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Tech giant Google is planning to shut down its popular messaging app Hangouts by next year.
NEW DELHI: While it might not be as popular as some of the other services from Google, Hangouts is still widely used. So this piece of news might come as a shock to those who use the service. It has been reported that Google plans to shut down Hangouts app for consumers sometime in 2020.
As per a report from 9to5Google, the Alphabet-owned company had stopped the development of the app about a year ago which means that next year – 2019 – will be the last of its existence.
While it's not clear why Hangouts will be shut down, it has been hinted by Google that it wants users to shift to its RCS chat feature within Android messages. RCS stands for Rich Communication Services and Google has been keen on making it the default service for its messages across platforms.
The chat feature interestingly hasn't been launched yet and is expected to see the light of the day sometime in 2019. Once that app is launched, it is being suggested that Hangouts for individuals will cease to exist. The details, however, are still unclear and Google hasn't confirmed anything about either of the apps.
Hangouts came into existence sometime in 2013 as a replacement to GChat app. Google has made a lot of alterations to the Hangouts app since its launch. For instance, in 2017 it announced the addition of Hangouts Meet for enterprises. Meet is a video chat platform within Hangouts which is mainly used by organisations.
Since Meet was launched sometime last year, it's clear that Google won't kill that off, which means Hangouts might still exist but only for organisations. For individual consumers, chances are that the app will cease to exist.
Earlier this year, Google had announced that it is "pausing" investment and development in another of its messaging app called Allo. The app still exists and hasn't been killed completely. Maybe Hangouts too will have a similar fate but we will have wait and see what Google has in store for the app.
Messaging is much better than Hangout.


text:
Walentin Wasilew Bożiczkow (bułg. Валентин Василев Божичков, ur. 2 maja 1958 w Samokowie) – bułgarski skoczek narciarski, reprezentant klubu Army Club Septemwirsko Znamje, uczestnik Zimowych Igrzysk Olimpijskich 1984.

Kariera
Bożiczkow wziął udział w konkursie indywidualnym w skokach narciarskich na igrzyskach olimpijskich w Sarajewie. Oddał skoki na 79,5 oraz 76 metrów, dzięki czemu z notą 170,8 punktu został sklasyfikowany na 37. miejscu.

Startował także w zawodach Pucharu Świata, jednak nigdy nie zdobył punktów, a jego najlepszymi miejscami były 47. pozycje w konkursach w Oberstdorfie i Garmisch-Partenkirchen w sezonie 1982/1983. Dwukrotnie brał udział w mistrzostwach świata w narciarstwie klasycznym. W 1985 roku w Seefeld in Tirol zajął 54. i 44. miejsce, a w 1987 w Oberstdorfie był 36. i 61. Startował na mistrzostwach świata w lotach narciarskich w 1983, gdzie zajął 31. miejsce (najlepsze w swojej karierze).

Zwyciężył w Turnieju Schwarzwaldzkim w 1985.

Przypisy

Bibliografia

Urodzeni w 1958
Bułgarscy skoczkowie narciarscy
Bułgarscy olimpijczycy
Ludzie urodzeni w Samokowie

redpajama_set_name: RedPajamaWikipedia


text:
\section{Introduction}
The study of discrete nonlinear Schr\"odinger equation is extremely important since the seventies. The equation appeared for the first time in the biophysical context \cite{davy} and
since then the topic developed rapidly in nonlinear optics, Bose-Einstein condensates, etc. Initially, it was used for dynamics of wave guide arrays \cite{chr} and then in the study of
discrete diffraction, Peierls barriers, dispersion management, etc. \cite{eis}. It was shown afterwards that nonlinear localized waves in optically-induced lattices and photo refractive
media are accurately described by discrete nonlinear Schr\"odinger equation (NLS).
However, these phenomena are more or less described by the non-integrable variant of discrete NLS. But quite recently it was shown that motion of discrete curves
(modelling polymer's motions) can be modelled by the completely integrable variant of discrete NLS, namely Ablowitz-Ladik \cite{doliwa} and Hirota-Tsujimoto \cite{kaji} equations.
Integrable coupled semi-discrete and discrete equations represent a topic which is still not well developed.  They have a very interesting phenomenology because of the
extra-degrees of freedom coming from the matrix structure. Coupled soliton equations have been proposed by several authors \cite{yon,ohta,iwao,svin}. Integrable
discretization of such systems is in general quite complicated. The Hirota bilinear formalism \cite{hirbook} turns out to be very effective inasmuch as the
bilinear form can be discretized straightforwardly and multisoliton solution or B\"acklund transformations can be obtained directly.

In this paper we are discussing a generalisation of the Ablowitz-Ladik model \cite{AL} to the multicomponent (matrix) case, but with many branches of
dispersion relation (being different from the well known discrete Manakov system \cite{barbara}). We will show complete integrability of this system using
Hirota bilinear formalism and displaying the Lax pair. The main feature of such system is the structure of dispersion relation (having multiple branches)
and phases of the components parametrised by the order $N$ roots of unity. The existence of many branches of the dispersion relation allows more freedom in
soliton interaction. We do the analysis directly on the system using Hirota bilinear formalism and we provide also the Lax pairs proving thus the complete
integrability. Furthermore we will show how starting from a {\it diagonal} Ablowitz-Ladik (having a diagonal one-dimensional evolution in two discrete independent
variables) one can obtain all the results, in a very simple way, using a periodic reduction on the second discrete independent variable. The paper is organised
as follows: in the first and second chapter we discuss the soliton solutions and Lax pairs for two and three component system and in the third one we discuss the general case and the periodic reduction.

\section{Coupled Ablowitz-Ladik equations}
The general differential-difference Ablowitz-Ladik system with branched dispersion is:

\begin{equation}\label{sys}
i\frac{d}{dt}Q_n=(1+Q_n Q_n^{*})(E_{\sigma_1}Q_{n+1}E_{\sigma_2}+E_{\sigma_2}Q_{n-1}E_{\sigma_1}),
\end{equation}
where $Q(n,t)$ is a diagonal matrix of complex functions $q_{i}(n,t)$ given by:
$$
Q_n=\left(
\begin{array}{ccccccc}
q_1(n,t)&0&0&......&0\\
0&q_2(n,t)&0&......&0\\
0&0&q_3(n,t)&......&0\\
....&....&....&.....&......&\\
0&0&0&......&q_N(n,t)\\
\end{array}\right)
$$
and $E_{\sigma_i}$  are permutation matrices corresponding to the following permutations:
$$
\sigma_1=\left(
\begin{array}{ccccccc}
1&2&.&.&.&.&N\\
2&3&.&.&.&.&1\\
\end{array}\right),\quad
\sigma_2=\left(
\begin{array}{ccccccc}
1&2&.&.&.&.&N\\
N&1&2&.&.&.&N-1\\
\end{array}\right).
$$
On the components, system (\ref{sys}) has the following expression:
\begin{eqnarray}\label{sys1}
i\dot q_1&=&(1+|q_1|^2)(\overline{q_2}+\underline{q_N})\nonumber \\
i\dot q_2&=&(1+|q_2|^2)(\overline{q_3}+\underline{q_1})\nonumber \\
&.....&\nonumber \\
i\dot q_{N-1}&=&(1+|q_{N-1}|^2)(\overline{q_N}+\underline{q_{N-2}})\\
i\dot q_N&=&(1+|q_N|^2)(\overline{q_1}+\underline{q_{N-1}})\nonumber
\end{eqnarray}
where $q_{\mu}=q_{\mu}(n,t)$ and $\overline{q_{\mu}}=q_{\mu}(n+1,t), \underline{q_{\mu}}=q_{\mu}(n-1,t)$ for any $\mu=1,...,N$.

\vspace{0.2cm}
For $N=1$, system (\ref{sys1}) reduces to the classical Ablowitz-Ladik (AL) equation:
\begin{eqnarray}\label{1 system-NLS}
i \dot q_1&=&(1+|q_1|^2)(\overline {q_{1}}+\underline {q_{1}}),
\end{eqnarray}
for which the Lax pair and the multisoliton solutions are known  \cite{AL}.

For $N=2$, we get the following system:
\begin{eqnarray}\label{2 system-NLS}
i \dot q_1&=&(1+|q_1|^2)(\overline {q_{2}}+\underline {q_{2}})\nonumber \\
i \dot q_2&=&(1+|q_2|^2)(\overline {q_{1}}+\underline {q_{1}})
\end{eqnarray}
which, to our knowledge, was very little investigated. The main goal of this paper is to investigate the integrability and the multisoliton solutions for the general case.
 There are many methods for investigating the integrability of a dynamical system. One of them consists in a direct computation of the conserved quantities \cite{RC1}.
 Another one is based on the computation of Lie symmetries of the system \cite{ RC2}, \cite{ DA}. Also very interesting results
 appeared about symmetries and conservation laws on (semi)discrete nonlinear Schr\"odinger equation in \cite{Zhang}, \cite{wei}

 In this paper we are going to use the Hirota bilinear formalism and we will provide the Lax pairs as well.

\subsection{Soliton solutions and Lax pairs for $N=2$}

We consider the coupled Ablowitz-Ladik with two equations (\ref{2 system-NLS}), for which we apply the Hirota bilinear formalism.
Using the nonlinear substitutions $q_1=G_1/F_1$ and $q_2=G_2/F_2$, we cast (\ref{2 system-NLS}) into the bilinear form:
\begin{eqnarray}\label{2 system-NLS bilinear}
i {\bf D_t} G_1 \cdot F_1&=&\overline{G_2} \underline{F_2} + \underline{G_2} \overline{F_2} \nonumber\\
i {\bf D_t} G_2 \cdot F_2&=&\overline{G_1} \underline{F_1} + \underline{G_1} \overline{F_1} \nonumber\\
F_1^2+|G_1|^2&=&\overline{F_2} \underline{F_2} \nonumber\\
F_2^2+|G_2|^2&=&\overline{F_1} \underline{F_1}
\end{eqnarray}
where $F_1, F_2$ are real functions, while $G_1, G_2$ and complex valued functions. In order to solve this bilinear system we take for the one soliton solution the following ansatz:
$$G_1=\alpha_1 e^{\eta_1}, \quad G_2=\alpha_2 e^{\eta_1}, \quad F_1=1+e^{\eta_1+\eta_1^*+\phi_1}, \quad F_2=1+e^{\eta_1+\eta_1^*+\phi_2}$$
where amplitudes are given by the components of the ``polarization vector'' $(\alpha_1,\alpha_2)$. From the first two bilinear
equations we get an homogeneous algebraic system with the unknowns $\alpha_1,\alpha_2$. Its compatibility condition gives the dispersion relation:
$$ \omega_1=-2 i \epsilon \cosh(k_1), \quad \epsilon \in \{\pm1\}.$$
Next, two bilinear equations give the following two possible relations $e^{\phi_1}=e^{\phi_2}$ and $e^{\phi_1}=-e^{\phi_2}$.
The last one gives $\alpha_1=\alpha_2=0$ so remains only $\phi_1=\phi_2\equiv \phi_{12}$. Also, from the first bilinear equations we get
$\epsilon\alpha_1=\alpha_2$. Now fixing $\alpha_1=1$ we can write down the one soliton solution in the simplest form:
\begin{eqnarray}
G_1=e^{\eta_1}, \quad G_2=\epsilon e^{\eta_1}, \quad F_1=F_2=1+e^{\eta_1+\eta_1^*+\phi_{12}} \nonumber
\end{eqnarray}
where:
$$e^{\phi_{12}}=\frac{1}{2[\cosh(k_1+k_1^*)-1]}$$
and
$$\eta_1=k_1 n+ \omega_1 t+\eta_1^{(0)}.$$
Explicitly, the 1-soliton solution of (\ref{2 system-NLS}) is:
\begin{eqnarray}
q_1=\frac{e^{k_1 n-2 i \epsilon \cosh(k_1) t+\eta_1^{(0)}}}{1+\frac{1}{2(\cosh(k_1+k_1^*)-1)} e^ {(k_1+k_1^*) n -2 i  \epsilon [\cosh(k_1)-\cosh(k_1^*)] t+\eta_1^{(0)}+\eta_1^{(0)*}}} \nonumber\\
q_2=\frac{\epsilon e^{k_1 n-2 i \epsilon \cosh(k_1) t+\eta_1^{(0)}}}{1+\frac{1}{2(\cosh(k_1+k_1^*)-1)} e^ {(k_1+k_1^*) n -2 i  \epsilon [\cosh(k_1)-\cosh(k_1^*)] t+\eta_1^{(0)}+\eta_1^{(0)*}}} \nonumber
\end{eqnarray}
{\bf Remark:} Because physically the interest is in the square modulus amplitudes, here they are fixed (unlike the case of Manakov system where the polarization vector is free) and only the propagation direction can be different.

