Differential-difference Kadomtsev-Petviashvili equation

Differential-difference Kadomtsev-Petviashvili equation: properties and integrability

We pre~c~l t a mum%, on ccmn m e p b h t y prupcmes of dlfFmtid41fference Kndomtsev-Petv~ashvh (DAKP) eqi~auon Wc cxplm Ihc d~iCe~csUul-d~ffe~cz~cc vusluil of Salo U m r y unddmve Ulr DAKFequstmn d.; 1 tiril nonmv- lai nrelnki m the ringlecompanellt KT inrml) in (his pmccss. we exploit tho Sato theoq to o b w coiservatmn l w b and general~sed yrnmetnb of the same Wc funhrr shon~ that thc Wronshan form of h N-sohton sob~tions and t l . i

tmd ruluUon\ Iollow nalurally from Unr upproilsh S ~ u ~ l a n t y reducuon and Punlcvc-nmgulanry confincmcnt vnalyi~s are ycrfoimd We Yeso d~scuir a gauge rqruvdmce of rheDdKP equahon and study cmltnm n l e p b t h t y properties oi themnlnng system as well

1. Introduction

Modem nonlmear dynamlcs swrted with a fundamental question-when is the gven model mtegrable? Can one define ~t prec~sely? The answer is obv~ously oo. But, at the same trnlc, we try to find a closc deiimbon of mtegrability with the meanlng of integrabng and find die solu- boniexistence theory hke L~ouvi l le . '~~ In contrast to the linear theory ~t is well known thai there ia nu general theory to handle norhear systems and get theu solubons However, the present status 1s not bad, thanks to the dlscovety of Inverse Scattering Transfom (KT) method3.' through aluch a large class of norilmear partial differenual equauons (NPDEs) had been solved and spccial solutio~is called soltons obtamed At first, Zabusky and Kruskal observed soliton soluuon, numerically m Korteweg-de Vnes (KdV) equauon.' Gardner er d6 developed IST method to solve the imbal value problem for the KdV equabon Tlus discovery becomes vltal smce, after KdV equabon was solved exphcitiy, many more nonlinear systems of physical importance were also treated u'itli IS1 Again, this method was generalized to matrix formalism, notably by Zakharov and Shabar (zs)' and thcn by Ablowltz, Kaup, Newell and Segur (AKNS) These developments extended the applrcability of IST to handle equations with complex potentmls and coupled systems as well Lax fml reformulated these settings of IST m terms of linear operators. now popularly called Lax p a r or L M These operators sahsfy c e m n linear eigenvalue problem Obtanmg a suitable G M p a for a given nonlinear svstem is eaulvalent to sav that the eiven NPDE ha^ been heanzed and thus solvabihh, is feasible usmg IST Those equations solvable through IST are called integrable systems and

they possess soliton solutions, in fact N of them. After these pioneer~ng works, maly mathe- mahcal twls such as Hmta's b h e a formalism!M2 Panlev6 testsP3"' recursion operator, Lie-Backlund symmehles and were developed to idenOfy the integrable systems and study further analytical and algebraic propeItIes of them

In the case of infil~te &mensional systems (IDES), the system 1s considered integrable (worlung definihon) if rt satisfies one ofthe following cntena.

1 The system rs linearized through suitable variable transfonnatron.

2. The system is solvable through IST method,' after findrng suitable Lax p m or elgenvaiue problems.

3 The system possesses rnfiito number of conserved quantlhes

4 In vrew of the symmetry apptuach, "An equation is mtegrable lf it possesses lnfinltely many tnne-mdepndent non-Lx pomt syrnmewes" These symmetries are called generalized symnctnes or Lzc-BacHund transfonnahons."

5 The system IS hl- or tnhneanwble through smrable dependent vanahle transformahons and a h i s N-soliton solut~ons.

6. The system whrch passes the so-called PainlevC test is a good candidate of an rnlegrable system In this cornemon, Ablowio, Kamani and Segur (ARS) Iomulated the Panlev6 conjecture, "Every ordinary d~fferentral equamn whrch mseF as a reduction of a completely integrable system i? of ParnlcvB type (perhaps after a transfonnahon of vanab~es)".~ Ths conjecblre pmvrded a most useful mlegrabity detector. Foflomg this conjecture, A M proposed an dgonthm whch tests thc ParnJevb property llus property sdtes that a system of ODES saUsfies the necessaiy condIbon for the Painlevt- property, t.e having no movable crihcal porn$ other than poles, if all Its solutions can be expanded in the Laurent series n m eveq one of then movable singulariOes Th~s test was exlended by Weiss el aid' to PDEs

In the folloivmg, we briefly discuss vanous methods used to mdy the rntegrable system?

1.2. lar method

The idea behind ttus approach is to denve the nonlinear evoluhon equaoon whrch anses as the wmpahbihty cond~tlon of two hear equations assonated wlh a spectral problem The spectral pmblem and the time evoluuon of the eigenfunction m given hy the eqnaaon$

LY= av (1)

and

2Y = My, b (2)

mpechvely Assume that the spectral parameter k 1s mdependent of t The compatlbi11N con- &tion of these two equations givcs us

whcre L and Mare linear operaton. Equallon (1) IS called Lax equailon. JAX' explans how to derrvc operator ,I4 for glven L Once this goal rs xhreved we can fn~d the solution of the glveri nonhnear equauon through IST method Nunerous generalrzatlons of the e~genvalue problems such as Zakharov and shahat,' Ahlowtr, r.1 a1 ,8 Ablowitl and ~aherman,'%au~ and new^ ell," Wadnu, et a1 ,'" " Shrmzo and Wadah," 1sh1mnn,i4.~" Wadau and Sogo:' Konno and Jeffrey '' ase ava~lable m lllerature curennp a wde range of evolution equations

Ibrnta mtrodiiied a more illrect to derive soliton solut~ons of nonlinear equatluns By mtioducing a suitable dependent vvanble transforn~aoon, the solmn equahon becomes br- Imca Apply~ng the perturbation technque to the rcwlt~ne hilmcar cquatlon one can qystem- ahcall) construct lhc h"so11ton ?elution. In fact, if thr system admls Id-sulilon solullon lhc senes expansion m terms u l small parameter In the ahovc perturbatwe analys~s truncates auto- matlcally In thls process, we obtain a class of lincm p m a l illfferenlldl equations whrch can be ~olvud success~velv to obiam the exact aoluuon The solrton aoluhon obtmed throunh t h ~ s techmqui. IS a polynomal In exponenual funcuons. It 1s also noted thar the N-sol~ton soluttans of Iltroh's bilmar equauon can be wrltten In thc fonn of Wronskran and Grammron de lem-

and p[alfian, 21 2'. 28-36 The lalter iormnl~sm is more compact and eaty to handle Apart from findlng N-sohtoi~ soluhon using thls techmqire. we can also obtain the rational soluuons in a d~recr way Y3-y6 Recently, hllmear approach bar been extended ta oinltrlmear fonn, In par- l~culai Lo tnlrncdr 's Many intcrcsiing ri~tcgrahlc systcnls have been bnught In ih~s franework In aomc cases. Ihc tnlmear [oms can be wnllen In bil~near forms by iutroducrng exira 7-funct~ons

1.4 Conse~vat~on laws and ,yenerahied symmernes

For a given nonhnear evolut~on equaoon of the form

u, = F(u, s. a, 1, (6) a consendon law is defined in the f o m ~

T,+X,=O (7)

whsh 1s saticfied by all solutions of ( G ) , where Tts the conserved density and X the flux. Here, T, denotes the total derivative of T wlrh respect to I, bkew~se, X, denotes the same for X w~th respect to x It IS noted thar Tmvolves u and its s derivaoves only and the terms like u,, u,, . are replaced by u, u , u, appropmtely usrng 16). the grven equation. The existence of mfi- mls number ~ i c u n r r v e d . ,U~IN:IITS tt111.11 ase IP n i i ( I u ~ ~ . ~ n nilh r:ip~;.I 10 .I \u~lible I 'OI , , .~ 1ucL.e~ I I W L I I I C ~ ~ ~ I L I \ d ' a ~ : t ~ v z n ~ \ W I I J . " '* " ' '- I t I , 3 u d l c ~ t ~ l ~ ~ l i : . I ?XI U ~ I - , - , there is a close connection between the symmetries and conserved quantlhes through the sym- plectlc operators used In Pomon brackets?"

