# Distilled Non-Semantic Speech Embeddings with Binary Neural Networks for
Low-Resource Devices

Harlin Lee, and Aaqib Saeed H. Lee<EMAIL_ADDRESS>is with the
Department of Mathematics, University of California, Los Angeles, USA. Her
work is supported by NSF DMS-1952339. A. Saeed<EMAIL_ADDRESS>is
with Philips Research, Eindhoven, The Netherlands. Philips infrastructure is
not used in any manner for this project, neither for data processing nor for
model development, training or evaluation.

###### Abstract

This work introduces BRILLsson, a novel binary neural network-based
representation learning model for a broad range of non-semantic speech tasks.
We train the model with knowledge distillation from a large and real-valued
TRILLsson model with only a fraction of the dataset used to train TRILLsson.
The resulting BRILLsson models are only $2$MB in size with a latency less than
$8$ms, making them suitable for deployment in low-resource devices such as
wearables. We evaluate BRILLsson on eight benchmark tasks (including but not
limited to spoken language identification, emotion recognition, health
condition diagnosis, and keyword spotting), and demonstrate that our proposed
ultra-light and low-latency models perform as well as large-scale models.

###### Index Terms:

speech representations, knowledge distillation, paralinguistic tasks, binary
neural networks, digital health, internet-of-things

##  I Introduction

Representation learning takes advantage of large amounts of unlabeled data to
learn features that can be used for a variety of downstream signal processing
and machine learning tasks. This has demonstrated especially impressive
performance in speech and audio processing [1, 2, 3], as the ubiquity of smart
phones, watches, and home appliances has made it easy and inexpensive to
collect a wealth of unlabeled audio signals. In particular, there is growing
interest in using representation learning to build general-purpose models for
non-semantic speech tasks, which are problems related to human speech other
than its meaning, such as spoken language identification [4], emotion
recognition [5], heath condition diagnosis [6], keyword spotting [7] and more.

However, the large sizes of the trained models and the amount of computational
resources that are required to run them on newly acquired data have stalled
the real-world deployment of existing models for non-semantic speech
applications. These assumptions are critical in mobile computing, edge
computing, internet-of-things (IoT), and tiny machine learning (tinyML)
settings, which are where many paralinguistic speech applications actually
take place, e.g., in small wearable for healthcare or with voice-controlled
artificial intelligence (AI) assistants in smart devices. Although models such
as FRILL [8] and TRILLsson [3] were recently proposed to reduce the complexity
and size of the deep learning models, a large gap still remains between highly
effective (i.e., accurate) and highly efficient (i.e., light enough to be run
on devices as small as smartwatches) representation learning models for
paralinguistic speech tasks.

To this end, we design and evaluate BRILLsson, binary neural networks (BNNs)
[9] that are small and fast enough to be deployed in devices with limited
memory and computational resources. BNNs have weights of only +1 or -1, which
make them ideal compact architectures, especially in conjunction with co-
designed machine learning hardware that one may see in modern IoT
applications. Furthermore, we employ knowledge distillation [10, 11] from
TRILLsson to BRILLsson, and show that distillation can be achieved using data
that is slightly unrelated or smaller than the one used to train the original
model, which is beneficial when original data is not available or so large
that it requires extensive computing power. Finally, we illustrate that
BRILLsson achieves performance on many non-semantic speech benchmark and other
tasks that is comparable to that of much larger models.

In summary, our main contributions are:

  * •

We propose BRILLsson, ultra-light and fast models for representation learning
that are suitable for low-resource devices. BRILLsson’s size is only $2$MB,
and its latency is less than 8ms.

  * •

Perform successful knowledge transfer via embedding distillation from a large-
scale real-valued model (i.e., EfficientNet-v2) to binary neural networks.
While similar approaches have been explored in image classification [12] and
speech separation [13], ours is the first in the context of general-purpose
representation learning for non-semantic speech.

  * •

Demonstrate that despite their compressed size, our BNNs perform comparably to
TRILL, FRILL, and TRILLsson on eight different benchmark datasets.

  * •

Our models are ideal for continuous on-device inference for privacy-preserving
health monitoring (e.g., coughing, sneezing) due to its low-computational
footprint.

##  II Methods

Our objective is to create extremely compact general-purpose audio models that
1) generate informative embeddings (or representations) for a broad range of
audio recognition tasks, and 2) can run efficiently on-device with low latency
for low-resource devices, e.g., wearables without constant connection to the
cloud. We use knowledge distillation to transfer learned representations of a
large-scale pre-trained teacher model to smaller BNN-based student models that
are otherwise difficult to pre-train with self-supervised learning (or a
similar strategy) due to their limited capacity. We would like to highlight
that while distillation has been successfully leveraged for knowledge transfer
for non-semantic speech before, to the best of our knowledge, this work is the
first attempt at utilizing it for ultra-compact binary neural networks. Figure
1 provides high-level illustration of our approach, and the following
subsections describe its essential building blocks.

###  II-A Embedding Distillation

Distillation is a technique to create a model with smaller size and less
computational load without sacrificing its effectiveness [10]. It transfers
information from a large supervising model —the teacher ($\mathcal{F}_{t}$)—
to a relatively small model —the student ($\mathcal{F}_{s}$)— with the goal of
compression for efficient inference. The teacher is generally a fixed pre-
trained network learned with a massive amount of high-quality data, in other
words a privileged model. In contrast, the student is a low capacity network
that is guided to imitate the output of the teacher. The information-rich
signal from the teacher enables a compact student network to learn important
aspects of the data that would otherwise be missed when solely minimizing a
task-specific objective.

In the seminal paper, Hinton et al. [10] proposed to use softened class
probabilities from the teacher to provide supervision, which acts as targets
for the student model to optimize for. Here, as our teacher model provides
$1024$-dimensional embeddings, we instead leverage mean-squared-error loss and
a linear layer ($\mathcal{F}_{r}$) of the same dimensionality on top of
student model for distillation purpose [11]. We use one-second audio clips as
inputs to teacher-student models to get outputs of $1024$ dimensions, on top
of which the following loss function is computed:

$\mathcal{L}(\theta_{s},\theta_{r})=\frac{1}{2}\parallel\mathcal{F}_{t}(\mathbf{x};\theta_{t})-\mathcal{F}_{r}(\mathcal{F}_{s}(\mathbf{x};\theta_{s});\theta_{r})\parallel^{2}.$
(1)

$\mathbf{x}$ is the training data, $\theta$ represents the respective model
parameters, and $\mathcal{F}_{r}$ is the linear layer model representing a
regressor function to match teacher’s dimensionality that is discarded after
distillation. We use a batch size of $512$ and a fixed learning rate of
$0.001$ with an Adam optimizer [14] to train for approximately $234$K steps
with a single NVIDIA RTX3090 GPU.

Figure 1: Illustration of distilling binary neural models for non-semantic
speech.

###  II-B Teacher: Large EfficientNet-V2 Model

For our teacher model $\mathcal{F}_{t}$, we use EfficientNet-v2 [15] from
TRILLsson [3], which has achieved exceptional performance on several Non-
Semantic Speech Benchmark [2] (NOSS) and other related tasks. This model was
trained using a combination of two large-scale speech datasets (Speech
AudioSet $4.9$K and Libri-light $53$K) and the CAP12 model (with non-public
$606$M parameters) via the teacher-student distillation framework. We choose
version three of the EfficientNet-v2 model with $21.5$M parameters and access
it directly from TensorFlow Hub (TFHub)111https://tfhub.dev during training.
This model has a front-end based on log-magnitude Mel spectrograms with $80$
bins ranging from $125$Hz to $7500$Hz, uses window length of $25$ms and hop
length of $10$ms, and is initially trained with frame width of $2$s of audio.

There are larger models available within the TRILLsson family, but we choose
the EfficientNet variant as it is both high-performing and less
computationally demanding to accommodate a modest compute budget (e.g.,
hardware with a single GPU system). This allows us to train, or distill,
longer in a short period of time with a large batch size, and demonstrate our
central point that we can extract compact binary models purely with knowledge
distillation. Further, EfficientNet is a mobile friendly architecture
discovered by neural architecture search for image classification tasks with a
large capacity. However, we do note that having access to more computing
resources may allow one to leverage even bigger models with better
supervision, which we leave for future work.

###  II-C Student: Light-weight Binary Neural Networks

Our student models $\mathcal{F}_{s}$ are binary neural network (BNN)
architectures with single-bit weights and activations. Because neurons in BNNs
can have only two possible states, BNNs provide extreme compression and speed-
up gains compared to real-valued artificial neural networks. These fast and
energy efficient BNNs are well-suited for deployment on low-resource devices
with limited memory and battery power. Specifically, we use binary
convolutional networks: a binary DenseNet-28 [16] and MeliusNet [17] with
around $4.5$M and $6.4$M parameters, respectively. Their sizes are less than
$2$MB in floating-point format, and $1.03$MB for DenseNet-28 and $1.25$MB for
MeliusNet in quantized form, which are several folds smaller compared to
models utilized in [2, 3, 8]. We use the Larq [18] framework for the
implementation of BNNs.

