# Secrecy Outage Probability Analysis for Downlink NOMA with Imperfect SIC at
Untrusted Users

Sapna Thapar1, Insha Amin1, Deepak Mishra2, and Ravikant Saini1 1Department of
Electrical Engineering, Indian Institute of Technology Jammu, India
2School of Electrical Engineering and Telecommunications, University of New
South Wales, Australia
Emails<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Non-orthogonal multiple access (NOMA) has come to the fore as a spectrally
efficient technique for fifth-generation networks and beyond. At the same
time, NOMA faces severe security issues in the presence of untrusted users due
to successive interference cancellation (SIC)-based decoding at receivers. In
this paper, to make the system model more realistic, we consider the impact of
imperfect SIC during the decoding process. Assuming the downlink mode, we
focus on designing a secure NOMA communication protocol for the considered
system model with two untrusted users. In this regard, we obtain the power
allocation bounds to achieve a positive secrecy rate for both near and far
users. Analytical expressions of secrecy outage probability (SOP) for both
users are derived to analyze secrecy performance. Closed-form approximations
of SOPs are also provided to gain analytical insights. Lastly, numerical
results have been presented, which validate the exactness of the analysis and
reveal the effect of various key parameters on achieved secrecy performance.

##  I Introduction

Non-orthogonal multiple access (NOMA) has been perceived as a promising
enabling technology for fifth-generation (5G) wireless networks and beyond, as
it entertains the possibility of serving more users in limited available
resources [1]. At the same time, the broadcast nature of NOMA poses a security
problem as the signal is vulnerable to eavesdropping. The use of physical
layer security (PLS) has sparked widespread interest in solving the security
concerns of the information-carrying signal in wireless communication.
Therefore, achieving secure NOMA communication by utilizing the potential of
PLS is a promising area of research [2].

###  I-A Related Works

Based on the concept of PLS, existing works have the prior objective of
securing the information-carrying signal against external eavesdroppers [2].
Additionally, the users who share the same resource block in NOMA may be
untrusted, thereby making it compulsory to provide secrecy against internal
eavesdropping. An untrusted users’ scenario is a hostile situation where no
users have mutual trust amongst each other, and therefore, they focus on
securing their data from others [3], [4]. In this regard, [5], [6] have
considered the secrecy issue of only near user against the far untrusted user.
However, a robust NOMA system should be designed such that even the far user
is provided secrecy against the near untrusted user. Based on this, [7]
proposed a PLS design for NOMA with a stronger near untrusted user. In [8], a
directional demodulation approach is followed to protect the data of weak user
from an untrusted strong user. In [9], a linear precoding technique is
proposed to prevent NOMA users from eavesdropping on each other. In [4], [10],
a novel secure decoding order is suggested to provide positive secrecy rate
for both strong and weak users. In [11], feasible secure decoding orders are
investigated to ensure a positive secrecy rate for all users in an $N$-user
system.

###  I-B Research Gap and Motivation

A common assumption in [7]-[10] is that perfect successive interference
cancellation (SIC) is performed by the receivers. Here, the interference from
the decoded users is cancelled altogether while decoding later users. However,
this might not be a realistic approach due to practical implementation
problems such as decoding errors and complexity scaling [12]. Consequently,
imperfect SIC, where the residual interference (RI) from incorrectly decoded
users remains while decoding later users, would be a practical model [12],
[13]. In NOMA literature, researchers have assumed either a fixed value of RI
[14], [15] or considered RI as a linear function of the interfering power
[12], [13]. However, in the direction of untrusted NOMA security, imperfect
SIC has not received much focus yet. In [4], secrecy outage probability (SOP)
of an untrusted NOMA system has been analyzed with a fixed RI value, which is
a strong unrealistic assumption. In contrast, the linear model can more
effectively represent the relationship between RI and power of the received
signal. In [11], though the linear SIC model has been considered, no SOP
analysis has been done. _Hence, to analyze the realistic impact of imperfect
SIC, we investigate the secrecy performance analysis for a two-user untrusted
NOMA system with linear SIC model, which to the best of our knowledge, has not
been explored yet._

###  I-C Key Contributions

The key contributions of this work are summarized below: (1) Considering the
impact of RI with linear imperfect SIC model in a two-user untrusted NOMA
system, the power allocation (PA) bounds to achieve a positive secrecy rate
for both users are investigated. (2) To analyze secrecy performance, the
analytical expressions of SOP for both near and far users are derived. (3) The
exact closed-form approximations of SOPs have also been obtained to attain
analytical insights. (4) Numerical results have been provided to validate the
analytical expressions, followed by insightful discussions on the impact of
different key parameters on the system performance.

Figure 1: Illustration of a downlink two-user untrusted NOMA system.

##  II NOMA Transmission among Untrusted Users

Here we first present the system model. Then, the possible decoding orders for
an untrusted NOMA system are discussed.

###  II-A System Model and NOMA Principle

Downlink of a NOMA system is considered, where one base station (BS)
communicates with two untrusted users (Fig. 1). We denote the $n$-th user by
$U_{n}$, where $n\in\mathcal{N}=\\{1,2\\}$. All the nodes in the network are
assumed to be equipped with one antenna. The Rayleigh fading channel gain
coefficient from BS to $U_{n}$ is denoted by $h_{n}$. The channel power gains
$|h_{n}|^{2}$ obeys an exponential distribution with mean parameter
$\lambda_{n}=L_{p}d_{n}^{-e}$, where $d_{n}$ denotes the distance between BS
and $U_{n}$, $L_{p}$ is the path loss constant, and $e$ indicates the path
loss exponent.Without loss of generality, we assume that
$|h_{1}|^{2}>|h_{2}|^{2}$, and thus, near and far user, i.e., $U_{1}$ and
$U_{2}$ could be regarded as strong and weak user, respectively. The BS
superposes information signals of users and broadcasts the superimposed signal
with a total BS transmission power $P_{t}$. The fraction of $P_{t}$ allocated
for $U_{1}$ is denoted by $\alpha$. The remaining fraction $(1-\alpha)$ is
alloted to $U_{2}$. At the receiver side, each user performs SIC wherein
inter-user interference imposed by the superposition is cancelled out to
extract the desired signal [16]. Without loss of generality, received additive
white Gaussian noise is assumed with mean $0$ and variance $\sigma^{2}$ at
both users. We consider an imperfect SIC scenario where RI from inaccurately
decoded signals exists while decoding later users. $\beta$, $0\leq\beta\leq
1$, denotes the RI factor, where $\beta=1$ corresponds to the scenario of
maximum interference, and $\beta=0$ indicates perfect SIC [12], [13].

###  II-B Decoding Orders for Untrusted NOMA

In an untrusted NOMA system, during the SIC process, each user can decode its
own signal and other users’ signal as well [4], [16]. This SIC process is
performed in a certain sequence, which is known as the ”decoding order” of the
system. In a two users’ scenario, the total possible decoding orders are $4$
[4]. Let us denote the decoding order as a $2\times 2$ matrix
$\mathbf{D}_{o}$, where $o\in\\{1,2,3,4\\}$ represents the index of $o$-th
decoding order. Here $m$-th column of matrix $\mathbf{D}_{o}$ is specified by
a column vector $\mathbf{d}_{m}$ of size $2\times 1$, which depicts the SIC
sequence observed by $U_{m}$, where $m\in\mathcal{N}$. To be more explicit,
$[\mathbf{d}_{m}]_{k}=n$ signifies that $U_{m}$ decodes data of $U_{n}$ at
$k$-th stage, where $n,k\in\mathcal{N}$ and
$[\mathbf{d}_{m}]_{1}\neq[\mathbf{d}_{m}]_{2}$. Thus, the $4$ possible
decoding orders can be written as $\mathbf{D}_{1}=[2,1;2,1]$,
$\mathbf{D}_{2}=[2,1;1,2]$, $\mathbf{D}_{3}=[1,2;2,1]$, and
$\mathbf{D}_{4}=[1,2;1,2]$. In [4, Theorem 2], it is proved that the optimal
decoding order with respect to providing maximum secrecy rate at both users is
$\mathbf{D}_{2}$. So, all further investigations will be carried out for
$\mathbf{D}_{2}$.

##  III PA Bounds from Secrecy Perspective

With the objective of securing one user’s data from another, first we
investigate the feasible PA bounds that ensure positive secrecy rate for both
users. In $\mathbf{D}_{2}$, both near and far users first decode signals of
other user, perform SIC, and then decode their own signal [4]. Thereby, using
linear SIC model [11]-[13], the achievable signal-to-interference-plus-noise-
ratio (SINR) $\Gamma_{nm}$ at $U_{m}$, when $U_{n}$ is decoded by $U_{m}$,
where $m,n\in\mathcal{N}$, is given as

$\displaystyle\Gamma_{21}=\frac{(1-\alpha)|h_{1}|^{2}}{\alpha|h_{1}|^{2}+\frac{1}{\rho_{t}}},\quad$
$\displaystyle\Gamma_{12}=\frac{\alpha|h_{2}|^{2}}{(1-\alpha)|h_{2}|^{2}+\frac{1}{\rho_{t}}},$
$\displaystyle\Gamma_{11}=\frac{\alpha|h_{1}|^{2}}{(1-\alpha)\beta|h_{1}|^{2}+\frac{1}{\rho_{t}}},\quad$
$\displaystyle\Gamma_{22}=\frac{(1-\alpha)|h_{2}|^{2}}{\alpha\beta|h_{2}|^{2}+\frac{1}{\rho_{t}}},$
(1)

where $\rho_{t}\stackrel{{\scriptstyle\Delta}}{{=}}\frac{P_{t}}{\sigma^{2}}$
denotes the BS transmit signal-to-noise ratio (SNR). The achievable secrecy
rates $R_{s1}$ and $R_{s2}$ at $U_{1}$ and $U_{2}$, respectively, can be given
as [2]

$R_{s1}=R_{11}-R_{12},\quad R_{s2}=R_{22}-R_{21},$ (2)

where $R_{nm}=\log_{2}(1+\Gamma_{nm})$ denotes the data rate at $U_{m}$ as
given by Shannon’s Theorem. To achieve positive secrecy rate for a given user,
the rate of the legitimate channel has to be higher than that of the
eavesdropper’s channel. Thus, for $U_{1}$, the positive secrecy rate
condition, i.e., $R_{11}>R_{12}$, which simplifies to
$\Gamma_{11}>\Gamma_{12}$, must be appeased. This gives

$\alpha<1+\frac{|h_{1}|^{2}-|h_{2}|^{2}}{|h_{1}|^{2}|h_{2}|^{2}\rho_{t}(1-\beta)}.$
(3)

Similarly, $\Gamma_{22}>\Gamma_{21}$ to obtain positive $R_{s2}$ for $U_{2}$
gives

$\alpha>\frac{|h_{1}|^{2}-|h_{2}|^{2}}{|h_{1}|^{2}|h_{2}|^{2}\rho_{t}(1-\beta)}.$
(4)

From (3) and (4), we can easily infer that in decoding order $\mathbf{D}_{2}$,
positive secrecy rate can be obtained at both the users, provided
$\frac{|h_{1}|^{2}-|h_{2}|^{2}}{|h_{1}|^{2}|h_{2}|^{2}\rho_{t}(1-\beta)}<\alpha<1+\frac{|h_{1}|^{2}-|h_{2}|^{2}}{|h_{1}|^{2}|h_{2}|^{2}\rho_{t}(1-\beta)}$.

