# TERMinator: A system for scientific texts processing

Elena Bruches
A.P. Ershov Institute of Informatics Systems / Russia
Novosibirsk State University / Russia
<EMAIL_ADDRESS>
Olga Tikhobaeva
Novosibirsk State University / Russia
<EMAIL_ADDRESS>
Yana Dementyeva
Novosibirsk State University / Russia
<EMAIL_ADDRESS>
Tatiana Batura
A.P. Ershov Institute of Informatics Systems / Russia
<EMAIL_ADDRESS>
###### Abstract

This paper is devoted to the extraction of entities and semantic relations
between them from scientific texts, where we consider scientific terms as
entities. In this paper, we present a dataset that includes annotations for
two tasks and develop a system called TERMinator for the study of the
influence of language models on term recognition and comparison of different
approaches for relation extraction. Experiments show that language models pre-
trained on the target language are not always show the best performance. Also
adding some heuristic approaches may improve the overall quality of the
particular task. The developed tool and the annotated corpus are publicly
available at https://github.com/iis-research-team/terminator and may be useful
for other researchers.

##  1 Introduction

Nowadays the amount of scientific publications is constantly growing. In this
regard, the processing of scientific texts becomes especially relevant in
relation to rapidly developing scientific fields, for example, computer
science. Information extraction from scientific texts can be useful in domain-
specific areas, for completion of knowledge graphs, in search and question-
answering systems. This paper describes the study on entity recognition and
relation extraction from scientific texts on computer science in Russian.

Currently, there are a number of datasets with annotations of entities and
relations in a general domain Doddington et al. (2004); Roth and Yih (2004);
Loukachevitch et al. (2021), biomedical domain Kim et al. (2003); Gurulingappa
et al. (2012); Li et al. (2016), or even multi-domains Terryn et al. (2020).
Still it is more difficult to find a publicly available dataset such as SciERC
Luan et al. (2018) for scientific fields other than biomedical, and especially
in languages other than English.

Despite that the named entity recognition task is well studied, it still faces
multiple challenges Li et al. (2022a), namely, NER in domain-specific areas
Weber et al. (2021), NER from noisy data Derczynski et al. (2017), code-mixed
data Fetahu et al. (2021), and detection of fine-grained and nested named
entities Kim and Kim (2021); Ringland et al. (2019); Loukachevitch et al.
(2021). This is caused by several issues: defining boundaries of compound
terms; recognition of whether a lexical unit is part of a compound term;
identification of a lexical unit as a term depending on the context and topic
of the text in which this lexical unit is used etc. The relation extraction
task also remains an unsolved problem, as it often requires the use of
knowledge outside the text (for example, from knowledge bases or obtained in
another way), and also due to the lack of a large amount of labeled data in
different languages.

Selection of the most appropriate language model, which is able to provide the
best quality for extraction of terms and relations from scientific texts is
one of the relevant issues. Our experimental results not only show the
usefulness of the proposed dataset, but also provide baselines for further
research.

We make the following contributions:

  * •

Provide a new dataset for both tasks (term recognition and relation
extraction) for Russian scientific texts and develop a TERMinator tool for
further research experiments.

  * •

Study of influence of language models (without additional information, with
heuristics and dictionaries) on term extraction.

  * •

Compare three approaches for relation extraction (based on lexical patterns,
classification with a CLS-vector, and combination of them).

##  2 Related Work

Entity recognition and relation extraction are the main tasks in information
extraction. There are various approaches to solve them.

A traditional approach includes two stages: extracting n-grams which
potentially may be terms, and then classification whether this n-gram is a
term or not. In Stanković et al. (2016) authors proposed to use dictionaries
and morphological and syntax information. There are some works which use pre-
defined ontologies for terms extraction Ivanisenko et al. (2020). Another idea
is to solve this task as a sequence labeling Kucza et al. (2018). It allows to
implement terms extraction in one stage and take into account syntax and
semantic information from the context. For terms extraction the main challenge
is to identify the exact term boundaries. In Zhu and Li (2022) authors
proposed to use boundary smoothing as a regularization technique to overcome
this problem.

Relation extraction is usually considered as a classification problem: for two
given terms one needs to determine whether there is a semantic relation
between them or not, and if they are related then to define its type. Some
works describe the use of knowledge bases for relation extraction Li et al.
(2019); Baldini Soares et al. (2019). With the spread of transformers-based
architectures, different pre-trained language models are used to solve this
task Shi and Lin (2019). Some researchers try to make use of incorporating
external data sources in the model, for example the list of hand-written
syntax patterns Tao et al. (2019), information about sentence syntax tree
Ningthoujam et al. (2019); Nayak and Ng (2019).

Recently, special attention has been paid to systems which solve terms
recognition and relation extraction jointly. The authors propose an
architecture that sequentially extracts entities and relationships between
them, but in end-to-end settings Eberts and Ulges (2020); Ji et al. (2020);
Huang et al. (2019); Miwa and Bansal (2016). Another idea is to train a model
with two outputs: one output is for term extraction, and the other is for
relation extraction Xue et al. (2019). However such approaches require quite a
lot of annotated data to find hidden regularities.

##  3 Data Preparation

For the experiments we create an annotated dataset, which consists of
abstracts of scientific papers on Information Technology in Russian.

As entities we consider nouns or noun groups, which are terms in this
particular domain. Terms that we recognize as entities may consist of one or
several tokens (“ _software_ ”, “ _non-preemptive multitasking_ ”),
abbreviations (“ _CPU_ ”, “ _DLL_ ”), names of programming languages (“
_Python_ ”, “ _C++_ ”) and libraries (“ _Pytorch_ ”, “ _SpaCy_ ”), hyphenated
concepts containing Latin characters (” _n-gram_ ”, ” _web-service_ ”).
Thereby we consider all possible chains of tokens that can be terms, except
for those that are recursive or overlap. The entities are marked in the BIO
format: each token is assigned a B-TERM tag if it is the initial tag for an
entity, I-TERM if it is inside a term, or O if it is outside any entity.

Statistics for our dataset is presented in Table 1.

Unit | train | test
---|---|---
`texts` | 136 | 80
`tokens` | 12 809 | 11 157
`terms` | 2 028 | 2 027
`relations` | 356 | 620
Table 1: Dataset statistics

The list of relations is selected based on the following criteria. At first, a
relation should be monosemantic (for example, we don’t consider a semantic
relation _< Entity-Destination>_ because it has indirect meaning as well).
Secondly, a relation should link scientific terms (for example, in relation _<
Communication-Topic>_ (an act of communication is about topic) the actants are
not scientific terms). Thus, six semantic relations were selected. Types of
relations in a corpus, their meanings and distribution by train and test sets
are presented in Table 2.