In constructing the two-soliton solution we have to take into account that all types of solitons must be considered (in this case both directions of propagation).
Straightforward calculation gives 2-soliton solution  that describes interaction of two solitons for any propagation direction:
\begin{eqnarray*}
G_1\!\!\!&\!=\!\!\!\!&e^{\eta_1}+e^{\eta_2}+e^{\eta_1+\eta_2+\eta_2^*+\phi_{12}+\phi_{14}+\phi_{24}} +e^{\eta_1+\eta_2+\eta_1^*+\phi_{12}+\phi_{13}+\phi_{23}}\nonumber\\
G_2\!\!\!&\!=\!\!\!\!&\epsilon_1 e^{\eta_1}+\epsilon_2 e^{\eta_2}+\epsilon_1 e^{\eta_1+\eta_2+\eta_2^*+\phi_{12}+\phi_{14}+\phi_{24}} +\epsilon_2 e^{\eta_1+\eta_2+\eta_1^*+\phi_{12}+\phi_{13}+\phi_{23}}\nonumber\\
F_1\!\!\!&\!=\!\!\!\!&1+e^{\eta_1+\eta_1^*+\phi_{13}}+e^{\eta_2+\eta_2^*+\phi_{24}}+e^{\eta_1+\eta_2^*+\phi_{14}}+ e^{\eta_2+\eta_1^*+\phi_{23}}+ e^{\eta_1+\eta_1^*+\eta_2+\eta_2^*+\sum\limits^4_{1\leq i<j}\phi_{ij}} \nonumber\\
F_2\!\!\!&\!=\!\!\!\!&1+e^{\eta_1+\eta_1^*+\phi_{13}}+e^{\eta_2+\eta_2^*+\phi_{24}}+\frac{\epsilon_1}{\epsilon_2}e^{\eta_1+\eta_2^*+\phi_{14}}+\frac{\epsilon_2}{\epsilon_1} e^{\eta_2+\eta_1^*+\phi_{23}}+ e^{\eta_1+\eta_1^*+\eta_2+\eta_2^*+\sum\limits^4_{1\leq i<j}\phi_{ij}} \nonumber
\end{eqnarray*}
where:
$$e^{\phi_{ij}} = \begin{cases} \frac{1}{2} \left[\epsilon_i \epsilon_j\cosh(k_i+k_j^*)-1\right]^{-1}, & \mbox{if } i=1,2 \mbox{ and } j=3,4; \vspace{0.3cm}\\
2\left[\epsilon_i \epsilon_j\cosh(k_i-k_j)-1\right], & \mbox{if } i=1,2 \mbox{ and } j=1,2\\
& \mbox{or } i=3,4 \mbox{ and } j=3,4; \end{cases}$$
\begin{equation}\label{coef}
\eta_j=k_j n+ \omega_j t+\eta_j^{(0)}, \quad k_{j+2}=k_j^*
\end{equation}
$$ \omega_j=-2 i \epsilon_j \cosh(k_j), \quad \epsilon_j \in\{-1,1\}, \quad j=1,2.$$
Long but straightforward calculations show the three-soliton solution in the form:
\begin{eqnarray*}
G_1\!\!\!&\!=\!\!\!\!&e^{\eta_1}+e^{\eta_2}+e^{\eta_3}+\nonumber\\
&+& e^{\eta_1+\eta_2+\eta_2^*+\phi_{12}+\phi_{15}+\phi_{25}} +e^{\eta_2+\eta_1+\eta_1^*+\phi_{12}+\phi_{14}+\phi_{24}}+e^{\eta_3+\eta_1+\eta_1^*+\phi_{13}+\phi_{14}+\phi_{34}}+\nonumber\\
&+&e^{\eta_1+\eta_3+\eta_3^*+\phi_{13}+\phi_{16}+\phi_{36}}+e^{\eta_2+\eta_3+\eta_3^*+\phi_{23}+\phi_{26}+\phi_{36}} +e^{\eta_3+\eta_2+\eta_2^*+\phi_{23}+\phi_{25}+\phi_{35}}+\nonumber\\
&+&e^{\eta_1+\eta_2+\eta_3^*+\phi_{12}+\phi_{16}+\phi_{26}} +e^{\eta_1+\eta_3+\eta_2^*+\phi_{13}+\phi_{15}+\phi_{35}}+e^{\eta_2+\eta_3+\eta_1^*+\phi_{23}+\phi_{24}+\phi_{34}}+\nonumber\\
&+&e^{\eta_1+\eta_2+\eta_3+\eta_1^*+\eta_2^*+\phi_{12}+\phi_{13}+\phi_{14}+\phi_{15}+\phi_{23}+\phi_{24}+\phi_{25}+\phi_{34}+\phi_{35}+\phi_{45}}+ \nonumber\\
&+&e^{\eta_1+\eta_2+\eta_3+\eta_1^*+\eta_3^*+\phi_{12}+\phi_{13}+\phi_{14}+\phi_{16}+\phi_{23}+\phi_{24}+\phi_{26}+\phi_{34}+\phi_{36}+\phi_{46}}+ \nonumber\\
&+&e^{\eta_1+\eta_2+\eta_3+\eta_2^*+\eta_3^*+\phi_{12}+\phi_{13}+\phi_{15}+\phi_{16}+\phi_{23}+\phi_{25}+\phi_{26}+\phi_{35}+\phi_{36}+\phi_{56}} \nonumber
\end{eqnarray*}
\begin{eqnarray*}
G_2\!\!\!&\!=\!\!\!\!&e^{\eta_1}+e^{\eta_2}+e^{\eta_3}+\nonumber\\
&+& \epsilon_1 e^{\eta_1+\eta_2+\eta_2^*+\phi_{12}+\phi_{15}+\phi_{25}} +\epsilon_2 e^{\eta_2+\eta_1+\eta_1^*+\phi_{12}+\phi_{14}+\phi_{24}}+\epsilon_3 e^{\eta_3+\eta_1+\eta_1^*+\phi_{13}+\phi_{14}+\phi_{34}}+\nonumber\\
&+&\epsilon_1 e^{\eta_1+\eta_3+\eta_3^*+\phi_{13}+\phi_{16}+\phi_{36}}+\epsilon_2 e^{\eta_2+\eta_3+\eta_3^*+\epsilon_3 \phi_{23}+\phi_{26}+\phi_{36}} +\epsilon_3 e^{\eta_3+\eta_2+\eta_2^*+\phi_{23}+\phi_{25}+\phi_{35}}+\nonumber\\
&+&\frac{\epsilon_1 \epsilon_2}{\epsilon_3}e^{\eta_1+\eta_2+\eta_3^*+\phi_{12}+\phi_{16}+\phi_{26}} +\frac{\epsilon_1 \epsilon_3}{\epsilon_2}e^{\eta_1+\eta_3+\eta_2^*+\phi_{13}+\phi_{15}+\phi_{35}}+\frac{\epsilon_2 \epsilon_3}{\epsilon_1}e^{\eta_2+\eta_3+\eta_1^*+\phi_{23}+\phi_{24}+\phi_{34}}+\nonumber\\
&+&\epsilon_1 e^{\eta_1+\eta_2+\eta_3+\eta_2^*+\eta_3^*+\phi_{12}+\phi_{13}+\phi_{15}+\phi_{16}+\phi_{23}+\phi_{25}+\phi_{26}+\phi_{35}+\phi_{36}+\phi_{56}}+ \nonumber\\
&+&\epsilon_2 e^{\eta_1+\eta_2+\eta_3+\eta_1^*+\eta_3^*+\phi_{12}+\phi_{13}+\phi_{14}+\phi_{16}+\phi_{23}+\phi_{24}+\phi_{26}+\phi_{34}+\phi_{36}+\phi_{46}}+ \nonumber\\
&+&\epsilon_3 e^{\eta_1+\eta_2+\eta_3+\eta_1^*+\eta_2^*+\phi_{12}+\phi_{13}+\phi_{14}+\phi_{15}+\phi_{23}+\phi_{24}+\phi_{25}+\phi_{34}+\phi_{35}+\phi_{45}}\nonumber\\
\end{eqnarray*}
\begin{eqnarray*}
F_1\!\!\!&\!=\!\!\!\!&1+e^{\eta_1+\eta_1^*+\phi_{14}}+e^{\eta_2+\eta_2^*+\phi_{25}}+e^{\eta_3+\eta_3^*+\phi_{36}}+e^{\eta_1+\eta_2^*+\phi_{15}}+ e^{\eta_1+\eta_3^*+\phi_{16}}+ \nonumber\\
&+&e^{\eta_2+\eta_1^*+\phi_{24}}+e^{\eta_2+\eta_3^*+\phi_{26}}+e^{\eta_3+\eta_1^*+\phi_{34}}+e^{\eta_3+\eta_2^*+\phi_{35}}+\nonumber\\
&+&e^{\eta_1+\eta_2+\eta_1^*+\eta_2^*+\phi_{12}+\phi_{14}+\phi_{15}+\phi_{24}+\phi_{25}+\phi_{45}}+e^{\eta_1+\eta_2+\eta_1^*+\eta_3^*+\phi_{12}+\phi_{14}+\phi_{16}+\phi_{24}+\phi_{26}+\phi_{46}}+\nonumber\\
&+&e^{\eta_1+\eta_2+\eta_2^*+\eta_3^*+\phi_{12}+\phi_{15}+\phi_{16}+\phi_{25}+\phi_{26}+\phi_{56}}+e^{\eta_1+\eta_3+\eta_1^*+\eta_2^*+\phi_{13}+\phi_{14}+\phi_{15}+\phi_{34}+\phi_{35}+\phi_{45}}+\nonumber\\
&+&e^{\eta_1+\eta_3+\eta_1^*+\eta_3^*+\phi_{13}+\phi_{14}+\phi_{16}+\phi_{34}+\phi_{36}+\phi_{46}}+e^{\eta_1+\eta_3+\eta_2^*+\eta_3^*+\phi_{13}+\phi_{15}+\phi_{16}+\phi_{35}+\phi_{36}+\phi_{56}}+\nonumber\\
&+&e^{\eta_2+\eta_3+\eta_1^*+\eta_2^*+\phi_{23}+\phi_{24}+\phi_{26}+\phi_{34}+\phi_{35}+\phi_{45}}+e^{\eta_2+\eta_3+\eta_1^*+\eta_3^*+\phi_{23}+\phi_{24}+\phi_{26}+\phi_{34}+\phi_{36}+\phi_{46}}+\nonumber\\
&+&e^{\eta_2+\eta_3+\eta_2^*+\eta_3^*+\phi_{23}+\phi_{25}+\phi_{26}+\phi_{35}+\phi_{36}+\phi_{56}}+e^{\eta_1+\eta_2+\eta_3+\eta_1^*+\eta_2^*+\eta_3^*+\sum\limits^6_{1\leq i<j}\phi_{ij}} \nonumber
\end{eqnarray*}
\begin{eqnarray*}
F_2\!\!\!&\!=\!\!\!\!&1+e^{\eta_1+\eta_1^*+\phi_{14}}+e^{\eta_2+\eta_2^*+\phi_{25}}+e^{\eta_3+\eta_3^*+\phi_{36}}+\frac{\epsilon_1}{\epsilon_2}e^{\eta_1+\eta_2^*+\phi_{15}}+ \frac{\epsilon_1}{\epsilon_3}e^{\eta_1+\eta_3^*+\phi_{16}}+ \nonumber\\
&+&\frac{\epsilon_2}{\epsilon_1}e^{\eta_2+\eta_1^*+\phi_{24}}+\frac{\epsilon_2}{\epsilon_3}e^{\eta_2+\eta_3^*+\phi_{26}}+\frac{\epsilon_3}{\epsilon_1}e^{\eta_3+\eta_1^*+\phi_{34}}+\frac{\epsilon_3}{\epsilon_2}e^{\eta_3+\eta_2^*+\phi_{35}}+\nonumber\\
&+&e^{\eta_1+\eta_2+\eta_1^*+\eta_2^*+\phi_{12}+\phi_{14}+\phi_{15}+\phi_{24}+\phi_{25}+\phi_{45}}+\frac{\epsilon_2}{\epsilon_3}e^{\eta_1+\eta_2+\eta_1^*+\eta_3^*+\phi_{12}+\phi_{14}+\phi_{16}+\phi_{24}+\phi_{26}+\phi_{46}}+\nonumber\\
&+&\frac{\epsilon_1}{\epsilon_3}e^{\eta_1+\eta_2+\eta_2^*+\eta_3^*+\phi_{12}+\phi_{15}+\phi_{16}+\phi_{25}+\phi_{26}+\phi_{56}}+\frac{\epsilon_3}{\epsilon_2}e^{\eta_1+\eta_3+\eta_1^*+\eta_2^*+\phi_{13}+\phi_{14}+\phi_{15}+\phi_{34}+\phi_{35}+\phi_{45}}+\nonumber\\
&+&e^{\eta_1+\eta_3+\eta_1^*+\eta_3^*+\phi_{13}+\phi_{14}+\phi_{16}+\phi_{34}+\phi_{36}+\phi_{46}}+\frac{\epsilon_1}{\epsilon_2}e^{\eta_1+\eta_3+\eta_2^*+\eta_3^*+\phi_{13}+\phi_{15}+\phi_{16}+\phi_{35}+\phi_{36}+\phi_{56}}+\nonumber\\
&+&\frac{\epsilon_3}{\epsilon_1}e^{\eta_2+\eta_3+\eta_1^*+\eta_2^*+\phi_{23}+\phi_{24}+\phi_{26}+\phi_{34}+\phi_{35}+\phi_{45}}+\frac{\epsilon_2}{\epsilon_1}e^{\eta_2+\eta_3+\eta_1^*+\eta_3^*+\phi_{23}+\phi_{24}+\phi_{26}+\phi_{34}+\phi_{36}+\phi_{46}}+\nonumber\\
&+&e^{\eta_2+\eta_3+\eta_2^*+\eta_3^*+\phi_{23}+\phi_{25}+\phi_{26}+\phi_{35}+\phi_{36}+\phi_{56}}+e^{\eta_1+\eta_2+\eta_3+\eta_1^*+\eta_2^*+\eta_3^*+\sum\limits^6_{1\leq i<j}\phi_{ij}} \nonumber
\end{eqnarray*}
where:
$$e^{\phi_{ij}} = \begin{cases} \frac{1}{2} \left[\epsilon_i \epsilon_j\cosh(k_i+k_j^*)-1\right]^{-1}, & \mbox{if } i=1,2,3 \mbox{ and } j=4,5,6; \vspace{0.3cm}\\
2\left[\epsilon_i \epsilon_j\cosh(k_i-k_j)-1\right], & \mbox{if } i=1,2,3 \mbox{ and } j=1,2,3\\
& \mbox{or } i=4,5,6 \mbox{ and } j=4,5,6; \end{cases}$$
\begin{equation}\label{coef}
\eta_j=k_j n+ \omega_j t+\eta_j^{(0)}, \quad k_{j+3}=k_j^*
\end{equation}
$$ \omega_j=-2 i \epsilon_j \cosh(k_j), \quad \epsilon_j \in\{-1,1\}, \quad j=1,2,3.$$
\vspace{0.5cm}

