in our empirical evaluation, with a specific focus on the accuracy of our
model.

Evaluation protocol.

To assess the accuracy of our AI model, we adopted the following evaluation
protocol. We compared our model with seven baseline methods using three
datasets: SemEval, AllSides-S, and AllSides-L: five text-based methods
(Mikolov et al., 2013; Pennington et al., 2014; Peters et al., 2018; Kenton
and Toutanova, 2019; Liu et al., 2019) and two knowledge-based political
stance prediction methods (Feng et al., 2021a; Zhang et al., 2022). Word2Vec
(Mikolov et al., 2013), GloVe (Pennington et al., 2014), and ELMo (Peters et
al., 2018) are general language models that aim to capture context from text.
We also use pre-trained BERT (Kenton and Toutanova, 2019) and RoBERTa (Liu et
al., 2019) by fine-tuning on training datasets. KGAP (Feng et al., 2021a) is a
knowledge-aware approach that leverages a political knowledge graph with graph
neural networks. KCD (Zhang et al., 2022), the state-of-the-art model,
considers knowledge walks from a political knowledge graph and textual cues in
political stance prediction. To ensure consistency, we (1) implemented the
five baseline language models using their available source code, incorporating
a softmax layer for the final political stance prediction. Subsequently, we
(2) applied the baseline methods to the SemEval, AllSides-S, and AllSides-L
datasets, (3) calculated their political stance prediction accuracies, and (4)
reported the obtained results. For the two knowledge-based methods (i.e., KGAP
(Feng et al., 2021a) and KCD (Zhang et al., 2022)), however, we were unable to
obtain their results due to the unavailability of certain parts of the source
code. Specifically, the source code for generating cue embeddings in KCD and
the source code for generating knowledge embeddings in KGAP were not provided,
respectively. Thus, we used the results of KGAP and KCD on the SemEval and
AllSides-S datasets, reported in (Zhang et al., 2022). 555KGAP:
https://github.com/BunsenFeng/news_stance_detection, KCD:
https://github.com/Wenqian-Zhang/KCD

Results and analysis.

We train our AI model on the three real-world political news article datasets
for 50 epochs with varying knowledge embedding methods (RotatE (Sun et al.,
2018a), HAKE (Zhang et al., 2020), and ModE (Zhang et al., 2020)). For the
SemEval and AllSides-L datasets, we apply k-fold cross validations (10-fold
for SemEval and 3-fold for AllSides-S) and then report the averaged accuracy
of k-fold cross validations. For AllSides-L, we train the model on the
training set (647k articles) and measure the model accuracy using the
validation set (71.9k articles), which is not used in the training process.
Tables 7 presents the results. Our AI model consistently outperforms all
baseline methods in terms of the model accuracy, regardless of knowledge
embedding methods. Specifically, Our model improves the state-of-the-art
method, KCD (Zhang et al., 2022) by 4.77% and 5.92% in SemEval and AllSides-S
datasets, respectively. These improvements over KCD are significant, given
that KCD has already achieved quite high accuracies in those datasets. We also
evaluate our model on a large-scale dataset (AllSides-L) which is $48\times$
larger and has more classes (i.e., more difficult to predict) than AllSides-S.
our model still significantly outperforms all baseline methods although the
accuracy of our model decreases to some extent in AllSides-L, compared with
the other datasets. In addition, we have conducted the t-tests with a 95%
confidence level and verified that the improvement of our model over all
baseline methods are statistically significant (i.e., the p-values are below
0.05). As a result, considering the consistent superiority of our model to
overall baseline methods, our experimental results convincingly demonstrate
the superiority of our model in comparison to existing political stance
prediction methods.