$\displaystyle s_{1}$
$\displaystyle=\text{Pr}\\{R_{s1}<R_{s1}^{th}\\}\\!=\\!\text{Pr}\bigg{\\{}\frac{1+\Gamma_{11}}{1+\Gamma_{12}}<\Pi_{1}\bigg{\\}}\\!=\\!\text{Pr}\bigg{\\{}|h_{1}|^{2}D_{1}<N_{1}\bigg{\\}}=\text{Pr}\bigg{\\{}|h_{1}|^{2}<\frac{N_{1}}{D_{1}},D_{1}>0\bigg{\\}}\\!+\\!\text{Pr}\bigg{\\{}|h_{1}|^{2}\geq\frac{N_{1}}{D_{1}},D_{1}\leq
0\bigg{\\}},$
$\displaystyle=\text{Pr}\bigg{\\{}|h_{1}|^{2}<\frac{N_{1}}{D_{1}},|h_{2}|^{2}<T_{1}\bigg{\\}}+\text{Pr}\bigg{\\{}|h_{1}|^{2}\geq\frac{N_{1}}{D_{1}},|h_{2}|^{2}\geq
T_{1}\bigg{\\}},$
$\displaystyle=\begin{cases}\int_{0}^{T_{1}}\bigg{(}1-\exp\bigg{\\{}\frac{-N_{1}}{D_{1}\lambda_{1}}\bigg{\\}}\bigg{)}f_{|h_{2}|^{2}}(y_{1})dy_{1}+\int_{T_{1}}^{\infty}1\times
f_{|h_{2}|^{2}}(y_{1})dy_{1},&\alpha_{1a}<\alpha<\alpha_{1b}\\\
\int_{0}^{\infty}\bigg{(}1-\exp\bigg{\\{}\frac{-N_{1}}{D_{1}\lambda_{1}}\bigg{\\}}\bigg{)}f_{|h_{2}|^{2}}(y_{1})dy_{1},&\alpha\geq\alpha_{1b}\\\
\int_{0}^{\infty}1\times
f_{|h_{2}|^{2}}(y_{1})dy_{1},&\text{otherwise}\end{cases}$
$\displaystyle=\begin{cases}1-\frac{1}{\lambda_{2}}\int_{0}^{T_{1}}\exp\bigg{\\{}\frac{-((\Pi_{1}-1)((1-\alpha)y_{1}\rho_{t}+1)+\alpha
y_{1}\rho_{t}\Pi_{1})}{(\alpha\rho_{t}((1-\alpha)y_{1}\rho_{t}+1)-(\Pi_{1}-1)((1-\alpha)y_{1}\rho_{t}+1)\beta(1-\alpha)\rho_{t}-\Pi_{1}\beta\alpha(1-\alpha)\rho_{t}^{2}y_{1})\lambda_{1}}-\frac{y_{1}}{\lambda_{2}}\bigg{\\}}dy_{1},&\alpha_{1a}<\alpha<\alpha_{1b}\\\
1-\frac{1}{\lambda_{2}}\int_{0}^{\infty}\exp\bigg{\\{}\frac{-((\Pi_{1}-1)((1-\alpha)y_{1}\rho_{t}+1)+\alpha
y_{1}\rho_{t}\Pi_{1})}{(\alpha\rho_{t}((1-\alpha)y_{1}\rho_{t}+1)-(\Pi_{1}-1)((1-\alpha)y_{1}\rho_{t}+1)\beta(1-\alpha)\rho_{t}-\Pi_{1}\beta\alpha(1-\alpha)\rho_{t}^{2}y_{1})\lambda_{1}}-\frac{y_{1}}{\lambda_{2}}\bigg{\\}}dy_{1},&\alpha\geq\alpha_{1b}\\\
1,&\text{otherwise}\end{cases}$ (5)

$\displaystyle s_{2}$
$\displaystyle=\text{Pr}\\{R_{s2}<R_{s2}^{th}\\}=\text{Pr}\Big{\\{}\frac{1+\Gamma_{22}}{1+\Gamma_{21}}<\Pi_{2}\Big{\\}}=\text{Pr}\bigg{\\{}|h_{2}|^{2}D_{2}<N_{2}\bigg{\\}}\\!=\\!\text{Pr}\bigg{\\{}|h_{2}|^{2}<\frac{N_{2}}{D_{2}},D_{2}>0\bigg{\\}}+\text{Pr}\bigg{\\{}|h_{2}|^{2}\geq\frac{N_{2}}{D_{2}},D_{2}\leq
0\bigg{\\}},$
$\displaystyle=\text{Pr}\bigg{\\{}|h_{2}|^{2}<\frac{N_{2}}{D_{2}},|h_{1}|^{2}<T_{2}\bigg{\\}}+\text{Pr}\bigg{\\{}|h_{2}|^{2}\geq\frac{N_{2}}{D_{2}},|h_{1}|^{2}\geq
T_{2}\bigg{\\}},$
$\displaystyle=\begin{cases}\int_{0}^{T_{2}}\bigg{(}1-\exp\bigg{\\{}\frac{-N_{2}}{D_{2}\lambda_{2}}\bigg{\\}}\bigg{)}f_{|h_{1}|^{2}}(y_{2})dy_{2}+\int_{T_{2}}^{\infty}1\times
f_{|h_{1}|^{2}}(y_{2})dy_{2},\quad&\alpha_{2a}<\alpha<\alpha_{2b}\\\
\int_{0}^{\infty}\bigg{(}1-\exp\bigg{\\{}\frac{-N_{2}}{D_{2}\lambda_{2}}\bigg{\\}}\bigg{)}f_{|h_{1}|^{2}}(y_{2})dy_{2},\quad\quad\quad\quad\quad\quad\quad\quad&\alpha\leq\alpha_{2a}\\\
\int_{0}^{\infty}1\times
f_{|h_{1}|^{2}}(y_{2})dy_{2},&\text{otherwise}\end{cases}$
$\displaystyle=\begin{cases}1-\frac{1}{\lambda_{1}}\int_{0}^{T_{2}}\exp\bigg{\\{}\frac{-((\Pi_{2}-1)(\alpha
y_{2}\rho_{t}+1)+(1-\alpha)y_{2}\rho_{t}\Pi_{2})}{((1-\alpha)\rho_{t}(\alpha
y_{2}\rho_{t}+1)-(\Pi_{2}-1)(\alpha
y_{2}\rho_{t}+1)\beta\alpha\rho_{t}-\Pi_{2}\beta\alpha(1-\alpha)\rho_{t}^{2}y_{2})\lambda_{2}}-\frac{y_{2}}{\lambda_{1}}\bigg{\\}}dy_{2},\quad&\alpha_{2a}<\alpha<\alpha_{2b}\\\
1-\frac{1}{\lambda_{1}}\int_{0}^{\infty}\exp\bigg{\\{}\frac{-((\Pi_{2}-1)(\alpha
y_{2}\rho_{t}+1)+(1-\alpha)y_{2}\rho_{t}\Pi_{2})}{((1-\alpha)\rho_{t}(\alpha
y_{2}\rho_{t}+1)-(\Pi_{2}-1)(\alpha
y_{2}\rho_{t}+1)\beta\alpha\rho_{t}-\Pi_{2}\beta\alpha(1-\alpha)\rho_{t}^{2}y_{2})\lambda_{2}}-\frac{y_{2}}{\lambda_{1}}\bigg{\\}}dy_{2},&\alpha\leq\alpha_{2a}\\\
1,&\text{otherwise}\end{cases}$ (6)

$\displaystyle
s_{1[C]}\\!=\\!\\!\begin{cases}1\\!-\\!\frac{1}{\lambda_{2}}\int_{0}^{T_{1[C]}}\exp\bigg{\\{}\frac{-((\Pi_{1}-1)(\beta(1-\alpha)y_{1}\rho_{t}+1)+\alpha
y_{1}\rho_{t}\Pi_{1})}{(\alpha\rho_{t}(\beta(1-\alpha)y_{1}\rho_{t}+1)-(\Pi_{1}-1)(\beta(1-\alpha)y_{1}\rho_{t}+1)\beta(1-\alpha)\rho_{t}-\Pi_{1}\beta\alpha(1-\alpha)\rho_{t}^{2}y_{1})\lambda_{1}}-\frac{y_{1}}{\lambda_{2}}\bigg{\\}}dy_{1},\quad&\alpha>\alpha_{1[C]}\\\
1,&\text{otherwise}\end{cases}$ (7)

$\displaystyle
s_{2[C]}=\begin{cases}1-\frac{1}{\lambda_{1}}\int_{0}^{T_{2[C]}}\exp\bigg{\\{}\frac{-((\Pi_{2}-1)(\alpha
y_{2}\rho_{t}+1)+(1-\alpha)y_{2}\rho_{t}\Pi_{2})}{((1-\alpha)\rho_{t}(\alpha
y_{2}\rho_{t}+1)-(\Pi_{2}-1)(\alpha
y_{2}\rho_{t}+1)\alpha\rho_{t}-\Pi_{2}\alpha(1-\alpha)\rho_{t}^{2}y_{2})\lambda_{2}}-\frac{y_{2}}{\lambda_{1}}\bigg{\\}}dy_{2},\quad&\alpha<\alpha_{2[C]}\\\
1,&\text{otherwise}\end{cases}$ (8)

