text:
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boost::gil::PixelDereferenceAdaptorConcept

PixelDereferenceAdaptorConcept Struct Template Reference

[Concepts]

#include <gil_concept.hpp>

List of all members.

Detailed Description

template<typename D>

 struct boost::gil::PixelDereferenceAdaptorConcept< D >

Represents a unary function object that can be invoked upon dereferencing a pixel iterator.

DEREFERENCE ADAPTOR CONCEPTS This can perform an arbitrary computation, such as color conversion or table lookup

concept PixelDereferenceAdaptorConcept<boost::UnaryFunctionConcept D>
  : DefaultConstructibleConcept<D>, CopyConstructibleConcept<D>, AssignableConcept<D>  {
    typename const_t;         where PixelDereferenceAdaptorConcept<const_t>;
    typename value_type;      where PixelValueConcept<value_type>;
    typename reference;         // may be mutable
    typename const_reference;   // must not be mutable
    static const bool D::is_mutable;

    where Convertible<value_type,result_type>;
};

Public Member Functions

void 	constraints ()

Public Attributes

D 	d

The documentation for this struct was generated from the following file:

gil_concept.hpp

<address style="text-align: right;">Generated on Sat May 2 13:50:17 2009 for Generic Image Library by

 1.5.6 </address>

redpajama_set_name: RedPajamaGithub


This SG700-S is new upgraded version of SG700, which features some new features like 1080P wide angle camera, gesture controlled videoing and palm manipulation. It is foldable, easy to carry in. Optical flow positioning provides more accurate and stable hovering fly.Black and white, two colors for your own option. Just come and get it!
With foldable transmitter, great design and easy to carry. 3D eversion function for you to experience cool flying play.
Optical flow positioning: this new tech guarantees a more accurate and stable hovering during the fly.
Palm manipulation: within about 1 meter, you can use palm to manipulate the aircraft to go up and down.
Altitude hold mode: with built in air barometer, the drone will stay at a certain height.
Gesture control mode: allows you to order the camera to take a photo or record a video by just doing relative hand gesture.
Follow me: smart portrait recognition of the camera will let the drone automatically follow the object you selected.
Please do not use in situation of many WiFi options, there may be signal interference. If you have trouble to connect the drone's wifi, please try to turn it off and on again for several times. Any other Wifi problem, please contact us as soon as possible.
Pay attention that the optical flow positioning will be influenced by lighting condition and other environmental factors, please read the optical flow sensor instruction for correct operation.
Do not continue flying over 3-4 batteries without taking a break, which is no good for the life of the drone.
Don't over-charge, or over-discharge batteries. Don't put it beside the high temperature condition. Don't throw it into fire.Don't throw it into water.
The phone in the picture is not included. There could be some deviations due to manual measurement or slight color difference owing to photographing condition. Thank you for your understand.


Despite the market trends in Indianapolis slowly shifting, it's still very much a buyer's market. With so much property available, you could find yourself waiting months or longer to get a fair price on your home.
One option that many homeowners are turning to is selling to a real estate investor like Indy Home Offer to buy your house from you.
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If you need to sell your Indianapolis home very quickly, with little hassle, selling to a Indianapolis real estate investor is a great option.


text:
Cutry (Aisne)
 Cutry (Meurthe-et-Moselle)

redpajama_set_name: RedPajamaWikipedia


It's natural to feel like we should cut back on our calories if we want to shed some pounds, but the reality is, it's the quickest way to get your body to store fat.
Why is that? Our bodies recognize distress signals fast, and hunger is a very important one. It's genetic. In times of hunting and gathering, food wasn't so readily available, so the body knew to store food as fat for times of famine.
When you keep your body from having food at reasonable intervals, it triggers the body to store future calories as fat. It's easy to avoid this, simply eat every three to four hours. Most importantly, eat whole foods: fruits, vegetables, nuts, meats, eggs and nothing processed.
Eating whole foods that are rich in all the nutrients and minerals your body needs keeps you from craving the foods that you don't need. Our bodies don't crave food when we support our system nutritionally. Eating sugar makes you crave sugar. Eating carbs makes you crave carbs. Why? Because your body is incredibly intuitive and it understands that it did not get what it was looking for with those last few calories you gave it, so it triggers hunger because it needs more. More calories? No, more nutrition.
So give your body what it needs and do it at regular intervals. Never go more than four hours without eating something healthy and full of vitamins and nutrients. Have an apple or a handful of nuts, keep it simple and notice how you feel. As you stabilize your blood sugar levels, your energy levels will increase, your cravings will diminish and those extra pounds you're not so fond of will fall right off.
Send me your thoughts and ideas on how to improve your health, lose weight and look better. It's never too late and it's almost always simple.


text:
Search - List of Books by Chris Taylor
Christopher Thornton "Chris" Taylor (born 15 July 1973) is an Australian comedian, writer and former radio host from Sydney. As a member of The Chaser, he is best known for co-writing and appearing on satirical ABC Television shows CNNNN (2002—2003) and The Chaser's War on Everything (2006 — July 2009). He formerly co-hosted the drive radio show Today Today (2004—2005) on Triple J with fellow Chaser member Craig Reucassel, and in 2007, he wrote the musical comedy Dead Caesar. With his Chaser colleague Andrew Hansen, Taylor recently made a musical comedy series for Triple J titled The Blow Parade, which became the number one podcast in the country.
Early Years more less
Taylor grew up on Sydney's Northern Beaches and North Shore and, (as with all The Chaser team except Craig Reucassel), attended a private school. Talking about his schooling at the Shore School in an interview once, he said that "It's not a school I have much respect for but in many ways it was the making of me" and that "there's nothing like a private school for breeding a good satirist". Taylor is also an atheist.
After completing secondary school, Taylor undertook an arts degree at the University of Sydney, and also studied playwrighting at NIDA. But at 21, he abandoned playwrighting and went to back to university to study journalism at University of Technology, Sydney. In 1997 he moved to Melbourne after the ABC offered him a journalism cadetship there.
For five years, Taylor worked as an ABC journalist, including two years as a court reporter. He covered cases including the murder of toddler Jaidyn Leskie, the fatal Longford gas explosion and the unsuccessful defamation action by the then Victoria premier Jeff Kennett against The Australian.Taylor then moved back to Sydney in 2001 to work full-time for The Chaser's newspaper, and ultimately became a Chaser member.
↑ Hide "Early Years" Section ↑
CNNNN more less
In 2001, Andrew Denton got hold of one of The Chaser's newspapers and gave them the opportunity to produce a television show, hence The Chaser team produced their first ABC Television program The Election Chaser, which covered the 2001 Australian Federal Election. They then went on to create other television shows for ABC TV, including two series of the Logie Award-winning CNNNN in 2002—03 (in which Taylor portrayed a chief commentator), and The Chaser Decides in 2004, which covered the 2004 Australian Federal Election.
↑ Hide "CNNNN" Section ↑
Triple J more less
In addition to his roles on television, Taylor has teamed up with Chaser colleague Andrew Hansen to write and star in a musical comedy series, The Blow Parade, on Australian radio station Triple J from 14 April 2010 until 26 May 2010. The program will also be available to download free of charge for a limited time.
Taylor also united with fellow Chaser Craig Reucassel to present the Triple J drive slot, Today Today in 2004—2005.
As part of his role on Today Today, Taylor wrote and voiced a weekly Coma FM sketch, satirising the formats and style of commercial radio. He also co-hosted the radio program Bloody Sunday with Reucassel in 2006 and 2007. During the shows Chris would regularly ask listeners to "go for broke" and vandalise Wikipedia pages. Tara Reid, Peter Overton and Wikipedia's own Wikipedia page were chosen targets. And in 2008 Taylor and Reucassel returned to present The Race Race, covering the US Presidential election. The show quickly became the number one podcast in the country.
Taylor also has the distinction of being Roy Slaven's official understudy at Triple J, broadcasting alongside HG Nelson on This Sporting Life and a Twenty20 cricket game on the couple of instances when Roy Slaven was absent.
In a notable instance while working at Triple J, Taylor agreed to streak across the grounds of the Sydney Big Day Out if the theme song from Media Watch polled in the 'Triple J Hottest 100' Countdown. It was announced that in terms of votes counted, the Media Watch theme had in fact come in at number 7, and Taylor proceeded to do his nude run in front of a large group of spectators.
↑ Hide "Triple J" Section ↑
the Chaser's War on Everything more less
Taylor, who's often referred to as "Tayls" by his colleagues, is one of the main presenters on The Chaser's War on Everything. He is also the program's script editor and principal writer behind many of the show's segments, sketches and songs.
Appearance on Sunrise
In the The Chaser's War on Everything skit "Will You Divorce Me?", Taylor made a guest appearance on Sunrise, a live morning talk show on Channel 7. He said he had an announcement for his wife and ended up yelling "Get the fuck out of my life" at the camera. The appearance of authenticity of this clip made it appear as though the segment went live to air on Sunrise, but it only ever appeared on The Chaser's War on Everything. Taylor and the hosts admitted it was a staged event, filmed on the Sunrise set after one of their shows but the latter said the dialogue was improvised and were shocked by the profane punchline. A clip of this segment became an Internet phenomenon
The Tim Freedman Parody
Taylor sparked controversy with Tim Freedman, singer and songwriter of indie rock band The Whitlams. On 7 October 2007, Freedman posted a message on the band's official website, citing a parody of him which was written by Taylor, and had aired on 3 October 2007 episode of The War. The song was performed by fellow Chaser Andrew Hansen, and the lyrics implied that Freedman was obsessed with the Sydney suburb of Newtown, and that he continually referred to the area of Newtown in his own songs. In his message, Freedman alluded that the lyrics were indeed written by Taylor, and that the parody had been motivated by jealousy over a girl they had both dated Anna Skellern who was a non-contributing cast member in the first season of CNNNN. Taylor denied this, and said that the song had nothing to do with Skellern. Andrew Hansen in the commentary track for the season 2 DVD revealed that Taylor and Freedman had sorted the issue out and that everything between them was all good. In the final episode of Season 3, Freedman even made a friendly cameo appearance on the show in a mock oral-sex scene with Taylor.
The Eulogy Song
On 17 October 2007 episode of The War, Andrew Hansen performed The Eulogy Song, which was a song written by Taylor which satirised the lives of several deceased celebrities, including Peter Brock, Princess Diana, Donald Bradman, Steve Irwin, Stan Zemanek, John Lennon, Jeff Buckley and Kerry Packer, expressing the view that people with flaws during life are often disproportionately hailed as "top blokes" after death. The song became the target of significant media attention, with several radio and television personalities saying the song was in "bad taste", as well as then Prime Minister of Australia John Howard and then Opposition Leader Kevin Rudd expressing negative views.
In response to the attention, Taylor defended the song, stating that it was a legitimate skit about the way the media air-brushes celebrities in death. He also revealed it was a "watered down" version of the song which was previously performed on stage in the musical Dead Caesar. The original had a verse referencing Rene Rivkin, but it was replaced with a verse about Stan Zemanek.
↑ Hide "the Chaser's War on Everything" Section ↑
Dead Caesar more less
As well as writing for The Chaser's War on Everything, Taylor wrote the play called "Dead Caesar" which premiered at the Sydney Theatre Company on 1 February 2007, and then had a one-month run in July 2007. The play was a musical comedy based on the life of Julius Caesar, featuring original music composed by Andrew Hansen, who also appeared in the play as Mark Antony and Lucius. The now infamous Eulogy Song was originally written for this production.
↑ Hide "Dead Caesar" Section ↑
This author page uses material from the Wikipedia article "Chris Taylor", which is released under the Creative Commons Attribution-Share-Alike License 3.0
Total Books: 99
Available Only List
2020 - Understanding and Treating Incels Case Studies Guidance and Treatment of VIolence Risk in the Involuntary Celibate Community (Paperback)
Genres: Health, Fitness & Dieting, Law, Medical Books
2019 - The New Pie Modern Techniques for the Classic American Dessert (Hardcover)
Genre: Cookbooks, Food & Wine
2017 - Extended Essay for the Ib Diploma Skills for Success (Paperback)
Genres: Reference, Education & Teaching
2016 - Company Law Uk Edition [Law Express] (Paperback)
Genres: Reference, Law
2016 - Asean3 People Business Travel (Hardcover)
Genres: Business & Money, Travel
2015 - The Perfect Husband [The Sydney Harbour Hospital Series - Volume 1] (Paperback)
2015 - How Star Wars Conquered the Universe the Past Present and Future of a Multibillion Dollar Franchise (Audio CD) → Paperback, Hardcover
Genres: Biographies & Memoirs, Business & Money, Humor & Entertainment
2015 - Practical Guide to Evidence (Paperback)
2015 - Twenty Dinners (Hardcover)
2014 - How Star Wars Conquered the Universe the Past Present and Future of a Four Billion Dollar Franchise (Hardcover)
2014 - Law Express Company Law (Paperback)
2014 - Law Express Evidence [Revision Guide] (Paperback)
2014 - The Predator [Munro Family, Bk 3] (Paperback)
Genres: Mystery, Thriller & Suspense, Romance
2014 - The Investigator [Munro Family, Bk 2] (Paperback)
2014 - The Profiler [The Munro Family Series - Volume 1] (Paperback)
Genres: Literature & Fiction, Romance
2013 - Riding the Dragon a Journey Through Every Chinese Province (Paperback)
2012 - Empathic Care for Children with Disorganized Attachments a Model for Mentalizing Attachment and Trauma-informed Care (Paperback)
Genres: Health, Fitness & Dieting, Reference, Medical Books
2012 - Law Express Constitutional and Administrative Law Revision (Paperback)
2010 - Harvest Season a Novel (Paperback)
Genre: Literature & Fiction
2010 - Leatherwork a Practical Guide (Hardcover)
2010 - A Practical Guide to Caring for Children and Teenagers with Attachment Difficulties (Paperback)
Genres: Parenting & Relationships, Nonfiction, Science & Math
2010 - China (Other)
2010 - Evidence [Law Express] (Paperback)
2010 - Constitutional Administrative Law Uk Edition [Law Express] (Paperback)
2009 - Textbook on Constitutional and Administrative Law (Paperback)
2009 - Granted a Teacher S Guide to Writing Winning Classroom Grants (Paperback)
2009 - Faewolf (Paperback)
2009 - Land Arts of the American West (Hardcover)
2009 - The Formula for Success in Network Marketing (Paperback)
2009 - Constitutional and Administrative Law in the Uk [Law Express] (Paperback)
2009 - Elizabeth (Paperback)
2008 - Incubo Atacama Lab [English and Spanish Edition] (Paperback)
2008 - Law Express and Constitutional and Administrative Law 14th Edition Supplement Constritutional and Administrative Law (Paperback)
2008 - Creativity Never Rests the Ronnie Duncan Collection (Paperback)
2008 - Constitutional and Administrative Law and Law Express Constitutional and Administrative Law (Paperback)
2008 - Statistical Models of Shape Optimisation and Evaluation (Hardcover)
2008 - Foundations of Playwork (Paperback)
2008 - A Pocketful of Wisdom Life Lessons of 21st Century Achievers (Hardcover)
2008 - Constitutional Administrative Law [Law Express] (Paperback)
2007 - Constitutional and Administration Law and Law Express Constitutional and Administrative Law (Paperback)
2004 - Combining Methods in Educational Research [Conducting Educational Research] (Paperback)
2003 - Schools Markets and Choice Policies (Hardcover)
2002 - Opening Up a New World (Paperback)
2002 - Traveler's Companion VIetnam Laos Cambodia 2nd [Traveler's Companion Series] (Paperback)
2001 - Traveler's Companion Japan 2nd [Traveler's Companion Series] (Paperback)
2001 - Parks and Gardens of Britain (Paperback)
2001 - Traveler's Companion China 2nd [Traveler's Companion Series] (Paperback)
2000 - Size Matters Does School Choice Lead to Spirals of Decline - Measuring Markets - the Case of the Era 1988 (Paperback)
2000 - The Welsh Bus Handbook (Paperback)
1999 - The Beautiful Game Journey Through Latin American Football (Paperback)
1999 - Traveler's India Companion [Traveler's Companion] (Paperback)
1999 - Traveler's Companion India (Paperback)
1999 - Lonely Planet Tibet [4th ed] (Paperback)
1998 - Core Java Web Server (Paperback)
1998 - Indonesia [Traveler's Companion] (Paperback)
1998 - Traveler's Companion Indonesia (Paperback)
1998 - Traveler's Companion China 98-99 (Paperback)
1998 - The Beautiful Game (Paperback)
1997 - Traveler's Nepal Companion (Paperback)
1997 - Japan [Lonely Planet] (Paperback)
1997 - Lonely Planet Indonesia (Paperback)
1997 - Lonely Planet Southeast Asia on a Shoestring [Lonely Planet on a Shoestring Series] (Paperback)
1996 - Lonely Planet Malaysia Singapore Brunei a Travel Survival Kit [6th ed] (Paperback)
1996 - Lonely Planet Cambodia a Travel Survival Kit [2nd ed] (Paperback)
1996 - Lonely Planet China a Travel Survival Kit [Lonely Planet China] (Paperback)
1996 - Change the Context Change the Text (Paperback)
1995 - Lonely Planet Tokyo [Lonely Planet Travel Survival Kit] (Paperback)
1994 - Lonely Planet Japan (Paperback)
Geography of the New Education Market Secondary School Choice in England and Wales (Hardcover)
Traveler's Companion Indonesia 2nd [Traveler's Companion Series] (Paperback)
Traveler's Companion India 2nd [Traveler's Companion] (Paperback)
Lonely Planet Chinese Phrasebook [Lonely Planet Language Survival Kit] (Paperback)
Lonely Planet Tokyo City Guide [Lonely Planet Tokyo] (Paperback)
Lonely Planet Seoul City Guide [Lonely Planet City Guide] (Paperback)
Lonely Planet North East Asia [Lonely Planet North-East Asia on a Shoestring] (Paperback)
Lonely Planet Mandarin Phrasebook [Lonely Planet Language Survival Kit] (Paperback)
Compliance and the Art of Customer Communication a Handbook for the Working Veterinary Practice (Paperback)
Law for Business Students and Law Express Company Law (Paperback)
India [Traveler's Companion] (Paperback)
The Compleat Design Experiment From Soup to Nuts [Research Paper Series 2006] (Paperback)
Combating Copying Teaching Enquiry Research Skills in History [Devon curriculum advice history] (Paperback)
Specialist Schools in England Track Record and Future Prospect [Occasional paper series] (Paperback)
Rcbn Consultation Exercise the Stakeholder Report [Cardiff University, School of Social Sciences, Occasional Paper S.] (Paperback)

