# Modeling Document-level Temporal Structures for Building Temporal Dependency
Graphs

Prafulla Kumar Choubey
Salesforce Research
<EMAIL_ADDRESS>
&Ruihong Huang
Texas A&M University
<EMAIL_ADDRESS>
Work done while at Texas A&M University

###### Abstract

We propose to leverage news discourse profiling to model document-level
temporal structures for building temporal dependency graphs. Our key
observation is that the functional roles of sentences used for profiling news
discourse signify different time frames relevant to a news story and can,
therefore, help to recover the global temporal structure of a document. Our
analyses and experiments with the widely used knowledge distillation technique
show that discourse profiling effectively identifies distant inter-sentence
event and (or) time expression pairs that are temporally related and otherwise
difficult to locate111Code is available at
https://github.com/prafulla77/Discourse_TDG_AACL2022.

##  1 Introduction

Grounding all events and time expressions to a reference timeline is
fundamental to text understanding. Recently, Yao et al. (2020) proposed a new
task and dataset for building temporal dependency graph (TDG)222The dataset
was obtained from
https://github.com/Jryao/temporal_dependency_graphs_crowdsourcing. TDG is
based on the notion of narrative time and temporal anaphora, and references
each timex to a timex or a meta node and each event to a timex and maybe an
event. The reference timex of an event is either the smallest time (when
identifiable) that encloses the event or the document creation time (DCT).
Similarly, the reference event is selected such that it gives the most precise
temporal interpretation for a child event.

Because each event and timex is referenced to only one timex (or additionally
an event), identified temporal relations represent the most salient relations
that can potentially be used to infer additional temporal relations through
transitivity or commonsense reasoning Yao et al. (2020). This makes
identifying reference timex and reference event more challenging, especially
when they are mentioned across sentences. Human evaluations by Yao et al.
(2020) also found that identifying the appropriate reference timex and
reference event was the most challenging aspect of their annotation.

Figure 1: Temporal structures induced by different content types from the News
Discourse Profiling.

In this work, we focus on improving cross-sentence reference timex and event
mentions identification by exploring discourse-level temporal cues. We choose
the news discourse profiling structure (DP) Choubey et al. (2020). DP
classifies sentences in a news document into one of eight content types,
defined based on the functional role of a sentence in describing the main news
story Teun A (1986); Van Dijk (1988a, b); Choubey et al. (2020), and provides
an event-based functional interpretation of sentences. The eight content types
include main, consequence, previous event, current context, historical,
anecdotal, evaluation and expectation.

As shown in Figure 1, different content types induce different time frames
relevant to a news story that can be beneficial for the global interpretation
of temporal orders among event and timex mentions. For instance, mentions in
historical sentences have a temporal adjacency with other mentions in
historical sentences but are likely to be distant from mentions in other
content types. Similarly, mentions in previous event sentences may have a
temporal adjacency with mentions from one of the previous event, main event or
current-context sentences but are likely to be separated from mentions in any
of the historical, expectation or consequence sentences.

We first summarize the distributional association between the position of
reference mentions and discourse content types in $\S$2.3. Then, we propose a
knowledge distillation-based method to incorporate discourse knowledge into
the TDG system. We experiment with the BERT Devlin et al. (2019) and RoBERTa
Liu et al. (2019) pre-trained languages models and find that the proposed
knowledge distillation-based TDG system is effective in using discourse-level
cues and achieves improved performance on identifying cross-sentence reference
mentions while retaining performance on the intra-sentence mention pairs.

##  2 Background and Analysis

###  2.1 News Discourse Profiling (DP)

Following the news content schemata proposed by Van Dijk Teun A (1986); Van
Dijk (1988a, b), DP (Choubey et al., 2020) defines eight content types. Each
content type describes the functional role of a sentence in describing the
main news event. Main event (M1) sentence describes the major events and
subjects of the news article. Consequence (M2) describes events that are
triggered by the main event. Previous Event (C1) describes recent events that
are a possible cause of the main event. Current Context (C2) describes
remaining contextual information. Historical Event (D1) describes past events
that precede the main events in months and years, Anecdotal Event (D2)
describes unverifiable facts, Evaluation (D3) describes opinionated contents
from immediate participants, experts or journalists, and Expectation (D4)
describes speculations or possible consequences of the main or context events.

Figure 2: An example TDG.

###  2.2 Temporal Dependency Graph (TDG)

TDG (Yao et al., 2020) is a directed edge-labeled graph in which each node is
either an event, a timex, or a meta node (e.g. document creation time). The
reference for each timex/event node is another timex node or a meta node.
Optionally, the temporal position of some events can be more precisely
determined by referencing them to another event, and thus they can also have a
reference event node. For instance, in Figure 2, the event incident can only
be temporally positioned with respect to the timex August 23 while the
temporal order of event broke can be determined with respect to both the timex
later and the event occurred. The edges between event/ timex node pairs are
labeled with one of the overlap, after, before and included temporal relations
while the edges between a timex node and a meta node is assigned a generic
depend-on label. In this work, we focus exclusively on identifying the
reference timex (and event) for each timex (event) without predicting the
temporal relations between them.

###  2.3 Analysis of TDG Structures w.r.t. DP Sentence Types

As illustrated in Figure 1, discourse roles have temporal interpretations that
are useful to locate event and timex relations in a document. Therefore, we
use the recently proposed discourse profiling system by Choubey and Huang
(2021)333The discourse profiling system was obtained from
https://github.com/prafulla77/Discoure_Profiling_RL_EMNLP21Findings. to assign
content type labels to all sentences in the training data and analyze the
distribution of reference timex and event mentions across different content
types. Note that our analyses are based on a neural network model-predicted
discourse content types which are noisy. Additionally, a sentence often
contains more than one event and timex mentions and its content type can only
provide a broad temporal ordering for constituent mentions.

First, we observe that reference timex for both timex (66% to 100%) and event
(54% to 80%) mentions from all content types, except the historical, is
majorly the DCT. Further, among the events from non-historical sentences that
are not referenced to DCT, we observe that majority (71% to 89%) of them are
referenced to a time expression from main, current-context, or previous-event
sentences that overlaps with the DCT. On the other hand, roughly 66% of the
timex mentions in historical sentences are not referenced to any timex mention
but to a meta-node. Similarly, over 52% of event mentions in historical
sentences are referenced to a timex mention within the same sentence. This is
expected given historical sentences describe events from the distant past that
are not easily referable to current timex antecedents.

Second, we observe that a significant proportion of cross-sentence event-event
relations (45% to 84%) have references in either sentence of the same content
type or current context sentences. This can be accounted to the anaphoric
nature of TDG representation that only selects reference event which provides
the most precise temporal interpretation for a given event. Since sentences
with the same content types describe temporally adjacent events, they are
conducive to including the most temporally salient related references for all
events. The exact distribution of all timex and event mentions across
different content types are tabulated in the appendix B.

##  3 Empirical Evaluations and Results

Based on our observations in $\S$2.3, we perform empirical evaluations to
demonstrate the effectiveness of news discourse profiling for building TDG.

###  3.1 Models

Following recent works on temporal relation identification Ballesteros et al.
(2020) and temporal dependency parsing Ross et al. (2020), we experiment with
pre-trained language models, BERT Devlin et al. (2019) and RoBERTa Liu et al.
(2019). We model TDG as a ranking problem Yao et al. (2020), where we add a
meta node each for reference timex and event. Then for each event and timex,
we obtain the reference timex by selecting the one with the highest score.
Similarly, we perform ranking over events to obtain the reference event for
each event. To build a TDG from ranking scores, we adopt the technique used by
Ross et al. (2020) and iteratively select the highest-ranked reference that
does not form a cycle. Within the ranking framework, we develop three models
based on each of the BERT and RoBERTa to analyze the role of news discourse
structure in building TDG.

#### Baseline

: Given the sentences ($x^{1}_{1}$,..,$m_{1}$,..,$x^{n_{1}}_{1}$ and
$x^{1}_{2}$,..,$m_{2}$,..,$x^{n_{2}}_{2}$) corresponding to two mentions
($m_{1}$ and $m_{2}$), we first enclose both mentions in special symbols
($\$m_{1}\$$ and $@m_{2}@$) and follow standard language model tokenization
step to obtain the context representation sequence (e.g. for RoBERTa, we get
$<$s$>$,$x^{1}_{1}$,..,$\$m_{1}\$$,..,$x^{n_{1}}_{1}$,$<$/s$>$,
$<$/s$>$,$x^{1}_{2}$,..,$@m_{2}@$,..,$x^{n_{2}}_{2}$,$<$/s$>$). Then, we use
the pre-trained model to obtain the context representation followed by a
linear neural layer to obtain the final score. Note that the context sequence
follows the textual order of sentences in a document.

#### DP-Feature

: In addition to the context pre-processing used for the baseline model, it
appends special symbols to each sentence corresponding to its discourse
content type (e.g. the context for a mention in the main sentence is
represented as $x_{1}$,..$m_{1}$,..$x_{n}$,$\\#$M1$\\#$). Besides that, it
mimics the baseline model.

#### DP-Distillation

: It uses the distillation technique Hinton et al. (2015) to introduce news
discourse knowledge into our ranking system. We consider the DP model (Choubey
and Huang, 2021) as the teacher network and the language model component from
the baseline model as the student network. The teacher model generates hard
labels for sentences using the argmax function. Using the language model, we
first obtain embeddings for all sentences in a document and then use a linear
neural layer to predict their discourse content types. During training, we
perform iterative gradient updates where we first update parameters based on
the discourse profiling loss followed by gradient updates based on the
temporal ranking loss in each batch. We observe that the order of gradient
updates is important. Performing joint gradient updates or switching the order
of gradient updates may significantly lower the validation performance.

Model | Valid | Test
---|---|---
Yao et al. (2020) | 69.0* | 79.0*
BERT
Baseline | 71.90 | 76.69
DP-Feature | 72.04 | 76.76
DP-Distillation | 72.20 | 78.30
RoBERTa
Baseline | 74.63 | 77.26
DP-Feature | 74.70 | 77.30
DP-Distillation | 75.03 | 78.93
Table 1: Accuracy of different systems on the validation and test datasets. *Results for Yao et al. (2020) are directly taken from the paper and correspond to the single best run.  | Intra-Sentence | Cross-Sentence | No-Parent
---|---|---|---
Model | P | R | F1 | P | R | F1 | P | R | F1
Valid
Baseline | 81.03 | 84.66 | 82.8 | 70.60 | 65.36 | 67.86 | 70.03 | 79.16 | 74.3
DP-Distillation | 81.90 | 83.66 | 82.76 | 72.00 | 68.86 | 68.76 | 67.4 | 82.90 | 74.33
Test
Baseline | 80.6 | 85.86 | 83.16 | 75.30 | 70.56 | 72.83 | 76.20 | 80.93 | 78.43
DP-Distillation | 80.53 | 86.13 | 83.20 | 79.90 | 71.96 | 75.70 | 74.23 | 86.16 | 79.70
Table 2: Precision, recall and F1 scores for RoBERTa-based baseline and DP-
distillation models on intra-sentence, cross-sentence and no-parent subsets
from the validation datasets.