Our student models are paired with an audio processing front-end based on log-
magnitude Mel spectrograms, and can directly consume raw audio waveform. Our
front-end uses window size of $25$ms, hop size of $10$ ms, and $64$ Mel-spaced
frequency bins in the range of $60$Hz to $7800$Hz for $98$ frames,
corresponding to $980$ms. These inputs to the BNNs are then mapped to latent
vectors of size $576$ for DenseNet-28 and $512$ for MeliusNet. A
GlobalMaxPooling layer on top reduces the size of the bottleneck layer by
computing the maximum of all pixels in each output feature map. Finally, these
compact output embeddings are provided to classifiers for downstream tasks. We
note that to match TRILLsson’s embedding dimensions, we add an additional
binary fully-connected layer with $1024$ hidden units $\mathcal{F}_{r}$, which
is discarded after the training phase.

##  III Experimental setup

###  III-A Distillation Dataset

We perform knowledge distillation with open-source Libri-light dataset [19],
which is derived from public audio books in the LibriVox project. It is the
largest publicly available, unlabeled semi-supervised audio dataset to date.
From this, we use a medium subset of the dataset with around $5193$ hours of
speech (approximately $321$GB in size) due to our modest compute budget. We
split each audio clip into non-overlapping one-second segments for training,
resulting in around $12$M examples. It is important to note that TRILLsson
models are trained with $58$K hours of speech data originating from both
Audioset and Libri-light. Also, TRILLsson is trained with a teacher of massive
scale, i.e., CAP12, that is in turn trained on $900$K hours of audio from YT-U
data [20].

TABLE I: Overview of the downstream evaluation datasets involving non-semantic
speech.

Dataset | Task | Samples | Classes
---|---|---|---
MUSAN [21] | Speech, Music, and Noise | 2,016 | 3
ESC-50 (HS) [22] | Human Sounds | 400 | 10
Voxforge [4] | Spoken Language | 176,436 | 6
SpeechCommands [7] | Commands Recognition | 100,503 | 12
CREMA-D [5] | Emotion Detection | 7,438 | 6
MSWC-(Micro EN) [23] | Keyword Spotting | 96,099 | 31
MSWC-(Micro ES) [23] | 28,039 | 20
Vocalsound [6] | Health Condition Monitoring | 21,024 | 6

###  III-B Downstream Speech Sensing Tasks and Evaluation

We evaluate the effectiveness of our method on a broad range of tasks varying
from spoken language identification, keyword spotting, accent recognition,
identifying emotion, to health condition monitoring. Table I provides an
overview and key characteristics of the $8$ datasets. We use MUSAN [21] to
evaluate the detection of music, speech and noise in audio clips. Voxforge [4]
is used for identifying spoken English, Spanish, French, German, Russian, and
Italian. We use SpeechCommands [7] with $12$ classes for spoken commands and
CREMA-D with $6$ classes (anger, disgust, fear, happy/joy, neutral, sad) for
emotion recognition. For human sounds task, we utilize $10$ classes subset
from ESC-50 [22], same as FRILL [8]. We also use microsets from multilingual
word corpus (MSWC) for keyword spotting in English (EN) and Spanish (ES), each
task with $31$ and $20$ classes, respectively. Lastly, Vocalsound [6] contains
audio recording for detection of laughter, sighs, coughs, throat clearing,
sneezes, and sniffs.

We follow standard train and test splits of the datasets except for MUSAN,
where we randomly split the data into training (80%) and test (20%) sets. In
case of ESC-50 (HS), following FRILL, we use first four folds as training set
and the last fold as a test set. We evaluate the quality of learned
representations with a linear classifier trained on top frozen feature
extractor or encoder in a similar manner as prior work [2, 3]. The classifier
is trained with a batch size of $64$ (except for CREMA-D, where we use batch
size of $32$ due to relatively small size of the datasets) with learning rate
of $0.001$ with Adam optimizer [14] for $100$ epochs. We use a randomly
selected one-second segment from each audio clip in the training set, and
evaluate the performance on the entire audio clip during testing.

TABLE II: Generalization performance of BRILLsson on a range of non-semantic
speech representations tasks. We train a linear classifier on top of distilled
frozen models to assess the quality of learned embeddings. RN-50 is ResNet-50,
EN-v2 is EfficientNet-v2, DN is DenseNet-28, and MN is MeliusNet. T denotes a
tiny model based on a intermediate layer of DenseNet-28.

Method | MUSAN | ESC-50 (HS) | Voxforge | SpeechCommands | CREMA-D | MSWC-EN | MSWC-ES | Vocalsound | Size (MB) | Latency (ms)
---|---|---|---|---|---|---|---|---|---|---
TRILL | 98.2 | 86.4 | 84.5 | 81.9 | 66.2 | 81.3 | 88.0 | 88.2 | 98.1 | 275.3
TRILL-Distilled | 98.5 | 87.9 | 80.0 | 80.2 | 70.2 | 74.4 | 87.9 | 85.8 | 107.1 | 22.5
FRILL | 98.2 | 86.4 | 76.9 | 79.7 | 70.9 | 79.1 | 87.6 | 86.7 | 38.5 | 8.5
TRILLsson (RN-50) | 98.5 | 60.0 | 98.6 | 91.2 | 81.3 | 91.4 | 94.5 | 87.2 | 22.0 | -
TRILLsson (EN-v2) | 98.7 | 87.5 | 99.2 | 93.2 | 83.2 | 87.2 | 93.9 | 89.0 | 99.0
BRILLsson (DN) | 93.0 | 85.0 | 70.8 | 88.7 | 65.3 | 87.6 | 88.6 | 80.2 | 2.0 | 6.4
BRILLsson (MN) | 91.5 | 80.0 | 70.1 | 89.2 | 63.8 | 88.5 | 89.1 | 83.2 | 2.1 | 7.6
BRILLsson (T) | 90.5 | 73.7 | 73.0 | 88.5 | 54.6 | 87.2 | 88.4 | 78.4 | 0.65 | 6.1

###  III-C Latency Benchmarking of Binary Neural Networks

We use Larq Compute Engine [18] for latency benchmarking of our BNN models. To
align with prior work, we create a float-$32$ TFLite format model and run it
for $150$ runs in a single thread to get an averaged inference time on a
device equipped with Snapdragon 855.

###  III-D Baseline Models for Comparison

We compare our approach with five methods: TRILL, TRILL-Distilled, FRILL,
teacher model TRILLsson3 (EfficientNet-V2), and TRILLsson1 (ResNet-50). TRILL
[2] is a TRIpLet-Loss Network that is pre-trained with large amount of speech
data from Audioset. It has shown to learn powerful representations for non-
semantic speech tasks and achieved state-of-the-art performance on some of
them when it was published in 2020. It uses a ResNet-50 network architecture,
and its layer $19$ has shown to provide the most useful features with
dimensionality of $12288$. TRILL-Distilled [2] is a smaller MobileNet-based
model with $2048$-dimensional embeddings that is trained with distillation to
predict TRILL’s embeddings. Along a similar line, FRILL [8] uses a MobileNetV3
model that is designed to be a fast variant of TRILL specifically focusing on
mobile devices. It is trained with distillation to mimic the TRILL layer $19$
representations. In our work, we use Small 2.0 GAP model with
$2048$-dimensional embeddings due to its best performance. Finally, we compare
against the teacher model $\mathcal{F}_{t}$, as well as a smaller model in
TRILLsson family, i.e., a ResNet-50 model number one with $5$M parameters.

We access all the baseline models from TFHub and use them as frozen feature
extractors. We use default audio front-end that comes along with the model
from TFHub. Similar to our method, we only add a linear classification layer
for evaluating performance on downstream tasks as explained in Section III-B.
In cases where baseline models provide predictions per-time step, we average
them to compute final prediction. Furthermore, we train and evaluate baseline
models for tasks and datasets that were not presented in the prior work to
establish fair comparison; for the rest, we use accuracy score as reported in
[3, 8]. The baseline latency values when available are taken from FRILL [8].
Importantly, unlike previous works that trained multiple linear classifiers
with different techniques, we only train and evaluate a logistic regressor
implemented with a linear dense layer. An exhaustive search over
classification methods may yield further improvement.

##  IV Results and discussion

We evaluate the performance of our BRILLsson approach, and contextualize how
well binary models generalize on a broad range of speech sensing tasks as
compared to large-scale models. Table II presents results of BRILLsson along
with five large-scale baselines models. First, we notice that the teacher
model TRILLsson (EN-v2) overall perform well in comparison to other floating-
point based models, which highlights the usefulness of this model as teacher
for distillation. Our DenseNet (DN) based binary model demonstrate excellent
performance on all considered tasks even with its small size. Note that DN is
only $2$MB including the audio front-end, whereas the teacher model has size
of around $99$MB. Similarly, our MeliusNet (MN) has similar or slightly better
performance than DN, in particular on Vocalsound where it achieves accuracy of
$83.2$%. This is only 2% less than the acuracy of TRILL-Distilled.
Interestingly, BRILLsson has superior generalization on keyword spotting
tasks, achieving $89.2$% accuracy on SpeechCommands and $88.5$% on MSWC-EN. We
once again emphasize that our BRILLsson models have latency of less than $8$ms
with size of merely around $2$MB. Furthermore, we add a linear classifier
after batch-normalization-12 intermediate layer in DN model to experiment with
creating an even smaller model, labeled BRILLsson (T) in Table II. The
resulting model has size of $0.65$MB and latency of $6.1$ms. Interestingly, on
Voxforge the tiny model achieves $73.0$% while having only $1.4$M parameters.
These results demonstrate the usefulness of representations learned with BNNs
and that a single BNN model can be used as a feature extractor on low-resource
devices for multiple downstream tasks.