redpajama_set_name: RedPajamaCommonCrawl


text:
Student Gets Nod From Coldplay For Video That Took 5 Years To Make
Filed Under:Coldplay, college
Coldplay has been nominated for: Best Pop Duo/Group Performance. (Credit: Cornelius Poppe/AFP/Getty Images)
By Chelsea Lacey-Mabe
PHILADELPHIA (CBS) — Want to see someone age five years in under five minutes?
It seems impossible but University of Virginia student, Sam Reid did it by taking a picture of himself every day for five years and turning the photos into a video set to Coldplay's "Life in Technicolor ii."
"It took me five years to sing this song," the video description reads.
Meanwhile, it took him just a few days to get the attention of thousands who have since watched, liked and shared his video, including Coldplay themselves who tweeted it.
This Life In Technicolor ii video is amazing! (And the T-shirt budget must have been substantial.) A https://t.co/Nq1v4XwZ7u
— Coldplay (@coldplay) May 8, 2016
Reid, who is also a cross country coach at a Christian day school in Charlottesville, was inspired by fellow YouTuber, Noah Kalina, known for a stop-motion video of pictures over 12.5 years.
"I was inspired by other "photo a day" projects, but I wanted to add my own twist to it, so I came up with the idea to lip sync the lyrics over the course of the video," Reid explained.
Reid started his video project in February 2011, when he was a junior in high school and all throughout college where he studied Classics with a concentration in Ancient Greek. He finally finished the video last month, ending on a picture of himself in a cap and gown–a preview of his upcoming graduation which was on Saturday.


text:
\section{Introduction}

More than 40
years have passed since
the Korteweg--de Vries (KdV)
equation
was
solved
by Gardner {\it et al.}~\cite{GGKM}
using the inverse scattering method
based on
its Lax pair~\cite{Lax}.
The number of
known integrable
systems
following
the KdV equation,
particularly
partial differential equations (PDEs)
in \mbox{$1+1$} space-time dimensions,
has
increased
enormously,
and
various techniques to
study them
have been
developed.
Recently the center of researchers' interest
has
shifted from
continuous
PDEs
to
differential-difference or partial difference equations
wherein at least one of the independent variables
takes discrete values.
A major problem in
this
trend
is
how to find a suitable
difference
analogue of a
given
differential equation.
The suitable discretization
of an integrable continuous system
is generally
required to retain
the integrability~\cite{Suris03},
but
that is not sufficient if
the original
continuous
system has
some essential
internal symmetries.
This becomes conspicuous if we
consider
integrable discretizations of
nonlinear Schr\"odinger (NLS)-type systems,
i.e.,
two-component
systems of second order
that allow
the complex conjugation reduction
between the two dependent variables.
As a prototypical example, we
discuss
the
cubic NLS system in \mbox{$1+1$} dimensions~\cite{AKNS},
\begin{subequations}
\label{nonredNLS}
\begin{align}
& \mathrm{i} q_t + q_{xx} - 2q^2 r = 0,
\label{NLS1}
\\
& \mathrm{i} r_t - r_{xx} + 2r^2 q = 0.
\label{NLS2}
\end{align}
\end{subequations}
Note that
the reduction
\mbox{$r=\sigma q^\ast$}
with a real constant $\sigma$
simplifies
the two-component system
(\ref{nonredNLS}) to
the scalar NLS equation~\cite{ZS1,ZS2}.
In addition,
(\ref{nonredNLS}) is invariant under the space reflection
\mbox{$x \to -x$}
as well as
the time reflection
\mbox{$t \to -t$} with \mbox{$q \leftrightarrow r$}.
The
suitable
and
elegant
space discretization of
the NLS system (\ref{nonredNLS}) was
proposed by
Ablowitz and Ladik~\cite{AL1}
in the form
\begin{subequations}
\label{sd-AL}
\begin{align}
& \mathrm{i} q_{n,t} + (q_{n+1} + q_{n-1} -2 q_n)
        - q_n r_n (q_{n+1} + q_{n-1}) =0, \\
& \mathrm{i} r_{n,t} - (r_{n+1} + r_{n-1} -2 r_n)
        + r_n q_n (r_{n+1} + r_{n-1}) =0.
\end{align}
\end{subequations}
Indeed,
system (\ref{sd-AL}) is
integrable
and,
with a rescaling of
variables,
reduces to (\ref{nonredNLS})
in
the continuous
space limit.
Moreover, (\ref{sd-AL})
allows the complex conjugation reduction
between
$q_n$ and $r_n$
and
possesses
the same invariance properties
with respect to the space/time reflection
as the continuous system (\ref{nonredNLS}).
Although
35 years have already passed since
their work,
the Ablowitz--Ladik
discretization
(\ref{sd-AL})
is still
a rare example
of success.
Indeed,
even now,
only a
small number of suitable
space discretizations of integrable
NLS-type systems are known (see, {\it e.g.},~\cite{Tsuchi02});
they are all
\mbox{$(1+1)$}-dimensional
systems with only one discrete spatial variable.
The problem of
how to discretize the continuous
time variable in
such
systems
is
an
interesting
topic~\cite{2010JPA},
but we
do
not discuss it in this paper.

The
main objective of this paper is to provide
the {\it first} example of a suitable
discretization
of an NLS-type system in \mbox{$2+1$} dimensions.
In particular,
we
consider
the discretization
of {\it both} spatial variables in
a \mbox{$(2+1)$}-dimensional NLS system known as
the Davey--Stewartson system~\cite{DS74}
(also see~\cite{Benney69}).
Note that
the Davey--Stewartson system
is integrable~\cite{Hab75,Morris77,Anker,Ab78,Cornille}
and
appears to be
the only genuinely \mbox{$(2+1)$}-dimensional
generalization of the NLS system (\ref{nonredNLS})
(cf.~the Calogero--Degasperis system~\cite{Calo76}).
Moreover,
even if we
include
other types of
integrable systems,
the
list of known
systems with two discrete
and one continuous
independent variables
is still
very
short.
Thus, it is a highly nontrivial and
challenging
task
to obtain
the suitable
space
discretization
of the Davey--Stewartson system.
To solve
this problem,
we first propose
a
natural
\mbox{$(2+1)$}-dimensional generalization
of
the Ablowitz--Ladik lattice
(cf.~(\ref{sd-AL}))
by constructing its Lax pair.
This \mbox{$(2+1)$}-dimensional Ablowitz--Ladik lattice
certainly
reduces to the Davey--Stewartson system
in the continuous
space limit.
A relevant Lax pair
as well as
the resulting system
was previously
studied
by other authors~\cite{Hu06}
(also see~\cite{Hu07}),
but the time part of our
Lax pair
is essentially more general than
the previously known one~\cite{Hu06}.
As a result,
the time evolution of
our
system
is
a linear
combination
of four elementary time evolutions, two of which
were
previously unknown.
Moreover, it can be shown that the
four time evolutions
are mutually
commutative.
Thus, the \mbox{$(2+1)$}-dimensional Ablowitz--Ladik lattice
is general enough and appears to be
promising.
However, it
does not allow the complex conjugation reduction
directly
and thus is not
a suitable space
discretization of
the Davey--Stewartson system
in
its
present form.
To fix
this shortcoming,
we only have to
consider
a
certain nonlocal
transformation of dependent variables,
which
symmetrizes the
equations of motion.
Thus,
we
obtain
the suitable
space discretization of the
Davey--Stewartson system
that indeed
allows the complex conjugation reduction
between the new
variables
after the transformation.
In addition,
the invariance properties of the continuous
Davey--Stewartson system with respect to space/time reflections
turn out to be properly
incorporated in
our
space-discrete Davey--Stewartson system.

This paper is organized as follows.
In section 2, we propose a \mbox{$(2+1)$}-dimensional version
of the Ablowitz--Ladik lattice by considering an appropriate
generalization of the
Lax pair
for the
original Ablowitz--Ladik lattice.
To uncover
how the complex conjugation reduction
can be imposed
as an NLS-type system,
we
consider
a nonlocal change
of dependent variables;
it can
turn the \mbox{$(2+1)$}-dimensional Ablowitz--Ladik lattice
into
the suitable space discretization of
the Davey--Stewartson system
that
indeed
allows the complex
conjugation
reduction
between the new
dependent
variables.
In section 3,
we elucidate how
the general
time evolution considered
can be
decomposed into
four
elementary
time
evolutions corresponding to the
four
directions
on the two-dimensional lattice.
On the basis of this decomposition
and
using the Hirota
bilinear method~\cite{Hirota04},
we
construct
some
exact solutions
of
the \mbox{$(2+1)$}-dimensional Ablowitz--Ladik lattice
and the space-discrete Davey--Stewartson system.
In particular,
multidromion solutions
are presented explicitly.
The last section, section 4,
is devoted to concluding remarks.

\section{Derivation based on Lax pairs}

\subsection{\mbox{$(2+1)$}-dimensional Ablowitz--Ladik lattice}
\label{2.1}

As a generalization of the
Lax pair
introduced
by
Ablowitz and Ladik~\cite{AL1},
we consider
the following
linear system
on
the
two-dimensional lattice:
\begin{subequations}
\label{DS1-Lax}
\begin{align}
& \boldsymbol{\Delta}_n^+ \psi_{n,m} :=
\psi_{n+1,m} -\psi_{n,m} = q_{n,m} \phi_{n,m},
\label{DS1-Lax1}
\\
& \boldsymbol{\Delta}_m^+ \phi_{n,m} :=
\phi_{n,m+1} -\phi_{n,m} = r_{n,m} \psi_{n,m},
\label{DS1-Lax2}
\\
& \frac{\partial \psi_{n,m}}{\partial t}
  =a \psi_{n,m+1} + A_{n,m} \psi_{n,m} + C_{n,m} \psi_{n,m-1}
   + b q_{n-1,m} \phi_{n,m} - q_{n,m} D_{n,m} \phi_{n-1,m},
\label{DS1-Lax3}
\\
& \frac{\partial \phi_{n,m}}{\partial t}
  =b \phi_{n+1,m} + B_{n,m} \phi_{n,m} + D_{n,m} \phi_{n-1,m}
   + a r_{n,m-1} \psi_{n,m} - r_{n,m} C_{n,m} \psi_{n,m-1}.
\label{DS1-Lax4}
\end{align}
\end{subequations}
Here, $\boldsymbol{\Delta}_n^+$ and $\boldsymbol{\Delta}_m^+$
denote the forward difference operators in
each
spatial
direction, and
the parameters
$a$ and $b$ are arbitrary constants.
The time dependence of the functions
is
usually
suppressed.
The compatibility conditions
\mbox{$\partial_t \boldsymbol{\Delta}_n^+ \psi_{n,m}=
	\boldsymbol{\Delta}_n^+ \partial_t \psi_{n,m}$}
and \mbox{$ \partial_t \boldsymbol{\Delta}_m^+ \phi_{n,m}=
	\boldsymbol{\Delta}_m^+ \partial_t \phi_{n,m}$}
for the linear system (\ref{DS1-Lax})
provide
the
\mbox{$(2+1)$}-dimensional Ablowitz--Ladik lattice,
\begin{subequations}
\label{DS1}
\begin{align}
 \frac{\partial q_{n,m}}{\partial t}
 &+ q_{n+1,m} D_{n+1,m} + b q_{n-1,m} - a q_{n,m+1}
  - C_{n+1,m} q_{n,m-1}
\nonumber \\[-0.5mm] &
 + q_{n,m} B_{n,m} - A_{n+1,m} q_{n,m}=0,
\label{DS1-q}
\\[1mm]
 \frac{\partial r_{n,m}}{\partial t}
 &+ r_{n,m+1} C_{n,m+1} + a r_{n,m-1} - b r_{n+1,m} - D_{n,m+1} r_{n-1,m}
\nonumber \\[-0.5mm] &
 + r_{n,m} A_{n,m} - B_{n,m+1} r_{n,m}=0,
\label{DS1-r}
\\[1mm]
& \hspace{-10mm}
A_{n+1,m} -A_{n,m} = -a (q_{n,m+1} r_{n,m} -q_{n,m} r_{n,m-1}),
\label{fdAL1}
\\[1mm]
& \hspace{-10mm}
B_{n,m+1} -B_{n,m} = -b (r_{n+1,m} q_{n,m} -r_{n,m} q_{n-1,m}),
\label{fdAL2}
\\[1mm]
& \hspace{-10mm}
(1- q_{n,m} r_{n,m}) C_{n,m} =C_{n+1,m}(1- q_{n,m-1} r_{n,m-1}),
\label{fdAL3}
\\[1mm]
& \hspace{-10mm}
(1- r_{n,m} q_{n,m}) D_{n,m} =D_{n,m+1}(1- r_{n-1,m} q_{n-1,m}).
\label{fdAL4}
\end{align}
\end{subequations}
Indeed,
if
all the functions
depend on $n$ and $m$ only through
\mbox{$n+m$},
(\ref{DS1}) reduces to
the original
Ablowitz--Ladik lattice~\cite{AL1};
the latter contains
the integrable discrete NLS system (\ref{sd-AL}) as a special case.
To restore
the \mbox{$(1+1)$}-dimensional
Lax pair
involving
the spectral parameter $z$, we
set
\mbox{$\psi_{n,m}= z^{m-n} \psi'_{n,m}$}
and \mbox{$\phi_{n,m}= z^{m-n-1} \phi'_{n,m}$},
rewrite the linear problem (\ref{DS1-Lax})
in terms of \mbox{$\psi'_{n,m}$} and \mbox{$\phi'_{n,m}$},
and then consider
the dimensional reduction.