Panlev6 analysis is used to study the s~nylaril)' structure of the grven nodloear equat~on." '~~ If only the movable cntrcal s~ngulanues of all soluhon oC the grven system are poles, then we say that the system passes the PalnlevC test and hence it could be a good cand~datc of an inte-

314 K M TAMIZHMANI AND S KANAOA VEL

gable system In pracuce, this anaaysls is a very effective tool to class~fy possible ~ntegrable systems In a systemauc way In recent times, the nature of smgulantles also &crates the form of the r-funchon which 1s used to bllineanze the gwen system "9.99.1W lu addihon, there IS some evidence to show that this method can also be used to obtam Lax pars and Backlund transfor- maoons?' However, lt is not yet estabhshed completely that one can o h m the last mentloned propemes systemaucally through tlus approach.

The theory of Lle pomt symmetnes goes back to the 19th century Sophus Lie stuthed the mvanance properties of the symmeQ goups and used them to solveiclass~fy the dlfferenhal equahon. Using one-parameter Lie group of symmetnes one can systemaucally reduce the order of the ordmary differenhd equahon by one F~ndmg this group of symmetries whch leaves the system lnvaciant IS tedious and computatlonally complex, neverthcless, it is very much algorithmic and systemauc Due to thc a l g o n t h c nature, b s techmque has been used widely to find special soluhons to reduce the dmension of the independent variables espe- cially m NF'DEs, to ldenhfy mtegable systems to one of the SIX PamlevC equahons By this method we can understand the underlymg algebraic and geometric ShuCNre of the gven system,S3TJ 101-110

1.7 Sato theov

Various methods developed so far to lnveshgate the sollton equations inmcate the nch mathe- matical shucmre of the sohton systems It is Sat0 who unveiled the algebrax ssrmcue behnd them usmg the method of algebwc analys~s I". 'IZ He nooced that the r-funchon of the Kadomtsev-Petviashvili (KP) equahon IS connected with the Plucker coordinates appeanng in the theory of Grassmann manfolds He also nouced that the tutality of soluuons of the KP equauon as well as 1 s generahation consutute an infimte-dunensional Grassmann mandold Ohta et ai 'I3 presented a clear descrtpt~on of Sato theory m an elementxy way S m g fram the pseudo-differenhal operator they conshucl a h e a r homogeneous ordinary dfferenual equation and explicitly explam the conuechon between the coefficients and the soluhons of the same 'lbey lntioduce an infimte number of tune vanables In the coefticientc and impose certam tune dependence on the soluhons and denve Sato, Lax. and ZS equatlons and the h e a r e~genvdue problem associated to the generahed Lax equauon. As a consequence, the KP luer- achy was denved in a systemaoc manner and vanous reductions of ~t have been presented. They brought out the connection between the z-fnnchon and the billnear f o m uslng Young diagram After the development of this grand theory, Date et al.!", 'I5 and limbo and Ivbwaci6 extended Sato's idea and developed the theory of transformahon groups for sol~ton equatlons whch essenually e x p l u s the group-theoretical foundation of Huota's method and Sato theory. The man m of h s theory is to reveal the intimate relat~on between the KP herarchy and the lntinrte dimensional Lie algebra gl+) usmg the language of free Femon operators They indeed had a hlg p r o m to class~fy sol~ton herarches written in b~linear form accordlug to vanous realizations of Lie algebras Unng Sato theory, recently a mce method was developed to derive the generalized symmetnes and conserved quantrhes of KP h~erarchy "7ii9 Hence, n is clear that Sato theory is the most powerful method through whch one can o b m systemati-

DFFmENTIAC,-DIFm,RFNCE KADOMTSEWPETVIASHVli.1 EQUATION 315

cally 1ntegrabihQ properties such as Lax pair, soliton solu:ions, conservation laws and symmetries of the ultegrahle systcms m an u m k d way " i~1 '3~L1 '~1L9 Usmg multrcomponenr verslon of thrq theory one can amw at Davey-Stewatson equahon and nonllncar Schrddinger equauon in 2 + l d~mensions ' I q , '2'', "'

1 8 Diflerennai-ditferencefiamework

In contrast to the conhnuous equanons wherem enormous amount of research has been done to mnvesngate vanous aspects of ~ntegrablllty, differential-difference or fully discrete systems have not heen srudled m depth However, some attempts had been made by Hirota In looklng at the bhnear formalism for many known soliton eqnahons m differentm-difference act- nngs '"I9 Also, Ablowltz and Ladik moduced dfferenud-difference analogue of AKNS scheme and ohtamed many different~d-difference soliton systems like nonhnear Schrodinger equahon They also extended the IST method lo dfferential-dfference case In addtion, many more important developments have taken place tn this area."'. 12C1" Usmg group- theorehc techn~ques, Date eta^."^ proposed a method and denved a large class of conunuous, sem-continuous, discrete sollton equations More recently, there was a remarkable discovery of proposing smgulanty structure analysls for nonhnear difference equanon hy Grammahcos et a1 lS6 %IS technque, now popularly called slngnlar~ty confinement, 19 very powerful In ~dentifylng d~acrete integrabie systems In particular, 11 1s mtereshng to see that the discrelc versions of P a n l e d equations have been obta~ned through thls lechmque and other propeaies have been stumed '" 17' Soon after the d~scovery of smgulanty confinemeni, Raman1 er a1 13' synthes~zed both the classical Panlev6 analys~s and smgulanty confinement together and proposed the singularity confinement approach to test the nature of sinylanty m ddfcrcnt~al- dlficrcncc equatms Agan. t h ~ s has been successfully implemented for several diffennbal- dlilercncc systems n~cludmg integm-d~fierenual eqaauons "' In the same penod, Collowmg the idea of ~ a e d a , " ~ ' ~ ' Lev1 and Winternltz proposcd Lie bymmctly analysls for differenual- d~fference systems"~'"' whlch was stndIed by others also 14"'50 As m the conhnuous case, the existence of Lie point symmetnes obtained through Lie's one-parameter tran~formalion group for dlfferent~al-dffererence cquat~oiis agam becomes very important. Using this theory, as 111 the contlnmus case, one could iind sirmlanty soluhoos and use them for reductions Though t h ~ s method is shll m the early stage many mteresting results have already heen obtamed.

