_{xx}-2\sigma|u|^{2}u+4\beta^{2}|u|^{4}u+4i\beta\sigma(|u|^{2})_{x}u=0,$
(3)

with $u(x,t)$ being the complex potential function of spatial $x$ and temporal
$t$ $u(x,t):\mathbb{R}^{2}\mapsto\mathbb{C}$. This equation is called the
defocusing KE equation as $\sigma=1$ and focusing KE equation as $\sigma=-1$.
The KE equation (3) was firstly put forward by Kundu to investigate the
Landau-Lifshitz equations and derivative nonlinear Schr$\ddot{o}$dinger type
equations [5]. For the special case $\beta=0$, the KE equation (3) reduces to
nonlinear Schr$\ddot{o}$dinger equation. In recent years, the KE equation (3)
has been investigated via different methods, for example, the long time
asymptotic [7, 8, 6, 9], the higher-order rogue wave solutions [10], rogue
waves in a chaotic wave field [11], the Darboux transformation [12, 13],
integrable discretizations [14]. Recently, the KE equation with zero boundary
conditions was investigated by using Riemann-Hilbert method [8].

To our knowledge, there is still no work on investigating KE equation (3) with
nonzero boundary conditions by using inverse scattering transformation or
Riemannn-Hilbert method. In this paper, we would like to investigate the
soliton solution of focusing KE equation (3) with the following nonzero
boundary conditions

$\lim_{x\rightarrow\pm\infty}u(x,t)=u_{\pm}e^{2it(q_{0}^{2}-2\beta^{2}q_{0}^{4})+i\theta_{\pm}},$
(4)

where $|u_{\pm}|=u_{0}>0$ and $u_{0}$ and $\theta_{\pm}$ are constants,
$\theta_{\pm}$ is the arguments of $q_{\pm}$.

As we all known, the inverse scattering transform method plays an important
role for finding the exact solutions of completely integrable systems [15].
The Riemann-Hilbert method as a new version of inverse scattering transform
method streamline the research process and preferred by researchers. And
Riemann-Hilbert method can be used to investigate the soliton solutions [16]
and the long-time asymptotic of integrable systems [7]. Especially, in recent
years, it has become a hot topic to investigate integrable systems with
nonzero boundary conditions [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28].

This work is organized as follows. In section 2, we analyze the spectral
problem via introduce a transformation. And we introduced a appropriate
Riemann surface for the single-valued function of the spectral parameter, that
is $k$-plane mapped into $z$-plane. Section 3-5, we obtained the asymptotic,
analyticity and symmetries of Jost solution and scattering matrix, which are
used to get a Riemann-Hilbert problem in section 6. Section 7, we analyze the
discrete spectrum and residue conditions which are used to solve the Riemann-
Hilbert problem. Section 8, we establish the connection between the solution
of KE equation and the solution of Rimann-Hilbert problem. A formula of the
$N$-soliton solution of KE equation is obtained by using the Rimann-Hilbert
problem. A conclusion is given in section 9.

##  2 Spectral Analysis

It is well-known that the focusing KE equation ($\sigma=-1$) admits Lax pair

$(\partial_{x}-\mathcal{U})\phi=0,\ \ (\partial_{t}-\mathcal{V})\phi=0,$ (5)

where

$\displaystyle\mathcal{U}=-ik\sigma_{3}+i\beta|q|^{2}\sigma_{3}+U,\hskip
14.22636ptU=\left(\begin{array}[]{cc}0&u\\\
-\overline{u}&0\end{array}\right),\ \ \
\sigma_{3}=\left(\begin{array}[]{cc}1&0\\\ 0&-1\end{array}\right)$ (10)
$\displaystyle\begin{split}\mathcal{V}=&-2ik^{2}\sigma_{3}+2kU+2\beta|u|^{2}U+i\left(|u|^{2}+4\beta^{2}|u|^{4}\right)\sigma_{3}-iU_{x}\sigma_{3}\\\
&-\beta\left(UU_{x}-U_{x}U\right),\end{split}$

where $k$ is the spectral parameter and $\overline{u}$ denotes the complex
conjugation of $u$.

For convenience of using Riemann-Hilbert method, we first deal with Lax par
(5) and the boundary condition (4). By making transformation

$q(x,t)=u(x,t)e^{-2it(q_{0}^{2}-2\beta^{2}q_{0}^{4})},$

the focusing KE equation can be reduced to the form

$iq_{t}+q_{xx}+2(|q|^{2}+2\beta^{2}|q|^{4}-q_{0}^{2}-2\beta^{2}q_{0}^{4})q-4i\beta(|q|^{2})_{x}q=0,\quad\beta\in\mathbb{R},$
(11)

and corresponding boundary condition (4) becomes

$\lim_{x\rightarrow\pm\infty}q(x,t)=q_{\pm}=q_{0}e^{i\theta_{\pm}},$

where we have denote $u_{\pm}$ and $u_{0}$ as $q_{\pm}$ and $q_{0}$
respectively.

The equation (11) admits the following Lax pair

$\displaystyle(\partial_{x}-\mathcal{U})\phi=0,$ (12a)
$\displaystyle(\partial_{t}-\mathcal{V})\phi=0,$ (12b)

where

$\displaystyle\mathcal{U}=-ik\sigma_{3}+i\beta|q|^{2}\sigma_{3}+Q,\hskip
14.22636ptQ=\left(\begin{array}[]{cc}0&q\\\
-\overline{q}&0\end{array}\right),$ (15)
$\displaystyle\begin{split}\mathcal{V}=&-2ik^{2}\sigma_{3}+2kQ+2\beta|q|^{2}Q+i\left(|q|^{2}+4\beta^{2}|q|^{4}-q_{0}^{2}-2\beta^{2}q_{0}^{4}\right)\sigma_{3}\\\
&-iQ_{x}\sigma_{3}-\beta\left(QQ_{x}-Q_{x}Q\right).\end{split}$

Let $x\rightarrow\pm\infty$ and by using boundary condition (4), we obtain the
following asymptotic Lax pair

$\displaystyle(\partial_{x}-\mathcal{U}_{\pm})\phi=0,\quad\mathcal{U}_{\pm}=\lim_{x\rightarrow\pm\infty}\mathcal{U}=\left(-ik+i\beta
q_{0}^{2}\right)\sigma_{3}+Q_{\pm},$ (16a)
$\displaystyle(\partial_{t}-\mathcal{V}_{\pm})\phi=0,\quad\mathcal{V}_{\pm}=\lim_{x\rightarrow\pm\infty}\mathcal{V}=\left(-2ik^{2}+2i\beta^{2}q_{0}^{4}\right)\sigma_{3}+\left(2k+2\beta
q_{0}^{2}\right)Q_{\pm},$ (16b)

where

$\displaystyle
Q_{\pm}=\lim_{x\rightarrow\pm\infty}Q=\left(\begin{array}[]{cc}0&q_{\pm}\\\
-\overline{q}_{\pm}&0\end{array}\right).$ (19)

it is obvious that $\mathcal{V}_{\pm}$ and $\mathcal{U}_{\pm}$ possess the
linear relationship

$\mathcal{V}_{\pm}=\left(2k+2\beta q_{0}^{2}\right)\mathcal{U}_{\pm}.$

The fundamental matrix solution of asymptotic spectral problem (16) can be
obtained as

$\phi_{\pm}(x,t,k)=\Xi_{\pm}(k)e^{-i\theta(x,t,k)\sigma_{3}},\quad\quad
k\neq\beta q_{0}^{2}\pm iq_{0},$ (20)

where

$\theta(x,t,k)=\lambda\left[x+(2k+2\beta
q_{0}^{2})t\right],\quad\lambda=\sqrt{(k-\beta q_{0}^{2})^{2}+q_{0}^{2}},$
$\displaystyle\Xi_{\pm}=\left(\begin{array}[]{cc}1&\frac{-iq_{\pm}}{\lambda+(k-\beta
q_{0}^{2})}\\\ \frac{-i\bar{q}_{\pm}}{\lambda+(k-\beta
q_{0}^{2})}&1\end{array}\right).$ (23)

Since $\lambda$ is doubly branched with branch points are $k=\beta
q_{0}^{2}\pm iq_{0}$, it is necessary to introduce a two-sheeted Riemann
surface to such that $\lambda$ is a single-valued function on each sheet.