# CT Liver Segmentation via PVT-based Encoding and Refined Decoding

Debesh Jha, Nikhil Kumar Tomar, Koushik Biswas, Gorkem Durak, Alpay
Medetalibeyoglu,
Matthew Antalek, Yury Velichko, Daniela Ladner, Amir Borhani, Ulas Bagci
Machine & Hybrid Intelligence Lab, Department of Radiology, Northwestern
University

###### Abstract

Accurate liver segmentation from CT scans is essential for computer-aided
diagnosis and treatment planning. Recently, Vision Transformers achieved a
competitive performance in computer vision tasks compared to convolutional
neural networks due to their exceptional ability to learn global
representations. However, they often struggle with scalability, memory
constraints, and computational inefficiency, particularly in handling high-
resolution medical images. To overcome scalability and efficiency issues, we
propose a novel deep learning approach, PVTFormer, that is built upon a
pretrained pyramid vision transformer (PVT v2) combined with advanced residual
upsampling and decoder block. By integrating a refined feature channel
approach with hierarchical decoding strategy, PVTFormer generates high quality
segmentation masks by enhancing semantic features. Rigorous evaluation of the
proposed method on Liver Tumor Segmentation Benchmark (LiTS) 2017 demonstrates
that our proposed architecture not only achieves a high dice coefficient of
86.78%, mIoU of 78.46%, but also obtains a low HD of 3.50. The results
underscore PVTFormer’s efficacy in setting a new benchmark for state-of-the-
art liver segmentation methods. The source code of the proposed PVTFormer is
available at https://github.com/DebeshJha/PVTFormer.

_K_ eywords First keyword $\cdot$ Second keyword $\cdot$ More

##  1 Introduction

The liver is the largest solid organ in the human body and plays a crucial
role in metabolic functions and digestive processes. Liver cancer ranks as the
third leading cause of cancer-related deaths worldwide, highlighting its
significant impact on public health [1]. The liver is also a common site for
metastases from various abdominal cancers, such as colon, rectum, and
pancreas, as well as distant cancers like breast and lung. Therefore, accurate
segmentation of the liver is crucial for targeted therapies and surgical
planning [2]. With the help of medical imaging technologies such as computed
tomography (CT) and magnetic resonance imaging (MRI), we can now visualize and
segment the liver with precision, leading to more accurate diagnosis and
treatment strategies [3].

CT exams are widely available and cost-effective. They are essential for
examining liver diseases and detecting liver tumors, making them a common
choice for clinical trials and preoperative planning [4]. However, manual
liver segmentation is a challenging task. It is time-consuming, operator-
dependent, and lacks reproducibility. The liver varies in size and shape
within the population and is located in close proximity to many organs due to
its central location and size [3]. Therefore, computer-aided diagnosis methods
are necessary to improve patient care in examining liver diseases and
detecting tumors, utilizing CT’s high-resolution imaging and contrast
sensitivity. Ongoing research and technological advances in liver segmentation
hold the promise of making therapeutic treatments more precise and effective.

Figure 1: Overview of the proposed PVTFormer architecture: Input images are
passed via PVT v2 encoder, generating three diverse feature maps which are
passed through a series of ($1\times 1$ Conv, BN, and ReLU) in the CBR layers
before entering the Up block and the decoder block. The Up block is passed
through bilinear upsampling to upscale feature map dimensions to that of the
input image. Next, a residual block fine-tunes the upsampled feature map,
enhancing its ability to learn better representation.

Recently, there have been numerous works on liver segmentation [5, 6, 7, 8].
Zhang et al. [7] proposed a deep learning model that was based on 3D
convolution and convolutional long short-term memory (C-LSTM) for
hepatocellular carcinoma (HCC) lesion segmentation. The proposed model
utilized 4D data from dynamic contrast-enhanced (DCE) magnetic resonance
imaging (MRI) images to segment liver tumors. Similarly, Rahman et al. [8]
proposed a novel algorithm for automatically segmenting liver tumors from CT
images using a hybrid ResUNet [9] model. The model was built on a combination
of ResNet and U-Net [10] networks and involved processes such as
preprocessing, image augmentation, feature extraction and selection, reason of
interest selection, and passing it through the ResUNet architecture for ROI
tumor segmentation.

Li et al. [11] proposed a densely connected UNet framework (H-DenseUNet) for
liver and tumor segmentation. The architecture comprises a 2-D DenseUNet
focused on efficiently extracting intra-slice features with a 3D component
designed for hierarchically aggregating volumetric contexts. The dual element
for the architecture addressed the limitations of 2D convolutions that ignore
the volumetric information and the limitation of 3D convolutions that suffer
from high computational costs. Vorontsov et al. [12] proposed an architecture
that consists of two fully convolutional networks, connected in tandem and
trained together end-to-end for liver and tumor segmentation. The first FCN
(FCN 1) generates a liver segmentation mask, while the second (FCN 2) uses the
latent output from the first and the original input as an additional input to
precisely delineate liver lesions.