1 9 Present work

In Section 2, wc d~scurs the denvations of tlk d~fferential-d~fference Kadomtsev-Petviashvili @AKP) equahon, conservauon laws, generaked symnmctnes and solut~ons.'~'. li2 Next. we bnefly mention smylant). stmctures and Lle symmetry analysls of DAKP. Detalls w ~ l l bc published e l s e ~ h e r e . ' ~ ' ~ ' ~ ' Rnally, we also dwuss a gauge equvalence of the DAKP equaon and study certam lntegrabllity properties of t h ~ s system.'53~'"

lo Section 3, prchrmnary definitions and results uceded to develop the differential- drfference Salo theo~y are presented. Mso, the Sato equahon, generdlzed Lax equation, Zak- harov-Shabat euuatlon. and DAKP luerarchv are denved. m e associated eieenvalue oroblem is . . u .

considered and the conserved quantities a~id generalrzed symmetnes of DAKP equatlon are

316 K. M T- AND S KANAGA VEL

obtained systematically. In Section 4, N-soliton soluuon and the ratlonal solutions of DAKP and its hierarchy are given m terms of a Wronslaan determinant In Secuon 5, ke-pomt symmetry analyns ~s performed for the Werenhal-difference KP equauon and the Veselov- Shabat e q ~ t l o n ' ~ is o b m e d as a slrmlanty reducuon. Also, srnguhty structure of the solution of DAKP equation a analysed using Pamlev6-singuhty confmement method for the dif- femal-d~fference equation. In Section 6 , we denve a gauge eqmvalence of DAKP equahon and study certam integrability pmperhes such as Lax pau, wnservatmn laws and genedzed symmetries of the resulting system and perform Lie symmetry and singularity structure analysis

2. Differentinl-diflerence Sato theory

2.1 Introduchon

The search for dimete or sem-&mete mtegrable equations started alter the ldenhficatlcm of sohmns m Toda lamce Toda lamce IS a prototype model for the Merentlal-erence sohton equation wiuch possesses all mteggrab~hty propemes such as Lax par representauon, existence of a t e number of conserved quanubes and N-sohton solutson, etc as other soliton equations in continuous case Thus, various methods used to identify the Integrable systems m the continuous case were extended to sem-d~screte case too POI examplc, the IST metbod by Ablowltz and ~ a d k . ' ~ , " ~ discrete bllmear forms by h t a , L Z Z Q group-theoretlc method by Date et al."" I" and Jmbo and m ~ a , " ~ Lax method by ~upershmidl,"~ Lre symmetry method of Mae& er a1 '"la, Lev1 and coworkers,'"-'" Quispel and others'"-149 and ~aeta"" the Panlev.! method by Raman1 et a1 13"'~' Srnce Sato theory urnties all these approaches in the confinuous case, it IS natural to expect that the Sato theory plays the same mle for Merentsal-dBerence case also. llus mouvates us to look for the Sato theory for dlf- ferenbal4fference mtegrable equahons. l 3 we formulated a suitable framework to treat the differenhal4fference equahons. In fact, using this approach we have obmed the lax par, consematson laws and generalued symmetries of the D m equahon systematically 'jl

2 2 Prelmmnes

We stvt fmm the definibon of the forward difference operator A and the shift operator E glven by

Mn) =An + 1) -An) Efin) =& t 1) ( 8 )

for all values of n (real or complex). Here the step sue 1s taken to be one. From (8) it 1s clear that A = E - 1. The Leibniz rule for the difference of product of two FuncUons e gwen by

DiWEREWIAL-!JFFERENCE KADOMTSEV-PEWLASHVIU EQUATION 311

for all integers m Using the Lembniz ru le (9) for the &fference set-up we amve at negahve and posihve powers of Am the form

( f g ) = ( E - ~ f ) ~ - ~ ~ - 3 ( E 4 4 ) A 4 g + ~ ( E - ~ A ' f)A-'g+..

A-' ( f g ) = (E-2 f ) ~ - ' ~ - Z ( E - ~ A ~ ) A - ~ ~ + 3(E4A2 f ) ~ + ~ + . .

A-'(fg) = (E-' f)A-'g - ( E ' ~ A ~ ) A - ~ ~ +(E-~A' f ) ~ - ~ g + . .

A ( f d = (Ef )& + (Af )E

& f d = ( E ' ~ ~ ~ + ~ ( E A ~ ) A E + ( A ' ~ ) E

h 3 ( f g ) = ( E 3 f ) ~ ~ g + 3 ( ~ ~ ~ f ~ ~ + 3 ( ~ ~ ~ f ) A g t ( ~ ~ f ) ~

Throughout tlus paper, we use the followmng convenhon

where 1, m and k are mtegers Now we define the formal mner product of the gmen functions ~(n), v(n) lo such a way that

c u(n), v(n)> = A-'(u(n)v(n)). (12)

Also, we assume that u(n), v(n) 4 0 as n + m The formal adjomnt of the d~fference operator 1s defmed by

( q W m p(4) = ( - l Y " p W T q ( n ) (13)

for all funchons p(n) and q(n) Throughout this paper we assume that the bfference operator A and the Werentlal operator kcommutes.

2 3. Pseudo-dzfference operator

In the continuous case, the pseudo-d~fferential operator plays a fundamental mle in developing Sato theory.'" BY proper mmpulatlon of tbts operator one can & m e Lax par, conserved

318 K M T V AND S LWAGA VEL

quantiucs and symmemes m a systematic way. So, by analogy, with contmuous case it is natural and wonh to start w~th the pseudo-dffercnce operator W, glven by

where w,, j = 1, 2: are h c u o n s of n We expect that the inverse of the pseudo-d~fference operator W n also of the same €om and is given by

W' = I + V,S' + Y~A-' + . (13

where v,, J = I, 2, are funchons of n. Smce W and U" are formal inverse to each other, we h a v e W W ' = ~ ~ ~ = l .

Using the expressions m eqns (14) and (15) in WW' = 1 and remanglng the terms and comparing the lke powers of A on both sides we get an infimite number of equations for v,s m terms of w?, 8 . 1 = 1, 2; whlch gwe the relationslnp between v,s and w,s, i, J = 1, 2, . We llst the first few of them below.

For convenience, we restncr the operator W to only a finrte number of terns say m and thus conslder the mth ordcr linear homogeneous ord~nary drfference equduon glven by

ULA'Xn)= (Am t wlAm-' +w2Aa2 + . + w,,)f(n) = 0 (17) which has m Imearly mdependent soluUons say, f "in), f "in), . f"'(n) Smcc thescj%k we soluhons of eqn (17) and hence we have a system of m l i n w equabons in rn unknowns iq , wi, -, w,, w e n by

Solvmg this systcrn of algebnic eyuabuns uslng Crarner's rule (th~s a possibie bccause the determinant of the coefficlenl matnx oi the above system (18) IS nothmg but thc Casorati de- terminait whlch is nonzero, due to thc faa that 4:'s are Inearly independent), we mive at

(19)

for] = 1,2, m. Substituting the values of w,s m (17) and smplifymg, we get

In eqn (20), the operator A3, J = 1, 2,. ., m has to be put m the nghtmost posltion when we evaluate the determinant of the numerator.