Along similar lines, we evaluate the quality of representations from
intermediate layers of the distilled model using MN on SpeechCommands. For
each intermediate layer, a classification head is added on top and trained in
the same manner as previous experiments. The rest of the model is fixed during
this phase. Then, we convert each model to TFLite format, evaluate its
accuracy and latency, and report the results in Figure 2. We see that
conversion to TFLite format does not result in a significant performance
degradation. Also, we observe a trade-off between accuracy and speed, as
expected; for instance, the model built on the layer section-2-transition-pw
has the highest accuracy of $86.1$%, but has relatively high latency of
$7.3$ms.

Figure 2: Performance of intermediate layers’ representations on
SpeechCommands for MeliusNet TFLite format model. Each point corresponds to
the accuaracy and latency of a model where we added a classifier after an
intermediate layer. For example, a model with classifier after
section-1-transition-pw layer achieves $81$% accuracy with latency of $7$ms.
stpw is an abbreviation of section-transition-pw.

##  V Conclusions

We have designed, developed, and publicly released BRILLsson— an extremely
compact, fast, and flexible model for non-semantic speech representation
learning. We used embedding distillation to transfer knowledge from an
existing TRILLsson model to small binary neural networks. Our approach
significantly reduced the model size while keeping the performance on par with
large-scale real-valued counterparts, which is valuable for low-resource
devices. While this work focused on utilizing existing neural architectures,
we would like to explore neural architecture search methods in the future to
design even more light-weight BNN models that are suitable for micro-
controllers.

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# Integrable Local and Non-local Vector Non-linear Schrödinger Equation with
Balanced loss and Gain

Debdeep Sinha111email<EMAIL_ADDRESS>
(Theory Division
Saha Institute of Nuclear Physics
1/AF Bidhannagar, Calcutta - 700064, India.)

###### Abstract

The local and non-local vector Non-linear Schrödinger Equation (NLSE) with a
general cubic non-linearity are considered in presence of a linear term
characterized, in general, by a non-hermitian matrix which under certain
condition incorporates balanced loss and gain and a linear coupling between
the complex fields of the governing non-linear equations. It is shown that the
systems posses a Lax pair and an infinite number of conserved quantities and
hence integrable. Apart from the particular form of the local and non-local
reductions, the systems are integrable when the matrix representing the linear
term is pseudo hermitian with respect to the hermitian matrix comprising the
generic cubic non-linearity. The inverse scattering transformation method is
employed to find exact soliton solutions for both the local and non-local
cases. The presence of the linear term restricts the possible form of the
norming constants and hence the polarization vector. It is shown that for
integrable vector NLSE with a linear term, characterized by a pseudo-hermitian
matrix, the inverse scattering transformation selects a particular class of
solutions of the corresponding vector NLSE without the linear term and map it
to the solution of the integrable vector NLSE with the linear term via a
pseudo unitary transformation, for both the local and non-local cases.

##  1 Introduction

The NLSE being an integrable and exactly solvable model has always been a
focal point of mathematical investigation to understand nonlinear evolution
equations and over the years many methods have been developed exploring the
intriguing richness of such systems. Further importance of this model arises
due to the fact that NLSE and its various generalizations are capable of
describing many physical phenomena encompassing a diverse branches of physics
that include wave propagation in non-linear media [1], Bose-Einstein
condensation (BEC) [2], plasma physics [3], gravity waves [4], $\alpha$-helix
protein dynamics [5] etc. The initial studies on NLSE was confined to
homogeneous and autonomous systems where it describes the wave propagation in
non-linear optical media. As soon as it is realized that various
generalizations of NLSE can also be integrable, its application enhances to
include inhomogeneous and non-autonomous solitons. Such systems describe many
interesting physical phenomenon that find applications in the study of BEC,
soliton lasers, ultrafast soliton switches, and logic gates [6, 7, 8, 9, 10,
11]. Recently, the emergence of Parity-Time (${\cal PT}$) symmetric systems
and its growing relevance in the nonlinear optics necessitate the study of
various generalizations of NLSE in the realm of ${\cal PT}$\- symmetric
systems as well [12, 13]. In case of nonlinear optics the confining potential
is made to be ${\cal PT}$\- symmetric and a great deal of investigation to
find soliton solution is devoted in this direction that involves, some
interesting study such as, the ${\cal PT}$\- symmetry breaking as a function
of system parameter and the behavior of soliton near the exceptional points
[14, 15, 16]. A note worthy advancement in the possible form of NLSE in the
realm of ${\cal PT}$\- symmetric systems is the introduction of a new type of
non-local reduction for NLSE which is shown to be integrable [17] with an
interesting feature of admitting both bright and dark soliton solutions with
the same kind of attractive interaction. The corresponding vector non-local
NLSE is also shown to be integrable [18]. A great deal of investigation has
also been carried out with non-local NLSE [19, 20, 21, 22, 23, 24, 25, 26].
Another variety of NLSE, in case of ${\cal PT}$\- symmetric systems, is the
balanced loss and gain systems with linear coupling that appears in the study
of coupled and multi coupled wave guide systems described by multi component
NLSE. Recent development in this direction may be found in the following Refs.
[27, 28, 29, 30, 31].

The purpose of the present paper is to investigate the integrable properties
of local and non-local vector NLSE with generic cubic non-linearity and with a
linear term characterized, in general, by a non-hermitian matrix which under
certain condition incorporates balanced loss and gain with a linear coupling
between the components of the complex fields corresponding to the vector
nonlinear equations. Recently, a systematic investigation of NLSE with
balanced loss and gain and with linear coupling term is presented in Ref.
[32]. It is shown that the system can be solved exactly via a non-unitary
transformation. In particular, a vector NLSE with balanced loss-gain, linear
coupling and a general form of cubic non-linearity is considered [32]. A non-
unitary transformation is used to map the system to the same equation without
the loss-gain and linear coupling terms with a modified time-modulated non-
linear interaction. Further, the non-linear term remains invariant for the
special case of pseudo unitary transformation. The non-unitary transformation
considered in this case may be viewed as a gauge transformation involving
complex scalar gauge potential. However, since the original system and mapped
system are connected via non-unitary transformation the observables have
different time dependence and the two systems are not gauge equivalent.
Further investigation is also made to construct exactly solvable models for
the case of time dependent loss and gain and space time modulated non-linear
interaction [33]. It should be mentioned here that the NLSE with balanced loss
and gain arises in the study of non-linear optics and admits bright as well as
dark soliton solutions[29, 30, 34], rouge waves [35], exceptional points [36,
37] and breathers solitons [36]. Some of the models involving balanced loss
and gain are also investigated from the viewpoint of exactly solvable models
[30, 38]. Numerical investigation has also been made to exhibit the efficiency
in the stabilization of solitons and soliton switching in case of NLSE with
time dependent balanced loss and gain.

It should be mentioned that the study of the integrable properties of vector
NLSE incorporating balanced loss and gain and linear coupling is rather
limited, albeit, some exactly solvable models are obtained. Therefore, it is
affirmative to study the integrable properties of such systems. In the present
work investigation is made to study the integrable properties of vector NLSE
with a linear term characterized, in general, by a non-hermitian matrix. In
particular, a system of vector NLSE with general form of cubic interaction
that includes self-phase and cross-phase modulation along with four wave
mixing is considered with a linear term characterized by a non-hermitian
matrix. A particular choice of this matrix generates the balanced loss and
gain system with a linear coupling between the complex fields of the governing
non-linear equation. Further, a non-local vector NLSE is also considered in
presence of a linear term. This may be viewed as a two fold generalization of
the model considered in Ref. [18]. In particular, it generalizes the cubic
interaction to include the self-phase and cross-phase modulation along with
four wave mixing. Further, the integrability of this model is investigated in
presence of a linear term characterized by a non-hermitian matrix which under
certain condition represents balanced loss and gain with a linear coupling
between the complex fields of the governing non-linear equation. The Lax-pair
of the systems are constructed for both local and non-local cases and an
infinite number of conserved quantities are found which confirm the
integrability of the systems. Apart from the particular form of the
reductions, the systems are integrable when a specific condition is satisfied.
In particular, the systems are integrable only when the matrix representing
the linear term is pseudo hermitian with respect to the hermitian matrix
governing the cubic non-linearity of the systems. This is an important result
in the realm of integrability of vector NLSE in presence of a generic linear
term. It is interesting to note that the condition of pseudo hemitisity on the
matrix representing the generic linear term of the vector NLSE, also appears
as a condition for the system to be mapped to the same equation without the
loss-gain and linear coupling terms while the non-linear term remains
invariant [32]. The inverse scattering transformation method is employed to
find exact soliton solutions for the local and non-local vector NLSE in
presence of a linear term characterized, in general, by a non-hermitian
matrix. An important finding of the present investigation is that the the
presence of the linear term restricts the possible form of the norming
constants and hence the polarization vector. It is shown that for integrable
vector NLSE with a linear term characterized, by a pseudo hermitian matrix,
the inverse scattering transformation selects a particular class of solutions
of the corresponding vector NLSE without the linear term and map it to the
solution of the integrable vector NLSE with the linear term via a pseudo
unitary transformation, for both the local and non-local cases. It should be
mentioned that the mapping of the solution of a particular system to that of
an integrable one does not necessarily imply the integrability of the system
under consideration. Two integrable systems are said to be gauge equivalent if
the corresponding Lax-pairs are related by a gauge transformation. For the
present case this gauge transformation to the integrable vector NLSE is not
apparent. Further, the transformation performed in Ref. [32] to map the system
to an integrable one is not unitary. Therefore, the study of integrability of
the local and non-local vector NLSE with generic cubic non-linearity in
presence of a linear term characterized, in general, by a non-hermitian
matrix, is an important addition to the existing list of integrable systems.