This solution describes interaction of three solitons for any propagation direction.
Although the existence of three-soliton solution in Hirota form for NLS-type equations is a strong indicator for the complete integrability \cite{jarmo}, we also give here the Lax pair:

\begin{eqnarray}\label{Lax 2 cuplate}
L_{n}=\left(
\begin{array}{cccc}
0&z&0&q_2\\
z&0&q_1&0\\
0&-q_2^*&0&z^{-1}\\
-q_1^*&0&z^{-1}&0\\
\end{array}\right),
\end{eqnarray}

$$B_n=\left(
\begin{array}{cccccc}
\!\!\!i(Z-q_1\underline{q_2^*})\!\!&\!\!0\!\!&\!\!i(-zq_1+z^{-1}\underline{q_2})\!\!&\!\!0\!\!\\
\!\!\!0\!&\!\!i(Z-q_2\underline{q_1^*})\!\!&\!\!0\!\!&\!\!i(-zq_2+z^{-1}\underline{q_1})\!\!\\
\!\!\!i(z\underline{q^*_2}-z^{-1}q_1^*)\!\!&\!\!0\!\!&\!\!-i(Z-q_1^*\underline{q_2})\!\!&\!\!0\!\!\\
\!\!\!0&\!\!i(z\underline{q^*_1}-z^{-1}q_2^*)\!\!&\!\!0\!\!&\!\!-i(Z-q_2^*\underline{q_1})\!\!\\
\end{array}\right),$$
where  $z$ is the spectral parameter and $Z=-\frac{1}{2}\left(z^2+z^{-2}\right)$. The compatibility condition  $\partial_t L+LB-\overline{B}L=0$ gives precisely system  (\ref{2 system-NLS}).

\vspace{0.5cm}
{\bf Remark:} The above spectral Lax operator can be put in a block Ablowitz-Ladik spectral operator as in {\cite{16a}}

\vspace{0.5cm}

Because the system is completely integrable we can write easily the $N$-soliton solution:

\begin{eqnarray*}
G_1&=&\sum\limits_{\mu=0,1}D_2(\underline{\mu})\exp\left({\sum\limits_{i=1}^{2N}\mu_i \eta_i+\sum\limits^{2N}_{1\leq i<j}\mu_i\mu_j\phi_{ij}}\right)\\
G_2&=&\sum\limits_{\mu=0,1}D_2(\underline{\mu})\exp\left({\sum\limits_{i=1}^{2N}\mu_i [\eta_i+\log(\epsilon_i)]+\sum\limits^{2N}_{1\leq i<j}\mu_i\mu_j\phi_{ij}}\right)\\
F_1&=&\sum\limits_{\mu=0,1}D_1(\underline{\mu})\exp\left({\sum\limits_{i=1}^{2N}\mu_i \eta_i+\sum\limits^{2N}_{1\leq i<j}\mu_i\mu_j\phi_{ij}}\right)\\
F_2&=&\sum\limits_{\mu=0,1}D_1(\underline{\mu})\exp\left({\sum\limits_{i=1}^{2N}\mu_i [\eta_i+\log(\epsilon_i)]+\sum\limits^{2N}_{1\leq i<j}\mu_i\mu_j\phi_{ij}}\right)\\
\end{eqnarray*}
where:
$$e^{\phi_{ij}} = \begin{cases} \frac{1}{2} \left[\epsilon_i \epsilon_j\cosh(k_i+k_j^*)-1\right]^{-1}, & \mbox{if } i=1,...N \mbox{ and } j=N+1,...,2N; \vspace{0.3cm}\\
2\left[\epsilon_i \epsilon_j\cosh(k_i-k_j)-1\right], & \mbox{if } i=1,...N  \mbox{ and } i=1,...,N \\
& \mbox{or } i=N+1,...,2N \mbox{ and } j=N+1,...,2N; \end{cases}$$
\begin{eqnarray*}
\eta_j=k_j n+ \omega_j t+\eta_j^{(0)}, \quad \eta_{j+N}=\eta_j^*, \quad k_{j+N}=k_j^*,\\
\omega_j=-2 i \epsilon_j \cosh(k_j), \quad \omega_{j+N}=\omega_j^*,  \quad j=1,...,N\\
\log(\epsilon_{j+N})=\log(\epsilon_{j})^*, \quad \log(\epsilon_j)\in\{\pm \pi i\},
\end{eqnarray*}
and where:
$$D_1(\underline{\mu}) = \begin{cases} 1, & \mbox{when } \sum\limits_{i=1}^N \mu_i=\sum\limits_{i=1}^N \mu_{i+N}; \\
0 & \mbox{otherwise}; \end{cases}$$
$$D_2(\underline{\mu}) = \begin{cases} 1, & \mbox{when } \sum\limits_{i=1}^N \mu_i=1+\sum\limits_{i=1}^N \mu_{i+N}; \\
0 & \mbox{otherwise}. \end{cases}$$

\subsection{Soliton solutions and Lax pairs for $N=3$}

We consider the coupled Ablowitz-Ladik with three equations:
\begin{eqnarray}\label{3 system-NLS}
i \dot q_1&=&(1+|q_1|^2)(\overline {q_{2}}+\underline {q_{3}})\nonumber \\
i \dot q_2&=&(1+|q_2|^2)(\overline {q_{3}}+\underline {q_{1}})\nonumber \\
i \dot q_3&=&(1+|q_3|^2)(\overline {q_{1}}+\underline {q_{2}})
\end{eqnarray}
The system is again a completely integrable one. The Lax pair is the following

\begin{eqnarray}\label{Lax 3 cuplat}
L_n=\left(
\begin{array}{cccccc}
0&0&z&0&0&q_3\\
z&0&0&q_1&0&0\\
0&z&0&0&q_2&0\\
0&0&-q_3^*&0&0&z^{-1}\\
-q_1^*&0&0&z^{-1}&0&0\\
0&-q_2^*&0&0&z^{-1}&0\\
\end{array}\right), \quad B_n=
\end{eqnarray}
$$
\left(
\begin{array}{cccccc}
\!\!\!\!i(Z_+\!-\!q_1\underline{q_3^*})\!\!&\!\!0\!\!&\!\!0\!\!&\!\!i(-zq_1\!+\!\frac{\underline{q_3}}{z})\!\!&\!\!0\!\!&\!\!0\!\!\!\!\\
\!\!\!\!0\!\!&\!\!i(Z_+\!-\!q_2\underline{q_1^*})\!\!&\!\!0\!\!&\!\!0&\!\!i(-zq_2\!+\!\frac{\underline{q_1}}{z})\!\!&\!\!0\!\!\!\!\\
\!\!\!\!0\!\!&\!\!0\!&\!\!i(Z_+\!-\!q_3\underline{q_2^*})\!\!&\!\!0\!\!&\!\!0\!\!&\!\!i(-zq_3\!+\!\frac{\underline{q_2}}{z})\!\!\!\!\\
\!\!\!\!i(z\underline{q^*_3}\!-\!\frac{q_1^*}{z})\!\!&\!\!0\!\!&\!\!0\!\!&\!\!-i(Z_-\!-\!q_1^*\underline{q_3})\!\!&\!\!0\!&\!0\!\!\!\!\\
\!\!\!\!0\!\!&\!\!i(z\underline{q^*_1}\!-\!\frac{q_2^*}{z})\!\!&\!\!0\!\!&\!\!0\!&\!\!-i(Z_-\!-\!q_2^*\underline{q_1})\!\!&\!\!0\!\!\!\!\\
\!\!\!\!0\!\!&\!\!0\!\!&\!i(z\underline{q^*_2}\!-\!\frac{q_3^*}{z})\!\!&\!0\!\!\!&\!\!0\!&\!-i(Z_-\!-\!q_3^*\underline{q_2})\!\!\!\!\\
\end{array}\right)
$$
where $Z_+=1-z^2, Z_-=-1-z^{-2}$.

The compatibility condition $\partial_t L+LB-\overline{B}L=0$ gives precisely our system (\ref{3 system-NLS}). In the last section we will show how these Lax pairs were found.

In order to see the solutions we build the Hirota bilinear form, considering the nonlinear substitutions: $q_1=G_1/F_1$, $q_2=G_2/F_2$ and $q_3=G_3/F_3$:
\begin{eqnarray*}\label{3 system-NLS bilinear}
i {\bf D_t} G_1 \cdot F_1&=&\overline{G_2} \underline{F_3} + \underline{G_3} \overline{F_2} \nonumber\\
i {\bf D_t} G_2 \cdot F_2&=&\overline{G_3} \underline{F_1} + \underline{G_1} \overline{F_3} \nonumber\\
i {\bf D_t} G_3 \cdot F_3&=&\overline{G_1} \underline{F_2} + \underline{G_2} \overline{F_3} \nonumber\\
F_1^2+|G_1|^2&=&\overline{F_2} \underline{F_3} \nonumber\\
F_2^2+|G_2|^2&=&\overline{F_3} \underline{F_1} \nonumber\\
F_3^2+|G_3|^2&=&\overline{F_1} \underline{F_2}
\end{eqnarray*}
where $F_1, F_2, F_3$ are real functions, while $G_1, G_2, G_3$ and complex functions.
Doing the same machinery as in the case of $N=2$, we can start with the following ansatz $G_i=\alpha_i e^{\eta}, \quad F_i=1+e^{\eta_i+\eta_i^*+\phi_i}$ for $i=1,2,3$. Plugging into
the bilinear equations, we will find that $\phi_1=\phi_2=\phi_3\equiv \phi_{12}$ and $\alpha_1,\alpha_2,\alpha_3$ are ordered as the cubic roots of unity. Accordingly, the 1-soliton solution is:
\begin{eqnarray*}
G_1=e^{\eta_1},\quad G_2=\epsilon e^{\eta_1}, \quad G_3=\epsilon^2 e^{\eta_1}, \quad F_1=F_2=F_3=1+ e^{\eta_1+\eta_1^*+\phi_{12}}
\end{eqnarray*}
where:
$$\epsilon \in \left \{1,-\frac{1}{2}\pm i \frac{\sqrt3}{2}\right \}, \quad e^{\phi_{12}}=\frac{1}{2[\cosh(k_1+k_1^*)-1]}$$
and
$$\eta_1=k_1 n+ \omega_1 t+\eta_1^{(0)}$$
with three branches of dispersion:
\begin{eqnarray}
\omega_1=-2 i \cosh(k_1) \hspace{1.9cm} & \mbox{for} & \epsilon=1 \nonumber \\
\omega_1= i \cosh(k_1)+\sqrt3 \sinh(k_1) & \mbox{for} & \epsilon = -\frac{1}{2}+i \frac{\sqrt 3}{2} \nonumber \\
\omega_1= i \cosh(k_1)-\sqrt3 \sinh(k_1) & \mbox{for} & \epsilon = -\frac{1}{2}-i \frac{\sqrt 3}{2} \nonumber
\end{eqnarray}
\vspace{0.1cm}