###  3.2 Experimental Settings

We use the training, validation and test splits from Yao et al. (2020) for all
our experiments. Since our goal here is to evaluate the performance of a model
on predicting reference timex and event mentions, we use the gold annotations
for event and timex mentions. All three models are trained using AdamW
optimizer Loshchilov and Hutter (2017) for a maximum of 15 epochs and we use
the epoch yielding the best validation performance. We use the batch size of 5
documents and the learning rate of 0.0001 with linear scheduling and warmup
steps equivalent to 5 epochs. We search learning rate and warmup steps from
[5e-4, 1e-4, 5e-6] and [3, 5, 7] respectively using the baseline model. Then,
both the learning rate and warmup steps are kept constant for all models. Each
training run takes $\sim$12 hours for the baseline and DP-feature models and
$\sim$15 hours for the DP-distillation model. RoBERTa or BERT model is fine-
tuned during the training. We run each model 3 times with random seeds and
report the average performance to reduce the influence of randomness in
training.

All experiments are performed on two NVIDIA-RTX-3090-24GB using PyTorch
1.7.1+cu110 Paszke et al. (2019) and HuggingFace Transformer (v 4.0.1)
libraries Wolf et al. (2019). We use gradient accumulation to fit a batch on 2
GPUs.

###  3.3 Result and Analysis

Table 1 shows the results from our experiments and the previous best-
performing model Yao et al. (2020). The average accuracy of the baseline
model, which relies on the pre-trained RoBERTa (BERT), is 5.63% (2.90%) higher
than the best performing neural model from Yao et al. (2020) on the validation
dataset. Surprisingly, on the test dataset, our RoBERTa (BERT)-based baseline
model achieves 1.74% (2.31%) lower average accuracy.

Next, using discourse content types as a feature in the input sequence brings
negligible improvement over the baseline for both RoBERTa and BERT-based
models. We suspect that special symbols used to represent each content type
are unaware of the temporal associations between different content types.
Thus, the DP-feature model is only capable of modeling co-occurrences of
different content types with reference event and timex mentions. Additionally,
the pre-training of the BERT/ RoBERTa model did not consider special content
types symbols which leads to inconsistent interpretation of their
corresponding tokens during the pre-training and the fine-tuning steps.

DP-distillation method using the RoBERTa (BERT) model, on the other hand,
improves the average accuracy of Baseline by 0.4% (0.3%) and 1.66% (1.61%) on
validation and test datasets respectively. Training with the distillation
technique enables the transfer of DP knowledge directly from the teacher DP
model into the student RoBERTa/ BERT model, unlike the DP-feature model which
is unaware of DP knowledge unless specified through features. Further, the DP-
distillation model learns to predict content type labels, while being
validated over performance on ranking true reference mentions, which provides
it with higher flexibility to distill and retain directly relevant knowledge.

#### Why Discourse Profiling helps?

Since DP provides temporal cues at the sentence level, we mainly expect the
performance improvement to come from cross-sentence event/ timex pairs. To
verify that, we partition our validation and test datasets into three subsets:
1) intra-sentence that includes pairs with both given mention and reference
mention from the same sentence, 2) cross-sentence that includes pairs with
given mention and reference mention from different sentences, and 3) no-parent
that includes mentions which are referenced to a meta node. We compare the
RoBERTa based-baseline and DP-distillation models, which perform better than
the corresponding BERT-based models, on three data partitions in Table 2. As
expected, we found that both the baseline and DP-distillation models achieve
comparable performance on the same-sentence subset. For the no-parent subset,
we observe higher recall and lower precision for the DP-distillation model.
Intuitively, the model learns to link more event and timex mentions to a meta
node. Note that timex mentions from historical sentences are majorly linked to
a meta node ($\S$2.3), which may be partly responsible for this behavior.

On the cross-sentence subset, we observe consistent improvement on all
precision, recall and F1 scores for the DP-distillation model. This is
consistent with our hypothesis that discourse profiling can be used to induce
document-level temporal structures and help in identifying references for
event/timex mentions that require cross-sentence temporal cues.

##  4 Related Work

Most previous works Mani et al. (2006); Bethard and Martin (2007); Kolomiyets
et al. (2012); D’Souza and Ng (2013); Bethard (2013); Ng et al. (2013);
Laokulrat et al. (2013); Mirza and Tonelli (2014); Choubey and Huang (2017);
Yao et al. (2017); Dai et al. (2017); Yao and Huang (2018); Ballesteros et al.
(2020) treat temporal relation extraction as a pair-wise classification
problem and most widely used datasets follow the same pair-wise schema for
annotating temporal relations between event/ timex pairs Graff (2002);
Pustejovsky et al. (2003b, a); Cassidy et al. (2014); UzZaman et al. (2013);
Ning et al. (2018). However, as discussed by Zhang and Xue (2018b, a); Ross et
al. (2020); Yao et al. (2020), pairwise annotations as well as classification
models suffer from quadratic complexity, partial annotations and inconsistent
predictions. Recently, Zhang and Xue (2018b) proposed to build a dependency
tree (TDT) structure to address the above three problems with pair-wise
annotations and modeling and later extended that to temporal dependency graph
Yao et al. (2020). We use the most recent temporal dependency graph dataset
that improves the expressiveness of previous TDT datasets Zhang and Xue
(2018b, 2019) and follow their neural ranking modeling approach. However,
different from the previous work, we explore news discourse profiling to
explicitly focus on improving the performance of a neural ranking model on
cross-sentence event/ timex pairs.

Ng et al. (2013) were the first to show the effectiveness of several discourse
analysis frameworks, including rhetorical structure theory (RST) Mann and
Thompson (1988), PDTB-style discourse relations Prasad et al. (2008) and
topical text segmentation Hearst (1994) for temporal relation extraction.
Different from the above three discourse structures, discourse profiling is a
functional Webber and Joshi (2012) structure and has global event-centric
interpretations. Secondly, Ng et al. (2013) focused on classifying temporal
relations between a given pair of temporally related events. In contrast, our
goal is to identify the most salient reference for every event/ timex mention
that determines its most precise location on the timeline.

##  5 Conclusion

We have shown that news discourse profiling can be used to incorporate
document-level temporal structures when building temporal dependency graphs.
Through analyses, we have shown the distributional association between
discourse content types and positions of reference and child mentions.
Further, empirical evaluation using the knowledge distillation technique shows
that discourse profiling is effective in identifying cross-sentence reference-
child mention pairs. In the future, we will explore new linguistics structures
and modeling techniques to incorporate document-level temporal structures for
building TDG.

##  6 Acknowledgements

We gratefully acknowledge support from National Science Foundation via the
awards IIS-1942918. We would also like to thank the anonymous reviewers for
their feedback.

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  * Yao and Huang (2018) Wenlin Yao and Ruihong Huang. 2018.  Temporal event knowledge acquisition via identifying narratives.  In _Proceedings of the 56th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers)_ , pages 537–547, Melbourne, Australia. Association for Computational Linguistics.
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##  Appendix A Responsible NLP Research Checklist

###  A.1 Limitations and Risks

Our proposed method relies on news articles’ specific functional discourse
structure, called news discourse profiling. This limits the applicability of
the method to the news domain only. We run all experiments on the dataset in
the English language. While we expect the method to work well for other
languages, provided we have a dataset/ model for constructing the news
discourse profiling structure in the target language, we have not verified
this experimentally. Our results are based on the average of 3 runs with
random seeds. We do not expect any potential risk from the proposed method.

###  A.2 Artifacts

We use two publicly available datasets, TDG corpus Yao et al. (2020) and
NewsDiscourse corpus Choubey et al. (2020), for our experiments and analyses.
Our implementations are based on the HuggingFace transformers Wolf et al.
(2019) (Apache license 2.0) and we will release our code under the BSD 3
license.

##  Appendix B Distributional Analysis

| DCT | Meta-node
---|---|---
M1 | 86.5 | 8.5
M2 | 88.9 | 4.4
C1 | 81.9 | 9.0
C2 | 79.8 | 14.6
D1 | 30.9 | 66.1
D2 | 100.0 | -
D3 | 88.8 | 8.8
D4 | 88.4 | 10.9
NA | 66.7 | 25.0
Table 3: Distribution of timex and their reference timex mentions, for each content type. | DCT | Intra-sentence
---|---|---
M1 | 58.4 | 30.4
M2 | 60.1 | 17.5
C1 | 54.3 | 28.7
C2 | 63.6 | 17.1
D1 | 34.4 | 52.3
D2 | 73.5 | 6.0
D3 | 80.8 | 7.1
D4 | 75.6 | 15.8
NA | 69.0 | 20.0
Table 4: Distribution of event and their reference timex mentions, for each content type. | M1 | M2 | C1 | C2 | D1 | D2 | D3 | D4 | NA
---|---|---|---|---|---|---|---|---|---
M1 | 39.8 | 1.1 | 17.2 | 29.0 | 6.5 | - | 1.1 | 2.2 | 3.2
M2 | 69.7 | 4.5 | 7.6 | 13.6 | 4.5 | - | - | - | -
C1 | 31.7 | - | 43.9 | 10.1 | 9.4 | 1.0 | 3.6 | 1.0 | -
C2 | 36.8 | 1.1 | 17.1 | 29.2 | 9.3 | - | 3.5 | 1.8 | 1.3
D1 | 7.2 | - | 39.6 | 10.8 | 36.0 | - | 4.5 | - | 1.8
D2 | 61.3 | - | 9.7 | 19.4 | 6.5 | - | 3.2 | - | -
D3 | 27.0 | 1.9 | 17.1 | 31.4 | 8.8 | 1.0 | 5.0 | 3.6 | 4.7
D4 | 39.7 | 2.9 | 10.3 | 22.1 | 5.9 | - | 4.4 | 11.8 | 2.9
NA | - | - | - | 81.8 | 18.2 | - | - | - | -
Table 5: Distribution of event and their reference timex mentions over different content types, when the reference timex is not the DCT. We can see that majority (71% to 89%) of the events from non-historical sentences are referenced to a time expression from main, current-context, or previous-event sentences that overlaps with the DCT. | M1 | M2 | C1 | C2 | D1 | D2 | D3 | D4 | NA
---|---|---|---|---|---|---|---|---|---
M1 | 43.9 | 1.8 | 10.5 | 21.1 | 5.3 | - | 12.3 | 1.8 | 3.5
M2 | 19.6 | 17.4 | 6.5 | 52.2 | 2.2 | - | - | 2.2 | -
C1 | 15.8 | 1.7 | 35.0 | 30.8 | 10.0 | 0.8 | 3.3 | 1.7 | 0.8
C2 | 8.7 | 2.8 | 7.9 | 62.1 | 5.6 | 0.2 | 9.2 | 3.3 | 0.1
D1 | 3.4 | 0.7 | 4.7 | 31.8 | 46.6 | - | 10.8 | 2.0 | -
D2 | 57.6 | 3.0 | - | 15.2 | 3.0 | - | 18.2 | - | 3.0
D3 | 1.7 | 0.6 | 3.5 | 30.3 | 3.5 | 0.6 | 54.1 | 4.5 | 1.1
D4 | 5.9 | 1.7 | 5.9 | 33.1 | 6.8 | - | 33.1 | 12.7 | 0.8
NA | - | 0.1 | - | 0.2 | 0.1 | 0.1 | 0.3 | - | 0.3
Table 6: Distribution of cross-sentence event and their reference event
mentions over different content types, where the event and its reference event
are from different sentences. We can see that significant proportion of cross-
sentence event-event temporal links (45% to 84%) have references in either
sentences of the same content type or current context sentences.