Under appropriate boundary conditions at spatial infinity,
we can
use (\ref{fdAL1})--(\ref{fdAL4}) recursively
to express the auxiliary
fields
$A_{n,m}$, $B_{n,m}$, $C_{n,m}$, and $D_{n,m}$
globally in terms of $q_{n,m}$ and $r_{n,m}$.
Thus, they can be considered as the defining relations
for the auxiliary fields,
and
the \mbox{$(2+1)$}-dimensional Ablowitz--Ladik lattice
(\ref{DS1}) has intrinsically nonlocal nonlinearity.
Note that
relations
(\ref{fdAL1})--(\ref{fdAL4})
already appeared in
the literature on
an integrable time discretization
of the
\mbox{$(1+1)$}-dimensional
Ablowitz--Ladik lattice~\cite{AL77,Chiu77,Suris03}.
Incidentally, in the stationary case of
\mbox{$\partial_t q_{n,m} = \partial_t r_{n,m}=0$},
(\ref{DS1})
reduces to a nontrivial system of partial difference equations
in \mbox{$1+1$} dimensions.

Using the simple transformation
\[
q_{n,m} = q'_{n,m} \mathrm{e}^{\gamma t},\hspace{5mm}
r_{n,m} = r'_{n,m} \mathrm{e}^{-\gamma t},
\]
where $\gamma$ is a constant, and
omitting the prime,
we can introduce the terms $+\gamma q_{n,m}$ and $-\gamma r_{n,m}$
in (\ref{DS1-q}) and (\ref{DS1-r}), respectively.
Clearly,
these terms can be absorbed by
constant shifts of $A_{n,m}$ and $B_{n,m}$.

\subsection{Continuum limit}

By choosing the parameters appropriately
and taking
the continuous space limit,
we can reduce system (\ref{DS1})
to the continuous
Davey--Stewartson system.
To see this,
we first
shift
the auxiliary fields
as
\begin{align}
& A_{n,m} = b-a + \widehat{A}_{n,m},\hspace{5mm}
  B_{n,m} = a-b + \widehat{B}_{n,m},
\nonumber \\
& C_{n,m} = a + \widehat{C}_{n,m},\hspace{5mm}
  D_{n,m} = b + \widehat{D}_{n,m},
\nonumber
\end{align}
and
rewrite (\ref{DS1}) as
\begin{subequations}
\label{DS-limit}
\begin{align}
\frac{\partial q_{n,m}}{\partial t}
& + b (q_{n+1,m} + q_{n-1,m}-2q_{n,m})
 - a (q_{n,m+1} +q_{n,m-1}-2q_{n,m})
\nonumber \\ &
 + q_{n+1,m} \widehat{D}_{n+1,m}
 - \widehat{C}_{n+1,m} q_{n,m-1}
 + q_{n,m} \widehat{B}_{n,m} - \widehat{A}_{n+1,m} q_{n,m}=0,
\label{}
\\
 \frac{\partial r_{n,m}}{\partial t}
& + a (r_{n,m+1} +r_{n,m-1} -2r_{n,m})
  - b (r_{n+1,m}+ r_{n-1,m}-2r_{n,m})
 \nonumber \\ &
  + r_{n,m+1} \widehat{C}_{n,m+1} - \widehat{D}_{n,m+1} r_{n-1,m}
  + r_{n,m} \widehat{A}_{n,m} - \widehat{B}_{n,m+1} r_{n,m}=0,
\label{}
\\
& \hspace{-10mm}
	\widehat{A}_{n+1,m} -\widehat{A}_{n,m}
	= -a (q_{n,m+1} r_{n,m} -q_{n,m} r_{n,m-1}),
\label{}
\\
& \hspace{-10mm}
	\widehat{B}_{n,m+1} -\widehat{B}_{n,m}
	= -b (r_{n+1,m} q_{n,m} -r_{n,m} q_{n-1,m}),
\label{}
\\
& \hspace{-10mm}
 \widehat{C}_{n+1,m} - \widehat{C}_{n,m}
 = - a(q_{n,m} r_{n,m} - q_{n,m-1} r_{n,m-1})
 +\widehat{C}_{n+1,m} q_{n,m-1} r_{n,m-1}- q_{n,m} r_{n,m} \widehat{C}_{n,m},
\\
& \hspace{-10mm}
 \widehat{D}_{n,m+1} - \widehat{D}_{n,m}
 = - b (r_{n,m} q_{n,m}-r_{n-1,m} q_{n-1,m})
 + \widehat{D}_{n,m+1} r_{n-1,m} q_{n-1,m} - r_{n,m} q_{n,m}\widehat{D}_{n,m}.
\label{}
\end{align}
\end{subequations}
Subsequently, we rescale
the
variables
and
parameters as
\begin{align}
& q_{n,m}=\varDelta x \cdot q (x, y), \hspace{5mm}
 r_{n,m}=\varDelta y \cdot r (x,y), \hspace{5mm}
 x:=n \, \varDelta x , \hspace{5mm} y:=m\,\varDelta y,
\nonumber \\
& \widehat{A}_{n,m}= A (x,y), \hspace{5mm}
  \widehat{B}_{n,m}= B (x,y), \hspace{5mm}
  \widehat{C}_{n,m}= C (x,y), \hspace{5mm}
  \widehat{D}_{n,m}= D (x,y),
\nonumber
\end{align}
wherein the time dependence is suppressed
and
\begin{align}
a =
\frac{\alpha}{(\varDelta y)^2}, \hspace{5mm}
b= \frac{\beta}{(\varDelta x)^2}.
\nonumber
\end{align}
Thus, in the continuum limit
\mbox{$\varDelta x, \varDelta y \to 0$},
(\ref{DS-limit}) reduces to the continuous
Davey--Stewartson system~\cite{Ab78,Morris77,Cornille,Nizh80,Nizh82},
\begin{subequations}
\label{DS-conti}
\begin{align}
& q_t + \beta q_{xx}
 - \alpha q_{yy}- (A+C) q + q (B+D) =0,
\label{}
\\
& r_t + \alpha r_{yy} - \beta r_{xx} + r (A+C) - (B+D) r =0,
\label{}
\\
& A_{x} =C_x = -\alpha (q r)_y,
\label{}
\\
& B_y = D_y = -\beta (rq)_x.
\label{}
\end{align}
\end{subequations}
Note that the Davey--Stewartson system (\ref{DS-conti})
is a linear combination
of the
two commuting flows
corresponding to \mbox{$\alpha=0$}, \mbox{$\beta \neq0$}
and \mbox{$\alpha \neq 0$}, \mbox{$\beta=0$}~\cite{Kaji90,MY97}
(also see~\cite{Nizh82}).
In subsection~\ref{subs_commut},
we present
its discrete analogue,
that is,
the \mbox{$(2+1)$}-dimensional Ablowitz--Ladik lattice (\ref{DS1})
is a linear combination of four
commuting flows.
This
is
a
quite
natural
result because
(i) each of the two
Davey--Stewartson flows
provides
an asymmetric
\mbox{$(2+1)$}-dimensional generalization of
the
NLS system and
(ii) the
Ablowitz--Ladik
discretization
of the NLS system
is actually
a sum
of two
elementary
flows
(and
one
trivial
flow)
in the same
hierarchy~\cite{AL1,Suris03,Kako,Kulish,GIK84}.

\subsection{Noncommutative
extension}

Actually,
the \mbox{$(2+1)$}-dimensional Ablowitz--Ladik lattice
(\ref{DS1}) is integrable
in
the general case
where the dependent variables take their values in matrices,
as long as
the operations such as
addition and
multiplication
make sense.
In that case,
``$1$'' in (\ref{fdAL3})
and (\ref{fdAL4}) should be interpreted
as the identity matrix.

We can further
generalize
it to a variable-coefficient
system
wherein
the parameters $a$ and $b$ become
arbitrary matrix-valued
functions of
one
spatial variable
as
\mbox{$a_m:=a(m)$} and \mbox{$b_n:=b(n)$}.
To obtain
such an
extension,
we
consider the
following
generalization of
the
linear system (\ref{DS1-Lax}):
\begin{subequations}
\label{nonDS1-Lax}
\begin{align}
\psi_{n+1,m} &= z \psi_{n,m} + q_{n,m} \phi_{n,m},
\label{nonDS1-Lax1}
\\[1mm]
\phi_{n,m+1} &= z^{-1} \phi_{n,m} + r_{n,m} \psi_{n,m},
\label{nonDS1-Lax2}
\\[1mm]
 \frac{\partial \psi_{n,m}}{\partial t}
  &= z a_m \psi_{n,m+1} + A_{n,m} \psi_{n,m}
  +z^{-1} C_{n,m} \psi_{n,m-1}
\nonumber \\[-0.5mm] & \hphantom{=} \;
  +z^{-1} q_{n-1,m} b_{n-1} \phi_{n,m} - q_{n,m} D_{n,m} \phi_{n-1,m},
\label{nonDS1-Lax3}
\\[1mm]
 \frac{\partial \phi_{n,m}}{\partial t}
  & =z^{-1} b_n \phi_{n+1,m} + B_{n,m} \phi_{n,m}
	+ z D_{n,m} \phi_{n-1,m}
\nonumber \\[-0.5mm] & \hphantom{=} \;
   + z r_{n,m-1} a_{m-1} \psi_{n,m} - r_{n,m} C_{n,m} \psi_{n,m-1}.
\label{nonDS1-Lax4}
\end{align}
\end{subequations}
Note that the
``spectral parameter''
$z$ is
nonessential in the \mbox{$(2+1)$}-dimensional case
and can be
fixed at $1$
as described in subsection~\ref{2.1}.
The compatibility conditions for
the linear system (\ref{nonDS1-Lax})
indeed provide
the
noncommutative
system with
site-dependent
coefficients,
\begin{subequations}
\label{nonDS1}
\begin{align}
\frac{\partial q_{n,m}}{\partial t}
 & + q_{n+1,m} D_{n+1,m} + q_{n-1,m} b_{n-1} - a_m q_{n,m+1}
  - C_{n+1,m} q_{n,m-1}
\nonumber \\ &
 + q_{n,m} B_{n,m} - A_{n+1,m} q_{n,m}=0,
\label{}
\\[1mm]
\frac{\partial r_{n,m}}{\partial t}
& +r_{n,m+1} C_{n,m+1} +r_{n,m-1}a_{m-1} - b_n r_{n+1,m} - D_{n,m+1} r_{n-1,m}
\nonumber \\ &
 + r_{n,m} A_{n,m} - B_{n,m+1} r_{n,m}=0,
\label{}
\\[1mm]
& \hspace{-10mm}
A_{n+1,m} -A_{n,m} = -a_m q_{n,m+1} r_{n,m} + q_{n,m} r_{n,m-1}a_{m-1},
\label{}
\\[1mm]
& \hspace{-10mm}
B_{n,m+1} -B_{n,m} = -b_n r_{n+1,m} q_{n,m} +r_{n,m} q_{n-1,m}b_{n-1},
\label{}
\\[1mm]
& \hspace{-10mm}
(I- q_{n,m} r_{n,m}) C_{n,m} =C_{n+1,m}(I- q_{n,m-1} r_{n,m-1}),
\label{I-e}
\\[1mm]
& \hspace{-10mm}
(I- r_{n,m} q_{n,m}) D_{n,m} =D_{n,m+1}(I- r_{n-1,m} q_{n-1,m}).
\label{I-f}
\end{align}
\end{subequations}
If $q_{n,m}$ and $r_{n,m}$ are rectangular
matrices,
the identity matrix $I$ in (\ref{I-e}) and that
in (\ref{I-f})
have unequal
sizes.
In the commutative case of the parameters,
the site-dependent
nature of (\ref{nonDS1})
is nonessential if both
\mbox{$\prod_{m=-\infty}^\infty a_m$} and
\mbox{$\prod_{n=-\infty}^\infty b_n$} take nonzero
finite values.
Indeed, if
we change the variables as
\begin{align}
& q_{n,m}=
\left( \prod_{j=-\infty}^{m-1} a_j \right)^{-1}
 \left( \prod_{k=-\infty}^{n-1} b_k\right)
	\widetilde{q}_{n,m}, \hspace{5mm}
r_{n,m}=
\left( \prod_{j=-\infty}^{m-1} a_j \right)
 \left( \prod_{k=-\infty}^{n-1} b_k\right)^{-1}
	\widetilde{r}_{n,m}, \hspace{5mm}
\nonumber \\[1mm]
& C_{n,m} = a_{m-1}^{-1}\widetilde{C}_{n,m},
\hspace{5mm} D_{n,m} = b_{n-1}^{-1}\widetilde{D}_{n,m},
\nonumber
\end{align}
the site-dependent
parameters
can be normalized to
$1$.
We can
also
obtain a similar result
in the noncommutative case.

\subsection{Appropriate
change of dependent
variables}
\label{2.4}

For simplicity, in the following discussion,
we consider only the
commutative and constant-coefficient
case
wherein the parameters $a$ and $b$ are constants
and all the quantities are scalar.
Thus,
the lowest-order conservation law for (\ref{DS1}) is given by
\begin{align}
 \frac{\partial \log ( 1- q_{n,m}r_{n,m} )}{\partial t}
&= \boldsymbol{\Delta}_n^+
\left[ -b q_{n-1,m}r_{n,m}
 + D_{n,m}(1- q_{n-1,m}r_{n-1,m} )^{-1}q_{n,m} r_{n-1,m} \right]
\nonumber \\ &
+ \boldsymbol{\Delta}_m^+
\left[ - a q_{n,m} r_{n,m-1}
 +  C_{n,m}(1- q_{n,m-1} r_{n,m-1})^{-1}q_{n,m-1}r_{n,m} \right].
\label{DScons}
\end{align}
The existence of an ultralocal conserved density
\mbox{$\log ( 1- q_{n,m}r_{n,m} )$}
implies that
a
nonlocal
transformation involving infinite products
of \mbox{$(1- q_{n,m}r_{n,m})^\delta$}
with \mbox{$\delta \neq 0$}
could be
applied (cf.~\cite{GI2});
this is indeed the case as we
will
see below.