The 2-soliton solution has the following form:
\begin{eqnarray*}
G_1\!\!\!&\!=\!\!\!\!&e^{\eta_1}+e^{\eta_2}+e^{\eta_1+\eta_2+\eta_2^*+\phi_{12}+\phi_{14}+\phi_{24}} +e^{\eta_1+\eta_2+\eta_1^*+\phi_{12}+\phi_{13}+\phi_{23}}\nonumber\\
G_2\!\!\!&\!=\!\!\!\!&\epsilon_1 e^{\eta_1}+\epsilon_2 e^{\eta_2}+\epsilon_1 e^{\eta_1+\eta_2+\eta_2^*+\phi_{12}+\phi_{14}+\phi_{24}} +\epsilon_2 e^{\eta_1+\eta_2+\eta_1^*+\phi_{12}+\phi_{13}+\phi_{23}}\nonumber\\
G_3\!\!\!&\!=\!\!\!\!&\epsilon_1^2 e^{\eta_1}+\epsilon_2^2 e^{\eta_2}+\epsilon_1^2 e^{\eta_1+\eta_2+\eta_2^*+\phi_{12}+\phi_{14}+\phi_{24}} +\epsilon_2^2 e^{\eta_1+\eta_2+\eta_1^*+\phi_{12}+\phi_{13}+\phi_{23}}\nonumber\\
F_1\!\!\!&\!=\!\!\!\!&1+e^{\eta_1+\eta_1^*+\phi_{13}}+e^{\eta_2+\eta_2^*+\phi_{24}}+e^{\eta_1+\eta_2^*+\phi_{14}}+ e^{\eta_2+\eta_1^*+\phi_{23}}+ e^{\eta_1+\eta_1^*+\eta_2+\eta_2^*+\sum\limits^4_{1\leq i<j}\phi_{ij}} \nonumber\\
F_2\!\!\!&\!=\!\!\!\!&1+e^{\eta_1+\eta_1^*+\phi_{13}}+e^{\eta_2+\eta_2^*+\phi_{24}}+\frac{\epsilon_1}{\epsilon_2}e^{\eta_1+\eta_2^*+\phi_{14}}+\frac{\epsilon_2}{\epsilon_1} e^{\eta_2+\eta_1^*+\phi_{23}}+ e^{\eta_1+\eta_1^*+\eta_2+\eta_2^*+\sum\limits^4_{1\leq i<j}\phi_{ij}} \nonumber\\
F_3\!\!\!&\!=\!\!\!\!&1+e^{\eta_1+\eta_1^*+\phi_{13}}+e^{\eta_2+\eta_2^*+\phi_{24}}+\left(\frac{\epsilon_1}{\epsilon_2}\right)^2e^{\eta_1+\eta_2^*+\phi_{14}}+\left(\frac{\epsilon_2}{\epsilon_1}\right)^2 e^{\eta_2+\eta_1^*+\phi_{23}}+\nonumber\\
 &&+e^{\eta_1+\eta_1^*+\eta_2+\eta_2^*+\sum\limits^4_{1\leq i<j}\phi_{ij}} \nonumber
\end{eqnarray*}
where:
$$e^{\phi_{ij}} = \begin{cases} \frac{1}{2} \left(\frac{\epsilon_i^2+\epsilon_j^2}{2\epsilon_i\epsilon_j}\cosh(k_i+k_j^*)+\frac{\epsilon_i^2-\epsilon_j^2}{2\epsilon_i\epsilon_j}\sinh(k_i+k_j^*)-1\right)^{-1}, \mbox{if } i=1,2 \mbox{ and } j=3,4; \vspace{0.5cm}\\
2\left(\frac{\epsilon_i^2+\epsilon_j^2}{2\epsilon_i\epsilon_j}\cosh(k_i-k_j)+\frac{\epsilon_i^2-\epsilon_j^2}{2\epsilon_i\epsilon_j}\sinh(k_i-k_j)-1\right),\mbox{if } i=1,2  \mbox{ and } j=1,2 \vspace{0.2cm}\\
\hspace{8.7cm} \mbox{or } i=3,4 \mbox{ and } j=3,4; \end{cases}$$
and:
\begin{eqnarray*}
&\eta_j=k_j n+ \omega_j t+\eta_j^{(0)}, \quad k_{j+2}=k_j^*, \quad j=1,2\\
&\log(\epsilon_{j+2})=\log(\epsilon_{j})^*, \quad \log(\epsilon_j)\in\{\frac{2 \pi}{3}i, \frac{4 \pi}{3}i, 2\pi i \},
\end{eqnarray*}
with the three branches of dispersion ($j=1,2$):
 \begin{eqnarray}
\omega_j=-2 i \cosh(k_j) \hspace{1.9cm} & \mbox{for} & \epsilon_j=1 \nonumber \\
\omega_j= i \cosh(k_j)+\sqrt3 \sinh(k_j) & \mbox{for} & \epsilon_j = -\frac{1}{2}+i \frac{\sqrt 3}{2} \nonumber \\
\omega_j= i \cosh(k_j)-\sqrt3 \sinh(k_j) & \mbox{for} & \epsilon_j = -\frac{1}{2}-i \frac{\sqrt 3}{2}. \nonumber
\end{eqnarray}
$3$-soliton solution can be constructed easily and this validates the Lax integrability.
\begin{eqnarray*}\label{3ss-3}
G_1&=&\sum\limits_{\mu=0,1}D_2(\underline{\mu})\exp\left({\sum\limits_{i=1}^{6}\mu_i \eta_i+\sum\limits^{6}_{1\leq i<j}\mu_i\mu_j\phi_{ij}}\right)\\
G_2&=&\sum\limits_{\mu=0,1}D_2(\underline{\mu})\exp\left({\sum\limits_{i=1}^{6}\mu_i [\eta_i+\log(\epsilon_i)]+\sum\limits^{6}_{1\leq i<j}\mu_i\mu_j\phi_{ij}}\right)\\
G_3&=&\sum\limits_{\mu=0,1}D_2(\underline{\mu})\exp\left({\sum\limits_{i=1}^{6}\mu_i [\eta_i+2\log(\epsilon_i)]+\sum\limits^{6}_{1\leq i<j}\mu_i\mu_j\phi_{ij}}\right)\\
F_1&=&\sum\limits_{\mu=0,1}D_1(\underline{\mu})\exp\left({\sum\limits_{i=1}^{6}\mu_i \eta_i+\sum\limits^{6}_{1\leq i<j}\mu_i\mu_j\phi_{ij}}\right)\\
F_2&=&\sum\limits_{\mu=0,1}D_1(\underline{\mu})\exp\left({\sum\limits_{i=1}^{6}\mu_i [\eta_i+\log(\epsilon_i)]+\sum\limits^{6}_{1\leq i<j}\mu_i\mu_j\phi_{ij}}\right)\\
F_3&=&\sum\limits_{\mu=0,1}D_1(\underline{\mu})\exp\left({\sum\limits_{i=1}^{6}\mu_i [\eta_i+2\log(\epsilon_i)]+\sum\limits^{6}_{1\leq i<j}\mu_i\mu_j\phi_{ij}}\right)\\
\end{eqnarray*}
where:
$$e^{\phi_{ij}} = \begin{cases} \frac{1}{2} \left(\frac{\epsilon_i^2+\epsilon_j^2}{2\epsilon_i\epsilon_j}\cosh(k_i+k_j^*)+\frac{\epsilon_i^2-\epsilon_j^2}{2\epsilon_i\epsilon_j}\sinh(k_i+k_j^*)-1\right)^{-1}\vspace{0.2cm}\\
\hspace{3cm} \mbox{if } i=1,2,3 \mbox{ and } j=4,5,6; \vspace{0.5cm}\\
2\left(\frac{\epsilon_i^2+\epsilon_j^2}{2\epsilon_i\epsilon_j}\cosh(k_i-k_j)+\frac{\epsilon_i^2-\epsilon_j^2}{2\epsilon_i\epsilon_j}\sinh(k_i-k_j)-1\right),\vspace{0.2cm}\\
\hspace{3cm} \mbox{if } i=1,2,3  \mbox{ and } i=1,2,3 \vspace{0.2cm}\\
\hspace{3cm} \mbox{or } i=4,5,6 \mbox{ and } j=4,5,6; \end{cases}$$
and:
\begin{eqnarray*}
\eta_j=k_j n+ \omega_j t+\eta_j^{(0)}, \quad \eta_{j+3}=\eta_j^*, \quad k_{j+3}=k_j^*, \quad j=1,...,3\\
\log(\epsilon_{j+3})=\log(\epsilon_{j})^*, \quad \omega_{j+3}=\omega_j^*, \quad \log(\epsilon_j)\in\{\frac{2 \pi}{3}i, \frac{4 \pi}{3}i, 2\pi i \}.
\end{eqnarray*}
\begin{eqnarray}\label{coeficienti generali}
D_1(\underline{\mu}) = \begin{cases} 1, & \mbox{when } \sum\limits_{i=1}^3 \mu_i=\sum\limits_{i=1}^3 \mu_{i+3}; \\
0 & \mbox{otherwise}; \end{cases}\nonumber\\
D_2(\underline{\mu}) = \begin{cases} 1, & \mbox{when } \sum\limits_{i=1}^3 \mu_i=1+\sum\limits_{i=1}^3 \mu_{i+3}; \\
0 & \mbox{otherwise}. \end{cases}
\end{eqnarray}
Since $\epsilon_j \in \left\{ 1,  -\frac{1}{2}\pm i \frac{\sqrt 3}{2} \right\}$, each of the three solitons can have any of the following three branches of dispersion:
\begin{eqnarray}
\omega_j=-2 i \cosh(k_j) \hspace{1.9cm} & \mbox{for} & \epsilon_j=1 \nonumber \\
\omega_j= i \cosh(k_j)+\sqrt3 \sinh(k_j) & \mbox{for} & \epsilon_j = -\frac{1}{2}+i \frac{\sqrt 3}{2} \nonumber \\
\omega_j= i \cosh(k_j)-\sqrt3 \sinh(k_j) & \mbox{for} & \epsilon_j = -\frac{1}{2}-i \frac{\sqrt 3}{2} \nonumber
\end{eqnarray}

Extension to $N$-soliton solution is straightforward. Formulas are the same except that the sums in (\ref{3ss-3}) are taken from 1 to $2N$ and all the definitions with $i+3$ and $j+3$  will turn into $i+N$ and $j+N$

\section{General case and periodic reduction}

Starting from the general system of coupled Ablowitz-Ladik with $N$ coupled equations, given in (\ref{sys1}):
\begin{eqnarray}
i \dot q_1&=&(1+|q_1|^2)(\overline {q_{2}}+\underline {q_{N}})\nonumber \\
i \dot q_2&=&(1+|q_2|^2)(\overline {q_{3}}+\underline {q_{1}})\nonumber \\
i \dot q_3&=&(1+|q_3|^2)(\overline {q_{4}}+\underline {q_{2}}) \nonumber \\
.....&....&.......................\nonumber \\
i \dot q_{N-1}&=&(1+|q_{N-1}|^2)(\overline {q_{N}}+\underline {q_{N-2}})\nonumber \\
i \dot q_N&=&(1+|q_N|^2)(\overline {q_{1}}+\underline {q_{N-1}})\nonumber
\end{eqnarray}
and using  the nonlinear substitutions: $q_{\nu}=G_{\nu}/F_{\nu}$, ${\nu}=1,...,N$, we obtain the Hirota bilinear form:
\begin{eqnarray}\label{N-bil}
i {\bf D_t} G_{\nu} \cdot F_{\nu}&=&\overline{G_{\nu+1}} \underline{F_{\nu-1}} + \underline{G_{\nu-1}} \overline{F_{\nu+1}} \nonumber\\
F_{\nu}^2+|G_{\nu}|^2&=&\overline{F_{\nu+1}} \underline{F_{\nu-1}}.
\end{eqnarray}
The 3-soliton solution for (\ref{sys1}) has the expressions for $G_{\nu}$, $F_{\nu}$, $(\nu=1,...,N)$:
\begin{eqnarray}\label{solutie generalaa}
G_{\nu}&=&\sum\limits_{\mu=0,1}D_2(\underline{\mu})\exp\left({\sum\limits_{i=1}^{6}\mu_i [\eta_i+(\nu-1)\log(\epsilon_i)]+\sum\limits^{6}_{1\leq i<j}\mu_i\mu_j\phi_{ij}}\right)\nonumber\\
F_{\nu}&=&\sum\limits_{\mu=0,1}D_1(\underline{\mu})\exp\left({\sum\limits_{i=1}^{6}\mu_i [\eta_i+(\nu-1)\log(\epsilon_i)]+\sum\limits^{6}_{1\leq i<j}\mu_i\mu_j\phi_{ij}}\right)
\end{eqnarray}
where $D_1(\underline{\mu})$, $D_2(\underline{\mu})$, $\eta_i$, $\phi_{ij}$ are given in (\ref{coeficienti generali})
 with the difference that now we have $N$ branches of dispersion for each soliton ($k_j$ is the wave number of the $j$-soliton where $j=1,2,3$):
\begin{equation}
\omega_j(k_{j})\!\!=\!\!-2 i \left[\frac{\epsilon_j^2+1}{2\epsilon_j}\cosh k_{j}+\frac{\epsilon_j^2-1}{2\epsilon_j}\sinh k_{j}\right]\!\!, \: \epsilon_j \in \left\{\!e^{l \frac{2 \pi i}{N}\!} \right\}\!,l=1,..N,
j=1, 2, 3. \nonumber
\end{equation}
So the branches of dispersion are labelled by the index $l$. So the parameter $\epsilon_j$ which characterizes the $j$-soliton ($j=1,2,3$) can have $N$ values, the order $N$ roots of unity.