The plan of the paper is as follows. In the next section the models are
introduced and the Lax-pair is constructed for the local and non-local vector
NLSE in presence of a generic linear term. In section-3, general results
concerning the inverse scattering transformation are presented and the time
evolution of the scattering data is considered. The one soliton solution and
the conserved quantities for the local and non-local cases are respectively
presented in section-4 and in section-5. In the last section, we make a
summary and discuss the results.

##  2 Introduction to the models and the construction of the Lax-pair

In this section, we construct the Lax-pair for the local and non-local vector
NLSE with a generic form of cubic non-linearity in presence of a linear term.
A natural generalization of the vector NLSE may be expressed in the following
manner

$\displaystyle i{\bf Q}_{t}-{\bf Q}_{xx}-{\bf AQ}+2{\bf Q}{\bf RQ}=0$ (1)

where ${\bf Q}_{t}=\frac{d{\bf Q}}{dt}$, ${\bf Q}_{xx}=\frac{d^{2}{\bf
Q}}{dx^{2}}$ and ${\bf Q}=(q_{1}(x,t),q_{2}(x,t))^{T}$, with $T$ denoting the
transpose of a matrix, is a two component column vector with $q_{1}(x,t)$ and
$q_{2}(x,t)$ being two complex fields with real arguments $x$ and $t$. The two
component row vector has the form ${\bf R}=(r_{1}(x,t),r_{2}(x,t))$ where
$r_{1}(x,t)$ and $r_{2}(x,t)$ are two complex fields with real arguments $x$
and $t$. The specific nature of ${\bf R}$ determines the form of the nonlinear
interaction. For example, we shall be considering two specific choices for
${\bf R}$ corresponding to local and non-local reduction respectively,
involving the cross-phase and self-phase modulation along with four-wave
mixing. The matrix ${\bf A}$ is a constant matrix that determines the linear
interaction between the components of complex vector field ${\bf Q}$, with the
diagonal terms are responsible for the self-coupling and the off-diagonal
terms are responsible for the cross coupling. The interesting case of balanced
loss and gain occurs when the diagonal elements of A become complex with the
imaginary parts of the diagonal elements add to zero. For ${\bf A}=0$, the
system described by Eq. (1) reduces to the ordinary vector NLSE of specific
form depending on choice of the reduction for ${\bf R}$. The local vector NLSE
with a linear term arises for the reduction:

$\displaystyle{\bf R}=\mp({\bf GQ})^{\dagger}$ (2)

where ${\bf G}$ is a non-singular hermitian matrix of constant arguments and
${}^{\prime}\dagger^{\prime}$ denotes the transpose with complex conjugation.
Eq. (1) with the reduction of Eq. (2) gives the local vector NLSE with a
linear term and have the following form:

$\displaystyle i{\bf Q}_{t}-{\bf Q}_{xx}-{\bf AQ}\mp 2\left({\bf
Q}^{\dagger}{\bf GQ}\right){\bf Q}=0.$ (3)

The hermitian matrix ${\bf G}$ includes generic cubic interaction with the
diagonal terms involving the cross-phase and self-phase modulation while the
off-diagonal terms are denoting the four-wave mixing. The integrable
properties of the system for the special case of ${\bf A}=0$, i.e, the vector
NLSE with cross-phase and self-phase modulation and four-wave mixing without
the linear term has been considered in Ref.[39]. It may be mentioned here,
that the integrability of vector NLSE without the four-wave mixing has been
considered earlier for the case when the self-phase coefficient is equal to
the cross-phase coefficient [1, 40, 41] and also when the self-phase
coefficient has the opposite sigh to that of the cross-phase coefficient [42,
43]. The recent applications of vector NLSE includes the case of coupled
${\cal PT}$-symmetric optical wave guide systems [27, 28, 29], the case of
multi-core wave guide structures [30], Bloch-wave packets in a periodic system
[44], spinor Bose–Einstein condensates [45, 46] etc.. The minus and plus sign
before the non-linear interaction term in Eq. (3) respectively correspond to
attractive and repulsive type of interactions. In Ref. [32] it is shown that
the model (3) is exactly solvable for autonomous as well as non autonomous
case when the matrix ${\bf G}$ includes specific time dependence. The model is
also exactly solvable for the specific time dependence of the coupling
parameters and space-time modulated nonlinear interaction term [33]. In this
paper, the integrable properties of the autonomous system (3) is discussed.
The integrable properties of the non-autonomous version is considered in Ref.
[47].

The non-local vector NLSE with a linear term arises for the reduction:

$\displaystyle{\bf R}=\mp({\bf GQ})^{P}$ (4)

where $P$ denotes the transpose with complex conjugation plus the parity
transformation of the spatial argument. In particular, ${\bf
Q}^{P}=(q^{*}_{1}(-x,t),q^{*}_{2}(-x,t))$ with ${}^{\prime}*^{\prime}$ denotes
the complex conjugation. Eq. (1) with the reduction of Eq. (4) gives the non-
local vector NLSE with a linear term of the following form:

$\displaystyle i{\bf Q}_{t}-{\bf Q}_{xx}-{\bf AQ}\mp 2\left({\bf Q}^{P}{\bf
GQ}\right){\bf Q}=0.$ (5)

It should be noted that in contrast to the vector NLSE, here the argument in
the potential term of the complex field is $-x$ instead of $x$. This
introduces a non-local nature in the potential. The system corresponding to
the particular case of ${\bf A}=0$ and ${\bf G}={\bf I}$, where ${\bf I}$
denotes a $2\times 2$ unit matrix, has been considered in Ref. [18] which is
shown to be integrable and soliton solution is obtained via inverse scattering
transformation. Eq. (5) may be considered as a twofold generalization of the
model considered earlier in Ref. [18]. One, in incorporating cross-phase and
self-phase modulation and four-wave mixing term in the nonlinear interaction
and other in considering the linear interaction between the components of the
components of the complex field ${\bf Q}$. Here, also the minus and plus sing
before the potential term in Eq. (5), respectively correspond to the
attractive and repulsive type of interaction.

Eqs. (3) and (5) yield a Lax-pair that reduces the non-linear equations into a
pair of linear equations, one of which gives the scattering problem and other
gives the time evolution of the scattering data, and may be represented as

$\displaystyle v_{x}$ $\displaystyle=$ $\displaystyle{\bf X}v,\ \ {\bf
X}=\begin{pmatrix}-ik{\bf I_{2}}&{\bf Q}\\\ {\bf R}&ik{\bf I_{1}}\\\
\end{pmatrix},$ (6) $\displaystyle v_{t}$ $\displaystyle=$ $\displaystyle{\bf
T}v,\ \ {\bf T}=\begin{pmatrix}2ik^{2}{\bf I_{2}}+i{\bf Q}{\bf R}-i{\bf
A}&-2k{\bf Q}-i{\bf Q}_{x}\\\ -2k{\bf R}+i{\bf R}_{x}&-2ik^{2}{\bf
I_{1}}-i{\bf R}{\bf Q}\\\ \end{pmatrix}.$ (7)

Eq. (6) is an eigenvalue equation with a constant parameter $k$ and Eq. (7)
gives the time evolution of the corresponding fields. In this case $v(x,t)$ is
a three component column vector and it is assumed that the fields
$\\{q_{1}(x,t),q_{2}(x,t)\\}$ and $\\{r_{1}(x,t),r_{2}(x,t)\\}$ vanish rapidly
as $|x|$ $\rightarrow$ $\infty$, and ${\bf I}_{n}$ denotes $n\times n$ square
matrices . The important result is that the compatibility condition $v_{xt}$
$=$ $v_{tx}$ between Eqs. (6) and (7) give rise to the local and non-local
vector NLSE with the linear term of the form of Eq. (3) and Eq. (5) under the
symmetry reduction of Eq. (2) and Eq. (4) respectively, only when the
following condition is satisfied:

$\displaystyle{\bf A}^{\dagger}={\bf GAG^{-1}},$ (8)

i.e, the matrix ${\bf A}$ is ${\bf G}$-pseudo-hermitian [48]. This is a note
worthy condition as an integrable criteria for a class of NLSE with a linear
term. So far the matrix ${\bf A}$ and ${\bf G}$, in Eqs. (3) and (5), which
appear to be independent, now constrained by the condition of Eq. (8) in order
that the systems are integrable. Some observations are in order:

i) ${\bf A}$ is hermitian: In this case condition (8) implies that the systems
are integrable only when the matrix ${\bf A}$ and ${\bf G}$ commute, i.e,
$[{\bf A},{\bf G}]=0$. In that case there will be no loss and gain.

ii) ${\bf G}$ is a unit matrix: In this case ${\bf A}$ becomes hermitian and
again there will be no loss and gain.

iii) The interesting case arises when the matrix ${\bf A}$ is non-hermitian.
Further, the physically motivated case of balanced loss and gain occurs when
the diagonal elements of ${\bf A}$ become complex with the imaginary parts of
the diagonal elements add to zero. In particular, the matrix ${\bf A}$ may be
decomposed as a sum of two hermitian matrices ${\bf B,C}$, i.e, ${\bf A}={\bf
B}+i{\bf C}$ with ${\bf C}$ being a traceless diagonal matrix representing the
strength of loss/gain. It should be noted that the system is still integrable
if $\text{Tr}(C)\neq 0$ as far as condition (8) is satisfied. However,
unbalanced loss and gain leads to unstable solution and physically motivated
cases arise when loss and gain are equally balanced [32]. The condition (8)
also appears in Ref. [32] as a condition for the system to be mapped to the
same equation without the loss-gain and linear coupling terms while the non-
linear term remains invariant and exact solutions are obtained. For the rest
of the discussions, we will consider the case for which ${\bf A}$ is ${\bf
G}$-pseudo-hermitian.