The \mbox{$(2+1)$}-dimensional Ablowitz--Ladik lattice (\ref{DS1})
is invariant under a
space reflection
\mbox{$(n,m)
\to (-m,-n)$}
with a minor redefinition
of the parameters and
the auxiliary fields.
However, (\ref{DS1})
does not
allow
the complex conjugation reduction
between
$q_{n,m}$ and $r_{n,m}$
in the local form.
Therefore,
we need to
identify
new ``conjugate'' variables
instead of $q_{n,m}$ and $r_{n,m}$ and
rewrite
(\ref{DS1}) in
a more symmetric form
using
the new
variables.
For this purpose,
we
consider a gauge transformation
so that the spatial part of the Lax representation
obtains
invariance
with respect to the combined space reflection
\mbox{$(n,m) \to (-n,-m)$}
or, equivalently, a
180 degree
rotation
around the origin.
Thus, we apply
the gauge transformation
\begin{equation}
\psi_{n,m} = X_{n,m}
\Psi_{n,m},
\hspace{5mm}
\phi_{n,m} = Y_{n,m}
\Phi_{n,m}
\label{gauge1}
\end{equation}
to (\ref{DS1-Lax1}) and (\ref{DS1-Lax2})
and
change the dependent variables as
\begin{equation}
u_{n,m} = \frac{Y_{n,m}
}{X_{n,m}
}q_{n,m}, \hspace{5mm}
v_{n,m} = \frac{X_{n,m}
}{Y_{n,m}
} r_{n,m}.
\label{trans1}
\end{equation}
Here, $X_{n,m}$ and $Y_{n,m}$ are defined as
\begin{align}
\label{XY}
& X_{n,m}:=\frac{1}{h_m} \prod_{j=-\infty}^{n-1} \sqrt{1- q_{j,m} r_{j,m}},
\hspace{5mm}
Y_{n,m}:= \frac{1}{l_n} \prod_{k=-\infty}^{m-1} \sqrt{1- q_{n,k} r_{n,k}}.
\end{align}
The
norming
functions
$h_m (t)$ and $l_n (t)$ are introduced to
realize the complex conjugation reduction
between $u_{n,m}$ and $v_{n,m}$; they
will be
determined
later.
One can also use
\mbox{$\left(
\prod_{k=m}^{+\infty} \sqrt{1- q_{n,k} r_{n,k}} \right)^{-1}$}
instead of \mbox{$\prod_{k=-\infty}^{m-1} \sqrt{1- q_{n,k} r_{n,k}}$}
to maintain the invariance
under the space reflection
\mbox{$(n,m)
\to (-m,-n)$}.
This modification
causes no essential difference in the following discussion,
so
the transformed system
can become
ideally symmetric
with respect to
space reflections.
Here and hereafter,
we assume that
\mbox{$\abs{q_{n,m} r_{n,m}} \ll 1$}
so that \mbox{$\sqrt{1- q_{n,m} r_{n,m}}$}
and its inverse as well as their infinite products as considered
above
are
uniquely
and
well defined.
For example, we
consider that
\mbox{$q_{n,m}
=
O (\varDelta x)$} and \mbox{$r_{n,m} = O (\varDelta y)$}
(cf.~(\ref{DS1-Lax1}) and (\ref{DS1-Lax2})),
and
\mbox{$\sqrt{1- q_{n,m} r_{n,m}}$}
is defined
as
the Maclaurin series
in \mbox{$q_{n,m} r_{n,m} = O(\varDelta x \hspace{1pt}\varDelta y)$}.
Note that \mbox{$u_{n,m} v_{n,m}=q_{n,m} r_{n,m}$}, so that
the inverse transformation of (\ref{trans1}) can be obtained immediately.
Thus, the spatial part of the Lax representation
acquires the form
\begin{equation}
\left[
\begin{array}{c}
 \Psi_{n+1,m} \\
 \Phi_{n,m+1} \\
\end{array}
\right]
= \frac{1}{\sqrt{1- u_{n,m} v_{n,m}}}
\left[
\begin{array}{cc}
1 &  u_{n,m} \\
v_{n,m} & 1 \\
\end{array}
\right]
\left[
\begin{array}{c}
 \Psi_{n,m} \\
 \Phi_{n,m} \\
\end{array}
\right].
\label{DS2-Lax1}
\end{equation}
Very recently,
D.~Zakharov has
considered essentially the same scattering
problem
in~\cite{Dmitry09-1,Dmitry09-2}.
However,
this is an accidental coincidence, because
the first author
arrived at
this Lax representation as well as its generalization
implied
in
subsection~\ref{2.5}
independently
before
the papers~\cite{Dmitry09-1,Dmitry09-2} appeared.
The invariance
under the combined space reflection
\mbox{$(n,m) \to (-n,-m)$}
can be
easily
seen if we shift
the indices of the linear wavefunction by $1/2$, i.e.,
\[
\left[
\begin{array}{c}
 \widehat{\Psi}_{n+\frac{1}{2},m} \\
 \widehat{\Phi}_{n,m+\frac{1}{2}} \\
\end{array}
\right]
= \frac{1}{\sqrt{1- u_{n,m} v_{n,m}}}
\left[
\begin{array}{cc}
1 &  u_{n,m} \\
v_{n,m} & 1 \\
\end{array}
\right]
\left[
\begin{array}{c}
 \widehat{\Psi}_{n-\frac{1}{2},m} \\
 \widehat{\Phi}_{n,m-\frac{1}{2}} \\
\end{array}
\right],
\]
where \mbox{$\Psi_{n,m}=:\widehat{\Psi}_{n-\frac{1}{2},m}$}
and \mbox{$\Phi_{n,m} =:\widehat{\Phi}_{n,m-\frac{1}{2}}$}.
Indeed, because the determinant of
the spatial Lax matrix above
is unity,
we obtain
\[
\left[
\begin{array}{c}
 \widehat{\Psi}_{n-\frac{1}{2},m} \\
 - \widehat{\Phi}_{n,m-\frac{1}{2}} \\
\end{array}
\right]
= \frac{1}{\sqrt{1- u_{n,m} v_{n,m}}}
\left[
\begin{array}{cc}
1 &  u_{n,m} \\
v_{n,m} & 1 \\
\end{array}
\right]
\left[
\begin{array}{c}
 \widehat{\Psi}_{n+\frac{1}{2},m} \\
 -\widehat{\Phi}_{n,m+\frac{1}{2}} \\
\end{array}
\right].
\]

It should be possible to apply
the transformation
(\ref{trans1}) with (\ref{XY})
directly
to
the \mbox{$(2+1)$}-dimensional Ablowitz--Ladik lattice
(\ref{DS1})
and
derive the transformed
equations of motion
with the aid of the conservation law (\ref{DScons}).
However,
the
nonlocal nature of (\ref{DS1})
makes such a computation
rather complicated and difficult.
Thus,
as an
alternative,
we
apply
the gauge transformation (\ref{gauge1})
with (\ref{trans1}) and (\ref{XY})
to the time
part of the Lax representation,
(\ref{DS1-Lax3}) and (\ref{DS1-Lax4}), and determine
the time evolution of
the
gauge-transformed
wavefunction,
$\partial_t \Psi_{n,m}$ and $\partial_t \Phi_{n,m}$.
Its compatibility
with
the scattering problem
(\ref{DS2-Lax1})
can
provide
the transformed
equations of motion.

\subsection{Space-discrete
Davey--Stewartson system}
\label{2.5}

Before applying
the
transformation described
in subsection~\ref{2.4},
we
fix
the
boundary conditions for
the \mbox{$(2+1)$}-dimensional Ablowitz--Ladik lattice
(\ref{DS1})
as
\begin{subequations}
\label{ALbc1}
\begin{align}
& \lim_{n \to - \infty} (q_{n,m}, r_{n,m}) =
\lim_{m \to - \infty} (q_{n,m}, r_{n,m}) = \vt{0},
\label{qr-bc}
\\[1mm]
& \lim_{n \to - \infty} C_{n,m} = c \left( \frac{h_{m-1}}{h_{m}}\right)^2,
\hspace{5mm}
\lim_{m \to - \infty} D_{n,m} = d \left( \frac{l_{n-1}}{l_{n}}\right)^2.
\label{ALbc1-CD}
\end{align}
\end{subequations}
However,
we do not fix
\mbox{$\lim_{n \to - \infty} A_{n,m}$}
and
\mbox{$\lim_{m \to - \infty} B_{n,m}$}
in order
to obtain interesting solutions such as dromion solutions;
they
can
also depend on
the remaining spatial variable and
time $t$.
In (\ref{qr-bc}), the dynamical variables
$q_{n,m}$ and $r_{n,m}$ are assumed to approach zero
sufficiently rapidly.
In (\ref{ALbc1-CD}),
$c$ and $d$ are constants.
The defining
relations
(\ref{fdAL3}) and (\ref{fdAL4})
enable $C_{n,m}$ and $D_{n,m}$ to be expressed globally
as
\[
C_{n,m}= c \left(\frac{X_{n,m}}{X_{n,m-1}}\right)^2,
\hspace{5mm}
D_{n,m}= d \left(\frac{Y_{n,m}}{Y_{n-1,m}}\right)^2.
\]
Thus, the gauge transformation (\ref{gauge1})
with (\ref{trans1}) and (\ref{XY})
changes (\ref{DS1-Lax3}) and (\ref{DS1-Lax4}) to
\begin{align}
 \frac{\partial \Psi_{n,m}}{\partial t}
 &=  a\frac{X_{n,m+1}}{X_{n,m}} \Psi_{n,m+1}
 + \widetilde{A}_{n,m} \Psi_{n,m}
 + c \frac{X_{n,m}}{X_{n,m-1}} \Psi_{n,m-1}
\nonumber \\[1mm] & {\hphantom =}\;
 + b \frac{Y_{n,m}}{Y_{n-1,m+1}} u_{n-1,m}\Phi_{n,m}
-d \frac{Y_{n,m}}{Y_{n-1,m}} u_{n,m}\Phi_{n-1,m}
\nonumber
\end{align}
and
\begin{align}
 \frac{\partial \Phi_{n,m}}{\partial t}
 &=b\frac{Y_{n+1,m}}{Y_{n,m}} \Phi_{n+1,m} +\widetilde{B}_{n,m} \Phi_{n,m}
 +d\frac{Y_{n,m}}{Y_{n-1,m}}\Phi_{n-1,m}
\nonumber \\[1mm] & {\hphantom =}\;
 +a \frac{X_{n,m}}{X_{n+1,m-1}}v_{n,m-1} \Psi_{n,m}
- c \frac{X_{n,m}}{X_{n,m-1}} v_{n,m}\Psi_{n,m-1},
\nonumber
\end{align}
where
\begin{align}
\label{trans2}
& \widetilde{A}_{n,m} := A_{n,m}
	 - \frac{\partial_t X_{n,m}}{X_{n,m}},\hspace{5mm}
\widetilde{B}_{n,m} := B_{n,m} - \frac{\partial_t Y_{n,m}}{Y_{n,m}}.
\end{align}
Recalling that \mbox{$q_{n,m} r_{n,m}=u_{n,m} v_{n,m}$},
the above relations
combined with (\ref{DS2-Lax1})
comprise
the Lax representation for the transformed system.
To express it
in a concise form,
we introduce
the quantities
\begin{subequations}
\label{trans3}
\begin{align}
& w_{n,m}:= \frac{1}{\sqrt{1- q_{n,m} r_{n,m}}}
	= \frac{1}{\sqrt{1- u_{n,m} v_{n,m}}},
\label{} \\[2mm]
& f_{n,m}:= \frac{X_{n,m+1}}{X_{n,m}}
 = \frac{h_{m} \prod_{j=-\infty}^{n-1} \sqrt{1- u_{j,m+1} v_{j,m+1}}}
        {h_{m+1} \prod_{j=-\infty}^{n-1} \sqrt{1- u_{j,m} v_{j,m}}},
\label{} \\[2mm] &
g_{n,m}:= \frac{Y_{n+1,m}}{Y_{n,m}}
 = \frac{l_n \prod_{k=-\infty}^{m-1} \sqrt{1- u_{n+1,k} v_{n+1,k}}}
 {l_{n+1} \prod_{k=-\infty}^{m-1} \sqrt{1- u_{n,k} v_{n,k}}}.
\end{align}
\end{subequations}
Thus,
we obtain
the Lax representation in the form,
\begin{subequations}
\label{dDS-Lax}
\begin{align}
 \Psi_{n+1,m} &= w_{n,m} \Psi_{n,m} + w_{n,m} u_{n,m} \Phi_{n,m},
\label{dDS-Lax1}
\\[1mm]
 \Phi_{n,m+1} &= w_{n,m} \Phi_{n,m} + w_{n,m} v_{n,m} \Psi_{n,m},
\label{dDS-Lax2}
\\[1mm]
 \frac{\partial \Psi_{n,m}}{\partial t}
 &= a f_{n,m} \Psi_{n,m+1} + \widetilde{A}_{n,m} \Psi_{n,m}
	+c f_{n,m-1} \Psi_{n,m-1}
\nonumber \\[-1mm] & \hphantom{=}\;\hspace{1pt}
 + b w_{n-1,m} g_{n-1,m} u_{n-1,m}\Phi_{n,m}- d g_{n-1,m}u_{n,m}\Phi_{n-1,m},
\label{dDS-Lax3}
\\[1mm]
 \frac{\partial \Phi_{n,m}}{\partial t}
 & =b g_{n,m} \Phi_{n+1,m} + \widetilde{B}_{n,m} \Phi_{n,m}
	+ d g_{n-1,m} \Phi_{n-1,m}
\nonumber \\[-1mm] & \hphantom{=}\;\hspace{1pt}
 +a w_{n,m-1}f_{n,m-1}v_{n,m-1} \Psi_{n,m} - c f_{n,m-1} v_{n,m} \Psi_{n,m-1}.
\label{dDS-Lax4}
\end{align}
\end{subequations}
The corresponding
boundary conditions are given by
\begin{align}
& \lim_{n \to - \infty} (u_{n,m}, v_{n,m}) =
\lim_{m \to - \infty} (u_{n,m}, v_{n,m}) = \vt{0},
\nonumber
\\[1mm]
& \lim_{n \to - \infty} f_{n,m} = \frac{h_m}{h_{m+1}}, \hspace{5mm}
\lim_{m \to - \infty} g_{n,m} = \frac{l_n}{l_{n+1}}.
\nonumber
\end{align}

Actually,
we can generalize
(\ref{dDS-Lax}) to
a more general
form
wherein the spatial part
is
given by
\begin{align}
& \Psi_{n+1,m} = w_{n,m} \Psi_{n,m} + q_{n,m} \Phi_{n,m},
\nonumber
\\
& \Phi_{n,m+1} = s_{n,m} \Phi_{n,m} + r_{n,m} \Psi_{n,m},
\nonumber
\end{align}
with four independent functions
$w_{n,m}$, $s_{n,m}$, $q_{n,m}$, and $r_{n,m}$.
Thus, it is possible to start with this general
Lax representation and then consider the
reduction.
However, we
skip such a discussion
to
maintain an
easy-to-read flow of the paper.

The compatibility conditions
for the linear system (\ref{dDS-Lax}) with
\mbox{$ w_{n,m}=\left( 1- u_{n,m} v_{n,m} \right)^{-\frac{1}{2}}$}
provide the
time evolution equations
for $u_{n,m}$ and $v_{n,m}$.
They can be
written in a
natural
compact form
using
new auxiliary fields
$\alpha_{n,m}$ and $\beta_{n,m}$ defined
as
\begin{subequations}
\label{trans4}
\begin{align}
& \widetilde{A}_{n,m}
 =: \frac{1}{2} w_{n-1,m} g_{n-1,m}\left( b u_{n-1,m}v_{n,m}
	-d u_{n,m} v_{n-1,m} \right)-\frac{1}{2} \alpha_{n,m},
\label{} \\[1mm]
& \widetilde{B}_{n,m}
 =: \frac{1}{2} w_{n,m-1} f_{n,m-1} \left( a u_{n,m}v_{n,m-1}
	-c u_{n,m-1} v_{n,m}\right) -\frac{1}{2} \beta_{n,m}.
\label{}
\end{align}
\end{subequations}
Thus,
we finally
arrive at
the desired system,
\begin{subequations}
\label{sym-dDS}
\begin{align}
& \frac{\partial u_{n,m}}{\partial t}
 + (1-u_{n,m}v_{n,m}) \left( d w_{n,m} g_{n,m}u_{n+1,m}
 + b w_{n-1,m} g_{n-1,m} u_{n-1,m} \right.
\nonumber \\ &
\hspace{27mm}
\left. \mbox{}- a w_{n,m} f_{n,m} u_{n,m+1} - c w_{n,m-1}f_{n,m-1} u_{n,m-1}
\right)
\nonumber \\ &
\hspace{10mm}
+\frac{1}{2} u_{n,m}\left[ \alpha_{n,m} -w_{n,m-1} f_{n,m-1}
 (a u_{n,m} v_{n,m-1} +c u_{n,m-1} v_{n,m}) \right.
\nonumber \\ &
\hspace{27mm}
\left. \mbox{}- \beta_{n,m} + w_{n-1,m} g_{n-1,m} (b u_{n-1,m} v_{n,m}
 +d u_{n,m}v_{n-1,m})\right]=0,
\label{symm-1}
\\[1mm]
& \frac{\partial v_{n,m}}{\partial t}
 + (1-u_{n,m}v_{n,m}) \left( c w_{n,m} f_{n,m} v_{n,m+1}
 + a w_{n,m-1}f_{n,m-1} v_{n,m-1}
\right.
\nonumber \\ &
\hspace{27mm}
\left. \mbox{} -b w_{n,m} g_{n,m}v_{n+1,m}
 -d w_{n-1,m} g_{n-1,m} v_{n-1,m}
\right)
\nonumber \\ &
\hspace{10mm}
-\frac{1}{2} v_{n,m}\left[ \alpha_{n,m} -w_{n,m-1} f_{n,m-1}
 (a u_{n,m} v_{n,m-1} +c u_{n,m-1} v_{n,m}) \right.
\nonumber \\ &
\hspace{27mm}
\left. \mbox{}- \beta_{n,m} + w_{n-1,m} g_{n-1,m} (b u_{n-1,m} v_{n,m}
 +d u_{n,m}v_{n-1,m})\right]=0,
\label{symm-2}
\\[1mm]
& w_{n,m}f_{n,m}=w_{n,m+1}f_{n+1,m} \hspace{3mm} \mathrm{if} \;
	(a,c) \neq \vt{0},
\label{symm-3}
\\[1mm]
& w_{n,m}g_{n,m}=w_{n+1,m}g_{n,m+1} \hspace{3mm} \mathrm{if} \;
	(b,d) \neq \vt{0},
\label{symm-4}
\\[1mm]
& \boldsymbol{\Delta}_n^+ \alpha_{n,m}
= \boldsymbol{\Delta}_m^+ \left[ w_{n,m-1} f_{n,m-1}
 (a u_{n,m} v_{n,m-1} + c u_{n,m-1}v_{n,m})\right],
\label{symm-5} \\[1mm]
& \boldsymbol{\Delta}_m^+ \beta_{n,m}
= \boldsymbol{\Delta}_n^+ \left[ w_{n-1,m} g_{n-1,m}
 (b u_{n-1,m} v_{n,m} + d u_{n,m}v_{n-1,m})\right].
\label{symm-6}
\end{align}
\end{subequations}
Here, $a$, $b$, $c$, and $d$ are constants and
\mbox{$ w_{n,m}=\left( 1- u_{n,m} v_{n,m} \right)^{-\frac{1}{2}}$}.
In the same way as (\ref{DS1}),
(\ref{sym-dDS})
also admits
a dimensional reduction
to the
Ablowitz--Ladik lattice~\cite{AL1}.
Using
(\ref{symm-3})--(\ref{symm-6}),
we can
rewrite
(\ref{symm-1}) and (\ref{symm-2})
in
a more symmetric
form
with respect to
space reflections.
When \mbox{$c= -a^\ast$} and \mbox{$d= -b^\ast$},
the \mbox{$(2+1)$}-dimensional system (\ref{sym-dDS}) allows
the complex conjugation reduction
\mbox{$v_{n,m}=\sigma u_{n,m}^\ast$} with a real constant $\sigma$;
in this reduction,
the auxiliary fields $f_{n,m}$ and $g_{n,m}$
become
real-valued,
while the auxiliary fields $\alpha_{n,m}$ and $\beta_{n,m}$
become
purely imaginary.
In particular,
(\ref{sym-dDS}) with
purely imaginary $a$, $b$,
\mbox{$c\hspace{1pt}(= -a^\ast)$}, and \mbox{$d\hspace{1pt}(= -b^\ast)$}
provides the suitable space discretization
of the Davey--Stewartson system (cf.~(\ref{DS-conti})).