$\displaystyle\widehat{s}_{1}$
$\displaystyle=\begin{cases}1-\bigg{(}\big{(}1-\exp\big{\\{}\frac{-T_{1}}{\lambda_{2}}\big{\\}}\big{)}\times\exp\bigg{\\{}\frac{-((\Pi_{1}-1)(1-\alpha)\rho_{t}+\alpha\rho_{t}\Pi_{1})}{(\alpha\rho_{t}^{2}(1-\alpha)-(\Pi_{1}-1)(1-\alpha)^{2}\beta\rho_{t}^{2}-\Pi_{1}\beta\alpha(1-\alpha)\rho_{t}^{2})\lambda_{1}}\bigg{\\}}\bigg{)},\quad&\alpha_{1a}<\alpha<\alpha_{1b}\\\
1-\exp\bigg{\\{}\frac{-((\Pi_{1}-1)(1-\alpha)\rho_{t}+\alpha\rho_{t}\Pi_{1})}{(\alpha\rho_{t}^{2}(1-\alpha)-(\Pi_{1}-1)(1-\alpha)^{2}\beta\rho_{t}^{2}-\Pi_{1}\beta\alpha(1-\alpha)\rho_{t}^{2})\lambda_{1}}\bigg{\\}},&\alpha\geq\alpha_{1b}\\\
1,&\text{otherwise}\end{cases}$ (9)

$\displaystyle\widehat{s}_{2}$
$\displaystyle=\begin{cases}1-\bigg{(}\big{(}1-\exp\big{\\{}\frac{-T_{2}}{\lambda_{1}}\big{\\}}\big{)}\times\exp\bigg{\\{}\frac{-((\Pi_{2}-1)\alpha\rho_{t}+(1-\alpha)\rho_{t}\Pi_{2})}{((1-\alpha)\rho_{t}^{2}\alpha-(\Pi_{2}-1)\alpha^{2}\rho_{t}^{2}\beta-(1-\alpha)\Pi_{2}\rho_{t}^{2}\alpha\beta)\lambda_{2}}\bigg{\\}}\bigg{)},\quad&\alpha_{2a}<\alpha<\alpha_{2b}\\\
1-\exp\bigg{\\{}\frac{-((\Pi_{2}-1)\alpha\rho_{t}+(1-\alpha)\rho_{t}\Pi_{2})}{((1-\alpha)\rho_{t}^{2}\alpha-(\Pi_{2}-1)\alpha^{2}\rho_{t}^{2}\beta-(1-\alpha)\Pi_{2}\rho_{t}^{2}\alpha\beta)\lambda_{2}}\bigg{\\}},&\alpha\leq\alpha_{2a}\\\
1,&\text{otherwise}\end{cases}$ (10)

$\displaystyle\widehat{s}_{1[C]}$
$\displaystyle=\begin{cases}1-\bigg{(}\big{(}1-\exp\big{\\{}\frac{-T_{1[C]}}{\lambda_{2}}\big{\\}}\big{)}\times\exp\bigg{\\{}\frac{-((\Pi_{1}-1)\beta(1-\alpha)\rho_{t}+\alpha\rho_{t}\Pi_{1})}{(\beta\alpha\rho_{t}^{2}(1-\alpha)-(\Pi_{1}-1)(1-\alpha)^{2}\beta^{2}\rho_{t}^{2}-\Pi_{1}\beta\alpha(1-\alpha)\rho_{t}^{2})\lambda_{1}}\bigg{\\}}\bigg{)},\quad&\alpha>\alpha_{1[C]}\\\
1,&\text{otherwise}\end{cases}$ (11)

$\displaystyle\widehat{s}_{2[C]}$
$\displaystyle=\begin{cases}1-\bigg{(}\big{(}1-\exp\big{\\{}\frac{-T_{2[C]}}{\lambda_{1}}\big{\\}}\big{)}\times\exp\bigg{\\{}\frac{-((\Pi_{2}-1)\alpha\rho_{t}+(1-\alpha)\rho_{t}\Pi_{2})}{(\alpha\rho_{t}^{2}(1-\alpha)-(\Pi_{2}-1)\alpha^{2}\rho_{t}^{2}-\Pi_{2}\alpha(1-\alpha)\rho_{t}^{2})\lambda_{2}}\bigg{\\}}\bigg{)},\quad&\alpha<\alpha_{2[C]}\\\
1,&\text{otherwise}\end{cases}$ (12)

##  IV Secrecy Performance Analaysis

In this section, we derive the analytical expressions of SOP to explore
secrecy performance for $\mathbf{D}_{2}$. Asymptotic approximations are also
provided to gain analytical insights.

###  IV-A Secrecy Outage Probability

The SOP is defined as the probability that the maximum achievable secrecy rate
at a user is less than a threshold secrecy rate [5]. Let us denote SOP for
$U_{n}$ as $s_{n}$, where $n\in\mathcal{N}$.

####  IV-A1 Near User

Assuming target and achievable secrecy rate, respectively, for $U_{1}$ as
$R_{s1}^{th}$ and $R_{s1}$, the SOP $s_{1}$ is given as (III). Here
$\text{Pr}\\{.\\}$ denotes the probability measure,
$\Pi_{1}\stackrel{{\scriptstyle\Delta}}{{=}}2^{R_{s1}^{th}}$,
$N_{1}=(\Pi_{1}-1)((1-\alpha)|h_{2}|^{2}\rho_{t}+1)+\alpha|h_{2}|^{2}\rho_{t}\Pi_{1}$,
$D_{1}=\alpha\rho_{t}((1-\alpha)|h_{2}|^{2}\rho_{t}+1)-(\Pi_{1}-1)((1-\alpha)|h_{2}|^{2}\rho_{t}+1)\beta(1-\alpha)\rho_{t}-\Pi_{1}\beta\alpha(1-\alpha)\rho_{t}^{2}|h_{2}|^{2}$,
$T_{1}=\frac{\alpha-(\Pi_{1}-1)\beta(1-\alpha)}{(1-\alpha)\rho_{t}((\Pi_{1}-1)\beta(1-\alpha)+\alpha\Pi_{1}\beta-\alpha)}$,
$\alpha_{1a}=\frac{\beta(\Pi_{1}-1)}{1+\beta(\Pi_{1}-1)}$,
$\alpha_{1b}=\frac{\beta(\Pi_{1}-1)}{1-\beta}$, and $f_{\mid
h_{2}\mid^{2}}(x)$ is the probability density function (PDF) of $|h_{2}|^{2}$.
Note that $\alpha_{1b}>\alpha_{1a}$, as this simplifies to $\beta\Pi_{1}>0$,
which is always true.

####  IV-A2 Far user

Considering $R_{s2}^{th}$ and $R_{s2}$, respectively, as target and achievable
secrecy rate, of $U_{2}$, $s_{2}$ is given as (III), where
$\Pi_{2}\stackrel{{\scriptstyle\Delta}}{{=}}2^{R_{s2}^{th}}$,
$N_{2}=(\Pi_{2}-1)(\alpha|h_{1}|^{2}\rho_{t}+1)+(1-\alpha)|h_{1}|^{2}\rho_{t}\Pi_{2}$,
$D_{2}=(1-\alpha)\rho_{t}(\alpha|h_{1}|^{2}\rho_{t}+1)-(\Pi_{2}-1)(\alpha|h_{1}|^{2}\rho_{t}+1)\beta\alpha\rho_{t}-\Pi_{2}\beta\alpha(1-\alpha)\rho_{t}^{2}|h_{1}|^{2}$,
$T_{2}=\frac{(1-\alpha)-(\Pi_{2}-1)\beta\alpha}{\alpha\rho_{t}((\Pi_{2}-1)\beta\alpha+(1-\alpha)\Pi_{2}\beta-(1-\alpha))}$,
$\alpha_{2a}=\frac{1-\Pi_{2}\beta}{1-\beta}$,
$\alpha_{2b}=\frac{1}{1+\beta(\Pi_{2}-1)}$, and $f_{|h_{1}|^{2}}(x)$ is PDF of
$|h_{1}|^{2}$. Here $\alpha_{2b}>\alpha_{2a}$, as it gives $\Pi_{2}>1$, which
holds true.

Similarly, we can obtain SOP expressions for other decoding orders. As a
special case, the SOPs, $s_{1[C]}$ and $s_{2[C]}$ for conventional decoding
order $\mathbf{D}_{1}$, are given in (7) and (8), respectively. Here
$T_{1[C]}=\frac{\alpha-(\Pi_{1}-1)\beta(1-\alpha)}{(1-\alpha)\rho_{t}\beta((\Pi_{1}-1)\beta(1-\alpha)+\alpha(\Pi_{1}-1))}$,
$\alpha_{1[C]}=\frac{(\Pi_{1}-1)\beta}{1+\beta(\Pi_{1}-1)}$,
$T_{2[C]}=\frac{(1-\alpha)-(\Pi_{2}-1)\alpha}{\alpha\rho_{t}((\Pi_{2}-1)\alpha+(1-\alpha)(\Pi_{2}-1))}$
and $\alpha_{2[C]}=\frac{1}{\Pi_{2}}$. $[C]$ stands for conventional decoding
order.

###  IV-B Asymptotic Approximations

Next, to provide analytical insights, we present closed-form approximations of
SOPs at both users $U_{1}$ and $U_{2}$ for $\mathbf{D}_{2}$.

####  IV-B1 Near User

The exact closed-form expression of $s_{1}$, i.e., $\widehat{s}_{1}$, obtained
by using $((1-\alpha)\rho_{t}y_{1}+1)$ $\approx$ $(1-\alpha)\rho_{t}y_{1}$ for
$\rho_{t}\gg 1$ in (III) is given in (9).

####  IV-B2 Far User

The closed-form asymptotic approximation $\widehat{s}_{2}$ of $s_{2}$, which
we obtain by setting $(\alpha\rho_{t}y_{2}+1)$ $\approx$ $\alpha\rho_{t}y_{2}$
for high $\rho_{t}$ in (III), is provided in (10).

Similar to above approximations, closed-form SOP expressions for other
decoding orders can also be obtained. For conventional decoding order
$\mathbf{D}_{1}$, the asymptotic SOP expressions, $\widehat{s}_{1[C]}$ and
$\widehat{s}_{2[C]}$, are given in (11) and (12), respectively.

