# No Intruder, no Validity: Evaluation Criteria for
Privacy-Preserving Text Anonymization

Maximilian Mozes1,2,3 Bennett Kleinberg4,1
1Department of Security and Crime Science, University College London
2Dawes Centre for Future Crime, University College London
3Department of Computer Science, University College London
4Department of Methodology and Statistics, Tilburg University
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

For sensitive text data to be shared among NLP researchers and practitioners,
shared documents need to comply with data protection and privacy laws. There
is hence a growing interest in automated approaches for text anonymization.
However, measuring such methods’ performance is challenging: missing a single
identifying attribute can reveal an individual’s identity. In this paper, we
draw attention to this problem and argue that researchers and practitioners
developing automated text anonymization systems should carefully assess
whether their evaluation methods truly reflect the system’s ability to protect
individuals from being re-identified. We then propose TILD, a set of
evaluation criteria that comprises an anonymization method’s technical
performance, the information loss resulting from its anonymization, and the
human ability to de-anonymize redacted documents. These criteria may
facilitate progress towards a standardized way for measuring anonymization
performance.

##  1 Introduction

Vast developments in natural language processing have led to an increased
interest in the application of text processing approaches across fields,
including the biomedical Armengol-Estapé et al. (2019), law enforcement Yang
and Chow (2014) and business sectors Davenport (2018). Part of the drive for
applying methods beyond research contexts is the enormous availability and
increasing relevance of digital text data as a rich source of information.
Textual corpora can now be analyzed with quantitative methods, for example, to
extract information and harness methods from machine learning for predictive
tasks (e.g., text classification and natural language generation). However,
the great potential of text data for researchers, businesses and policymakers,
rests on a key precondition: access to and the sharing of text data.

Researchers could further push NLP boundaries and set out to address topical
problems if data are more easily shareable. Similarly, businesses would
benefit as data flow becomes less constrained (e.g., by providing access to
real-world data in tender procedures). Lastly, cross-sector relationships,
where academic research feeds into policy-making or business processes—or vice
versa—rely on access to text data. Especially in contexts where the text data
are sensitive by nature (e.g., police reports, health care dossiers) or
protected by privacy laws (e.g., the General Data Protection Regulation;
GDPR), data sharing is currently either non-existent or requires extensive
data-sharing agreement negotiations. Consequently, academic research often
resorts to easy-to-get data that is removed from an actual problem context,
and public and private sector organizations who possess sensitive data only
rarely harness the powerful methods developed at academic institutions.

Example statement | Explanation
---|---
PERSON1, nicknamed the “Iron Lady”, served as LOCATION1 OCCUPATION1 from DATE1 to DATE2. |  "Iron Lady" likely to reveal Margaret Thatcher’s identity
OTHER1 produced and recorded a variety of musical hits such as OTHER2, “Bohemian Rhapsody” and OTHER3. |  “Bohemian Rhapsody” likely to reveal Queen’s identity
PERSON1 known for PRONOUN role as PERSON2 struggles to say the word penguin. |  The penguin reference may reveal Benedict Cumberbatch111https://www.vanityfair.com/hollywood/2014/09/benedict-cumberbatch-penguins
Table 1: Illustration of failure cases in the anonymization process where
missing a single personally-identifying attribute invalidates the success of
the text anonymization.

The common solution to reconcile privacy protection with the sharing of text
data is text anonymization. While the idea of redacting documents is not new,
the volume of the data now available and needed by many NLP approaches makes
manual efforts futile. A solution is automated text anonymization which,
broadly, aims to build systems that automatically redact personally-
identifying information from text, to make the anonymized text fit for sharing
without violating the privacy of individuals identifiable from the text. In
this paper, we draw attention to the problems of current evaluation standards
of text anonymization systems and propose TILD, a set of evaluation criteria
that we recommend any suchlike system to be tested against.

##  2 Requirements for text anonymization

Text anonymization can serve two purposes, each of which determines the
sophistication of the anonymization approach. First, one may decide to fully
redact documents by removing any trace of personally-identifying information.
Such removal of information may apply to government documents released to the
public without any secondary analysis envisioned. The second purpose is the
sharing of text data to enable secondary analysis based on the shared
documents. While these two purposes both share the aim of removing personally-
identifying information, the critical difference is what the original
information (e.g., names, locations, attributes) are replaced with. Simply
crossing out the relevant information might render the documents useless for
secondary analyses. One of the criteria we outline below postulates that the
difference in findings based on the original and anonymized documents should
be minimal. Current anonymization systems lack a framework that can be used
for evaluation, making it difficult to compare approaches and leaves an
ambiguous picture for stakeholders and data owners deciding which
anonymization system works best for their purpose.

Criterion | Description | How to test it?
---|---|---
Technical tool evaluation |  Measures how well the system detects relevant information |  Reporting of common performance metrics (detection accuracy, precision)
Information loss |  Reports the difference in prediction tasks (utility loss) or linguistic variables (construct loss) between original and anonymized texts |  Statistical testing of the loss deviation from a tolerance value
De-anonymization |  Assesses whether a motivated intruder can re-identify individuals in anonymized text |  Pentesting, game-like studies with human participants tasked to identify individuals
Table 2: Overview of the three TILD evaluation criteria technical tool
evaluation, information loss and de-anonymization.

###  2.1 Evaluation problems

Recently, several works dealing with text anonymization for multiple languages
have been put forward (e.g., Mamede et al., 2016; Kleinberg and Mozes, 2017;
Di Cerbo and Trabelsi, 2018; Hassan et al., 2018; Adams et al., 2019; Berg and
Dalianis, 2019; Francopoulo and Schaub, 2020). For example, Adams et al.
(2019) proposed an automated text anonymization system based on named entity
recognition and coreference solution modules. The proposed method identifies
sensitive phrases in a document and replaces them with generic symbols or
entity-specific replacement tokens (e.g., LOCATION or ENGLISH_CITY for
London). Francopoulo and Schaub (2020) developed an anonymization system in
the context of Customer Relationship Management consisting of a named entity
recognition model, an entity linker and a substitution method. While these
approaches include a learning-based component for the automated anonymization
of text data, Strathern et al. (2020) presented a tool that assists
practitioners in manually anonymizing texts. Some other tools are available
commercially, but these are typically not evaluated against a peer-reviewed
standard.

Existing works to automated text anonymization (e.g., Adams et al., 2019;
Francopoulo and Schaub, 2020) predominantly use fully automated precision- and
recall-based methods to assess the performance of a system. While such an
evaluation approach measures a system’s ability to extract relevant
information from text, we here argue that it is not sufficient to assess a
text anonymization system’s performance. Our argumentation against the use of
purely automated methods for evaluating a text anonymization system is
twofold.

First, missing a single but critical attribute in a text invalidates any other
performance or evaluation metric that explains which fraction of the required
entities have been identified and anonymized. This is illustrated in Table 1.
Here, an anonymization method correctly redacts all but one of the personally-
identifying attributes in a given text and would score high on purely
quantitative methods. Yet missing a single detail would reveal the
individual’s or group’s identity.

Second, while identifying and redacting personally-identifying information is
a necessary criterion for successful anonymization, it is not a sufficient
one. For anonymized texts to be useful for data analysis purposes, it needs to
be ensured that anonymization methods preserve the context of a text to the
degree that the transformed text retains its usefulness for text mining tasks
(e.g., sentiment analysis and topic modelling). We refer to this below as the
preservation of information (see Feyisetan et al., 2020).

##  3 The TILD evaluation criteria

While some attention has been paid to the development of technical solutions
to the anonymization problem, little attention has been given to how we can
assess the success of text anonymization efforts. That evaluation question is
the most fundamental one from a data protection point of view and thus the
central aspect which decision-makers need to ask. What is currently used as an
evaluation approach falls short of the fundamental aim of text anonymization.

To address this problem, we propose three criteria against which any
anonymization approach should be evaluated. We denote this set of criteria
with TILD, comprising a model’s technical tool evaluation, the information
loss resulting from the anonymization, and the human ability to de-anonymize
the redacted documents (see Table 2). We also illustrate how TILD can be put
into practice to evaluate anonymization systems. The requirements should guide
research efforts to pay attention to these aspects and equip decision-makers
(academic, business, policymaking) with a set of criteria to look out for.

Of the three criteria comprised by TILD, two are currently somewhat present in
text anonymization work (i.e., the technical evaluation and the preservation
of information). The third criterion (person re-identification) has thus far
been ignored but sits at the heart of what anonymization is supposed to do.

###  3.1 Technical tool evaluation

A simple technical tool evaluation is frequently contained in assessments of a
system’s performance in detecting (tagging) relevant information in text data
(e.g., the accuracy or the precision). For example, a system might be
evaluated by the percentage of locations, persons and dates it detects in a
text. For anonymization systems, meeting this criterion is necessary but by no
means sufficient: it makes no statement about how much information is lost
during anonymization, and it inherently assumes that removing specific pieces
of information does indeed render persons unidentifiable from the text. Thus,
the technical evaluation cannot be used as a sole criterion for the goodness
of anonymization efforts.

###  3.2 Information loss

A system that performs well on the technical evaluation level can be assessed
on the next level: preserving utility. This aspect is critical for secondary
data analyses conducted on the anonymized text data. Information loss is
defined broadly here as the difference in some quantitative measure between
the original text and its anonymized version. Information loss can be assessed
statistically by testing whether the difference between original vs anonymized
text is markedly larger than can be tolerated (or strictly: larger than zero).
We differentiate between two kinds of information loss.

#### Utility loss.

The utility loss Feyisetan et al. (2020) has been used to assess the
difference in performance (e.g., on a benchmark sentiment classification task)
that is obtained when using the original texts versus the anonymized texts.
High utility loss implies that the performance differs so that the
anonymization distorts any secondary analysis results.

#### Construct loss.

The construct loss is not focused on performance differences. Instead, it
measures the differences between original and anonymized text on some higher-
order linguistic measure (e.g., those derived with dictionary tools frequently
used in socio- and psycholinguistic research). High construct loss is
undesirable because it would indicate that the anonymization procedure
affected the text to the extent that findings from the original do not
replicate in the anonymized one. Both information loss criteria address
divergent results in anonymized texts compared to original ones, but they
alone are yet insufficient for the core anonymization purpose.

###  3.3 De-anonymization

The core purpose of text anonymization is to protect the privacy of
individuals. Consequently, any text anonymization effort should be tested
about its ability to not reveal the identity of individuals. Since an
individual can be identified by more than just their name, address or age (see
Table 1), it may not suffice to test whether directly identifying and
protected attributes are leaking. Purely automated evaluation metrics (i.e.,
technical and information loss criteria) are, therefore, insufficient for an
assessment of an anonymization system.

Instead, a more viable approach is de-anonymization penetration testing by
focusing on two possible outcomes: successful or failed anonymization. Such an
evaluation can be realized via a motivated intruder test Office (2012). The
UK’s Information Commissioner’s Office suggests evaluating the quality of any
anonymization system (i.e., not just text) by tasking an individual human
being (the intruder) with the de-anonymization of any persons from an
anonymized piece of information. Such a test puts a human intruder—who is
allowed and encouraged to use external resources—into a game-like scenario
with the goal of re-identifying individuals. Crucially, the intruder does not
have privileged information, nor do they have any specialist knowledge beyond
what is available publicly. This provides a more realistic demand on the
anonymization because it excludes scenarios with highly privileged and hence
directly revealing information (e.g., a physician knowing that a condition X
only occurs in patient A). If an intruder can de-anonymize a text, the
anonymization failed.