Denote $h=k-\beta q_{0}^{2}$ and let

$h+iq_{0}=\rho_{1}e^{i\theta_{1}},\quad h-iq_{0}=\rho_{2}e^{i\theta_{2}},\ \
-\frac{\pi}{2}\leq\theta_{1},\theta_{2}\leq\frac{3\pi}{2},$

one can rewrite $\lambda$ on each sheet as

$\lambda_{\mathrm{I}}=\sqrt{\rho_{1}\rho_{2}}e^{i\frac{\theta_{1}+\theta_{2}}{2}},\quad\lambda_{\mathrm{II}}=-\lambda_{\mathrm{I}}=\sqrt{\rho_{1}\rho_{2}}e^{i(\frac{\theta_{1}+\theta_{2}}{2})+i\pi}.$

And the branch cut of the Riemann surface is the segment $[-iq_{0},iq_{0}]$ in
the complex $h$-plane.

Now we introduce a uniformization variable

$z=\lambda+h,$ (24)

then its inverse transformation gives

$\lambda=\frac{1}{2}\big{(}z+q_{0}^{2}/z\big{)},\quad
h=\frac{1}{2}\big{(}z-q_{0}^{2}/z\big{)},\quad
k=\frac{1}{2}\big{(}z-q_{0}^{2}/z\big{)}+\beta q_{0}^{2}.$ (25)

Further we can show the following relations between the Riemann surface, the
$h$-plane and the $z$-plane.

  * -

The region where $\mathrm{Im}\lambda>0$ come from the upper-half plane of the
sheet-$\mathrm{I}$ and the lower-half plane of the sheet-$\mathrm{II}$. The
region where $\mathrm{Im}\lambda<0$ come from the upper-half plane of the
sheet-$\mathrm{II}$ and the lower-half plane of the sheet-$\mathrm{I}$.

  * -

On the sheet-$\mathrm{I}$, $z\rightarrow\infty$ as $h\rightarrow\infty$, and
on the sheet-$\mathrm{II}$, $z\rightarrow 0$ as $h\rightarrow\infty$.

  * -

The real $\lambda$ (real $k$) axes is mapped into the real $z$ axes.

  * -

The branch cut $[-iq_{0},iq_{0}]$ is mapped into the circle $C_{0}$ of the
radius $q_{0}$ in $z$-plane.

  * -

The sheet-$\mathrm{I}$ and sheet-$\mathrm{II}$, except for the branch cut, are
mapped into the exterior and the interior of $C_{0}$, respectively.

The jump contour in the complex $z$-plane is denoted by $\Sigma=\mathbb{R}\cup
C_{0}$. The gray and white regions in Fig.1 denote $D^{+}$ and $D^{-}$,
respectively

$\begin{split}&D^{+}=\\{z\in\mathbb{C}|\mathrm{Im}\lambda=\left(|z|^{2}-q_{0}^{2}\right)\mathrm{Im}z>0\\},\\\
&D^{-}=\\{z\in\mathbb{C}|\mathrm{Im}\lambda=\left(|z|^{2}-q_{0}^{2}\right)\mathrm{Im}z<0\\}.\end{split}$

(a) (b)

(c) (d)

Based on the above results, we can rewrite the fundamental matrix solution
(20) as

$\phi_{\pm}(x,t,z)=\Xi_{\pm}(z)e^{-i\theta(x,t,z)\sigma_{3}},\quad z\neq
iq_{0}$

where

$\displaystyle\begin{split}&\Xi_{\pm}(z)=\left(\begin{array}[]{cc}1&\frac{-iq_{\pm}}{z}\\\
\frac{-i\bar{q}_{\pm}}{z}&1\end{array}\right)=\mathrm{I}-\frac{i}{z}\sigma_{3}Q_{\pm},\\\
&\theta(x,t,z)=\frac{1}{2}\big{(}z+\frac{q_{0}^{2}}{z}\big{)}\left\\{x+\Big{[}\big{(}z-\frac{q_{0}^{2}}{z}\big{)}+4\beta
q_{0}^{2}\Big{]}t\right\\}.\\\ \end{split}$

Direct calculation shows that

$\begin{split}&\det\Xi_{\pm}=1+\frac{q_{0}^{2}}{z^{2}}\triangleq\gamma\neq 0,\
\ \
\Xi_{\pm}^{-1}=\frac{1}{\gamma}\big{(}\mathrm{I}+\frac{i}{z}\sigma_{3}Q_{\pm}\big{)}.\\\
\end{split}$

##  3 Jost Solutions

The Lax pair (12) can be rewrite as

$\displaystyle(\partial_{x}-\mathcal{U}_{\pm}-\Delta\hat{Q}_{\pm})\phi=0,$
(26a)
$\displaystyle(\partial_{t}-\mathcal{V}_{\pm}-\Delta\hat{R}_{\pm})\phi=0,$
(26b)

where

$\displaystyle\begin{split}&\Delta\hat{Q}_{\pm}=i\beta\left(|q|^{2}-q_{0}^{2}\right)\sigma_{3}+\Delta
Q_{\pm},\quad\Delta\hat{R}_{\pm}=\hat{R}-\hat{R}_{\pm},\\\ &\Delta
Q_{\pm}=Q-Q_{\pm},\quad\hat{R}_{\pm}=2kQ_{\pm}+2\beta
q_{0}^{2}Q_{\pm}+2i\beta^{2}q_{0}^{4}\sigma_{3},\\\
&\begin{split}\hat{R}=&2kQ+2\beta|q|^{2}Q+i\left(|q|^{2}+4\beta^{2}|q|^{4}-q_{0}^{2}-2\beta^{2}q_{0}^{4}\right)\sigma_{3}-iQ_{x}\sigma_{3}\\\
&-\beta\left(QQ_{x}-Q_{x}Q\right).\end{split}\end{split}$

Now, one can define the Jost solutions as the simultaneous solutions of Lax
pair (12) such that

$\Psi_{\pm}(x,t,z)\sim\Xi_{\pm}(z)e^{-i\theta(x,t,z)\sigma_{3}},\quad
z\in\Sigma,\quad x\rightarrow\pm\infty,$ (27)

and the modified Jost solutions

$\displaystyle\begin{split}e^{\int_{-\infty}^{x}i\beta\left(|q|^{2}-q_{0}^{2}\right)dy\hat{\sigma}_{3}}\mu_{\pm}(x,t,z)e^{-\int_{x}^{+\infty}i\beta\left(|q|^{2}-q_{0}^{2}\right)dy\sigma_{3}}=&\Psi_{\pm}(x,t,z)e^{i\theta(x,t,z)\sigma_{3}}\\\
=&\eta_{\pm}(x,t,z),\end{split}$ (28)