\subsection{The periodic reduction}
One could obtain the above results starting from a general ``diagonal'' equation in two dimensions and performing periodic reduction.
This idea has been used for the first time in \cite{toki-fane} where coupled semidiscrete KdV equations were obtained modelling a modular genetic network.
The idea is to consider that the independent discrete variable of ordinary Ablowitz-Ladik equation is in fact a {\it diagonal} in a two-dimensional (or $d$-dimensional) lattice.
Imposing periodic reduction on the one such coordinate in that 2D-lattice, then we will obtain {\it coupled systems} of Ablowitz-Ladik equations. In order to see it clearly let us start with the following equation:
\begin{eqnarray}\label{general system-NLS}
i \frac{d}{dt} Q_{n,m}(t)&=&(1+|Q_{n,m}(t)|^2)(Q_{n+1,m+1}(t)+Q_{n-1,m-1}(t)).
\end{eqnarray}
This equation is of course completely integrable. It has the following Lax pair:
\begin{eqnarray}\label{Lax general}
L_{n,m}=\left(
\begin{array}{cc}
z&Q_{n,m}(t)\\
-Q^*_{n,m}(t)&1/z\\
\end{array}\right),
\end{eqnarray}
$$
B_{n,m}=\left(
\begin{array}{cc}
i(Z-Q_{n,m}(t)Q^*_{n-1,m-1}(t))&i(-zQ_{n,m}(t)+z^{-1}Q_{n-1,m-1}(t))\\
i(-z^{-1}Q^*_{n,m}(t)+zQ^*_{n-1,m-1}(t))&-i(Z-Q^*_{n,m}(t)Q_{n-1,m-1}(t))
\end{array}\right),
$$
where  $Z=-\frac{1}{2}\left(z^2+z^{-2}\right)$, the compatibility condition $\partial_t L_{n,m}+L_{n,m}B_{n,m}-B_{n+1,m+1}L_{n,m}=0$ gives exactly (\ref{general system-NLS}).

Now lets consider the periodic 2-reduction on the $m$ direction (meaning that $Q(n,m)$ is a periodic function only with respect to $m$ and the period is 2). This means
that $Q(n,m)\equiv q_1(n), \: Q(n,m+1)\equiv q_2(n), \: Q(n,m+2)\equiv q_1(n), \: Q(n,m-1)\equiv q_2(n), \: etc.$ Introducing this reduction in (\ref{general system-NLS}) we get precisely:
\begin{eqnarray*}
i \dot q_1&=&(1+|q_1|^2)(\overline {q_{2}}+\underline {q_{2}})\nonumber \\
i \dot q_2&=&(1+|q_2|^2)(\overline {q_{1}}+\underline {q_{1}}).
\end{eqnarray*}

In the same way, if we impose periodic-3 reduction, we get the system with three equations (\ref{3 system-NLS}) and so on.
Accordingly, the general system can be obtained from this diagonal Ablowitz-Ladik equation (\ref{general system-NLS}), but choosing a periodic $N$-reduction on $m$.

The Hirota bilinear form can be obtained by $Q_{n,m}(t)=G^m_n/F^m_n$ (in this notation $m$ is not exponent). Then we cast (\ref{general system-NLS}) into:
\begin{eqnarray}\label{2-bil}
i {\bf D_t} G^m_n \cdot F^m_n&=&G^{m+1}_{n+1} F^{m-1}_{n-1}+G^{m-1}_{n-1} F^{m+1}_{n+1}\nonumber\\
(F^m_n)^2+|G^m_n|^2&=&F^{m+1}_{n+1} F^{m-1}_{n-1},
\end{eqnarray}
where $F_n^m$ is real function, while $G_n^m$ is complex valued function.\\
The 1-soliton solution is:
$$q^m_n=\frac{G^m_n}{F^m_n}=\frac{e^{\eta_1}}{1+\frac{1}{2[\cosh(k_1+p_1+k_1^*+p_1^*)-1]}e^{\eta_1+\eta_1^*}}, \quad \eta_1=k_1 n+p_1 m +\omega_1 t+\eta_1^{(0)},$$
where:
$$ \omega_1=-2 i \cosh(k_1+p_1), \quad (\forall)\quad k_1,  p_1 \in \mathbb{C}.$$\\
For 2-soliton solution we obtain:
\begin{eqnarray*}
G^m_n\!\!\!&\!=\!\!\!\!&e^{\eta_1}+e^{\eta_2}+e^{\eta_1+\eta_2+\eta_2^*+\phi_{12}+\phi_{14}+\phi_{24}} +e^{\eta_1+\eta_2+\eta_1^*+\phi_{12}+\phi_{13}+\phi_{23}}\nonumber\\
F^m_n\!\!\!&\!=\!\!\!\!&1+e^{\eta_1+\eta_1^*+\phi_{13}}+e^{\eta_2+\eta_2^*+\phi_{24}}+e^{\eta_1+\eta_2^*+\phi_{14}}+ e^{\eta_2+\eta_1^*+\phi_{23}}+ e^{\eta_1+\eta_1^*+\eta_2+\eta_2^*+\sum\limits^4_{1\leq i<j}\phi_{ij}} \nonumber
\end{eqnarray*}
where:
$$e^{\phi_{ij}} = \begin{cases} \frac{1}{2} \left[\cosh(k_i+p_i+k_j^*+p_j^*)-1\right]^{-1}, & \mbox{if } i=1,2 \mbox{ and } j=3,4; \vspace{0.3cm}\\
2\left[\cosh(k_i+p_i-k_j-p_j)-1\right], & \mbox{if } i=1,2 \mbox{ and } j=1,2\\
& \mbox{or } i=3,4 \mbox{ and } j=3,4; \end{cases}$$
\begin{equation}\label{coeficienti redusi}
\eta_j=k_j n+p_j m+ \omega_j t+\eta_j^{(0)}, \quad j=1,2
\end{equation}
$$ \omega_j=-2 i \cosh(k_j+p_j), \quad k_{j+2}=k_j^*, \quad p_{j+2}=p_j^*.$$
The three-soliton solution is given by:
\begin{eqnarray*}
G^m_n\!\!\!&\!=\!\!\!\!&e^{\eta_1}+e^{\eta_2}+e^{\eta_3}+\nonumber\\
&+& e^{\eta_1+\eta_2+\eta_2^*+\phi_{12}+\phi_{15}+\phi_{25}} +e^{\eta_2+\eta_1+\eta_1^*+\phi_{12}+\phi_{14}+\phi_{24}}+e^{\eta_3+\eta_1+\eta_1^*+\phi_{13}+\phi_{14}+\phi_{34}}+\nonumber\\
&+&e^{\eta_1+\eta_3+\eta_3^*+\phi_{13}+\phi_{16}+\phi_{36}}+e^{\eta_2+\eta_3+\eta_3^*+\phi_{23}+\phi_{26}+\phi_{36}} +e^{\eta_3+\eta_2+\eta_2^*+\phi_{23}+\phi_{25}+\phi_{35}}+\nonumber\\
&+&e^{\eta_1+\eta_2+\eta_3^*+\phi_{12}+\phi_{16}+\phi_{26}} +e^{\eta_1+\eta_3+\eta_2^*+\phi_{13}+\phi_{15}+\phi_{35}}+e^{\eta_2+\eta_3+\eta_1^*+\phi_{23}+\phi_{24}+\phi_{34}}+\nonumber\\
&+&e^{\eta_1+\eta_2+\eta_3+\eta_1^*+\eta_2^*+\phi_{12}+\phi_{13}+\phi_{14}+\phi_{15}+\phi_{23}+\phi_{24}+\phi_{25}+\phi_{34}+\phi_{35}+\phi_{45}}+ \nonumber\\
&+&e^{\eta_1+\eta_2+\eta_3+\eta_1^*+\eta_3^*+\phi_{12}+\phi_{13}+\phi_{14}+\phi_{16}+\phi_{23}+\phi_{24}+\phi_{26}+\phi_{34}+\phi_{36}+\phi_{46}}+ \nonumber\\
&+&e^{\eta_1+\eta_2+\eta_3+\eta_2^*+\eta_3^*+\phi_{12}+\phi_{13}+\phi_{15}+\phi_{16}+\phi_{23}+\phi_{25}+\phi_{26}+\phi_{35}+\phi_{36}+\phi_{56}} \nonumber\\
F^m_n\!\!\!&\!=\!\!\!\!&1+e^{\eta_1+\eta_1^*+\phi_{14}}+e^{\eta_2+\eta_2^*+\phi_{25}}+e^{\eta_3+\eta_3^*+\phi_{36}}+e^{\eta_1+\eta_2^*+\phi_{15}}+ e^{\eta_1+\eta_3^*+\phi_{16}}+ \nonumber\\
&+&e^{\eta_2+\eta_1^*+\phi_{24}}+e^{\eta_2+\eta_3^*+\phi_{26}}+e^{\eta_3+\eta_1^*+\phi_{34}}+e^{\eta_3+\eta_2^*+\phi_{35}}+\nonumber\\
&+&e^{\eta_1+\eta_2+\eta_1^*+\eta_2^*+\phi_{12}+\phi_{14}+\phi_{15}+\phi_{24}+\phi_{25}+\phi_{45}}+e^{\eta_1+\eta_2+\eta_1^*+\eta_3^*+\phi_{12}+\phi_{14}+\phi_{16}+\phi_{24}+\phi_{26}+\phi_{46}}+\nonumber\\
&+&e^{\eta_1+\eta_2+\eta_2^*+\eta_3^*+\phi_{12}+\phi_{15}+\phi_{16}+\phi_{25}+\phi_{26}+\phi_{56}}+e^{\eta_1+\eta_3+\eta_1^*+\eta_2^*+\phi_{13}+\phi_{14}+\phi_{15}+\phi_{34}+\phi_{35}+\phi_{45}}+\nonumber\\
&+&e^{\eta_1+\eta_3+\eta_1^*+\eta_3^*+\phi_{13}+\phi_{14}+\phi_{16}+\phi_{34}+\phi_{36}+\phi_{46}}+e^{\eta_1+\eta_3+\eta_2^*+\eta_3^*+\phi_{13}+\phi_{15}+\phi_{16}+\phi_{35}+\phi_{36}+\phi_{56}}+\nonumber\\
&+&e^{\eta_2+\eta_3+\eta_1^*+\eta_2^*+\phi_{23}+\phi_{24}+\phi_{26}+\phi_{34}+\phi_{35}+\phi_{45}}+e^{\eta_2+\eta_3+\eta_1^*+\eta_3^*+\phi_{23}+\phi_{24}+\phi_{26}+\phi_{34}+\phi_{36}+\phi_{46}}+\nonumber\\
&+&e^{\eta_2+\eta_3+\eta_2^*+\eta_3^*+\phi_{23}+\phi_{25}+\phi_{26}+\phi_{35}+\phi_{36}+\phi_{56}}+e^{\eta_1+\eta_2+\eta_3+\eta_1^*+\eta_2^*+\eta_3^*+\sum\limits^6_{1\leq i<j}\phi_{ij}} \nonumber
\end{eqnarray*}
where:
$$e^{\phi_{ij}} = \begin{cases} \frac{1}{2} \left[\cosh(k_i+p_i+k_j^*+p_j^*)-1\right]^{-1}, & \mbox{if } i=1,2,3 \mbox{ and } j=4,5,6; \vspace{0.3cm}\\
2\left[\cosh(k_i+p_i-k_j-p_j)-1\right], & \mbox{if } i=1,2,3 \mbox{ and } j=1,2,3\\
& \mbox{or } i=4,5,6 \mbox{ and } j=4,5,6; \end{cases}$$
\begin{equation}
\eta_j=k_j n+p_j m+ \omega_j t+\eta_j^{(0)}, \quad j=1,2,3
\end{equation}
$$ \omega_j=-2 i \cosh(k_j+p_j), \quad k_{j+3}=k_j^*, \quad p_{j+3}=p_j^*.$$
The $N$-soliton solution has the following form for $G^m_n$ and $F^m_n$ (being essentially the multisoliton solution of Ablowitz-Ladik):
\begin{eqnarray*}
G^m_n&=&\sum\limits_{\mu=0,1}D_2(\underline{\mu})\exp\left({\sum\limits_{i=1}^{2N}\mu_i \eta_i+\sum\limits^{2N}_{1\leq i<j}\mu_i\mu_j\phi_{ij}}\right)\\
F^m_n&=&\sum\limits_{\mu=0,1}D_1(\underline{\mu})\exp\left({\sum\limits_{i=1}^{2N}\mu_i \eta_i+\sum\limits^{2N}_{1\leq i<j}\mu_i\mu_j\phi_{ij}}\right)\\
\end{eqnarray*}
where:
$$e^{\phi_{ij}} = \begin{cases} \frac{1}{2} \left[\cosh(k_i+p_i+k_j^*+p_j^*)-1\right]^{-1}, \\
\hspace{3cm} \mbox{if } i=1,...,N \mbox{ and } j=N+1,...,2N; \vspace{0.3cm}\\
2\left[\cosh(k_i+p_i-k_j-p_j)-1\right], \\
\hspace{3cm} \mbox{if } i=1,...,N  \mbox{ and } i=1,...,N \\
\hspace{3cm} \mbox{or } i=N+1,...,2N \mbox{ and } j=N+1,...,2N; \end{cases}$$
\begin{equation}
\eta_j=k_j n+p_j m+ \omega_j t+\eta_j^{(0)}, \quad \eta_{j+N}=\eta_j^*, \quad j=1,...,N
\end{equation}
$$ \omega_j=-2 i \cosh(k_j+p_j), \quad \omega_{j+N}=\omega_j^*,  \quad k_{j+N}=k_j^*, \quad p_{j+N}=p_j^*.$$
and where:
$$D_1(\underline{\mu}) = \begin{cases} 1, & \mbox{when } \sum\limits_{i=1}^N \mu_i=\sum\limits_{i=1}^N \mu_{i+N}; \\
0 & \mbox{otherwise}; \end{cases}$$
$$D_2(\underline{\mu}) = \begin{cases} 1, & \mbox{when } \sum\limits_{i=1}^N \mu_i=1+\sum\limits_{i=1}^N \mu_{i+N}; \\
0 & \mbox{otherwise}. \end{cases}$$

Now, all the results are coming straightforward from the diagonal Ablowitz-Ladik (\ref{general system-NLS}).
This can be seen immediately looking at the two bilinear systems (\ref{2-bil}) and (\ref{N-bil}).
The systems are the same (considering the bar to be the up-shift/down-shift in $m$-direction).
In the case $N=2$, the $m$-dependence is dropped, $p$ from the definition of $\eta$ or $\omega(k,p)$ will be $-i \pi$  or $+i \pi$,
making the dispersion relation to have two branches (allowing solitons to move either in the same direction or in the opposite one).
In the case $N=3$, we have similarly $p$, its exponentials are the cubic roots of the unity.
The Lax pairs can be obtained in the same way but imposing periodicity on the spectral functions  namely:
$$
\left(
\begin{array}{c}
\psi_1(n+1,m+1)\\
\psi_2(n+1,m+1)\\
\end{array}\right)=L_{n,m}\left(
\begin{array}{c}
\psi_1(n,m)\\
\psi_2(n,m)\\
\end{array}\right)
$$
$$
\partial_t\left(
\begin{array}{c}
\psi_1(n,m)\\
\psi_2(n,m)\\
\end{array}\right)=B_{n,m}\left(
\begin{array}{c}
\psi_1(n,m)\\
\psi_2(n,m)\\
\end{array}\right).
$$
Imposing, let us say $N=3$ periodicity, we get $\psi_i(n,m)\equiv u^i_{1}(n), \: \psi_i(n,m+1)=u^i_{2}(n), \:
\psi_i(n,m+2)=u^i_{3}(n), \: \psi_i(n,m+3)=u^i_1(n), \: etc.$ for any $i=1,2$. We will obtain 6 equations and $6\times 6$ Lax pair matrices that we have already found above (\ref{Lax 3 cuplat}).