Relation type | Meaning | train | test
---|---|---|---
`CAUSE` | x is the cause of y | 19 | 19
`ISA` | x is y | 96 | 93
`PART_OF` | x is part of y | 23 | 87
`SYNONYMS` | x is the same as y | 35 | 22
`TOOL` | x allows to create/etc. y | 54 | 38
`USAGE` | x is used for/in y | 126 | 330
Table 2: Types of relations

Here is a sample sentence where two terms and the relation between them are
highlighted: _Pokazany preimushchestva primeneniya <e1>mul’timedijnyh
tekhnologij</e1> v <e2>uchebnom processe</e2> i effektivnost’ ih
ispol’zovaniya vo vremya lekcij i seminarov_. (The advantages of using
multimedia technologies in the educational process and the effectiveness of
their use during lectures and seminars are shown.). The relation between _e1_
and _e2_ is _USAGE_.

The dataset is available for other researchers111https://github.com/iis-
research-team/ruserrc-dataset.

##  4 Influence of language models on term extraction

###  4.1 Models without additional information

The experimental methodology is as follows: texts are fed as input; during the
vectorization procedure, each text is divided into spans (in our case these
are BPE tokens), each of which is assigned to a vector. Initially, the model
learns to match tokens with labels by using the training data; then based on
the revealed regularities the model makes predictions on the validation data.
In this way, the metrics are fixed after each training epoch. The output is a
set of labels associated by the system with each word from the input text.

We experimented with two models: multilingual BERT Devlin et al. (2018) and
BERT pre-trained on Russian texts Kuratov and Arkhipov (2019).

In the first stage of the experiment each pre-trained language model was fine-
tuned on the train set described above. The optimal learning rate was chosen
as $10^{-6}$, and the batch size was 12. Such values prevent overfitting and
obtain the best results. At this stage, the metrics show that on partial match
both models give the same performance, while in exact match the model pre-
trained on Russian-language texts gives better scores.

Then we extended a train set by adding 212 texts with a pseudo labeling
method. We collected a dictionary (list of scientific terms) in a semi-
automatic way:

  * •

We extracted 2-, 3- and 4-grams from the scientific papers and manually
filtered phrases, which potentially can be terms.

  * •

We extracted all titles of articles from Wikipedia, which are included in a
subgraph of category “Science”, and then manually selected words and phrases,
which potentially can be terms.

Thus we obtained a list of 17 252 terms, which we used for pseudo labeling.
This technique is useful for rapidly changing areas of knowledge, when it is
difficult to have dictionaries of terms and keep them up to date.

Due to the less detailed checking of the markup of the corpus, even with its
comparatively large size, the metrics received with the models trained on it
turned out to be lower than those of the same models trained on the manual
annotated texts.

###  4.2 Models with heuristics

Experimentally we found out that in order to improve the quality of term
extraction, we need to improve the definition of term boundaries. The task of
defining terms’ boundaries is more challenging than classifying a token as a
“term” type. To improve the recognition of term boundaries, we apply some
heuristics to handle such cases as removing a preposition as a first token of
a term and some others. From the results shown in Table 3, it can be seen that
the heuristics improve the quality of term extraction on the exact match.

Train set | Model | F-M P | F-M R | F-M F1 | P-M P | P-M R | P-M F1
---|---|---|---|---|---|---|---
Manually labeled | mBERT | 0.40 | 0.46 | 0.43 | 0.89 | 0.88 | 0.88
`mBERT + h` | 0.49 | 0.45 | 0.47 | 0.86 | 0.86 | 0.86
`mBERT + d + h` | 0.47 | 0.50 | 0.48 | 0.86 | 0.87 | 0.87
`ruBERT` | 0.48 | 0.50 | 0.49 | 0.89 | 0.88 | 0.88
`ruBERT + h` | 0.52 | 0.47 | 0.49 | 0.89 | 0.88 | 0.88
`ruBERT + d + h` | 0.49 | 0.51 | 0.50 | 0.86 | 0.87 | 0.87
Pseudo labeled | mBERT | 0.33 | 0.34 | 0.34 | 0.80 | 0.75 | 0.75
`mBERT + h` | 0.42 | 0.38 | 0.40 | 0.80 | 0.79 | 0.79
`mBERT + d + h` | 0.41 | 0.39 | 0.40 | 0.80 | 0.80 | 0.80
`ruBERT` | 0.32 | 0.32 | 0.33 | 0.78 | 0.76 | 0.75
`ruBERT + h` | 0.40 | 0.37 | 0.38 | 0.79 | 0.79 | 0.79
`ruBERT + h + d` | 0.38 | 0.39 | 0.38 | 0.79 | 0.74 | 0.76
Table 3: Metrics for full match (F-M) and partial match (P-M) terms
extraction; _d_ is for dictionary, _h_ is for heuristics.

###  4.3 Models with heuristics and dictionaries

It the third stage, the system extracts terms not only with the trained model,
but also with the use of the dictionary described above. Heuristics are also
applied.

Table 3 shows that the ruBERT model fine-tuned on the manually annotated train
set extracts the terms in half of the cases, which is the best result of a
exact match. On the pseudo-labeled train set, this combined method gives good
results for the multilingual BERT both for exact and partial matches.

Both models solve the task of term recognition with a high quality, which can
be seen from the good results on a partial match. The best result on an exact
match pertains to the RuBERT model supported by a dictionary and heuristics.
The results of the models on the exact match are expectedly lower than the
results on a partial match, which again draws our attention to the task of
defining terms’ boundaries in texts.

The markup quality significantly affects the quality, as we can note from a
comparison of the results of the first and second stages of the experiment
(see Sections 4.1, 4.2). Fine-tuning on the manually-annotated training set
gives better performance than fine-tuning on the pseudo-labeled training set,
even though its size is larger than the size of the manually-annotated one. It
is hard to compare results with other researchers as this is the first corpora
for scientific texts in Russian as far as we know. But for the similar dataset
in English SciERC Luan et al. (2018) the authors Eberts and Ulges (2020)
reported F1-measure to be 0.70.

We observed that for the term extraction task the model mistakes in
recognizing the exact term boundaries. Another problem arises when a term is
divided by other words or signs in the sentence, for example, _"Morphological
and syntax analysis"_. Probably, it should be solved at a post-processing
stage.

##  5 Comparison of approaches for relation extraction

###  5.1 Using lexical patterns

At first, we applied an approach for relation identification based on lexical
patterns. It consists in the following: for texts with tagged terms, we
extract a context between each pair of terms, lemmatize it, and compare it
with the lexical patterns. If they match, these two terms are connected by
this relation. The length of the context should not exceed six words. This
value was obtained experimentally by changing it and comparing the quality of
the model. The obtained metrics for this approach are shown in Table 4. We
used 111 patterns; examples of patters and their distribution by relations are
presented in Table 5.