We assume that the set of linearly Independent soluuons~'(n), j = 1, 2, ., m of the mth order h e a r M e ~ e n c e equation (17) are analytw and hence can be expanded by usmg Newton- Gregory formula,

where A ~ ' ( O ) = ~ ~ ) and n("=n(n - 1)-(n-rt 1) Usmg (18) and (21) we can wnte the system of h e a r equations (17) as

where

1s an 1 x - matrix and @ is an - x m mahix Let A be the shtft matnx

given by

Usmg the above matnx, we can wnte

K M TAMZffMAM AND S W A G A VEL

Now. we define,

(26)

The de temant formed by ihe fmt m rows of H(4 IS nothlng but the denominator in W, wbch is the Casofab detemant for the soluuons of the difference equation (17).

h iius sechon, we dncusa the impact of hme dependence m the coeffic~ents of the ordinary d~ffcrence equatlon (17). We lnboduce an infkte number of one variables t = (1%. t,, ) m w,

as w, = w,in;r),j = 1, 2, As a consequence of this, we have

We cassfdet the ame evolumn of H(n) In the form

where q(t, A) .- ~ b , t ~ k We wnte fondly

Expandmg the above exprelsron (29) and comparing the coefficients of lrke powers of A on both sldes, we get

These polynomials are analogues to Schur polynormals in the conhnuous case They have a specla1 property

We use fhe above property (31) to express the funcnon H(n,t) in terms of P6. whxh u wntten m Ule form

4 p2 ... *!I *@I . 56") H(n;r)=I; i;? :: :: ::]

It is easy to ider fmm (33) and (34) that h(n;t) = h t l (n ; t), and j = 1,2, ., m are solutions of a set of hear partla1 difEmennal4fference equabous

K M TAMUHMAW AND S KANAGA VEL

wlth the i m t d value h(n,O) =f"(n) Hence, the lmear d~fference equation (17) becomes,

~,A'"$)(n,l)=(d'" +w,A~- ' +w2~m-2+ . . +w,,,)F$)(n;t)=0, 1 = 1,2, . m. (36)

Solvmg these system of equations (36) as esher, we get,

and hence

Now w, and W, are completely p e n m terms of drfferent1al4Sference analogues of Schur polynomah P's u m g (30)

2.5. Sato, Lar, Zakharov and Shahat equatiom

It is well known that the mcrgrability of the n o h e a r systems is associated wrth the findmg of appropndte Lax or ZakhsovShabat equations As m the continuous the differenhal- dfierence version of Sato theory provides the Sato, Law and Zakharov-Shabat equations naN- rally. To actueve this goal we pmceed as follows: d~fferentlatmg eqn (36) with respect to tk, we obtain

smce A and % commutes Usmg the relations (34), m (39). we get

UIETRkN1lAL-DWFERL<NCF K A D O M T S E V P S EQUATION 323

whe~e Bt 1s a Mh-order d~fference operator. 8,s can be obmned by applying bm w;' from the

nght of eqn i4 l )

B, , s w ; l + W,A~W;~ &

(42)

From cqn (42), a c can obtarn by lnultlplyrng Wm from nght,

Hence the tlme evolution of the pseudo-diiference operator WJn, I ) la goiemed by

whch 1s the differentwl-difference version of the famous Sam equauon I]' The Bcs m the Sato equauon can be computed from W using the follonmg relation

where ( )+ denotes the nonnegahve powers of A only. We have d~scarded the iirrt term of eqn (42). because it involves only negatwc powers ol A, whereas Bk wrisists only nonnegative powers of A Usmng (45) we can denve the Bts exphcltly We list below a first few of them:

Next, we will derive the generalized Lax equahon, involving mfinlte number of nme variables. For tlus, we &fine

L = WAW~. (47)

Substituhng the values of W and W' and rearranging the terms we can wnte

L=A+%+u,A-I+u&+ -. (48)

324 K M T- AND S KANAGA VEL

where y s are expressed in terms of w,s r = 0, 1, 1 = 1,2, We present some u,s

uo =-Awl

q =-Awl -Awl +w,Aw, (49)

u, =-AW,-AW,+AW,E-b, +W,AW, + E - ~ ~ , A W , - ~ , A w ~ E - ' w ,

D8ferenOahng eqn (47) mth respect to tt, we get

The first term on the nght-hand side of the above expressloon (50) will be replaced by the Sato

equation (44) whereas for the second term we have to find 5 For thls, dfferentlamg

WW1 = 1 wth respect to 4 we have

Operabng W' from the left of the above expresslon (51) and rearranMing the terms we have

Subsuiuhng the value of from the Saw equanon (49, we have

= B,WAW-I - WAI+' W" - WAW"B~ + WA~+'W-'

= B,L-LBk

= [ B ~ , L ~

Tbs, we have the generahzed Lax equation

JL -= [Bk, L], k = 1,2,,,. dk

It 1s noted from (47) that

and hence

B, = (L~)+ .

From Bk = (L')~, it is now immediate that

B,=A+uo

B, = A ~ + ( Z U ~ + A I L ~ ) A + ( A U ~ + U ~ + ~ U ~ + h , )

(58) We can show that usmg (55) and (561,

1s also hue Now, we wlll obain the Zakharov-Shabat equation. From (59) we can show that

holds huc Wc denotc B,' = B, - Lk w h ~ h contarns only terms with A-', I > 0 Employmg h s

relatlon In eqn (HI), we amve at

But from Lk = B, - B;, we have

Equahng the difference pat on both sldes of (62), we find that JBm JBk - ,& dt, at", [ I

which is the ZakhamvShabat equation "3

2 6. Di#erentiaMr.ference KP equanon

In the previous section, we denved the Sato, Lax and Zakharov-Shabat equahons ~n a sys- temahe way. In this secaon, we obtain a hierarchy of differential-difference soliton equations usmg Sato theory Since the fmt non-tnv~al member 10 this hemchy a DAKP equauon, we call tlus as DAKP hie~archy. Cons& the L operator and the operatom, k = 1, 2, Uslng the generalued Lax equanon (55). for a given BE, we can denve a set of mnhte number of equahons involving a, u,, So, it IS possible to generate infinlte set of lnfin~te number of equahons for uo, 111, . By appropriately choosing the equaoons ln drfferent sets, we can denve mtegrable n o h e a r dIBerentlaI4fference equations We wlsh to renund the reader that not evay member m these seb e ind~v~dually lotegrable. For example, takmg k= 1 m eqn (55) we get an infinite set of equanons gven by

Also, fork= 2, we have

&=A2ui +2An2 +AZu2 +uIAu0 +2uOAuI + A u O h I +uOul - U , E ~ ~ I ~ %

% = A ' ~ , + z h 2 +2~'u, + 2 h 3 + d u 3 + ~ u , ~ u , +Auoa, +2uou2 a2

+u2Auo + 2 u 0 4 +thu2Au, +ulAuo +up: +u: + u I h i (65)

-u,E-'u, + U ~ E ~ ' Y ~ - U ~ ( E ~ ' U ~ ) I - u I ~ d u , - u ~ E - ~ u ~ - U ~ E - ' U ~

Now, we consider tbe first two equations hom the set of equahons gven m (64) and the first equatlon in (65). Solving these equahons for we arrive at the DAKP equation

DIPFERENTLALDIFF%RFNCB KADOMTSEV-PErVIASWILI EQUATION 327

Tins equation was first denved by Date et a1 through group of transformations approa~h."~