#### Anonymization intrusion in practice.

A possible way of putting the de-anonymization criterion of the proposed TILD
guidelines to the test would be via online crowdsourcing. Potential intruders
could be recruited on crowdsourcing platforms such as Amazon’s MTurk or
Prolific Academic. In an online behavioral study, participants would be asked
to read an anonymized piece of information and then tasked with re-identifying
any individual mentioned or described in the text. The intruder should be
encouraged (and potentially even incentivized) to use any external resources
available to them (such as a web search engine). The ratio of successful de-
anonymizations (i.e., where intruders correctly revealed the identities of
individuals) to all attempts serves as the key evaluation metric for a text
anonymization system.

##  4 Conclusion

In this paper, we argued that relying on a technical evaluation is
insufficient to measure the true capabilities of automated text anonymization
systems. We proposed TILD, a set of more comprehensive evaluation criteria. In
addition to technical evaluation metrics, TILD measures an anonymization
system’s ability to preserve task-relevant linguistic concepts and the degree
to which it truly preserves the privacy of individuals mentioned in a text. We
encourage researchers and practitioners to adopt more diverse evaluation
criteria for anonymization systems to provide an assessment that more
rigorously reflects the potential of their proposed methods.

## Acknowledgements

We would like to thank Toby Davies for helpful discussions about this work.

## References

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  * Armengol-Estapé et al. (2019) Jordi Armengol-Estapé, Felipe Soares, Montserrat Marimon, and Martin Krallinger. 2019.  Pharmaconer tagger: a deep learning-based tool for automatically finding chemicals and drugs in spanish medical texts.  _Genomics & informatics_, 17(2).
  * Berg and Dalianis (2019) Hanna Berg and Hercules Dalianis. 2019.  Augmenting a de-identification system for Swedish clinical text using open resources and deep learning.  In _Proceedings of the Workshop on NLP and Pseudonymisation_ , pages 8–15, Turku, Finland. Linköping Electronic Press.
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  * Feyisetan et al. (2020) Oluwaseyi Feyisetan, Borja Balle, Thomas Drake, and Tom Diethe. 2020.  Privacy-and utility-preserving textual analysis via calibrated multivariate perturbations.  In _Proceedings of the 13th International Conference on Web Search and Data Mining_ , pages 178–186.
  * Francopoulo and Schaub (2020) Gil Francopoulo and Léon-Paul Schaub. 2020.  Anonymization for the gdpr in the context of citizen and customer relationship management and nlp.  In _workshop on Legal and Ethical Issues (Legal2020)_ , pages 9–14. ELRA.
  * Hassan et al. (2018) Fadi Hassan, Josep Domingo-Ferrer, and Jordi Soria-Comas. 2018.  Anonymization of unstructured data via named-entity recognition.  In _Modeling Decisions for Artificial Intelligence_ , pages 296–305, Cham. Springer International Publishing.
  * Kleinberg and Mozes (2017) Bennett Kleinberg and Maximilian Mozes. 2017.  Web-based text anonymization with node. js: Introducing netanos (named entity-based text anonymization for open science).  _Journal of Open Source Software_ , 2(14):293.
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#  Compound WKI-SP hierarchy and multiloop soliton solutions

Xiaorui Hu  Tianle Xu  Junyang Zhang  Shoufeng Shen  Department of Applied
Mathematics, Zhejiang University of Technology, Hangzhou 310023, Zhejiang,
China

###### Abstract

The generalized hierarchies of compound WKI-SP (Wadati-Konno-Ichikawa and
short pulse) equations are presented. The proposed integrable nonlinear
equations include the WKI-type equations, the SP-type equations and the
compound generalized WKI-SP equations. A chain of hodograph transformations
are established to relate the compound WKI-SP equations with the MKdV-SG
(modified Korteweg-de Vries and sine-Gordon) equations. As applications, the
multiloop soliton solutions of one compound WKI-SP (named WKI-SP(1,1))
equation are obtained. We emphasize on showing abundant solitonic behaviors of
two loop solitons. The role of each parameter plays in the movement of two-
loop solion are shown detailedly in a table.

###### keywords:

WKI-SP equation, MKdV-SG equation, hodograph transformation, loop soliton.

††journal: XXX

##  1 Introduction

A remarkable development in our understanding of a certain class of nonlinear
integrable equations known as the consistency condition for a system of linear
differential equations has taken place in the past decade. This zero curvature
equations on real Lie algebras such as sl(2,$\mathbb{R}$)and
so(3,$\mathbb{R}$) lay the foundation for constructing soliton hierarchies.
The most famous one is the Ablowitz-Kaup-Newell-Segur (AKNS) hierarchy,
including the fundamental Korteweg-de Vries (KdV) equation, the the modified
KdV (MKdV) equation, the sine-Gordon (SG) equation, the nonlinear Schrödinger
(NLS) equation and so on.

In 1979, Wadati, Konno and Ichikawa (WKI) [1, 2] constructed a new series of
integrable nonlinear evolution equations, such as

$\displaystyle u_{t}+\bigg{[}\frac{u_{x}}{(1+u^{2})^{3/2}}\bigg{]}_{xx}=0.$
(1)

The nonlinear terms of Eq.(1) have saturation effects. Eq.(1) describes
nonlinear transverse oscillations of elastic beams under tension [3]. In the
many past decades, lots of research have been done for this WKI equation (1).
For example, an infinite number of conservation laws and the Hamiltonian form
was found by Wadati et al [1]. The one-soliton and two-soliton solutions were
obtained by using inverse scattering transform method in Ref.[4]. Qu and Zhang
[5] derived the WKI model from motion of curves in E3 and gave the
corresponding group-invariant solutions. The Darboux transformation is
proposed by Zhang et. al. [6] and the direct scattering problem with box-like
initial value was considered by Tu and Xu [7].

Recently, the systems of short pulse type have attracted considerable
attention due to their important applications in physics. The very first such
equation may be the short pulse (SP) equation

$\displaystyle u_{xt}=u+\frac{1}{6}(u^{3})_{xx},$ (2)

which possesses a Lax pair of the WKI type. The SP equation (2) is proposed by
Schäfer and Wayne [8, 9] as an alternative (to the NLS equation) model for
approximating the evolution of ultrashort intense infrared pulses in silica
optics. It is shown in [9] by numerical simulations that the SP equation can
be successfully used for describing pulses with broad spectrum. It turns out
that the SP equation made its first appearance in Rabelo’s paper [10] in his
study pseudospherical surfaces. The SP and two-component SP (real or complex)
equations are proposed as special integrable cases in the negative WKI
hierarchy for the first time in Refs.[11, 12]. Multi-soliton solutions and the
Cauchy problem for a two-component SP system are given in Ref.[13]. The SP
equation is proved integrable [15, 16] since it admits Lax pair, recursion
operator, and bi-Hamiltonian structure and various exact solutions. Different
approaches have been employed to construct solutions for the SP equation. In
particular, Anton Sakovich and Sergei Sakovich [17] found an exact nonsingular
solitary wave solution from the breather solution of the SG equation by means
of a transformation between these two integrable equations. Kuetche et al [18]
calculated the two-loop soliton solution by use of Hirota bilinear method and
Hodnett-Moloney approach. Matsuno [19, 20] obtained the multiloop soliton,
multibreather and the periodic solutions. Liu et al [21] constructed the
$N$-fold Darboux transformation given by determinants. The Riemann-Hilbert
approach [22, 23] was applied to the SP equation and the long-time behavior of
the solution was studied. Liu and Mao [24] established the $N$-Bäcklund
transformation and worked out various solutions including loop solitons,
breather solutions, and their interaction solutions. Feng et al [25, 26]
studied its integrable discretizations and both semidiscrete and full-discrete
systems were obtained.

In this paper, we focus on the construction of the generalized hierarchies of
compound WKI-SP equations from the consistency condition of two linear
differential system. It is known that both the WKI equation and the SP
equation admit loop soliton solutions [4, 27]. It is noted that they can be
transformed to certain soliton equations which admit smooth soliton solutions
through hodograph transformations[28]. For example, the WKI elastic beam
equation is transformed to the potential mKdV equation. The SP equation can be
transformed into the integrable SG equation. Hence we want to seek for a
connection between the compound MKdV-SG equations (given by Gu in Ref.[29,
30]) and the WKI-SP equations. As applications, we will calculate multiloop
soliton solutions for one WKI-SP equation.

##  2 Compound equations of the WKI-SP type

Many of the hardest problems and most interesting phenomena being studied by
mathematicians, engineers and physicists are nonlinear in nature. Often, these
phenomena can be modeled (and there is good reason to believe that these
models are accurate) by nonlinear partial differential equations and to be
sure, it will be many years to come before we have the mathematical
sophistication to handle these equations completely. In the last decades, the
inverse scattering transform (IST) has been employed to solve many physically
significant equations. Due to the similarity of the method itself to Fourier
transforms, this theory can be considered a natural extension of Fourier
analysis to nonlinear problems. Along with the IST, a systematic method has
developed which allows one to identify certain important classes of evolution
equations which can be solved by the method of inverse scattering. These
evolution equations are expressed as the consistency condition

$\displaystyle U_{t}-V_{x}+[U,V]=0,\quad[U,V]=UV-VU$ (3)

for a system of linear differential equations

$\displaystyle\Phi_{x}=U(x,t;\lambda)\Phi,\quad\quad\Phi_{t}=V(x,t;\lambda)\Phi,$
(4)

where $U$, $V$ and $\Phi$ are complex $N\times N$ matrixes. The matrixes $U$
and $V$ are usually rational functions of the parameter $\lambda$.

###  2.1 AKNS-type and WKI-type Lax pair

To increase the readability of the article, we firstly recall the similar
results for the known AKNS hierarchy and the WKI hierarchy.