Relation type | Precision | Recall | F1
---|---|---|---
`CAUSE` | 0.07 | 0.05 | 0.06
`ISA` | 0.18 | 0.19 | 0.19
`PART_OF` | 0.17 | 0.14 | 0.15
`SYNONYMS` | 0.23 | 0.82 | 0.35
`TOOL` | 0.06 | 0.08 | 0.07
`USAGE` | 0.21 | 0.39 | 0.27
`NO-RELATION` | 0.96 | 0.92 | 0.94
`macro-average` | 0.27 | 0.37 | 0.29
Table 4: Metrics for lexical pattern’s approach Relation type |  Examples of patterns (transliteration) |  Examples of patterns (translation) |  N
---|---|---|---
`CAUSE` |  Uvelichivsheesya potreblenie rafinirovannyh produktov pitaniya yavlyaetsya prichinoj mnozhestva takih zabolevanij. |  Increased consumption of refined foods is the cause of many diseases. |  23
`ISA` |  Odnim iz samyh tochnyh i effektivnyh sposobov upravleniya zhestami yavlyaetsya upravlenie aktivnost’yu myshc. |  One of the most accurate and effective ways to control gestures is to control muscle activity. |  13
`PART_OF` |  Process referirovaniya sostoit iz pyati osnovnyh shagov. |  The referencing process consists of five main steps. |  5
`SYNONYMS` |  Stat’ya posvyashchena issledovaniyu vertikal’nogo poleta robota s mashushchim krylom, takzhe nazyvaemogo ornitopterom. |  The article is devoted to the study of the vertical flight of a robot with a flap wing, also called an ornithopter. |  5
`TOOL` |  V stat’e predstavlen opyt razrabotki informacionnoj sistemy, avtomatiziruyushchej process raspredeleniya studentov po bazam praktik. |  The article presents the experience of development of the information system, which automates the process of distribution of students by the bases of practice. |  29
`USAGE` |  V nastoyashchee vremya aktivno razvivaetsya napravlenie, svyazannoe s proektirovaniem nejronnyh setej dlya ispol’zovaniya v mobil’nyh ustrojstvah. |  Currently, the development of neural networks for use on mobile devices is growing rapidly. |  36
Table 5: Examples of lexical patterns

###  5.2 Classification task with a CLS-vector

The second approach we used for the relation extraction is similar to R-BERT,
and is used by other authors Hosseini et al. (2022); Aldahdooh et al. (2021);
Li et al. (2022b). We consider the task of relation extraction as a
classification task (with 7 classes of relations: CAUSE, ISA, PART-OF,
SYNONYMS, TOOL, USAGE, NO-RELATION). We take the vector of a special token CLS
(it is considered as a vector of the input text) and the vector of two terms
(connected by the relation). These three vectors are concatenated and the
resulting vector is fed to the classifier Wu and He (2019). We tried to use
three different language models: mBERT, ruBERT Kuratov and Arkhipov (2019) and
cointegrated/rubert-tiny2.

In addition, some features of the training are noteworthy. Firstly, to train
the models, we used the corpus of Russian texts without dividing it into
training and validation sets, and the most appropriate number of epochs was
selected experimentally, because there are very few examples for some
relations, and therefore the validation set would be unrepresentative to
determine the quality of the model. Secondly, to reduce the imbalance between
the number of examples in the classes, we added only 50% of the randomly
selected pairs of terms to the training set, excluding those with the distance
between tokens more than 10.

Finally, we implemented an ensemble which includes both approaches: model and
lexical patterns. All metrics are presented in Table 6. F1-score for all types
of relations for combined approach are presented in Table 7. For comparison,
the state-of-the-art result achieved on SciERC with the SpERT (using SciBERT)
method is 50.84% for relation extraction Eberts and Ulges (2020). Our results
may also be related to insufficient data, as Russian is morphologically rich,
which additionally complicates the work of the language model. Moreover, error
analysis of relation extraction revealed that relations are often present
implicitly between terms and one can recognize them if one only knows these
particular terms. Quite many terms in IT texts are abstract (for example,
_"program implementation"_ , _"testing"_ , etc.) and it can be difficult to
define, whether there is any semantic relation between them or not. We plan to
study this aspects in the future.

Model | Precision | Recall | F1
---|---|---|---
`mBERT` | 0.26 | 0.32 | 0.26
`ruBERT` | 0.26 | 0.34 | 0.27
`rubert-tiny2` | 0.22 | 0.23 | 0.22
`mBERT + p` | 0.26 | 0.41 | 0.29
`ruBERT + p` | 0.29 | 0.35 | 0.28
`rubert-tiny2 + p` | 0.29 | 0.24 | 0.24
Table 6: Metrics for different language models and combined approach; _p_ is for patterns Relation type | mBERT | ruBERT | ruBERT-tiny2
---|---|---|---
`CAUSE` | 0.06 | 0.09 | 0.10
`ISA` | 0.30 | 0.28 | 0.14
`PART_OF` | 0.14 | 0.04 | 0.00
`SYNONYMS` | 0.32 | 0.33 | 0.38
`TOOL` | 0.04 | 0.07 | 0.00
`USAGE` | 0.27 | 0.22 | 0.11
`NO-RELATION` | 0.93 | 0.95 | 0.94
`macro-average` | 0.29 | 0.28 | 0.24
Table 7: F1-score for all types of relations for combined approach

##  6 Conclusion

In this paper, we built a new dataset and study several methods for term
recognition and relation extraction from computer science texts in Russian. We
conducted several experiments with different pre-trained language models for
both tasks. The results of our experiments show that language models pre-
trained on the target language are not always show the best performance. Also
adding some heuristic approaches may improve the overall quality for the
particular task.

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# Domain walls in fractional media

Shatrughna Kumar1, Pengfei Li2,3, and Boris A. Malomed1,4 1Department of
Physical Electronics, School of Electrical Engineering, Faculty of
Engineering, and Center for Light-Matter Interaction, Tel Aviv University,
P.O.B. 39040, Tel Aviv, Israel
2Department of Physics, Taiyuan Normal University, Jinzhong, 030619, China
32Institute of Computational and Applied Physics, Taiyuan Normal University,
Jinzhong, 030619, China
4Instituto de Alta Investigación, Universidad de Tarapacá, Casilla 7D, Arica,
Chile

###### Abstract

Currently, much interest is drawn to the analysis of optical and matter-wave
modes supported by the fractional diffraction in nonlinear media. We predict a
new type of such states, in the form of domain walls (DWs) in the two-
component system of immiscible fields. Numerical study of the underlying
system of fractional nonlinear Schrödinger equations demonstrates the
existence and stability of DWs at all values of the respective Lévy index
($\alpha<2$) which determines the fractional diffraction, and at all values of
the XPM/SPM ratio $\beta$ in the two-component system above the immiscibility
threshold. The same conclusion is obtained for DWs in the system which
includes the linear coupling, alongside the XPM interaction between the
immiscible components. Analytical results are obtained for the scaling of the
DW’s width. The DW solutions are essentially simplified in the special case of
$\beta=3$, as well as close to the immiscibility threshold. In addition to
symmetric DWs, asymmetric ones are constructed too, in the system with unequal
diffraction coefficients and/or different Lévy indices of the two components.