##  V Numerical Results

Downlink NOMA system is considered with one BS and two untrusted users. Near
user is assumed to be at a distance of $d_{1}=50$ meter from BS, and for far
user, distance $d_{2}=100$ meter is adopted. Noise power is set to $-90$ dBm
with noise signal following Gaussian distribution at all users. Small scale
fading is presumed to obey an exponential distribution with a $1$ mean value
at both links [5]. Simulations are averaged over $10^{6}$ randomly generated
channel realizations by using Rayleigh distribution for both users. The value
of $L_{p}$ and $e$, respectively, are taken to be $1$ and $3$. $\rho_{r}$ is
assumed as the received SNR in decibels (dB) at $U_{2}$. The value of $\beta$
is taken to be $0.1$. Simulation, analytical, and asymptotic results are,
respectively, marked as ‘Sim’, ‘Ana’, and ‘Asy’.

Figure 2: Validation of SOPs at $U_{1}$ and $U_{2}$ for $\mathbf{D}_{2}$,
$\alpha=0.33$. Figure 3: Validation of the accuracy of the proposed closed-
form asymptotic approximations of SOPs for $\mathbf{D}_{2}$ with $\alpha=0.5$.

###  V-A Validation of Analysis

Here in Fig. 2, the validation of SOPs, $s_{1}$ with $R_{s1}^{th}$ and $s_{2}$
with $R_{s2}^{th}$ for different values of $\rho_{r}$ are shown. The perfect
agreement between simulated and analytical curves confirms the exactness of
$s_{1}$ and $s_{2}$ analysis. It can be visualized from the results that
$s_{1}$ and $s_{2}$ increase with the increase in threshold rates
$R_{s1}^{th}$ and $R_{s2}^{th}$, respectively. Because outage occurs when the
maximum achievable secrecy rate drops below a threshold rate, it is clear that
increasing threshold secrecy rates at the user will, in turn, increase SOP. It
can also be observed that an increase in $\rho_{r}$ decreases both $s_{1}$ and
$s_{2}$. This happens because the secrecy rates achieved at users increase by
an increase in SNR, and so, for a given threshold secrecy rate, SOP decreases.
From Fig. 3, we can visualize that analytical results match with asymptotic
results at high SNR, and it confirms the exactness of asymptotic expressions.

Figure 4: Verification of the optimality of the decoding order
$\mathbf{D}_{2}$ among $4$ possibilities {$\mathbf{D}_{1}$, $\mathbf{D}_{2}$,
$\mathbf{D}_{3}$, $\mathbf{D}_{4}$}, for $U_{1}$ in (a) and $U_{2}$ in (b)
with $\rho_{r}=10$ dB, $R_{s1}^{th}=0.5$ and $R_{s2}^{th}=0.1$. Figure 5:
Insights on optimal power allocation $\alpha$ that minimizes SOPs, $s_{1}$ in
(a) and $s_{2}$ in (b), for different values of $d_{2}$, for $\mathbf{D}_{2}$,
$d_{1}=40$ meter, $\rho_{t}=70$ dB, $R_{s1}^{th}=1$ and $R_{s2}^{th}=0.5$.

###  V-B Validation of Optimal Decoding Order

Here we have sketched a plot to validate that the optimal decoding order for
providing the highest secrecy rate for both near and far users is
$\mathbf{D}_{2}$, as given in [4, Theorem 2]. In Fig. 4, SOPs at both $U_{1}$
and $U_{2}$ for all $4$ decoding orders with respect to $\alpha$ are shown.
The results corroborate that better SOP performance is obtained for
$\mathbf{D}_{2}$, and hence it is optimal.

###  V-C Impact of Relative Distance between Users

In Fig. 5(a), we notice the effect of varying the distance $d_{2}$ from BS on
achievable SOPs. $d_{1}$ is fixed at $40$ meters. It can be seen that $s_{1}$
decreases with an increase in $d_{2}$. This happens because an increase in
distance $d_{2}$ causes a drop in achievable data rate at $U_{2}$, which in
turn provide better secrecy rate at $U_{1}$, thereby decreasing the SOP at
$U_{1}$. Conversely, as shown in Fig. 5(b), a decrease in data rate at $U_{2}$
signifies a reduction in secrecy rate at $U_{2}$, which further increases the
SOP for $U_{2}$. Thus, it can be observed that increasing the distance from BS
to $U_{2}$ has a contradictory effect on $s_{1}$ and $s_{2}$. Fig. 5(a) and
Fig. 5(b) also confirm the existence of an optimal PA that minimizes the SOP
performance for both users $U_{1}$ and $U_{2}$.

##  VI Concluding Remarks

We have focused on the practical but adverse problem of SIC being imperfect in
a secure NOMA system. Considering RI, the PA bounds are calculated to provide
a positive secrecy rate for users. Analytical and asymptotic expressions of
SOP are derived. Numerical results are provided to validate the analytical
expressions and exhibit the effects of different key parameters on
performance. Future work includes extending the study of SOP in a multi-user
untrusted NOMA scenario.

## Acknowledgement

This work was partially supported by the Faculty Funded Postdoctoral Research
Fellowship support of EET School at UNSW Sydney.

## References

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# Miura type transformations
for integrable lattices in 3D

I. T. Habibullin, A. R. Khakimova and A. U. Sakieva

###### Abstract

The article studies a class of integrable semidiscrete equations with one
continuous and two discrete independent variables. Miura type transformations
are obtained that relate the equations of the class. A new integrable chain of
this type is found, for which the Lax pair is presented. Integrable in the
sense of Darboux reductions of the chain are discussed, for small order
reductions complete sets of integrals are constructed. Continuum limits for
the chain are discussed. A method for finding particular solutions of chains
based on integrable in sense of Darboux reductions is proposed. The
effectiveness of the method is illustrated by an example.

Keywords: integrability, differential-difference lattices, Miura
transformation, continuum limit, Lax pair, local conservation laws, explicit
solutions.

##  1 Introduction

Nonlinear integrable differential-difference equations with one continuous and
two discrete variables are intermediate objects between the class of
integrable nonlinear equations of dimension 1 + 2, such as the Toda chain, the
Kadomtsev-Petviashvili equation, the Davey-Stewartson equation, etc. (see [1],
[2]), and the class of fundamental discrete Hirota-Miwa type equations (see
[3], [4], [5]). They also have applications in the theory of Laplace
transformations in the class of linear differential-difference equations of
hyperbolic type[9]. More precisely, the sequence of Laplace invariants
satisfies some integrable differential-difference equation of the form (1.1),
related by a point transformation to equation 7) from the list below. Such
chains can find application in studying the nonlinear dynamics of localized
magnetic inhomogeneities in such magnetically ordered substances as
ferromagnets, antiferromagnets with weak ferromagnetism, magnetoelastic and
magnetoelectric interactions [10], [11]. Equations of the type under
consideration have potential applications in the problem of describing
dislocations in media with a microstructure, as well as in the nonlinear
theory of elastic and nonelastic deformations accompanied with deep
reconstruction of an initially ideal lattice: switching of interatomic bonds,
changing of the class of symmetry, formation of new phases, singular defects
and heterogeneities, fragmentation of the lattice (see, for instance, [12],
[13]).

All known integrable differential-difference equations with one continuous and
two discrete independent variables can be reduced to the following form

$u^{j}_{n+1,x}=f(u^{j}_{n,x},u^{j+1}_{n},u^{j}_{n},u^{j}_{n+1},u^{j-1}_{n+1}),$
(1.1)

where the sought function $u=u^{j}_{n}(x)$ depends on the continuous $x$ and
discrete $n$ and $j$. The subscript $x$ denotes the derivative
$u_{x}=\frac{d}{dx}u$. Below we give a list of currently known examples of
integrable equations of the form (1.1):

  1. 1)

$u^{j}_{n+1,x}=u^{j}_{n,x}+e^{u^{j}_{n+1}-u^{j+1}_{n}}-e^{u^{j-1}_{n+1}-u^{j}_{n}},$

  2. 2)

$u^{j}_{n+1,x}=u^{j}_{n,x}+e^{u^{j-1}_{n+1}-u^{j}_{n+1}}-e^{u^{j-1}_{n+1}-u^{j}_{n}}-e^{u^{j}_{n}-u^{j+1}_{n}}+e^{u^{j}_{n+1}-u^{j+1}_{n}},$

  3. 3)

$u^{j}_{n+1,x}=u^{j}_{n,x}\frac{{(u^{j}_{n+1})^{2}}}{{u^{j-1}_{n+1}}u^{j+1}_{n}},$

  4. 4)

$u^{j}_{n+1,x}=u^{j}_{n,x}\frac{u^{j}_{n+1}-u^{j+1}_{n}}{u^{j-1}_{n+1}-u^{j}_{n}},$

  5. 5)

$u^{j}_{n+1,x}=u^{j}_{n,x}\frac{u^{j}_{n+1}(u^{j}_{n+1}-u^{j+1}_{n})}{u^{j+1}_{n}(u^{j-1}_{n+1}-u^{j}_{n})},$

  6. 6)

$u^{j}_{n+1,x}=u^{j}_{n,x}\frac{(u^{j-1}_{n+1}-u^{j}_{n+1})(u^{j}_{n+1}-u^{j+1}_{n})}{(u^{j-1}_{n+1}-u^{j}_{n})(u^{j}_{n}-u^{j+1}_{n})},$

  7. 7)

$u^{j}_{n+1,x}=u^{j}_{n,x}+e^{{-u^{j+1}_{n}+u^{j}_{n}}+u^{j}_{n+1}-u^{j-1}_{n+1}},$

  8. 8)

$u^{j}_{n+1,x}=u^{j}_{n,x}-e^{u^{j+1}_{n}}+e^{u^{j}_{n}}+e^{u^{j}_{n+1}}-e^{u^{j-1}_{n+1}},$

  9. 9)

$u^{j}_{n+1,x}=u^{j}_{n,x}+(u^{j}_{n+1}-u^{j}_{n})(u^{j+1}_{n}-u^{j}_{n+1}-u^{j}_{n}+u^{j-1}_{n+1}).$

Equation 1) was derived in [14] on the basis of iterations of the Bäcklund
transformation for the two-dimensional Toda lattice. Equations 7) and 8) were
found in [9] while studying the Laplace invariants of linear differential-
difference equations of hyperbolic type with variable coefficients. Equations
2)-6) were obtained in [15] by an original method based on a preliminary study
of the dispersionless limits of chains for integrability, followed by a return
transition to the class of chains (1.1) as a result of dispersion
deformations. Equation 9) is new: it is obtained in the present work.