\section{Time discretization}
We are going to perform fully integrable discretzation of our system ($t$ discretization) using again the Hirota bilinear formalism \cite{hir}.
It is easy to discretize the bilinear equation. Just replace the continuous bilinear operator with a discrete one and impose the
gauge-invariance (i.e. invariance with respect to the multiplication with exponential of linears) \cite{hir}:
\begin{equation*}
{\bf D_t} a \cdot b \rightarrow \frac{\tilde a b-a \tilde b}{\delta},
\end{equation*}
where $\delta$ is the discrete step.
\begin{equation*}
{\bf D_t} a \cdot b=a_t b-a b_t=\frac{\tilde a -a}{\delta} b-a \frac{\tilde b -b}{\delta}=\frac{1}{\delta}(\tilde a b-a \tilde b),
\end{equation*}
where $\tilde a=a((m+1)\delta,n)$ and $t$ becomes $m \delta$.

So, our general system (\ref{sys1}), as mentioned in (\ref{N-bil}), has the following bilinear form:
\begin{eqnarray*}
i {\bf D_t} G_\mu \cdot F_\mu&=&\overline{G_{\mu+1}} \underline{F_{\mu-1}} + \underline{G_{\mu-1}} \overline{F_{\mu+1}} \nonumber\\
F_\mu^2+|G_\mu|^2&=&\overline{F_{\mu+1}} \underline{F_{\mu-1}},
\end{eqnarray*}
where $\mu=1,...,N, \: F_0=F_N, \: F_{N+1}=F_1,\: G_0=G_N, \: G_{N+1}=G_1$.

Discretizing the first bilinear equation we get:
\begin{eqnarray*}
i (\widetilde G_\mu F_\mu -G_\mu \widetilde F_\mu)=\delta(\widetilde {\overline{G_{\mu+1}}} \underline{F_{\mu-1}}+\underline{G_{\mu-1}}\widetilde {\overline{F_{\mu+1}}} ) \nonumber,
\end{eqnarray*}
where $F_\mu=F_\mu(m\delta,n)$, $G_\mu=G_\mu(m\delta,n)$, $\widetilde {\overline{G_{\mu}}}=G_\mu((m+1)\delta,n+1)$ and  $\widetilde {\overline{F_\mu}}=F_\mu((m+1)\delta,n+1)$.

Using Hirota-Tsujimoto approach \cite{hir}, \cite{DNLS} we are not changing the second bilinear equation (otherwise the bilinear system will not have two-soliton solution). Our fully discrete bilinear system will be:
\begin{eqnarray}
i (\widetilde G_\mu F_\mu-G_\mu \widetilde F_\mu)&=&\delta(\widetilde {\overline{G_{\mu+1}}} \underline{F_{\mu-1}}+\underline{G_{\mu-1}}\widetilde {\overline{F_{\mu+1}}} ) \label{dis}\\
F_\mu^2+|G_\mu|^2&=&\overline{F_{\mu+1}} \underline{F_{\mu-1}}. \label{con}
\end{eqnarray}

Integrability of the bilinear system (\ref{con}), (\ref{dis}) can be seen from the existence of 3-soliton solutions, which has the same form as (\ref{solutie generalaa}), namely:

\begin{eqnarray}\label{solutie generalaa}
G_{\mu}&=&\sum\limits_{\nu=0,1}D_2(\underline{\nu})\exp\left({\sum\limits_{i=1}^{6}\nu_i [\eta_i+(\mu-1)\log(\epsilon_i)]+\sum\limits^{6}_{1\leq i<j}\nu_i\nu_j\phi_{ij}}\right)\nonumber\\
F_{\mu}&=&\sum\limits_{\nu=0,1}D_1(\underline{\nu})\exp\left({\sum\limits_{i=1}^{6}\nu_i [\eta_i+(\mu-1)\log(\epsilon_i)]+\sum\limits^{6}_{1\leq i<j}\nu_i\nu_j\phi_{ij}}\right)
\end{eqnarray}
where $D_1(\underline{\nu})$, $D_2(\underline{\nu})$, $\eta_i$, $\phi_{ij}$ are given in (\ref{coeficienti generali})
with the difference that now we have $N$ branches of dispersion for each soliton  with the following fully discrete form:
\begin{equation}
\exp{\delta \: \omega_j(k_j)}=\frac{i-\delta \epsilon_j^{-1} e^{-k_j}}{i+\delta \epsilon_j e^{k_j}}, \quad \epsilon_j \in \left\{e^{l \frac{2 \pi i}{N}} \right\},  \: l=1,...,N \: j=1,2,3 \nonumber
\end{equation}
where  $k_j$ is the wave number and $j$ is the index of the soliton. The Lax pair of this systems is unknown to us. However because we have three soliton solution we expect the system to be completely integrable.

Dividing (\ref{dis}) by $F_\mu \widetilde F_\mu$ and putting $q_\mu=G_\mu/F_\mu$ and $\Gamma_\mu={\widetilde {\overline{F_{\mu+1}}} \underline {F_{\mu-1}} }/{F_\mu \widetilde F_\mu}$, we get:
\begin{eqnarray}
i (\widetilde{q_\mu} -q_\mu)&=&\delta (\widetilde {\overline{q_{\mu+1}}}+\underline{q_{\mu-1}}) \Gamma_\mu \label{comp1}\\
1+|q_\mu|^2&=&\frac{{\overline{F_{\mu+1}}} \underline {F_{\mu-1}} }{F_\mu^2}. \label{comp2}
\end{eqnarray}

But, using (\ref{comp2}) we get:
\begin{equation*}\label{comp3}
\frac{\overline{\Gamma_{\mu+1}}}{\Gamma_\mu}=\frac{\widetilde{\overline{\overline{F_{\mu+2}}}} \: \:F_{\mu}}{\overline{F_{\mu+1}} \:\widetilde{\overline{F_{\mu+1}}} } \:\:  \frac{F_\mu \:\:\widetilde F_\mu}{\widetilde {\overline{F_{\mu+1}}} \: \:\underline{F_{\mu-1}} }
=\left(  \frac{\widetilde{\overline{\overline{F_{\mu+2}}}} \: \: \widetilde{F_{\mu}}} { \:\widetilde{\overline{F_{\mu+1}}}^2 } \right) \left ( \frac{F_\mu ^2 }{\overline{F_{\mu+1}} \: \: \underline{F_{\mu-1}} }\right)= \frac {1+|\widetilde{\overline{q_{\mu+1}}}|^2}{1+|q_\mu|^2}.
\end{equation*}

Finally, our fully discrete Ablowitz-Ladik is  the following system:
\begin{eqnarray*}
i (\widetilde{q_\mu} -q_\mu)&=&\delta (\widetilde {\overline{q_{\mu+1}}}+\underline{q_{\mu-1}}) \Gamma_\mu\\
\frac{\overline{\Gamma_{\mu+1}}}{\Gamma_\mu}&=& \frac {1+|\widetilde{\overline{q_{\mu+1}}}|^2}{1+|q_\mu|^2}, \quad \mu=1,...,N.
\end{eqnarray*}
Now, eliminating $\Gamma_\mu$, we will obtain the following (higher order) fully discrete Ablowitz-Ladik system:

\begin{equation}
\left(\frac{\widetilde{\overline{q_{\mu+1}}}-\overline{q_{\mu+1}}}{\widetilde{\overline{\overline{q_{\mu+2}}}}+q_\mu}\right)\left(\frac{\widetilde{\overline{q_{\mu+1}}}+\underline{q_{\mu-1}}}{\widetilde{q_{\mu}}-q_\mu}\right) \left(\frac{1+|q_\mu|^2}{1+|\widetilde{\overline{q_{\mu+1}}}|^2} \right)=1, \quad \mu=1,..N.
\end{equation}

\section{Conclusions}
In this paper we studied coupled Ablowitz-Ladik equations with branched dispersion relations. The main motivation was to see that the integrability survives and how to study collisions of solitons in different directions. It was shown by Hirota bilinear formalism and Lax pairs that the system is integrable and moreover it was shown that with a very simple periodic reduction of an 2D integrable Ablowitz-Ladik equation all the results are recovered. This simple method was applied initially in \cite{toki-fane} but we think that it can be widely extended to a lot of semidiscrete and discrete equations to obtain integrable matrix systems with branched dispersions. In fact, for any completely integrable semidiscrete equation
$F(\dot u_n, u_n,u_{n+1},u_{n-1})=0$, its 2D variant $F(\dot u_{n,m} u_{n,m}u_{n+1,m+1},u_{n-1,m-1})=0$, if it is integrable, then by various periodic reduction in $m$, one can obtain integrable matrix systems.

From the point of view of applications we do not know any physical application. However, because the semidiscrete NLS can model motion of discrete curves \cite{doliwa} we expect that the systems described in this paper can describe motion of
{\it modular} polymers (containing different types of segments), exactly as in the case of modular gene networks which can be modelled \cite{toki-fane} by semidiscrete KdV systems with branched dispersion.
We intend to do this in a future publication.

\vskip 0.5cm
{\bf Acknowledgements:} The work is supported by the project PN-II-ID-PCE-2011-3-0083, Romanian Ministery of Education and Research. ASC is supported by the project PN-II-ID-PCE-2011-3-0137,
Romanian Ministery of Education and Research.
Also we are very grateful to the referees for all the obervations.

redpajama_set_name: RedPajamaArXiv


In her paintings, Liza takes inspiration from her surroundings in coastal New England. She lives in Waltham and frequents Gloucester to paint, working abstractly and representationally, depending on the mood.
Liza has studied under artists such as Richard Raiselis, Dana Clancy, Harold Reddicliffe, Rokhaya Waring and Caleb Stone. She has completed many commissions and her paintings are held in private collections internationally and in the United States.
Liza teaches K-5 Art at the Hemenway Elementary School in Framingham, MA. She previously served as the lower school art teacher for three years at Glen Urquhart School in Beverly, MA. Liza attended Boston University where she earned a BFA in Painting and an MA in Art Education.


text:
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Andrew Olmsted built the first FBReader packages for Harmattan, after tweaking the build system a bit. The desktop version of FBReader already used Qt 4, and ran almost unmodified, but with some bugs (segfault on task switch) and ugly UI.
I started with the Ubuntu packages for FBReader, since they used a more sane build system for .debs (compared to upstream's funky shell script). Some tweaks were needed to make it build in Scratchbox: since GTK+ and Hildon libraries aren't available on Harmattan, I had to disable the building of -gtk and -maemo versions of libzlui. I also got to learn a new tool — quilt.
Fixing the segfault took a couple of days of debugging, studying the source code of both FBReader and Qt itself, and asking for help on IRC. Turns out FBReader was holding an active QPainter instance for too long, and its backing pixmap got destroyed (or, rather, converted from an OpenGL texture to a plain X11 pixmap) during a task switch, causing the crash. I'm probably describing this wrong BTW, but, in any case, adding QPainter::begin() and QPainter::end() calls in the paintEvent handler fixed the segfault.
Next, a small tweak in the .desktop file to make FBReader a single-instance application: change Exec=FBReader to Exec=single-instance /usr/bin/FBReader (I'm paraphrasing slightly).
Then, a more ambitious goal: making FBReader intercept volume keys and use them for scrolling. Google gave me a pointer to QmKeys, which was the wrong API to use here, but gave me a lead to qmkeyd2, which appears to be an open source daemon, which gave me a lead to sysuid, another open source daemon, which in turn gave me a lead to libresourceqt, and that was the right API at last.
Volume keys generate regular key events for XF86AudioRaiseVolume and XF86AudioLowerVolume, but they're also intercepted by qmkeyd2, which tells all subscribers (and sysuid is one) about them. Which subscriber gets to react is determined by the resource policy framework. So what I needed to do in FBReader was acquire the ScaleButtonResource when FBReader starts up or gets focus, and release it when FBReader quits or goes into background. That also required some IRC help until I discovered installEventFilter() and the ApplicationActivate/ApplicationDeactivate events. And QApplication::instance().
The various tools available in the developer firmware were invaluable: openssh, gdb, valgrind, strace, xev, xprop, lsof, netstat. Also, I would not have achieved my second goal without being able to look at the sources of Meego system components (qmkeyd, sysuid). Yay open source!
Here are my changes to the source code. You can find my modified Debian packaging files, as well as prebuilt binary packages (with full debug info, for gdb goodness), in my experimental harmattan apt repository. The UI is still ugly and non-native, but it doesn't matter much in fullscreen mode :) .
Note to self: when next building fbreader, make sure the 2 megabyte tags file doesn't end up in the .diff.gz. And speaking of crud in source packages, the vim package I built for Harmattan the other day contains the entire 50 meg .hg in the .orig.tar.gz. I need to figure out how to tell dh_make to omit it.