# Algebraic solitons in the massive Thirring model

Jiaqi Han  Department of Mathematics and Statistics, McMaster University,
Hamilton, Ontario, Canada, L8S 4K1<EMAIL_ADDRESS>,  Cheng He  School of
Mathematics and Statistics, Ningbo University, Ningbo 315211, People’s
Republic of China<EMAIL_ADDRESS>and  Dmitry E. Pelinovsky  Department
of Mathematics and Statistics, McMaster University, Hamilton, Ontario, Canada,
L8S 4K1<EMAIL_ADDRESS>
###### Abstract.

We present exact solutions describing dynamics of two algebraic solitons in
the massive Thirring model. Each algebraic soliton corresponds to a simple
embedded eigenvalue in the Kaup–Newell spectral problem and attains the
maximal mass among the family of solitary waves traveling with the same speed.
By coalescence of speeds of the two algebraic solitons, we find a new solution
for an algebraic double-soliton which corresponds to a double embedded
eigenvalue. We show that the double-soliton attains the double mass of a
single soliton and describes a slow interaction of two identical algebraic
solitons.

##  1\. Introduction

Algebraic solitons are traveling solitary waves with the power rather than
exponential decay rate at infinity. Such solutions are common for integrable
nonlinear equations with nonlocal terms such as the Benjamin–Ono and
Kadomtsev–Petviashvili equations, where they are associated with isolated
eigenvalues of the linear Lax equations [1]. However, algebraic solitons are
special for local integrable nonlinear equations since they arise as the
limiting points in the family of exponential solitons and they are associated
with embedded eigenvalues in the continuous spectrum of the linear Lax
equations [16, 21]. Physically relevant examples of the algebraic solitons as
special limits of exponential solitons appear in the modified Korteweg–de
Vries equation [7, 24], the derivative nonlinear Schrödinger equation [10, 25,
31], and the nonlinear Dirac equation [12].

This work is devoted to the algebraic solitons in the massive Thirring model
(MTM) written in laboratory coordinates as

$\left\\{\begin{array}[]{l}\mathrm{i}(u_{t}+u_{x})+v=|v|^{2}u\\\
\mathrm{i}(v_{t}-v_{x})+u=|u|^{2}v\end{array}\right.$ (1)

where $(u,v)\in\mathbb{C}^{2}$ and subscripts denote partial derivatives in
$(x,t)\in\mathbb{R}^{2}$. The MTM system (1) is a prototypical Dirac equation
which belongs to the class of integrable equations associated with the
Kaup–Newell (KN) spectral problem [15, 17].

Stability of algebraic solitons is a notoriously difficult mathematical
problem, where every method of nonlinear analysis known in the theory of
integrable systems fails. Coercivity of the energy function required for the
proof of Lyapunov stability holds for exponential solitons [23] but fails for
algebraic solitons because the spectral gap between the zero eigenvalue and
the continuous spectrum in the linearized MTM system closes up in the limit to
the algebraic soliton. Stability of exponential solitons in the MTM system can
be proven with the Darboux transformation [4] which allows us to construct
exponential solitons from an isolated eigenvalue of the KN spectral problem.
However, the Darboux transformation does not allow us to obtain algebraic
solitons because the embedded eigenvalue has to be defined inside the
continuous spectrum of the KN spectral problem, where both eigenfunctions are
bounded. Finally, the inverse scattering transform (IST) method requires fast
spatial decay of solutions of the MTM system at infinity in order to ensure
smoothness properties of the scattering data and solvability of the associated
Riemann–Hilbert problems [13, 22]. Algebraic solitons decay too slowly and
violate the requirements of the fast spatial decay.

Due to these limitations, the main purpose of this paper is to explore direct
methods of solutions of the MTM system (1) and to study interaction of two
algebraic solitons. Hirota’s bilinear formulation of the MTM system (1) was
recently developed in [5] to obtain exponential multi-solitons. By using the
analytical expressions for two exponential solitons, we obtain the exact
solutions for two algebraic solitons which scatter fast from each other with
two different wave speeds. In the limit when the wave speeds coincide, we
obtain the double-soliton solution which describes a slow interaction of two
identical algebraic solitons.

There has been a recent spike in the study of rational solutions of the
integrable systems in the context of rogue wave dynamics [26, 27, 28]. Similar
rational solutions of the MTM system for rogue waves were studied in [6, 11,
29], where they appear on the constant, modulationally unstable background.
Compared to these solutions, our rational solution for the algebraic double-
soliton is not a rogue wave since it describes two algebraic solitons at the
trivial, modulationally stable background. To the best of our knowledge, the
double-soliton solution is derived in the MTM system (1) for the first time.
Its existence suggests that a hierarchy of higher-order rational solutions
exists in the MTM system (1) which has not been explored yet. Similar
solutions for the algebraic double solitons in the derivative NLS equation
were constructed in [10] by using generalized Darboux transformation and in
[31] by taking the limit of the exponential double-solitons obtained from the
double-fold Darboux transformation.

Double solitons traveling with nearly the same speed correspond to double
eigenvalues of the KN spectral problem. The IST method was recently employed
to construct exponential double-solitons of the derivative NLS equation and
related equations [30, 32, 33, 34]. Such solutions are related to double
isolated eigenvalues or, equivalently, to double poles in solutions of the
Riemann–Hilbert problems. The method does not work for the double embedded
eigenvalues. In another work [19], we will obtain the exponential double-
solitons of the MTM system (1) by using the IST method and show that these
solutions degenerate into the algebraic double-solitons in the limit when the
double isolated eigenvalue becomes embedded in the continuous spectrum of the
KN spectral problem. This provides the first nontrivial example of embedded
eigenvalues of higher algebraic multiplicity for the KN spectral problem, the
possibility of which was first predicted 20 years ago in [16].

Our results suggest stability of the traveling algebraic solitons in the time
evolution of the MTM system (1). This conclusion agrees with the perturbation
theory for embedded eigenvalues of the KN spectral problem developed in [16,
Sections 6-7], which suggests that the simple embedded eigenvalues for
algebraic solitons are generally ejected from the continuous spectrum to
become simple isolated eigenvalues for the exponential solitons with nearly
selected speed and frequency. A rigorous proof of orbital stability of the
traveling algebraic soliton is still an open problem for the MTM system (1).

The rest of this paper is organized as follows. The main results are presented
in Section 2. The computational proofs are elaborated in Section 3 where we
obtain a new parameterization of the exponential two-soliton solutions of the
MTM system (1) and then take the limits to the algebraic two-soliton solutions
with different speeds and to the algebraic double-soliton of the same speed.
Section 4 emphasizes further directions which may be undertaken from the
outcomes of our work.

##  2\. Main results

To simplify presentation of soliton solutions of the MTM system (1), we shall
use the basic symmetries of this Hamiltonian system. These include the
translational and rotational symmetries

$\left[\begin{matrix}u(x,t)\\\
v(x,t)\end{matrix}\right]\quad\mapsto\quad\left[\begin{matrix}u(x+x_{0},t+t_{0})e^{\mathrm{i}\theta_{0}}\\\
v(x+x_{0},t+t_{0})e^{\mathrm{i}\theta_{0}}\end{matrix}\right],\qquad
x_{0},t_{0},\theta_{0}\in\mathbb{R},$ (2)

as well as the Lorentz symmetry

$\left[\begin{matrix}u(x,t)\\\
v(x,t)\end{matrix}\right]\;\;\mapsto\;\;\left[\begin{matrix}\left(\frac{1-c}{1+c}\right)^{1/4}u\left(\frac{x+ct}{\sqrt{1-c^{2}}},\frac{t+cx}{\sqrt{1-c^{2}}}\right)\\\
\left(\frac{1+c}{1-c}\right)^{1/4}v\left(\frac{x+ct}{\sqrt{1-c^{2}}},\frac{t+cx}{\sqrt{1-c^{2}}}\right)\end{matrix}\right],\qquad
c\in(-1,1).$ (3)

Without loss of generality, each solution of the MTM system (1) can be
extended with three translational parameter in (2) and the speed parameter
$c\in(-1,1)$ in (3).

A normalized family of exponential solitons of the MTM system (1) is given by
the standing wave solutions of the form

$\left[\begin{matrix}u_{\rm sol}(x,t)\\\ v_{\rm
sol}(x,t)\end{matrix}\right]=\sin\gamma\left[\begin{matrix}{\rm
sech}\left(x\;\sin\gamma+\frac{\mathrm{i}\gamma}{2}\right)\\\ {\rm
sech}\left(x\;\sin\gamma-\frac{\mathrm{i}\gamma}{2}\right)\end{matrix}\right]e^{\mathrm{i}t\cos\gamma},\quad\gamma\in(0,\pi).$
(4)

A general family with two translational parameters and the speed parameter
$c\in(-1,1)$ is obtained from the translational and Lorentz symmetry given by
(2) and (3).