A distinctive feature of nonlinear chains with three independent variables is
that they allow reductions in the form of systems of differential-difference
equations with complete sets of characteristic integrals. Such systems of
equations are called Darboux integrable. As a rule it is possible to find in
an explicit form general solution to Darboux integrable systems at least for
the case of small orders. This circumstance allows one finding localized
particular solutions for the lattices in 3D via the Darboux integrable
reductions. In our articles [17], [18], [19], [20], an algorithm for
classifying integrable equations of the form (1.1) is developed, where the
presence of a hierarchy of the Darboux integrable reductions is taken as a
classification criterion.

One can check that the equations from the list above are not related to each
other by point changes of variables. However, the equations could be
transformed from one to the other using more complex irreversible Miura type
transformations, which would be useful in finding explicit solutions to the
equations (see, for instance, [6], [7], [8]). Therefore, the task of searching
for such transformations is in demand. The study of this problem is the aim of
the present work.

The results of working with Miura type transformations for the class of
lattices (1.1) exceeded our expectations. They led to the discovery of a new
integrable chain 9). This equation is unique in some extent, since it has a
polynomial (quadratic) right hand side, while the right hand sides of the
other chains are either rational or exponential.

Let us briefly discuss on the content of the article. In §2 we suggest an
algorithm for searching differential or/and difference substitutions generated
by local conservation laws. In §3 we apply the algorithm to equations 1)–9).
We have established connections between the equations. In §4 we give the Lax
pair for the chain 9) and calculate for it the continuum limits of two types.
We also consider examples of the small order Darboux integrable reductions
admitted by the chain. For the reduced systems the complete sets of the
characteristic integrals are constructed. An explicit particular solution to
chain 9) is presented obtained via the reduced equation.

##  2 Local conservation laws and Miura type transformations.

To find the Miura transformation that converts an equation of a given class
(1.1) into some other equation of the same class, we use the following scheme.
Let’s first represent the equation as a local conservation law of the form:

$(D_{n}-1)A=(D_{j}-1)B$ (2.1)

or the form

$D_{x}A=(D_{j}-1)B,$ (2.2)

where $A$ and $B$ are functions depending on the sought function
$u=u^{j}_{n}(x)$, as well as on its derivatives with respect to $x$ and shifts
with respect to discrete variables $j$ and $n$. Here $D_{n}$ and $D_{j}$
denote operators for shifting a function argument in $n$ and $j$, for example,
$D_{n}y(n)=y(n+1),\quad D_{j}y(j)=y(j+1)$, and $D_{x}$ stands for the operator
of the total differentiation with respect to $x$. If the representation (2.1)
takes place, we define a new function $v=v^{j}_{n}(x)$ by introducing a
potential, i.e. assuming

$(D_{j}-1)v^{j}_{n}=A.$ (2.3)

Then substituting instead of the function $A$ its representation (2.3) into
relation (2.1) we arrive at the equality

$(D_{j}-1)(D_{n}-1)v^{j}_{n}=(D_{j}-1)B,$ (2.4)

from which, applying the operator inverse to $(D_{j}-1)$, we find another
important relation connecting the variables $u^{j}_{n}$ and $v^{j}_{n}$:

$(D_{n}-1)v^{j}_{n}=B+C.$ (2.5)

Since we do concentrate only on autonomous transformations we assume that $C$
is a constant. In the examples considered below, it is usually possible to
completely exclude the variable $u^{j}_{n}$ from the pair of relations (2.3),
(2.5). In such a case, we arrive at some equation for the new function
$v^{j}_{n}$.

The second way to look for the desired substitution uses representation (2.2).
Here we introduce a new function ${\omega}^{j}_{n}$ due to the relation:

$D_{x}{\omega}^{j}_{n}=B.$ (2.6)

Now the equation yields a relation

$D_{x}A=D_{x}\left(D_{j}-1\right){\omega}^{j}_{n}.$ (2.7)

Integrating the latter, we find, in addition to (2.6), the following
constraint that the variable ${\omega}^{j}_{n}$ should satisfy,

$\left(D_{j}-1\right){\omega}^{j}_{n}=A+\hbox{const}.$ (2.8)

To obtain the lattice equation for ${\omega}^{j}_{n}$ it is necessary to get
rid the variable $u^{j}_{n}$ by means of the relations (2.6) and (2.8).

As one can see from the examples below, this approach very often leads to
success. In particular, it is in this way that we managed to find Miura type
transformations for each of the equations from the list under consideration.

##  3 Searching for the transformations

In this section we apply the algorithm discussed above to the lattices 1)-9).
It is interesting to note that the set of equations falls into two non-
overlapping subclasses. Most of the equations belong to the first subclass,
namely 1), 3), 4), 7), 8), 9). Only three equations fall into the second
class: 2), 5), 6).

###  3.1 Connection between equations 1) and 9)

Obviously equation 1) can be rewritten as follows

$(D_{n}-1)u^{j}_{n,x}=(D_{j}-1)e^{u^{j-1}_{n+1}-u^{j}_{n}}.$ (3.1)

According to the rule above we introduce a new variable $v^{j}_{n}$ by setting

$u^{j}_{n,x}=v^{j+1}_{n}-v^{j}_{n}=(D_{j}-1)v^{j}_{n},$ (3.2)

then formulas (3.1), (3.2) imply

$(D_{j}-1)(D_{n}-1)v^{j}_{n}=(D_{j}-1)e^{u^{j-1}_{n+1}-u^{j}_{n}}$

or, the same

$(D_{n}-1)v^{j}_{n}=e^{u^{j-1}_{n+1}-u^{j}_{n}}+C.$ (3.3)

Let us differentiate (3.3) with respect to $x$ and replace derivatives
$u^{j-1}_{n+1,x}$, $u^{j}_{n,x}$ due to (3.2). As a result we arrive at the
equation

$v^{j}_{n+1,x}-v^{j}_{n,x}=(v^{j}_{n+1}-v^{j-1}_{n+1}-v^{j+1}_{n}+v^{j}_{n})(v^{j}_{n+1}-v^{j}_{n}-C),$
(3.4)

that is reduced to 9) by a point transformation $x=-\bar{x}$,
$v^{j}_{n}=u^{j}_{n}-Cn$.

###  3.2 Connection between equations 2) and 5)

Now we derive a substitution generated by the equation 5). To this end we
first simplify 5) by the change of variables $u^{j}_{n}=e^{v^{j}_{n}}$:

$v^{j}_{n+1,x}=v^{j}_{n,x}\frac{e^{v^{j}_{n+1}-v^{j+1}_{n}}-1}{e^{v^{j-1}_{n+1}-v^{j}_{n}}-1}.$

Then we represent the obtained equation in the form of the local conservation
law:

$(D_{n}-1)\ln{v^{j}_{n,x}}=(D_{j}-1)\ln\left(e^{v^{j-1}_{n+1}-v^{j}_{n}}-1\right).$
(3.5)

Let us introduce a new dependent variable ${\omega}^{j}_{n}$ as

$\left(D_{j}-1\right){\omega}^{j}_{n}=\ln v^{j}_{n,x}$ (3.6)

or

$v^{j}_{n,x}=e^{{\omega}^{j+1}_{n}-{\omega}^{j}_{n}}.$

Then from (3.5), (3.6) we obtain the equation

$\left(D_{j}-1\right)\left(D_{n}-1\right){\omega}^{j}_{n}=\left(D_{j}-1\right)\ln\left(e^{v^{j-1}_{n+1}-v^{j}_{n}}-1\right),$

which implies:

$\left(D_{n}-1\right){\omega}^{j}_{n}=\ln\left(e^{v^{j-1}_{n+1}-v^{j}_{n}}-1\right)+C.$
(3.7)

One can easily verify that the variable $v^{j}_{n}$ is completely eliminated
from formulas (3.6), (3.7) and their consequences in a rather simple way.
Indeed, by differentiating (3.7) we get

${\omega}^{j}_{n+1,x}-{\omega}^{j}_{n,x}=\frac{e^{v^{j-1}_{n+1}-v^{j}_{n}}\left(v^{j-1}_{n+1,x}-v^{j}_{n,x}\right)}{e^{v^{j+1}_{n+1}-v^{j}_{n}}-1}.$
(3.8)

The right hand side of the formula is greatly simplified due to equalities
(3.6), (3.7) and as a result the equation reduces to the form

${\omega}^{j}_{n+1,x}-{\omega}^{j}_{n,x}=\left(1+e^{{\omega}^{j}_{n}-{\omega}^{j}_{n+1}+C}\right)\left(e^{{\omega}^{j}_{n+1}-{\omega}^{j-1}_{n+1}}-e^{{\omega}^{j+1}_{n}-{\omega}^{j}_{n}}\right).$
(3.9)

Up to the linear transformation ${\omega}^{j}_{n}=-u^{j}_{n}+Cj$ equation
(3.9) coincides with equation 2).

###  3.3 Connection between equations 3) and 1)

Let us turn to equation 3). Taking the logarithm of both sides of the equation
we get an identity

$\ln u^{j}_{n+1,x}-\ln u^{j}_{n,x}=2\ln u^{j}_{n+1}-\ln u^{j-1}_{n+1}-\ln
u^{j+1}_{n},$ (3.10)

that implies a local conservation law of the form

$\left(D_{n}-1\right)\ln\frac{u^{j}_{n,x}}{u^{j}_{n}}=\left(D_{j}-1\right)\ln\frac{u^{j-1}_{n+1}}{u^{j}_{n}}.$
(3.11)

Having introduced a new variable $h^{j}_{n}$ by setting

$\left(D_{j}-1\right)h^{j}_{n}=\ln\frac{u^{j}_{n,x}}{u^{j}_{n}},$ (3.12)

we deduce from (3.11) the equality

$\left(D_{j}-1\right)\left(D_{n}-1\right)h^{j}_{n}=\left(D_{j}-1\right)\ln\frac{u^{j-1}_{n+1}}{u^{j}_{n}},$
(3.13)

that gives rise to

$\left(D_{n}-1\right)h^{j}_{n}=\ln\frac{u^{j-1}_{n+1}}{u^{j}_{n}}+C,$ (3.14)

where $C$ is assumed to be constant. Now we differentiate (3.14) respect to
$x$ and find

$h^{j}_{n+1,x}-h^{j}_{n,x}=\frac{u^{j-1}_{n+1,x}}{u^{j-1}_{n+1}}-\frac{u^{j}_{n,x}}{u^{j}_{n}},$

which is easily simplified due to (3.12). As a result we obtain the lattice

$h^{j}_{n+1,x}-h^{j}_{n,x}=e^{h^{j}_{n+1}-h^{j-1}_{n+1}}-e^{h^{j+1}_{n}-h^{j}_{n}}.$

The latter coincides with the discrete Toda lattice 1) up to the linear change
of the independent discrete variables $j=-\bar{j},n=\bar{j}+\bar{n}$.