It may be noted that two integrable systems having the Lax-pairs $({\bf
X},{\bf T})$ and $({\bf\tilde{X}},{\bf\tilde{T}})$ and admit the linear
equations of the form $v_{x}={\bf X}v,v_{t}={\bf T}v$ and
$u_{x}={\bf\tilde{X}}u,u_{t}={\bf\tilde{T}}u$ respectively, are said to be
gauge equivalent, if the Lax-pairs are related by a gauge transformation ${\bf
g}$, in the following manner:

$\displaystyle{\bf X}={\bf g}{\bf\tilde{X}}{\bf g}^{-1}+{\bf g}_{x}{\bf
g}^{-1},\ \ \ \ {\bf T}={\bf g}{\bf\tilde{T}}{\bf g}^{-1}+{\bf g}_{t}{\bf
g}^{-1}$ (9)

where $v={\bf g}u$. For example, $({\bf\tilde{X}},{\bf\tilde{T}})$ may be
chosen to represent the Lax-pair of the vector NLSE without the linear term,
i.e, ${\bf A}=0$, where $({\bf X},{\bf T})$ is given by Eqs. (6) and (7). In
this case $({\bf X},{\bf T})$ and $({\bf\tilde{X}},{\bf\tilde{T}})$ are
related by the following expressions:

$\displaystyle{\bf X}={\bf\tilde{X}},\ \ \ {\bf
T}={\bf\tilde{T}}+{\bf\tilde{S}},\ \ \ {\bf\tilde{S}}=\begin{pmatrix}-i{\bf
A}&{\bf 0}\\\ {\bf 0}&0\end{pmatrix}$ (10)

where ${\bf\tilde{S}}$ is a $3\times 3$ matrix. In this case, Eqs. (9) and
(10), yield the following set of equations for the gauge transformation matrix
${\bf g}$:

$\displaystyle{\bf g}_{x}=[{\bf\tilde{X}},{\bf g}],\ \ \ \ {\bf
g}_{t}=[{\bf\tilde{T}},{\bf g}]+{\bf\tilde{S}}{\bf g}.$ (11)

However, the solution of (11) is not apparent. It is interesting to note that
the solution of the nonlinear equation represented by the Lax-pair
$({\bf\tilde{X}},{\bf\tilde{T}})$ are related to the solution of the nonlinear
equation represented by the Lax-pair $({\bf X},{\bf T})$ via a pseudo unitary
transformation depending on the specific form of ${\bf A}$ [32].

##  3 The inverse scattering transformation

In this section, we present the general results concerning the inverse
scattering transformation and obtain the time evolution of the scattering
data. In order to solve the scattering problem of Eq. (6), we define the
following Jost functions satisfying the constant boundary conditions at
$|x|\rightarrow\infty$:

$\displaystyle M(x,t)=exp(ikx)\phi(x,k),\
\bar{M}(x,k)=exp(-ikx)\bar{\phi}(x,k),$ $\displaystyle
N(x,t)=exp(-ikx)\psi(x,k),\ \bar{N}(x,k)=exp(ikx)\bar{\psi}(x,k),$ (12)

where $\phi,\bar{\psi}$ and hence M and $\bar{N}$ are $3\times 2$ and
$\bar{\phi},\psi$ and hence $\bar{M}$ and $N$ are $3\times 1$ matrices. The
functions $(\phi,\bar{\phi})$ are the solutions of the scattering problem of
Eq. (6) satisfying the boundary conditions as $x\rightarrow-\infty$ and
$(\psi,\bar{\psi})$ are the solutions satisfying the boundary conditions as
$x\rightarrow\infty$. The asymptotic form of the functions $(\phi,\bar{\phi})$
and $(\psi,\bar{\psi})$ may be obtained by using the vanishing boundary
condition ${\bf Q},{\bf R}\rightarrow 0$ as $|x|\rightarrow\infty$ in Eq. (6),
with the following results:

$\displaystyle\phi(x,k)\sim\begin{pmatrix}{\bf I}_{2}\\\
0\end{pmatrix}e^{(-ikx)},\ \ \ \ \bar{\phi}(x,k)\sim\begin{pmatrix}0\\\ {\bf
I}_{1}\end{pmatrix}e^{(ikx)}\ \ \text{as}\ \ \ x\rightarrow-\infty$
$\displaystyle\psi(x,k)\sim\begin{pmatrix}0\\\ {\bf
I}_{1}\end{pmatrix}e^{(ikx)},\ \ \ \ \bar{\psi}(x,k)\sim\begin{pmatrix}{\bf
I}_{2}\\\ 0\end{pmatrix}e^{(-ikx)}\ \ \text{as}\ \ \ x\rightarrow\infty.$ (13)

By computing the Wronskian and using the asymptotic from of Eqs. (13) as
$x\rightarrow\mp\infty$, it can be shown that the functions
$(\phi,\bar{\phi})$ and $(\psi,\bar{\psi})$ are two sets of linearly
independent solutions of the scattering problem satisfying the boundary
conditions at $x\rightarrow\mp\infty$ respectively. Further, it can be shown
that if ${\bf Q},{\bf R}\in L^{1}({\mathbb{R}})$, then the functions $M,N$ are
analytic in the upper half whereas $\bar{M},\bar{N}$ are analytic in the lower
half of the complex k-plane [49]. An integral expression for the Jost
functions [49] with the constant boundary conditions induced by Eqs. (13) can
be obtained by using the expressions of Eqs. (12) in the scattering problem of
Eq. (6). Since $\\{\phi,\bar{\phi}\\}$ and $\\{\psi,\bar{\psi}\\}$ are the two
sets of linearly independent solutions of the scattering problem one can
express one set of solutions in terms of the linear combinations of the other.
Thus we can write

$\displaystyle\begin{pmatrix}\phi(x,k)&\bar{\phi}(x,k)\end{pmatrix}=\begin{pmatrix}\bar{\psi}(x,k)&\psi(x,k)\end{pmatrix}\begin{pmatrix}{\bf
a}&{\bf\bar{b}}\\\ {\bf b}&{\bf\bar{a}}\end{pmatrix}$ (14)

where ${\bf{b}}(k)$ is $1\times 2$, ${\bf{a}}(k)$ is $2\times 2$,
$\bar{\bf{a}}(k)$ is $1\times 1$ and $\bar{\bf{b}}(k)$ is $2\times 1$ matrix.

The inverse scattering problem deals with the reconstruction of the potentials
by using the RH (Riemann-Hilbert) approach. Using Eq. (14) one can obtain, by
exploiting the analytic properties of the Jost functions, the following
equations for ${\bf N}(x,k)$ and ${\bf\bar{N}}(x,k)$

$\displaystyle{\bf N}(x,k)=\begin{pmatrix}{\bf 0}\\\ {\bf
I_{1}}\end{pmatrix}+\sum^{\bar{J}}_{j=1}\frac{e^{-2i{\bar{k}_{j}}x}{\bf\bar{N}_{j}}(x){\bf\bar{C}_{j}}}{(k-{\bar{k}_{j}})},\
\ \ \ \bar{{\bf N}}(x,k)=\begin{pmatrix}{\bf I_{2}}\\\ {\bf
0}\end{pmatrix}+\sum^{J}_{j=1}\frac{e^{2ik_{j}x}{\bf N_{j}}(x){\bf
C_{j}}}{(k-k_{j})}$ (15)

where ${\bf N_{j}}(x)={\bf N}(x,k_{j})$,
${\bf\bar{N}_{j}}(x)={\bf\bar{N}}(x,\bar{k}_{j})$ and $k_{j}$ and
$\bar{k}_{j}$ are the proper eigenvalues of the scattering problem in upper
and lower half of the complex $k$-plane respectively. The proper eigenvalues
of the scattering problem is defined as the complex value of $k$ for which the
scattering problem of Eq. (6) admits a bounded solution that decays as
$x\rightarrow\pm\infty$. The norming constants are given by the expressions
${\bf C}_{j}=\frac{{\bf b}(k_{j}){\bf\alpha}(k_{j})}{(\det{{\bf
a})^{\prime}}(k_{j})}$,
${\bf\bar{C}}_{j}=\frac{{\bf\bar{b}(\bar{k}_{j})\bar{\alpha}(\bar{k}_{j})}}{(\det{\bar{{\bf
a}})}^{\prime}{(\bar{k}_{j})}}$, where ′ denotes the derivative with respect
to $k$ and $\alpha$ and $\bar{\alpha}$ are respectively the transpose of
cofactor of $\bf a$ and $\bf\bar{a}$. Further, in the present case, it is
assumed that ${\bf ba^{-1}}={\bf\bar{b}\bar{a}^{-1}}=0$. In order to obtain
the solution for ${\bf Q},{\bf R}$, the asymptotic expressions for $N,\bar{N}$
of Eq. (15) are compared to that of the asymptotic expressions of the same
functions obtained by solving the scattering problem for large $k$. In
particular, the following results are obtained:

$\displaystyle{\bf
Q}=2i\sum^{\bar{J}}_{j=1}e^{-2i\bar{k}_{j}x}\begin{pmatrix}\bar{N}_{11}\bar{C}_{1}+\bar{N}_{12}\bar{C}_{2}\\\
\bar{N}_{21}\bar{C}_{1}+\bar{N}_{22}\bar{C}_{2}\end{pmatrix},\ \ {\bf
R}=-2i\sum^{{J}}_{j=1}e^{2i{k}_{j}x}{\bf C}N_{3}$ (16)

where ${\bar{N}_{ij}},i,j=1,2$ and $N_{3}$ are respectively the components of
the respective matrices.

In order to obtain the time dependence of the scattering data, the following
time dependent functions are defined

$\displaystyle\Phi$ $\displaystyle=$
$\displaystyle\phi(x,t)e^{(2ik^{2}{\bf{I}}_{2}-i{\bf A)}t},\ \
\bar{\Phi}=e^{-2ik^{2}t}\bar{\phi}(x,t)$ $\displaystyle\Psi$ $\displaystyle=$
$\displaystyle e^{-2ik^{2}t}\psi(x,t),\ \
\bar{\Psi}=\bar{\psi}(x,t)e^{(2ik^{2}{\bf I}_{2}-i{\bf A})t}$ (17)

as solutions of the time evolution Eq. (7). Therefore, the time dependence of
the sets $\\{\phi,\bar{\phi}\\}$ and $\\{\psi,\bar{\psi}\\}$ can now be
obtained by substituting Eq. (17) in Eq. (7), with the following results

$\displaystyle\phi_{t}$ $\displaystyle=$ $\displaystyle
T\phi-\phi\left(2ik^{2}{\bf I_{2}}-i{\bf A}\right),\ \ \ \
\bar{\phi}_{t}=T\bar{\phi}+2ik^{2}\bar{\phi}$ $\displaystyle\psi_{t}$
$\displaystyle=$ $\displaystyle T\psi+2ik^{2}\psi,\ \ \ \
\bar{\psi}_{t}=T\bar{\psi}-\bar{\psi}\left(2ik^{2}{\bf I_{2}}-i{\bf
A}\right).$ (18)

Taking the time derivative of both side of Eq. (14), using the results of (18)
and employing the boundary conditions that $\\{\psi,\bar{\psi}\\}$ satisfy as
$x\rightarrow\infty$, the following time dependence of the scattering
constants are obtained

$\displaystyle{\bf a}_{t}$ $\displaystyle=$ $\displaystyle-i[{\bf A},{\bf
a}],\ \ \ \ \ \ {\bf{b}}_{t}={{\bf{b}}(-4ik^{2}{\bf I_{2}}+i{\bf A})}$
$\displaystyle{\bf\bar{a}}_{t}$ $\displaystyle=$ $\displaystyle 0,\ \ \ \ \ \
\ \ \ \ \ {\bf{\bar{b}}}_{t}=(4ik^{2}{\bf I_{2}}-i{\bf A}){\bf{\bar{b}}}.$
(19)

It is convenient to chose ${\bf a}$ as a constant matrix commutating with
${\bf A}$. This is an important condition in the study the integrability of
vector NLSE incorporation generic form of linear interaction in presence of a
linear term characterized by the matrix ${\bf A}$, and may be expressed as

$\displaystyle[{\bf A},{\bf a}]=0.$ (20)

With this choice, matrix ${\bf a}$, ${\bf\bar{a}}$ become constant. For vector
NLSE without the linear term, ${\bf A}=0$, and condition (20) is automatically
satisfied. However, for vector NLSE with a linear term, condition (20) has
important consequences in fixing the possible form of the norming constants as
will be seen later. The solutions for ${\bf{b}}$ and ${\bf\bar{b}}$ can
readily be obtained from Eq. (19) with the results, ${\bf
b}={\bf{b}}(0)\exp{[(-4ik^{2}{\bf I_{2}}+i{\bf A})t]}$,
${\bf\bar{b}}=\exp{[(4ik^{2}{\bf I_{2}}-i{\bf A})t]}{\bf{\bar{b}}}(0)$, where
${\bf{b}}(0)$, ${\bf{\bar{b}}}(0)$ are constant vectors of appropriate
dimensions. The time dependence of the norming constants are given by

$\displaystyle{\bf C}_{j}={\bf C}_{j}(0)e^{(-4ik_{j}^{2}{\bf I}_{2}+i{\bf
A})},\ \ \ \ {\bf\bar{C}_{j}}=e^{(4i\bar{k}_{j}^{2}{\bf I}_{2}-i{\bf
A})}{\bf\bar{C}_{j}}(0),$ (21)

where ${\bf C}_{j}(0)$ and ${\bf\bar{C}_{j}}(0)$ are the integration
constants. For vector NLSE without the linear term, the constants ${\bf
C}_{j}(0)$ and ${\bf\bar{C}_{j}}(0)$ remain arbitrary. However, for vector
NLSE involving a linear term, condition (20) fixes the constants ${\bf
C}_{j}(0)$ and ${\bf\bar{C}_{j}}(0)$ as will be seen during the construction
of the conserved quantities.

##  4  One soliton solution for the reduction ${\bf R}=-({\bf GQ})^{\dagger}$

In this section, the one soliton solution for the reduction ${\bf R}=-({\bf
GQ})^{\dagger}$ is presented and the conserved quantities are obtained. The
reduction ${\bf R}=-({\bf GQ})^{\dagger}$ induces a symmetry in the scattering
data with the results $\bar{k}=k^{*}$ and ${\bf C}=-({\bf
G\bar{C}})^{\dagger}$. The one soliton solution is obtained by taking
$J=\bar{J}=1$ and closing the system by putting $k=k_{1}$ and $k=\bar{k}_{1}$
in Eq. (15). The expressions of ${\bf N}$ and ${\bf\bar{N}}$, thus obtained,
can now be used in Eq. (16) together with the time dependence of the norming
constants as given by Eq. (21) to obtain the following expressions for the
potentials (omitting the subscript for conveniences):

$\displaystyle{\bf Q}=\frac{2ie^{-2ik^{*}x}e^{\\{4i(k^{*})^{2}{\bf
I}_{2}-i{\bf
A}\\}t}{\bf\bar{C}}(0)}{1-\frac{e^{2i(k-k^{*})x}e^{-4i\\{k^{2}-(k^{*})^{2}\\}t}||{\bf\bar{C}}||^{2}}{(k-k^{*})^{2}}},\
\ \ \ {\bf R}=\frac{-2ie^{2ikx}{\bf C}(0)e^{(-4ik^{2}{\bf I}_{2}+i{\bf
A})t}}{1-\frac{e^{2i(k-k^{*})x}e^{-4i\\{k^{2}-(k^{*})^{2}\\}t}||{\bf\bar{C}}||^{2}}{(k-k^{*})^{2}}}$
(22)

where ${||{\bf\bar{C}}||^{2}=\bf\bar{C}}^{\dagger}(0){\bf G}{\bf\bar{C}}(0)$.
It should be noted that $||{\bf\bar{C}}||^{2}$ is not necessarily positive
definite for any ${\bf G}$ as in the case when ${\bf G}$ is a unit matrix. In
case $||{\bf\bar{C}}||^{2}<0$, the solution for ${\bf Q}$ encounters
singularity. Therefore, we take $||{\bf\bar{C}}||={2\eta}e^{2\delta}>0$. The
expression for ${\bf Q}$ can be further simplified by taking
$k=\xi+i\eta,\xi,\eta\in\mathbb{R}\ \ \text{and}\ \ \eta>0$, in this case the
solution for ${\bf Q}$ reduces to the following form

$\displaystyle{\bf Q}=e^{-i{\bf A}t}2\eta e^{-i\\{2\xi
x-4(\xi^{2}-\eta^{2})t-\frac{\pi}{2}\\}}\text{Sech}{(2\eta x-8\xi\eta
t-2\delta)}{\bf P},$ (23)

where ${\bf P}=\frac{{\bf\bar{C}}(0)}{||{\bf\bar{C}}||}$ and should be fixed,
subjected to the condition of Eq. (20) and is not arbitrary (as will be
discussed during the construction of the conserved quantities) in contrast to
the corresponding vector NLSE without the linear term characterized by the
matrix ${\bf A}$. It should be noted that the solution for ${\bf Q}$ is
exactly of the form ${\bf Q}=e^{-i{\bf A}t}{\bf\tilde{Q}}$ with ${\bf
U=e^{-i{\bf A}t}}$ being a pseudo-unitary transformation as discussed in Ref.
[32]. It is easy to check that ${\bf\tilde{Q}}$ is the solution of Eq. (3)
with ${\bf A}=0$, and have the following form

$\displaystyle{\bf\tilde{Q}}=2\eta e^{-i\\{2\xi
x-4(\xi^{2}-\eta^{2})t-\frac{\pi}{2}\\}}\text{Sech}{(2\eta x-8\xi\eta
t-2\delta)}{\bf P}.$ (24)

However, it should be noted that ${\bf\tilde{Q}}$ represents a particular
class of solutions of the vector NLSE without the linear term, in which case
the polarization vector is determined by the condition of Eq. (20). Therefore,
we have the important conclusion that for integrable vector NLSE with a linear
term characterized, in general, by a non-hermitian matrix, the inverse
scattering transformation selects a particular class of solutions of the
corresponding vector NLSE without the linear term and map it to the solution
of the integrable vector NLSE with the linear term via a pseudo unitary
transformation through the relation ${\bf Q}=e^{-i{\bf A}t}{\bf\tilde{Q}}$.