In 1973, a wide class of nonlinear evolution equations were presented by
Ablowitz, Kaup,Newell and Segur (AKNS) [31], where $U$ and $V$ read

$\displaystyle U=\left(\begin{array}[]{cc}-i\lambda&q(x,t)\cr
r(x,t)&i\lambda\end{array}\right),\quad
V=\left(\begin{array}[]{cc}A(x,t;\lambda)&B(x,t;\lambda)\cr
C(x,t;\lambda)&-A(x,t;\lambda)\end{array}\right).$ (9)

The corresponding consistency condition (3) leads to the flowing equations

$\displaystyle A_{x}=qc-rB,$ (10) $\displaystyle q_{t}-B_{x}-2i\lambda
B-2qA=0,$ (11) $\displaystyle r_{t}-C_{x}+2i\lambda C+2rA=0.$ (12)

For example, AKNS set

$\displaystyle A=\sum_{j=0}^{N}a_{j}\lambda^{N},\quad
B=\sum_{j=0}^{N}b_{j}\lambda^{N},\quad C=\sum_{j=0}^{N}c_{j}\lambda^{N}.$ (13)

To illustrate, take $N=3$. By solving Eqs.(10), AKNS found

$\displaystyle
A=a_{3}\lambda^{3}+a_{2}\lambda^{2}+(\frac{1}{2}a_{3}qr+a_{1})\lambda+\frac{1}{2}a_{2}qr-\frac{i}{4}a_{3}(qr_{x}-q_{x}r)+a_{0},$
(14) $\displaystyle
B=ia_{3}q\lambda^{2}+(ia_{2}q-\frac{1}{2}a_{3}q_{x})\lambda+(ia_{1}q+\frac{i}{2}a_{3}q^{2}r-\frac{1}{2}a_{2}q_{x}-\frac{i}{4}a_{3}q_{xx}),$
(15) $\displaystyle
C=ia_{3}r\lambda^{2}+(ia_{2}r+\frac{1}{2}a_{3}r_{x})\lambda+(ia_{1}r+\frac{i}{2}a_{3}qr^{2}+\frac{1}{2}a_{2}r_{x}-\frac{i}{4}a_{3}r_{xx})$
(16)

together with the evolution equations

$\displaystyle
q_{t}+\frac{i}{4}a_{3}(r_{xxx}-6qrq_{x})+\frac{1}{2}a_{2}(q_{xx}-2q^{2}r)-ia_{1}q_{x}-2a_{0}q=0,$
(17) $\displaystyle
r_{t}+\frac{i}{4}a_{3}(r_{xxx}-6qrr_{x})-\frac{1}{2}a_{2}(r_{xx}-2qr^{2})-ia_{1}r_{x}+2a_{0}r=0.$
(18)

_Case I._ As a special case, let $r=-q$ and

$\displaystyle a_{0}=a_{1}=a_{2}=0,\quad a_{3}=-4i.$ (19)

Then AKNS got the modified mKdV equation

$\displaystyle q_{t}+6q^{2}q_{x}+q_{xxx}=0.$ (20)

_Case II._ In the same way, by taking

$\displaystyle A=\frac{a(x,t)}{\lambda},\quad
B=\frac{b(x,t)}{\lambda},C=\frac{c(x,t)}{\lambda},$ (21)

AKNS found

$\displaystyle q_{xt}=-4iaq,\quad r_{xt}=-4iar,\quad
a_{x}=\frac{i}{2}(qr)_{t}.$ (22)

As a special but important case, we list the following SG equation

$\displaystyle u_{xt}=\sin u$ (23)

with

$\displaystyle a=\frac{i}{4}\cos u,\quad b=c=\frac{i}{4}\sin u,\quad
r=-q=\frac{1}{2}u_{x}.$ (24)

In 1979, Wadati, Konno and Ichikawa [1, 2] proposed a generalization of the
inverse scattering formalism (especially $U$) and found a new series of
integrable nonlinear evolution equations. In fact, they considered

$\displaystyle U=\left(\begin{array}[]{cc}-i\lambda&\lambda q\cr\lambda
r&i\lambda\end{array}\right),\quad V=\left(\begin{array}[]{cc}A&B\cr
C&-A\end{array}\right).$ (29)

The corresponding consistency condition (3) leads to the flowing equations

$\displaystyle A_{x}=\lambda(qC-rB),$ (30) $\displaystyle\lambda
q_{t}-B_{x}-2i\lambda B-2\lambda qA=0,$ (31) $\displaystyle\lambda
r_{t}-C_{x}+2i\lambda C+2\lambda rA=0.$ (32)

_Case III._ As a special case, by choosing

$\displaystyle
A=-\frac{4i}{\sqrt{1-rq}}\lambda^{3}+\frac{rq_{x}-qr_{x}}{(1-rq)^{3/2}}\lambda^{2},$
(33) $\displaystyle
B=\frac{4q}{\sqrt{1-rq}}\lambda^{3}+\frac{2iq_{x}}{(1-rq)^{3/2}}\lambda^{2}-\bigg{(}\frac{q_{x}}{(1-rq)^{3/2}}\bigg{)}_{x}\lambda,$
(34) $\displaystyle
C=\frac{4r}{\sqrt{1-rq}}\lambda^{3}-\frac{2ir_{x}}{(1-rq)^{3/2}}\lambda^{2}-\bigg{(}\frac{r_{x}}{(1-rq)^{3/2}}\bigg{)}_{x}\lambda,$
(35)

they obtained

$\displaystyle q_{t}+\bigg{(}\frac{q_{x}}{(1-rq)^{3/2}}\bigg{)}_{xx}=0,$ (36)
$\displaystyle r_{t}+\bigg{(}\frac{r_{x}}{(1-rq)^{3/2}}\bigg{)}_{xx}=0.$ (37)

For $r=-q$, above equation is reduced to the typical WKI equation (1).

_Case IV._ Moreover, as another special case, by choosing $r=q=iu_{x}$ and

$\displaystyle A=\frac{i\lambda}{2}u^{2}+\frac{1}{4i\lambda},\quad
B=\frac{i\lambda}{2}u^{2}u_{x}-\frac{1}{2}u,\quad
C=\frac{i\lambda}{2}u^{2}u_{x}+\frac{1}{2}u,$ (38)

one can get the known SP equation (2). This Lax pair for (2) was firstly
discovered by Sakovich A and Sakovich S [15] in 2005.

###  2.2 WKI-SP type integrable equations

In this paper, we focus on a series of compound equations of the WKI-SP type,
which will be derived on the basis of the linear system (4). For convenience,
we rewrite $U$ as

$\displaystyle U=\left(\begin{array}[]{cc}\lambda&\lambda p(x,t)\cr\lambda
p(x,t)&-\lambda\end{array}\right).$ (41)

Eqs.(30)-(32) become

$\displaystyle A_{x}=\lambda p(C-B),$ (42) $\displaystyle\lambda
p_{t}-B_{x}+2\lambda B-2\lambda pA=0,$ (43) $\displaystyle\lambda
p_{t}-C_{x}-2\lambda C+2\lambda pA=0.$ (44)

Consequently, we have

$\displaystyle C-B=\frac{A_{x}}{\lambda p},$ (45) $\displaystyle
C+B=-\frac{(C-B)_{x}}{2\lambda}+2pA,$ (46) $\displaystyle
p_{t}=\frac{(C+B)_{x}}{2\lambda}+(C-B).$ (47)

Different from the previous references, we select $A$ as follows:

$\displaystyle A=\sum_{j=0}^{n+m}a_{j}\lambda^{2n-2j+1},\quad\quad(n\geq
0,m\geq 0).$ (48)

Substituting (45) and (46) into (47) and then expanding it into a power series
of $\lambda$, we obtain

$\displaystyle
p_{t}=-\frac{1}{4}\bigg{(}\frac{a_{n-1,x}}{p}\bigg{)}_{xx}+(pa_{n})_{x}+\frac{a_{nx}}{p}$
(49)

together with the following two sets of recursive formulas

$\displaystyle(pa_{0})_{x}+\frac{a_{0x}}{p}=0,$ (50)
$\displaystyle(pa_{j})_{x}+\frac{a_{j,x}}{p}=\frac{1}{4}\bigg{(}\frac{a_{j-1,x}}{p}\bigg{)}_{xx},\quad(j=1,2,\ldots,n-2,n-1),$
(51)

and

$\displaystyle-\frac{1}{4}\bigg{(}\frac{a_{n+m,x}}{p}\bigg{)}_{xx}=0,$ (52)
$\displaystyle\frac{1}{4}\bigg{(}\frac{a_{j,x}}{p}\bigg{)}_{xx}=(pa_{j+1})_{x}+\frac{a_{j+1,x}}{p},\quad(j=n+m-1,\ldots,n+1,n).$
(53)

It is seen that $\\{a_{0},a_{1},\ldots,a_{n-1}\\}$ and
$\\{a_{n},a_{n+1},\ldots,a_{n+m-1},a_{n+m}\\}$ are determined by Eqs.(50)-(51)
and Eqs.(52)-(53), respectively. For convenience, we denote the defined
equation in (49) as WKI-SP(n,m) equation for the concrete $n$ and $m$.

Furthermore, by solving Eqs.(50)-(51), we obtain

$\displaystyle a_{0}=\frac{a_{0}^{0}(t)}{\sqrt{1+p^{2}}},$ (54) $\displaystyle
a_{j}=\frac{1}{4\sqrt{1+p^{2}}}\partial^{-1}\bigg{(}\frac{p}{\sqrt{1+p^{2}}}\bigg{(}\frac{a_{j-1,x}}{p}\bigg{)}_{xx}\bigg{)},\quad(j=1,2,\ldots,n-1),$
(55)

which imply

$\displaystyle
a_{1}=-\frac{a_{0}^{0}(t)\biggl{[}(p^{3}+p)p_{xx}-\frac{1}{2}(6p^{2}+1)p_{x}^{2}\biggr{]}}{4(1+p^{2})^{\frac{7}{2}}},$
(56)
$\begin{split}&a_{2}=-\frac{a_{0}^{0}(t)}{32(1+p^{2})^{\frac{13}{2}}}\bigg{[}2p(1+p^{2})^{3}p_{xxxx}-2(1+p^{2})^{2}(15p^{2}+1)p_{x}p_{xxx}-(1+p^{2})^{2}(20p^{2}-1)p_{xx}^{2}\\\
&+21p(1+p^{2})(10p^{2}-1)p_{x}^{2}p_{xx}-\frac{21}{4}(40p^{4}-16p^{2}-1)p_{x}^{4}\bigg{]},\\\
&\quad\vdots\end{split}$

By solving Eqs.(52)-(53), we have

$\displaystyle a_{n+m}=a_{n+m}^{0}(t),$ (57) $\displaystyle
a_{j}=4\partial^{-1}\bigg{(}p\partial^{-2}\bigg{(}\frac{a_{j+1,x}}{p}\bigg{)}\bigg{)}+4\partial^{-1}(p\partial^{-1}(pa_{j+1})),$
(58) $\displaystyle(j=n+m-1,n+m-2,\ldots,n),$

which imply

$\displaystyle a_{n+m}=a_{n+m}^{0}(t),$ (59) $\displaystyle
a_{n+m-1}=2a_{n+m}^{0}(t)(\partial^{-1}p)^{2},$ (60) $\displaystyle
a_{n+m-2}=a_{n+m}^{0}(t)\bigg{[}\frac{2}{3}(\partial^{-1}p)^{4}+16(\partial^{-1}p)(\partial^{-3}p)-8(\partial^{-2}p)^{2}\bigg{]},$
(61)
$\displaystyle\begin{split}&a_{n+m-3}=a_{n+m}^{0}(t)\bigg{\\{}\frac{4}{45}(\partial^{-1}p)^{6}+\frac{32}{3}\biggl{[}(\partial^{-1}p)^{3}(\partial^{-3}p)+(\partial^{-1}p)\bigg{(}\partial^{-2}(\partial^{-1}p)^{3}\bigg{)}-(\partial^{-2}p)\bigg{(}\partial^{-1}(\partial^{-1}p)^{3}\bigg{)}\\\
&-\frac{3}{2}\partial^{-1}\bigg{(}p(\partial^{-2}p)^{2}\bigg{)}\biggr{]}+64\bigg{[}(\partial^{-1}p)(\partial^{-5}p)-(\partial^{-2}p)(\partial^{-4}p)+\frac{1}{2}(\partial^{-3}p)^{2}\bigg{]}\bigg{\\}},\end{split}$
(62) $\displaystyle\quad\vdots$

In order to express the fomulas clearly, we write $\partial^{-1}$ instead of
$\int^{x}\cdot dx$ sometimes.