##  I Introduction

The Schrödinger equation with fractional spatial dispersion was originally
derived for the wave function of particles moving by Lévy flights, using the
Feynman-integral formulation of fundamental quantum mechanics [1, 2]. While
experimental realization of fractional quantum mechanics has not been reported
yet, it was proposed to emulate it in terms of classical photonics, using the
commonly known similarity of the Schrödinger equations and equations for the
paraxial diffraction of optical beams [3, 4]. A universal method for the
emulation of the fractional diffraction is to use the basic 4f configuration,
which makes it possible to perform the spatial Fourier transform of the beam,
apply the phase shift, which is tantamount to the action of the fractional
diffraction, by means of an appropriate phase plate, and finally transform the
beam back from the Fourier space [3]. In addition to that, implementations of
the fractional Schrödinger equations were proposed in Levy crystals [5] and
polariton condensates [6].

Theoretical studies initiated by the above-mentioned scheme were developed in
various directions, including the interplay of the fractional diffraction with
parity-time ($\mathcal{PT}$) symmetric potentials [7]-[10], propagation of
Airy waves in the fractional geometry [11, 12], and adding the natural Kerr
nonlinearity to the underlying setting, thus introducing fractional nonlinear
Schrödinger equations (FNLSEs). The work with nonlinear models has produced
many predictions, such as the modulational instability of continuous waves
(CWs) [13] and diverse types of optical solitons [14]-[44]. These are quasi-
linear “accessible solitons” [17, 18], gap solitons maintained by lattice
potentials [23]-[27], self-trapped vortices [28, 29], multi-peak [30]-[33] and
cluster [34] modes, fractional solitons in discrete systems [35], localized
states featuring spontaneously broken symmetry [8, 37, 38], solitons in dual-
core couplers [40, 9], solitary states supported by the quadratic nonlinearity
[41, 42], and dark modes [36]. Also studied were dissipative solitons in the
fractional version of the complex Ginzburg-Landau equation [43]. Many of these
results were reviewed in Ref. [44].

The objective of the present work is to introduce one-dimensional settings for
binary immiscible fields under the action of the fractional diffraction. The
immiscibility naturally gives rise to stable patterns in the form of domain
walls (DW), alias grain boundaries, which separate half-infinite domains
filled by the immiscible field components. In areas of traditional physical
phenomenology, DWs are well known as basic patterns in thermal convection
[45]-[49]. Grain boundaries of a different physical origin occur in various
condensed-matter settings [50]-[55]. In optics, DWs were predicted and
experimentally observed in bimodal light propagation in fibers [56, 57].
Similar states were predicted in binary Bose-Einstein condensates (BECs),
provided that the inter-component repulsion is stronger than the self-
repulsion of each component, which provides for the immiscibility [58]-[61].

The interplay of the two-component immiscibility, that maintains DWs, with
fractional diffraction may naturally appear in optics, considering the
fractional bimodal propagation of light in a self-defocusing spatial
waveguide. A similar model, based on a system of fractional Gross-Pitaevskii
equations (FGPEs) [44], may also naturally emerge in a binary BEC composed of
repulsively interaction particles which move by Lévy flights. We construct DW
solutions for coupled FNLSEs and verify their stability by means of numerical
methods. Some results – in particular, scaling relations which determine the
DW’s width as a function of basic parameters of the system – are obtained in
an analytical form.

The paper is organized as follows. The model is formulated in Section 2, which
also includes analytical expressions for CW, i.e., spatially uniform states,
that may be linked by DW patterns, thus supporting them. Analytical results
for the DWs are collected in Section 3. Numerical results are reported in
Section 4, and the paper is concluded by Section 5.

##  II The model and CW states

###  II.1 Basic equations

In terms of the optical bimodal propagation in the spatial domain, the scaled
system of coupled FNLSEs for amplitudes of copropagating electromagnetic waves
$u\left(x,z\right)$ and $v\left(x,z\right)$ with orthogonal polarizations is

$\displaystyle i\frac{\partial u}{\partial z}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(-\frac{\partial^{2}}{\partial
x^{2}}\right)^{\alpha/2}u+(|u|^{2}+\beta|v|^{2})u-\lambda v,$ $\displaystyle
i\frac{\partial v}{\partial z}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(-\frac{\partial^{2}}{\partial
x^{2}}\right)^{\alpha/2}v+(|v|^{2}+\beta|u|^{2})v-\lambda u,$ (1)

where $z$ is the propagation distance, $x$ is the transverse coordinate, and
the cubic terms, with normalized coefficients $1$ and $\beta>0$, represent,
respectively, the defocusing nonlinearity of the self-phase-modulation (SPM)
and cross-phase-modulation (XPM) types. The optical self-defocusing occurs, in
particular, in semiconductor waveguides [62]. In the BEC model, the SPM and
XPM terms represent repulsive interactions between two atomic states in the
binary condensate. In the latter case, the system of scaled FGPEs is written
in the form of Eq. (1), with $z$ replaced by the temporal variable, $t$.

In optics, two natural values of the XPM coefficient are $\beta=2$ for
components $u$ and $v$ representing circular polarizations of light, or
$\beta=2/3$ in the case of linear polarizations [63]. The value of $\beta$ may
be varied in broader limits (in particular, the case of $\beta=3$ plays an
essential role below) in photonic crystals [64, 65]. In binary BEC, the
effective XPM coefficient can be readily adjusted by means of the Feshbach
resonance [66, 67].

In the case of orthogonal linear polarizations in optics [corresponding to
$\beta=2/3$ in Eq. (1)], the nonlinear interaction between the components
includes, in addition to the XPM terms, also the four-wave mixing (FWM),
represented by terms $\left(1/3\right)v^{2}u^{\ast}$ and
$\left(1/3\right)u^{2}v^{\ast}$ in FNLSEs (1) for $u$ and $v$ (where $\ast$
stands for complex conjugate), although these terms are usually suppressed by
the phase-velocity-birefringence effect [63]. In any case, the FWM terms
appearing in the optical system with orthogonal linear polarizations are not
relevant in the present context, as the condition of the immiscibility of the
two components holds only for $\beta>1$ [see Eq. (20) below], eliminating the
case of $\beta=2/3$. The optical system with orthogonal circular polarizations
corresponds, as said above, to $\beta=2$, which admits the immiscibility, but
the FWM terms do not appear in the latter case. Normally, they do not appear
either in the BEC model based on the system of coupled FGPEs, therefore FWM
terms are not considered here.