SANFORD, FL—Out of jail after posting a $1 million bond, George Zimmerman told reporters Monday he remained firmly committed to community safety and had no intention of letting a single unpleasant episode prevent him from fulfilling his regular neighborhood watch duties. "Just because I had one little setback doesn't mean I'm suddenly going to forget my pledge to keep the Retreat at Twin Lakes safe and crime-free," said Zimmerman, adding that his arrest on charges of murdering 17-year-old Trayvon Martin was just a "single crummy day," and he was eager to get back to patrolling his gated subdivision of townhouses for suspicious activity. "So I had a bad experience and it spooked me. So what? I've got to dust myself off and try again. After all, they need me out there." At press time, Zimmerman reported that being back behind the wheel of his SUV and quietly scanning his darkened neighborhood streets again "feels good. Feels right."


How do you make a good first impression, and how do you make a very good first impression?
Does it start with hello?
Logically you say yes of course; like Hello, Hi, Hola, Ola, Bon Jour, Dias Buenas and many other greetings are usually the first sound and words we express to others.
In fact, the first impression is long before the first word is published.
If this study is accurate then there is a huge 55% – 93% anyone will "pick up" on you intuition, emotional or energy level before you even have the opportunity to talk to them. Remember, when you have a smile in a perfect stranger just because …?
How does body language and ultra-emotional and emotional intelligence give people the opportunity to get to know you intuitively and decide if they like you or even if they want to say hello … without knowing you?
Making a good first impression, it's amazing and amazing how much we can perceive each other through our presentation, our beliefs, how we dress and expressions we have on our face. You might say "I just had this feeling …"
How can you tell if someone had a bad day? What's to say rumors?
The tumors decreased, head hanging low, sad or angry looking at their faces. There are people who see, when they enter the room, as they come with a "black cloud" of darkness. They may be busy, not paying attention or busy on a call or just "lost in their own thoughts."
A person who is happy, charismatic or interested gives a very different character and you will get a very different first vision. What are the characteristics of this person?
Hopefully, happy, attractive and present at the moment?
Jumping in the step, the bright smile, even if they are in the phone, they are pleasant to others by talking directly to them. Do you know someone who has the ability to spice up the whole room when they come in … just by the energy caused by their appearance or new or healthy glow.
What first impression do you want the project to affect your family, friends, customers, and members each time you enter the room and before you say hello?
What kind of energy, enthusiasm and emotional intelligence are you looking for in relationships and the team you build around you?
You may have heard this testimony from Ralph Waldo Emerson – "Whoever you speak so loudly, I don't hear what you say."
You can read that when emotions are high, then the intelligence is low and the inverse is true, when the information component IQ is high emotional intelligence EQ, is low.
How to get a good balance of intellectual and emotional intelligence so that you can be passionate, brilliant, quick, charismatic and create a very good first impression. What skills do you have the ability to be interested, consistent, receptive and non-responsive?
So you always have a good first impression when you're excited and charismatic before you say hello!
To become aware, alert and mindful of the impressions and effects people make on each other, it is useful to understand the different levels of energy and emotional information and how it affects your world.
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Ontario Mayor's Court
Ontario Mayor's Court includes only charges involving fines totaling $150 or more. The sentences and fines listed here were handed down between Aug. 1 and 14.
Justin L. Carroll, 709 Park Avenue East, Mansfield; non compliance; $400 fine.
Shirley M. Dillen, 941 Oakland Park Ave., Columbus; DUS license forfeiture; $200 fine.
Rodney A. Martin, 710 Clay St. Lot 63, Galion; DUS license forfeiture; $300 fine.
William Townsend Jr., 50 Carlton Ave., Shelby; DUS license forfeiture; $200 fine.
Robert R. Adkins, 100 N. Third St. Apt. 21, Shelby; petty theft; $400 fine; 30 days in jail, suspended.
Craig S. Henry, 1097 W. Longview Apt. 12, Mansfield; petty theft; $375 fine; 30 days in jail, suspended.
James R. Montgomery, 124½ W. Third St., Mansfield; (4) petty theft; (4) $400 fine; (4) 30 days in jail, suspended.


redefinition of the powers and tasks, logistics.
intelligent and green future transport.
of injustice, to solve in consultation and dialogue.
pubic affairs, procedures improvement, etc.
and evaluation of policies in the Health sector.
ICE EEIG was established to maximise the strengths of its members (know-how, experts' network, financial capacity) ensuring the availability of a broad range of skills to the challenges of large Technical Assistance projects worldwide.
ICE (International Consulting Expertise) is based in Brussels and the group has other permanent branch offices (including in IPA and ACP countries) as well as several project offices around the world.
Christian Linck is the Managing Director, who reports to and liaises with ICE EEIG's Management Board which includes the Managing Directors of the members.
Together we work, enjoy and bring the change.
ICE objective is to use different strengths (expertise, experience, network, financial capacity) of several international organisations in joining forces for bidding on large scale technical assistance funded projects and in implementing and managing the awarded projects.
We permanently employ an average of 150 experts on short-term and long-term assignments, we have several permanent offices (Ankara, Belgrade, Brussels (headquarters), Bucharest, Marseille and Paris), tens of project offices worldwide and an average annual turnover of more than 15 million euros. We have implemented more than 1000 projects in 179 countries, partnered with more than 300 companies worldwide and have developed a sophisticated interactive experts' database of more than 12,000 CVs.


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Sido and Happy enjoy each other's company at Nyaru Menteng, Kalimantan, Borneo
Maybe I'm too idealistic, but I like to look for sanctuaries in this troubled world, places that are safe and apart from the influence of the world and that let you heal and grow.
For the last seven days I have been in a place like that, at the Nyaru Menteng sanctuary for orphan orangutans near Palangkaraya, in Kalimantan, Borneo.
I am here as I start work on a new book about morality in apes and human beings, after leading a small eco-tour with my friend Gary Shapiro to see orangutans roaming freely at Camp Leakey, in Tanjung Puting national park in Kalimantan.
And I don't think I could be in a better place to observe morality in apes and humans than in a sanctuary like Nyaru Menteng, originally set up under the Borneo Orangutan Survival foundation by the visionary Danish angel Lone Droscher-Neilsen, whom I met here in 2004.
The current manager here, a 37-year-old Javanese man named Anton Nurcahyo, says that I am still in the honeymoon phase of being at Nyaru Menteng, but that is because he is too modest and practical (unlike me) about has accomplishments after two years here.
Having visited most of the orangutan centres, I can say that this one is well managed because of the joy I see in both the orangutans and the staff, which you don't see in the places where people and primates feel tension and pressure. The staff laugh and joke with each other and the orangutans approach them as companions would, unafraid.
Two-year-old Miko at Nyaru Menteng
Some mornings I go out in the jungle with Hani, the staff and a group of about 10 orangutans, for the orangutans to get some jungle time in the trees, which, for them, is the most interesting place to be. These orangutans are orphans who lost their mother because of human beings and need to learn how to function in the jungle. Nyaru Menteng has more than 600 orangutans, with a staff of 200, on 60 hectares of buildings, grounds and forest, plus some additional patches of land on a few islands and in other locations.
After the morning jungle romp, I return for the end-of-the-day meal and playtime from 3-5 p.m. Some of the orangutans still have some energy left from the trees and roll, stretch and play with their friends.
I spend the evening and sleep in a staff cabin where I have to dodge a vicious macaque on the narrow walkway over the swamp and watch for the renegade female orangutan who borrowed the cushions I put on the rattan couch on the deck. I am happy if I could make her life more comfortable.
I have to say that being here in Borneo with the orangutans I have forgotten about my other life -- except for my partner Wendy, of course, and my children, Pearce and Caitlin -- and it takes some effort to remind myself that I live in North American and have responsibilities there.
One day I was even drifting off in a hammock in the jungle with the orangutans, watching the sky and the trees for entertainment as they would, oblivious like the orangutans to the babble of Indonesian around me -- when the voice of Hani cut through asking, "WHAT ARE YOU DOING, MR?" "Thinking," I said, which seemed like what a writer and university professor should be doing. Since then, whenever anyone seems to be catnapping for a moment, it is called "thinking" in honour of me.
But that's the magic of a sanctuary too. It satisfies you and fulfills you and gives you the space to think unobstructed. Too many abstractions and theories can blind you. I think that being in places like Nyaru Menteng will help me understand in a practical way how we can establish a moral purpose that resists the destructiveness of the world and works to maintain a healthy community of people and apes.
There are so many different theories of what is good and moral, without agreement of what actually works, so Nyaru Menteng for me could be a living model of how these things can function.
I hope that Nyaru Menteng will last many years. I suspect it will seem much the same to me as today when the honeymoon is ended. At least, that's want we need to believe in a world like ours. We need to think that this place is possible, that, while it would be too idealistic to think that it is perfect, it still survives all the temptations of a difficult world with its basic integrity intact, that it deserves the name of "sanctuary."
Posted by Unknown on Friday, June 01, 2012


The debate over whether it's still worth to major in the humanities becomes louder during difficult economic times when many people, especially recent college graduates, are suffering. Critics of the humanities cite higher earnings and better employment statistics for people who majored in technical fields like STEM (science, technology, engineering, and mathematics) and healthcare. Proponents of humanities counsel patients patience, saying that general education provides a broad range of skills, such as critical thinking, offering individuals long-term flexibility in a changing labor market.
The truth lies somewhere in the middle. Engineering and other technical majors provide education and training that link directly to relevant job skills, which makes the transition into the labor market easier: STEM majors earn more and are employed at higher rates initially after graduating. For liberal arts and humanities majors, on the other hand, the benefits of their degree often aren't fully realized until later in life. Humanities and liberal arts majors also serve as one of the most traveled gateway to graduate education (43% of these majors get a graduate degree boosting median lifetime earnings from $1.6M to $2M (27%) over the prime age career (25-54). Furthermore, humanities majors with a graduate degree who become managers end up with median earnings of $103,000 – cutting the earnings gap between the humanities and engineers nearly in half.


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Q: Convert time to local time in Objective-C I'm having this issue where the correct time is passed but once it hits the [formatter setDateFormat:date] it comes out in a different time zone. Here is a snippet
NSDate* date = user.startDate;
NSDateFormatter *format = [[NSDateFormatter alloc] init];
NSLog(@"-------> (1) time is %@", date);
NSLog(@"-------> (2) time is %@", date);
[format setDateFormat:@"h:mm a"];
NSLog(@"-------> (3) time is %@", date);
NSLog(@"-------> (4) time is %@", [format stringFromDate:date]);

Here is the debugger output for the log messages
-------> (1) time is 2016-06-30 09:25:17 +0000
-------> (2) time is 2016-06-30 09:25:17 +0000
-------> (3) time is 2016-06-30 09:25:17 +0000
-------> (4) time is 2:25 AM

I'm expecting the output for (4) to be time is 9:25 AM

A: Here's what is going on: the log of a date will always be GMT. (I don't think it makes sense for me to get into details as there is a great explanation here.) That means that log 1, 2 and 3 will be "wrong" (not in your timezone) and that should be expected.
If, for instance, you want to log the string as GMT, you can change the property timeZone of your date formatter.
Dealing with dates can be extremely hard. Please take the time to read if not all, some important parts of this document (fundamentals/basics).