####  2.2.1 SP-hierarchy for $n=0$

If $n=0$, (48) becomes

$\displaystyle
A=\sum_{j=0}^{m}a_{j}\lambda^{-2j+1}=a_{0}\lambda+a_{1}\lambda^{-1}+a_{2}\lambda^{-3}+\cdots+a_{m}\lambda^{-(2m-1)}.$
(63)

Then we obtain the generalized hierarchy of SP equations

$\displaystyle p_{t}=(pa_{0})_{x}+\frac{a_{0x}}{p}.$ (64)

Here $a_{0}$ is determined by the recursive formulas (57) and (58) with $n=0$.
Now we give some special cases of the SP hierarchy.

_Case 1._ For $m=1$ and $p=u_{x}$, we have

$\displaystyle a_{1}=a^{0}_{1}(t),\quad a_{0}=2a_{1}^{0}(t)u^{2},$ (65)

which exactly give rise to the short pulse equation

$\displaystyle u_{xt}=4a_{1}^{0}(t)\bigg{(}\frac{1}{6}(u^{3})_{xx}+u\bigg{)}.$
(66)

According to the convention above, we label Eq.(66) as WKI-SP(0,1) equation.

_Case 2._ For $m=2$ and $p=u_{xxx}$, we have

$\displaystyle a_{2}=a^{0}_{2}(t),\quad a_{1}=2a_{2}^{0}(t)u^{2}_{xx},\quad
a_{0}=\frac{2}{3}a_{2}^{0}(t)u_{xx}^{4}+16a_{2}^{0}(t)uu_{xx}-8a_{2}^{0}(t)u_{x}^{2},$
(67)

which yields a high-order short pulse (i.e.WKI-SP(0,2)) equation

$\displaystyle
u_{xxxt}=\frac{2}{15}a_{2}^{0}(t)(u_{xx}^{5})_{xx}+a_{2}^{0}(t)[u_{xxx}(16uu_{xx}-8u_{x}^{2})]_{x}+\frac{8}{3}a_{2}^{0}(t)u_{xx}^{3}+16a_{2}^{0}(t)u.$
(68)

This equation is just the local form of the one given in Ref.[32].

_Case 3._ For $m=3$ and $p=u_{xxxxx}$, we have

$a_{3}=a^{0}_{3}(t),\quad a_{2}=2a_{3}^{0}(t)u_{xxxx}^{2},\quad
a_{1}=\frac{2}{3}a_{3}^{0}(t)u_{xxxx}^{4}+16a_{3}^{0}(t)u_{xx}u_{xxxx}-8a_{3}^{0}(t)u_{xxx}^{2},$
(69)
$\begin{split}a_{0}=&\frac{4}{45}a_{3}^{0}(t)u_{xxxx}^{6}+32a_{3}^{0}(t)\bigg{[}\frac{1}{3}u_{xx}u_{xxxx}^{3}-\frac{1}{2}u_{xxx}^{2}u_{xxxx}^{2}+2uu_{xxxx}-2u_{x}u_{xxx}+u_{xx}^{2}+u_{xxxx}\partial^{-1}(u_{xxxx}^{2}u_{xxx})\\\
&+\frac{32}{3}u_{xxxx}\partial^{-2}u_{xxxx}^{3}\bigg{]},\end{split}$ (70)

which yields the WKI-SP(0,3) equation

$\begin{split}&u_{xxxxxt}=a_{3}^{0}(t)\bigg{\\{}u_{xxxxxx}\bigg{[}\frac{4}{45}u_{xxxx}^{6}+32\bigg{(}\frac{1}{3}u_{xx}u_{xxxx}^{3}-\frac{1}{2}u_{xxx}^{2}u_{xxxx}^{2}+u_{xxxx}\partial^{-1}(u_{xxxx}^{2}u_{xxx})+\frac{32}{3}u_{xxxx}\partial^{-2}(u_{xxxx}^{3})\\\
&+64\bigg{(}uu_{xxxx}-u_{x}u_{xxx}+\frac{1}{2}u_{xx}^{2}\bigg{)}\bigg{)}\bigg{]}+(u_{xxxxx}+{u^{-1}_{xxxxx}})\bigg{[}\frac{8}{15}u_{xxxx}^{5}u_{xxxxx}+32u_{xxxxx}\\\
&\bigg{(}u_{xx}u_{xxxx}^{2}-u_{xxx}^{2}u_{xxxx}+\partial^{-1}(u_{xxxx}^{2}u_{xxx})+\frac{32}{3}u_{xxxxx}\partial^{-2}(u_{xxxx}^{3})\bigg{)}+64uu_{xxxxx}\bigg{]}\bigg{\\}}.\end{split}$
(71)

####  2.2.2 WKI-hierarchy for $m=0$

If $m=0$, (48) is rewritten as

$\displaystyle
A=\sum_{j=0}^{n}a_{j}\lambda^{2n-2j+1}=a_{0}\lambda^{2n+1}+a_{1}\lambda^{2n-1}+a_{2}\lambda^{-3}+\cdots+a_{n}\lambda.$
(72)

Then we obtain the generalized hierarchy of WKI equations

$\displaystyle
p_{t}=-\frac{1}{4}\bigg{(}\frac{a_{n-1,x}}{p}\bigg{)}_{xx}+a_{n}^{0}(t)p_{x},$
(73)

with the recursive formulas (54) and (55). Some special equations in the WKI
hierarchy are given in the follows.

_Case 4._ For $n=1$ and $p=u_{x}$, Eq.(73) with (56) is exactly the WKI
equation

$\displaystyle
u_{xt}=\frac{a_{0}^{0}(t)}{4}\left[\frac{u_{xx}}{(1+u_{x}^{2})^{\frac{3}{2}}}\right]_{xx}+a^{0}_{1}(t)u_{xx}.$
(74)

_Case 5._ For $n=2$ and $p=u_{x}$, Eq.(73) with (2.2) is the high-order WKI
(i.e. WKI-SP(2,0)) equation

$\begin{split}&u_{xt}=\frac{a_{0}^{0}(t)}{16\left(1+u_{x}^{2}\right)^{\frac{13}{2}}}\Bigg{[}u_{xxxxxx}\left(1+u_{x}^{2}\right)^{4}-20u_{x}u_{xx}u_{xxxxx}\left(1+u_{x}^{2}\right)^{3}-35u_{xxxx}\Bigg{(}u_{xxx}\left(u_{x}^{3}+u_{x}\right)-\frac{39}{7}u_{xx}^{2}\\\
&(u_{x}^{2}-\frac{1}{6})\Bigg{)}\left(1+u_{x}^{2}\right)^{2}+270u_{xx}\Bigg{(}\left(1+u_{x}^{2}\right)^{2}(u_{x}^{2}-\frac{1}{6})u_{xxx}^{2}-\frac{77}{18}u_{x}u_{xxx}u_{xx}^{2}(u_{x}^{2}-\frac{1}{2})\left(1+u_{x}^{2}\right)\\\
&+\frac{28}{9}u_{xx}^{4}(u_{x}^{4}-u_{x}^{2}+\frac{1}{16})\Bigg{)}\Bigg{]}+\frac{1}{4}a_{1}^{0}\left(t\right)\left(\frac{u_{x}}{\sqrt{1+u_{x}^{2}}}\right)_{xxx}+a_{3}^{0}(t)u_{xx}.\end{split}$
(75)

####  2.2.3 Compound WKI-SP hierarchy for $n\neq 0,m\neq 0$

For $n\neq 0$ and $m\neq 0$, we have the generalized compound WKI-SP(n,m)
equations (49). Here, we give three examples.

_Case 6._ For $n=1,m=1$ and $p=u_{x}$, Eq.(49) gives the WKI-SP(1,1) equation

$\displaystyle
u_{xt}=\frac{a_{0}^{0}(t)}{4}\left[\frac{u_{xx}}{\bigg{(}1+u_{x}^{2}\bigg{)}^{\frac{3}{2}}}\right]_{xx}+a_{1}^{0}(t)u_{xx}+4a_{2}^{0}(t)\bigg{(}\frac{1}{6}(u^{3})_{xx}+u\bigg{)}.$
(76)

_Case 7._ For $n=1,m=2$ and $p=u_{xxx}$, we obtain the following WKI-SP(1,2)
equation

$\displaystyle
u_{xxxt}=\frac{a_{0}^{0}(t)}{4}\left[\frac{u_{xxxx}}{(1+u_{xxx}^{2})^{\frac{3}{2}}}\right]_{xx}+a_{2}^{0}(t)\bigg{(}\frac{2}{15}(u_{xx}^{5})_{xx}+[u_{xxx}(16uu_{xx}-8u_{x}^{2})]_{x}+\frac{8}{3}u_{xx}^{3}+16u\bigg{)}.$
(77)

_Case 8._ For $n=2,m=1$ and $p=u_{x}$, we obtain the following WKI-SP(2,1)
equation

$\begin{split}&u_{xt}=\frac{a_{0}^{0}(t)}{16\left(1+u_{x}^{2}\right)^{\frac{13}{2}}}\Bigg{[}u_{xxxxxx}\left(1+u_{x}^{2}\right)^{4}-20u_{x}u_{xx}u_{xxxxx}\left(1+u_{x}^{2}\right)^{3}-35u_{xxxx}\Bigg{(}u_{xxx}\left(u_{x}^{3}+u_{x}\right)-\frac{39}{7}u_{xx}^{2}\\\
&(u_{x}^{2}-\frac{1}{6})\Bigg{)}\left(1+u_{x}^{2}\right)^{2}+270u_{xx}\Bigg{(}\left(1+u_{x}^{2}\right)^{2}(u_{x}^{2}-\frac{1}{6})u_{xxx}^{2}-\frac{77}{18}u_{x}u_{xxx}u_{xx}^{2}\left(u_{x}^{2}-\frac{1}{2}\right)(1+u_{x}^{2})\\\
&+\frac{28}{9}u_{xx}^{4}(u_{x}^{4}-u_{x}^{2}+\frac{1}{16})\Bigg{)}\Bigg{]}+\frac{1}{4}a_{1}^{0}\left(t\right)\left(\frac{u_{x}}{\sqrt{1+u_{x}^{2}}}\right)_{xxx}+4a_{3}^{0}(t)\left(u+\frac{1}{6}\left(u^{3}\right)_{xx}\right).\end{split}$
(78)

##  3 Hodograph transformations between MKdV-SG and WKI-SP eqautions

In Ref.[29, 30], Gu obtained the compound equations of the MKdV-SG type

$\displaystyle
q_{t}-\frac{1}{4}\Bigg{[}4a_{n}q+\bigg{(}\frac{a_{nx}}{q}\bigg{)}_{x}\Bigg{]}_{x}+\frac{a_{n+1,x}}{q}=0$
(79)

together with the following two sets of recursive formulas

$\displaystyle a_{0x}=0,$ (80) $\displaystyle
a_{hx}=\frac{1}{4}q\Bigg{[}4a_{h-1}q+\bigg{(}\frac{a_{h-1,x}}{q}\bigg{)}_{x}\Bigg{]}_{x},h=1,2,...\,,n,$
(81)

and

$\displaystyle\Bigg{[}4a_{l+m}q+\bigg{(}\frac{a_{l+m,x}}{q}\bigg{)}_{x}\Bigg{]}_{x}=0,$
(82)
$\displaystyle\Bigg{[}4a_{h-1}q+\bigg{(}\frac{a_{h-1,x}}{q}\bigg{)}_{x}\Bigg{]}_{x}=4\frac{a_{h,x}}{q},h=n+m,...\,,n+2.$
(83)

In this section, we would build the hodograph transformations between some
members in the above MKdV-SG type equations and the WKI-SP(n,m) equation. To
see it clearly, we denote the concrete equation among (79) as MKdV-SG(n,m)
equation for specific $n$ and $m$.