The linear-coupling terms with coefficient $\lambda\geq 0$ in Eq. (1) account
for mixing between the optical modes, or between the two atomic states in BEC.
In the former case, the linear mixing between circular polarizations may be
imposed by the birefringence [63], and in the latter case the mutual
conversion of atomic states in BEC may be driven by resonant radiofrequency
radiation [68].

The fractional-diffraction operator with a positive Lévy index (LI) $\alpha$
is defined as the Riesz derivative [69, 70, 71],

$\displaystyle\left(-\frac{\partial^{2}}{\partial
x^{2}}\right)^{\alpha/2}u(x)\equiv$
$\displaystyle=\frac{1}{2\pi}\int_{-\infty}^{+\infty}|p|^{\alpha}dp\int_{-\infty}^{+\infty}d\xi
e^{ip(x-\xi)}u(\xi)\equiv\frac{1}{\pi}\int_{0}^{+\infty}p^{\alpha}dp\int_{-\infty}^{+\infty}d\xi\cos\left(p(x-\xi\right))u(\xi),$
(2)

which is built as the juxtaposition of the direct and inverse Fourier
transform, with the fractional diffraction acting at the intermediate stage.
While there are different definitions of fractional derivatives, this one
naturally appears in quantum mechanics [1, 2] and optics [3]. Normally, the LI
takes values $1<\alpha\leq 2$, but, in the case of the self-defocusing sign of
the nonlinearity, when the system is not subject to the wave collapse
(implosion driven by self-attraction), it is also possible to consider values
$0<\alpha\leq 1$. The usual (non-fractional) diffraction naturally corresponds
to $\alpha=2$ in Eq. (1).

Stationary solutions to Eqs. (1) with propagation constant $k<0$ are looked
for as

$\left\\{u\left(x,z\right),v\left(x,z\right)\right\\}=e^{ikz}\left\\{U(x),V(x)\right\\},$
(3)

where $U(x)$ and $V(x)$ are real functions which satisfy the following system
of equations:

$\displaystyle kU+\frac{1}{2}\left(-\frac{\partial^{2}}{\partial
x^{2}}\right)^{\alpha/2}U+(U^{2}+\beta V^{2})U-\lambda V$ $\displaystyle=$
$\displaystyle 0,$ $\displaystyle
kV+\frac{1}{2}\left(-\frac{\partial^{2}}{\partial
x^{2}}\right)^{\alpha/2}V+(V^{2}+\beta U^{2})V-\lambda U$ $\displaystyle=$
$\displaystyle 0.$ (4)

The energy (Hamiltonian) of the stationary state (4) with the Riesz
derivatives defined as per Eq. (2) is

$\displaystyle
E=\frac{1}{4\pi}\int_{-\infty}^{+\infty}dx\int_{-\infty}^{+\infty}d\xi\int_{0}^{+\infty}p^{\alpha}dp\int_{-\infty}^{+\infty}d\xi\cos\left(p(x-\xi\right))\left[U(x)U(\xi)+V(x)V(\xi)\right]$
$\displaystyle+\int_{-\infty}^{+\infty}dx\left[\frac{1}{4}\left(U^{4}+V^{4}+2\beta
U^{2}V^{2}\right)-\lambda UV\right].$ (5)

Stability of stationary solutions, obtained in the form of expression (3),
against small perturbations was investigated by means of the usual approach,
looking for the perturbed solution as

$\displaystyle u\left(x,z\right)$ $\displaystyle=$ $\displaystyle
e^{ikz}\left[U(x)+e^{\gamma z}a(x)+e^{\gamma^{\ast}z}b^{\ast}(x)\right],$
$\displaystyle v\left(x,z\right)$ $\displaystyle=$ $\displaystyle
e^{ikz}\left[V(x)+e^{\gamma z}c(x)+e^{\gamma^{\ast}z}d^{\ast}(x)\right],$ (6)

where $\left\\{a(x),b(x),c(x),d(x)\right\\}$ are components of an eigenmode of
infinitesimal perturbations, and $\gamma$ is the respective eigenvalue (which
may be a complex number). The substitution of the perturbed expression (6) in
Eq. (1) and linearization leads to the system of coupled equations,

$\displaystyle\left(-k+i\gamma\right)a$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(-\frac{\partial^{2}}{\partial
x^{2}}\right)^{\alpha/2}a+\left(2U^{2}+\beta V^{2}\right)a+U^{2}b+\beta
UV(c+d)-\lambda c,$ $\displaystyle\left(-k-i\gamma\right)b$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(-\frac{\partial^{2}}{\partial
x^{2}}\right)^{\alpha/2}b+\left(2U^{2}+\beta V^{2}\right)b+U^{2}a+\beta
UV(c+d)-\lambda d,$ $\displaystyle\left(-k+i\gamma\right)c$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(-\frac{\partial^{2}}{\partial
x^{2}}\right)^{\alpha/2}c+\left(2U^{2}+\beta V^{2}\right)c+U^{2}d+\beta
UV(a+b)-\lambda a,$ $\displaystyle\left(-k-i\gamma\right)d$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(-\frac{\partial^{2}}{\partial
x^{2}}\right)^{\alpha/2}d+\left(2U^{2}+\beta V^{2}\right)d+U^{2}c+\beta
UV(a+b)-\lambda b.$

The underlying DW solution is stable if numerical solution of Eq.
(LABEL:linearized) yields solely imaginary eigenvalues, with zero real parts.

###  II.2 Continuous-wave (CW) solutions and the immiscibility condition

The spatially uniform version of Eq. (4), with $U,V=\mathrm{const}$, gives
rise to two asymmetric (partly immiscible, with $U\neq V$) CW solutions,
labeled by subscripts $+$ and $-$, which are mirror images of each other:

$\displaystyle\left\\{\begin{array}[]{c}U_{+}\\\ V_{+}\end{array}\right\\}$
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left\\{\begin{array}[]{c}\sqrt{-\frac{k}{2}+\frac{\lambda}{\beta-1}}+\sqrt{-\frac{k}{2}-\frac{\lambda}{\beta-1}}\\\
\sqrt{-\frac{k}{2}+\frac{\lambda}{\beta-1}}-\sqrt{-\frac{k}{2}-\frac{\lambda}{\beta-1}}\end{array}\right\\},$
(12) $\displaystyle\left\\{\begin{array}[]{c}U_{-}\\\
V_{-}\end{array}\right\\}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left\\{\begin{array}[]{c}\sqrt{-\frac{k}{2}+\frac{\lambda}{\beta-1}}-\sqrt{-\frac{k}{2}-\frac{\lambda}{\beta-1}}\\\
\sqrt{-\frac{k}{2}+\frac{\lambda}{\beta-1}}+\sqrt{-\frac{k}{2}-\frac{\lambda}{\beta-1}}\end{array}\right\\}.$
(17)