2.7 Conserved pannnes

Once we have the L a pair, it 1s nahlral to ask for the eltlstence of rnfinite number of conservation laws. a basic orooertv of mteerable svstems Aean we u b the Sato's framework to de- , . . . rive them Matsukidaua et a1 "' developed a me&d to denve conservation laws of (he KP equation through Sato theory and the same melhod was implemented"Y to denve the conserved qwmtles oiToda lattice. We follow a slrnilar approach and derive the rnfinrte number of conserved quannhes for DAKP equation through Sato theoiy. For t h s purpose, we first cons~der the linear ergenvalue problem associated with the generahzed Lax equation (55)

and A,, = 0 Uslng Bt = Lk + 8;. we rewnte eqn (67), as

We recall that B,i conslsts only the negahve powers of A. Now we wdl express A*. j = 1. 2, In twms of L-'. For ths pulpose first we fmd L-' We assume that L-' 1s of the form

L-I (69)

Using L-'L = 1 we can determme q,s and we list some of them:

q2 = -E"u,

q3 =E-'u, - E - % ~ -E-~U, + E - ~ U ~ E - ~ U ~

qr =-E"U, +2E"u, -E4uo +E-'u1 -E%, -E-'u2 - ~ E ~ ~ u ~ E ~ ~ u ~

+ E ~ ' ~ ~ E ~ ~ U , + E"U, E - ~ U , + E ~ ~ ~ , E ~ ~ U , + E"U,E"U,

328 K M TAMPnrmANl AND S KAN.4G.4 VEI.

Usmg the cxpressiuns fur I>' and (70), we can express A-!,I = 1,2, :

Hence, eqn (68) becomes

(73)

where oy's are functions of u,s for all j, k = 1,2, , 1 = 0, 1,. .. On using IIw= 21ym (73). we

obtain

From th~s, we get

We denote a(,) = z ; , ~ ~ A - J and hence eqn (75) becomes

Dlfferentiatmg eqn (76) with respect to the time variable I,, we will amve at the conservation laws

Nohct that d" and wrrespond to consewed quanhty and flux, respectively. We list the a.

fust few of the uj"s :

'q) =-ul

a!) =-u,E%, -uZ (78)

a!) = -u,(~"u,)Z + u, E%, -U,E-~U, -u I~ - 'uI - U ~ E - ~ U ,

-u2E"u0 - u,

It IS known from Section 2.6 that the Lax equation w ~ t h k = 1 gwes

hr, = AX' a,

%=h, t Au, tu,v, -u,E-'u, a1

F~om the above equations (79), we cao express u,, u2, u3, m terms of w, and we list the fmt few of r,s for1 = 1,2,- .

- A-1 % I - a,

+A-~(?'u&~ $]-2A-(uo ?$..I+ AP1[uoBB~UO $1

Now snbsntutlng these values in oj", we obtain the conserved demtles of the D A D equa-

uon (66). We West below some of them:

(81)

J Z - ~ - l dU,E-l*-1hr,-E-21'0A* Uo+E-2Uo,1 4 Jfl a, a: &I

2.8 Generalzed symmelnes

Another imponant feature of the Integrable system is that it admits mfimtely many tune- mdepsndent non-LI~ point symmetries-called ~ e n e r a h e d symmelnes Agiun ~t IS s&le to denve the eeneralued svmmetties mine rhe theon, of Sato. In fact. Maisukidaira et oro- posed a method to denve the genera!xed symmeuies of the KT equation us= Sato theory They exphcitly ubtiuned the eigenfunctlon of the associated linear e~genvalue problem and showed that the squared e~genfunction generates generalued qmnernes. We show that tlus strategy can also be adopted here and denve the generahzed symmetries for DhKP equation Before dolng so, we gwe a bnef review of the basic notions in this theory. We cons~der an evolnaon equatlon

where K 1s a funchonal of u. We call the functlonal S(u) a symmeby of (82) d ~t satisfies the l~neanzed equatlon grven by

where the Fr6chet derivahve i€(u) is defined by

It can be shown that a symmeuy Smust satufy

332 K M T&h&%MANl AUD S KANAGA YEL

Tbis means that any symmetry S commutes w~th K(u)

We will show that the e~genfunction of the hear eigenvalne problem (67) and 1ts adjoint generate the symmetries of the DbKP equation Nonce that L = WAW'. Hence, we rewrite eqn (67) in the form

LY= a ~ ,

I e. W A W ' ~ = k y (86)

Applying W' on both sides of (86), we have

A W ~ V = wlay= awlw By- taking yo= w ' y i o the above equation ( 8 8 )

A % = &

The above equation (88) 1s just a fuat-order ordmary lmear Merence equaUon, whose soluuon is gven by

YO =g(i,,t,- , a ) ( i + a y (89)

where g(i1, tz, ; A) 1s an integrahon function. From tius result ~t follows that

~ ~ y r = % = ~ ( ~ t ~ , t ~ , ) ( ~ + a r (90)

and hence the e~genfuncuon is given by

To find the eigenfvnction we need the value of Am'(l + A)')". For t h s purpose, first we daive A-'(l+ M". Now we compute A(l +A)".

Operating )A-I on both sides of (92), we arrive at

Applylng A-' repeatedly on (931, we have

DIFFERENTLLLDIEEXENCE KADOMTSEV-PElVIASWUI EQUATION 333

B U ~ , we have,

? Y = ( L ~ + & ~ +* ,~-=+ . . . ) y i &k

(96)

where es are appropriatefunctions of A, ti, t?, . On usmng L ' p X v , j = l , 2 , ,we obtain

%s is uue for any integer k > 0. On integrang the set of equatrons, we finally find,

when 6 1s agan appropnate functions Compariog eqn (99) w ~ t h (95) at t, = 0, V] = 1, 2,. . we get F, = w,, Vj= 1.2. . . , a d

Hence

We wdl ohtarn the formal adjolnt of Was

To find the value of y* we have lo determine w; , Vj= 1,2, ..For dns purpose, we cons~der

Zw'= anr*

K M TAMIZHMA'II 4ND S KANAGA VEL

( w ~ ) ' ( - A , - ~ ) ( w ' ) ~ ' = ay' =l~',,,'

Talung w ' ~ ' = y; we amve at a linear difference equatlon

-a~- ly ; -Aw;

Solving the abovc equat~on (104) we amie at

= W'y' =h(t, , t2, - ,1)(1+A)-"

and hence we have

y ' - ( ~ ~ ' ) * h ( t , , t , ; ,A)(l+l)-"

Usmg (13) and(l5) ~n (106), we haw

C* =(i+v,d +V,A.?+ )'h(t,,t,, - , ~ j ( l t ~ ) - , ~

= ( I - A ~ ' E , , - ~ A " E ~ ~ , + . .) h(t,,r, , . . . , l)(l+I)- ' (107)

Expanding the above equaoan (107). using Lelbmz d c (lo), we have

I),' = (1-E(v,hl - A V , E A - ~ + A ~ V , E ~ A - ~ + ) +E~(v,L\-~ - 2 d p 2 ~ ? + . . . ) (108)

-E~(v,A-~+ )+...)h(t,,f 2 , ..A)(] +A)-*

v* = (I-EU,EA-' +(EA", + E ? V , ) E ~ A - ~

-(.Eh2r, + ~ A E ' Y ~ + E ~ V ~ ) E ~ A - ' + )h(t,,i2 .,nj(l+n)-" (109)

Now. wc coroputc &(I i~ A) " . A,? I ( ; + /l)-" -A([ + A)+

"1 ( i + ;,)-"'I

= (i +A)-" (1 -(I t A)) (1 10)

= - a ( ~ - ~ a ) - ~

Operatlng -I A-'E on both sides of (1 lo), we have

and hence

A-'E~ (1 t ,I)-n = - IJ

(112)

Using the above result (1 12) m (log), we have.