###  3.1 Hodograph transformation between MKdV-SG(1,1) and WKI-SP(1,1)
equation

By taking $n=1,m=1$ and $q=-\frac{\theta_{s}}{2}$ in Eq.(79), the author gave
the MKdV-SG(1,1) equation

$\displaystyle\theta_{ts}-\frac{1}{4}\alpha_{0}(t)\biggl{(}\frac{3}{2}\theta_{s}^{2}\theta_{ss}+\theta_{ssss}\biggr{)}-\alpha_{1}(t)\theta_{ss}=\alpha_{2}(t)\sin{\theta},$
(84)

which is related to the motion of a nonlinear one-dimensional lattice of
atoms.

Integrating (84) with respect to $s$ leads to

$\displaystyle\theta_{t}-\frac{1}{4}\alpha_{0}(t)\Bigg{(}\frac{1}{2}\theta_{s}^{3}+\theta_{sss}\Bigg{)}-\alpha_{1}(t)\theta_{s}=\alpha_{2}(t)\int^{s}\sin{\theta}(s_{1},t)\mathrm{d}s_{1}.$
(85)

A conservation law of Eq.(85) is given by

$\displaystyle(\cos\theta)_{t}+\Bigg{[}\frac{1}{4}\alpha_{0}(t)\bigg{(}\theta_{ss}\sin{\theta}-\frac{1}{2}\theta_{s}^{2}\cos{\theta}\bigg{)}-\alpha_{1}(t)\cos{\theta}+\frac{1}{2}\alpha_{2}(t)\bigg{(}\int^{s}\sin{\theta}(s_{1},t)\mathrm{d}s_{1}\bigg{)}^{2}\Bigg{]}_{s}=0.$
(86)

Consider the hodograph transformation

$\displaystyle
x(s,t)=\int^{s}\cos\theta(s_{1},t){\rm{d}}s_{1}+x_{0},\quad\quad\quad\quad
t^{\prime}(s,t)=t,$ (87)

which leads to

$\displaystyle\frac{\partial}{\partial s}=\cos\theta\frac{\partial}{\partial
x},\quad\quad\frac{\partial}{\partial
t}=\Bigg{[}\frac{1}{4}\alpha_{0}(t)\bigg{(}\frac{1}{2}\theta_{s}^{2}\cos{\theta}-\theta_{ss}\sin{\theta}\bigg{)}+\alpha_{1}(t)\cos{\theta}-\frac{1}{2}\alpha_{2}(t)\bigg{(}\int^{s}\sin{\theta}(s_{1},t)\mathrm{d}s_{1}\bigg{)}^{2}\Bigg{]}\frac{\partial}{\partial
x}+\frac{\partial}{\partial t^{\prime}}.$ (88)

Applying (88) to (86), we obtain

$\displaystyle(\tan\theta)_{t^{\prime}}-\frac{1}{4}\alpha_{0}(t)(\sin{\theta})_{xxx}-\frac{\alpha_{2}(t)}{2}\bigg{(}\frac{(\int^{s}\sin{\theta}(s_{1},t)\mathrm{d}s_{1})^{2}}{\cos{\theta}}\bigg{)}_{x}\frac{1}{\sin{\theta}}=0,$
(89)

with

$\displaystyle\theta_{s}=(\cos\theta)\theta_{x}=(\sin\theta)_{x},\quad\theta_{ss}=(\cos\theta)(\theta_{s})_{x}=(\cos\theta)(\sin\theta)_{xx}.$
(90)

Introduce a new dependent variable

$\displaystyle u(s,t)=\int^{s}\sin\theta(s_{1},t){\rm{d}}s_{1},$ (91)

which gives

$\displaystyle u_{s}=\sin\theta.$ (92)

Applying (88) to (91), we obtain

$\displaystyle u_{s}=(\cos\theta)u_{x}.$ (93)

Combing (92) with (93) gives

$\displaystyle\tan\theta=u_{x},$ (94)

then it follows

$\displaystyle\sin\theta=\frac{u_{x}}{\sqrt{1+u_{x}^{2}}},\quad\quad\cos\theta=\frac{1}{\sqrt{1+u_{x}^{2}}}.$
(95)

Substituting (94) and (95) into (89) yields

$\displaystyle
u_{xt^{\prime}}-\frac{1}{4}\alpha_{0}(t)\bigg{(}\frac{u_{x}}{\sqrt{1+u_{x}^{2}}}\bigg{)}_{xxx}-\alpha_{2}(t)\bigg{(}u+\frac{1}{6}(u^{3})_{xx}\bigg{)}=0,$
(96)

which is just the WKI-SP(1,1) equation given by (76) with $a^{0}_{1}(t)=0$.

###  3.2 Hodograph transformation between MKdV-SG(2,1) and WKI-SP(2,1)
equation

For $n=2,m=1$, the MKdV-SG(2,1) equation is given by

$\displaystyle
q_{t}-\frac{1}{4}\bigg{[}\bigg{(}\frac{a_{2,s}}{q}\bigg{)}_{s}+4a_{2}q\bigg{]}_{s}+\frac{a_{3,s}}{q}=0.$
(97)

By solving (81), one obtain

$\displaystyle a_{0}=\alpha_{0}(t),$ (98) $\displaystyle
a_{1}=\frac{1}{2}\alpha_{0}(t)q^{2}+\alpha_{1}(t),$ (99) $\displaystyle
a_{2}=\alpha_{0}(t)\bigg{(}\frac{1}{4}qq_{ss}-\frac{1}{8}q_{s}^{2}+\frac{3}{8}q^{4}\bigg{)}+\frac{1}{2}\alpha_{1}(t)q^{2}+\alpha_{2}(t).$
(100)

Let $q=-\frac{\theta_{s}}{2}$ and choose the following special solution for
$a_{3}$ in (83)

$\displaystyle a_{3}=\frac{1}{4}\alpha_{3}(t)\cos\theta.$ (101)

The MKdV-SG(2,1) equation (97) is expressed by

$\begin{split}&\theta_{st}+\bigg{[}\frac{1}{128}\alpha_{0}(t)\bigg{(}-8\theta_{ssssss}-20\theta_{ss}^{3}-80\theta_{s}\theta_{ss}\theta_{sss}-20\theta_{s}^{2}\theta_{ssss}-15\theta_{s}^{4}\theta_{ss}\bigg{)}+\alpha_{1}(t)(-\frac{1}{4}\theta_{ssss}-\frac{3}{8}\theta_{s}^{2}\theta_{ss})-\alpha_{2}(t)\theta_{ss}\bigg{]}\\\
&-\alpha_{3}(t)\sin\theta=0.\end{split}$ (102)

Integrating Eq.(102) with respect to $s$ leads to

$\displaystyle\theta_{t}-\bigg{[}\frac{1}{128}\alpha_{0}(t)\bigg{(}8\theta_{sssss}+20\theta_{s}\theta_{ss}^{2}+20\theta_{s}^{2}\theta_{sss}+3\theta_{s}^{5}\bigg{)}-\frac{1}{8}\alpha_{1}(t)\bigg{(}2\theta_{sss}+\theta_{s}^{3}\bigg{)}-\alpha_{2}(t)\theta_{s}\bigg{]}=\alpha_{3}(t)\int^{s}\sin\theta(s_{1},t)\mathrm{d}s_{1}.$
(103)

Multiplying both sides of Eq.(103) by $-\sin\theta$, one conservation law is
given by

$\displaystyle\begin{split}&(\cos\theta)_{t}+\bigg{[}\frac{1}{128}\alpha_{0}(t)\bigg{(}12\theta_{s}^{2}\theta_{ss}\sin\theta+8\theta_{ssss}\sin\theta+4\theta_{ss}^{2}\cos\theta-3\theta_{s}^{4}\cos\theta-8\theta_{s}\theta_{sss}\cos\theta\bigg{)}\\\
&+\frac{1}{8}\alpha_{1}(t)\bigg{(}2\theta_{ss}\sin\theta-\theta_{s}^{2}\cos\theta\bigg{)}-\alpha_{2}(t)\cos\theta+\frac{1}{2}\alpha_{3}(t)\bigg{(}\int^{s}\sin\theta(s_{1},t)\mathrm{d}s_{1}\bigg{)}^{2}\bigg{]}_{s}=0.\end{split}$
(104)

Consider the same hodograph transformation (87). It leads to

$\displaystyle\frac{\partial}{\partial s}=\cos\theta\frac{\partial}{\partial
x},$ (105) $\displaystyle\begin{split}&\frac{\partial}{\partial
t}=\bigg{[}-\frac{1}{128}\alpha_{0}(t)\bigg{(}12\theta_{s}^{2}\theta_{ss}\sin\theta+8\theta_{ssss}\sin\theta+4\theta_{ss}^{2}\cos\theta-3\theta_{s}^{4}\cos\theta-8\theta_{s}\theta_{sss}\cos\theta\bigg{)}\\\
&\qquad-\frac{1}{8}\alpha_{1}(t)\bigg{(}2\theta_{ss}\sin\theta-\theta_{s}^{2}\cos\theta\bigg{)}+\alpha_{2}(t)\cos\theta-\frac{1}{2}\alpha_{3}(t)\bigg{(}\int^{s}\sin\theta(s_{1},t)\mathrm{d}s_{1}\bigg{)}^{2}\bigg{]}\frac{\partial}{\partial
x}+\frac{\partial}{\partial t^{\prime}}.\end{split}$ (106)

We have

$\displaystyle\theta_{s}=(\cos\theta)\theta_{x}=(\sin\theta)_{x},\quad\theta_{ss}=\cos\theta(\theta_{s})_{x}=\cos\theta(\sin\theta)_{xx},$
(107)
$\displaystyle\theta_{sss}=\cos^{2}\theta\sin\theta_{xxx}+{\cos\theta}(\cos{\theta})_{x}(\sin\theta)_{xx},$
(108)
$\displaystyle\begin{split}&\theta_{ssss}=2\cos^{2}\theta(\cos\theta)_{x}(\sin\theta)_{xxx}+\cos^{3}\theta(\sin{\theta})_{xxxx}+\cos{\theta}(\cos\theta)_{x}^{2}(\sin\theta)_{xx}\\\
&+\cos^{2}\theta(\cos\theta)_{xx}(\sin\theta)_{xx}+\cos^{2}\theta(\cos\theta)_{x}(\sin\theta)_{xxx}.\end{split}$
(109)

Applying (105) and (106) to Eq.(104), we obtain

$\displaystyle\begin{split}&(\tan\theta)_{t^{\prime}}+\frac{\alpha_{0}(t)}{128}\bigg{[}24(\sin\theta)_{x}(\sin\theta)_{xx}^{2}+12(\sin\theta)_{x}^{2}(\sin\theta)_{xxx}+16(\sin\theta)_{xxx}(\cos\theta)_{x}^{2}+16(\sin\theta)_{xxx}\cos{\theta}(\cos\theta)_{xx}\\\
&+32(\sin\theta)_{xxxx}\cos{\theta}(\cos\theta)_{x}+8(\sin\theta)_{xxxxx}\cos^{2}\theta+16(\sin\theta)_{xx}(\cos\theta)_{x}(\cos\theta)_{xx}+8(\sin\theta)_{xxx}(\cos\theta)_{x}^{2}\\\
&+8(\sin\theta)_{xx}(\cos\theta)_{x}(\cos\theta)_{xx}+8(\sin\theta)_{xx}\cos\theta(\cos\theta)_{xxx}+8(\sin\theta)_{xxx}\cos\theta(\cos\theta)_{xx}+8(\sin\theta)_{xxx}(\cos\theta)_{x}^{2}\\\
&+8(\sin\theta)_{xxx}(\cos\theta)_{x}^{2}+8(\sin\theta)_{xxx}\cos\theta(\cos\theta)_{xx}+8(\sin\theta)_{xxxx}\cos\theta(\cos\theta)_{x}\bigg{]}-\frac{1}{4}\alpha_{1}(t)(\sin\theta)_{xxx}\\\
&-\frac{\alpha_{3}(t)}{2}\frac{1}{\sin\theta}\bigg{[}\frac{\bigg{(}\int^{s}\sin\theta(s_{1},t)\mathrm{d}s_{1}\bigg{)}^{2}}{\cos\theta}\bigg{]}_{x}=0.\end{split}$
(110)