The only parameter $\gamma\in(0,\pi)$ in (4) defines the frequency parameter
$\omega:=\cos(\gamma)$ of the exponential solitons. The frequency $\omega$ is
chosen in the gap $(-1,1)$ of the frequency spectrum of the linear Dirac
operator

$\mathcal{D}:=\left[\begin{matrix}\mathrm{i}\partial_{x}&1\\\
1&-\mathrm{i}\partial_{x}\end{matrix}\right],$

which determines the time evolution of the MTM system (1). This is one of the
reasons why Dirac solitons are sometimes called the gap solitons [20].

The limits $\omega\to\pm 1$ are referred to as the nonrelativistic limits of
the MTM system (1). It is well-known (see, e.g., [2, 3, 8]) that the nonlinear
Dirac equations such as the MTM system (1) can be reduced to the focusing NLS
equation as $\omega\to 1$ and to the defocusing NLS equation as $\omega\to-1$.
The normalized form for the two NLS equations is given by

$\mathrm{i}\psi_{t}+\psi_{xx}+\sigma|\psi|^{2}\psi=0,\qquad\sigma={\rm
sgn}(\omega)=\pm 1.$ (5)

The family (4) reduces to the small-amplitude, long-scale, ${\rm sech}$-shaped
soliton of the focusing NLS equation (5) with $\sigma=+1$ as $\omega\to 1$
($\gamma\to 0$) and to the finite-amplitude, finite-scale, algebraic soliton

$\gamma=\pi:\quad\left[\begin{matrix}u_{\rm alg}(x,t)\\\ v_{\rm
alg}(x,t)\end{matrix}\right]=\left[\begin{matrix}\displaystyle\frac{2}{1+2\mathrm{i}x}\\\
\displaystyle\frac{2}{1-2\mathrm{i}x}\end{matrix}\right]e^{-\mathrm{i}t}$ (6)

as $\omega\to-1$ ($\gamma\to\pi$). Note that the NLS equations (5) holds in
the limit of small amplitudes and hence the algebraic soliton (6) does not
satisfy the reduction to the defocusing NLS equation (5) with $\sigma=-1$ as
$\omega\to-1$.

The algebraic soliton (6) has the largest mass among the exponential solitons
in the family (4), where the mass for the MTM system (1) is defined by

$Q(u,v):=\int_{\mathbb{R}}(|u|^{2}+|v|^{2})dx.$ (7)

It follows from (4) that

$|u(x,t)|^{2}+|v(x,t)|^{2}=\frac{4\sin^{2}\gamma}{\cos\gamma+\cosh(2x\;\sin\gamma)},$

which implies that $Q(u_{\rm sol},v_{\rm sol})=4\gamma$ with the largest mass
at $Q(u_{\rm alg},v_{\rm alg})=4\pi$.

Let us now present the main results of this paper. First, we have obtained the
exact formula for the algebraic two-soliton solution of the MTM system (1).
The exact formula can be written in the form:

$u(x,t)=-\frac{2\mathrm{i}\delta_{1}^{-1/2}e^{-\mathrm{i}T_{1}}\left[2X_{2}+\mathrm{i}-\frac{4\mathrm{i}\delta_{1}}{\delta_{1}-\delta_{2}}\right]+2\mathrm{i}\delta_{2}^{-1/2}e^{-\mathrm{i}T_{2}}\left[2X_{1}+\mathrm{i}+\frac{4\mathrm{i}\delta_{2}}{\delta_{1}-\delta_{2}}\right]}{(2X_{1}-\mathrm{i})(2X_{2}-\mathrm{i})+\frac{4\sqrt{\delta_{1}\delta_{2}}}{(\delta_{1}-\delta_{2})^{2}}\left[\sqrt{\delta_{1}}e^{\frac{\mathrm{i}}{2}(T_{1}-T_{2})}-\sqrt{\delta_{2}}e^{-\frac{\mathrm{i}}{2}(T_{1}-T_{2})}\right]^{2}}$
(8)

and

$v(x,t)=\frac{2\mathrm{i}\delta_{1}^{1/2}e^{-\mathrm{i}T_{1}}\left[2X_{2}-\mathrm{i}-\frac{4\mathrm{i}\delta_{2}}{\delta_{1}-\delta_{2}}\right]+2\mathrm{i}\delta_{2}^{1/2}e^{-\mathrm{i}T_{2}}\left[2X_{1}-\mathrm{i}+\frac{4\mathrm{i}\delta_{1}}{\delta_{1}-\delta_{2}}\right]}{(2X_{1}+\mathrm{i})(2X_{2}+\mathrm{i})+\frac{4\sqrt{\delta_{1}\delta_{2}}}{(\delta_{1}-\delta_{2})^{2}}\left[\sqrt{\delta_{1}}e^{-\frac{\mathrm{i}}{2}(T_{1}-T_{2})}-\sqrt{\delta_{2}}e^{\frac{\mathrm{i}}{2}(T_{1}-T_{2})}\right]^{2}}$
(9)

where

$\displaystyle X_{j}=\frac{x+c_{j}t}{\sqrt{1-c_{j}^{2}}}+x_{j},\quad
T_{j}=\frac{t+c_{j}x}{\sqrt{1-c_{j}^{2}}}+t_{j},\quad
c_{j}=\frac{\delta_{j}^{2}-1}{\delta_{j}^{2}+1}$

with the parameters $\delta_{1,2}>0$ such that $\delta_{1}\neq\delta_{2}$ and
translational parameters $x_{1,2}\in\mathbb{R}$ and $t_{1,2}\in\mathbb{R}$.

Figure 1 shows the solution surfaces which suggest that the algebraic two-
soliton solution given by (8) and (9) describes scattering of two algebraic
solitons. When the wave speeds $c_{1}$ and $c_{2}$ are very different from
each other (top panels), the scattering is fast and the trajectories of the
two solitons are almost straight lines. When the wave speeds approach to each
other (bottom panels), the scattering becomes slow and the trajectories of the
two solitons are curved near the soliton overlaping regions.

Figure 1. The solution surface for $|u|^{2}+|v|^{2}$ versus $(x,t)$ for the
family (8) and (9) with $x_{1}=x_{2}=t_{1}=t_{2}=0$ and
$\delta_{1}=1+\varepsilon$, $\delta_{2}=1-\varepsilon$ with $\varepsilon=0.75$
(top left), $\varepsilon=0.5$ (top right), $\varepsilon=0.25$ (bottom left),
and $\varepsilon=0.01$ (bottom right).

In the limit $\delta_{1},\delta_{2}\to 1$ of the algebraic two-soliton
solution given by (8) and (9), we derived the following new rational solution
to the MTM system (1):

$\left[\begin{matrix}u_{\rm double}(x,t)\\\ v_{\rm
double}(x,t)\end{matrix}\right]=\left[\begin{matrix}\displaystyle\frac{4(-3+6\mathrm{i}x-12x^{2}-8\mathrm{i}x^{3}-12t(2x-\mathrm{i})-\mathrm{i}\beta)}{3+24\mathrm{i}x-24x^{2}+32\mathrm{i}x^{3}-16x^{4}+48t^{2}+2\beta(2x-\mathrm{i})}\\\
\displaystyle\frac{4(-3-6\mathrm{i}x-12x^{2}+8\mathrm{i}x^{3}+12t(2x+\mathrm{i})+\mathrm{i}\beta)}{3-24\mathrm{i}x-24x^{2}-32\mathrm{i}x^{3}-16x^{4}+48t^{2}+2\beta(2x+\mathrm{i})}\end{matrix}\right]e^{-\mathrm{i}t}.$
(10)

where $\beta\in\mathbb{R}$ is a free parameter of the family. A general family
with three translational parameters and the speed parameter $c\in(-1,1)$ is
obtained from the translational and Lorentz symmetry given by (2) and (3).

Figure 2. The solution surface for $|u|^{2}+|v|^{2}$ versus $(x,t)$ for the
family (25) with $\beta=0$ (top left), $\beta=1$ (top right), $\beta=10$
(bottom left), and $\beta=100$ (bottom right).

The algebraic double-soliton given by (10) describes a slow scattering of two
identical algebraic solitons. The parameter $\beta$ describes the distance
between the two solitons. Figure 2 illustrates the solution surface for
$|u|^{2}+|v|^{2}$ versus $(x,t)$ for the family of solutions (10) with
$\beta=0,1,10,100$. The solution with $\beta=0$ is symmetric with the global
maximum at $(0,0)$. Since $|u(0,0)|^{2}+|v(0,0)|^{2}=32$ for (10) and
$|u(0,0)|^{2}+|v(0,0)|^{2}=8$ for (6), the double-soliton has the quadruple
magnification factor for the squared amplitudes compared to the single
algebraic soliton.

As $\beta$ increases, the symmetry is broken and the magnification factor
becomes smaller. For sufficiently large $\beta$, the two solitons do not
overlap but slowly scatter at a distance from each other. As $\beta\to\infty$,
one soliton goes to infinity and the other soliton is located near the origin.
Indeed, the family of solutions (10) converges as $\beta\to\infty$ to a single
algebraic soliton (6). We will prove that

$Q(u_{\rm double},v_{\rm double})=8\pi=2Q(u_{\rm alg},v_{\rm alg}),$ (11)

which implies that the double-soliton (10) has a double mass compared to the
single algebraic soliton (6).

Figure 3. The contour plots for the solution surfaces from Fig. 1 with
$\varepsilon=0.5$ (left) and from Fig. 2 with $\beta=0$ (right). The red lines
show the straight lines $x+c_{1}t=0$ and $x+c_{2}t=0$ (left) and the parabolas
$x^{2}=\pm 3t$ (right).

For the fast scattering of two algebraic solitons given by (8) and (9), the
algebraic solitons move along straight lines before and after interaction in
the overlapping region. No phase shift arises as a result of the soliton
interaction, which is a standard feature of algebraic multi-soliton solutions,
see [9, 14]. This is illustrated on the contour plot of Figure 3 (left panel),
where we showed the solution from Figure 1 with $\varepsilon=0.5$ together
with the straight lines $x+c_{1}t=0$ and $x+c_{2}t=0$. On the other hand, the
slow scattering of two identical solitons given by (10) results in the
solitons propagating along a curve on the $(x,t)$ plane. Figure 3 (right
panel) shows the solution from Figure 2 with $\beta=0$ together with the
parabolas $x^{2}=\pm 3t$. The free algebraic solitons would be standing waves
with $c=0$ but their slow interaction results in the dynamics along the
trajectories at $x\approx\pm\sqrt{3t}$ as $t\to\infty$ with nonzero but
asymptotically vanishing velocities $\pm\frac{\sqrt{3}}{2\sqrt{t}}\to 0$ as
$t\to+\infty$.