###  3.4 Connection between equations 3) and 7)

We note that for equation 3) one can derive one more substitution. To this end
we use the equation (3.10) rewritten as

$\left(D_{n}-1\right)\ln{u^{j}_{n,x}}=\left(2D_{n}-D_{n}D^{-1}_{j}-D_{j}\right)\ln
u^{j}_{n}.$ (3.15)

Assuming that $\ln u^{j}_{n}=\left(D_{n}-1\right)v^{j}_{n}$, we obtain

$u^{j}_{n}=e^{v^{j}_{n+1}-v^{j}_{n}}.$ (3.16)

Then combining (3.15) and (3.16) we arrive at

$u^{j}_{n,x}=e^{2v^{j}_{n+1}-v^{j-1}_{n+1}-v^{j+1}_{n}}$ (3.17)

and by differentiating (3.16) we find a similar relation

$u^{j}_{n,x}=\left(v^{j}_{n+1,x}-v^{j}_{n,x}\right)e^{v^{j}_{n+1}-v^{j}_{n}}.$
(3.18)

Comparing the last two representations for $u^{j}_{n,x}$ one can get the
equation

$v^{j}_{n+1,x}-v^{j}_{n,x}=e^{-v^{j+1}_{n}+v^{j}_{n}+v^{j}_{n+1}-v^{j-1}_{n+1}},$
(3.19)

which is nothing but equation 7).

###  3.5 Connection between equations 4) and 1)

Now we study equation 4) in the list. It can easily be represented as a local
concervation law of the form

$\left(D_{n}-1\right)\ln
u^{j}_{n,x}=\left(D_{j}-1\right)\ln\left(u^{j-1}_{n+1}-u^{j}_{n}\right).$
(3.20)

We introduce a new unknown $s^{j}_{n}$ by taking

$\left(D_{j}-1\right)\ln s^{j}_{n}=\ln u^{j}_{n,x}.$ (3.21)

Then from (3.20) it follows that

$\left(D_{j}-1\right)\left(D_{n}-1\right)\ln
s^{j}_{n}=\left(D_{j}-1\right)\ln\left(u^{j-1}_{n+1}-u^{j}_{n}\right).$

By applying to the latter equation the operator $\left(D_{j}-1\right)^{-1}$ we
obtain the second part of the searched substitution:

$\left(D_{n}-1\right)\ln s^{j}_{n}=\ln\left(u^{j-1}_{n+1}-u^{j}_{n}\right)+\ln
C,\quad C=\hbox{const}.$ (3.22)

Let’s simplify (3.22) a little, after which, applying the operator $D_{x}$ to
both its sides, we obtain a nonlinear lattice

$\frac{s^{j}_{n+1,x}}{s^{j}_{n}}-\frac{s^{j}_{n+1}s^{j}_{n,x}}{\left(s^{j}_{n}\right)^{2}}=C\left(\frac{s^{j}_{n+1}}{s^{j-1}_{n+1}}-\frac{s^{j+1}_{n}}{s^{j}_{n}}\right),$

that is reduced to the discrete Toda lattice 1) by the point transformation
$s^{j}_{n}=e^{-u^{j}_{n}},\quad x\mapsto{Cx}$.

###  3.6 Connection between equations 6) and 2)

Let us look for the Miura type transformations generated by equation 6). By
taking logarithm of both sides of 6) we get a relation

$\left(D_{n}-1\right)\ln
u^{j}_{n,x}=\left(D_{n}-D_{j}\right)\ln\left(u^{j-1}_{n}-u^{j}_{n}\right)+\left(D_{j}-1\right)\ln\left(u^{j-1}_{n+1}-u^{j}_{n}\right).$

Afterwards by using the decomposition
$D_{n}-D_{j}=D_{n}-1-\left(D_{j}-1\right)$ one can rewrite the relation as a
local conservation law of the form:

$\left(D_{n}-1\right)\ln\frac{u^{j}_{n,x}}{u^{j-1}_{n}-u^{j}_{n}}=\left(D_{j}-1\right)\ln\frac{u^{j-1}_{n+1}-u^{j}_{n}}{u^{j-1}_{n}-u^{j}_{n}}.$
(3.23)

Then we introduce a new variable $h^{j}_{n}$ by the rule

$\left(D_{j}-1\right)\ln
h^{j}_{n}=\ln\frac{u^{j}_{n,x}}{u^{j-1}_{n}-u^{j}_{n}}.$ (3.24)

Applying the reasoning above to (3.23), (3.24) one can derive:

$\left(D_{n}-1\right)\ln
h^{j}_{n}=\ln\frac{u^{j-1}_{n+1}-u^{j}_{n}}{u^{j-1}_{n}-u^{j}_{n}}+\ln C.$
(3.25)

Let us rewrite Miura transformation (3.24), (3.25) in a more convenient form

$\frac{u^{j}_{n,x}}{u^{j-1}_{n}-u^{j}_{n}}=\frac{h^{j+1}_{n}}{h^{j}_{n}},\qquad
C\frac{u^{j-1}_{n+1}-u^{j}_{n}}{u^{j-1}_{n}-u^{j}_{n}}=\frac{h^{j}_{n+1}}{h^{j}_{n}},$
(3.26)

which implies that

$\frac{1}{C}\frac{u^{j}_{n,x}}{u^{j-1}_{n+1}-u^{j}_{n}}=\frac{h^{j+1}_{n}}{h^{j}_{n+1}}.$
(3.27)

Differentiating (3.25) we find a relation

$\frac{h^{j}_{n+1,x}}{h^{j}_{n+1}}-\frac{h^{j}_{n,x}}{h^{j}_{n}}=\frac{u^{j-1}_{n+1,x}-u^{j}_{n,x}}{u^{j-1}_{n+1}-u^{j}_{n}}-\frac{u^{j-1}_{n,x}-u^{j}_{n,x}}{u^{j-1}_{n}-u^{j}_{n}}=\frac{u^{j-1}_{n+1,x}}{u^{j-1}_{n+1}-u^{j}_{n}}-C\frac{h^{j+1}_{n}}{h^{j}_{n+1}}-\frac{u^{j-1}_{n,x}}{u^{j-1}_{n}-u^{j}_{n}}+\frac{h^{j+1}_{n}}{h^{j}_{n}}.$
(3.28)

We simplify some of the terms in (3.28) due to the lattice 6) and get:

$\displaystyle\frac{u^{j-1}_{n+1,x}}{u^{j-1}_{n+1}-u^{j}_{n}}-\frac{u^{j-1}_{n,x}}{u^{j-1}_{n}-u^{j}_{n}}$
$\displaystyle=\frac{u^{j-1}_{n,x}}{u^{j-1}_{n}-u^{j}_{n}}\left(\frac{u^{j-2}_{n+1}-u^{j-1}_{n+1}}{u^{j-2}_{n+1}-u^{j-1}_{n}}-1\right)=$
$\displaystyle=C\frac{h^{j}_{n}}{h^{j-1}_{n+1}}\frac{u^{j-1}_{n}-u^{j-1}_{n+1}}{u^{j-1}_{n}-u^{j}_{n}}=C\frac{h^{j}_{n}}{h^{j-1}_{n+1}}\left(1-\frac{u^{j-1}_{n+1}-u^{j}_{n}}{u^{j-1}_{n}-u^{j}_{n}}\right)=C\frac{h^{j}_{n}}{h^{j-1}_{n+1}}\left(1-\frac{1}{C}\frac{h^{j}_{n+1}}{h^{j}_{n}}\right).$

Finally we obtain a lattice

$\frac{h^{j}_{n+1,x}}{h^{j}_{n+1}}-\frac{h^{j}_{n,x}}{h^{j}_{n}}=\frac{h^{j+1}_{n}}{h^{j}_{n}}-C\frac{h^{j+1}_{n}}{h^{j}_{n+1}}+C\frac{h^{j}_{n}}{h^{j-1}_{n+1}}-\frac{h^{j}_{n+1}}{h^{j-1}_{n+1}},$

which is related to the lattice 2) by a point transformation
$h^{j}_{n}=C^{n}e^{-p^{j}_{n}}$:

$p^{j}_{n+1,x}-p^{j}_{n,x}=e^{p^{j-1}_{n+1}-p^{j}_{n+1}}-e^{p^{j-1}_{n+1}-p^{j}_{n}}-e^{p^{j}_{n}-p^{j+1}_{n}}+e^{p^{j}_{n+1}-p^{j+1}_{n}}.$

###  3.7 Connection between equations 8) and 1)

Let us look for a substitution that would connect lattices 8) and 1). At first
we rewrite equation 8) in the form of a local conservation law:

$\left(D_{n}-1\right)u^{j}_{n,x}=\left(D_{j}-1\right)\left(e^{u^{j-1}_{n+1}}-e^{u^{j}_{n}}\right).$
(3.29)

Supposing that

$u^{j}_{n,x}=\left(D_{j}-1\right)p^{j}_{n,x},$ (3.30)

we introduce a new dependent variable $p^{j}_{n}$. Then obviously it follows
from (3.29), (3.30) that

$\left(D_{j}-1\right)\left(D_{n}-1\right)p^{j}_{n,x}=\left(D_{j}-1\right)\left(e^{u^{j-1}_{n+1}}-e^{u^{j}_{n}}\right).$
(3.31)

From the latter one can derive the second part of the Miura type
transformation

$\left(D_{n}-1\right)p^{j}_{n,x}=e^{u^{j-1}_{n+1}}-e^{u^{j}_{n}}+C_{1}.$
(3.32)

The expression $u^{j}_{n}=\left(D_{j}-1\right)p^{j}_{n}+C_{2}$ is an evident
consequence of (3.30), therefore we have

$e^{u^{j}_{n}}=e^{C_{2}}e^{p^{j+1}_{n}-p^{j}_{n}}.$ (3.33)

Let us eliminate the variables $u^{j}_{n},u^{j-1}_{n+1}$ from (3.32) due to
(3.33) and obtain a lattice of the form:

$p^{j}_{n+1,x}-p^{j}_{n,x}=e^{C_{2}}\left(e^{p^{j}_{n+1}-p^{j-1}_{n+1}}-e^{p^{j+1}_{n}-p^{j}_{n}}\right)+C_{1}.$

Changing the variables by setting $p^{j}_{n}=q^{j}_{n}+C_{1}xn$ we arrive at
the lattice

$q^{j}_{n+1,x}-q^{j}_{n,x}=e^{C_{2}}\left(e^{q^{j}_{n+1}-q^{j-1}_{n+1}}-e^{q^{j+1}_{n}-q^{j}_{n}}\right),$

that converts to the discrete Toda lattice 1) under replacement
$j=-\bar{j},n=\bar{j}+\bar{n},x=e^{-C_{2}}\bar{x}$.