Introduce a new dependent variable
$v(s,t)=\int^{s}\sin\theta(s_{1},t){\rm{d}}s_{1}$ (note $\tan\theta=v_{x}$,
$\sin\theta=\frac{v_{x}}{\sqrt{1+v_{x}^{2}}},\cos\theta=\frac{1}{\sqrt{1+v_{x}^{2}}}$).
Then Eq.(110) is transformed to

$\begin{split}&v_{xt^{\prime}}-\frac{\alpha_{0}\left(t\right)}{16\left(1+v_{x}^{2}\right)^{\frac{13}{2}}}\Bigg{[}v_{xxxxxx}\left(1+v_{x}^{2}\right)^{4}-20v_{x}v_{xx}v_{xxxxx}\left(1+v_{x}^{2}\right)^{3}-35v_{xxxx}\Bigg{(}v_{xxx}\left(v_{x}^{3}+v_{x}\right)-\frac{39}{7}v_{xx}^{2}\\\
&(v_{x}^{2}-\frac{1}{6})\Bigg{)}\left(1+v_{x}^{2}\right)^{2}+270v_{xx}\Bigg{(}\left(1+v_{x}^{2}\right)^{2}(v_{x}^{2}-\frac{1}{6})v_{xxx}^{2}-\frac{77}{18}v_{x}v_{xxx}v_{xx}^{2}(v_{x}^{2}-\frac{1}{2})\left(1+v_{x}^{2}\right)\\\
&+\frac{28}{9}v_{xx}^{4}\left(v_{x}^{4}-v_{x}^{2}+\frac{1}{16}\right)\Bigg{)}\Bigg{]}-\frac{1}{4}\alpha_{1}\left(t\right)\left(\frac{v_{x}}{\sqrt{1+v_{x}^{2}}}\right)_{xxx}-\alpha_{3}(t)\left(v+\frac{1}{6}\left(v^{3}\right)_{xx}\right)=0,\end{split}$
(111)

which is just the WKI-SP(2,1) equation (78).

###  3.3 Hodograph transformation between MKdV-SG(1,2) and WKI-SP(1,2)
equation

For $n=1,m=2$, the MKdV-SG(1,2) equation is given by

$\displaystyle
q_{t}-\frac{1}{4}\bigg{[}\bigg{(}\frac{a_{1,s}}{q}\bigg{)}_{s}+4a_{1}q\bigg{]}_{s}+\frac{a_{2s}}{q}=0.$
(112)

By solving the recursive formulas (81) and (83) with
$q=-\frac{\theta_{s}}{2}$, we get

$\displaystyle a_{0}=\alpha_{0}(t),\quad
a_{1}=\frac{1}{2}\alpha_{0}(t)q^{2}+\alpha_{1}(t),\quad
a_{3}=\frac{1}{4}\alpha_{3}(t)\cos{\theta},\quad$ (113)

and

$\begin{split}&\frac{a_{2s}}{q}=C_{1}\sin{\theta}+C_{2}\cos{\theta}+2\sin{\theta}\int^{s}\sin\theta(s,t)\left(\int^{s_{1}}\sin\theta(s_{2},t)\mathrm{d}s_{2}\right)\mathrm{d}s_{1}\\\
&+2\cos{\theta}\int^{s}\cos\theta(s_{1},t)\left(\int^{s_{1}}\sin\theta(s_{2},t)\mathrm{d}s_{2}\right)\mathrm{d}s_{1},\end{split}$
(114)

with $q=-\frac{\theta_{s}}{2}$.

Then the MKdV-SG(1,2) equation (97) is rewritten as

$\begin{split}&\theta_{t}=\frac{1}{4}\alpha_{0}(t)\theta_{sss}+\frac{1}{8}\alpha_{0}(t)\theta_{s}^{3}+\alpha_{1}(t)\theta_{s}+2C_{1}\int^{s}\sin{\theta}(s_{1},t)\mathrm{d}s_{1}\\\
&+2C_{2}\int^{s}\cos{\theta}(s_{1},t)\mathrm{d}s_{1}+\frac{2}{3}\biggl{(}\int^{s}\sin{\theta}(s_{1},t)\mathrm{d}s_{1}\biggr{)}^{3}\\\
&+4\int^{s}\cos{\theta}(s_{1},t)\biggl{[}\int^{s_{1}}\sin{\theta}(s_{2},t)\biggl{(}\int^{s_{2}}\sin{\theta}(s_{3},t)\mathrm{d}s_{3}\biggr{)}\mathrm{d}s_{2}\biggr{]}\mathrm{d}s_{1}.\end{split}$
(115)

The corresponding conservation law of Eq.(115) is

$\begin{split}&(\cos\theta)_{t}=\Bigg{[}-\frac{1}{8}\alpha_{0}(t)\left(2\theta_{ss}-\theta_{s}^{2}\cos{\theta}\right)+\alpha_{1}(t)\cos{\theta}-C_{1}\left(\int^{s}\sin{\theta}(s_{1},t)\mathrm{d}s_{1}\right)^{2}\\\
&-\frac{1}{6}\left(\int^{s}\sin{\theta}(s_{1},t)\mathrm{d}s_{1}\right)-\int^{s}A\mathrm{d}s_{1}\Bigg{]}_{s}\end{split}$
(116)

with

$\begin{split}&A=2C_{2}\sin{\theta}\int^{s}\cos{\theta}(s_{1},t)\mathrm{d}s_{1}+4\sin{\theta}\int^{s}\cos{\theta}(s_{1},t)\left[\int^{s_{1}}\cos{\theta}(s_{2},t)\left(\int^{s_{2}}\cos{\theta}(s_{3},t)\mathrm{d}s_{3}\right)\mathrm{d}s_{2}\right]\mathrm{d}s_{1}.\end{split}$
(117)

The same hodograph transformation (87) leads (116) to

$\displaystyle\frac{\partial}{\partial s}=\cos\theta\frac{\partial}{\partial
x},$ (118) $\displaystyle\begin{split}&\frac{\partial}{\partial
t}=\Biggl{[}-\frac{1}{8}\alpha_{0}(t)(2\theta_{ss}\sin{\theta}-\theta_{s}^{2}\cos{\theta})+\alpha_{1}(t)\cos{\theta}-C_{1}\biggl{(}\int^{s}\sin{\theta}(s_{1},t)\mathrm{d}s_{1}\biggr{)}-\frac{1}{6}\biggl{(}\int^{s}\sin{\theta}(s_{1},t)\mathrm{d}s_{1}\biggr{)}^{4}\\\
&-\int^{s}A\mathrm{d}s\Biggr{]}\frac{\partial}{\partial
x}+\frac{\partial}{\partial t^{\prime}}.\end{split}$ (119)

Applying (118) and (119) to Eq.(116), we obtain

$\displaystyle(\tan(\theta))_{t^{\prime}}-\frac{1}{4}(\sin{\theta})_{xxx}-\Biggl{[}\frac{C_{1}\left(\int^{s}\sin{\theta}(s_{1},t)\mathrm{d}s_{1}\right)^{2}+\frac{1}{6}\left(\int^{s}\sin{\theta}(s_{1},t)\mathrm{d}s_{1}\right)^{4}-\int^{s}A\mathrm{d}s_{1}}{\cos{\theta}}\Biggr{]}_{x}\frac{1}{\sin{\theta}}=0.$
(120)

Introduce a new dependent variable
$v(s,t)=\int^{s}\sin\theta(s_{1},t){\rm{d}}s_{1}$ (note $\tan\theta=v_{x}$,
$\sin\theta=\frac{v_{x}}{\sqrt{1+v_{x}^{2}}},\cos\theta=\frac{1}{\sqrt{1+v_{x}^{2}}}$).
Then Eq.(120) is transformed to

$\begin{split}&v_{xt^{\prime}}=\frac{1}{4}\alpha_{0}(t)\left(\frac{v_{x}}{\sqrt{1+v_{x}^{2}}}\right)_{xxx}+C_{1}v^{2}v_{xx}+2C_{2}xv_{xx}+2C_{2}v_{xx}\int^{x}v\mathrm{d}x^{\prime}+4v_{xx}\int^{x}v^{\prime}_{x}\left[\left(\int^{x^{\prime}}v\mathrm{d}x^{\prime\prime\prime}\right)\mathrm{d}x^{\prime\prime}\right]\mathrm{d}x^{\prime}\\\
&+2C_{1}v(1+v_{x}^{2})+\frac{2}{3}v^{3}(1+v_{x}^{2})+\frac{1}{6}v^{4}v_{xx}+2C_{2}x(1+v_{x}^{2})+4(1+v_{x}^{2})\int^{x}\left(\int^{x^{\prime}}v\mathrm{d}x^{\prime\prime}\right)\mathrm{d}x^{\prime}.\end{split}$
(121)

We set $v=u_{xx}$ with $C_{1}=C_{2}=0$ and obtain

$\begin{split}&u_{xxxt^{\prime}}=\frac{1}{4}\alpha_{0}(t)\left(\frac{u_{xxx}}{\sqrt{1+u_{xxx}^{2}}}\right)_{xxx}+\frac{1}{30}(u_{xx}^{5})_{xx}+\left[(4uu_{xx}-2u_{x}^{2})u_{xxx}\right]_{x}+\frac{2}{3}u_{xx}^{3}+4u.\end{split}$
(122)

Now Eq.(122) is just the WKI-SP(1,2) equation.

Remark 1: We believe that the hodograph transformation defined by (87) and
(91) is not only true of the three pairs of equations given above, but also
valid for the the whole MKdV-SG and the WKI-SP hierarchy.

##  4 Loop soliton solutions of the WKI-SP(1,1) equation

For simplicity, we rewrite the MKdV-SG(1,1) equation and the WKI-SP(1,1)
equation as

$\displaystyle\theta_{ts}+\alpha\bigg{[}\frac{3}{2}\theta_{s}^{2}\theta_{ss}+\theta_{ssss}\bigg{]}=\beta\sin(\theta)$
(123)

and

$\displaystyle
u_{xt}+\alpha\bigg{(}\frac{u_{x}}{\sqrt{1+u_{x}^{2}}}\bigg{)}_{xxx}=\beta\bigg{(}u+\frac{1}{6}(u^{3})_{xx}\bigg{)}$
(124)

with $\alpha$ and $\beta$ being arbitrary real constants.