Similarly to the continuous case
(cf.~\cite{Bogdanov1,Bogdanov2,Boiti88}),
when
\mbox{$c= -a$} and \mbox{$d= -b$},
we can consider the
reduction of
\mbox{$v_{n,m}=\sigma u_{n,m}$}
and
\mbox{$\alpha_{n,m}=\beta_{n,m}=0$}
to obtain
a \mbox{$(2+1)$}-dimensional analogue of
the modified Volterra lattice~\cite{Hiro73},
\begin{subequations}
\label{2DmLV}
\begin{align}
& \frac{\partial u_{n,m}}{\partial t} =
  \left( 1-\sigma u_{n,m}^2 \right) \left( b w_{n,m} g_{n,m}u_{n+1,m}
 - b w_{n-1,m} g_{n-1,m} u_{n-1,m} \right.
\nonumber \\ &
\hspace{27mm}
\left. \mbox{}+ a w_{n,m} f_{n,m} u_{n,m+1} -a w_{n,m-1}f_{n,m-1} u_{n,m-1}
\right),
\label{}
\\[1mm]
& w_{n,m}f_{n,m}=w_{n,m+1}f_{n+1,m} \hspace{3mm} \mathrm{if} \;
	a \neq 0,
\label{}
\\[1mm]
& w_{n,m}g_{n,m}=w_{n+1,m}g_{n,m+1} \hspace{3mm} \mathrm{if} \;
	b \neq 0.
\label{}
\end{align}
\end{subequations}
Here, \mbox{$ w_{n,m}=\left( 1- \sigma u_{n,m}^2 \right)^{-\frac{1}{2}}$}.
It would be interesting to
look
for a relationship between
(\ref{2DmLV}) and
the discrete
modified Nizhnik--Veselov--Novikov hierarchy
in~\cite{Dmitry09-1} (also see~\cite{HuJPA05,HuJMAA05,Kri10}).

\section{Solutions
by the Hirota
method}

In this section, we
discuss
how to
construct
exact solutions of the
discrete Davey--Stewartson system (\ref{sym-dDS})
using the Hirota bilinear method~\cite{Hirota04}.
Because of
the
complexity
and irrationality
of the equations of motion,
it would be
too hard
to
solve (\ref{sym-dDS}) directly,
so
we take
an alternative approach.
First,
we
bilinearize
the \mbox{$(2+1)$}-dimensional Ablowitz--Ladik lattice (\ref{DS1}).
Subsequently, we
consider
the effect of
the
nonlocal
transformation (\ref{trans1})
with (\ref{XY}),
(\ref{trans2}), (\ref{trans3}), and (\ref{trans4})
in the bilinear formalism.
The
infinite products
appearing in the
nonlocal transformation
can essentially be
expressed locally
in terms of
a ``tau function''.
Thus,
we can
obtain
exact solutions of (\ref{DS1})
and (\ref{sym-dDS}) concurrently
from the same set of bilinear equations.

\subsection{Decomposition into
four commutative
flows}
\label{subs_commut}

Before applying
the Hirota bilinear method,
we demonstrate that
the \mbox{$(2+1)$}-dimensional Ablowitz--Ladik lattice (\ref{DS1})
can be decomposed into the four elementary flows.
In view of (\ref{fdAL1}),
(\ref{fdAL2}), and
(\ref{ALbc1-CD}),
we
rescale the auxiliary fields
as
\begin{equation}
\label{aux-scale}
A_{n,m}=: a A^{(0)}_{n,m}, \hspace{5mm}
B_{n,m}=: b B^{(0)}_{n,m}, \hspace{5mm}
C_{n,m}=: c \hspace{1pt}C^{(0)}_{n,m}, \hspace{5mm}
D_{n,m}=: d D^{(0)}_{n,m}.
\end{equation}
The corresponding boundary conditions are
\begin{align}
& \lim_{n \to - \infty} (q_{n,m}, r_{n,m}) =
\lim_{m \to - \infty} (q_{n,m}, r_{n,m}) = \vt{0},
\nonumber
\\[1mm]
& \lim_{n \to - \infty} C_{n,m}^{(0)} =
\left( \frac{h_{m-1}}{h_{m}}\right)^2, \hspace{5mm}
\lim_{m \to - \infty} D_{n,m}^{(0)} =
\left( \frac{l_{n-1}}{l_{n}}\right)^2.
\nonumber
\end{align}
Thus,
considering the
simplest
cases where only one of the
parameters $a$, $b$, $c$, and $d$
does not vanish, we
obtain the four elementary systems:
\\
\vspace{-3mm}
\\
$\bullet$ $a$-system
\begin{subequations}
\label{a-system}
\begin{align}
& \partial_{t_a} q_{n,m} =  q_{n,m+1} + A_{n+1,m}^{(0)} q_{n,m},
\label{}
\\
& \partial_{t_a} r_{n,m} =-  r_{n,m-1} - r_{n,m} A_{n,m}^{(0)},
\label{}
\\
& A^{(0)}_{n+1,m} -A^{(0)}_{n,m} =
 -\left( q_{n,m+1} r_{n,m} -q_{n,m} r_{n,m-1} \right),
\label{}
\end{align}
\end{subequations}
$\bullet$ $b$-system
\begin{subequations}
\label{b-system}
\begin{align}
& \partial_{t_b} q_{n,m} =
  - q_{n-1,m} - q_{n,m} B_{n,m}^{(0)},
\label{}
\\
& \partial_{t_b} r_{n,m}
  = r_{n+1,m} + B_{n,m+1}^{(0)} r_{n,m},
\label{}
\\
& B_{n,m+1}^{(0)} -B_{n,m}^{(0)}
 = -\left( r_{n+1,m} q_{n,m} -r_{n,m} q_{n-1,m} \right),
\label{}
\end{align}
\end{subequations}
$\bullet$ $c$-system
\begin{subequations}
\label{c-system}
\begin{align}
 & \partial_{t_c} q_{n,m} = C_{n+1,m}^{(0)} q_{n,m-1},
\label{}
\\
& \partial_{t_c} r_{n,m} =- r_{n,m+1} C_{n,m+1}^{(0)},
\label{}
\\
& (1- q_{n,m} r_{n,m}) C_{n,m}^{(0)} =C_{n+1,m}^{(0)}(1- q_{n,m-1} r_{n,m-1}),
\label{}
\end{align}
\end{subequations}
$\bullet$ $d$-system
\begin{subequations}
\label{d-system}
\begin{align}
 & \partial_{t_d} q_{n,m} =- q_{n+1,m} D_{n+1,m}^{(0)},
\label{}
\\
 & \partial_{t_d} r_{n,m} = D_{n,m+1}^{(0)} r_{n-1,m},
\label{}
\\
& (1- r_{n,m} q_{n,m}) D_{n,m}^{(0)}=D_{n,m+1}^{(0)}(1- r_{n-1,m} q_{n-1,m}).
\label{}
\end{align}
\end{subequations}
Clearly,
the time evolution in
(\ref{DS1}) is a linear combination of these four
time evolutions, that is,
\mbox{$\partial_t = a\partial_{t_a}+b\partial_{t_b}
	+c\partial_{t_c}+d\partial_{t_d}$}.
In fact, they
are
mutually
commutative,
so
the above
four systems
belong to the same integrable hierarchy
as the original system (\ref{DS1}).
To check
the commutativity conditions
\mbox{$\partial_{t_\alpha} \partial_{t_\beta} q_{n,m}
= \partial_{t_\beta} \partial_{t_\alpha} q_{n,m}$} and
\mbox{$\partial_{t_\alpha} \partial_{t_\beta} r_{n,m}
= \partial_{t_\beta} \partial_{t_\alpha} r_{n,m}$}
for \mbox{$\{\alpha, \beta \} \subset \{ a, b, c, d \}$},
we
need
to
know
how to
express
time derivatives of the auxiliary fields.
Using (\ref{a-system})--(\ref{d-system}),
we can
obtain all
necessary expressions
in
the
local forms,
{\it e.g.},
\begin{align}
& \partial_{t_b}A_{n,m}^{(0)}=
	-\left( q_{n-1,m+1} r_{n,m} -q_{n-1,m} r_{n,m-1} \right),
\nonumber \\
& \partial_{t_c}A_{n,m}^{(0)}= -\left( C_{n,m+1}^{(0)}-C_{n,m}^{(0)} \right),
\nonumber \\
& \partial_{t_d}A_{n,m}^{(0)}=  q_{n,m+1} D_{n,m+1}^{(0)} r_{n-1,m}
	- q_{n,m} D_{n,m}^{(0)} r_{n-1,m-1},
\hspace{3mm} \mathrm{etc.}
\nonumber
\end{align}
Here, we assumed
that all ``integration constants'' etc.\
can be set equal to zero.
With these local expressions,
we can check
the commutativity
of the four flows
by direct computations.
Note that
\mbox{$\lim_{n \to - \infty} \partial_{t_c}A_{n,m}^{(0)}$},
\mbox{$\lim_{m \to - \infty} \partial_{t_d}B_{n,m}^{(0)}$},
\mbox{$\lim_{n \to - \infty} \partial_{t_a}C_{n,m}^{(0)}$}, and
\mbox{$\lim_{m \to - \infty} \partial_{t_b}D_{n,m}^{(0)}$}
do not vanish in general.
Thus,
the $t_a$-flow
can change the boundary value
of the auxiliary field
in the $t_c$-flow
and vice versa; the same applies for
the $t_b$-flow and
$t_d$-flow.

\subsection{Bilinearization}

Because the four
systems
(\ref{a-system})--(\ref{d-system})
are compatible, in the sense that their flows mutually commute,
we will consider here their common solution
denoted as
\mbox{$q_{n,m}(t_a, t_b, t_c, t_d)$},
\mbox{$r_{n,m}(t_a, t_b, t_c, t_d)$}, etc.
Here, $t_a$, $t_b$, $t_c$, and $t_d$ are
independent arguments.
Thus,
the solution of
the
original system (\ref{DS1})
is obtained by setting
\begin{equation}
\label{t-t}
t_a = at, \;\;\; t_b= bt, \;\;\; t_c= ct, \;\;\; t_d=dt,
\end{equation}
which indeed
implies
the relation \mbox{$\partial_t = a\partial_{t_a}+b\partial_{t_b}
+c\partial_{t_c}+d\partial_{t_d}$}.

We assume
a solution
expressible
in the form,
\begin{subequations}
\label{bi-trans1}
\begin{align}
& q_{n,m} = \frac{G_{n,m}}{F_{n+1,m}}, \hspace{5mm}
r_{n,m} = \frac{H_{n,m}}{F_{n,m+1}}, \hspace{5mm}
\label{FGH} \\[1mm]
&
A^{(0)}_{n,m}= \partial_{t_a}
	\log \left( \frac{F_{n,m+1}}{F_{n,m}}\right)
 = \boldsymbol{\Delta}_m^+ \left(
	\frac{\partial_{t_a} F_{n,m}}{F_{n,m}}\right),
\label{A0} \\[1mm]
& B^{(0)}_{n,m}=\partial_{t_b}
	\log \left( \frac{F_{n+1,m}}{F_{n,m}}\right)
 = \boldsymbol{\Delta}_n^+ \left(
	\frac{\partial_{t_b} F_{n,m}}{F_{n,m}}\right),
\label{B0} \\[1mm]
& C^{(0)}_{n,m}=\frac{F_{n,m+1}F_{n,m-1}}{\left( F_{n,m}\right)^{2}},
\hspace{5mm}
D^{(0)}_{n,m}=\frac{F_{n+1,m}F_{n-1,m}}{\left( F_{n,m}\right)^{2}},
\label{C0D0}
\end{align}
\end{subequations}
and bilinearize the four
systems (\ref{a-system})--(\ref{d-system})
in terms of the
``tau functions'' $F_{n,m}$, $G_{n,m}$, and $H_{n,m}$
as follows:
\\
\vspace{-3mm}
\\
$\bullet$ $a$-system
\begin{subequations}
\label{a-system2}
\begin{align}
& F_{n+1,m+1} \partial_{t_a} G_{n,m} -G_{n,m} \partial_{t_a} F_{n+1,m+1}
 = F_{n+1,m} G_{n,m+1},
\label{F-ta-G}
\\
& F_{n,m}\partial_{t_a} H_{n,m} -H_{n,m}\partial_{t_a} F_{n,m}
  =-F_{n,m+1} H_{n,m-1},
\label{F-ta-H}
\\
& F_{n,m}\partial_{t_a} F_{n+1,m} -F_{n+1,m}\partial_{t_a} F_{n,m}
  =-G_{n,m} H_{n,m-1},
\label{F-ta-F}
\end{align}
\end{subequations}
$\bullet$ $b$-system
\begin{subequations}
\label{b-system2}
\begin{align}
& F_{n,m}\partial_{t_b} G_{n,m} -G_{n,m}\partial_{t_b} F_{n,m}
  =-F_{n+1,m} G_{n-1,m},
\label{F-tb-G}
\\
& F_{n+1,m+1} \partial_{t_b} H_{n,m} -H_{n,m} \partial_{t_b} F_{n+1,m+1}
 = F_{n,m+1} H_{n+1,m},
\label{F-tb-H}
\\
& F_{n,m}\partial_{t_b} F_{n,m+1} -F_{n,m+1}\partial_{t_b} F_{n,m}
  =-G_{n-1,m} H_{n,m},
\label{F-tb-F}
\end{align}
\end{subequations}
$\bullet$ $c$-system
\begin{subequations}
\label{c-system2}
\begin{align}
& F_{n+1,m}\partial_{t_c} G_{n,m} -G_{n,m}\partial_{t_c} F_{n+1,m}
  =F_{n+1,m+1} G_{n,m-1},
\label{}
\\
& F_{n,m+1} \partial_{t_c} H_{n,m} -H_{n,m} \partial_{t_c} F_{n,m+1}
 = -F_{n,m} H_{n,m+1},
\label{}
\\
& F_{n+1,m}F_{n,m+1} -F_{n+1,m+1} F_{n,m} = G_{n,m} H_{n,m},
\label{FF-GH1}
\end{align}
\end{subequations}
$\bullet$ $d$-system
\begin{subequations}
\label{d-system2}
\begin{align}
& F_{n+1,m}\partial_{t_d} G_{n,m} -G_{n,m}\partial_{t_d} F_{n+1,m}
  =-F_{n,m} G_{n+1,m},
\label{}
\\
& F_{n,m+1} \partial_{t_d} H_{n,m} -H_{n,m} \partial_{t_d} F_{n,m+1}
 = F_{n+1,m+1} H_{n-1,m},
\label{}
\\
& F_{n+1,m}F_{n,m+1} -F_{n+1,m+1} F_{n,m} = G_{n,m} H_{n,m}.
\label{FF-GH2}
\end{align}
\end{subequations}
To be
precise,
each
triplet of
bilinear equations
gives
a sufficient condition
for the
corresponding
original system.
Note that for
the $c$-system and
$d$-system,
the bilinear forms
as well as some exact solutions
were
studied
in~\cite{Hu06}.