The next section contains derivation of (8)–(9) and (10) as well as the proof
of (11).

##  3\. Proof of the main results

As a starting point, the MTM system (1) is transformed to a system of bilinear
equations by the following transformation [5],

$u=\frac{g}{\bar{f}},\qquad v=\frac{h}{f},$ (12)

where $\bar{f}$ is complex conjugate of $f$. Substituting (12) into (1) yields
the following system of bilinear equations for $f$, $h$, and $g$:

$\left.\begin{array}[]{r}\mathrm{i}f(g_{t}+g_{x})-\mathrm{i}g(f_{t}+f_{x})+h\bar{f}=0,\\\
\mathrm{i}\bar{f}(h_{t}-h_{x})-\mathrm{i}h(\bar{f}_{t}-\bar{f}_{x})+gf=0,\\\
\mathrm{i}\bar{f}(f_{x}+f_{t})-\mathrm{i}f(\bar{f}_{t}+\bar{f}_{x})-|h|^{2}=0,\\\
\mathrm{i}f(\bar{f}_{t}-\bar{f}_{x})-\mathrm{i}\bar{f}(f_{t}-f_{x})-|g|^{2}=0.\end{array}\right\\}$
(13)

It was proven in [5] that the system (13) is satisfied by the following two-
soliton solutions in the general form:

$\left\\{\begin{array}[]{l}f=1+c_{11}e^{\zeta_{1}+\bar{\zeta}_{1}}+c_{12}e^{\zeta_{1}+\bar{\zeta}_{2}}+c_{21}e^{\bar{\zeta}_{1}+\zeta_{2}}+c_{22}e^{\zeta_{2}+\bar{\zeta}_{2}}+c_{1212}e^{\zeta_{1}+\bar{\zeta}_{1}+\zeta_{2}+\bar{\zeta}_{2}},\\\
h=\bar{\alpha}_{1}e^{\zeta_{1}}+\bar{\alpha}_{2}e^{\zeta_{2}}+c_{121}e^{\zeta_{1}+\zeta_{2}+\bar{\zeta}_{1}}+c_{122}e^{\zeta_{1}+\zeta_{2}+\bar{\zeta}_{2}},\\\
g=\frac{\mathrm{i}\bar{\alpha}_{1}}{p_{1}}e^{\zeta_{1}}+\frac{\mathrm{i}\bar{\alpha}_{2}}{p_{2}}e^{\zeta_{2}}-\frac{\mathrm{i}\bar{p}_{1}}{p_{1}p_{2}}c_{121}e^{\zeta_{1}+\zeta_{2}+\bar{\zeta}_{1}}-\frac{\mathrm{i}\bar{p}_{2}}{p_{1}p_{2}}c_{122}e^{\zeta_{1}+\zeta_{2}+\bar{\zeta}_{2}},\end{array}\right.$
(14)

where

$\zeta_{j}=\frac{1}{2}\left(p_{j}+\frac{1}{p_{j}}\right)x+\frac{1}{2}\left(p_{j}-\frac{1}{p_{j}}\right)t$

and

$\displaystyle c_{ij}$
$\displaystyle=-\frac{\mathrm{i}p_{i}\bar{\alpha}_{i}\alpha_{j}}{(p_{i}+\bar{p}_{j})^{2}},$
$\displaystyle c_{12j}$
$\displaystyle=(p_{1}-p_{2})\bar{p}_{j}\left[\frac{\bar{\alpha}_{2}c_{1j}}{p_{1}(p_{2}+\bar{p}_{j})}-\frac{\bar{\alpha}_{1}c_{2j}}{p_{2}(p_{1}+\bar{p}_{j})}\right],$
$\displaystyle c_{1212}$
$\displaystyle=|p_{1}-p_{2}|^{2}\left[\frac{c_{11}c_{22}}{(p_{1}+\bar{p}_{2})(p_{2}+\bar{p}_{1})}-\frac{c_{12}c_{21}}{(p_{1}+\bar{p}_{1})(p_{2}+\bar{p}_{2})}\right],$

whereas parameters $p_{1},p_{2},\alpha_{1},\alpha_{2}\in\mathbb{C}$ are
arbitrary.

We are going to obtain new solutions of the MTM system (1) in the form (8)–(9)
and (10) by using a new parameterization of the exponential two-soliton
solutions (14) and by taking the limits to the algebraic two-soliton
solutions.

###  3.1. New parameterization of the exponential two-soliton solutions

In order to represent the two-soliton solutions (14) in the meaningful way
where each soliton resembles the exponential soliton given by (4), we will use
the following parameterization:

$p_{j}=\mathrm{i}\delta_{j}e^{-i\gamma_{j}},\quad\alpha_{j}=2\sqrt{\delta_{j}}\sin\gamma_{j}e^{\frac{\mathrm{i}\gamma_{j}}{2}+\sin\gamma_{j}x_{j}-i\cos\gamma_{j}t_{j}},\quad
j=1,2,$ (15)

with arbitrary parameters $\gamma_{j}\in(0,\pi)$, $\delta_{j}>0$, and
$(x_{j},t_{j})\in\mathbb{R}^{2}$. By using the parameterization (15) for
$p_{j}$, we obtain

$\zeta_{j}=\sin\gamma_{j}\left[\frac{1}{2}(\delta_{j}+\delta_{j}^{-1})x+\frac{1}{2}(\delta_{j}-\delta_{j}^{-1})t\right]+\mathrm{i}\cos\gamma_{j}\left[\frac{1}{2}(\delta_{j}-\delta_{j}^{-1})x+\frac{1}{2}(\delta_{j}+\delta_{j}^{-1})t\right].$

This representation resembles the Lorentz transformation (3) with

$\frac{1}{2}(\delta_{j}+\delta_{j}^{-1})=\frac{1}{\sqrt{1-c_{j}^{2}}},\quad\frac{1}{2}(\delta_{j}-\delta_{j}^{-1})=\frac{c_{j}}{\sqrt{1-c_{j}^{2}}},$

where we have introduced the wave speeds

$c_{j}:=\frac{\delta_{j}^{2}-1}{\delta_{j}^{2}+1}\in(-1,1),\quad j=1,2.$ (16)

Due to parameterization (15), we obtain

$c_{jj}=e^{-\mathrm{i}\gamma_{j}+2\sin\gamma_{j}x_{j}},\quad j=1,2,$

and, more generally,

$c_{ij}=-\frac{4\sqrt{\delta_{i}\delta_{j}}\sin\gamma_{i}\sin\gamma_{j}\delta_{i}}{(\delta_{i}e^{-\frac{\mathrm{i}}{2}(\gamma_{i}+\gamma_{j})}-\delta_{j}e^{\frac{\mathrm{i}}{2}(\gamma_{i}+\gamma_{j})})^{2}}e^{-\frac{\mathrm{i}}{2}(\gamma_{i}+\gamma_{j})+\sin\gamma_{i}x_{i}+\sin\gamma_{j}x_{j}+\mathrm{i}\cos\gamma_{i}t_{i}-\mathrm{i}\cos\gamma_{j}t_{j}},$

so that we can introduce the following two real-valued coordinates

$\xi_{j}:=\sin\gamma_{j}\left(\frac{x+c_{j}t}{\sqrt{1-c_{j}^{2}}}+x_{j}\right),\quad\eta_{j}:=\cos\gamma_{j}\left(\frac{t+c_{j}x}{\sqrt{1-c_{j}^{2}}}+t_{j}\right),$
(17)

where $(x_{j},t_{j})\in\mathbb{R}^{2}$ play the role of translational
parameters in (2).

To derive the explicit expressions for $c_{12j}$ and $c_{1212}$, we use (15)
and obtain

$\displaystyle
c_{121}=(p_{1}-p_{2})\bar{p}_{1}\left[\frac{\bar{\alpha}_{2}c_{11}}{p_{1}(p_{2}+\bar{p}_{1})}-\frac{\bar{\alpha}_{1}c_{21}}{p_{2}(p_{1}+\bar{p}_{1})}\right]=\frac{\mathrm{i}\bar{p}_{1}|\alpha_{1}|^{2}\bar{\alpha}_{2}(p_{1}-p_{2})^{2}}{(p_{1}+\bar{p}_{1})^{2}(\bar{p}_{1}+p_{2})^{2}}$

that

$\displaystyle
c_{121}=\frac{(p_{1}-p_{2})^{2}}{(\bar{p}_{1}+p_{2})^{2}}\bar{\alpha}_{2}e^{i\gamma_{1}+2\sin\gamma_{1}x_{1}},\quad
c_{122}=\frac{(p_{1}-p_{2})^{2}}{(p_{1}+\bar{p}_{2})^{2}}\bar{\alpha}_{1}e^{i\gamma_{2}+2\sin\gamma_{2}x_{2}}.$

Similarly, we obtain

$\displaystyle
c_{1212}=e^{-\mathrm{i}\gamma_{1}-\mathrm{i}\gamma_{2}+2\sin\gamma_{1}x_{1}+2\sin\gamma_{2}x_{2}}A_{12},$

where

$\displaystyle A_{12}$
$\displaystyle=\frac{|p_{1}-p_{2}|^{2}}{(p_{1}+\bar{p}_{2})(p_{2}+\bar{p}_{1})}\left[1-\frac{16\delta_{1}^{2}\delta_{2}^{2}\sin^{2}\gamma_{1}\sin^{2}\gamma_{2}}{(p_{1}+\bar{p}_{1})(p_{2}+\bar{p}_{2})(p_{1}+\bar{p}_{2})(\bar{p}_{1}+p_{2})}\right]$
$\displaystyle=-\frac{|p_{1}-p_{2}|^{2}}{(p_{1}+\bar{p}_{2})^{2}(p_{2}+\bar{p}_{1})^{2}}\left[(\delta_{1}e^{-\mathrm{i}\gamma_{1}}-\delta_{2}e^{\mathrm{i}\gamma_{2}})(\delta_{2}e^{-\mathrm{i}\gamma_{2}}-\delta_{1}e^{\mathrm{i}\gamma_{1}})+4\delta_{1}\delta_{2}\sin\gamma_{1}\sin\gamma_{2}\right]$
$\displaystyle=\left(\frac{\delta_{1}^{2}+\delta_{2}^{2}-2\delta_{1}\delta_{2}\cos(\gamma_{1}-\gamma_{2})}{\delta_{1}^{2}+\delta_{2}^{2}-2\delta_{1}\delta_{2}\cos(\gamma_{1}+\gamma_{2})}\right)^{2}.$