The above work shows that there has been a significant effort for efficient
liver and tumor segmentation. The literature primarily focuses on 2D or 3D
convolutional networks, which can lead to a trade-off between computational
load and the ability to capture intricate features essential for precise liver
segmentation. Our model harmonizes these aspects by utilizing PVT v2 as a
backbone, enriching feature representation through residual learning and
effectively maintaining crucial information without the computational expense
associated with 3D convolutions. This enables the architecture to capture
detailed semantic features with more significant precision than traditional
CNN-based models, which might not fully utilize the hierarchical nature of
image features or demand for the increased computational demand.

The main contributions of the work are as follows:

  * •

PVTFormer architecture: We propose a novel encoder-decoder based architecture,
PVTFormer, that utilizes PVT v2 as a backbone. This model takes advantage of
residual learning for enhanced feature representation. By combining PVT v2, Up
block, and Decoder block, we construct an end-to-end segmentation pipeline
that optimizes computational resources while preserving essential information,
significantly improving segmentation performance.

  * •

Hierarchical decoding strategy: Our approach includes a novel hierarchical
decoding strategy that incorporates specialized upscaling in the Up block with
effective multi-scale feature fusion in the Decoder block. This approach
significantly enhances the network’s ability to delineate detailed semantic
features, which is important for precise liver segmentation.

  * •

Systematic evaluation: We have evaluated PVTFormer with the eight existing
state-of-the-art methods to show the method’s superiority. Our proposed
architecture obtained a high dice coefficient of 86.78%, and a mean IoU of
78.46%, and low H.D of 3.50, underscoring the architecture’s precision and
reliability in segmenting healthy liver tissue.

##  2 Methodology

###  2.1 Encoder:

Figure 1 shows the block diagram of the PVTFormer. The PVTFormer is an
encoder-decoder based framework that leverages Pyramid Vision Transformer (PVT
v2 b3) [13] as the pre-trained encoder. The main motivation behind adopting
PVT v2 is to reduce the computational complexity and significantly improve
medical image segmentation tasks. The distinctive design of PVT v2
incorporates linear spatial reduction attention, overlapping patch embeddings,
and a convolutional feed-forward network. The PVT v2 enhances the local
continuity of images and feature map coherence while effectively handling
variable-resolution input images with linear computational complexity compared
to CNNs [13]. The input images are passed through the PVT v2 encoder, which
captures hierarchical image features through overlapped patch embeddings and
successive hierarchical transformer blocks, producing multi-scale
representations. The encoder transforms input images into feature-rich
representations and extracts three diverse feature maps at varying resolutions
of 64, 128, 320. These representations undergo a sequential transformation
involving $1\times 1$ Convolution, Batch Normalization (BN), and Rectified
Linear Unit (ReLU) activation operations. This is referred as CBR in the block
diagram. This effectively reduces feature channels to $64$. This reduction
optimizes computational efficiency and ensures the preservation of essential
anatomical information crucial for precise liver segmentation.

###  2.2 Up block:

Figure 2 (left) shows the block diagram of the Up block. The Up block acts as
a scaling unit to increase the spatial dimensions of feature maps. It
comprises an upsampling layer followed by a residual block. Within the Up
block, the input feature map is first passed through a bilinear upsampling to
upscale the feature map’s height and width to that of the original input
image. The residual block, which consists of two convolutional operations with
a skip connection, refines the upscaled features, enabling the network to
learn a more robust representation.

###  2.3 Decoder block and output generation:

The Decoder block (Figure 2 (right)) fuses multi-scale features using skip
connections and upsampling layer. It first upsamples the lower-resolution
feature map and then concatenates it with a corresponding higher-resolution
feature map from the encoder. This concatenated feature map is passed through
another residual block to refine the combined features and enhance the
segmentation accuracy for the liver structures. To produce the final
segmentation output, the feature maps from both the Up blocks and Decoder
blocks are concatenated to integrate multi-level contextual information
(Figure 1). We apply a final residual block to this integrated feature map to
ensure uniform scale refinement. The refined feature map is then passed
through a $1\times 1$ Conv2D layer followed by a sigmoid activation to
generate the binary segmentation map.