\subsection{General solution formulas}

Once a
solution
of
the
bilinear equations
(\ref{a-system2})--(\ref{d-system2}) is
obtained,
formula (\ref{bi-trans1}) with (\ref{aux-scale}) and (\ref{t-t})
provides the solution of
the \mbox{$(2+1)$}-dimensional Ablowitz--Ladik lattice (\ref{DS1}).
We assume that it satisfies the boundary conditions (\ref{ALbc1}).
Thus, by applying
the
nonlocal
transformation (\ref{trans1})
with (\ref{XY}),
(\ref{trans2}), (\ref{trans3}), and (\ref{trans4}),
we
can also obtain
the solution of the discrete Davey--Stewartson system (\ref{sym-dDS}).
To evaluate
the effect of
this nonlocal transformation,
we
use
(\ref{FGH})
and (\ref{FF-GH1}) (or (\ref{FF-GH2}))
to rewrite
the
infinite products as
\begin{align}
&{\displaystyle \prod_{j=-\infty}^{n-1} \sqrt{1- q_{j,m} r_{j,m}}}
={\displaystyle \prod_{j=-\infty}^{n-1}
        \sqrt{\frac{F_{j+1,m+1} F_{j,m}}{F_{j+1,m}F_{j,m+1}}}}
= {\displaystyle \sqrt{\frac{F_{n,m+1}}{F_{n,m}}
\lim_{j \to -\infty}\frac{F_{j,m}}{F_{j,m+1}}}}
\hspace{1pt},
\nonumber \\[2mm]
& \displaystyle \prod_{k=-\infty}^{m-1} \sqrt{1- q_{n,k} r_{n,k}}
= \displaystyle \prod_{k=-\infty}^{m-1}
        \sqrt{\frac{F_{n+1,k+1} F_{n,k}}{F_{n+1,k}F_{n,k+1}}}
= \displaystyle \sqrt{\frac{F_{n+1,m}}{F_{n,m}}
\lim_{k \to - \infty}\frac{F_{n,k}}{F_{n+1,k}}
}\hspace{1pt}.
\nonumber
\end{align}
Because we assumed \mbox{$\abs{q_{n,m} r_{n,m}} \ll 1$},
the value of
\mbox{$(F_{n+1,m+1} F_{n,m})/(F_{n+1,m}F_{n,m+1})$}
is always restricted to
the neighborhood of
$1$.
For simplicity,
in considering the solution of (\ref{sym-dDS}),
we also assume that
$F_{n,m}$ is positive
(or, at least, \mbox{$\left| \arg F_{n,m}\right|$} is sufficiently small);
the positivity condition
\mbox{$F_{n,m}>0$} can fully
justify the use
of the formulas for the square root,
such as
\mbox{$\sqrt{X^2}=X$} and
\mbox{$\sqrt{X/Y}=\sqrt{X}/\sqrt{Y}$}.
We set the norming functions
$h_m$ and $l_n$
in (\ref{XY}) as
\[
h_m = \sqrt{\lim_{j \to -\infty}\frac{F_{j,m}}{F_{j,m+1}}}\hspace{1pt},
\hspace{5mm}
l_n= \sqrt{\lim_{k \to - \infty}\frac{F_{n,k}}{F_{n+1,k}}}\hspace{1pt}.
\]
Thus,
we obtain
\begin{equation}
\label{XY-local}
X_{n,m}= {\displaystyle \sqrt{\frac{F_{n,m+1}}{F_{n,m}}}}
\hspace{1pt}, \hspace{5mm}
Y_{n,m}={\displaystyle \sqrt{\frac{F_{n+1,m}}{F_{n,m}}}}\hspace{1pt}.
\end{equation}
After all,
we can
express the transformation from
(\ref{DS1}) to
(\ref{sym-dDS}) locally
in terms of the ``tau function'' $F_{n,m}$.
Combining
(\ref{trans1}), (\ref{trans2}), (\ref{trans3}),
(\ref{aux-scale}),
(\ref{bi-trans1}), and (\ref{XY-local}),
we arrive at
general solution formulas
for the discrete Davey--Stewartson system (\ref{sym-dDS})
in the form,
\begin{subequations}
\begin{align}
\label{}
& u_{n,m}  = \frac{G_{n,m}}{\sqrt{F_{n+1,m} F_{n,m+1}}}, \hspace{5mm}
v_{n,m} = \frac{H_{n,m}}{\sqrt{F_{n+1,m} F_{n,m+1}}},
\label{FGH2}
\\[2mm] &
\widetilde{A}_{n,m}=\frac{1}{2}
\left( a \partial_{t_a} -b\partial_{t_b}
-c\partial_{t_c}-d\partial_{t_d} \right)
        \log \left( \frac{F_{n,m+1}}{F_{n,m}}\right),
\\[2mm] &
\widetilde{B}_{n,m}=\frac{1}{2}
\left( b\partial_{t_b} -a\partial_{t_a}
-c\partial_{t_c}-d\partial_{t_d} \right)
        \log \left( \frac{F_{n+1,m}}{F_{n,m}}\right),
\\[2mm] &
f_{n,m}=  \frac{\sqrt{F_{n,m+2}F_{n,m}}}{F_{n,m+1}},
\hspace{5mm}
g_{n,m}= \frac{\sqrt{F_{n+2,m}F_{n,m}}}{F_{n+1,m}}.
\end{align}
\end{subequations}
Here,
the time variables are set
as in (\ref{t-t})
and
the auxiliary fields
$\alpha_{n,m}$ and $\beta_{n,m}$ are
determined from
$\widetilde{A}_{n,m}$ and $\widetilde{B}_{n,m}$
through (\ref{trans4}).
Using the bilinear
equations (\ref{F-ta-F}) and (\ref{F-tb-F})
and noting that
(\ref{symm-5}) and (\ref{symm-6}) are
identities in $a$, $b$, $c$, and $d$,
we obtain
compact expressions for $\alpha_{n,m}$ and $\beta_{n,m}$,
\begin{subequations}
\begin{align}
\label{}
& \alpha_{n,m}= \left( -a \partial_{t_a} +c\partial_{t_c} \right)
        \log \left( \frac{F_{n,m+1}}{F_{n,m}}\right),
\\[1mm] &
\beta_{n,m}= \left( -b\partial_{t_b} + d\partial_{t_d} \right)
        \log \left( \frac{F_{n+1,m}}{F_{n,m}}\right),
\end{align}
\end{subequations}
and
new bilinear equations,
\begin{align}
& F_{n,m}\partial_{t_c} F_{n+1,m} -F_{n+1,m}\partial_{t_c} F_{n,m}
  =G_{n,m-1} H_{n,m},
\label{new-bi1}
\\
& F_{n,m} \partial_{t_d} F_{n,m+1} -F_{n,m+1} \partial_{t_d} F_{n,m}
 = G_{n,m} H_{n-1,m}.
\label{new-bi2}
\end{align}
Note that (\ref{new-bi1}) and (\ref{new-bi2})
fill in the
piece
missing in (\ref{a-system2})--(\ref{d-system2}).
In the next subsection, we
construct
common
solutions to
all these bilinear equations.

As described below (\ref{sym-dDS}),
when \mbox{$c= -a^\ast$} and \mbox{$d= -b^\ast$},
we can impose the
reduction
\mbox{$v_{n,m}=\sigma u_{n,m}^{\ast}$}
with a real constant $\sigma$.
This reduction can be
realized
by requiring that
\mbox{$F_{n,m} >0 $} and \mbox{$H_{n,m} =\sigma G_{n,m}^{\hspace{1pt}\ast}$}.

\subsection{Solitons and dromions}

The set of
bilinear equations
(\ref{a-system2})--(\ref{d-system2})
together with
(\ref{new-bi1}) and (\ref{new-bi2}) is
not ideally
symmetric in its
present form.
In particular, it is not clear
why
reductions such as
\mbox{$F_{n,m}^{\hspace{1pt}\ast} = F_{n,m}$}
and \mbox{$H_{n,m} =\sigma G_{n,m}^{\hspace{1pt}\ast}$}
are allowed.
To restore the symmetry, we
need only to
rewrite
(\ref{F-ta-G}), (\ref{F-ta-H}),
(\ref{F-tb-G}), and (\ref{F-tb-H})
using (\ref{F-ta-F}), (\ref{F-tb-F}),
and (\ref{FF-GH1})
(or (\ref{FF-GH2})).
For example,
using (\ref{F-ta-F})
and then (\ref{FF-GH1}),
(\ref{F-ta-G}) can be rewritten as
\[
 F_{n,m+1} \partial_{t_a} G_{n,m} -G_{n,m} \partial_{t_a} F_{n,m+1}
 = F_{n,m} G_{n,m+1}.
\]
Thus, the full set of bilinear equations can be
reformulated in
the
symmetric form,
\begin{align}
& F_{n+1,m}F_{n,m+1} -F_{n+1,m+1} F_{n,m} = G_{n,m} H_{n,m},
\hspace{15mm}
\label{bilin1}
\end{align}
\vspace{-8mm}
\begin{subequations}
\label{a-system3}
\begin{align}
& F_{n,m+1} \partial_{t_a} G_{n,m} -G_{n,m} \partial_{t_a} F_{n,m+1}
 = F_{n,m} G_{n,m+1},
\label{bilin2}
\\
& F_{n+1,m}\partial_{t_a} H_{n,m} -H_{n,m}\partial_{t_a} F_{n+1,m}
  =-F_{n+1,m+1} H_{n,m-1},
\label{bilin3}
\\
& F_{n,m}\partial_{t_a} F_{n+1,m} -F_{n+1,m}\partial_{t_a} F_{n,m}
  =-G_{n,m} H_{n,m-1},
\label{bilin4}
\end{align}
\end{subequations}
\vspace{-8mm}
\begin{subequations}
\label{b-system3}
\begin{align}
& F_{n,m+1}\partial_{t_b} G_{n,m} -G_{n,m}\partial_{t_b} F_{n,m+1}
  =-F_{n+1,m+1} G_{n-1,m},
\label{bilin5}
\\
& F_{n+1,m} \partial_{t_b} H_{n,m} -H_{n,m} \partial_{t_b} F_{n+1,m}
 = F_{n,m} H_{n+1,m},
\label{bilin6}
\\
& F_{n,m}\partial_{t_b} F_{n,m+1} -F_{n,m+1}\partial_{t_b} F_{n,m}
  =-G_{n-1,m} H_{n,m},
\label{bilin7}
\end{align}
\end{subequations}
\vspace{-8mm}
\begin{subequations}
\begin{align}
& F_{n+1,m}\partial_{t_c} G_{n,m} -G_{n,m}\partial_{t_c} F_{n+1,m}
  =F_{n+1,m+1} G_{n,m-1},
\label{bilin8}
\\
& F_{n,m+1} \partial_{t_c} H_{n,m} -H_{n,m} \partial_{t_c} F_{n,m+1}
 = -F_{n,m} H_{n,m+1},
\label{bilin9}
\\
& F_{n,m}\partial_{t_c} F_{n+1,m} -F_{n+1,m}\partial_{t_c} F_{n,m}
  =G_{n,m-1} H_{n,m},
\label{bilin10}
\end{align}
\end{subequations}
\vspace{-8mm}
\begin{subequations}
\label{d-system3}
\begin{align}
& F_{n+1,m}\partial_{t_d} G_{n,m} -G_{n,m}\partial_{t_d} F_{n+1,m}
  =-F_{n,m} G_{n+1,m},
\label{bilin11}
\\
& F_{n,m+1} \partial_{t_d} H_{n,m} -H_{n,m} \partial_{t_d} F_{n,m+1}
 = F_{n+1,m+1} H_{n-1,m},
\label{bilin12}
\\
& F_{n,m} \partial_{t_d} F_{n,m+1} -F_{n,m+1} \partial_{t_d} F_{n,m}
 = G_{n,m} H_{n-1,m}.
\label{bilin13}
\end{align}
\end{subequations}
It is now clear that the $t_a$-flow and
$t_b$-flow
can be
identified with
the $t_c$-flow and
$t_d$-flow, respectively,
through
the complex conjugation reduction.
Moreover,
(\ref{a-system3}) and (\ref{b-system3})
correspond to each other
by the interchange
of $n$ and $m$, up to a redefinition of the variables.
With these symmetries in mind, we can
considerably
reduce the task of constructing
explicit solutions
to
the above 13
bilinear equations.

In
the same way as in
the continuous case
(see, {\it e.g.},
\cite{Hirota90}),
we can construct the one-soliton solution and
a two-soliton solution
straightforwardly.
The
one-soliton solution is given by
\begin{align}
& F_{n,m} = 1 -\frac{g \accentset{{\cc@style\underline{\mskip10mu}}}{g}}
{(1-p \accentset{{\cc@style\underline{\mskip10mu}}}{p}\hspace{1pt})(1-q \accentset{{\cc@style\underline{\mskip10mu}}}{q}\hspace{1pt})}
(p \accentset{{\cc@style\underline{\mskip10mu}}}{p}\hspace{1pt})^n (q \accentset{{\cc@style\underline{\mskip10mu}}}{q}\hspace{1pt})^m
\mathrm{e}^{\omega +\accentset{{\cc@style\underline{\mskip10mu}}}{\omega}},
\nonumber \\
& G_{n,m}= g \hspace{1pt}p^n q^m \mathrm{e}^{\omega}, \hspace{5mm}
  H_{n,m}= \accentset{{\cc@style\underline{\mskip10mu}}}{g} \hspace{1pt}\accentset{{\cc@style\underline{\mskip10mu}}}{p}\hspace{2pt}^n
	\accentset{{\cc@style\underline{\mskip10mu}}}{q}\hspace{2pt}^m
	\mathrm{e}^{\accentset{{\cc@style\underline{\mskip10mu}}}{\omega}},
\nonumber
\end{align}
where
\mbox{$\omega:=q \hspace{1pt}t_a -p^{-1}t_b + q^{-1} t_c-p \hspace{1pt}t_d$},
\mbox{$\accentset{{\cc@style\underline{\mskip10mu}}}{\omega} := -\accentset{{\cc@style\underline{\mskip10mu}}}{q}\hspace{1pt}^{-1} t_a
 +\accentset{{\cc@style\underline{\mskip10mu}}}{p}\hspace{2pt}t_b
 -\accentset{{\cc@style\underline{\mskip10mu}}}{q}\hspace{2pt} t_c+\accentset{{\cc@style\underline{\mskip10mu}}}{p}\hspace{1pt}^{-1}t_d$},
and $g$, $p$, $q$, etc.\ are
nonzero
constants.
The constant $q$ should not be confused with $q_{n,m}$.
Actually, we can shift
$t_a$, $t_b$, $t_c$, and $t_d$
in (\ref{t-t})
by arbitrary
constants,
but this freedom
can be absorbed by
rescaling $g$ and $\accentset{{\cc@style\underline{\mskip10mu}}}{g}$.
When \mbox{$c= -a^\ast$} and \mbox{$d= -b^\ast$}
(cf.~(\ref{t-t})),
we
set
\mbox{$\accentset{{\cc@style\underline{\mskip10mu}}}{p}= p^\ast$},
\mbox{$\accentset{{\cc@style\underline{\mskip10mu}}}{q}= q^\ast$},
and \mbox{$\accentset{{\cc@style\underline{\mskip10mu}}}{g}= \sigma g^\ast$}
with \mbox{$\sigma
(1-|p|^2)(1-|q|^2)<0$}.
Thus,
\mbox{$F_{n,m} >0 $} and \mbox{$H_{n,m} =\sigma G_{n,m}^{\hspace{1pt}\ast}$},
so
the complex conjugation reduction is realized.