This representation allows us to rewrite the component $f$ of the two-soliton
solution (14) in the explicit form:

$\displaystyle f$
$\displaystyle=1+e^{2\xi_{1}-\mathrm{i}\gamma_{1}}+e^{2\xi_{2}-\mathrm{i}\gamma_{2}}+A_{12}e^{2\xi_{1}+2\xi_{2}-\mathrm{i}\gamma_{1}-\mathrm{i}\gamma_{2}}-4\sqrt{\delta_{1}\delta_{2}}\sin\gamma_{1}\sin\gamma_{2}e^{\xi_{1}+\xi_{2}-\frac{\mathrm{i}}{2}\gamma_{1}-\frac{\mathrm{i}}{2}\gamma_{2}}$
$\displaystyle\quad\times\left[\frac{\delta_{1}e^{\mathrm{i}(\eta_{1}-\eta_{2})}}{(\delta_{1}e^{-\frac{\mathrm{i}}{2}(\gamma_{1}+\gamma_{2})}-\delta_{2}e^{\frac{\mathrm{i}}{2}(\gamma_{1}+\gamma_{2})})^{2}}+\frac{\delta_{2}e^{-\mathrm{i}(\eta_{1}-\eta_{2})}}{(\delta_{1}e^{\frac{\mathrm{i}}{2}(\gamma_{1}+\gamma_{2})}-\delta_{2}e^{-\frac{\mathrm{i}}{2}(\gamma_{1}+\gamma_{2})})^{2}}\right],$
(18)

where $\xi_{j}$ and $\eta_{j}$ are given by (17). The components $h$ and $g$
are written in the hybrid form for now:

$\displaystyle h$
$\displaystyle=\bar{\alpha}_{1}e^{\zeta_{1}}\left[1+\left(\frac{p_{1}-p_{2}}{p_{1}+\bar{p}_{2}}\right)^{2}e^{2\xi_{2}+\mathrm{i}\gamma_{2}}\right]+\bar{\alpha}_{2}e^{\zeta_{2}}\left[1+\left(\frac{p_{1}-p_{2}}{\bar{p}_{1}+p_{2}}\right)^{2}e^{2\xi_{1}+\mathrm{i}\gamma_{1}}\right]$
(19)

and

$\displaystyle g$
$\displaystyle=\frac{\mathrm{i}\bar{\alpha}_{1}}{p_{1}}e^{\zeta_{1}}\left[1+\left(\frac{p_{1}-p_{2}}{p_{1}+\bar{p}_{2}}\right)^{2}e^{2\xi_{2}+3\mathrm{i}\gamma_{2}}\right]+\frac{\mathrm{i}\bar{\alpha}_{2}}{p_{2}}e^{\zeta_{2}}\left[1+\left(\frac{p_{1}-p_{2}}{\bar{p}_{1}+p_{2}}\right)^{2}e^{2\xi_{1}+3\mathrm{i}\gamma_{1}}\right].$
(20)

The two-soliton solution corresponds to two exponential solitons propagating
according to their wave speeds $c_{1,2}$ obtained from $\delta_{1,2}$ by (16)
and having frequencies $\omega_{1,2}=\cos(\gamma_{1,2})$ obtained from
$\gamma_{1,2}$. The one-soliton solution appears from this formula by taking
$\xi_{2}\to-\infty$:

$\displaystyle
u=\lim_{\xi_{2}\to-\infty}\frac{g}{\bar{f}}=\frac{\mathrm{i}\bar{\alpha}_{1}e^{\zeta_{1}}}{p_{1}(1+e^{2\xi_{1}+\mathrm{i}\gamma_{1}})}=\sin\gamma_{1}\delta_{1}^{-1/2}{\rm
sech}\left(\xi_{1}+\frac{\mathrm{i}}{2}\gamma_{1}\right)e^{\mathrm{i}\eta_{1}}$

and similarly,

$\displaystyle
v=\lim_{\xi_{2}\to-\infty}\frac{h}{f}=\frac{\bar{\alpha}_{1}e^{\zeta_{1}}}{1+e^{2\xi_{1}-\mathrm{i}\gamma_{1}}}=\sin\gamma_{1}\delta_{1}^{1/2}{\rm
sech}\left(\xi_{1}-\frac{\mathrm{i}}{2}\gamma_{1}\right)e^{\mathrm{i}\eta_{1}},$

from which we recognize the exact solution (4) extended by the symmetry
transformations (2) and (3).

###  3.2. Limit to the algebraic two-soliton solutions

Each soliton in the two-soliton solution expressed in the Hirota form (12)
with (18), (19), (20) has four arbitrary parameters $\delta_{j}>0$,
$\gamma_{j}\in(0,\pi)$, and $(x_{j},t_{j})\in\mathbb{R}^{2}$ for $j=1,2$. In
order to get the algebraic two-soliton solutions, we need to take the limit
$\gamma_{j}\to\pi$ for each $j=1,2$. Hence, we set

$\gamma_{j}=\pi-\epsilon_{j},\quad j=1,2$

and expand to the leading order

$\sin\gamma_{j}=\epsilon_{j}+\mathcal{O}(\epsilon_{j}^{3}),\quad\cos\gamma_{j}=1+\mathcal{O}(\epsilon_{j}^{2}).$

We can then define

$X_{j}:=\frac{x+c_{j}t}{\sqrt{1-c_{j}^{2}}}+x_{j},\quad
T_{j}:=\frac{t+c_{j}x}{\sqrt{1-c_{j}^{2}}}+t_{j}$

and expand

$\displaystyle\left(\frac{p_{1}-p_{2}}{p_{1}+\bar{p}_{2}}\right)^{2}=\left(\frac{\delta_{1}e^{\mathrm{i}\epsilon_{1}}-\delta_{2}e^{\mathrm{i}\epsilon_{2}}}{\delta_{1}e^{\mathrm{i}\epsilon_{1}}-\delta_{2}e^{-\mathrm{i}\epsilon_{2}}}\right)^{2}=1-\frac{4\mathrm{i}\epsilon_{2}\delta_{2}}{\delta_{1}-\delta_{2}}+\mathcal{O}(\epsilon_{1}^{2},\epsilon_{2}^{2})$

and

$\displaystyle
A_{12}=\left(\frac{\delta_{1}^{2}+\delta_{2}^{2}-2\delta_{1}\delta_{2}\cos(\epsilon_{1}-\epsilon_{2})}{\delta_{1}^{2}+\delta_{2}^{2}-2\delta_{1}\delta_{2}\cos(\epsilon_{1}+\epsilon_{2})}\right)^{2}=1-\frac{8\delta_{1}\delta_{2}\epsilon_{1}\epsilon_{2}}{(\delta_{1}-\delta_{2})^{2}}+\mathcal{O}(\epsilon_{1}^{2}\epsilon_{2}^{2}).$

This yields the expansions:

$\displaystyle f$
$\displaystyle=1-e^{\epsilon_{1}(2X_{1}+\mathrm{i})+\mathcal{O}(\epsilon_{1}^{3})}-e^{\epsilon_{2}(2X_{2}+\mathrm{i})+\mathcal{O}(\epsilon_{2}^{3})}+A_{12}e^{\epsilon_{1}(2X_{1}+\mathrm{i})+\mathcal{O}(\epsilon_{1}^{3})+\epsilon_{2}(2X_{2}+\mathrm{i})+\mathcal{O}(\epsilon_{2}^{3})}$
$\displaystyle\quad+4\sqrt{\delta_{1}\delta_{2}}\epsilon_{1}\epsilon_{2}\frac{\delta_{1}e^{-\mathrm{i}(T_{1}-T_{2})}+\delta_{2}e^{\mathrm{i}(T_{1}-T_{2})}}{(\delta_{1}-\delta_{2})^{2}}+\mathcal{O}(\epsilon_{1}^{2}\epsilon_{2},\epsilon_{1}\epsilon_{2}^{2}),$
$\displaystyle h$
$\displaystyle=-2\mathrm{i}\delta_{1}^{1/2}\epsilon_{1}e^{-\mathrm{i}T_{1}}\left[1-\left(\frac{p_{1}-p_{2}}{p_{1}+\bar{p}_{2}}\right)^{2}e^{\epsilon_{2}(2X_{2}-\mathrm{i})+\mathcal{O}(\epsilon_{2}^{3})}\right]\left[1+\mathcal{O}(\epsilon_{1}^{2})\right]$
$\displaystyle\quad-2\mathrm{i}\delta_{2}^{1/2}\epsilon_{2}e^{-\mathrm{i}T_{2}}\left[1+\left(\frac{p_{1}-p_{2}}{\bar{p}_{1}+p_{2}}\right)^{2}e^{\epsilon_{1}(2X_{1}-\mathrm{i})+\mathcal{O}(\epsilon_{1}^{3})}\right]\left[1+\mathcal{O}(\epsilon_{2}^{2})\right],$
$\displaystyle g$
$\displaystyle=2\mathrm{i}\delta_{1}^{-1/2}\epsilon_{1}e^{-\mathrm{i}T_{1}}\left[1-\left(\frac{p_{1}-p_{2}}{p_{1}+\bar{p}_{2}}\right)^{2}e^{\epsilon_{2}(2X_{2}-3\mathrm{i})+\mathcal{O}(\epsilon_{2}^{3})}\right]\left[1+\mathcal{O}(\epsilon_{1}^{2})\right]$
$\displaystyle\quad+2\mathrm{i}\delta_{2}^{-1/2}\epsilon_{2}e^{-\mathrm{i}T_{2}}\left[1-\left(\frac{p_{1}-p_{2}}{\bar{p}_{1}+p_{2}}\right)^{2}e^{\epsilon_{1}(2X_{1}-3\mathrm{i})+\mathcal{O}(\epsilon_{1}^{3})}\right]\left[1+\mathcal{O}(\epsilon_{2}^{2})\right].$