Comparing eqns (113) and (101), we have

and the w; s are given by

w; =Ev,

wl = EAv, +E2vZ

w; =EA2v1 +2AEZv2 +E3vj

w; = t3E2A2vZ +3E3Av3 +E4vq

On usmg (16) En (115), we get

w; = -Ew,

WE =-EAw, +EW,E~W~ -fi2w2

w; =-EA~W, - ~ E ~ W ~ + ~ E ' W ~ E ~ W ~ +2i?'wZ - ~ E ~ W ~ E W ~ - E ~ W , E W ~ (116)

+ E ' W , E ~ ~ , - E ' ~ ~ E ~ ~ , E W ~ +E3w2Ew1 - E ~ +

336 K M TAMlZHMANi AND S KANAGA VEL

Now, we will show that the e~gen€unctlon and its adjoint m terms of w and w' can be used to generate generaked symmemes of the DAKE equahon. For tlus purpose, we adopt the prow dure developed in Matsuldm "' Usmg the elgenvalue problem (67)

*=BUw dl

dvldv l~2w (117)

and ~ t s adjomt etgenvalue problem

we have

* = ~ y r + u ~ y d l

- "' = - A ~ E - ~ ~ . +AF~(Z~, ,~ . + A ~ ~ ~ , ) h2

Uslng the d e h b o n of Wchet denvahve (83), the lmneanzed DAKP equation is glven by

From (120) and (121) lt 1s obvious that If s sausfies (120). then S=' satisfies the linearized 2,

DAKP equation (121) Hence, 1t n unmemate that $(w") satisfies the hneanzed DAKP

equation (121). Using (101) and (102), we have

where

w~ib wa = 1 and wi = 1 Since yy; is a polynomal expression In i, and ;l is independent of the

tune variables. we hsve

a S,=-s,, k=0,1,2;. (124)

a, whch are solutions of (121) and hcnce generalized symmcmes for the DAKP equation (66) We present below the fxa few generalized synmetnes.

3. Wronskian and rational solutions

3.1 Introduction

It 1s well knomn that many IST solvable nonhnear evolution eqwanons exhibd multisollton soiutlons When we use Hirota's b ~ h e a r method'-l4 these N-soliton solut~ons can be expressed as an Nth order polynomial m N exponenhals Perhapr, a more conveolent representa- tion of such a solution, however, 1s In t m s of the Wronslaam of N exponential functions. The N-soliton solution tor soliton equauons writtcn in the Wronskian form was first mtroduced by Sat~uma'~ and furrher developed by Freeman and ~ n n r n o ~ ~ and N~mmo and Freeman jU T h r proceduc has been applied to the KP,~"." the ~ o u s s i n e s ~ , ~ ~ ~ ~ ~ and other sohton systems ""'

338 K M TAWZHMANIAND S KANAGA VW.

It IS also known that the N-sohton solutron of KP luerarchy can be denved through Sam theory,"3 which is expressed by an appropnatc ?-funchon. This 7-hmcct~on can be expressed m the form of generalized Wronsklan detmnant defined on the mfinite-henslonal Grassmann d o l d . In this fmmework, %rota's b~llnear f o m anse natural!y as Pliicker relations. Using the Laplace expansion of thc determinant, we can easlly veniy that (he .r-function satisfies the gven H~rota's bilinear form.

It has been r e c o w e d that integrable systems, m the sense of IST. possess other class of s~htions as well, called rahonal soluhons 3,8'46 The rihonal soluuons of Ule sohlon equations can be obtamed through vanous means On the other hand, Sato theory promdes a systematic approad, to fmdrhe rahonal soluhons of KP hemchy The fundamental ones are represented In terms of Schur polynomials which sahsfy a certain set of h e a r ddferential equations

3.2 N-Soliton soluhon

Now, we cooslder the DAKP equahon (66) In the form

where u = u(t,, 11. n). Now, using the dependent vanable tmnsfonnahon

meqn (126), we amve at

We repwent this equahon m the Hirota's bilinear form, which can be wntten in terms of &- rota's brlinear operators These operators are defined by the followmg mle."

where m and k are ahhary nonnegative Integers. Usmg the ahove defimhon (129). we can find

DIFFERENi7.A.L-DIFWRNCE KADOMTSEV-PETVIASHVII.1 EQUATION 339

Now, we can easlly see that eqn (128) can be wntten m the bilinear form

(D,>+ZD,, -D;)z,+, 7 , =O. (131)

Next, we prove that the solutions of the DAKP equauon can be repmated In the form of Wronsluan (Casorah) determinant

The detemmant 5, in (132) 1s nothmg but Ihz denonmator in the expmslon (38) gwen m Scc-

tion 2. The entnes in the determinant (132) f;) = fO'(tl,t,,n). = 1,2,.. .N are the wlu-

hons of a set of lmeal- pwutral dlftercnual4fference equauons

One of tbepamcular solutions of (133) 1s read~ly given by

f ~ ) = ( ~ + ~ , ) ~ e x ~ ( ~ , r , + ~ ~ i , ) j=1,2 ,.., M (134)

To obtm N-sohton solutlon in the Wronsiaan fonn, it is well known that f,!' can be chosen in the form

$if) = cxp ql +expS, (135)

mlth 87, arid c,giveii hy

q, =p,i, t p,!i, +n!c- 4 I i y 1 1 + T ; ~

5, = q,tl +qji, +niog(~+y,j+:,:, (136)

Fokwlng Frccman and N ~ m o ' s n o t a h ~ n , ' ~ . ~ ~ we denote z, m (132) as

K M T A m I M A M AND S KANGA VEL

wherem the hnear equabons (133) are used in the derivat~on of the above results. Using Ibe above expressions (138) in (1281, w,e get

a2r, - - T - - ~ O Z N - 2 , . N - 2 (139) &I a;

-210,1.2,.. , ~ - 2 , ~ ~ l l , 2 , . ,N-l,N+lI

+210,1,1,. ,N-2,N+l/(l.2. .Ni

wh:ch IS he Ldplace expans:on of thc 2NX 2Ardetennani?

DTJFERENT14LDIFFFENCE KtWOMTSEV-I'FJnnIASHV(I.I EQUATION 341

where Nn-2 = 11, 2. , N-21 and 0 denotes the ( A - 2) x (N- 2) zero maim Sincc the above deiernunmt (140) is lero ~t ~ndeed venfies that 5, aahsfies the bihnear equation (125) rdenti- cally Thus we have proved that the z-function defined by (132) gives the ,A-soliton soluhon of the D A I 8 equatron (1 26)

In Uui secbon. our a m 1s to descnbs [he method at findlng a class of iauonal soluirons Tor thc DIKP equailon Foi !hs purpose, we consider the aet of h e a r pvtial dtfferenual-dfiiierence

equahons (133) wth (134) as pdrtlcular solullon Nome that f,Pi m (134) can be expressed as a formal power senes mp, and hence we havc

From (141). we have a set 01 polynonl~irls m the variables n, f I aid tz They can he expre~qed In a cornpact way m

where P, = 0, Vrn 5 0 These P,s are called the different~alilllierencc analogues ot Schw polynormals Also, one can see that they sahsfy the following equauons.