In Ref.[33, 34], the authors presented the following $N$-soliton solution for
the MKdV-SG(1,1) equation (123):

$\displaystyle\theta=2i\ln\frac{{f}^{*}}{f}$ (125)

with

$\displaystyle
f=f_{N}=\sum_{\mu=0,1}\exp\bigg{[}\sum_{j=1}^{N}\mu_{j}(\xi_{j}+\frac{\pi}{2}i)+\sum_{1\leq
j<l\leq N}^{N}\mu_{j}\mu_{l}A_{jl}\bigg{]},$ (126)
$\displaystyle\xi_{j}=k_{j}s+\omega_{j}t+\xi_{j0},\quad\quad\omega_{j}=\frac{\beta}{k_{j}}-\alpha
k_{j}^{3},\quad\quad{\rm{e}}^{A_{jl}}=\frac{(k_{j}-k_{l})^{2}}{(k_{j}+k_{l})^{2}},\quad\quad
k_{i}\neq\pm k_{j}.$ (127)

where the summation of $\mu\equiv\\{\mu_{1},\mu_{2},\ldots,\mu_{N}\\}$ should
be done for all permutations of $\mu_{j}=0,1,j=1,2,\ldots,N$. Here${f}^{*}$ is
the conjugate of $f$.

Due to the hodograph transformation (87) and (91), the loop soliton solutions
of the WKI-SP(1,1) equation are represented by the parametric form as

$\displaystyle
u=\frac{1}{\beta}[\theta_{t}+\alpha(\frac{1}{2}\theta_{s}^{3}+\theta_{sss})],\quad(\theta=2i\ln\frac{\bar{f}}{f}),$
(128) $\displaystyle
x=s-\frac{2}{\beta}\bigg{[}(\ln{f\bar{f}})_{t}+\alpha(\ln{f\bar{f}})_{sss}+6\alpha\int^{s}\bigg{(}(\ln
f)^{2}_{ss}+(\ln\bar{f})^{2}_{ss}\bigg{)}\mathrm{d}s\bigg{]}+x_{0},$ (129)

where $f$ satisfies (126).

###  4.1 One-loop soliton solution

In the case of $N=1$, we have

$\displaystyle
f=1+{\rm{ie}}^{\xi_{1}},\quad\xi_{1}=k_{1}s+\bigg{(}\frac{\beta}{k_{1}}-\alpha
k_{1}^{3}\bigg{)}t+\xi_{10}.$ (130)

Substituting (130) into (128) and (129) leads to

$\displaystyle u(s,t)=\frac{2}{k_{1}}{\rm{sech}}\xi_{1},$ (131) $\displaystyle
x(s,t)=s-\frac{2}{k_{1}}\tanh\xi_{1}+d+\frac{2}{k_{1}},$ (132)

which is the well-known one soliton solution and depicted in Figures 1 and 2.
The positive and negative $k_{1}$ correspond to loop soliton and antiloop
soliton, respectively. Most of the time, we call both of them loop soliton for
convenience.

Figure 1: a, b, and c are the profiles of x as a function of s, u as a
function of s, and one-loop soliton solution of the WKI-SP(1,1) equation with
parameters $\alpha=2$, $\beta=2$, $k_{1}=1.5$, $\xi_{10}=0$ and
$d_{1}=-\frac{4}{3}$ respectively.

Figure 2: a, b, and c are the profiles of x, as a function of s, u as a
function of s, and one-antiloop soliton solution of the WKI-SP equation with
parameters $\alpha=2$, $\beta=2$, $k_{1}=-1.5$, $\xi_{10}=0$ and
$d_{1}=\frac{4}{3}$ respectively.

To clarify the motion of the loop in the $(x,t)$ coordinate system, we rewrite
(132) into the form

$\displaystyle
x-c_{1}t+x_{10}=\frac{\xi_{1}}{k_{1}}-\frac{2}{k_{1}}\tanh\xi_{1}+d+\frac{2}{k_{1}},$
(133)

with $c_{1}={\alpha}k_{1}^{2}-\frac{\beta}{k_{1}^{2}}$ and
$x_{10}=\frac{\xi_{10}}{k_{1}}$.

Here, $c_{1}$ denotes the velocity of the loop soliton. For the given nonzero
$\alpha$ and $\beta$, it shows that the loop soliton could propagate to either
the left (i.e.,negative x direction) or the right (i.e., positive x
direction). This is different from the SP equation and the WKI equation. The
soliton of the SP equation and the WKI equation can only move in one direction
of $x$-axis. From (131), we see the amplitude (defined by $A_{1}$) of the loop
soliton is $\frac{2}{k_{1}}$. Then we have

$\displaystyle c_{1}=\frac{4\alpha}{A_{1}^{2}}-\frac{{\beta}A_{1}^{2}}{4}.$
(134)

Interestingly, the above expression shows the loop solitons of the WKI-SP(1,1)
equation have abundant solitonic behaviors. On the one hand, the large loop
can move more rapidly than the small loop. On the other hand, the small loop
can also move more rapidly than the large loop, which is different from the
typical solitonic movement.

###  4.2 Interaction of two loop solitons

For the two-loop soliton solution, (126) gives

$\displaystyle f=1+{\rm ie}^{\xi_{1}}+{\rm ie}^{\xi_{2}}-{\gamma}{\rm
e}^{\xi_{1}+\xi_{2}}$ (135)

with

$\displaystyle\gamma=\bigg{(}\frac{k_{1}-k_{2}}{k_{1}+k_{2}}\bigg{)}^{2}.$
(136)

Due to (128) and (129), the parametric representation of the solution reads

$\displaystyle
u(s,t)=\frac{2\sqrt{\gamma}}{k_{1}k_{2}}\frac{(k_{1}+k_{2})\cosh\Psi_{1}\cosh\Psi_{2}+(k_{1}-k_{2})\sinh\Psi_{1}\sinh\Psi_{2}}{\cosh^{2}\Psi_{1}+{\gamma}\sinh^{2}\Psi_{2}},$
(137) $\displaystyle x(s,t)=s+\frac{1}{k_{1}k_{2}}\frac{(k_{1}-k_{2})\sinh
2\Psi_{1}-\gamma(k_{1}+k_{2})\sinh
2\Psi_{2}}{\cosh^{2}\Psi_{1}+\gamma\sinh^{2}\Psi_{2}}+\frac{2(k_{1}+k_{2})}{k_{1}k_{2}}+d,$
(138)

where we have put

$\displaystyle\Psi_{1}=\frac{1}{2}(\xi_{1}-\xi_{2}),\quad\Psi_{2}=\frac{1}{2}(\xi_{1}+\xi_{2})+\frac{1}{2}\ln\gamma$
(139)

for simplicity.

The two-loop soliton has abundant solitonic behaviors. Two loops can move in
the same direction of the $x$-axes while they could move towards each other in
the opposite directions. It could be the fast small loop chasing the slow high
one, and it might be the fast high loop chasing the slow small one. To figure
out what each parameter does, we rewrite the amplitude and the velocity as
follows

$\displaystyle A_{i}=\frac{2}{k_{i}},\quad
c_{i}={\alpha}k_{i}^{2}-\frac{\beta}{k_{i}^{2}},\quad(i=1,2).$ (140)

A table is listed to intuitively show the chasing or the meeting movement.
Since $|k_{1}|\neq|k_{2}|$, we assume $|k_{1}|>|k_{2}|$ without generality. In
the following table, $v_{low}$ represents the speed of the small loop soliton
while $v_{high}$ represents the speed of the large loop soliton.

Table 1: The role of each parameter plays in the movement of two-loop soliton | one loop chases another loop |  two loops
---|---|---
both move to the positive $x$-axis |  both move to the negative $x$-axis |  $v_{low}>v_{high}$ |  $v_{low}<v_{high}$ |  meet each other
|  $\sqrt[4]{\frac{\beta}{\alpha}}<|k_{2}|<|k_{1}|$ |  $\checkmark$ $(\alpha>0,\beta>0)$ |  $\checkmark$ $(\alpha<0,\beta<0)$ |  $\checkmark$ |  |
$\alpha\beta>0$ |  $|k_{2}|<|k_{1}|<\sqrt[4]{\frac{\beta}{\alpha}}$ |  $\checkmark$ $(\alpha<0,\beta<0)$ |  $\checkmark$ $(\alpha>0,\beta>0)$ |  |  $\checkmark$ |
|  $|k_{2}|<\sqrt[4]{\frac{\beta}{\alpha}}<|k_{1}|$ |  |  |  |  |  $\checkmark$
|  $\frac{\sqrt{-\alpha\beta}}{|\alpha||k_{1}|}<|k_{2}|<|k_{1}|$ |  $\checkmark$ $(\alpha>0,\beta<0)$ |  $\checkmark$ $(\alpha<0,\beta>0)$ |  $\checkmark$ |  |
$\alpha\beta<0$ |  $|k_{2}|<min\\{|k_{1}|,\frac{\sqrt{-\alpha\beta}}{|\alpha||k_{1}|}\\}$ |  $\checkmark$ $(\alpha>0,\beta<0)$ |  $\checkmark$ $(\alpha<0,\beta>0)$ |  |  $\checkmark$ |

Some diverse figures are worked out to exhibit different solitonic movements.
In Figures 3 and 4, both the two loop solitons are moving to the right. There
are both a small but fast loop soliton chasing a larger and slower loop
soliton. Two loop solitons with dissimilar amplitudes are shown in Figures 3,
in which a small and fast loop soliton maybe observed traveling around a
larger and slower loop soliton. Two loop solitons with similar amplitudes
($k_{1}\approx k_{2}$) are shown in Figures 4, in which the loops do not over
lap and they just seems to exchange their amplitudes during the period of the
nonlinear interaction.

Figure 3: The interaction process for the pursuing of two loop solitons with
$\alpha=0.5$, $\beta=-0.5$, $k_{1}=2$, $k_{2}=0.6$, $\xi_{10}=-1.5$,
$\xi_{20}=0$ and $x_{0}=-5.5$.

Figure 4: The interaction process for the pursuing of two loop solitons with
$\alpha=-0.5$, $\beta=-0.5$, $k_{1}=0.9$, $k_{2}=0.6$, $\xi_{10}=1$,
$\xi_{20}=0$ and $x_{0}=-5$.

Figures 5 exhibit two loop solitons are moving to the left. Here a small and
slower loop soliton catches up with a larger and fast loop soliton.

Figure 5: The interaction process for the collision of two loop solitons with
$\alpha=-0.5$, $\beta=0.5$, $k_{1}=1.5$, $k_{2}=0.7$, $\xi_{10}=0$,
$\xi_{20}=0$ and $x_{0}=-18.5$.

Figures 6 illustrate one loop soliton and one antiloop soliton move towards
each other in the opposite directions. The antiloop travels to the right and
the loop travels to the left. As time goes, two solitons merge and then they
separate each other with leaving the original wave profiles.

Figure 6: The interaction process for the collision of one loop and one
antiloop soliton with $\alpha=0.5$, $\beta=0.5$, $k_{1}=-1.5$, $k_{2}=0.8$,
$\xi_{10}=-3.4$, $\xi_{20}=0$ and $x_{0}=0.3$.