Hence, we get the power expansions

$f=\epsilon_{1}\epsilon_{2}F+\mathcal{O}(\epsilon_{1}^{2}\epsilon_{2},\epsilon_{1}\epsilon_{2}^{2}),\quad
h=\epsilon_{1}\epsilon_{2}H+\mathcal{O}(\epsilon_{1}^{2}\epsilon_{2},\epsilon_{1}\epsilon_{2}^{2}),\quad
g=\epsilon_{1}\epsilon_{2}G+\mathcal{O}(\epsilon_{1}^{2}\epsilon_{2},\epsilon_{1}\epsilon_{2}^{2})$

with

$\displaystyle
F=(2X_{1}+\mathrm{i})(2X_{2}+\mathrm{i})+\frac{4\sqrt{\delta_{1}\delta_{2}}}{(\delta_{1}-\delta_{2})^{2}}\left[\sqrt{\delta_{1}}e^{-\frac{\mathrm{i}}{2}(T_{1}-T_{2})}-\sqrt{\delta_{2}}e^{\frac{\mathrm{i}}{2}(T_{1}-T_{2})}\right]^{2},$
(21) $\displaystyle
H=2\mathrm{i}\delta_{1}^{1/2}e^{-\mathrm{i}T_{1}}\left[2X_{2}-\mathrm{i}-\frac{4\mathrm{i}\delta_{2}}{\delta_{1}-\delta_{2}}\right]+2\mathrm{i}\delta_{2}^{1/2}e^{-\mathrm{i}T_{2}}\left[2X_{1}-\mathrm{i}+\frac{4\mathrm{i}\delta_{1}}{\delta_{1}-\delta_{2}}\right],$
(22)

and

$\displaystyle
G=-2\mathrm{i}\delta_{1}^{-1/2}e^{-\mathrm{i}T_{1}}\left[2X_{2}+\mathrm{i}-\frac{4\mathrm{i}\delta_{1}}{\delta_{1}-\delta_{2}}\right]-2\mathrm{i}\delta_{2}^{-1/2}e^{-\mathrm{i}T_{2}}\left[2X_{1}+\mathrm{i}+\frac{4\mathrm{i}\delta_{2}}{\delta_{1}-\delta_{2}}\right].$
(23)

The algebraic two-soliton solution of the MTM system (1) appears in the Hirota
form as

$u=\frac{G}{\bar{F}},\quad v=\frac{H}{F}$ (24)

and yields the exact solution (8)–(9). It describes two algebraic solitons
traveling with the speeds $c_{1,2}$ obtained from $\delta_{1,2}$ by (16). A
single algebraic solution appears by taking $X_{2}\to\infty$:

$\displaystyle
u=\lim_{X_{2}\to\infty}\frac{G}{\bar{F}}=\frac{2\delta_{1}^{-1/2}}{1+2\mathrm{i}X_{1}}e^{-\mathrm{i}T_{1}}$

and similarly,

$\displaystyle
v=\lim_{X_{2}\to\infty}\frac{H}{F}=\frac{2\delta_{1}^{1/2}}{1-2\mathrm{i}X_{1}}e^{-\mathrm{i}T_{1}},$

from which we recognize the exact solution (6) extended by the symmetry
transformations (2) and (3).

###  3.3. Limit to the algebraic double-soliton

Each algebraic soliton in the two-soliton solution expressed in the Hirota
form (24) with (21), (22), and (23) has three arbitrary parameters
$\delta_{j}>0$ and $(x_{j},t_{j})\in\mathbb{R}^{2}$ for $j=1,2$. We now take
the limit $\delta_{1}\to\delta_{2}$. Due to the Lorentz transformation (3), it
is sufficient to set

$\delta_{1}=1+\varepsilon,\quad\delta_{2}=1-\varepsilon$

and take the limit $\varepsilon\to 0$. This choice gives the algebraic double-
soliton with zero wave speed as in (10). Expanding $X_{1,2}$ and $T_{1,2}$ in
the first powers of $\varepsilon$, we write

$\displaystyle\left\\{\begin{array}[]{l}X_{1}=x+\varepsilon
t+\frac{1}{2}\varepsilon^{2}(x-t)-\frac{1}{2}\varepsilon^{3}(x-t)+x_{1}+\mathcal{O}(\varepsilon^{4}),\\\
X_{2}=x-\varepsilon
t+\frac{1}{2}\varepsilon^{2}(x-t)+\frac{1}{2}\varepsilon^{3}(x-t)+x_{2}+\mathcal{O}(\varepsilon^{4}),\\\
T_{1}=t+\varepsilon
x-\frac{1}{2}\varepsilon^{2}(x-t)+\frac{1}{2}\varepsilon^{3}(x-t)+t_{1}+\mathcal{O}(\varepsilon^{4}),\\\
T_{2}=t-\varepsilon
x-\frac{1}{2}\varepsilon^{2}(x-t)-\frac{1}{2}\varepsilon^{3}(x-t)+t_{2}+\mathcal{O}(\varepsilon^{4}).\end{array}\right.$

In view of the translational symmetry (2), it is also sufficient to set

$\displaystyle\left\\{\begin{array}[]{l}x_{1}=\varepsilon
a_{1}+\frac{1}{2}\varepsilon^{2}a_{2}-\frac{1}{2}\varepsilon^{3}a_{3},+\mathcal{O}(\varepsilon^{4}),\\\
x_{2}=-\varepsilon
a_{1}+\frac{1}{2}\varepsilon^{2}a_{2}+\frac{1}{2}\varepsilon^{3}a_{3},+\mathcal{O}(\varepsilon^{4}),\\\
t_{1}=\varepsilon
b_{1}-\frac{1}{2}\varepsilon^{2}b_{2}+\frac{1}{2}\varepsilon^{3}b_{3},+\mathcal{O}(\varepsilon^{4}),\\\
t_{2}=-\varepsilon
b_{1}-\frac{1}{2}\varepsilon^{2}b_{2}-\frac{1}{2}\varepsilon^{3}b_{3},+\mathcal{O}(\varepsilon^{4}),\end{array}\right.$

with arbitrary parameters $a_{1}$, $a_{2}$, $a_{3}$, $b_{1}$, $b_{2}$, and
$b_{3}$. This gives the algebraic double-soliton with zero translational
parameters for $(x,t)$ as in (10).

For expansion of $F$, we use

$\displaystyle(2X_{1}+\mathrm{i})(2X_{2}+\mathrm{i})$
$\displaystyle=(2x+\mathrm{i})^{2}+\varepsilon^{2}[2(2x+\mathrm{i})(x-t+a_{2})-4(t+a_{1})^{2}]+\mathcal{O}(\varepsilon^{4})$

and

$\displaystyle\quad\sqrt{\delta_{1}}e^{-\frac{\mathrm{i}}{2}(T_{1}-T_{2})}-\sqrt{\delta_{2}}e^{\frac{\mathrm{i}}{2}(T_{1}-T_{2})}$
$\displaystyle=-2\mathrm{i}\left(1-\frac{\varepsilon^{2}}{8}\right)\sin\left(\varepsilon(x+b_{1})+\frac{\varepsilon^{3}}{2}(x-t+b_{3})\right)+2\left(\frac{\varepsilon}{2}+\frac{\varepsilon^{3}}{16}\right)\cos\left(\varepsilon(x+b_{1})\right)+\mathcal{O}(\varepsilon^{5})$
$\displaystyle=\varepsilon(1-2\mathrm{i}(x+b_{1}))+\varepsilon^{3}\left[\frac{\mathrm{i}}{3}(x+b_{1})^{3}-\mathrm{i}(x-t+b_{3})+\frac{\mathrm{i}}{4}(x+b_{1})-\frac{1}{2}(x+b_{1})^{2}+\frac{1}{8}\right]+\mathcal{O}(\varepsilon^{5}).$

Substituting expansions into (21) yields
$F=F_{0}+\varepsilon^{2}F_{2}+\mathcal{O}(\varepsilon^{4})$ with

$\displaystyle F_{0}$ $\displaystyle=-4b_{1}(2x+\mathrm{i}+b_{1}),$
$\displaystyle F_{2}$
$\displaystyle=2(2x+\mathrm{i})(x-t+a_{2})-4(t+a_{1})^{2}-\frac{1}{2}(1-2\mathrm{i}(x+b_{1}))^{2}$
$\displaystyle\qquad+2(1-2\mathrm{i}(x+b_{1}))\left[\frac{\mathrm{i}}{3}(x+b_{1})^{3}-\mathrm{i}(x-t+b_{3})+\frac{\mathrm{i}}{4}(x+b_{1})-\frac{1}{2}(x+b_{1})^{2}+\frac{1}{8}\right].$

If $b_{1}\neq 0$, then the limit $\varepsilon\to 0$ recovers a single
algebraic soliton in the form (6). However, if $b_{1}=0$, then we get

$\displaystyle F_{2}$
$\displaystyle=(1-2\mathrm{i}x)\left[\frac{2\mathrm{i}}{3}x^{3}+3\mathrm{i}x-x^{2}-\frac{1}{4}+2\mathrm{i}(a_{2}-b_{3})\right]-4(t+a_{1})^{2}$
$\displaystyle=-\frac{1}{12}\left[3-24\mathrm{i}x-24x^{2}-32\mathrm{i}x^{3}-16x^{4}+48(t+a_{1})^{2}+24(b_{3}-a_{2})(2x+\mathrm{i})\right].$

For expansion of $H$, we use

$\displaystyle\quad\delta_{1}^{1/2}e^{-\mathrm{i}T_{1}}(2X_{2}-\mathrm{i})+\delta_{2}^{1/2}e^{-\mathrm{i}T_{2}}(2X_{1}-\mathrm{i})$
$\displaystyle=e^{-\mathrm{i}t+\frac{\mathrm{i}}{2}\varepsilon^{2}(x-t+b_{2})}\left\\{2(2x-\mathrm{i})+\varepsilon^{2}\left[2(x-t+a_{2})+2(t+a_{1})(2\mathrm{i}(x+b_{1})-1)\right]\right.$
$\displaystyle\qquad\left.-\varepsilon^{2}(2x-\mathrm{i})\left[(x+b_{1})^{2}+\mathrm{i}(x+b_{1})+\frac{1}{4}\right]\right\\}+\mathcal{O}(\varepsilon^{4})$

and

$\displaystyle\quad\delta_{2}^{1/2}e^{-\mathrm{i}T_{1}}-\delta_{1}^{1/2}e^{-\mathrm{i}T_{2}}$
$\displaystyle=e^{-\mathrm{i}t+\frac{\mathrm{i}}{2}\varepsilon^{2}(x-t+b_{2})}\;\left[-2\mathrm{i}\left(1-\frac{\varepsilon^{2}}{8}\right)\sin\left(\varepsilon(x+b_{1})+\frac{\varepsilon^{3}}{2}(x-t+b_{3})\right)\right.$
$\displaystyle\qquad\qquad\left.-2\left(\frac{\varepsilon}{2}+\frac{\varepsilon^{3}}{16}\right)\cos\left(\varepsilon(x+b_{1})\right)+\mathcal{O}(\varepsilon^{5})\right]$
$\displaystyle=e^{-\mathrm{i}t+\frac{\mathrm{i}}{2}\varepsilon^{2}(x-t+b_{2})}\left\\{-\varepsilon(1+2\mathrm{i}(x+b_{1}))\right.$
$\displaystyle\quad\left.+\varepsilon^{3}\left[\frac{\mathrm{i}}{3}(x+b_{1})^{3}+\frac{\mathrm{i}}{4}(x+b_{1})-\mathrm{i}(x-t+b_{3})+\frac{1}{2}(x+b_{1})^{2}-\frac{1}{8}\right]+\mathcal{O}(\varepsilon^{5})\right\\}.$