AP,"=P,-,

aprn - -- urn (143)

From the above equations (143). we see that the IJ,s are solutrons of the equahons in (133). But we have already shown that the Wronskian fonned by any solution of (1331, satisfies ihe bilinear fonn of D A D equahon (128) Thus P,s are also solutrons of bilinear DAKP equation

342 K M TAMlZHMANI AND S KANAOA VEI

(128) Therefore, the polynomials P, can be used to generate a class of rahonal solut~ons for (126). Consider the Wtonskian formed by the P,s

where I , , g, ., 1 , ~ are d~smct mtegers We ltst below first few ratlonal soluhons generated usmg (144):

P,=1

P 1 = n + i l

p2 =no+':+ntl+t2 21 2 '

Next, we construct a more general form of rational solutions For this purpose, we consider h e z-function gven by

cn = w(f,(L), f,(2',. , fp) (14.5)

where the f))s are gven by

md they aahsfy eqns (133) with

17(~,)=(n+n,) log( l+~)+p,( t~+i ; , ) + p 2 ~ ( 2 t +& ,) (147)

where n,, K, and G, are arbitrary phase constants From (146), we have

where

These polynomials Pm, (pi) are the d~fferent~al-difference analogues of the generalxed Schur

polynormals Agm, 11 should be noted that the Wronskm formed by these generalmd Schur polynomals are also rauonal solubons for the DAKP equauon (126). But thls Ume the entries m the determinant are a151tn.q h e a r combmatlons of the generdmd Schur polynomials (148) It 1s easy to dcnve the N-sohton soluuons and the ratlonal solubons of DAKP herarchy.

If we introduce the mfinrte number of time vanables In the funchons j,(" m such a way that

they sansfy the Imear equations

then the Wronshan formed by these functions 1s the N-sollton soluuon of the DAKP berarchy. The rauonal solutions of the DAKP h~erarchy can be obtamed as before.

4. Lie point symmetries and Painlevd-singularity continement analysis

In tlus secuon, we discuss the underlymg L I ~ pomt symmemes ofthe DAKP and also study the s111gnlmty saucmres of the soluuons of thls equauon. These two aspects played important role In mtegrable systems for many years. The f i t one helps us to find speaal class of solutlons m terms of new vanables called simlarity variables using these vanables we can also reduce the equauon to a lower hens lonal system. Furthemore, the shucture of the symmetries reveals the nature of the associated L I ~ algebra of symmey vector fields. The class&cahon of Lie algebras of symmetry vector fields in turn brings out the assaclated solunons As far as the sec- O I I ~ pm, il i, ucll r c l , g m d that the Pstn1et;-s~ngulmtv annlpsls played a r!ral role for ssv- srd s2an In ~ d a m h m e m,,~hle ~nlcuxahle ,!,terns uorli In OLWs and PDEs Th3 1s more dl- . -. - , rect and smple and yet a powerful approach to rdentify mtegrable systems though the nature of the singularihes appears m the solutlons. In the followmg discussion we give brief mtroducnon to both these methods and apply the techniques to DAKP. Demled analysis wdl be published el~ewhere.'~'

A Ltepoint symmetries

The wncept of symmey 1s extremely general, and the precise meaning of the term depends to a larger extent on the context we deal with When we use tb~s concept in connection wlth dif-

344 K M TAM1ZKMAM AND S KANAGA VEL

firenrial equaoons, we resene the word symmetry for group of tnnsfonni~t~ons which leave the sven system of differentd equanons Invanant For computing the symnletr~es we adopt mfinites~md analogues of the transfornarlon However, :here arc mcthuds to recover the full goup from ~nfinrtesimal symmetncs in pari~cuk, the mportance of Lic's invanance analysis hes m the fact that it is a systematic approach to d~scover a clasi of solut~ons. reductions to simpler equaoons through a new set of variables called similimi) tanahlea and smi~larity func. h o n ~ ' ~ . ' ~ Numerous equat~ons were analysed uslng thls powerfui fool Over the yeao, the method of Lie has been generalized m many direchons Though there was lntcnse activq on the symmetry malyas for conimuous systems, 11 is surpnsmg that untli recently this theory had no Impact on d~fferent~al-hfference systems and discrete equallons as well However, ~t 1s worth ment~on~ng that Maeda was the first one to apply the theory of symmetries to discrete systems m the vanauonal formalism '"m Due to rcsurgencc of lntcrcat In thc integrabil~ty of hscrete and d~fferentlal-dtfference systems, the symmetry approach agam becomes v~tal 10

look for symmemes, spend soluhons and reduchons In this bakgrnund, Lev1 aiid Winlemm, and h e r Quispel, Capel and Sahadevan deveioped rhe Lie symmelry method k r differentla!- d~fference equations 1*1~150 The Lie pomt symmctrjes for the fully diac~ele equat~ons were also ~mtiated Symmetry analysis for fully discrete systems is yet ru bc developed as ;an efficient tool as m the contmuous case In 1ww of the Importance of L e iheory ~tseli and the nontriv~al appl~cabllity. we denve the Lie polnl symmetnes of the DAKP equntlon In t h ~ s sect~on, md use them for reduction pmcess

Let us consldera fincuon u!x, n), u E R, x E P, n E Z We consider the diffcrent~al-differe~~ce equation of the form

F(x,n,u,,u, ,, u,~+l, ii,+i, ,. )=O (151)

where 1 E Z We say that the Lie point symmetry goup of transformalionl"

n l = n , i = h , ( x ) , i n = m g ( x , n . u n ) (152)

where g denotes the group parameters, and m, are mvertible mooth tunctions, 1s admitled by the system (151) if u,(x) is a soluhon of (151), then i,, (2) is also a solutlon of (151) The

power behlnd the Lie group of transfomatlon technque hes in the infinites~mal fornulauon of the group L~e's first fundamental theorem exphcitly gives the connect~on between the ~nfinl- tes~mal uansformauon and the Lie group of transfornat~on.~~ he mfimtes~mai one-parametlr Liepomt transfornation corresponding to (152) is given b)

a d the vector field comspondmg to the ?nf~nitesimal bansformdtion (153) 8s gvcir by

DIPPRRENTL4LDtWFRENCE KADOMTSEV-PEWlAS~~IL.1 EQUATION 345

The vector field (154) should be expanded to a larger space based on the order of the given equatlon (151) For example, d (151) is of order k then (154) should be extended (or pro-

longed) to p&') f defined by

where

denotes the total denvatwe operator as m fie conhnuous case." Now, the mvanance condtlon is given by

%.q,=, =O. (158)

The man difference between the continuous and duferenual-dfierence case IS the summa- bon over I in (155) The number of terms we have to keep depends on the thscrete order of the equahon. As In the contmuous case, eqn (151) is mvanant under the actlon of (152) I[ the cond~tron (158) holds good. Quation (158) gives the inyanant conhtlon from which we have to iind the ufinitesimal generators G+ the symmeig group (153) To do this. we expand eqn (1581, use eqn (151) and equate the coeffic~ents of the various denvahves of %+I to zero. Tius resuits m ao over-deterrmned system of linear equations for the mntimtes~rnal generators of the group. We can solve these detemmng equatrons in a closed form and obtain symmetries. These symmetries are then used to find s~mllanty solutions and reductions, etc In order to denve the smlarity soluhons of the system (151) we use the symmetries 5,s and @ In the cbarac- teristlc equation