Then the Lax pair (26) can be rewritten as

$\displaystyle\begin{split}&\left(\Xi_{\pm}^{-1}\eta_{\pm}(x,t,z)\right)_{x}=-i\lambda\left[\sigma_{3},\Xi_{\pm}^{-1}\eta_{\pm}(x,t,z)\right]+\Xi_{\pm}^{-1}\Delta\hat{Q}_{\pm}\eta_{\pm}(x,t,z),\\\
&\begin{split}\left(\Xi_{\pm}^{-1}\eta_{\pm}(x,t,z)\right)_{t}=&-i\lambda\left(2k+2\beta
q_{0}^{2}\right)\left[\sigma_{3},\Xi_{\pm}^{-1}\eta_{\pm}(x,t,z)\right]\\\
&+\Xi_{\pm}^{-1}\Delta\hat{R}_{\pm}\eta_{\pm}(x,t,z).\end{split}\end{split}$
(29)

It is easily known that Lax pair (29) can be written in full derivative form

$d\left(e^{i\theta(x,t,z)\hat{\sigma}_{3}}\Xi_{\pm}^{-1}\eta_{\pm}(x,t,z)\right)=e^{i\theta(x,t,z)\hat{\sigma}_{3}}\left(V_{1}dx+V_{2}dt\right),$
(30)

where

$\displaystyle\begin{split}&V_{1}=\Xi_{\pm}^{-1}\Delta\hat{Q}_{\pm}\eta_{\pm}(x,t,z),\\\
&V_{2}=\Xi_{\pm}^{-1}\Delta\hat{R}_{\pm}\eta_{\pm}(x,t,z).\end{split}$

We can obtain the Volterra integral equations

$\displaystyle\begin{split}&\begin{split}\mu_{-}(x,t,z)=\Xi_{-}+\int_{-\infty}^{x}&e^{-\int_{-\infty}^{x}i\beta\left(|q|^{2}-q_{0}^{2}\right)dy\sigma_{3}}\Xi_{-}e^{i\lambda(x^{\prime}-x)\hat{\sigma}_{3}}\\\
&\Xi_{-}^{-1}\Delta\hat{Q}_{-}e^{\int_{-\infty}^{x^{\prime}}i\beta\left(|q|^{2}-q_{0}^{2}\right)dy\sigma_{3}}\mu_{-}dx^{\prime},\end{split}\\\
&\begin{split}\mu_{+}(x,t,z)=\Xi_{+}-\int_{x}^{+\infty}&e^{-\int_{-\infty}^{x}i\beta\left(|q|^{2}-q_{0}^{2}\right)dy\sigma_{3}}\Xi_{+}e^{i\lambda(x^{\prime}-x)\hat{\sigma}_{3}}\\\
&\Xi_{+}^{-1}\Delta\hat{Q}_{+}e^{\int_{-\infty}^{x^{\prime}}i\beta\left(|q|^{2}-q_{0}^{2}\right)dy\sigma_{3}}\mu_{+}dx^{\prime}.\end{split}\end{split}$
(31)

It can be shown that the first column of $\mu_{-}$ is analytically extended to
$D^{+}$ and the second column of $\mu_{-}$ is analytically extended to
$D^{-}$. Similarly, the first column of $\mu_{+}$ can be analytically extended
to $D^{-}$ and the second column of $\mu_{+}$ can be analytically extended to
$D^{+}$. It can be summarized as follows

$\begin{split}D^{+}:\quad\mu_{-,1},\quad\mu_{+,2};\\\
D^{-}:\quad\mu_{-,2},\quad\mu_{+,1},\end{split}$

where the subscripts ‘1’ and ‘2’ identify the columns of matrix.

Consider the Laurent expansion of $\Xi_{\pm}^{-1}\eta_{\pm}(x,t,z)$ in system
(29)

$\displaystyle\Xi_{\pm}^{-1}\eta_{\pm}(x,t,z)=\alpha^{(0)}+\frac{\alpha^{(1)}}{z}+\frac{\alpha^{(2)}}{z^{2}}+O\big{(}\frac{1}{z^{3}}\big{)},\quad\quad
z\rightarrow\infty,$ (32a)
$\displaystyle\Xi_{\pm}^{-1}\eta_{\pm}(x,t,z)=\eta^{(0)}+\eta^{(1)}z+\eta^{(2)}z^{2}+O\left(z^{3}\right),\quad
z\rightarrow 0.$ (32b)

where $\alpha^{(0)},\alpha^{(1)},\alpha^{(2)},\cdots$ and
$\eta^{(0)},\eta^{(1)},\eta^{(2)},\cdots$are independent of $z$. Substituting
the above expansion into (29) and comparing the coefficients of $z^{n}(n=0,\pm
1,\pm 2,\cdots)$, we obtain the following equations

$\displaystyle\begin{split}&\left[\sigma_{3},\alpha^{(0)}\right]=0,\\\
&\alpha^{(0)}_{x}=-\frac{i}{2}\left[\sigma_{3},\alpha^{(1)}\right]+i\beta\left(|q|^{2}-q_{0}^{2}\right)\sigma_{3}\alpha^{(0)}+\Delta
Q_{\pm}\alpha^{(0)},\\\
&\begin{split}\alpha^{(1)}_{x}=&-\frac{i}{2}q_{0}^{2}\left[\sigma_{3},\alpha^{(0)}\right]-\frac{i}{2}\left[\sigma_{3},\alpha^{(2)}\right]+\beta\left(|q|^{2}-q_{0}^{2}\right)\left[\sigma_{3},\sigma_{3}Q_{\pm}\right]\alpha^{(0)}\\\
&+i\left[\sigma_{3}Q_{\pm},\Delta
Q_{\pm}\right]\alpha^{(0)}+i\beta\left(|q|^{2}-q_{0}^{2}\right)\sigma_{3}\alpha^{(1)}+\Delta
Q_{\pm}\alpha^{(1)},\end{split}\end{split}$
$\displaystyle\begin{split}&\left[\sigma_{3},\eta^{(0)}\right]=0,\\\
&\eta^{(0)}_{x}=-\frac{i}{2}q_{0}^{2}\left[\sigma_{3},\eta^{(1)}\right]+\frac{1}{q_{0}^{2}}i\beta\left(|q|^{2}-q_{0}^{2}\right)\sigma_{3}\eta^{(0)}+\frac{1}{q_{0}^{2}}\sigma_{3}Q_{\pm}\Delta
Q_{\pm}\sigma_{3}Q_{\pm}\eta^{(0)},\\\
&\begin{split}\eta^{(1)}_{x}=&-\frac{i}{2}\left[\sigma_{3},\eta^{(0)}\right]-\frac{i}{2}q_{0}^{2}\left[\sigma_{3},\eta^{(2)}\right]+\frac{1}{q_{0}^{2}}\beta\left(|q|^{2}-q_{0}^{2}\right)\left[\sigma_{3},\sigma_{3}Q_{\pm}\right]\eta^{(0)}\\\
&+\frac{i}{q_{0}^{2}}\left[\sigma_{3}Q_{\pm},\Delta
Q_{\pm}\right]\eta^{(0)}+\frac{1}{q_{0}^{2}}i\beta\left(|q|^{2}-q_{0}^{2}\right)\sigma_{3}Q_{\pm}^{2}\eta^{(1)}\\\
&+\frac{1}{q_{0}^{2}}\sigma_{3}Q_{\pm}\Delta
Q_{\pm}\sigma_{3}Q_{\pm}\eta^{(1)},\end{split}\end{split}$