##  3 Experiments

Figure 2: Components of the PVTFormer.

###  3.1 Datasets

We utilize the Liver Tumor Segmentation Benchmark (LiTS) [2] dataset for our
liver segmentation task. LiTs is a multi-center dataset collected from seven
clinical centers from Germany, Netherlands, Canada, France and three medical
centers from Israel. The dataset is entirely anonymous, and the personal
identifier has been removed. It contains 201 CT images of the abdomen, of
which 130 CT scans are made publicly available along with ground truth. Only
the training dataset is made publically available. Thus, we divide the
training dataset into three parts for our experimentation.

To avoid bias, we split the cases into independent training (70 patients),
validation (30 patients), and test (30 patients) sets. We resized the image to
$256\times 256$ pixels to optimize the trade-off between training time and
model complexity. The volumetric CT scans were processed pseudo-3D (slice-by-
slice) to fit into regular computer hardware (GPU). During preprocessing, we
extracted the healthy liver masks. Thus, we have 11684 slices in training
dataset, 2745 in validation and 4734 slices in the testing set. The difference
in slices in the validation and test is because the number of slices might
vary between different patients.

###  3.2 Implementation details

We have used the PyTorch [14] framework for all the experiments used for liver
segmentation tasks. All the experiments were performed on NVIDIA RTX A6000
GPU. The network is configured to train with a batch size 16 and a learning
rate set to 1e-4. We train all the models for 500 epochs to fine-tune the
network parameters adequately with an early stopping patience of 50. To
enhance the performance of our network, we have used a combination of binary
cross-entropy and dice loss and an Adam optimizer was chosen for parameter
updates.

###  3.3 Evaluation metrics

Liver segmentation performance is evaluated using overlap-based metrics (dice
coefficient (DSC), mean intersection over union (mIoU), recall, precision, and
F2 score) and distance-based metrics (HD). Overlap-based metrics provide
insight into the accuracy and reliability in terms of overlap, whereas HD
evaluates the differences between the segmented boundary and the actual liver
boundary.

Table 1: Model performance on the Liver Tumor Segmentation benchmark dataset [2]. Method | Publication | Dice (%) | mIoU (%) | Recall (%) | Precision (%) | F2 (%) | HD
---|---|---|---|---|---|---|---
U-Net [15] | MICCAI 2015 | 82.06 | 73.40 | 77.82 | 91.10 | 78.98 | 3.79
ResUNet++ [16] | IEEE ISM 2019 | 77.03 | 68.19 | 79.90 | 83.57 | 76.24 | 3.96
DoubleU-Net [17] | IEEE CBMS 2020 | 86.24 | 77.89 | 80.68 | 95.00 | 81.28 | 3.69
ColonSegNet [17] | IEEE Access 2021 | 80.87 | 71.71 | 80.07 | 87.50 | 79.22 | 3.84
NanoNet [18] | IEEE CBMS 2021 | 75.05 | 66.53 | 73.33 | 83.25 | 73.23 | 4.01
UNeXt [19] | MICCAI 2022 | 81.31 | 72.43 | 80.73 | 87.40 | 79.93 | 3.74
TransNetR [20] | MIDL 2023 | 86.11 | 77.95 | 80.16 | 96.34 | 82.30 | 3.54
TransResUNet [21] | IEEE CMBS 2023 | 86.38 | 78.23 | 80.85 | 95.77 | 82.83 | 3.54
PVTFormer | Proposed | 86.78 | 78.46 | 80.70 | 96.11 | 82.86 | 3.50

##  4 Results and Discussion

Table 1 shows the model performance of the state-of-the-art medical image
segmentation architecture on Liver Tumor Segmentation Benchmark (LiTS)
datasets. We have compared our proposed method with methods such as UNet [10],
ResUNet++ [16], DoubleU-Net [15], ColonSegNet [17], UNext [19], TransNetR
[20], NanoNet [18], and TransResUNet [21]. In our rigorous evaluation, it can
be observed that most of the models show strong performance for healthy liver
segmentation. DoubleUNet [15] that has VGG-19 [22] as backbone notably
achieved a competitive dice coefficient of 86.24% and mIoU of 77.89%,
outperforming the competitive benchmarks such as UNext [19], UNet[10],
ResUNet++[16], NanoNet[18] and ColonSegNet [17]. However, transformer-based
methods such as TransNetR [20] and TransResUNet [21] that used ResNet50 as the
backbone show either competitive or better performance than DoubleUNet. For
example, TransNetR obtained the highest precision of 96.34%, whereas
TransResUNet obtained the highest recall of 80.85%.