Note that the total density of solutions (12) and (17) is

$U_{+}^{2}+V_{+}^{2}=U_{-}^{2}+V_{-}^{2}=-k.$ (18)

While in the limit of $\lambda=0$ (no linear mixing), the obvious CW states
are completely immiscible, with $V_{+}=U_{-}=0$, the partly immiscible states
given by Eqs. (12) and (17) were found only recently in Ref. [49]. Parallel to
the asymmetric CW states (12) and (17) there is the mixed (symmetric) one,
with

$U_{0}=V_{0}=\sqrt{(\lambda-k)/(1+\beta)}.$ (19)

For given $k$, i.e., for given CW density [see Eq. (18)], CW states (12) and
(17) exist under the following condition:

$\beta-1>\left(\beta-1\right)_{\mathrm{immisc}}\equiv 2\lambda/|k|.$ (20)

In the absence of the linear mixing, $\lambda=0$, Eq. (20) amounts to the
commonly known immiscibility condition [58],
$\beta>\beta_{\mathrm{immisc}}=1$. At $\lambda>0$, Eq. (20) demonstrates that
the linear mixing pushes the immiscibility threshold to higher values, as was
first demonstrated in Ref. [72] under normalization condition $k=-1$.
Precisely at the threshold, i.e., at $\beta-1=2\lambda/|k|$, Eqs. (12), (17)
and (19) yield the following magnitude of the CW fields,

$U_{\mathrm{thresh}}=V_{\mathrm{thresh}}=\sqrt{\lambda/(\beta-1)}.$ (21)

As shown in Ref. [72], the meaning of immiscibility condition (20) can be
understood in terms of energy: at
$\beta-1>\left(\beta-1\right)_{\mathrm{immisc}}$, for given density of the CW
state [see Eq. (18)], the energy density of the partly immiscible state,
determined as per the second line of Eq. (5), is lower than that of the mixed
one (19), hence the asymmetric CW solutions (12) and (17) play the role of the
system’s ground state, while the mixed CW state (19) is unstable. On the other
hand, at $\beta-1<\left(\beta-1\right)_{\mathrm{immisc}}$ the mixed CW state
is the only existing one, being the (stable) ground state in that case.

##  III Domain-wall (DW) solutions: Analytical findings

DW states exist when Eq. (4) maintains the (partly) immiscible CW states, as
given by Eqs. (12) and (17). The DW solution links two different CW states,
that fill the space at $x\rightarrow\pm\infty$, according to the following
boundary conditions:

$\displaystyle\lim_{x\rightarrow+\infty}\left\\{\begin{array}[]{c}U(x)\\\
V(x)\end{array}\right\\}$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{c}U_{-}\\\ V_{-}\end{array}\right\\},$
(26) $\displaystyle\lim_{x\rightarrow-\infty}\left\\{\begin{array}[]{c}U(x)\\\
V(x)\end{array}\right\\}$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{c}U_{+}\\\ V_{+}\end{array}\right\\},$
(32)

An essential fact is that, in the particular case of $\beta=3$, the system of
two equations (4) can be exactly reduced to a single equation, by the
substitution of

$\left\\{\begin{array}[]{c}U(x)\\\
V(x)\end{array}\right\\}=\frac{1}{2}\left\\{\begin{array}[]{c}\sqrt{\lambda-k}-W(x),\\\
\sqrt{\lambda-k}+W(x),\end{array}\right\\}$ (33)

where $W(x)$ is a real odd function of $x$ satisfying the equation

$(k+\lambda)W+\frac{1}{2}\left(-\frac{\partial^{2}}{\partial
x^{2}}\right)^{\alpha/2}W+W^{3}=0,$ (34)

which is supplemented by the boundary conditions

$\lim_{x\rightarrow\pm\infty}W(x)=\pm\frac{1}{2}\sqrt{-k-\lambda}$ (35)

[note that it follows from Eq. (20) with $\beta=3$ that values
$\sqrt{-k-\lambda}$ in Eq. (35) are real].

In the case of the fractional diffraction (for $\alpha<2$), Eqs. (33) and (34)
represent a new result for $\beta=3$, while in the case of the usual
diffraction, i.e., at $\alpha=2$, the respective solution was recently found
in a fully explicit form [49]:

$\left\\{\begin{array}[]{c}U(x)\\\
V(x)\end{array}\right\\}_{\alpha=2,\beta=3}=\frac{1}{2}\left\\{\begin{array}[]{c}\sqrt{-k+\lambda}-\sqrt{-k-\lambda}\tanh\left(\sqrt{-k-\lambda}x\right)\\\
\sqrt{-k+\lambda}+\sqrt{-k-\lambda}\tanh\left(\sqrt{-k-\lambda}x\right)\end{array}\right\\}.$
(36)

The existence of the relevant solution to Eq. (34) with $\alpha<2$ is
corroborated by the numerical results for dark solitons as solutions of the
FNLSE, which were reported (in a different context) in Ref. [36]. Furthermore,
as $\left(k+\lambda\right)$ is the single control parameter in Eq. (34), an
exact property of the solutions with all values of $\alpha$ is a scaling
relation for the DW’s width, $L$:

$L\sim\left(-k-\lambda\right)^{-1/\alpha}.$ (37)

In particular, Eq. (37) agrees with the exact solution (36) in the case of
$\alpha=2$.

An approximate scaling relation for the DW can be constructed in the case when
propagation constant $k$ is taken close to the threshold value (20), i.e.,
setting

$k=-\frac{2\lambda}{\beta-1}-q,$ (38)

with

$0<q\ll 2\lambda/(\beta-1).$ (39)

In this case, Eqs. (LABEL:link) and (12), (17) show that the DW links boundary
values with a small difference between them,

$\left\\{U(x=\pm\infty),V(x=\pm\infty)\right\\}\approx\left\\{\sqrt{\lambda/(\beta-1)}\mp\sqrt{q}/2,\sqrt{\lambda/(\beta-1)}\pm\sqrt{q}/2\right\\},$
(40)

where the main term is the same as in Eq. (21). Further, straightforward
analysis of Eq. (4) demonstrates that, under condition (39), the DW’s width
scales with the variation of $q$ as

$L\sim q^{-1/\alpha},$ (41)

cf. Eq. (37).

Finally, in the case of $\lambda=0$ in Eq. (4), the near-threshold case is
defined, instead of Eq. (39), simply as

$0<\beta-1\ll 1,$ (42)

see Eq. (20). In this case, the analysis of Eq. (4) leads to the following
asymptotic scaling relation for the DW’s width,

$L\sim\left(\beta-1\right)^{-1/\alpha},$ (43)

cf. Eqs. (37) and (43).