from which, we can derive the following results

$\displaystyle\begin{split}&\alpha^{(0)}=e^{-\int_{x}^{+\infty}i\beta\left(|q|^{2}-q_{0}^{2}\right)dy\sigma_{3}},\\\
&\alpha^{(1)}_{o}=-i\sigma_{3}\Delta Q_{\pm}\alpha^{(0)}_{d},\\\
&\alpha^{(1)}_{d,x}=i\left[\sigma_{3}Q_{\pm},\Delta
Q_{\pm}\right]\alpha^{(0)}_{d}+i\beta\left(|q|^{2}-q_{0}^{2}\right)\sigma_{3}\alpha^{(1)}_{d}-i\Delta
Q_{\pm}\sigma_{3}\Delta Q_{\pm}\alpha^{(0)}_{d}.\end{split}$

and

$\displaystyle\begin{split}&\eta^{(0)}=e^{\int_{x}^{+\infty}i\beta\left(|q|^{2}-q_{0}^{2}\right)dy\sigma_{3}},\\\
&\begin{split}\eta^{(1)}_{d,x}=&\frac{i}{q_{0}^{2}}\left[\sigma_{3}Q_{\pm},\Delta
Q_{\pm}\right]\eta^{(0)}_{d}-i\beta\left(|q|^{2}-q_{0}^{2}\right)\sigma_{3}\eta^{(1)}_{d}\\\
&+\frac{1}{q_{0}^{4}}\sigma_{3}Q_{\pm}\Delta Q_{\pm}\sigma_{3}\Delta
Q_{\pm}\sigma_{3}Q_{\pm}\eta^{(0)}_{d}.\end{split}\end{split}$

And the asymptotic of the modified Jost solutions for $z\rightarrow\infty$ and
$z\rightarrow 0$ are respectively derived as

$\displaystyle\begin{split}&\begin{split}\mu_{\pm}=&I-\frac{i}{z}e^{-\int_{-\infty}^{x}i\beta\left(|q|^{2}-q_{0}^{2}\right)dy\hat{\sigma}_{3}}\sigma_{3}Q_{\pm}\\\
&+\frac{1}{z}e^{-\int_{-\infty}^{x}i\beta\left(|q|^{2}-q_{0}^{2}\right)dy\hat{\sigma}_{3}}\alpha^{(1)}e^{\int_{x}^{+\infty}i\beta\left(|q|^{2}-q_{0}^{2}\right)dy\sigma_{3}}+O\big{(}\frac{1}{z^{2}}\big{)},\quad
z\rightarrow\infty,\end{split}\\\
&\mu_{\pm}=-\frac{i}{z}\sigma_{3}Q_{\pm}+O\left(1\right),\quad z\rightarrow
0.\end{split}$ (33)

##  4 Scattering Matrix and Asymptotic

It is easy to check that $\mathrm{tr}\mathcal{U}=\mathrm{tr}\mathcal{V}=0$,
then by using Abel formula, we have

$(\det\Psi_{\pm})_{x}=(\det\Psi_{\pm})_{t}=0,$

which implies that

$\displaystyle\det\Psi_{\pm}=\det Y_{\pm}=\gamma$ (34)

by using asymptotic at $x\rightarrow\pm\infty$. Since $\Psi_{+}$ and
$\Psi_{-}$ are the fundamental solutions of the spectral problem (12), they
satisfy the following linear relationship

$\Psi_{+}(z)=\Psi_{-}(z)S(z),\quad z\in\Sigma\setminus\\{\pm iq_{0}\\},$ (35)

where $S(z)=(s_{ij}(z))_{2\times 2}$ is called spectral matrix. From (34), we
know that $\det S=1$. The relation formula (35) can be expanded as

$\displaystyle\Psi_{+,1}=s_{11}\Psi_{-,1}+s_{21}\Psi_{-,2},\quad\Psi_{+,2}=s_{12}\Psi_{-,1}+s_{22}\Psi_{-,2}.$
(36)

The reflection coefficients are defined as

$\rho(z)=\frac{s_{21}}{s_{11}},\quad\tilde{\rho}(z)=\frac{s_{12}}{s_{22}}.$
(37)

According to (36) the scattering coefficients have the following Wronskian
representations

$\displaystyle
s_{11}=\frac{\mathrm{Wr}\left(\Psi_{+,1},\Psi_{-,2}\right)}{\gamma},\quad
s_{12}=\frac{\mathrm{Wr}\left(\Psi_{+,2},\Psi_{-,2}\right)}{\gamma},$ (38a)
$\displaystyle
s_{21}=\frac{\mathrm{Wr}\left(\Psi_{-,1},\Psi_{+,1}\right)}{\gamma},\quad
s_{22}=\frac{\mathrm{Wr}\left(\Psi_{-,1},\Psi_{+,2}\right)}{\gamma}.$ (38b)

The equation (38) implies that $s_{11}$ is analytic in $D^{-}$ and $s_{22}$ is
analytic in $D^{+}$. However, $s_{12}$ and $s_{22}$ are just continuous on
$\Sigma$.

Proposition 1. The asymptotic behaviors of the scattering matrix $S(z)$ are
given as follows

$\displaystyle S(z)=I+O\big{(}\frac{1}{z}\big{)},\quad z\rightarrow\infty,$
(39a) $\displaystyle
S(z)=\mathrm{diag}\left(q_{-}/q_{+},q_{+}/q_{-}\right)+O\left(z\right),\quad
z\rightarrow 0.$ (39b)

###### Proof.

By using (28), (38) and the asymptotic behaviors of $\mu_{\pm}$, we can prove
that

As $z\rightarrow\infty$,

$\displaystyle\begin{split}s_{11}=&\frac{Wr\left(\Psi_{+,1},\Psi_{-,2}\right)}{\gamma}=\frac{1}{1+\frac{q_{0}^{2}}{z^{2}}}\det\left(\begin{array}[]{cc}\mu_{+,11}&\mu_{-,12}\\\
\mu_{+,21}&\mu_{-,22}\end{array}\right)\\\
=&\det\left(\begin{array}[]{cc}1+O(\frac{1}{z})&O(\frac{1}{z})\\\
O(\frac{1}{z})&1+O(\frac{1}{z})\end{array}\right)\big{(}1-\frac{q_{0}^{2}}{z^{2}}+\frac{q_{0}^{4}}{z^{4}}-\cdots\big{)}\\\
=&\big{(}1+O\big{(}\frac{1}{z}\big{)}\big{)}\big{(}1-\frac{q_{0}^{2}}{z^{2}}+\cdots\big{)}\\\
=&1+O\big{(}\frac{1}{z}\big{)}.\end{split}$

As $z\rightarrow 0$,

$\displaystyle\begin{split}s_{11}=&\frac{Wr\left(\Psi_{+,1},\Psi_{-,2}\right)}{\gamma}=\frac{1}{1+\frac{q_{0}^{2}}{z^{2}}}\det\left(\begin{array}[]{cc}\mu_{+,11}&\mu_{-,12}\\\
\mu_{+,21}&\mu_{-,22}\end{array}\right)\\\
=&\det\left(\begin{array}[]{cc}O(1)&-\frac{i}{z}q_{-}\\\
\frac{i}{z}\overline{q}_{+}&O(1)\end{array}\right)\frac{z^{2}}{q_{0}^{2}}\big{(}1-\frac{z^{2}}{q_{0}^{2}}+\cdots\big{)}\\\
=&\big{(}O(1)+\frac{1}{z^{2}}q_{-}\overline{q}_{+}\big{)}\big{(}\frac{z^{2}}{q_{0}^{2}}-\frac{z^{4}}{q_{0}^{4}}+\cdots\big{)}\\\
=&\frac{q_{-}\overline{q}_{+}}{q_{0}^{2}}+O\left(z\right)=\frac{q_{-}}{q_{+}}+O(z).\end{split}$

The asymptotic behaviors of $s_{22}$, $s_{12}$, and $s_{21}$ can also be
derived by the similar way. ∎

##  5 Symmetries

For the focusing KE equation with nonzero boundary conditions, the Jost
functions $\Psi_{\pm}(z)$ and spectral matrix $S(z)$ possess two kinds of
symmetries.