From the table, we can also observe that the proposed PVTFormer showcased
remarkable performance, demonstrating the highest dice coefficient of 86.78%,
mIoU of 78.46%, recall of 80.70%, precision of 96.11%, F2 score of 82.86%, and
a low HD score of 3.50. From the overall comparison, it can be demonstrated
that PVTFormer outperformed eight state-of-the-art medical image segmentation
architectures. Notably, our proposed method captures fine details and
contextually significant features, surpassing the CNN-based architectures like
ResUNet++[16], ColonSegNet [17], and NanoNet [18] and transformer based
approaches such as TransNetR [20] and TransResUNet [21]. The quantitative
results demonstrate the effectiveness of integrating residual connections with
Transformer mechanisms (PVT), presenting a robust framework for precise and
comprehensive healthy liver segmentation.

##  5 Conclusion

In this study, we proposed PVTFormer architecture by leveraging a pretrained
Pyramid Vision Transformer (PVT v2) as an encoder and incorporating Up block,
Decoder block and residual learning for accurate liver segmentation. The
hierarchical decoding strategy in decoder blocks enhances semantic features,
boosting the quality of output segmentation masks. The evaluation against
eight existing state-of-the-art methods showcased that PVTFormer achieves
excellent results and outperforms its competitor with a high dice coefficient
of 86.78%, a mIoU of 78.46% and a low Hausdorff distance of 3.50. The
performance of the PVTFormer demonstrates that it acts as an effective method
for precise healthy liver segmentation and can also be translated to other
medical applications. In the future, we plan to annotate the test dataset of
LiTs with the help of radiologists from our team and perform a more
comprehensive study on the multi-center dataset. Moreover, we plan to extend
the capabilities of PVTFormer for segmenting multi-orans in abdominal CT
scans.

### Compliance with Ethical Standards

This research study was conducted retrospectively using human subject data
made available in open access by Bilic et al. [2]. Ethical approval was not
required as confirmed by the license attached with the open-access data.

### Conflicts of Interest

The authors have no relevant financial or non-financial interests to disclose.

## Acknowledgments

The project is supported by the NIH funding: R01-CA246704, R01-CA240639,
U01-DK127384-02S1, and U01-CA268808.

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# Staggered and short period solutions of the Saturable Discrete Nonlinear
Schrödinger Equation

Avinash Khare1, Kim Ø. Rasmussen2, Mogens R. Samuelsen3, and Avadh Saxena2
1Institute of Physics, Bhubaneswar, Orissa 751005, India  2Theoretical
Division, Los Alamos National Laboratory, Los Alamos, New Mexico, 87545, USA
3Department of Physics, The Technical University of Denmark, DK-2800 Kgs.
Lyngby, Denmark

###### Abstract

We point out that the nonlinear Schrödinger lattice with a saturable
nonlinearity also admits staggered periodic as well as localized pulse-like
solutions. Further, the same model also admits solutions with a short period.
We examine the stability of these solutions and find that the staggered as
well as the short period solutions are stable in most cases. We also show that
the effective Peierls-Nabarro barrier for the pulse-like soliton solutions is
zero.