##  4 On a novel integrable chain of the form (1.1)

In this section, we focus on a detailed study of the found chain 9): we
represent the Lax pair, calculate the continuum limit, and study Darboux-
integrable reductions. For reduced systems of small dimensions, characteristic
integrals are found in both directions $n$ and $x$. We also briefly discuss on
the method of constructing particular solutions of the chain 9) by means of
its reduction.

Let us begin with the Lax pair. It is checked straightforwardly that system of
linear equations

$\displaystyle{\psi}^{j}_{n+1}=\left(v^{j}_{n}-v^{j}_{n+1}\right){\psi}^{j+1}_{n}+{\psi}^{j}_{n},$
(4.1)
$\displaystyle{\psi}^{j}_{n,x}=\left(v^{j}_{n}-v^{j-1}_{n}\right){\psi}^{j}_{n}-{\psi}^{j-1}_{n}$

is compatible if and only if function $v=v_{n}^{j}(x)$ is a solution to the
lattice 9).

###  4.1 Computation of the continuum limit for the lattice 9)

In this section, we will calculate the continuum limit of the chain 9) under
unbounded refinement of the difference grid in both directions $n$ and $j$. At
first we perform a scaling transformation $y=\delta^{2}x$ in the chain (3.4),
and represent it as follows:

$\delta^{2}\frac{d}{dy}\left(v^{j}_{n+1}-v^{j}_{n}\right)=\left(v^{j}_{n+1}-v^{j}_{n}\right)\left[\left(v^{j+1}_{n}-2v^{j}_{n}+v^{j-1}_{n}\right)+\left(v^{j-1}_{n+1}-v^{j-1}_{n}\right)-\left(v^{j}_{n+1}-v^{j}_{n}\right)\right].$
(4.2)

Let us assume that the sought function $v^{j}_{n}(x)$ is of the form
$v^{j}_{n}(x)=u(t,z,y)$, where $t=\delta^{2}n$, $z={\delta}j$ and $\delta$ is
a small positive parameter. Now we estimate the difference derivatives with
respect to the variables $n$ and $j$ appearing in (4.2) by virtue of the
Taylor formula

$v^{j}_{n+1}-v^{j}_{n}=u(t+\delta^{2},z,y)-u(t,z,y)=\delta^{2}u_{t}(t,z,y)+O(\delta^{4}),\qquad\delta\mapsto
0,$ (4.3) $\displaystyle v^{j-1}_{n+1}-v^{j-1}_{n}$
$\displaystyle=u(t+\delta^{2},z-\delta,y)-u(t,z-\delta,y)=$
$\displaystyle=u(t,z,y)+\delta^{2}u_{t}(t,z,y)-{\delta}u_{z}(t,z,y)+\frac{1}{2}\delta^{2}u_{zz}(t,z,y)-$
$\displaystyle-u(t,z,y)+\delta
u_{z}(t,z,y)-\frac{1}{2}\delta^{2}u_{zz}(t,z,y)+...+O(\delta^{3})=$
$\displaystyle=\delta^{2}u_{t}(t,z,y)+O(\delta^{3}),\qquad\delta\mapsto 0,$
$\displaystyle v^{j+1}_{n}-2v^{j}_{n}+v^{j-1}_{n}$
$\displaystyle=u(t,z+\delta,y)-2u(t,z,y)+u(t,z-\delta,y)=$
$\displaystyle=u(t,z,y)+{\delta}u_{z}(t,z,y)+\frac{1}{2}{\delta}^{2}u_{zz}(t,z,y)-2u(t,z,y)+$
$\displaystyle+u(t,z,y)-{\delta}u_{z}(t,z,y)+\frac{1}{2}{\delta}^{2}u_{zz}(z,t,y)+...=$
$\displaystyle={\delta}^{2}u_{zz}(t,z,y)+O(\delta^{3}),\qquad\delta\mapsto 0.$

Now we substitute the found asymptotic representations of finite differences
into the equation (4.2) and pass to the limit at $\delta\mapsto 0$. As a
result, we obtain the following dispersionless partial differential equation:

$u_{ty}=u_{t}u_{zz},$ (4.4)

which is integrable as well [16].

###  4.2 Intermediate continuum limit

Now we study the continuum limit of chain 9) assuming that sought function
$u^{j}_{n}(x)$ depends on a small parameter $\varepsilon>0$ as follows
$u^{j}_{n}(x)={\omega}^{j}(x,t)$ with $t=\varepsilon{n}$. Under the condition
$\varepsilon\mapsto 0$ lattice 9) tends to an equation of the Toda chain type
[7]

$\omega^{j}_{xt}+\omega^{j}_{t}\left(\omega^{j+1}-2\omega^{j}+\omega^{j-1}\right)=0$
(4.5)

having the Lax pair of the form

$\displaystyle\begin{aligned}
&\varphi^{j}_{t}=-\omega^{j}_{t}\varphi^{j+1}+\varphi^{j},\\\
&\varphi^{j}_{x}=\left(\omega^{j}-\omega^{j-1}\right)\varphi^{j}-\varphi^{j-1}\end{aligned}$
(4.6)

that is easily obtained from (4.1) under the relation
$\psi^{j}_{n}(x)=\varphi^{j}(x,t),$ when $\varepsilon\mapsto 0$. A remarkable
fact is that chains 9) and (4.5) have a close connection. Indeed, by comparing
the Lax pairs one can conclude that chain 9) realizes a sequence of the
Bäcklund transformations for lattice (4.5). More precisely any solution
$\omega^{j}$ of (4.5) is converted by the Bäcklund transformation

${\bar{\omega}}^{j}_{x}-\omega^{j}_{x}=\left({\bar{\omega}}^{j}-\omega^{j}\right)\left(-\omega^{j+1}+\omega^{j}+{\bar{\omega}}^{j}-{\bar{\omega}}^{j-1}\right)$
(4.7)

to another solution ${\bar{\omega}}^{j}$ of the lattice. We note that
eigenfunctions $\varphi^{j}$ and ${\bar{\varphi}}^{j}$ of the system (4.6) and
respectively of the system

${\bar{\varphi}}^{j}_{t}=-{\bar{\omega}}^{j}_{t}{\bar{\varphi}}^{j+1}+{\bar{\varphi}}^{j},\quad{\bar{\varphi}}^{j}_{x}=\left({\bar{\omega}}^{j}-{\bar{\omega}}^{j-1}\right){\bar{\varphi}}^{j}-{\bar{\varphi}}^{j-1}$

are connected with one another as follows

${\bar{\varphi}}^{j}=\left(\omega^{j}-{\bar{\omega}}^{j}\right)\varphi^{j+1}+\varphi^{j}.$
(4.8)

###  4.3 Darboux integrable reductions of the lattice 9)

Chain 9) admits the following cutting off boundary condition

${v^{j}_{n}(x)|}_{j=j_{0}}=C,\quad C=\hbox{const}$ (4.9)

preserving the integrability property. Imposing this kind of the conditions on
two points $j=0$ and $j=N$ we obtain a finite field reduction of lattice 9):

$\displaystyle\begin{aligned} &v^{-1}_{n}=C_{0},\\\
&v^{j}_{n+1,x}-v^{j}_{n,x}=\left(v^{j}_{n+1}-v^{j}_{n}\right)\left(-v^{j+1}_{n}+v^{j}_{n}+v^{j}_{n+1}-v^{j-1}_{n+1}\right),\qquad
0\leq{j}\leq{N-1},\\\ &v^{N}_{n}=C_{1},\end{aligned}$ (4.10)

admitting the Lax representation of the form

$\displaystyle\begin{cases}{\psi}^{j}_{n+1}=\left(v^{j}_{n}-v^{j}_{n+1}\right){\psi}^{j+1}_{n}+{\psi}^{j}_{n},\qquad{0}\leq{j}\leq{N-1},\\\
{\psi^{N}_{n+1}}={\psi}^{N}_{n},\end{cases}$ (4.11)
$\displaystyle\begin{cases}{\psi}^{0}_{n,x}=\left(v^{0}_{n}-C_{0}\right){\psi}^{0}_{n},\\\
{\psi}^{j}_{n,x}=\left(v^{j}_{n}-v^{j-1}_{n}\right){\psi}^{j}_{n}-{\psi}^{j-1}_{n},\qquad{1}\leq{j}\leq{N}\end{cases}$
(4.12)

deduced from (4.1) under the constraints ${\psi}^{-1}_{n}={\psi}^{N+1}_{n}=0$,
$v^{-1}_{n}=C_{0}$, $v^{N}_{n}=C_{1}$. Let us concentrate on the finite field
system (4.10). It is worth noting that (4.10) admits complete sets of
integrals in both directions $n$ and $x$. Recall that a function $I$ depending
of the dynamical variables is an $n$-integral of the differential-difference
system if it satisfies the relation $D_{n}I=I$ by virtue of the system.
Integrals of the form $I=I(x)$ and $J=J(n)$ depending on $x$ and $n$ only are
called trivial. We are only interested in non-trivial integrals. Let us give
examples of integrals.