To save space,
we omit
a rather
lengthy expression for
a two-soliton solution.
These
soliton
solutions
are direct
\mbox{$(2+1)$}-dimensional
analogues
of
the soliton solutions
of the \mbox{$(1+1)$}-dimensional systems
such as
(\ref{sd-AL}).
With an appropriate choice of the parameters,
they represent
straight
line solitons
in the physical variables
(cf.~(\ref{FGH})
or
(\ref{FGH2}))
and thus are not
localized.

In the following, we
obtain
more interesting
solutions,
that is,
dromion solutions;
dromions~\cite{FoSan}
are
spatially
localized
``solitons''
that
decay exponentially
in all
directions~\cite{Boiti88}
and
can
exhibit
nontrivial interaction
properties~\cite{Hirota90,GN91}.
More details as well as
an extensive list of
references
can be found in the review article~\cite{Boiti95}.
In analogy with the continuous case~\cite{Hirota90},
the
one-dromion solution is
obtained as
\begin{align}
 F_{n,m} & = 1 +\frac{\alpha_{11}}{1-p \accentset{{\cc@style\underline{\mskip10mu}}}{p}\hspace{1pt}}
 (p \accentset{{\cc@style\underline{\mskip10mu}}}{p}\hspace{1pt})^n\mathrm{e}^{\omega_1 +\accentset{{\cc@style\underline{\mskip10mu}}}{\omega}_1}
+\frac{\alpha_{22}}{1-q \accentset{{\cc@style\underline{\mskip10mu}}}{q}\hspace{1pt}}
 (q \accentset{{\cc@style\underline{\mskip10mu}}}{q}\hspace{1pt})^m \mathrm{e}^{\omega_2 +\accentset{{\cc@style\underline{\mskip10mu}}}{\omega}_2}
\nonumber \\
& \hphantom{=}\; + \frac{\alpha_{11}\alpha_{22}-\alpha_{12}\alpha_{21}}
{(1-p \accentset{{\cc@style\underline{\mskip10mu}}}{p}\hspace{1pt})(1-q \accentset{{\cc@style\underline{\mskip10mu}}}{q}\hspace{1pt})}
(p \accentset{{\cc@style\underline{\mskip10mu}}}{p}\hspace{1pt})^n (q \accentset{{\cc@style\underline{\mskip10mu}}}{q}\hspace{1pt})^m
 \mathrm{e}^{\omega_1 +\omega_2 +\accentset{{\cc@style\underline{\mskip10mu}}}{\omega}_1+\accentset{{\cc@style\underline{\mskip10mu}}}{\omega}_2},
\nonumber \\[2mm]
 G_{n,m} &= \alpha_{12}
p^n q^m \mathrm{e}^{\omega_1+\omega_2}, \hspace{5mm}
  H_{n,m}= \alpha_{21}\accentset{{\cc@style\underline{\mskip10mu}}}{p}\hspace{2pt}^n
	\accentset{{\cc@style\underline{\mskip10mu}}}{q}\hspace{2pt}^m
	\mathrm{e}^{\accentset{{\cc@style\underline{\mskip10mu}}}{\omega}_1+\accentset{{\cc@style\underline{\mskip10mu}}}{\omega}_2},
\nonumber
\end{align}
where
\[
\omega_1:= -p^{-1}t_b -p \hspace{1pt}t_d, \hspace{5mm}
\accentset{{\cc@style\underline{\mskip10mu}}}{\omega}_1 := \accentset{{\cc@style\underline{\mskip10mu}}}{p}\hspace{2pt}t_b
	+\accentset{{\cc@style\underline{\mskip10mu}}}{p}\hspace{2pt}^{-1}t_d, \hspace{5mm}
\omega_2 :=q \hspace{1pt}t_a + q^{-1} t_c, \hspace{5mm}
\accentset{{\cc@style\underline{\mskip10mu}}}{\omega}_2 :=
 -\accentset{{\cc@style\underline{\mskip10mu}}}{q}\hspace{2pt}^{-1} t_a -\accentset{{\cc@style\underline{\mskip10mu}}}{q}\hspace{2pt} t_c.
\]
When \mbox{$c= -a^\ast$} and \mbox{$d= -b^\ast$}
(cf.~(\ref{t-t})),
we
set
\mbox{$\accentset{{\cc@style\underline{\mskip10mu}}}{p}= p^\ast$},
\mbox{$\accentset{{\cc@style\underline{\mskip10mu}}}{q}= q^\ast$},
and \mbox{$\alpha_{21}= \sigma \alpha_{12}^\ast$}
so that
\mbox{$H_{n,m} =\sigma G_{n,m}^{\hspace{1pt}\ast}$}.
Moreover,
if
the
constant
coefficients
of the three
terms
in $F_{n,m}$
are
positive, then
\mbox{$F_{n,m} >0 $},
so
the complex conjugation reduction is realized.
Note that
the above
$F_{n,m}$
can be written
in a \mbox{$2 \times 2$} determinant form,
\begin{align}
 F_{n,m} & =
\det \left\{
I +
\left(
\begin{array}{cc}
\alpha_{11} & \alpha_{21} \\
\alpha_{12} & \alpha_{22} \\
\end{array}
\right)
\left(
\begin{array}{cc}
p^n\mathrm{e}^{\omega_1} & \\
 &  \accentset{{\cc@style\underline{\mskip10mu}}}{q}\hspace{2pt}^m \mathrm{e}^{\accentset{{\cc@style\underline{\mskip10mu}}}{\omega}_2}\\
\end{array}
\right)
\left(
\begin{array}{cc}
\frac{1}{1-p \accentset{{\cc@style\underline{\mskip10mu}}}{p}\hspace{1pt}} &  \\
 & \frac{1}{1-q \accentset{{\cc@style\underline{\mskip10mu}}}{q}\hspace{1pt}} \\
\end{array}
\right)
\left(
\begin{array}{cc}
\accentset{{\cc@style\underline{\mskip10mu}}}{p}\hspace{2pt}^n \mathrm{e}^{\accentset{{\cc@style\underline{\mskip10mu}}}{\omega}_1} & \\
 & q^m \mathrm{e}^{\omega_2} \\
\end{array}
\right)
\right\}.
\nonumber
\end{align}

Following the Gilson--Nimmo approach~\cite{GN91}
in the continuous case,
we construct the multidromion solution called
the
\mbox{$(M,N)$}-dromion solution.
The one-dromion solution corresponds
to the simplest case of \mbox{$M=N=1$}.
Hereinafter,
we
suppress
the subscripts
of the
functions
representing
their
dependence on
the spatial variables $n$ and $m$.
When
they
are shifted, we express it using
the shift operators
\[
(\vt{S}_n Z)_{n,m} := Z_{n+1,m}, \hspace{5mm}
(\vt{S}_m Z)_{n,m} := Z_{n,m+1}.
\]
We will consider
various \mbox{$(M+N) \times (M+N)$}
matrices; they
all have the same shape as
\mbox{$2 \times 2$} block matrices,
so
operations can be performed blockwise.
For simplicity,
off-diagonal zeros in
the block diagonal matrices
are omitted.
We
introduce two
diagonal matrices
as
\begin{align}
\nonumber
& \Xi :=
\left(
\begin{array}{cc}
\Xi_1 & \\
 & \Xi_2 \\
\end{array}
\right),
\hspace{5mm}
\Xi_1 := \mathrm{diag}(\varphi_1, \ldots, \varphi_M), \hspace{5mm}
\Xi_2 := \mathrm{diag}(\chi_1, \ldots, \chi_N),
\\[2mm]
& \nonumber
\accentset{{\cc@style\underline{\mskip10mu}}}{\Xi} :=
\left(
\begin{array}{cc}
\accentset{{\cc@style\underline{\mskip10mu}}}{\Xi}_1 & \\
 & \accentset{{\cc@style\underline{\mskip10mu}}}{\Xi}_2 \\
\end{array}
\right),
\hspace{5mm}
\accentset{{\cc@style\underline{\mskip10mu}}}{\Xi}_1 := \mathrm{diag}(\accentset{{\cc@style\underline{\mskip10mu}}}{\varphi}_1, \ldots,
	\accentset{{\cc@style\underline{\mskip10mu}}}{\varphi}_M), \hspace{5mm}
\accentset{{\cc@style\underline{\mskip10mu}}}{\Xi}_2 := \mathrm{diag}(\accentset{{\cc@style\underline{\mskip10mu}}}{\chi}_1,
	\ldots, \accentset{{\cc@style\underline{\mskip10mu}}}{\chi}_N),
\end{align}
where
\begin{align}
& \varphi_i(n) := p_i^n \mathrm{e}^{-p_i^{-1} t_b -p_i t_d}, \hspace{5mm}
\chi_i(m) := q_i^m \mathrm{e}^{-q_i^{-1} t_a -q_i t_c},
\nonumber \\[1mm]
& \accentset{{\cc@style\underline{\mskip10mu}}}{\varphi}_i(n) := \accentset{{\cc@style\underline{\mskip10mu}}}{p}_i^{\hspace{2pt}n}
 \mathrm{e}^{\accentset{{\cc@style\underline{\mskip10mu}}}{p}_i t_b
	+\accentset{{\cc@style\underline{\mskip10mu}}}{p}_i^{\hspace{1pt}-1} t_d}, \hspace{5mm}
\accentset{{\cc@style\underline{\mskip10mu}}}{\chi}_i(m) := \accentset{{\cc@style\underline{\mskip10mu}}}{q}_i^{\hspace{2pt}m}
\mathrm{e}^{\accentset{{\cc@style\underline{\mskip10mu}}}{q}_i t_a + \accentset{{\cc@style\underline{\mskip10mu}}}{q}_i^{\hspace{1pt}-1} t_c}.
\nonumber
\end{align}
We
also
introduce
\begin{align}
\nonumber
& R :=
\left(
\begin{array}{cc}
P & \\
 & Q \\
\end{array}
\right),
\hspace{5mm}
P := \mathrm{diag}(p_1, \ldots, p_M), \hspace{5mm}
Q := \mathrm{diag}(q_1, \ldots, q_N),
\\[2mm]
& \nonumber
\accentset{{\cc@style\underline{\mskip10mu}}}{R} :=
\left(
\begin{array}{cc}
\accentset{{\cc@style\underline{\mskip10mu}}}{P} & \\
 & \accentset{{\cc@style\underline{\mskip10mu}}}{Q} \\
\end{array}
\right),
\hspace{5mm}
\accentset{{\cc@style\underline{\mskip10mu}}}{P} := \mathrm{diag}(\accentset{{\cc@style\underline{\mskip10mu}}}{p}_1, \ldots,
 \accentset{{\cc@style\underline{\mskip10mu}}}{p}_M), \hspace{5mm}
\accentset{{\cc@style\underline{\mskip10mu}}}{Q} := \mathrm{diag}(\accentset{{\cc@style\underline{\mskip10mu}}}{q}_1, \ldots, \accentset{{\cc@style\underline{\mskip10mu}}}{q}_N).
\end{align}
Then, the following relations hold:
\begin{align}
& \vt{S}_n \Xi =
\Xi \left(
\begin{array}{cc}
P & \\
 & I\\
\end{array}
\right),
\hspace{5mm}
\vt{S}_m \Xi = \Xi \left(
\begin{array}{cc}
I & \\
 & Q\\
\end{array}
\right),
\label{}
\nonumber\\[2mm]
& \vt{S}_n \accentset{{\cc@style\underline{\mskip10mu}}}{\Xi} =
\left(
\begin{array}{cc}
\accentset{{\cc@style\underline{\mskip10mu}}}{P} & \\
 & I\\
\end{array}
\right) \accentset{{\cc@style\underline{\mskip10mu}}}{\Xi},
\hspace{5mm}
\vt{S}_m \accentset{{\cc@style\underline{\mskip10mu}}}{\Xi} = \left(
\begin{array}{cc}
I & \\
 & \accentset{{\cc@style\underline{\mskip10mu}}}{Q} \\
\end{array}
\right) \accentset{{\cc@style\underline{\mskip10mu}}}{\Xi},
\nonumber\\[2mm]
&
\partial_{t_a}
\Xi = -\Xi \left(
\begin{array}{cc}
O & \\
 & Q^{-1} \\
\end{array}
\right), \hspace{5mm}
\partial_{t_b}
\Xi = - \Xi \left(
\begin{array}{cc}
P^{-1} & \\
 & O \\
\end{array}
\right),\hspace{5mm}
\nonumber\\[2mm]
& \partial_{t_c} \Xi
= -\Xi\left(
\begin{array}{cc}
O & \\
 & Q \\
\end{array}\right),\hspace{5mm}
\partial_{t_d}
\Xi = -\Xi\left(
\begin{array}{cc}
P & \\
 & O \\
\end{array}
\right),
\nonumber\\[2mm]
&
\partial_{t_a}
\accentset{{\cc@style\underline{\mskip10mu}}}{\Xi} = \left(
\begin{array}{cc}
O & \\
 & \accentset{{\cc@style\underline{\mskip10mu}}}{Q}\\
\end{array}
\right) \accentset{{\cc@style\underline{\mskip10mu}}}{\Xi}, \hspace{5mm}
\partial_{t_b}
\accentset{{\cc@style\underline{\mskip10mu}}}{\Xi} = \left(
\begin{array}{cc}
\accentset{{\cc@style\underline{\mskip10mu}}}{P} & \\
 & O \\
\end{array}
\right) \accentset{{\cc@style\underline{\mskip10mu}}}{\Xi},\hspace{5mm}
\nonumber\\[2mm]
& \partial_{t_c}
\accentset{{\cc@style\underline{\mskip10mu}}}{\Xi} = \left(
\begin{array}{cc}
O & \\
 & \accentset{{\cc@style\underline{\mskip10mu}}}{Q}^{\hspace{1pt}-1} \\
\end{array}
\right) \accentset{{\cc@style\underline{\mskip10mu}}}{\Xi},\hspace{5mm}
\partial_{t_d}
\accentset{{\cc@style\underline{\mskip10mu}}}{\Xi} = \left(
\begin{array}{cc}
\accentset{{\cc@style\underline{\mskip10mu}}}{P}^{\hspace{1pt}-1} & \\
 & O \\
\end{array}
\right) \accentset{{\cc@style\underline{\mskip10mu}}}{\Xi}.
\nonumber
\end{align}
Note that the order
of
two diagonal matrices
on the right-hand side
can be changed
because they commute.
Moreover, we
introduce
\mbox{$M \times M $} matrices $K$ and $E_M$
as
\begin{align}
& (K)_{ij} := \frac{1}{1-p_i \accentset{{\cc@style\underline{\mskip10mu}}}{p}_j},
\hspace{5mm}
 (E_M)_{ij} := 1, \hspace{5mm} 1 \le i,j \le M,
\nonumber
\end{align}
and \mbox{$N \times N $} matrices $L$ and $E_N$ as
\begin{align}
& (L)_{kl} := \frac{1}{1-q_k \accentset{{\cc@style\underline{\mskip10mu}}}{q}_{\hspace{1pt}l}},
\hspace{5mm}
 (E_N)_{kl} := 1, \hspace{5mm} 1 \le k,l \le N.
\nonumber
\end{align}
They satisfy the relations,
\[
 K - P K \accentset{{\cc@style\underline{\mskip10mu}}}{P} =E_M, \hspace{5mm}
 L - Q L \accentset{{\cc@style\underline{\mskip10mu}}}{Q} =E_N.
\]
We
define
\mbox{$(M+N)$}-component
column
vectors
as
\begin{align}
\nonumber
\vt{e}_M := ( 1, \ldots, 1, 0, \ldots, 0 )^T,
\hspace{5mm}
\vt{e}_N := (0, \ldots, 0, 1, \ldots, 1)^T,
\end{align}
and
\begin{align}
\nonumber
& \vt{l} := \Xi \hspace{1pt}\vt{e}_M
=( \varphi_1, \ldots, \varphi_M, 0, \ldots, 0 )^T,
\hspace{5mm}
\vt{m} := \Xi \hspace{1pt}\vt{e}_N
 = ( 0, \ldots, 0, \chi_1, \ldots, \chi_N )^T,
\\[2mm]
&
\accentset{{\cc@style\underline{\mskip10mu}}}{\vt{l}} := \accentset{{\cc@style\underline{\mskip10mu}}}{\Xi}\hspace{1pt} \vt{e}_M
 =( \accentset{{\cc@style\underline{\mskip10mu}}}{\varphi}_1, \ldots, \accentset{{\cc@style\underline{\mskip10mu}}}{\varphi}_M, 0, \ldots, 0 )^T,
\hspace{5mm}
\accentset{{\cc@style\underline{\mskip10mu}}}{\vt{m}} := \accentset{{\cc@style\underline{\mskip10mu}}}{\Xi} \hspace{1pt}\vt{e}_N
 = ( 0, \ldots, 0, \accentset{{\cc@style\underline{\mskip10mu}}}{\chi}_1, \ldots, \accentset{{\cc@style\underline{\mskip10mu}}}{\chi}_N )^T.
\nonumber
\end{align}
Then, we can easily show the following relations:
\begin{align}
\nonumber
& \vt{S}_n \hspace{1pt}\vt{l} = R\hspace{2pt}\vt{l} , \hspace{5mm}
\vt{S}_m \hspace{1pt}\vt{l} = \vt{l} , \hspace{5mm}
\vt{S}_n \vt{m} = \vt{m}, \hspace{5mm}
\vt{S}_m \vt{m} = R \hspace{1pt}\vt{m},
\\[2mm]
& \vt{S}_n \hspace{1pt}\accentset{{\cc@style\underline{\mskip10mu}}}{\vt{l}}
 = \accentset{{\cc@style\underline{\mskip10mu}}}{R}\hspace{2pt}\accentset{{\cc@style\underline{\mskip10mu}}}{\vt{l}} , \hspace{5mm}
\vt{S}_m \hspace{1pt}\accentset{{\cc@style\underline{\mskip10mu}}}{\vt{l}} = \accentset{{\cc@style\underline{\mskip10mu}}}{\vt{l}}, \hspace{5mm}
\vt{S}_n \accentset{{\cc@style\underline{\mskip10mu}}}{\vt{m}} = \accentset{{\cc@style\underline{\mskip10mu}}}{\vt{m}}, \hspace{5mm}
\vt{S}_m \accentset{{\cc@style\underline{\mskip10mu}}}{\vt{m}} = \accentset{{\cc@style\underline{\mskip10mu}}}{R} \hspace{1pt}\accentset{{\cc@style\underline{\mskip10mu}}}{\vt{m}}.
\nonumber
\end{align}