A: Your time zone is apparently UTC-7000.
NSLog prints NSDate objects always in UTC, NSDateFormatter considers the current time zone. The printed dates are actually the same.
If you really need all dates in UTC, set the time zone of the formatter.
format.timeZone = [NSTimeZone timeZoneForSecondsFromGMT:0];

By the way: There are built-in methods to check if a date is in today or yesterday.

redpajama_set_name: RedPajamaStackExchange


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Home / Bible Commentaries / Wesley's Explanatory Notes/ Jeremiah
Wesley's Explanatory Notes
Jeremiah Jeremiah 52
For Israel hath not been forsaken, nor Judah of his God, of the LORD of hosts; though their land was filled with sin against the Holy One of Israel.
Forsaken — Not utterly forsaken.
Flee out of the midst of Babylon, and deliver every man his soul: be not cut off in her iniquity; for this is the time of the LORD's vengeance; he will render unto her a recompence.
Soul — By soul is meant life, and by iniquity the punishment of the Babylonian's iniquity.
Babylon hath been a golden cup in the LORD's hand, that made all the earth drunken: the nations have drunken of her wine; therefore the nations are mad.
Drunken — She had made all the nations about her drunken with the Lord's fury.
Mad — Through the misery they felt from her.
We would have healed Babylon, but she is not healed: forsake her, and let us go every one into his own country: for her judgment reacheth unto heaven, and is lifted up even to the skies.
We — The prophet seems to personate the mercenary soldiers, saying, they would have helped Babylon, but there was no healing for her.
The LORD hath brought forth our righteousness: come, and let us declare in Zion the work of the LORD our God.
Some — These words are spoken in the person of the Jews, owning the destruction of Babylon to be the mighty work of God, and an act of justice, revenging the wrongs of his people.
Set up the standard upon the walls of Babylon, make the watch strong, set up the watchmen, prepare the ambushes: for the LORD hath both devised and done that which he spake against the inhabitants of Babylon.
Set up — These seem to be the prophet's words to the Babylonians, rousing them out of their security. Historians tell us that the city was fortified by walls of fifty cubits high, and two hundred cubits broad, and by a very deep and large ditch.
O thou that dwellest upon many waters, abundant in treasures, thine end is come, and the measure of thy covetousness.
Waters — Babylon is said to dwell upon many waters, because the great river Euphrates, did not only run by it, but almost encompass it branching itself into many smaller rivers, which made several parts of the city, islands.
I will also break in pieces with thee the shepherd and his flock; and with thee will I break in pieces the husbandman and his yoke of oxen; and with thee will I break in pieces captains and rulers.
Break in pieces — The sense of all these three verses is the same; that God had made use, and was still making use of the Babylonians to destroy many nations, to spoil much people, wasting their goods, routing their armies, killing all sorts of their inhabitants.
Behold, I am against thee, O destroying mountain, saith the LORD, which destroyest all the earth: and I will stretch out mine hand upon thee, and roll thee down from the rocks, and will make thee a burnt mountain.
Mountain — Babylon was very high for its power, and greatness, and had very high walls and towers, that it looked at a distance like an high rocky mountain. They had destroyed many people.
Burnt — Thy cities and towers which appear like a mountain shall be burnt.
Set ye up a standard in the land, blow the trumpet among the nations, prepare the nations against her, call together against her the kingdoms of Ararat, Minni, and Ashchenaz; appoint a captain against her; cause the horses to come up as the rough caterpillers.
As caterpillars — The Median horses are compared to their insects, either with respect to their numbers, or in regard of the terror caused by them when they came, being a great plague to the places which they infected.
And the land shall tremble and sorrow: for every purpose of the LORD shall be performed against Babylon, to make the land of Babylon a desolation without an inhabitant.
The land — Babylon, or the land of Chaldea.
One post shall run to meet another, and one messenger to meet another, to shew the king of Babylon that his city is taken at one end,
At one end — Cyrus entered the city at one end, by the channel of the river, which he had drained, and surprized Belshazzar in the midst of his feast.
And that the passages are stopped, and the reeds they have burned with fire, and the men of war are affrighted.
The passages — The passages over the river Euphrates, and all the other passages by which the Babylonians might make their escape, were guarded with soldiers.
Reeds — On the border of the river Euphrates were vast quantities of great and tall reeds, which with the mud in which they stood, were as another wall to the city; but the Medes had burnt them so as the way was open.
For thus saith the LORD of hosts, the God of Israel; The daughter of Babylon is like a threshingfloor, it is time to thresh her: yet a little while, and the time of her harvest shall come.
Threshing floor — Babylon had been a threshing instrument, by which, and a threshing-floor in which God had threshed many other nations; God now intended to make it as a threshing-floor wherein he would thresh the Chaldeans.
Tread her — So they used to prepare their threshing-floors against the time of harvest.
The time — The harvest which the justice of God would have from the ruin of the Chaldeans.
Nebuchadrezzar the king of Babylon hath devoured me, he hath crushed me, he hath made me an empty vessel, he hath swallowed me up like a dragon, he hath filled his belly with my delicates, he hath cast me out.
Me — The prophet speaks this in the name of the Jews.
Cast me out — As beasts of prey eat what they please of other beasts they have preyed upon, and leave the rest in the field.
Therefore thus saith the LORD; Behold, I will plead thy cause, and take vengeance for thee; and I will dry up her sea, and make her springs dry.
Dry up — Alluding to what Cyrus did.
They shall roar together like lions: they shall yell as lions' whelps.
They — The Babylonians, upon the taking of their city.
In their heat I will make their feasts, and I will make them drunken, that they may rejoice, and sleep a perpetual sleep, and not wake, saith the LORD.
Heat — When they shall grow hot with wine, I will make them a feast of another nature. Interpreters judge that Belshazzar, Daniel 5:1, made a feast to a thousand of his Lords, when he and his wives, and concubines, drank wine in the vessels belonging to the temple, during which feast the city was taken.
And not awake — While they were merry with their wine, they fell into a sleep which they never awoke out of.
How is Sheshach taken! and how is the praise of the whole earth surprised! how is Babylon become an astonishment among the nations!
Sheshach — A name given to the city of Babylon.
The sea is come up upon Babylon: she is covered with the multitude of the waves thereof.
The sea — A multitude of enemies.
And I will punish Bel in Babylon, and I will bring forth out of his mouth that which he hath swallowed up: and the nations shall not flow together any more unto him: yea, the wall of Babylon shall fall.
Bel — Bel was the principal Babylonian idol.
Bring forth — All the vessels of the temple, 2 Chronicles 36:7, and whatever gifts the Babylonians had presented to him.
The wall — And the city of Babylon shall be also ruined.
My people, go ye out of the midst of her, and deliver ye every man his soul from the fierce anger of the LORD.
Go out of her — At all hazards escape for your lives.
Then the heaven and the earth, and all that is therein, shall sing for Babylon: for the spoilers shall come unto her from the north, saith the LORD.
Then — All the creatures in heaven and earth shall rejoice at the vengeance which God shall take upon Babylon.
As Babylon hath caused the slain of Israel to fall, so at Babylon shall fall the slain of all the earth.
Of all the earth — This term must be understood in a restrained sense; the Chaldeans coming up from all parts of Chaldea to help Babylon, were slain there, as by the means of Babylon the Israelites were slain that came from all parts of Judea to help Jerusalem.
Ye that have escaped the sword, go away, stand not still: remember the LORD afar off, and let Jerusalem come into your mind.
Ye — Ye Jews, leave Babylon as soon as liberty is proclaimed.
Remember — And remember in Judea the great things both of justice and mercy which God hath done.
We are confounded, because we have heard reproach: shame hath covered our faces: for strangers are come into the sanctuaries of the LORD's house.
We — We Jews are ashamed to hear the enemies reproaching us, for our God, or for our religion.
Strangers — Pagans that were strangers to the commonwealth of Israel, are come, not to worship, but to plunder, the sanctuaries of the Lord; even into the courts of the priests and of the Israelites; yea, into the most holy place.
Wherefore, behold, the days come, saith the LORD, that I will do judgment upon her graven images: and through all her land the wounded shall groan.
Wherefore — For this profanation of my holy place, I will be revenged not only upon their idols, but upon the worshippers of them, and cause a groaning of wounded men over all the country of the Chaldeans.
Because the LORD hath spoiled Babylon, and destroyed out of her the great voice; when her waves do roar like great waters, a noise of their voice is uttered:
The great voice — The noises caused from multitudes of people walking up and trafficking together.
A noise — The noise of her enemies that shall break in upon her shall be like the roaring of the sea.
Because the spoiler is come upon her, even upon Babylon, and her mighty men are taken, every one of their bows is broken: for the LORD God of recompences shall surely requite.
Because — Little more is said here than was before, only the words hint the taking of Babylon by a surprize when the king, and the inhabitants were not aware of it, which we had before also told us, verse39,40.
Requite — The wrongs done to his people.
And I will make drunk her princes, and her wise men, her captains, and her rulers, and her mighty men: and they shall sleep a perpetual sleep, and not wake, saith the King, whose name is the LORD of hosts.
Drunk — A plain allusion to the posture the king of Babylon, and the thousand of his lords were in, when their city was taken while they were drinking wine in the bowls that were brought from the temple at Jerusalem.
Thus saith the LORD of hosts; The broad walls of Babylon shall be utterly broken, and her high gates shall be burned with fire; and the people shall labour in vain, and the folk in the fire, and they shall be weary.
Weary — Though the people should labour to quench this fire, or to rebuild this city, yet it would be all lost labour.
The word which Jeremiah the prophet commanded Seraiah the son of Neriah, the son of Maaseiah, when he went with Zedekiah the king of Judah into Babylon in the fourth year of his reign. And this Seraiah was a quiet prince.
In the fourth year — This circumstance lets us know that this prophecy was many years before Babylon was destroyed; for it was seven years before Jerusalem was taken; so as it must be above sixty years before it was fulfilled in the first degree.
And Jeremiah said to Seraiah, When thou comest to Babylon, and shalt see, and shalt read all these words;
Shalt read — Probably to the Jews, that were in Babylon.
Then shalt thou say, O LORD, thou hast spoken against this place, to cut it off, that none shall remain in it, neither man nor beast, but that it shall be desolate for ever.
Shalt say — Thou shalt testify that thou believest what thou hast read.
And thou shalt say, Thus shall Babylon sink, and shall not rise from the evil that I will bring upon her: and they shall be weary. Thus far are the words of Jeremiah.
Weary — With that weight of judgment which shall be upon them.
The words — The prophetical words of Jeremiah; for the matter of the next chapter is historical, and the book of Lamentations is not prophetical.
Wesley, John. "Commentary on Jeremiah 51:4". "John Wesley's Explanatory Notes on the Whole Bible". http://odl.studylight.org/commentaries/wen/jeremiah-51.html. 1765.


Croatia and France will face off for the biggest prize in football as the two European nations meet in the 2018 FIFA World Cup final today.
France overcame Belgium 1-0 in the semi-finals, with Samuel Umtiti securing a 1-0 victory as the Red Devils were unable to find a way through the dogged defence of Les Bleus.
Didier Deschamps could write his name into the history books by becoming only the third man to win the World Cup as both a player and a coach after Brazil's Mario Zagallo and Germany's Franz Beckenbauer.
He captained France to their only World Cup victory to date in 1998, and it will be Les Bleus' third appearance at a World Cup final after their loss to Italy in 2006.
Croatia, meanwhile, showed great resolve to battle back from a goal down against England to eventually win 2-1 in extra-time courtesy of a Mario Mandzukic volley. This will be Croatia's first appearance in a World Cup final and they could become the eighth different country to win the competition.
Coach Zlatko Dalic says his players should be proud of their achievements.
Croatia only has five pitches that meet Uefa's international standard. The infrastructure is mostly appalling and the investment in grassroots football is basically non-existent.
Talented players are forced to leave their clubs early because of the bad financial situation most Croatian clubs face, which prompts them to sell their prized assets once they hit double-digit caps.
There are just two home-based players in Croatia's World Cup squad – Dinamo Zagreb goalkeeper Dominik Livakovic and Rijeka midfielder Filip Bradaric. Between them they have played less than half an hour in Russia.
Croatia have no major injury concerns despite having played three 120-minute matches.
Dalic said on Saturday: "We have some minor injuries but I hope we will overcome those today and all my players will be ready to play.
France also have no injury worries and are likely to stick with the same side that beat Belgium in the semi-finals.
The two nations have met five times, and Croatia have never beaten France.
Their first meeting came at the 1998 World Cup, with Les Bleus winning 2-1 in the semi-finals en route to lifting the famous trophy on home soil.
France then won friendlies in 1999 (3-0) and 2000 (2-0).
However, their last two meetings have ended in draws – a 2-2 draw at Euro 2004, and a goalless stalemate in 2011.
Neither side will have any players suspended for the final, while the French squad has a clean bill of health.
It may be a different matter for Croatia though, with plenty of players feeling the strain after every one of their knockout games has gone to extra time – Ivan Strinic may well be a doubt after limping off with what appeared to be a groin issue in extra time against England.
France and Croatia have faced five times previously, with France winning three of those games (in 1998, 1999 and 2000) and the other two games ending as draws (2004 and 2011).
Two of France and Croatia's five meetings have been at major tournaments. France won 2-1 against Croatia in the semi-final of the 1998 World Cup – Deschamps playing the full match – while their meeting at Euro 2004 ended in a 2-2 draw in the group stage.
The last three World Cup finals have all gone to extra-time. However, historically, only two of the showpiece games have ever been decided on penalties (1994, 2006).
France have trailed for just nine minutes and 12 seconds at the 2018 World Cup, while Croatia have been behind in all three of their knockout matches.
Antoine Griezmann has scored or assisted 11 goals in nine knockout games at major tournaments (World Cup + Euro), more than any other player for France over the last 50 years ahead of Zinedine Zidane (8) and Michel Platini (6).
France's Kylian Mbappe will be looking to become only the second teenager to score in a World Cup final after Pele, in 1958. Mbappe (3 goals) is already the highest scoring teenager in a World Cup tournament since Pele (6 in 1958).