In the case of the normal diffraction, $\alpha=2$, the situation valid under
condition (42) was considered in Ref. [56], where the scaling was obtained in
the form of $L\sim(\beta-1)^{-1/2}$, cf. Eq. (43). Also in agreement with Eq.
(43), DWs do not exist in the system with the Manakov’s nonlinearity [73],
$\beta=1$.

##  IV Numerical results

Numerical solutions of Eq. (4) for stationary DWs were produced by means of
the well-known Newton’s conjugate-gradient method [74]. The results are
presented below separately for the systems without and with the linear
coupling ($\lambda=0$ and $\lambda>0$, respectively), and also, in a brief
form for an asymmetric generalization of Eq. (4), with unequal diffraction
coefficients and/or values of the LI for components $U$ and $V$.

###  IV.1 The system without the linear coupling ($\lambda=0$)

First, we consider the basic system of stationary equations (4) with
$\lambda=0$. Results produced by the numerical solution of this system are
summarized in Fig. 1 for $\beta=3$, because this value of the XPM coefficient,
as shown above, simplifies the system, allowing one to reduce it to the single
equation (34). In this and similar figures, the propagation constant is fixed
as $k=-1$, which is always possible by means of rescaling. It is seen that the
DW patterns are truly robust ones, as the variation of the LI in broad limits,
from $\alpha=2$ up to $\alpha=0.2$, produces relatively mild changes in the
shape of the DWs, which persist, as solutions of Eq. (4), at all values of the
LI (numerical results are not presented for very small values of $\alpha$, as
the numerical method encounters technical problems in that case).

Figure 1: A set of stationary profiles of the two components of the
numerically generated DW (domain-wall) solutions of Eq. (4) with $\lambda=0$,
$\beta=3$, $k=-1$ and indicated values of the LI (Lévy index) varying from
$\alpha=2$ (which corresponds to the normal non-fractional diffraction) up to
$\alpha=0.2$.

Furthermore, the numerical solution of the system of linearized equations
(LABEL:linearized) for small perturbations produces completely stable spectra
of eigenvalues for all stationary DW patterns (not shown here in detail, as
they do not exhibit essential peculiarities). The stability of the DWs was
also corroborated by direct simulations of the system of underlying equations
(1), see a typical examples displayed in Fig. 2 for $\alpha=1$.

Figure 2: An example of the stable evolution of the two components of the DW
[shown in panels (a) and (b)] with $k=-1$, produced by simulations of Eq. (1)
with $\alpha=1$, $\beta=3$, and $\lambda=0$. The stability of the DW is
corroborated by a purely stable spectrum of eigenvalues produced, for the same
DW state, by numerical solution of Eq. (LABEL:linearized) (not shown here).

Similar results, i.e., a family of stable DW solutions, are produced by the
numerical analysis of Eqs. (1), (4), and (LABEL:linearized), for values of the
XPM coefficient $\beta\neq 3$, as shown by a set of profiles in Fig. 3 for
$\alpha=1$ (it is a typical value of the LI corresponding to the fractional
diffraction).

Figure 3: A set of DW solutions with $k=-1$, produced by Eq. (4) with
$\lambda=0$, $\alpha=1$ and values of XPM coefficient varying in interval
$1.05\leq\beta\leq 3$. All the solutions are stable, according to the
calculation of eigenvalues [see Eq. (LABEL:linearized)] and direct simulations
of Eq. (1).

The set of DW profiles is displayed in Fig. 3 for $\beta\geq 1.05$, as, for
very small values of $\beta-1$, the width of the DW diverges, in accordance
with the scaling relation given, for $\lambda=0$, by Eq. (43). For different
values of $\alpha$ the DW families are characterized by dependences of their
width $L$ on $(\beta-1)$, as shown in the top left panel of Fig. 4. The width
was identified, from the numerical solution, as the distance between points
where $U(x)$ and $V(x)$ take values $1/2$, i.e., half of the asymptotic values
$U(x\rightarrow-\infty)=V(x=+\infty)=1$ for $k=-1$, see Eqs. (12) and (17). In
the same figure 4, the numerically found dependences are compared to the
analytically predicted asymptotic scaling relations given by Eq. (43). It is
seen that the prediction is indeed very close to the numerical results for
sufficiently small values of $\left(\beta-1\right)$.

Figure 4: The top left panel: Width $L$ of the DW for $k=-1$, $\lambda=0$, and
three different values of the LI, $\alpha=2$ (which corresponds to the normal
non-fractional diffraction), $\alpha=1.5$, and $\alpha=1,$ vs. the proximity,
$\left(\beta-1\right)$, to the DW’s existence threshold, $\beta=1$. The other
panels show the comparison of the respective numerically found curves,
$L(\beta-1)$, to the asymptotic analytically predicted scaling relation (43).

###  IV.2 The system including the linear coupling ($\lambda>0$)

The inclusion of the linear coupling in Eq. (4) neither destroys DW solutions
nor destabilizes them, but makes their shapes more complex, even in the case
of $\beta=3$, when substitution (33) reduces the system of two equations (4)
to the single equation (34). First, Fig. 5(a) demonstrates that the linear
coupling with strength $\lambda=0.5$ produces a relatively weak effect on the
shape of the DW states if the LI takes values in the interval of
$1\leq\alpha\leq 2$. On the other hand, Fig. 5(b) demonstrates a more
conspicuous effect of the same linear coupling for smaller values of the LI,
viz., in the interval of $0.1\leq\alpha<1$: the corresponding DWs become
essentially broader, in comparison to their counterparts found at $\lambda=0$.
These results are summarized by Fig. 5(c), where, in a broader spatial domain,
it is shown that the DW solutions eventually converge to asymptotic values at
$|x|\rightarrow\infty$, which are, according to Eqs. (35) and (33),

$U_{\pm}=\left(\sqrt{0.725}\pm\sqrt{0.225}\right)/\sqrt{2}\approx\left\\{0.94,0.27\right\\}$
(44)

for the case of $k=1$, $\lambda=0.5$, $\beta=3$, although at small values of
$\alpha$ the convergence is very slow. These findings are readily explained by
the scaling relation (37), which predicts the growth of the DW’s width with
the increase of the linear-coupling constant $\lambda$ and decrease of the LI,
$\alpha$.

Figure 5: Shapes of stable DW solutions produced by Eqs. (33) and (34) for
$\beta=3$, $\lambda=0.5$, $k=1$, and the LI varying in intervals indicated in
the panels. (a) Narrow DWs for $1\leq\alpha\leq 2$; (b) broad DWs for
$0.1\leq\alpha<1$ [for the comparison’s sake, the DW profile with $\alpha=2$,
i.e., in the case of the normal diffraction, is also included in (b)]; (c)
both narrow and broad DWs, displayed in a much larger spatial domain for the
entire range of the values of the LI, $0.2\leq\alpha\leq 2$.