We set
the ``tau function'' $F_{n,m}$ as
\begin{align}
F
= \det {\cal F}.
\nonumber
\end{align}
Here, the \mbox{$(M+N) \times (M+N)$} matrix
${\cal F}$ is defined as
\begin{align}
{\cal F} := I + {\cal A}\hspace{2pt} \Xi
\left(
\begin{array}{cc}
 K & \\
 & L \\
\end{array}
\right)
\accentset{{\cc@style\underline{\mskip10mu}}}{\Xi},
\nonumber
\end{align}
where ${\cal A}$ is a constant \mbox{$(M+N) \times (M+N)$} matrix.
We also set
the other
``tau functions'' $G_{n,m}$
and $H_{n,m}$ as
\begin{align}
G
= \left( \accentset{{\cc@style\underline{\mskip10mu}}}{\vt{m}}^{T} {\cal F}^{-1} {\cal A}
	\hspace{2pt}\vt{l} \right) F, \hspace{5mm}
H
= \left(\hspace{1pt} \accentset{{\cc@style\underline{\mskip10mu}}}{\vt{l}}^{\hspace{2pt}T}
	{\cal F}^{-1} {\cal A} \hspace{1pt}\vt{m} \right) F.
\nonumber
\end{align}
Note that
the
one-dromion solution
is
reproduced
by setting \mbox{$M=N=1$}, up to
a minor redefinition of the parameters.
Let us check
that these ``tau functions'' indeed
satisfy
the only bilinear equation without time
derivatives,
(\ref{bilin1}).
With the aid of the previous relations, we have
\begin{align}
\vt{S}_n {\cal F} -{\cal F} & = {\cal A} \hspace{2pt}\Xi
\left(
\begin{array}{cc}
P & \\
 & I \\
\end{array}
\right)
\left(
\begin{array}{cc}
K & \\
 & L \\
\end{array}
\right)
\left(
\begin{array}{cc}
\accentset{{\cc@style\underline{\mskip10mu}}}{P} & \\
 & I \\
\end{array}
\right)
\accentset{{\cc@style\underline{\mskip10mu}}}{\Xi}
-{\cal A} \hspace{2pt}\Xi
\left(
\begin{array}{cc}
K & \\
 & L \\
\end{array}
\right) \accentset{{\cc@style\underline{\mskip10mu}}}{\Xi}
\nonumber \\
&= -{\cal A} \hspace{2pt}\Xi
\left(
\begin{array}{cc}
K-P K \accentset{{\cc@style\underline{\mskip10mu}}}{P} & \\
 & O \\
\end{array}
\right) \accentset{{\cc@style\underline{\mskip10mu}}}{\Xi}
\nonumber \\
&= -{\cal A} \hspace{2pt}\Xi \hspace{1pt}\vt{e}_M \hspace{1pt}
 \vt{e}_M^{\hspace{1pt}T} \hspace{1pt}\accentset{{\cc@style\underline{\mskip10mu}}}{\Xi}
\nonumber \\
&=-{\cal A} \hspace{2pt}  \vt{l} \hspace{2pt}\accentset{{\cc@style\underline{\mskip10mu}}}{\vt{l}}^{\hspace{2pt}T}.
\nonumber
\end{align}
Thus, we obtain
\begin{align}
\vt{S}_n  F &= \det \left( {\cal F} -{\cal A} \hspace{2pt}
	\vt{l} \hspace{2pt}\accentset{{\cc@style\underline{\mskip10mu}}}{\vt{l}}^{\hspace{2pt}T}\right)
\nonumber \\
&= \det \left( I - {\cal F}^{-1} {\cal A} \hspace{2pt}
	\vt{l} \hspace{2pt}\accentset{{\cc@style\underline{\mskip10mu}}}{\vt{l}}^{\hspace{2pt}T}\right) F
\nonumber \\
&= \left( 1 - \accentset{{\cc@style\underline{\mskip10mu}}}{\vt{l}}^{\hspace{2pt}T}
 {\cal F}^{-1} {\cal A} \hspace{2pt} \vt{l} \right) F.
\nonumber
\end{align}
Similarly, we
obtain
\begin{align}
\vt{S}_m  F &= \left( 1 - \accentset{{\cc@style\underline{\mskip10mu}}}{\vt{m}}^{T}
 {\cal F}^{-1} {\cal A} \hspace{1pt} \vt{m} \right) F.
\nonumber
\end{align}
To
compute
\mbox{$\vt{S}_n \vt{S}_m  F$}, we still need to know
\mbox{$\vt{S}_n {\cal F}^{-1}$}.
This can be expressed
as
\begin{align}
\vt{S}_n {\cal F}^{-1} &= \left( \vt{S}_n {\cal F} \right)^{-1}
\nonumber \\
&= \left( {\cal F} -{\cal A} \hspace{2pt}
	\vt{l} \hspace{2pt}\accentset{{\cc@style\underline{\mskip10mu}}}{\vt{l}}^{\hspace{2pt}T}\right)^{-1}
\nonumber \\
&= {\cal F}^{-1} + \frac{{\cal F}^{-1}{\cal A} \hspace{2pt}
	\vt{l} \hspace{2pt}\accentset{{\cc@style\underline{\mskip10mu}}}{\vt{l}}^{\hspace{2pt}T}{\cal F}^{-1}}
{1- \accentset{{\cc@style\underline{\mskip10mu}}}{\vt{l}}^{\hspace{2pt}T} {\cal F}^{-1}{\cal A} \hspace{2pt}\vt{l}}.
\nonumber
\end{align}
Here,
we
used the so-called Sherman--Morrison formula;
recall
that
\mbox{${\cal A} \hspace{2pt} \vt{l}$} is a column vector and
\mbox{$\accentset{{\cc@style\underline{\mskip10mu}}}{\vt{l}}^{\hspace{2pt}T}$} is a row vector.
Combining the above results, we obtain
\begin{align}
\left( \vt{S}_n \vt{S}_m  F \right) F
&=
\left[ 1 - \accentset{{\cc@style\underline{\mskip10mu}}}{\vt{m}}^{T}
 \left( \vt{S}_n {\cal F}^{-1}\right)
 {\cal A} \hspace{1pt} \vt{m} \right]
\left(\vt{S}_n F \right) F
\nonumber \\[1mm]
&= \left(\vt{S}_n F \right)
\left( 1 - \accentset{{\cc@style\underline{\mskip10mu}}}{\vt{m}}^{T}
 {\cal F}^{-1} {\cal A} \hspace{1pt} \vt{m} \right) F
 -\left( \accentset{{\cc@style\underline{\mskip10mu}}}{\vt{m}}^{T} {\cal F}^{-1} {\cal A}
        \hspace{2pt}\vt{l} \right) F
 \left(\hspace{1pt} \accentset{{\cc@style\underline{\mskip10mu}}}{\vt{l}}^{\hspace{2pt}T}
        {\cal F}^{-1} {\cal A} \hspace{1pt}\vt{m} \right) F
\nonumber
\nonumber \\[1mm]
&= \left(\vt{S}_n F \right) \left(\vt{S}_m F \right) -GH.
\nonumber
\end{align}
This completes the proof of (\ref{bilin1}).

It is a direct but lengthy calculation to check the
remaining
12 equations
(\ref{bilin2})--(\ref{bilin13})
involving time derivatives.
For example,
in order
to check (\ref{bilin3}),
we need the following intermediate formulas:
\begin{align}
\partial_{t_a} F &=
 -\left( \accentset{{\cc@style\underline{\mskip10mu}}}{\vt{m}}^{T}
 {\cal F}^{-1} {\cal A}\hspace{1pt} R^{-1}\vt{m} \right) F,
\nonumber \\[1mm]
\partial_{t_a} H &=
\left( \accentset{{\cc@style\underline{\mskip10mu}}}{\vt{l}}^{\hspace{2pt}T}
 {\cal F}^{-1} {\cal A}\hspace{1pt} R^{-1}\vt{m} \right)
\left( \accentset{{\cc@style\underline{\mskip10mu}}}{\vt{m}}^{T}
 {\cal F}^{-1} {\cal A} \hspace{1pt} \vt{m} -1 \right) F
 - \left(\hspace{1pt} \accentset{{\cc@style\underline{\mskip10mu}}}{\vt{l}}^{\hspace{2pt}T}
        {\cal F}^{-1} {\cal A} \hspace{1pt}\vt{m} \right)
 \left( \accentset{{\cc@style\underline{\mskip10mu}}}{\vt{m}}^{T}
 {\cal F}^{-1} {\cal A}\hspace{1pt} R^{-1}\vt{m} \right) F,
\nonumber
\nonumber \\[1mm]
\vt{S}_m^{-1} H
&=  \left( \accentset{{\cc@style\underline{\mskip10mu}}}{\vt{l}}^{\hspace{2pt}T}
 {\cal F}^{-1} {\cal A}\hspace{1pt} R^{-1}\vt{m} \right) F.
\nonumber
\end{align}
To
obtain
the first
formula, we first
compute \mbox{$\partial_{t_a} {\cal F}$} as
\begin{align}
\partial_{t_a} {\cal F} &
= - {\cal A} \hspace{2pt}\Xi
\left(
\begin{array}{cc}
O & \\
 & Q^{-1} \\
\end{array}
\right)
\left(
\begin{array}{cc}
K & \\
 & L \\
\end{array}
\right)
\accentset{{\cc@style\underline{\mskip10mu}}}{\Xi}
+ {\cal A} \hspace{2pt}\Xi
\left(
\begin{array}{cc}
K & \\
 & L \\
\end{array}
\right)
\left(
\begin{array}{cc}
O & \\
 & \accentset{{\cc@style\underline{\mskip10mu}}}{Q}\\
\end{array}
\right)
\accentset{{\cc@style\underline{\mskip10mu}}}{\Xi}
\nonumber \\
&= -{\cal A} \hspace{2pt}\Xi
\left(
\begin{array}{cc}
O & \\
 & Q^{-1} E_N \\
\end{array}
\right) \accentset{{\cc@style\underline{\mskip10mu}}}{\Xi}
\nonumber \\
&= -{\cal A} \hspace{1pt}R^{-1} \Xi \hspace{1pt}
 \vt{e}_N \hspace{1pt}
 \vt{e}_N^{\hspace{1pt}T} \hspace{1pt}\accentset{{\cc@style\underline{\mskip10mu}}}{\Xi}
\nonumber \\
&=-{\cal A} \hspace{1pt}R^{-1}
	\vt{m} \hspace{1pt}\accentset{{\cc@style\underline{\mskip10mu}}}{\vt{m}}^{T}.
\nonumber
\end{align}
Then, we multiply it
by
${\cal F}^{-1}$
and take the trace.

\section{Concluding remarks}

In this paper, we
have studied
a suitable space discretization
of the Davey--Stewartson system.
The Davey--Stewartson system is
an
integrable
NLS system in \mbox{$2+1$} dimensions,
which involves
two spatial variables
on an equal footing
and allows the complex conjugation reduction
between the
dependent
variables.
We started with
a natural
\mbox{$(2+1)$}-dimensional generalization
of the Ablowitz--Ladik lattice
and then
considered
a nonlocal
change
of dependent variables
to symmetrize the equations of motion.
Consequently,
we
obtained
the space-discrete Davey--Stewartson system
inheriting
most of the
important properties of the continuous system;
in particular,
it is integrable and
allows the complex conjugation reduction.
The price
to pay
is
the irrationality
of
the equations of motion
and their high degree of nonlocality,
which
are not seen
in the continuous
case.
Through
a simple reduction,
we
reduced the degree of nonlocality
and obtained a discrete
modified KdV-type system
in
two spatial
dimensions,
namely,
the \mbox{$(2+1)$}-dimensional
modified Volterra lattice (\ref{2DmLV}).

The \mbox{$(2+1)$}-dimensional Ablowitz--Ladik lattice,
as well as  the space-discrete Davey--Stewartson system,
is a superposition of four elementary
flows that are mutually commutative.
Naturally,
the number of
elementary flows
is
equal to the
number of
directions
on the square lattice.
Note also
that
both
the \mbox{$(1+1)$}-dimensional Ablowitz--Ladik lattice
and the continuous Davey--Stewartson system
can be written as a sum of
two
commuting flows.
We conjecture that
the \mbox{$(2+1)$}-dimensional Ablowitz--Ladik lattice
and the space-discrete Davey--Stewartson system
possess four infinite
sets
of higher
symmetries.
As in the original
Ablowitz--Ladik lattice~\cite{Vek02,Sada},
each
set of
symmetries
could be
generated from a single
discrete-time system
using the Maclaurin expansion in
the step-size
parameter.
It would be
interesting
to
provide
a more precise description
within
the framework of
the
Sato theory, {\it e.g.},
the discrete two-component KP hierarchy (cf.~\cite{Hu07}).

Using the Hirota bilinear method,
we
have constructed
exact solutions such as
the multidromion solutions
of the \mbox{$(2+1)$}-dimensional Ablowitz--Ladik lattice
and the space-discrete Davey--Stewartson system
concurrently.
Their solutions can be
obtained
from
the same set of bilinear equations,
although
their bilinearizing transformations
are
rather different
(cf.~(\ref{FGH}) and (\ref{FGH2})).
Note that (\ref{FGH2})
reflects the
irrationality
of
the space-discrete Davey--Stewartson system
that can, however, allow the complex conjugation reduction.
The solutions
were derived as the common solutions
of the four
elementary
flows.
On the level of the bilinear equations,
the
four flows look fully
symmetric
and stand on an equal footing.
Thus, it is relatively easy to
construct their common solutions
despite the high number of
bilinear
equations.

\section*{Acknowledgments}
The authors thank
Professor Folkert M\"uller-Hoissen
and
Dr.\ Ken-ichi
Maruno
for their useful comments.

\addcontentsline{toc}{section}{References}

redpajama_set_name: RedPajamaArXiv