Substituting expansions into (22) yields
$H=e^{-\mathrm{i}t+\frac{\mathrm{i}}{2}\varepsilon^{2}(x-t+b_{2})}\left[H_{0}+\varepsilon^{2}H_{2}+\mathcal{O}(\varepsilon^{4})\right]$
with

$\displaystyle H_{0}$ $\displaystyle=-8\mathrm{i}b_{1},$ $\displaystyle H_{2}$
$\displaystyle=2\mathrm{i}\left[2(x-t+a_{2})+2(t+a_{1})(2\mathrm{i}(x+b_{1})-1)-(2x-\mathrm{i})\left[(x+b_{1})^{2}+\mathrm{i}(x+b_{1})+\frac{1}{4}\right]\right]$
$\displaystyle\qquad+4\left[\frac{\mathrm{i}}{3}(x+b_{1})^{3}+\frac{\mathrm{i}}{4}(x+b_{1})-\mathrm{i}(x-t+b_{3})+\frac{1}{2}(x+b_{1})^{2}-\frac{1}{8}\right]+2\left[1+2\mathrm{i}(x+b_{1})\right].$

We confirm again that if $b_{1}\neq 0$, then the limit $\varepsilon\to 0$
recovers a single algebraic soliton in the form (6). However, if $b_{1}=0$,
then we get

$\displaystyle H_{2}$
$\displaystyle=2\mathrm{i}\left[2(a_{2}-b_{3})+2(t+a_{1})(2\mathrm{i}x-1)-\frac{4}{3}x^{3}-2\mathrm{i}x^{2}+x-\frac{\mathrm{i}}{2}\right]$
$\displaystyle=-\frac{1}{3}\left[-3-6\mathrm{i}x-12x^{2}+8\mathrm{i}x^{3}+12(t+a_{1})(2x+\mathrm{i})+12\mathrm{i}(b_{3}-a_{2})\right].$

Similarly for $G$, we obtain in the case of $b_{1}=0$ that
$G=\varepsilon^{2}e^{-\mathrm{i}t}G_{2}+\mathcal{O}(\varepsilon^{4})$ with

$\displaystyle
G_{2}=-\frac{1}{3}\left[-3+6\mathrm{i}x-12x^{2}-8\mathrm{i}x^{3}-12(t+a_{1})(2x-\mathrm{i})-12\mathrm{i}(b_{3}-a_{2})\right].$

The limit $\varepsilon\to 0$ in (24) yields a new solution for the algebraic
double-soliton in the form:

$\left[\begin{matrix}u(x,t)\\\
v(x,t)\end{matrix}\right]=\left[\begin{matrix}\displaystyle\frac{4(-3+6\mathrm{i}x-12x^{2}-8\mathrm{i}x^{3}-12(t+\alpha)(2x-\mathrm{i})-\mathrm{i}\beta)}{3+24\mathrm{i}x-24x^{2}+32\mathrm{i}x^{3}-16x^{4}+48(t+\alpha)^{2}+2\beta(2x-\mathrm{i})}\\\
\displaystyle\frac{4(-3-6\mathrm{i}x-12x^{2}+8\mathrm{i}x^{3}+12(t+\alpha)(2x+\mathrm{i})+\mathrm{i}\beta)}{3-24\mathrm{i}x-24x^{2}-32\mathrm{i}x^{3}-16x^{4}+48(t+\alpha)^{2}+2\beta(2x+\mathrm{i})}\end{matrix}\right]e^{-\mathrm{i}t}.$
(25)

where $\alpha:=a_{1}$ and $\beta:=12(b_{3}-a_{2})$ are two real-valued
parameters. Due to the symmetry transformation (2), the parameter $\alpha$ is
trivial and can be set to $0$ as is done in (10).

Note that we have confirmed validity of (25) by searching for polynomial
solutions of the bilinear equations (13) with $f$ being polynomial in $x$ of
degree $4$ and in $t$ of degree $2$ and with $h$ and $g$ being polynomials in
$x$ of degree $3$ and in $t$ of degree $1$. The only parameters of the
polynomial solutions were found to be $\alpha,\beta\in\mathbb{R}$ as in (25).

###  3.4. Mass of the algebraic double-soliton

We shall prove (11) here. It follows from (12) and (13) that

$|u|^{2}+|v|^{2}=\frac{|g|^{2}+|h|^{2}}{|f|^{2}}=2\mathrm{i}\left(\frac{f_{x}}{f}-\frac{\bar{f}_{x}}{\bar{f}}\right),$

where

$f=16x^{4}+32\mathrm{i}x^{3}+24x^{2}+24\mathrm{i}x-3-48t^{2}-2\beta(2x+\mathrm{i}).$

We claim that $f$ has no zeros on $\mathbb{R}$ in $x$ for every
$t\in\mathbb{R}$ and $\beta\in\mathbb{R}$. Indeed, if
$x,t,\beta\in\mathbb{R}$, zeros of $f$ must satisfy

$\left\\{\begin{array}[]{l}16x^{4}+2x^{2}-3-48t^{2}-4\beta x=0,\\\
32x^{3}+24x-2\beta=0.\end{array}\right.$

Expressing $\beta=16x^{3}+12x$ yields $-48x^{4}-24x^{2}-3-48t^{2}=0$, which
cannot be satisfied for $x,t\in\mathbb{R}$. Hence, there exist no roots of $f$
on $\mathbb{R}$ in $x$ for every $t\in\mathbb{R}$ and $\beta\in\mathbb{R}$.
This and the fast decay at infinity,

$\frac{f_{x}}{f}-\frac{\bar{f}_{x}}{\bar{f}}=\mathcal{O}\left(\frac{1}{|x|^{2}}\right)\quad\mbox{\rm
as}\;\;|x|\to\infty,$

justify the applications of Jordan’s lemma and the argument principle to
compute the integral on $\mathbb{R}$ with techniques of complex analysis:

$\displaystyle\int_{\mathbb{R}}(|u|^{2}+|v|^{2})dx$
$\displaystyle=\lim_{R\to\infty}\int_{[-R,R]\cup
C_{R}^{+}}(|u|^{2}+|v|^{2})dz$
$\displaystyle=2\mathrm{i}\lim_{R\to\infty}\int_{[-R,R]\cup
C_{R}^{+}}\left(\frac{f_{x}}{f}-\frac{\bar{f}_{x}}{\bar{f}}\right)dz$
$\displaystyle=4\pi(N_{\bar{f}}-N_{f}),$

where $C_{R}^{+}$ is a semicircle of radius $R$ in the upper half of the
complex extension of $x$ denoted by $\mathbb{C}_{+}$, $N_{f}$ is the number of
zeros of $f$ in $\mathbb{C}_{+}$ and $N_{\bar{f}}$ is the number of zeros of
$\bar{f}$ in $\mathbb{C}_{+}$. Since $f$ has no zeros on $\mathbb{R}$, we have

$N_{\bar{f}}=\deg(f)-N_{f}.$

Since $\deg(f)=4$, we only need to show that $N_{f}=1$ to obtain (11).
However, this is true for every $\beta\in\mathbb{R}$ as $|t|\to\infty$ due to
the representation of $f$ in the equivalent form

$f=(2x+\mathrm{i})^{4}+12(2x+\mathrm{i})^{2}-4\mathrm{i}(2x+\mathrm{i})-2\beta(2x+\mathrm{i})+4-48t^{2},$

from which we have

$(2x+\mathrm{i})=\sqrt[4]{12}\sqrt{|t|}e^{\frac{\mathrm{i}\pi
n}{2}}+\mathcal{O}\left(\frac{1}{\sqrt{|t|}}\right)\quad\mbox{\rm
as}\;\;|t|\to\infty,$

where $n=0,1,2,3$. There is only one root in $\mathbb{C}_{+}$ which
corresponds to $n=1$. Since the number $N_{f}$ cannot change in the
continuation of $f$ in $t\in\mathbb{R}$ for every $\beta\in\mathbb{R}$, we
have $N_{f}=1$ for every $t\in\mathbb{R}$ and $\beta\in\mathbb{R}$. Hence
$Q(u_{\rm double},v_{\rm double})=8\pi$ and (11) holds for every
$\beta\in\mathbb{R}$.

##  4\. Conclusion

We have constructed exact solutions of the MTM system (1) for fast and slow
scatterings of two algebraic solitons. Each algebraic soliton appears as the
zero-energy resonance of the nonlinear Dirac equations [12] and is supported
by the embedded eigenvalue in the KN spectral problem [16]. The fast
scattering of two algebraic solitons with different wave speeds $c_{1}\neq
c_{2}$ is described by the exact soluton (8) and (9). The slow scattering of
two identical solitons with zero speed is described by the exact solution
(10). The exact solutions suggest that the algebraic solitons are stable
coherent structures arising in a more complicated evolution of the MTM system
(1).

These discoveries lead to a number of open questions which can be addressed in
future research. First, the mathematical problem of proving orbital stability
of algebraic solitons is still open with only partial progress obtained within
the derivative NLS equation in [18]. Second, the algebraic double-soliton
solution (10) suggests existence of a hierarchy of higher-order rational
solutions of the MTM system (1) which has not been obtained in the previous
works [6, 11, 29]. Third, a similar algebraic double-soliton and a similar
hierarchy of higher-order rational solutions must exist in the other nonlinear
equations associated with the KN spectral problem, among which the most
significant model is the derivative NLS equation [10, 31]. Finally,
development of the IST methods and the generalized Darboux transformation
methods for the algebraic solitons associated with the embedded eigenvalues of
the KN spectral problem is still a challenging problem for future research.

Acknowledgements. The authors thank B. F. Feng, L. Ling, and J. He for
providing useful references.

Conflict of Interest: The authors have no conflict of interest.

Data availability statement: The paper does not contain new numerical data.

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