The effects of the linear coupling on the DW states at $\beta\neq 3$ are
qualitatively similar to those displayed in Fig. 5 for $\beta=3$. In all the
cases, the DW solutions remain stable if the linear coupling is incorporated.
Further, for comparison of the $L(\beta-1)$ dependences which are displayed
for $\lambda=0$ in Fig. 4, similar dependences for $\lambda=0.5$ are presented
in Fig. 6. In this case, the dependencies end close to $\beta-1=1$, in
accordance with Eq. (20), which gives
$\left(\beta-1\right)_{\mathrm{immisc}}=1$ for $\lambda=0.5$ and $k=-1$. The
fact that all the curves yield the same DWs’ width at $\beta=3$ is explained
by the above finding that this value plays a special role, simplifying the DW
solutions.

Figure 6: The numerically found dependences of width $L$ of the DW on
$\left(\beta-1\right)$ for $k=-1$, $\lambda=0.5$, and three different values
of the LI, $\alpha=2$ (which corresponds to the normal non-fractional
diffraction), $\alpha=1.5$, and $\alpha=1$.

###  IV.3 DWs in a system with unequal diffraction coefficients and different
values of the Lévy index

A relevant generalization of the system of coupled equations which give rise
to DW solutions is one with different diffraction coefficients, $D_{1}\neq
D_{2}$, in the two equations, as recently proposed in Ref. [49] (in the case
of the normal diffraction). In the optical system, unequal coefficients
$D_{1,2}\equiv\cos^{2}\theta_{1,2}$ are determined by different angles
$\theta_{1,2}$ between carrier wave vectors of the two components of the light
waves and the common propagation direction. The effective diffraction
coefficients are also unequal in the BEC model including two heteronuclear
components with different atomic masses, $m_{1,2}\sim 1/D_{1,2}$, but in the
latter case only the system with $\lambda=0$ is a physically relevant one (two
different atomic species cannot transform into each other).

The extension of Eq. (4) with $D_{1}\neq D_{2}$ takes the form of

$\displaystyle kU+\frac{D_{1}}{2}\left(-\frac{\partial^{2}}{\partial
x^{2}}\right)^{\alpha/2}U+(U^{2}+\beta V^{2})U-\lambda V$ $\displaystyle=$
$\displaystyle 0,$ $\displaystyle
kV+\frac{D_{2}}{2}\left(-\frac{\partial^{2}}{\partial
x^{2}}\right)^{\alpha/2}V+(V^{2}+\beta U^{2})V-\lambda U$ $\displaystyle=$
$\displaystyle 0.$ (45)

The numerical solution of Eq. (45) demonstrates that the asymmetry of the
diffraction coefficients makes the shapes of the two components of the DWs
mutually asymmetric, but does not destroy them. An example of the so deformed
shape of the DWs is displayed in Fig. 7, for both cases of $\lambda=0$ and
$\lambda>0$. This solution and all others produced by the asymmetric system
remain completely stable, as shown by the solution of the respectively
modified Eq. (LABEL:linearized), as well as by direct simulations of the
asymmetric version of Eq. (1) (not shown here in detail).

Figure 7: Examples of stable asymmetric DWs with $D_{1}\neq D_{2}$, produced
by the numerical solution of Eq. (45) with $\alpha=1$, $\beta=3$, $k=-1,$ and
two different values of the linear-coupling constant, $\lambda=0$ and
$\lambda=0.5$ in the top and bottom panels. respectively. The two field
components, $U$ and $V$, are displayed for three pairs of values $D_{1,2}$,
which are indicated, from top to bottom, in the panels.

Furthermore, it is also possible to consider the system with different values
of the LI, $\alpha_{1}\neq\alpha_{2}$, in the equations for the two
components:

$\displaystyle kU+\frac{D_{1}}{2}\left(-\frac{\partial^{2}}{\partial
x^{2}}\right)^{\alpha_{1}/2}U+(U^{2}+\beta V^{2})U-\lambda V$ $\displaystyle=$
$\displaystyle 0,$ $\displaystyle
kV+\frac{D_{2}}{2}\left(-\frac{\partial^{2}}{\partial
x^{2}}\right)^{\alpha_{1}/2}V+(V^{2}+\beta U^{2})V-\lambda U$ $\displaystyle=$
$\displaystyle 0,$ (46)

where we set $\alpha_{1}=2$ (non-fractional diffraction) and $\alpha_{2}=1$.
Such a system may be realized in terms of BEC, considering immiscible
components which represent usual particles ($U$) and those moving by the Lévy
flights ($V$). This system also supports stable DW states, as shown in Fig. 8.

Figure 8: Examples of stable asymmetric DWs with different values of the LI in
the two componnts, $\alpha_{1}=2$ and $\alpha_{2}=1$, produced by the
numerical solution of Eq. (46) with $\beta=3$, $k=-1,$ and $\lambda=0$. The
field components, $U$ and $V$, are displayed for three pairs of values of the
diffraction coefficients, $D_{1,2}$, which are indicated, from top to bottom,
in the plot.

##  V Conclusion

The objective of this work is to demonstrate that the variety of states
maintained by the interplay of the fractional diffraction and self-defocusing
cubic nonlinearity, which may be realized in optics and BEC, can be expanded
by predicting stable DWs (domain walls) in the two-component system of
immiscible fields. The numerical results clearly demonstrate that, in the
entire range of values of the respective LI (Lévy index), $\alpha<2$, that
determines the fractional diffraction, and at all values of the relative
XPM/SPM coefficient $\beta$, which exceed the immiscibility threshold, given
by Eq. (20), the DWs exist and are stable. The same is true for DWs in the
system including the linear mixing between the components, in addition to the
XPM interaction between them (which makes the immiscibility incomplete). The
main characteristic of DW structures is their width. The present analysis
demonstrates that the fractional diffraction essentially affects the scaling
which determines the dependence of the width on the system’s parameters.
Numerical results for the scaling corroborate analytical findings, represented
by Eqs. (37), (41), and (43). It is seen that the decrease of $\alpha$ leads
to steep increase of the scaling exponents $\sim 1/\alpha$, which is explained
by the fact that the fractional diffraction is represented by the nonlocal
operator [the Riesz derivative, defined as per Eq. (2)]. It is also
demonstrated that the DW solutions are essentially simplified in the special
case of $\beta=3$. Currently available techniques should make it possible to
create the predicted DWs patterns in the experiment that may be performed for
the bimodal light propagation in the temporal domain [75].

As extension of the present analysis, it may be natural to study static and
dynamical states built as multi-DW patterns. Another relevant direction is the
consideration of two-dimensional settings, such as radial DWs (also in static
and dynamical states) between immiscible light waves in bulk waveguides, cf.
Refs. [76]-[80].

## Acknowledgment

This work was supported, in part, by the Israel Science Foundation through the
grant No. 1695/22.

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