###  5.1 First Symmetry

Here we consider the symmetries between two points $z\mapsto\overline{z}$
(upper/lower half plane).
Proposition 2. For $z\in\Sigma$,
(1) The Jost solutions satisfy the symmetries

$\displaystyle\Psi_{\pm}(z)=\sigma_{2}\overline{\Psi_{\pm}(\overline{z})}\sigma_{2},$
(40a)
$\displaystyle\Psi_{\pm,1}(z)=i\sigma_{2}\overline{\Psi_{\pm,2}(\overline{z})},$
(40b)
$\displaystyle\Psi_{\pm,2}=-i\sigma_{2}\overline{\Psi_{\pm}(\overline{z})}.$
(40c)

(2) The scattering matrix satisfy the symmetries

$\displaystyle S(z)=\sigma_{2}\overline{S(\overline{z})}\sigma_{2},$ (41a)
$\displaystyle s_{11}(z)=\overline{s_{22}(\overline{z})},$ (41b)
$\displaystyle s_{12}(z)=-\overline{s_{21}(\overline{z})}.$ (41c)

(3) The reflection coefficient satisfy the symmetries

$\displaystyle\rho(z)=-\overline{\tilde{\rho}(\overline{z})}.$ (42)

###### Proof.

(1) The $\mathcal{U}$ and $\mathcal{V}$ in the Lax pair (12) with the
following symmetries on $z$-plane

$\overline{\mathcal{U}(\overline{z})}=\sigma_{2}\mathcal{U}(z)\sigma_{2},\quad\overline{\mathcal{V}(\overline{z})}=\sigma_{2}\mathcal{V}(z)\sigma_{2},$

by which, we can show that

$\displaystyle\begin{split}&\big{(}\sigma_{2}\overline{\Psi_{\pm}(\overline{z})}\sigma_{2}\big{)}_{x}=\mathcal{U}(z)\big{(}\sigma_{2}\overline{\Psi_{\pm}(\overline{z})}\sigma_{2}\big{)},\\\
&\big{(}\sigma_{2}\overline{\Psi_{\pm}(\overline{z})}\sigma_{2}\big{)}_{t}=\mathcal{V}(z)\big{(}\sigma_{2}\overline{\Psi_{\pm}(\overline{z})}\sigma_{2}\big{)},\end{split}$

Further, by suing asymptotic

$\Psi_{\pm}(z),\
\sigma_{2}\overline{\Psi_{\pm}(\overline{z})}\sigma_{2}\sim\Xi_{\pm}(z)e^{-i\lambda\left[x+(2k+2\beta
q_{0}^{2})t\right]\sigma_{3}},\ \ x\rightarrow\infty,$

we obtain the symmetry (40).
(2) On the basis of (35),

$\displaystyle\begin{split}S(z)=&\Psi_{-}^{-1}(z)\Psi_{+}(z)\\\
=&\sigma_{2}\overline{\Psi_{-}(\overline{z})}^{-1}\sigma_{2}\sigma_{2}\overline{\Psi_{+}(\overline{z})}\sigma_{2}\\\
=&\sigma_{2}\overline{\Psi_{-}(\overline{z})}^{-1}\overline{\Psi_{+}(\overline{z})}\sigma_{2}\\\
=&\sigma_{2}\overline{S(\overline{z})}\sigma_{2}.\end{split}$

Then (44) can be obtained.
(3)

$\rho(z)=\frac{s_{21}(z)}{s_{11}(z)}=-\frac{\overline{s_{12}(\overline{z})}}{\overline{s_{22}(\overline{z})}}=-\overline{\tilde{\rho}(\overline{z})}.$

∎

###  5.2 Second Symmetry

Here we consider the symmetries between two points
$z\mapsto-\frac{q_{0}^{2}}{z}$ (outside/inside of the circle $C_{0})$.
Proposition 3. For $z\in\Sigma$,
(1) The Jost solutions satisfy the following symmetries

$\displaystyle\Psi_{\pm}(z)=-\frac{i}{z}\Psi_{\pm}\big{(}-\frac{q_{0}^{2}}{z}\big{)}\sigma_{3}Q_{\pm},$
(43a)
$\displaystyle\Psi_{\pm,1}(z)=-\frac{i}{z}\overline{q}_{\pm}\Psi_{\pm,2}\big{(}-\frac{q_{0}^{2}}{z}\big{)},$
(43b)
$\displaystyle\Psi_{\pm,2}(z)=-\frac{i}{z}q_{\pm}\Psi_{\pm,1}\big{(}-\frac{q_{0}^{2}}{z}\big{)},$
(43c)

(2) The scattering matrix satisfies the symmetries

$\displaystyle
S(z)=\left(\sigma_{3}Q_{-}\right)^{-1}S\big{(}-\frac{q_{0}^{2}}{z}\big{)}\sigma_{3}Q_{+},$
(44a) $\displaystyle
s_{11}\big{(}-\frac{q_{0}^{2}}{z}\big{)}=\frac{q_{-}}{q_{+}}s_{22}(z),$ (44b)
$\displaystyle
s_{12}\big{(}-\frac{q_{0}^{2}}{z}\big{)}=\frac{q_{+}}{\overline{q}_{-}}s_{21}(z).$
(44c)

(3) The reflection coefficient satisfies the symmetries

$\rho(-\frac{q_{0}^{2}}{z})=-\frac{\overline{q}_{-}}{q_{-}}\overline{\rho(\overline{z})}.$
(45)

###### Proof.

The process of proof is similar with the Proposition 2. ∎

##  6 Riemann-Hilbert Problem

Based on the analytical and asymmetry properties of eigenfunctions $\mu_{\pm}$
and $S(z)$, we derive Riemann-Hilbert problem associated with KE equation with
nonzero boundary conditions.

Proposition 4. Define sectionally meromorphic matrix

$\displaystyle
M(x,t,z)=\begin{cases}\begin{split}&M^{+}(x,t,z)=\left(\begin{array}[]{cc}\mu_{-,1}&\frac{\mu_{+,2}}{s_{22}}\end{array}\right),\quad
z\in D^{+},\\\
&M^{-}(x,t,z)=\left(\begin{array}[]{cc}\frac{\mu_{+,1}}{s_{11}}&\mu_{-,2}\end{array}\right),\quad
z\in D^{-},\end{split}\end{cases}$ (46)

then we have the following Riemann-Hilbert problem

  * -

Analyticity: $M(x,t,z)$ is meromorphic in $D^{+}\cup D^{-}$ and has simple
poles.

  * -

Jump condition:

$M^{-}(x,t,z)=M^{+}(x,t,z)\left(I-G(x,t,z)\right),\quad z\in\Sigma,$

where

$\displaystyle
G(x,t,z)=e^{-i\theta\sigma_{3}}\left(\begin{array}[]{cc}\rho\tilde{\rho}&\tilde{\rho}\\\
-\rho&0\end{array}\right)e^{i\theta\sigma_{3}}.$ (49)

  * -

Asymptotic behavior:

$\displaystyle\begin{split}&M^{\pm}(x,t,z)=I+O\big{(}\frac{1}{z}\big{)},\quad
z\rightarrow\infty\\\ &M^{\pm}(x,t,z)=-\frac{i}{z}\sigma_{3}Q_{-}+O(1).\quad
z\rightarrow 0\end{split}$

###### Proof.

The analyticity can be derived from (36) and the analyticity of $\mu_{\pm}$.