Example 1. By setting $N=1$ in (4.10) we obtain a scalar equation for
$v:=v^{0}$

$v_{n+1,x}-v_{n,x}=\left(v_{n+1}-v_{n}\right)\left(v_{n+1}+v_{n}-C_{0}-C_{1}\right),$
(4.13)

which admits $n$-integral

$I=v_{x}-(v-C_{0})^{2}-(v-C_{1})(C_{0}-C_{1})$ (4.14)

and $x$-integral

$J=\frac{\left(v_{n}-v_{n-2}\right)\left(v_{n+1}-v_{n+3}\right)}{\left(v_{n}-v_{n+3}\right)\left(v_{n+1}-v_{n+2}\right)}.$
(4.15)

Example 2. For the choice $N=2$ we arrive at the system for $u_{n}=v^{0}_{n}$,
$v_{n}=v^{1}_{n}$

$\displaystyle\begin{aligned}
&u_{n+1,x}-u_{n,x}=\left(u_{n+1}-u_{n}\right)\left(-v_{n}+u_{n}+u_{n+1}-C_{0}\right),\\\
&v_{n+1,x}-v_{n,x}=\left(v_{n+1}-v_{n}\right)\left(-C_{1}+v_{n}+v_{n+1}-u_{n+1}\right).\end{aligned}$
(4.16)

The obtained system admits a complete set of $n$-integrals, i.e. a set of two
independent $n$-integrals

$\displaystyle
I_{1}=u_{n,x}+v_{n,x}-\left(u_{n}-C_{0}\right)^{2}-\left(u_{n}-C_{1}\right)\left(C_{0}-v_{n}\right)-\left(C_{1}-v_{n}\right)^{2},$
$\displaystyle
I_{2}=v_{n,xx}-2\left(v_{n}-C_{1}\right)v_{n,x}-\left(C_{0}-v_{n}\right)u_{n,x}-\left(v_{n}-C_{0}\right)\left(u_{n}-C_{1}\right)\left(u_{n}-v_{n}-C_{0}+C_{1}\right)$

and a complete set of $x$-integrals

$\displaystyle
J_{1}=\frac{\left(\tau^{0}_{n}\tau^{1}_{n}+\tau^{0}_{n+1}\tau^{1}_{n+1}+\tau^{0}_{n}\tau^{1}_{n+1}\right)(\tau^{0}_{n+1}\tau^{1}_{n+1}+\tau^{0}_{n+2}\tau^{1}_{n+2}+\tau^{0}_{n+1}\tau^{1}_{n+2})}{\tau^{0}_{n}\tau^{1}_{n+2}\left(\tau^{0}_{n+1}\tau^{1}_{n}+\tau^{0}_{n+2}\tau^{1}_{n+1}+\tau^{0}_{n+2}\tau^{1}_{n}\right)},$
$\displaystyle
J_{2}=\frac{\tau^{0}_{n+3}\tau^{1}_{n}\left(\tau^{0}_{n+1}\tau^{1}_{n+1}+\tau^{0}_{n+2}\tau^{1}_{n+2}+\tau^{0}_{n+1}\tau^{1}_{n+2}\right)}{\tau^{0}_{n+2}\tau^{1}_{n+1}\left(\tau^{0}_{n+1}\tau^{1}_{n}+\tau^{0}_{n+2}\tau^{1}_{n+1}+\tau^{0}_{n+2}\tau^{1}_{n}+\tau^{0}_{n+3}\tau^{1}_{n+2}+\tau^{0}_{n+3}\tau^{1}_{n+1}+\tau^{0}_{n+3}\tau^{1}_{n}\right)},$

where $\tau^{0}_{n}=u_{n+1}-u_{n}$, $\tau^{1}_{n}=v_{n+1}-v_{n}$ (about
complete set of integrals see, for instance, [22], [23]).

It can be proved that system (4.10) admits complete sets of $x$-integrals and
$n$-integrals for any choice of $N$. For $N\geq 3$ $x$-integrals have
extremely large expression. They are found by the method of characteristic
Lie-Rinehart algebras (see [19]). As can be seen from the above examples, the
$n$-integrals of the reduced system are much simpler. Note that the complete
set of $n$-integrals to the system (4.10) for an arbitrary natural $N\geq 1$
can be efficiently derived using the algorithm proposed in [19, Lemma 2]. An
alternative approach to the problem of looking for integrals is proposed in
[21].

###  4.4 Explicit solutions to the chain 9)

The presence of complete sets of integrals in both characteristic directions
$x$ and $n$ makes it possible to completely separate the variables. Indeed,
according to the definition, each solution $v=(v^{0},v^{1},\dots,v^{N-1})$ of
system (4.10) simultaneously satisfies two systems of equations, namely the
system of ordinary differential equations

$I_{1}={\varphi}_{1}(x),\quad I_{2}={\varphi}_{2}(x),\quad...,\quad
I_{N}={\varphi}_{N}(x)$ (4.17)

and the system of ordinary discrete equations

$J_{1}={\psi}_{1}(n),\quad J_{2}={\psi}_{2}(n),\quad...,\quad
J_{N}={\psi}_{N}(n),$ (4.18)

where ${\varphi}_{1}(x),{\varphi}_{2}(x),...,{\varphi}_{N}(x)$ are arbitrary
functions of $x$ and similarly ${\psi}_{1}(n),{\psi}_{2}(n),...,{\psi}_{N}(n)$
are arbitrary functions of the variable $n$.

In this case common solution $\left\\{v^{j}_{n}(x)\right\\}^{N-1}_{j=0}$ of
the systems provides a solution to the system (4.10). We remark that solution
of the reduced system (4.10) obviously can be prolonged to a particular
solution of the original lattice 9). However for arbitrary $N$ system (4.17)
is rather difficult to solve. Here we discuss only a comparatively simple case
$N=1$. Let us derive a formula describing general solution to the lattice
(4.13). Due to the formula $DI=I$ one can conclude that $I={\varphi}(x)$ i.e.
function $v_{n}(x)$ is a solution to the equation

$v_{n,x}-\left(v_{n}-C_{0}\right)^{2}-{\varepsilon}\left(v_{n}-C_{1}\right)={\varphi}(x),\quad\varepsilon=C_{0}-C_{1}.$
(4.19)

Since $\varphi(x)$ is arbitrary we can assume that $\varphi(x)$ is represented
as

$\varphi(x)=z_{x}-(z-C_{0})^{2}-\varepsilon(z-C_{1})$

for some function $z=z(x)$ that now is considered as a new functional
parameters. Let us change the variables in (4.19) by taking $v_{n}=z+s$ and
get a new equation with unknown $s$:

$s_{x}-s^{2}-s\left(\varepsilon+2\left(z-C_{0}\right)\right)=0,$ (4.20)

which has a form of the Bernoulli equation. Let us replace $s=\frac{1}{g}$ and
arrive at a linear equation

$g_{x}=-g\left(\varepsilon+2\left(z-C_{0}\right)\right)-1.$ (4.21)

We first solve the corresponding homogeneous equation

$g_{x}=-g\left(\varepsilon+2\left(z-C_{0}\right)\right).$

We pass from the functional parameter $z=z(x)$ to a new parameter $p(x)$ in
such a way:

$p_{x}=\varepsilon+2(z-C_{0}).$

Then the homogeneous equation

$g_{x}=-gp^{\prime}$

is easily integrated $g=Ce^{-p}$. Now supposing $C=C(x)$ we return to the
equation (4.21) and find $C_{x}=-e^{p}$. We change the parameter in the
equation again by setting $e^{p}={\omega}_{x}$, where $\omega$ is a new
parameter. As a result we get $C(x)=-\omega+K(n)$, where $K(n)$ is a constant
of integration that might depend on $n$. Therefore we have

$g=\frac{K(n)-\omega}{{\omega}_{x}}\quad\mbox{so that}\quad
s=\frac{{\omega}_{x}}{K(n)-\omega}.$

To write down the final form of $v_{n}(x)$ we need in expression of $z$ in
terms of $\omega$. To this end we exclude parameter $p(x)$ from the relations

$z=C_{0}+\frac{1}{2}(p_{x}-\varepsilon),\quad e^{p}={\omega}_{x}$

and find

$z=\frac{1}{2}\frac{{\omega}_{xx}}{{\omega}_{x}}+\frac{C_{0}+C_{1}}{2}.$

Now we substitute the found expressions into $v_{n}=s+z$ and obtain general
solution to the lattice (4.13):

$v_{n}(x)=\frac{1}{2}\frac{{\omega}_{xx}}{{\omega}_{x}}+\frac{{\omega}_{x}}{K(n)-\omega}+\frac{C_{0}+C_{1}}{2},$
(4.22)

where $\omega=\omega(x)$ is an arbitrary function of $x$, $K(n)$ is an
arbitrary function of $n$. Due to the fact that the termination conditions
$v^{-1}_{n}=C_{0}$, $v^{1}_{n}=C_{1}$ are fully consistent with dynamics, by
virtue of the original three-dimensional chain 9) the solution of the reduced
system (4.13) extends in an obvious way to the solution of the chain 9):

$\displaystyle v^{j}_{n}(x)=\begin{cases}C_{0}&\mbox{for}\quad j\leq-1,\\\
\displaystyle{\frac{{\omega}_{xx}}{2{\omega}_{x}}+\frac{{\omega}_{x}}{K(n)-\omega}+\frac{C_{0}-C_{1}}{2}}&\mbox{for}\quad
j=0,\\\ C_{1}&\mbox{for}\quad j\geq 1.\end{cases}$

## Conclusions

In the article integrable chains with three independent variables $x$, $n$,
$j$ are considered. A list of currently known examples is given. Differential
and difference substitutions linking equations of the list with each other are
presented. A new integrable chain is found that is investigated in more
detail. For this model the Lax pair is presented, the continuum limits are
computed. Degenerate cutoff condition is pointed out which reduces the chain
to a Darboux integrable system of differential-difference equations when it is
imposed on two different points $(x,n,j=0)$ and $(x,n,j=N)$. For the
reductions of small orders characteristic integrals are found. A method for
constructing explicit particular solutions to the chain via reductions is
proposed.

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