###### pacs:

61.25.Hq, 64.60.Cn, 64.75.+g

The saturable discrete nonlinear Schrödinger equation is increasingly finding
applications in various physical situations. Most notably it serves as a model
for optical pulse propagation in optically modulated photorefractive media
[1], and in this context the pulse dynamics it describes have been intensely
studied [2, 3, 4]. In addition to its important role for such applications the
saturable discrete nonlinear Schrödinger equation is also of interest from a
purely nonlinear science viewpoint [5, 6, 7]. This interest arises because the
saturable discrete nonlinear Schrödinger equation has been demonstrated [8] to
admit onsite and intersite soliton solutions, which have the same energy. This
is contrasted by the standard cubic nonlinear Schrödinger lattice where the
onsite solution always has lower energy than the intersite solution. This
phenomenon have often been characterized in terms of a so-called Peierls-
Nabarro (PN) barrier, which is the energy difference between these two
distinct solutions. The particular feature of the saturable discrete nonlinear
Schrödinger equation is thus that it allows the PN barrier to change sign and
specifically vanish for certain solutions. The vanishing of the PN barrier
have been associated with the ability of these solutions to translate
undisturbed through the lattice, which is impossible in the cubic discrete
nonlinear Schrödinger equation. Here we derive analytical solutions to the
saturable discrete nonlinear Schrödinger equation and demonstrate that the
localized soliton solutions have a zero PN barrier.

Recently, we obtained [9] two different temporally and spatially periodic
solutions to the saturable equation[10]

$i\dot{\psi}_{n}+(\psi_{n+1}+\psi_{n-1}-2\psi_{n})+\frac{\nu|\psi_{n}|^{2}}{1+\mu|\psi_{n}|^{2}}\psi_{n}=0\,,$
(1)

where $\psi_{n}$ is a complex valued ‘wavefunction’ at site $n$, while $\nu$
and $\mu$ are real parameters. In particular, the first solution is

$\psi_{n}^{I}=\frac{1}{\sqrt{\mu}}\frac{\mbox{sn}(\beta,m)}{\mbox{cn}(\beta,m)}\mbox{dn}([n+c]\beta,m)\exp\left(-i[\omega
t+\delta]\right),$ (2)

where the modulus of the elliptic functions $m$ must be chosen such that

$2-\omega=\frac{\nu}{\mu}=\frac{2\mbox{dn}(\beta,m)}{\mbox{cn}^{2}(\beta,m)},~{}~{}~{}\beta=\frac{2K(m)}{N_{p}}\,,$
(3)

and $c$ and $\delta$ are arbitrary constants. We only need to consider c
between 0 and $\frac{1}{2}$ (half the lattice spacing). Here $K(m)$ denotes
the complete elliptic integral of the first kind [11]. The second solution is

$\psi_{n}^{II}=\sqrt{\frac{m}{\mu}}\frac{\mbox{sn}(\beta,m)}{\mbox{dn}(\beta,m)}\mbox{cn}([n+c]\beta,m)\exp\left(-i[\omega
t+\delta]\right)\,,$ (4)

where the modulus $m$ is now determined such that

$2-\omega=\frac{\nu}{\mu}=\frac{2\mbox{cn}(\beta,m)}{\mbox{dn}^{2}(\beta,m)},~{}~{}~{}\beta=\frac{4K(m)}{N_{p}}\,.$
(5)

The integer $N_{p}$ denotes the spatial period of the solutions. In the limit
$N_{p}\rightarrow\infty$ ($m\rightarrow 1$), both the solutions $\psi_{n}^{I}$
and $\psi_{n}^{II}$ reduce to the same localized solution

$\psi_{n}^{III}=\frac{1}{\sqrt{\mu}}\frac{\sinh(\beta)}{\cosh([n+c]\beta)}e^{-i[\omega
t+\delta]},~{}~{}(N_{p}\rightarrow\infty),$ (6)

where $\beta$ is now given by

$2-\omega=\frac{\nu}{\mu}=2\mbox{cosh}\beta\,.$ (7)

In Ref. [9], we also developed the stability analysis and examined the linear
stability of these solutions to show that the solutions are linearly stable in
most cases.

The purpose of this note is to point out that the same model (1) also admits
the corresponding staggered solutions. In particular, using the identities for
the Jacobi elliptic functions [12], it is easily shown that the model admits
the following solutions