From (36), we can known that

$\displaystyle\begin{split}&\frac{\Psi_{+,1}}{s_{11}}=\left(1-\rho\tilde{\rho}\right)\Psi_{-,1}+\rho\frac{\Psi_{+,2}}{s_{22}},\\\
&\Psi_{-,2}=-\tilde{\rho}\Psi_{-,1}+\frac{\Psi_{+,2}}{s_{22}},\end{split}$

which lead to

$\displaystyle\left(\begin{array}[]{cc}\frac{\Psi_{+,1}}{s_{11}}&\Psi_{-,2}\end{array}\right)=\left(\begin{array}[]{cc}\Psi_{-,1}&\frac{\Psi_{+,2}}{s_{22}}\end{array}\right)\left(\begin{array}[]{cc}1-\rho\tilde{\rho}&-\tilde{\rho}\\\
\rho&1\end{array}\right).$ (54)

Then the jump condition can be derived as

$\displaystyle\begin{split}M^{-}(x,t,z)=&\left(\begin{array}[]{cc}\frac{\mu_{+,1}}{s_{11}}&\mu_{-,2}\end{array}\right)\\\
=&\left(\begin{array}[]{cc}\mu_{-,1}&\frac{\mu_{+,2}}{s_{22}}\end{array}\right)e^{-i\theta\sigma_{3}}\left(\begin{array}[]{cc}1-\rho\tilde{\rho}&-\tilde{\rho}\\\
\rho&1\end{array}\right)e^{i\theta\sigma_{3}}\\\
=&M^{+}(x,t,z)\left(I-G(x,t,z)\right).\end{split}$

Now we proof the asymptotic behavior. From (46), $M^{+}(x,t,z)$ with the
following form

$\displaystyle\begin{split}M^{+}(x,t,z)=&\left(\begin{array}[]{cc}\mu_{-,1}&\mu_{+,2}\end{array}\right)\left(\begin{array}[]{cc}1&0\\\
0&\frac{1}{s_{22}}\end{array}\right).\end{split}$

As $z\rightarrow\infty$,

$\displaystyle\begin{split}M^{+}(x,t,z)=&\left(I+O\big{(}\frac{1}{z}\big{)}\right)\left(\begin{array}[]{cc}1&0\\\
0&\frac{1}{1+O\left(\frac{1}{z}\right)}\end{array}\right)\\\
=&I+O\big{(}\frac{1}{z}\big{)}.\end{split}$

As $z\rightarrow 0$,

$\displaystyle\begin{split}M^{+}(x,t,z)=&\left(\begin{array}[]{cc}O(1)&-\frac{i}{z}q_{+}+O(1)\\\
-\frac{i}{z}\overline{q}_{-}+O(1)&O(1)\end{array}\right)\left(\begin{array}[]{cc}1&0\\\
0&\frac{1}{\frac{q_{+}}{q_{-}}+O(z)}\end{array}\right)\\\
=&-\frac{i}{z}\sigma_{3}Q_{-}+O(1).\end{split}$

The asymptotic behavior of $M^{-}(x,t,z)$ can be derived in the similar way. ∎

##  7 Discrete Spectrum and Residue Conditions

If $s_{22}(z_{n})=0$, we can derive from the forth equation of (38) that the
eigenfunctions $\Psi_{+,2}(x,t,z_{n})$ and $\Psi_{-,1}(x,t,z_{n})$ must be
proportional

$\Psi_{+,2}(x,t,z_{n})=b_{n}\Psi_{-,1}(x,t,z_{n}),$ (55)

where $b_{n}$ is independent of $x$, $t$ and $z$. Let $s_{22}$ has a finite
number of simple zeros $z_{1},z_{2},\cdots,z_{N}$ in
$D^{+}\cap\\{z\in\mathbb{C}|\mathrm{Im}z>0\\}$. According to the symmetry
properties of $S(z)$, we have that

$s_{22}(z_{n})=0\Longleftrightarrow
s_{11}(\overline{z}_{n})=0\Longleftrightarrow
s_{11}(-\frac{q_{0}^{2}}{z_{n}})=0\Longleftrightarrow
s_{22}(-\frac{q_{0}^{2}}{\overline{z}_{n}})=0.$

So the discrete spectrum accumulate the set

$\mathcal{Z}=\\{z_{n},\overline{z}_{n},-\frac{q_{0}^{2}}{z_{n}},-\frac{q_{0}^{2}}{\overline{z}_{n}}\\}_{n=1}^{N}.$

Now we study the residue condition that is very important to solve the
Riemann-Hilbert problem. The equation (55) can be rewritten as

$\mu_{+,2}(z_{n})=b_{n}\mu_{-,1}(z_{n})e^{-2i\theta(z_{n})}.$

Then

$\displaystyle\begin{split}\mathop{\mathrm{Res}}\limits_{z=z_{n}}\left[\frac{\mu_{+,2}(z)}{s_{22}(z)}\right]=&\frac{b_{n}\mu_{-,1}(z_{n})e^{-2i\theta(z_{n})}}{s_{22}^{{}^{\prime}}(z_{n})}=A_{n}\mu_{-,1}(z_{n})e^{-2i\theta(z_{n})},\end{split}$
(56)

where $A_{n}=\frac{b_{n}}{s_{22}^{{}^{\prime}}(z_{n})}$.

If $s_{11}(\overline{z}_{n})=0$, similarly, the first equation of (38) implies
that the eigenfunctions $\Psi_{+,1}(x,t,\overline{z}_{n})$ and
$\Psi_{-,2}(x,t,\overline{z}_{n})$ must be proportional

$\Psi_{+,1}(x,t,\overline{z}_{n})=\tilde{b}_{n}\Psi_{-,2}(x,t,\overline{z}_{n}),$
(57)

(57) also can be rewritten as

$\mu_{+,1}(\overline{z}_{n})=\tilde{b}_{n}\mu_{-,2}(\overline{z}_{n})e^{2i\theta(\overline{z}_{n})}.$

And derived that

$\displaystyle\begin{split}\mathop{\mathrm{Res}}\limits_{z=\overline{z}_{n}}\left[\frac{\mu_{+,1}(z)}{s_{11}(z)}\right]=&\frac{\tilde{b}_{n}\mu_{-,2}(\overline{z}_{n})e^{-2i\theta(\overline{z}_{n})}}{s_{11}^{{}^{\prime}}(\overline{z}_{n})}=\tilde{A}_{n}\mu_{-,2}(\overline{z}_{n})e^{2i\theta(\overline{z}_{n})},\end{split}$
(58)

where
$\tilde{A}_{n}=\frac{\tilde{b}_{n}}{s_{11}^{{}^{\prime}}(\overline{z}_{n})}$.
And it is easy to show that

$\tilde{A}_{n}=-\overline{A}_{n}.$ (59)

Combining (43), (55) and (57) we have the following relations

$\displaystyle\Psi_{+,2}\big{(}-\frac{q_{0}^{2}}{\overline{z}_{n}}\big{)}=\frac{q_{-}}{\overline{q}_{+}}\tilde{b}_{n}\Psi_{-,1}\big{(}-\frac{q_{0}^{2}}{\overline{z}_{n}}\big{)},$
(60a)
$\displaystyle\Psi_{+,1}\big{(}-\frac{q_{0}^{2}}{z_{n}}\big{)}=\frac{\overline{q}_{-}}{q_{+}}b_{n}\Psi_{-,2}\big{(}-\frac{q_{0}^{2}}{z_{n}}\big{)}.$
(60b)