Conserved quantities: As has been observed in section-2, that the scattering
coefficients ${\bf\bar{a}}$ and $\bf a$ are time independent. These
coefficients yield a Lorent series expression [49] in terms of the potentials
$\bf Q,R$ and their derivatives with each term being constant. Since the
Lorent series has an infinity number of terms, we get an infinite number of
conserved quantities in this case. Some of the conserved quantities may be
expressed, corresponding to the series of $\bf\bar{a}$, as:

$\displaystyle\bar{\Gamma_{1}}$ $\displaystyle=$
$\displaystyle-\int^{\infty}_{-\infty}{\bf
RQ}dx=\int^{\infty}_{-\infty}{\bf\tilde{Q}}^{\dagger}{\bf G}{\bf\tilde{Q}}dx,$
$\displaystyle\bar{\Gamma_{2}}$ $\displaystyle=$
$\displaystyle-\int^{\infty}_{-\infty}{\bf R}{\bf
Q}_{x}dx=\int^{\infty}_{-\infty}{\bf\tilde{Q}}^{\dagger}{\bf
G}{\bf\tilde{Q}_{x}}dx,$ $\displaystyle\bar{\Gamma_{3}}$ $\displaystyle=$
$\displaystyle-\int^{\infty}_{-\infty}[{\bf R}_{x}{\bf Q}_{x}-{(\bf
RQ})^{2}]dx=\int^{\infty}_{-\infty}[{\bf\tilde{Q}_{x}}^{\dagger}{\bf
G}{\bf\tilde{Q}_{x}}-({\bf\tilde{Q}}^{\dagger}{\bf G}{\bf\tilde{Q}})^{2}]dx.$
(25)

It should ne noted that the conserved quantities are not positive definite in
general and depend on the form of the matrix ${\bf G}$. Similar expression of
conserved quantities can be obtained for the expression of ${\bf a}$. However,
it should be noted that ${\bf a}$ is a $2\times 2$ matrix operator and is a
constant provided the condition of Eq. (20), i.e $[{\bf A},{\bf a}]=0$, is
satisfied. The Lorentz series expression of ${\bf a}$, [49] suggests that this
condition is satisfied provided ${\bf A}{\bf h}={\bf h}{\bf A}^{\dagger}$,
where ${\bf h}={\bf\bar{C}}(0){\bf\bar{C}}(0)^{\dagger}$ is a singular
hermitian matrix. Therefore, the integration constant ${\bf\bar{C}}(0)$, so
far appears to be arbitrary must be fixed properly. This is an important
result and appears solely due to the inclusion of the linear term
characterized by the matrix $\bf A$.

As an example, we may consider the case discussed in Ref. [32], and determine
the form of the norming constant. We take ${\bf A}$, ${\bf G}$ and
${\bf\bar{C}}(0)$ of the following form,

$\displaystyle{\bf
A}=\beta_{1}{\bf\sigma}_{1}+\beta_{2}{\bf\sigma}_{2}+i\Gamma{\bf\sigma}_{3},\
\ \ \ {\bf G}=\sum^{3}_{j=0}\alpha_{j}{\bf\sigma}_{j},\ \ \ \
{\bf\bar{C}}(0)=(\bar{C}_{1}e^{i\bar{\theta}_{1}},\bar{C}_{2}e^{i\bar{\theta}_{2}})^{T}$
(26)

where ${\bf\sigma}_{j},j=1,2,3$ are the Pauli matrices and ${\bf\sigma_{0}}$
is the $2\times 2$ identity matrix. The real parameters $\beta_{1,2}$ are
responsible for the coupling between the complex fields and $\Gamma$ measures
the loss/gain strength. The real parameters $\alpha_{0},\alpha_{3}$ determine
the self-phase and cross- phase modulation where the real parameters
$\alpha_{1},\alpha_{2}$ are responsible for the four-wave mixing. The matrix
${\bf A}$ is ${\bf G}$-pseudo hermitian for

$\displaystyle\alpha_{3}=0,\ \ \ \
\alpha_{0}=\frac{|\alpha||\beta|}{\Gamma}\sin{(\theta_{\alpha}-\theta_{\beta})}$
(27)

withe the choice $|\beta|>|\Gamma|$ with
$0<\theta_{\alpha}-\theta_{\beta}<\pi$ for $\Gamma>0$ and
$\pi<\theta_{\alpha}-\theta_{\beta}<2\pi$ for $\Gamma<0$ and
$\alpha\equiv\alpha_{1}+i\alpha_{2}=|\alpha|e^{i\theta_{\alpha}}$,
$\beta\equiv\beta_{1}+i\beta_{2}=|\beta|e^{\theta_{\beta}}$. Further, the
condition ${\bf A}{\bf h}={\bf h}{\bf A}^{\dagger}$ implies
$\bar{C}_{1}=\bar{C}_{2}=\bar{C}$,
$\sin{(\bar{\theta}_{12}+\theta_{\beta})}=\frac{\Gamma}{|\beta|}$, with
$\bar{\theta}_{12}=(\bar{\theta}_{1}-\bar{\theta}_{2})$ and with the choice
$|\beta|>|\Gamma|$ $0<\bar{\theta}_{12}+\theta_{\beta}<\pi$ for $\Gamma>0$ and
$\pi<\bar{\theta}_{12}+\theta_{\beta}<2\pi$ for $\Gamma<0$. Therefore, finally
we have the following expressions for ${\bf A}$, ${\bf G}$ and
${\bf\bar{C}}(0)$:

$\displaystyle{\bf
A}=\beta_{1}{\bf\sigma}_{1}+\beta_{2}{\bf\sigma}_{2}+i\Gamma{\bf\sigma}_{3},\
{\bf
G}=\frac{|\alpha||\beta|}{\Gamma}\sin{(\theta_{\alpha}-\theta_{\beta})}{\bf\sigma}_{0}+\alpha_{1}{\bf\sigma}_{1}+\alpha_{2}{\bf\sigma}_{2},\
{\bf\bar{C}}(0)=\bar{C}(e^{i\bar{\theta}_{1}},e^{i\bar{\theta}_{2}})^{T}.$
(28)

Thus the norming constant and hence the polarization vector ${\bf P}$ is fixed
depending on the possible form of the matrix ${\bf A}$. This is an important
result and solely appears due to the inclusion of the linear term,
characterized by the matrix $\bf A$, in case of vector NLSE.

##  5 One soliton solution for the reduction ${\bf R}=-({\bf GQ})^{P}$

In this section, the one soliton solution for the reduction ${\bf R}=-({\bf
GQ})^{P}$ is presented with the construction of the corresponding conserved
quantities. The condition ${\bf R}=-({\bf GQ})^{P}$ imposes a symmetry on the
scattering data and it may be observed from the Wronskian calculation of the
scattering data that if $k_{j}$ is an eigen value of the scattering data in
the upper half of $k$ plane then $k^{*}_{j}$ is also an eigen value [17, 18].
Similarly, if $\bar{k}_{j}$ is an eigen value of the scattering data in the
lower half of $k$ plane then $\bar{k}^{*}_{j}$ is also an eigen value. In the
subsequent discussion, without any loss of generality, we shall assume
$k_{j}=i\eta_{j}$ and $\bar{k}_{j}=-i\bar{\eta}_{j}$, i.e, the eigen values
are taken to be on the imaginary axis. The one soliton solution is obtained by
taking $J=\bar{J}=1$ and closing the system by putting $k=k_{1}$ and
$k=\bar{k}_{1}$ in Eq. (15). The expressions of ${\bf N}$ and ${\bf\bar{N}}$,
thus obtained, can now be used in Eq. (16) together with the time dependence
of the norming constants as given by Eq. (21) to obtain the following
expressions for the potentials (omitting the subscript for conveniences):

$\displaystyle{\bf Q}=2i\frac{e^{-2\bar{\eta}x}e^{-4i\bar{\eta}^{2}t}e^{-i{\bf
A}t}{\bf\bar{C}}(0)}{1+e^{i\theta}e^{-2x(\eta+\bar{\eta})}e^{-4i(\bar{\eta}^{2}-\eta^{2})t}},\
\ {\bf R}=-2i\frac{e^{-2\eta x}e^{4i\eta^{2}t}{\bf C}(0)e^{i{\bf
A}t}}{1+e^{i\theta}e^{-2x(\eta+\bar{\eta})}e^{-4i(\bar{\eta}^{2}-\eta^{2})t}}$
(29)

with the following conditions are satisfied due to the symmetry induced by
${\bf R}=-({\bf GQ})^{P}$:

$\displaystyle|{\bf C}(0){\bf\bar{C}}(0)|=(\eta+\bar{\eta})^{2},\ \ \ \
{\bf\bar{C}}^{\dagger}(0){\bf G}{\bf\bar{C}}(0)=(\eta+\bar{\eta})^{2}$ (30)

and we take ${\bf
C}(0){\bf\bar{C}}(0)=(\eta+\bar{\eta})^{2}e^{i(\theta+\pi)}$. In this case
also, the solution for ${\bf Q}$ can be expressed in the form ${\bf
Q}=e^{-{\bf A}t}{\bf\bar{Q}}$, with ${\bf U=e^{-i{\bf A}t}}$ being a pseudo-
unitary transformation. The vector ${\bf\bar{Q}}$ is the solution of the Eq.
(5) with ${\bf A}=0$ and have the following form:

$\displaystyle{\bf\bar{Q}}=2i\frac{e^{-2\bar{\eta}x}e^{-4i\bar{\eta}^{2}t}{\bf\bar{C}}(0)}{1+e^{i\theta}e^{-2x(\eta+\bar{\eta})}e^{-4i(\bar{\eta}^{2}-\eta^{2})t}}.$
(31)

This is an important result in study of integrable non-local vector NLSE with
generic non-linearity including self-phase and cross-phase modulation with
four wave mixing and it seems that the inverse scattering transformation for
this model has not been considered in the literature. The solution of Eq. (31)
encounters a singularity at a finite time $t=T$; at $x=0$, $T$ is given by

$\displaystyle T=\frac{(2n+1)\pi-\theta}{4(\eta^{2}-\bar{\eta}^{2})},\ \ \
n\in{\mathbb{Z}}.$ (32)

As in the corresponding local case the vector ${\bf\bar{C}}(0)$ is not
arbitrary and should be determined by the condition of Eq. (20). Therefore, we
have the important conclusion that for integrable non-local vector NLSE with a
linear term characterized, in general, by a non-hermitian matrix, the inverse
scattering transformation selects a particular class of solutions of the
corresponding non-local vector NLSE without the linear term and map it to the
solution of the integrable non-local vector NLSE with the linear term via a
pseudo unitary transformation.

Conserved quantities: The conserved quantities are obtained in a similar
manner as in the case of the local vector NLSE with the linear term. However,
in this case the hermitian conjugation is replaced by the hermitian
conjugation plus the parity transformation of the spatial coordinates that
incorporates the non-local effect. In this case also, we get an infinity
number of conserved quantities. Some of the constant quantities may be
expressed, corresponding to the series of $\bf\bar{a}$, as:

$\displaystyle\bar{\Gamma_{1}}$ $\displaystyle=$
$\displaystyle-\int^{\infty}_{-\infty}{\bf
RQ}dx=\int^{\infty}_{-\infty}{\bf\bar{Q}}^{\bf P}{\bf G}{\bf\bar{Q}}dx,$
$\displaystyle\bar{\Gamma_{2}}$ $\displaystyle=$
$\displaystyle-\int^{\infty}_{-\infty}{\bf R}{\bf
Q}_{x}dx=\int^{\infty}_{-\infty}{\bf\bar{Q}}^{\bf P}{\bf
G}{\bf\bar{Q}_{x}}dx,$ $\displaystyle\bar{\Gamma_{3}}$ $\displaystyle=$
$\displaystyle-\int^{\infty}_{-\infty}[{\bf R}_{x}{\bf Q}_{x}-{(\bf
RQ})^{2}]dx=\int^{\infty}_{-\infty}[{\bf\bar{Q}_{x}}^{\bf P}{\bf
G}{\bf\bar{Q}_{x}}-({\bf\bar{Q}}^{\bf P}{\bf G}{\bf\bar{Q}})^{2}]dx.$ (33)

It is interesting to note that none of the above conserved quantities are
hermitian even for a positive definite ${\bf G}$. This is due to the non-local
nature of the reduction of Eq. (4). However, some of the conserved quantities
can be shown to be real valued by splitting the fields as a sum of parity even
and parity odd terms [18, 47]. Similar expression of conserved quantities can
be obtained for the expression of ${\bf a}$. However, as in the corresponding
local case, the integration constant ${\bf\bar{C}}(0)$ should be fixed
subjected to the condition of Eq. (20), i.e $[{\bf A},{\bf a}]=0$. The Lorentz
series expression of ${\bf a}$, suggests that this condition is satisfied
provided ${\bf
A}\\{{\bf\bar{C}}(0){\bf\bar{C}}(0)^{\dagger}\\}=\\{{\bf\bar{C}}(0){\bf\bar{C}}(0)^{\dagger}\\}{\bf
A}^{\dagger}$, where ${\bf\bar{C}}(0){\bf\bar{C}}(0)^{\dagger}$ is a singular
hermitian matrix. Therefore, the integration constant ${\bf\bar{C}}(0)$, so
far appears to be arbitrary must be fixed properly. As an example, we may
chose the possible form of ${\bf A}$, ${\bf G}$ and ${\bf\bar{C}}(0)$ as given
by Eq. (26). The same calculation as in the corresponding local case yields
Eq. (28), however, in this case the condition (30) further specifies the
results with
$\bar{C}^{2}=\frac{(\eta+\bar{\eta})^{2}}{2[\alpha_{0}+|\alpha|\cos{(\bar{\theta}_{12}+\theta_{\alpha})}]}$,
with the choice $\alpha_{0}>|\alpha|$. Thus the norming constant and hence the
polarization of the solution is fixed depending on the possible form of the
matrix ${\bf A}$. This is an important result and solely arises due to the
inclusion of the linear term, characterized by the matrix $\bf A$, in the case
of non-local vector NLSE.

##  6 Summary and discussions

We have studied the integrable properties of local and non-local vector NLSE
with generic form of cubic non-linearity in presence of a linear term
characterized, in general, by a non-hermitian matrix which under certain
condition incorporates balanced loss and gain with a linear coupling between
the complex fields of the vector NLSE. The cubic interaction includes self-
phase and cross phase modulation along with four wave mixing. The non-local
model considered may be viewed as a two fold generalization of non-local
vector NLSE considered earlier in the literature [18]. In particular, it
generalizes the cubic interaction to include the self-phase and cross-phase
modulation along with four wave mixing. Further, the integrability of this
model has been investigated in presence of a linear term characterized, in
general, by a non-hermitian matrix which under particular situation
corresponds to balanced loss and gain with a linear coupling between the
complex fields of the non-local vector NLSE. An important corollary of the
present investigation is the study of the integrable properties of non-local
vector NLSE, only with generic cubic non-linearity involving the self-phase
and cross phase modulation along with four wave mixing and obtaining the exact
soliton solution by using inverse scattering method. It seems that the inverse
scattering transformation for non-local vector NLSE model with the generic
form of the cubic non-linearity has not been considered earlier in the
literature. The Lax-pair for vector NLSE with the linear term have been
constructed for both the local and non-local reductions and an infinite number
of conserved quantities are found. This confirms the integrability of the
systems. Apart from the particular form of the reduction, the systems are
shown to be integrable only when the matrix representing the linear term is
pseudo hermitian with respect to the matrix comprising the generic cubic non-
linearity. This is an important result in the realm of integrability of NLSE
with a linear term and having a generic cubic non-linearity. The inverse
scattering transformation method has been employed to find exact soliton
solutions for the vector NLSE with the linear term for the local as well as
non-local cases. Another important observation of the present investigation is
that the presence of the linear term restricts the possible form of the
norming constants and hence the polarization vector. It is shown that for
integrable vector NLSE with a linear term characterized, by a pseudo-hermitian
matrix, the inverse scattering transformation selects a particular class of
solutions of the corresponding vector NLSE without the linear term and map it
to the solution of the integrable vector NLSE with the linear term via a
pseudo unitary transformation, for both the local and non-local cases. It
should be mentioned that although some exactly solvable models are constructed
in the context of NLSE incorporating balanced loss and gain and linear
coupling, the integrable properties of such systems are not investigated
previously in the literature. Further, two integrable systems are said to be
gauge equivalent if the corresponding Lax-pairs are related via a gauge
transformation. For the present case this gauge transformation to the
corresponding integrable vector NLSE without the linear term is not apparent.
Within this background, the present study on the integrable properties of the
local and non-local vector NLSE in presence of a linear term characterized by
a non-hermitian matrix which under certain condition incorporates balanced
loss and gain with linear coupling between the complex fields of the vector
NLSE, is an important addition to the existing list of integrable systems.
Further, it is observed that the presence of the linear term characterized by
the non-hermitian matrix determines the possible form of the polarization
vector. Therefore, the study of the multi-soliton solution and the collision
dynamics between the solitons for the local and non-local vector NLSE with the
linear term characterized, in general, by a non-hermitian matrix will be very
much interesting [50].

##  7 Acknowledgements

DS acknowledges a research fellowship from CSIR (ACK No: 362103/2k19/1, File
No: 09/489(0125)
/2020-EMR-I).

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