$\psi_{n}^{IS}=(-1)^{n}\frac{1}{\sqrt{\mu}}\frac{\mbox{sn}(\beta,m)}{\mbox{cn}(\beta,m)}\mbox{dn}([n+c]\beta,m)\exp\left(-i[\omega
t+\delta]\right)\,,$ (8)

where the modulus $m$ must be chosen such that

$\omega-2=-\frac{\nu}{\mu}=\frac{2\mbox{dn}(\beta,m)}{\mbox{cn}^{2}(\beta,m)},~{}~{}~{}\beta=\frac{2K(m)}{N_{p}}\,.$
(9)
$\psi_{n}^{IIS}=(-1)^{n}\sqrt{\frac{m}{\mu}}\frac{\mbox{sn}(\beta,m)}{\mbox{dn}(\beta,m)}\mbox{cn}([n+c]\beta,m)\exp\left(-i[\omega
t+\delta]\right)\,,$ (10)

where the modulus $m$ is now determined such that

$\omega-2=-\frac{\nu}{\mu}=\frac{2\mbox{cn}(\beta,m)}{\mbox{dn}^{2}(\beta,m)},~{}~{}~{}\beta=\frac{4K(m)}{N_{p}}\,.$
(11)

In the limit $N_{p}\rightarrow\infty$ ($m\rightarrow 1$), both the solutions
$\psi_{n}^{IS}$ and $\psi_{n}^{IIS}$ reduce to the same localized staggered
solution

$\psi_{n}^{IIIS}=(-1)^{n}\frac{1}{\sqrt{\mu}}\frac{\sinh(\beta)}{\cosh([n+c]\beta)}e^{-i[\omega
t+\delta]},~{}~{}~{}(N_{p}\rightarrow\infty)\,,$ (12)

where $\beta$ is now given by

$\omega-2=-\frac{\nu}{\mu}=2\mbox{cosh}\beta\,.$ (13)

As an illustration we have plotted the exact solutions of the type IS and IIS
in Fig. 1. Here the period $N_{p}$ has to be even. We have shown two periods
for type IS and only one for type IIS.

There are, as expressed by Eqs. (9), (11), and (13), stringent conditions on
the parameters $\mu$ and $\nu$ for which these exact solutions exist. For
example, while the nonstaggered solutions are only valid for $\nu>0$ and hence
$\omega<2$, the staggered solutions are valid only if $\nu<0$ and hence
$\omega>2$. In the case IS the limitation is

$0~{}(m=1)<-\frac{2\mu}{\nu}<\cos^{2}\left(\frac{\pi}{N_{p}}\right)~{}(m=0)\,,$
(14)

while in the case IIS the limitation is

$0~{}(m=1)<-\frac{2\mu}{\nu}<\frac{1}{\cos(\frac{2\pi}{N_{p}})}~{}(m=0)\,.$
(15)

Similarly, the solution $\psi_{n}^{IIIS}$ exists only when $-\frac{2\mu}{\nu}$
is close to zero (m=1).

We have also examined the linear stability of these solutions and find that
the solutions are linearly stable in most cases. A single period $(N=N_{p}$i,
where $N$ is the lattice size) is always stable for both solutions IS and IIS.
A type IIS solution with more than one period ($N=jN_{p}$, where $j$ is an
integer larger than one) is also stable, while a type IS solution with more
than one period is always unstable. Thus, the first example in Fig. 1 is in
fact unstable.

Figure 1: Illustration of the exact solutions of two types. The parameters
are: $\nu=-1$, $\mu=0.4$, $\omega=4.5$, and $c=t=\delta=0$. $N_{p}=10$ for
$\psi_{n}^{IS}$ and 20 for $\psi_{n}^{IIS}$. The dashed curves represent the
solutions given by Eqs. (8) and (10) as if $n$ is a continuous variable. Lines
are guides to the eye.

For the solution IIIS, expressions for both the power and the Hamiltonian are
identical to those for the solution III and are given by Eqs. (13) and (14) of
Ref. [9]. Hence the PN barrier for the solutions III and IIIS is the same. We
would like to point out here that the calculation of PN barrier in I was not
quite correct. In particular, since both power P and the Hamiltonian H are
constants of motion, one must compute the energy difference between the
solutions when $c=0$ and $c=1/2$ in such a way that the power P is same in
both the cases. On using the expressions for P and H as given by Eqs. (13) and
(14) of I, we find that H for the solution III as well as IIIS is given by

$H=-\frac{4\sinh(\beta)}{\mu}+\frac{2\beta\nu}{\mu^{2}}+2\left(1-\frac{\nu}{2\mu}\right)P\,.$
(16)

Note that H is in fact independent of $c$, i.e. contrary to our claim in Ref.
[9], the PN barrier is in fact zero for our solution III (and hence also for
IIIS).