Using the symmetries of $s_{ij}$, we have results

$\displaystyle\mathop{\mathrm{Res}}\limits_{z=-\frac{q_{0}^{2}}{\overline{z}_{n}}}\left[\frac{\mu_{+,2}(z)}{s_{22}(z)}\right]=A_{N+n}\mu_{-,1}(-\frac{q_{0}^{2}}{\overline{z}_{n}})e^{-2i\theta(-\frac{q_{0}^{2}}{\overline{z}_{n}})},$
(61a)
$\displaystyle\mathop{\mathrm{Res}}\limits_{z=-\frac{q_{0}^{2}}{z_{n}}}\left[\frac{\mu_{+,1}(z)}{s_{11}(z)}\right]=\tilde{A}_{N+n}\mu_{-,2}(-\frac{q_{0}^{2}}{z_{n}})e^{2i\theta(-\frac{q_{0}^{2}}{z_{n}})}.$
(61b)

where

$A_{N+n}=\frac{q_{-}}{\overline{q}_{-}}\frac{q_{0}^{2}}{\overline{z}_{n}^{2}}\tilde{A}_{n},\quad\tilde{A}_{N+n}=\frac{\overline{q}_{-}}{q_{-}}\frac{q_{0}^{2}}{z_{n}^{2}}A_{n},$
(62)

and $\tilde{A}_{N+n}=-\overline{A}_{N+n}$.

##  8 N-soliton solutions of the KE equation

To obtain solution of KE equation with nonzero boundary conditions, we should
establish the connection between the solution of KE equation and the Riemann-
Hilbert problem.

###  8.1 Reconstruction formula

To solve the above Riemann-Hilbert problem, it is necessary to regularize it
by subtracting out the asymptotic and the pole contributions. For convenient,
we define $\zeta_{n}=z_{n}$ and
$\zeta_{N+n}=-\frac{q_{0}^{2}}{\overline{z}_{n}}$, $(n=1,\cdots,N)$ and

$\displaystyle\begin{split}&M^{-}-I+\frac{i}{z}\sigma_{3}Q_{-}-\mathop{\mathrm{\sum}}\limits_{n=1}^{2N}\frac{\mathop{\mathrm{Res}}\limits_{\overline{\zeta}_{n}}M^{-}}{z-\overline{\zeta}_{n}}-\mathop{\mathrm{\sum}}\limits_{n=1}^{2N}\frac{\mathop{\mathrm{Res}}\limits_{\zeta_{n}}M^{+}}{z-\zeta_{n}}\\\
=&M^{+}-I+\frac{i}{z}\sigma_{3}Q_{-}-\mathop{\mathrm{\sum}}\limits_{n=1}^{2N}\frac{\mathop{\mathrm{Res}}\limits_{\overline{\zeta}_{n}}M^{-}}{z-\overline{\zeta}_{n}}-\mathop{\mathrm{\sum}}\limits_{n=1}^{2N}\frac{\mathop{\mathrm{Res}}\limits_{\zeta_{n}}M^{+}}{z-\zeta_{n}}-M^{+}G\end{split}$
(63)

The left-hand side of this equation is analytic in $D^{-}$ and is
$O\left(\frac{1}{z}\right)$ as $z\rightarrow\infty$, and the sum of the first
four terms of the right-hand side is analytic in $D^{+}$ and is
$O\left(\frac{1}{z}\right)$ as $z\rightarrow\infty$. The asymptotic behavior
of the off-diagonal scattering coefficients implies that $G(x,t,z)$ is
$O\left(\frac{1}{z}\right)$ as $z\rightarrow\pm\infty$ and $O(z)$ as
$z\rightarrow 0$ along the real axis.

Now we introduce the Cauchy projectors $P_{\pm}$ over $\Sigma$

$P_{\pm}[f](z)=\frac{1}{2i\pi}\int_{\Sigma}\frac{f(\zeta)}{\zeta-(z\pm
i0)}d\zeta,$

where $\int_{\Sigma}$ denotes the integral along the oriented contour shown in
Fig.1, and the notation $z\pm i0$ indicates that when $z\in\Sigma$, the limit
is taken from the left(right) of it. Now recall Plemelj’s formulae: if
$f^{\pm}$ are analytic in $D^{\pm}$ and are $O\left(\frac{1}{z}\right)$ as
$z\rightarrow\infty$, one has $P^{\pm}f^{\pm}=\pm f^{\pm}$ and
$P^{+}f^{-}=P^{-}f^{+}=0$. Applying $P^{+}$ and $P^{-}$ to (63), then we have

$\displaystyle\begin{split}M(x,t,z)=&I-\frac{i}{z}\sigma_{3}Q_{-}+\mathop{\mathrm{\sum}}\limits_{n=1}^{2N}\frac{\mathop{\mathrm{Res}}\limits_{\overline{\zeta}_{n}}M^{-}}{z-\overline{\zeta}_{n}}+\mathop{\mathrm{\sum}}\limits_{n=1}^{2N}\frac{\mathop{\mathrm{Res}}\limits_{\zeta_{n}}M^{+}}{z-\zeta_{n}}\\\
&+\frac{1}{2i\pi}\int_{\Sigma}\frac{M^{+}(x,t,\zeta)G(x,t,\zeta)}{\zeta-z}d\zeta,\quad
z\in\mathbb{C}\backslash\Sigma.\end{split}$ (64)

From (64), we deriving the asymptotic behaviors of $M^{\pm}(x,t,z)$ as
$z\rightarrow\infty$ as

$\displaystyle\begin{split}M(x,t,z)=&I+\frac{i}{z}\Big{\\{}-\sigma_{3}Q_{-}+\mathop{\mathrm{\sum}}\limits_{n=1}^{2N}\mathop{\mathrm{Res}}\limits_{\overline{\zeta}_{n}}M^{-}+\mathop{\mathrm{\sum}}\limits_{n=1}^{2N}\mathop{\mathrm{Res}}\limits_{\zeta_{n}}M^{+}\\\
&-\frac{1}{2i\pi}\int_{\Sigma}M^{+}(x,t,\zeta)G(x,t,\zeta)d\zeta\Big{\\}}+O\big{(}\frac{1}{z^{2}}\big{)},\end{split}$
(65)

Comparing the (1,2) element of (65) with (33). Then we obtain the formula of
the potential $q(x,t)$

$q(x,t)=e^{2\int_{-\infty}^{x}i\beta(|q|^{2}-q_{0}^{2})dy}\left[q_{-}+i\Big{(}\mathop{\mathrm{\sum}}\limits_{n=1}^{2N}\mathop{\mathrm{Res}}\limits_{\zeta_{n}}M^{+}\Big{)}_{1,2}\\\
-\frac{1}{2\pi}\int_{\Sigma}(M^{+}(x,t,\zeta)G(x,t,\zeta))_{1,2}d\zeta\right].$
(66)

###  8.2 N-soliton solutions

In this subsection we consider the reflectionless potentials. In this case,
the reflection coefficient $\rho(z)=0$, the potential formula (66) reduced as

$q(x,t)=e^{2\int_{-\infty}^{x}i\beta(|q|^{2}-q_{0}^{2})dy}\left[q_{-}+i\mathop{\mathrm{\sum}}\limits_{n=1}^{2N}A_{n}e^{-2i\theta(\zeta_{n})}\mu_{-,11}(\zeta_{n})\right].$
(67)