###  4.3 Interaction of three loop solitons

For the three-loop soliton case, formulas (126) reduce to

$\displaystyle f=1+{\rm ie}^{\xi_{1}}+{\rm ie}^{\xi_{2}}+{\rm
ie}^{\xi_{3}}-{\gamma_{12}}{\rm e}^{\xi_{1}+\xi_{2}}-{\gamma_{13}}{\rm
e}^{\xi_{1}+\xi_{3}}-{\gamma_{23}}{\rm e}^{\xi_{2}+\xi_{3}}-{\rm
i}{\gamma_{123}}{\rm e}^{\xi_{1}+\xi_{2}+\xi_{3}}$ (141)

with

$\displaystyle\gamma_{ij}=\bigg{(}\frac{k_{i}-k_{j}}{k_{i}+k_{j}}\bigg{)}^{2},\quad\gamma_{123}=\gamma_{12}\gamma_{13}\gamma_{23}.$
(142)

By virtue of (128) and (129), the three-loop soliton in the parametric form is
provided as follows

$\displaystyle u(s,t)=\frac{2\Xi_{1}}{\Xi},$ (143) $\displaystyle
x(s,t)=s+\frac{2\Xi_{2}}{\Xi}+\frac{2(k_{1}k_{2}+k_{1}k_{3}+k_{2}k_{3})}{k_{1}k_{2}k_{3}}+d.$
(144)

where

$\displaystyle\begin{split}\Xi&=k_{1}k_{2}k_{3}[2(\gamma_{12}\sinh^{2}\Psi_{3}+\gamma_{13}\sinh^{2}\Psi_{2}+\gamma_{23}\sinh^{2}\Psi_{1})+2(1-\gamma_{13})\sqrt{\gamma_{12}\gamma_{23}}\cosh(\Psi_{1}+\Psi_{3})\\\
&+2(1-\gamma_{23})\sqrt{\gamma_{12}\gamma_{13}}\cosh(\Psi_{2}+\Psi_{3})+2(1-\gamma_{12})\sqrt{\gamma_{13}\gamma_{23}}\cosh(\Psi_{1}+\Psi_{2})\\\
&+\gamma_{123}\cosh(2(\Psi_{1}+\Psi_{2}+\Psi_{3}))+(\gamma_{12}+\gamma_{13}+\gamma_{23})]\end{split}$
(145)

$\begin{split}\Xi_{1}=&\sum_{1\leqslant m<n\leqslant
3}\gamma_{m,n}\bigg{(}\sum_{1\leqslant i<j\leqslant
3}k_{i}k_{j}-2k_{m}k_{n}\bigg{)}-\gamma_{123}\sum_{1\leqslant i<j\leqslant
3}k_{i}k_{j}\\\
&+2k_{1}k_{2}\sqrt{\gamma_{13}\gamma_{23}}\bigg{[}\gamma_{12}\cosh(\Psi_{1}+\Psi_{2}+2\Psi_{3})+\cosh(\Psi_{1}-\Psi_{2})\bigg{]}\\\
&+2k_{1}k_{3}\sqrt{\gamma_{12}\gamma_{23}}\bigg{[}\gamma_{13}\cosh(\Psi_{1}+2\Psi_{2}+\Psi_{3})+\cosh(\Psi_{1}-\Psi_{3})\bigg{]}\\\
&+2k_{2}k_{3}\sqrt{\gamma_{12}\gamma_{13}}\bigg{[}\gamma_{23}\cosh(2\Psi_{1}+\Psi_{2}+\Psi_{3})+\cosh(\Psi_{2}-\Psi_{3})\bigg{]}\end{split}$
(146)

$\begin{split}\Xi_{2}=&\gamma_{123}(k_{1}k_{2}+k_{1}k_{3}+k_{2}k_{3})\sinh(2(\Psi_{1}+\Psi_{2}+\Psi_{3}))+\gamma_{12}(-k_{1}k_{2}+k_{1}k_{3}+k_{2}k_{3})\sinh(2\Psi_{3})\\\
&+\gamma_{13}(k_{1}k_{2}-k_{1}k_{3}+k_{2}k_{3})\sinh(2\Psi_{2})+\gamma_{23}(k_{1}k_{2}+k_{1}k_{3}-k_{2}k_{3})\sinh(2\Psi_{1})\\\
&+k_{2}k_{3}(\gamma_{23}-1)\sqrt{\gamma_{12}\gamma_{13}}\sinh(\Psi_{2}+\Psi_{3})+k_{1}k_{3}(\gamma_{13}-1)\sqrt{\gamma_{12}\gamma_{23}}\sinh(\Psi_{1}+\Psi_{3})\\\
&+k_{1}k_{2}(\gamma_{12}-1)\sqrt{\gamma_{13}\gamma_{23}}\sinh(\Psi_{1}+\Psi_{2})\end{split}$
(147)

with

$\displaystyle\Psi_{1}=\frac{1}{2}(-\xi_{1}+\xi_{2}+\xi_{3})+\frac{1}{2}\ln(\gamma_{23}),$
(148)
$\displaystyle\Psi_{2}=\frac{1}{2}(\xi_{1}-\xi_{2}+\xi_{3})+\frac{1}{2}\ln(\gamma_{13}),$
(149)
$\displaystyle\Psi_{3}=\frac{1}{2}(\xi_{1}+\xi_{2}-\xi_{3})+\frac{1}{2}\ln(\gamma_{12}).$
(150)

Two examples of such solution are plotted below (see Figures 7 and 8). Figures
7 describe three loop solitons with different speed propagating to the
positive direction. It shows rich catch-up phenomenon. Figures 8 describes two
loop solitons colliding with one antiloop soliton.

Figure 7: The interaction process for three loop solitons with $\alpha=0.2$,
$\beta=0.01$, $k_{1}=0.5$, $k_{2}=1$, $k_{3}=1.5$, $\xi_{10}=0$,
$\xi_{20}=-0.9$, $\xi_{30}=0.3$ and $d=-11$.

Figure 8: The interaction process for two loops and one antiloop soliton with
$\alpha=0.2$, $\beta=0.2$, $k_{1}=1.2$, $k_{2}=2.5$, $k_{3}=-0.5$,
$\xi_{10}=-2.9$, $\xi_{20}=-5.9$, $\xi_{30}=0$ and $d=-1.1$.

_Remark 2._ Due to the multiloop soliton solutions obtained in this section,
the corresponding multibreather solutions and soliton molecules would be
constructed directly. Under certain condition imposed on the parameters, the
breather solution is shown to yield a nonsingular oscillating pulse solution
of the WKI-SP(1,1) equation, which we shall term the breather solution as
well. In fact, the $M$-breather solution of the WKI-SP(1,1) equation can be
constructed from the $M$-breather solution of the MKdV-SG(1,1) equation
(125)-(127) with $N=2M$ by choosing the parameters appropriately. So we
specify the parameters as

$\displaystyle k_{2j-1}=k_{2j}^{*},\quad\xi_{2j-1,0}=\xi_{2j,0}^{*}$ (151)

to satisfy

$\displaystyle\xi_{2j-1}=\xi_{2j}^{*}.$ (152)

The parametric solution (128) and (129) with (151) and (152) describes
multiple collisions of $M$ breathers.

Soliton molecules [35, 36, 37, 38] can be formed in some possible mechanisms
both theoretically and experimentally. Here, we can obtain loop soliton
molecules of the WKI-SP(1,1) equation by introducing the velocity resonance
conditions[39]

$\displaystyle\frac{k_{i}}{k_{j}}=\frac{\alpha
k_{i}^{3}-\frac{\beta}{k_{i}}}{\alpha k_{j}^{3}-\frac{\beta}{k_{j}}}.$ (153)

Two loop solitons are bounded to form a loop soliton molecule which are shown
in Fig.9(a). One can see that the distance between two loops will not be
changed as time goes on. Further, one can also immediately construct the
breather molecules by using the velocity resonant mechanism. Two breathers
constitute one breather molecule which is shown in Fig.9(b). It is worth
mentioning that such loop-type soliton molecules and nonsingular oscillating
pulse type breather molecules are firstly reported theoretically.

Figure 9: (a) A loop-type soliton molecule (b) a breather molecule

##  5 Conclusion

In this paper, we have constructed the generalized compound WKI-SP(n,m)
equations by virtue of two compatible linear spectral equations. By solving
two sets of recursive equations, we showed three SP-type equations (named WKI-
SP(0,1), WKI-SP(0,2) and WKI-SP(0,3)), two WKI-type equations (named WKI-
SP(1,0) and WKI-SP(2,0)) and three compound WKI-SP equations (called WKI-
SP(1,1), WKI-SP(1,2) and WKI-SP(2,1)). The classical WKI equation (1) is just
the WKI-SP(1,0) equation while the the SP equation equation (2) is the WKI-
SP(0,1) equation. The WKI-SP(0,2) and WKI-SP(0,3) equations are the high-order
short pulse equations.

It is known that there is a novel hodograph between the SP equation (2) and
the SG equation. And there also exits the hodograph transformation between the
WKI equation (1) and the (potential) MKdV equation. In this paper, with the
help of one conservation law, we have found one hodograph transformation which
successfully transformed the compound WKI-SP(1,1) equation into the compound
MKdV-SG(1,1) equation. Meanwhile, the same hodograph transformation could also
enable us to convert the compound WKI-SP(1,2) equation and the WKI-SP(2,1)
equation into the MKdV-SG(1,2) equation and the MKdV-SG(2,1) equation
respectively.

We constructed multiloop soliton solutions for the WKI-SP(1,1) equation as
applications. The loop (antiloop) soliton solutions arise from the kink
(antikink) solutions of the MKdV-SG(1,1) equation. Let $m$ and $N-m$ be the
number of positive and negative $k_{j}$, respectively. Then, the corresponding
soliton solution would include $m$ loop solitons and $N-m$ antiloop solitons.
Here, we also call antiloop as loop for convenience. The analytic solutions
for the one-loop, two-loop and three-loop solitons in the parametric forms
were shown. Especially, we stressed the description of two-loop soliton, which
exhibit abundant solitonic behaviors. On the one hand, two loop solitons can
either move in the same direction or in the opposite direction of the
$x$-axis. It is worth mentioned that there could be a small loop with a fast
speed chasing a large loop with a slow speed. Certainly, we also obtained the
typical solitonic behavior that the large loop moves more rapidly than the
small loop. On the other hand, there are two kinds of collision processes
depending on the ratio of the eigenvalues involved. In the first case of two
loop solitons with dissimilar amplitudes chasing each other, the small one
would travel around the larger one. The phase shift of the small loop soliton
can be seen to originate mainly from the delay caused by its travelling around
the large loop soliton. In the second case of the collision process for two
loop solitons with similar amplitudes, the slow loop soliton is pushed out
forward with a considerable phase shift when compared with that of the fast
loop. That is to say the slow loop soliton pass through the fast one.

It is also remarkable that all the models given by the WKI-SP(n,m) hierarchy
here could be with time-varying coefficients. Hence, their dispersion
relations will have a time dependent velocity and the solitons will
accelerate. These equations with variable coefficients may be nice candidates
in applications having accelerated ultra-short optical pulses. The method and
the results mentioned in this paper would be extended to the complex WKI-SP
hierarchy, the nonlocal WKI-SP hierarchy, the semi-discrete and the discrete
WKI-SP hierarchy, the multi-component WKI-SP hierarchy (and with their
corresponding complex, nonlocal, semi-discrete and discrete forms). These
problems are to be pursued in our near future work. The Darboux
transformation, Bäcklund transformation and the Riemann-Hilbert problem for
the compound WKI-SP equation are also in our near consideration.

## Acknowledgments

This work is supported by the national natural science foundation of China
(Grant No. 11771395 and No. 11871336) and the Zhejiang Provincial Natural
Science Foundation of China (Grant No. LY18A010034).

## References

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