Before completing this note, we would like to mention that the model (1) also
admits a few short period solutions.

Using the ansatz

$\psi_{n}(t)=\phi_{n}e^{-i(\omega t+\delta)}\,,$ (17)

in Eq. (1) it is easily checked that the only possible short period solutions
are to Eq. (1) are:

  1. 1.

Period 1 solution $\phi_{n}=(...,a,a,...)$ provided

$\omega=-\frac{\nu a^{2}}{1+\mu a^{2}}.$ (18)

  2. 2.

Period 2 solution $\phi_{n}=(...,a,-a,...)$ provided

$\omega=4-\frac{\nu a^{2}}{1+\mu a^{2}}.$ (19)

  3. 3.

Period 3 Solution $\phi_{n}=(...,a,0,-a,...)$ provided

$\omega=3-\frac{\nu a^{2}}{1+\mu a^{2}}.$ (20)

  4. 4.

Period 4 Solutions $\phi_{n}=(...,a,0,-a,0,...)$ and $(...,a,a,-a,-a,...)$
provided

$\omega=2-\frac{\nu a^{2}}{1+\mu a^{2}}.$ (21)

  5. 5.

Period 6 Solution $\phi_{n}=(...,a,a,0,-a,-a,0,...)$ provided

$\omega=1-\frac{\nu a^{2}}{1+\mu a^{2}}.$ (22)

Applying the stability analysis developed in Ref. [9] we have examined the
stability of these short period solutions and find that for a small
nonlinearity ($|\nu|<2\mu$) they are all stable. The period 4 solution
$(...,a,a,-a,-a,...)$ is always stable while all the other short period
solutions possess regions of instabilities at larger nonlinearity. For these
low period solutions the stability matrices given by Eqs. (20) and (21) of
Ref. [9] are simple and it is, for example, easy to see that the lowest non-
zero eigenvalue, $\lambda_{1}(a,\nu)$, of the stability problem for the period
1 solutions is given by ($\mu=1$)

$\lambda_{1}(a,\nu)=a^{4}+\left(2-\frac{2}{3}\nu\right)a^{2}+1.$ (23)

Similarly, we have for the period 2 solution

$\lambda_{2}(a,\nu)=a^{4}+\left(2+\frac{2}{3}\nu\right)a^{2}+1,$ (24)

and the period 4 solution

$\lambda_{4}(a,\nu)=a^{4}+(2-|\nu|)a^{2}+1.$ (25)

These expressions correspond to the relevant curves in Fig. 2.

It possible to derive similar expressions for the period 3 and period 6
solutions but the expressions are more complicated and will be omitted here.

Figure 2: Regions of stability for the short period solutions to Eq. (1) for
$\mu=1$. Period 1: thick full curve, period 2: dashed-dotted curve, period 3:
long-dashed curve, period 4: thin full curve, and period 6: short-dashed
curve. The instability occurs in the parameter region encompassed by the
respective curves.

Clearly the $p$ period solutions are unstable for the parameter values where
$\lambda_{p}(a,\nu)<0$, and we have illustrated these regions in Fig. 2.
Figure 2 shows the curves in the $(a,\nu)$-plane where $\lambda_{p}(a,\nu)=0$
so that the instability occurs in the regions that are encompassed by the
respective curves. A symmetry is apparent in this stability diagram and it is
easy to realize that this symmetry arises from the fact that the
transformation $(\nu,\phi_{n})\rightarrow(-\nu,(-1)^{n}\phi_{n})$ establishes
the following connection between the short period solutions: 1
$\leftrightarrow$ 2, 3 $\leftrightarrow$ 6, and 4 $\leftrightarrow$ 4.

In conclusion, we have obtained staggered as well as short period solutions of
the saturable discrete nonlinear Schrödinger equation. We also studied the
linear stability and found the solutions to be stable in certain parameter
ranges. Finally, we found that the Peierls-Nabarro barrier for the pulse
solutions is zero. Our results are relevant to optical soliton pulse
propagation in waveguides and photorefractive media [1].

Research at Los Alamos National Laboratory is carried out under the auspices
of the National Nuclear Security Administration of the U.S. Department of
Energy under Contract No. DE-AC52-06NA25396.

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