For convenient, we introduce the quantities

$\displaystyle
a_{j}(x,t,z)=\frac{\tilde{A}_{j}e^{2i\theta(x,t,\overline{\zeta}_{j})}}{z-\overline{\zeta}_{j}},\quad
j=1,\cdots,2N.$

Then from (64), we can derive the results

$\displaystyle\mu_{-,12}(\overline{\zeta}_{j})=-\frac{iq_{-}}{\overline{\zeta}_{j}}+\mathop{\mathrm{\sum}}\limits_{k=1}^{2N}\frac{A_{k}e^{-2i\theta(\zeta_{k})}}{\overline{\zeta}_{j}-\zeta_{k}}\mu_{-,11}(\zeta_{k})=-\frac{iq_{-}}{\overline{\zeta}_{j}}-\mathop{\mathrm{\sum}}\limits_{k=1}^{2N}\overline{a}_{k}(\zeta_{j})\mu_{-,11}(\zeta_{k})$
(68a)
$\displaystyle\mu_{-,11}(\zeta_{n})=1+\mathop{\mathrm{\sum}}\limits_{j=1}^{2N}\frac{\tilde{A}_{j}e^{2i\theta(\overline{\zeta}_{j})}}{\zeta_{n}-\overline{\zeta}_{j}}\mu_{-,12}(\overline{\zeta}_{j})=1+\mathop{\mathrm{\sum}}\limits_{j=1}^{2N}a_{j}(\zeta_{n})\mu_{-,12}(\overline{\zeta}_{j}).$
(68b)

Substituting the equation into the second one gives

$\mu_{-,11}(\zeta_{n})=1-iq_{-}\mathop{\mathrm{\sum}}\limits_{j=1}^{2N}\frac{a_{j}(\zeta_{n})}{\overline{\zeta}_{j}}-\mathop{\mathrm{\sum}}\limits_{j=1}^{2N}\mathop{\mathrm{\sum}}\limits_{k=1}^{2N}a_{j}(\zeta_{n})\overline{a}_{k}(\zeta_{j})\mu_{-,11}(\zeta_{k}).$
(69)

We write this system in matrix form. Let

$\mathbf{X}=(X_{1},\cdots,X_{2N})^{T},\quad\mathbf{B}=(B_{1},\cdots,B_{2N})^{T},$

where

$X_{N}=\mu_{-,11}(\zeta_{n}),\quad
B_{n}=1-iq_{-}\mathop{\mathrm{\sum}}\limits_{j=1}^{2N}\frac{a_{j}(\zeta_{n})}{\overline{\zeta}_{j}},\quad
n=1,\cdots,2N,$

and defining the $2N\times 2N$ matrix $A=(A_{n,k})$, where

$A_{n,k}=\mathop{\mathrm{\sum}}\limits_{j=1}^{2N}a_{j}(\zeta_{n})\overline{a}_{k}(\zeta_{j}),\quad
n,k=1,\cdots,2N,$ (70)

the system can be rewritten as $M\mathbf{X}=\mathbf{B}$, where
$M=I+A=(\mathbf{M}_{1},\cdots,\mathrm{M}_{2N})$. The system is simply

$X_{n}=\frac{\det{M_{n}^{ext}}}{\det{M}},\quad n=1,\cdots,2N,$

where

$M_{n}^{ext}=(\mathbf{M}_{1},\cdots,\mathbf{M}_{n-1},\mathbf{B},\mathbf{M}_{n+1},\cdots,\mathbf{M}_{2N}).$
(71)

Finally, upon substituting $X_{1},\cdots,X_{2N}$ into the reconstruction
formula (67), the $N$-soliton solution of KE equation can be written compactly
as

$q(x,t)=e^{2\int_{-\infty}^{x}i\beta(|r|^{2}-q_{0}^{2})dy}r(x,t),$ (72)

where

$r(x,t)=q_{-}-i\frac{\det M^{aug}}{\det M},$ (73)

and the augmented $(2N+1)\times(2N+1)$ matrix $M^{aug}$ is given by

$\displaystyle M^{aug}=\left(\begin{array}[]{cc}0&\mathbf{Y}^{t}\\\
\mathbf{B}&M\end{array}\right),\quad\mathbf{Y}=(Y_{1},\cdots,Y_{2N})^{T},$
(76) $Y_{n}=A_{n}e^{-2i\theta(x,t,\zeta_{n})},\quad n=1,\cdots,2N.$ (77)

###  8.3 one-Soliton solutions

In this subsection we mainly consider the one-soliton solution for which the
reflection coefficient $\rho=0$. Take $q_{0}=1$, eigenvalue $z_{1}=i\chi$
($\chi$=constant value and $\chi>1$) and $\tilde{A}_{1}=e^{\alpha-i\gamma}$
($\alpha,\gamma\in\mathbb{R}$). From the $N$-soliton solutions formula, we can
obtain the soliton solution

$q_{1}(x,t)=e^{2\int_{-\infty}^{x}i\beta(|r_{1}|^{2}-q_{0}^{2})dy}r_{1}(x,t),$
(78)

where $r_{1}(x,t)$ with the following trigonometric function form

$r_{1}(x,t)=\frac{\cosh\nu+\frac{1}{2}a_{1}(1+\frac{a_{2}^{2}}{a_{1}^{2}})\sin(\gamma+a_{1}a_{2}t)-ia_{2}\cos(\gamma+a_{1}a_{2}t)}{\cosh\nu+\frac{2}{a_{1}}\sin(\gamma+a_{1}a_{2}t)},$
(79)

and

$\displaystyle\begin{split}&\nu=(x+4\beta t)a_{2}+\alpha+a_{0},\\\
&e^{a_{0}}=\frac{a_{1}}{2\chi a_{2}},\\\ &e^{-a_{0}}=\frac{2\chi
a_{2}}{a_{1}}.\end{split}$

For $\beta=0$, soliton solution (78) convert into the one-soliton solution of
focusing nonlinear Schr$\ddot{o}$dinger equation Fig.2 (a). Fig.2 (b) displays
the $\beta\neq 0$ case and Fig.2 (c) displays the $\beta\neq 0$ and
$\chi\rightarrow 1$ case. The Fig.2 (d) displays the more general situation
that is the zero point $z_{1}=i\chi e^{i\alpha}$ where $\chi=2$ and
$\alpha=\frac{\pi}{4}$.

(a) (b)

(c) (d)

##  9 Conclusion

In this work, we investigated the focusing Kundu-Eckhaus equation with nonzero
boundary condition at infinity. For $\beta=0$, the soliton solutions are
reduced as the soliton solutions of focusing NLS with nonzero boundary
conditions. We introduced a transformation, such that the asymptotic spectral
problem with the linear relationship $\mathcal{V}_{\pm}=(2k+2\beta
q_{0}^{2})\mathcal{U}_{\pm}$. A appropriate Riemann surface for the single-
valued function of the spectral parameter was introduced. Then, the complex
$k$-plane transformed into the complex $z$-plane. Unlike the focusing
nonlinear Schr$\ddot{o}$dinger equation, the branch cut not on the
$\mathrm{Im}k$ axis, but shift $\beta q_{0}^{2}$ along the $x$ axis. The
orientation is up to $\beta$. The nonzero boundary conditions is different
from zero boundary conditions mainly reflected in the analytic region and the
zero points of $s_{11}$ and $s_{22}$. The analytic region not only involved
with the upper-half/lower-half plane, but also involved with the
inside/outside of the circle $C_{0}$.

## Acknowledgments

This work is supported by the National Science Foundation of China (Grant No.
11671095, 51879045).

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