After solving the above equauou we amve a t p - 1 new mdependent vmiaS1er called a~wlarity vwables. The new dependent vanable 1s the funct~on of similarity variables, called !he slmi-

346 K M TAMZfMANl AND S KANAGA VEl

lmty funcuon Subshtuhng the value of u, in terms of s l d a n r ) ~ fnnctIon In the system (151), and slmphfymg we amve at a new system which has the number of Independent vanables re. duced by one compzed to the gwen system. Obtairung lower dlmens~onal equations uslng this procedure IS called smlanty reduchon. We can also use Inflniteslmal generaton to classlfy the solutions It can be shown that vector fields assoc~ated with infmteslmal generators form Lie algebras

4.3 Symmemes andsnulanty reduchon of the DdKP eqUUtI0n

In ths secuon, we present the classical Lie polnt symmemes for the DAKP equation. Usmg these symmetries we find smlanty solnkon and sunilarity reduchon of the DAKP equation. As a sirmlanty reduchon we obtain Veselov-Shabat eq~ahon."~

For fms purpose, we start with DAKP equakon in the form

II, tun-2(1-ii)iix+2(1-u)u, -ii,+u: = O (160)

where we have used u, = u and ii,,, = ii Let us assume that the mfinlteslmal Lie group of transformauon as -

n = n

ii= u t E ~ ( n , x , t, u).

The vector field corresponding to (161) is given by

Smce the order of DAKP eqnauon 1s two we consrder the second prolongahon of the vector field (162), whch u given by

PPJ t = g a , + . r a , t ~ + p , + r x a , + p a , +$'a5 +pas , + p a , ,

+ p a , +$*a, + p a , + v a x v +pa,,. (163)

We get the invanant

on using the lovariant condhn (158) and applying (163) on eqn (160) To evaluate thls ex- Presslon we need o",?.$',S',$~,~" and we can explic~tly fmd ihem ustng (156). are hsted as below

DIFFEREWXLDEkTIWKE KADOM CSEV-PETYIASHVIIJ EQUATION 147

Now solve eqn (160) for < and hence we have

Io order to get the detemllung equations for the lnfiniteslmal generators we sr~bstltute the values of 1165) III (161) and usmg (166j. rcpiaclng ii, In the resultmg exprewon, we have an

expressron m ic,,u, ,<. $ .uU , uXr , i i .v,. Equatmg the coefficient of vanous powers of the de-

nvatlves of u and ii m rhe resnltrng expressmn to zero !be m v e at a I m m homogeneom system of parildi dlfferenual-d~fference equations. Solvlng tius overdetemned system we obta~a Ihe symetries

In order to perfom slmtuld$ reduction first we haw lo solve the charactemtlc equation

and denve tbe s d a n t y vanable and s ~ m l m t y function. On mtegmtrng (168), we relate u to the sim~lanty funct~on F(c, n) b o u g h

where the s~mlarity vanable CIS gven by

348 K hl TAMiiiMANI AND S KANACA VFI

Substihlung the value of a m DAKP equatloil we get the reduced equation

which is the VeselowShabat equanon 17' The above system can be denved using I-reduchon iechnque in Sato theory, and moreover this quahon can be b den ti tied wnh delay Parnlevt equat~ons "'. ' '' B PainlevCsinguianty coii(inen1ent nwaiysir

Even hef<ore the discovery of sohtons, we hd a remarkable theor). to tekt thc inregrabihty of ODES called singulmty-ar,hlysis first proposed by Roivalevsh "' The motlvatlon for hci &scovmy emerged from be fact that the cnfical sin@ulantles of a h e a t ODE are fixed?,'"" Tnis means that the Incation of smgulantiec of rhe soluhons of a h e a r ODE 1s determrned en- tirely by the coeftic~ents of the ODE Thn is cerla~nly not (lie cacc in nonlinear systems. The sfructure of the smgulanly In nonlinear equat~ons IS more coniplicatzd Wh!le the siogulanties are fixed for l~near ODEs, in the case of nonilnear differentla1 equouons, their locatlon (in the complex plane) depends on the mlhal condiuons These slngulantiea are called movable Pain- lev6 stated askrng for nonlinear ODES with tixed cnhcal sn~gnlwtles and attempted to class~fp all the second-order equattniis that belong to thrs class In put~culai, he exmlned rquat!ons oi the Conn

r"=f(w' , w , z) (172)

with f polynomial m w', ratlonal in w and analyt~c in 7 Tkrs class1ficstm was cornple~ed by Gambler iZ."6 Thus came the discovery of the famous six Palnlevd equat~ona."

Tbe Panlev6 equahons z e mtegrable tn pdnciplc. bowever, heir mtegratlon could not be performed wlth the methods available at that tlmc. Tha situation haa changed after the discov- eq of IST Ablowio and ~ e g u r ' ~ ' showed that the IST techn~que could be used to linearize the Panlev6 equations Soon afler, Ablowa, Ratnani and S e y r (ARS)" propowd the f o l l o ~ v ~ n ~ conjecture ' Z v q ODE whch anses as a reduchon of a coniple~cly ~ntegrable PDE Ir of Panlev6 type @erhaps after a bansformahon of varidbles)" The ~ntegrahle syslerrs also pos- ress what 1s called the Pamlevd propelty. If dl movable smgulanues of all solutions of an ODE are poles then we say that the system possesses Pa~nkvC property ARS also prov~ded an al- :or~thrn to tea this property for ODEs. The N ( S approach Umed out m be the most powerful ool to isolate good canhdales of rntegmble systems ' Improvements made to it by Welss ef 11 and Glbbon and ~ a b o r " ~ to treat PDEs d ~ e ~ t l y without the constraint of constdenng re- iuctions have resulted m soera1 new quauons The Panlev6 test IS undouhtedly powerful but t does not have the rigow of a theorern.

In recent years there has been a growing interest m the study of discrete equat~oni In mod- m sclencc discrete equakons play an importaot role "', '" With Lhe advancement of hgh- . peed computmg, dwehsation hecomes unavadable. Qurte often, dlscrete models are more Aishc than coutmuou ones to understand the physics of the problem bener Howeuer, are an clearly see a close parallel behaviour between the p~opcrt~es of the conhnuous systems and

DlFPERENTIALrDIFFEWNCE L4DOh4TSEV-PETVIASHC?LI EQUATION 349

their discrete analogues 16' At the same time, 1t 1s not obvlous to find the dlscrete analogues of all integrable equations In the past the focus was not much in this doman, but, has changed very recently due to the appearance of dlscrete Pmlev6 equahons '5"157

Although thcre was some progress in discrete systems, no singulanry-structure analysis ( P d e v C method) existed for such systems untll the discovery of smgulanty confinement (the equivalent of Panlev6 analysis for discrete systems) by Grammat~cos et a1 As in the Pan- lev6 method for contmuous systems. smgulanty confinement method becomes a powerful tool to detect posslble discrete integrable systems The alngulanty confinement was complemented by pre-lmage nonproliferation condibons whch means that at each point the mappmg wdl have a smgle pre-image In the case of mapping, if no unlque pre-mage exists then there is no need to use slngulanty confinements ' lo The most strlk~ng use of smgulanty confinement is the &s- covery of discrete Panlev6 equations T i also plays a v i a role in getbng other mtegrability properties of dlscrete systems 158wm

4 4 Algorithm

The pnnclple of singulanty confinement can be stated as follows In a rat~onal mapping, smgn- lanty may appear spontaneously due to a parl~cular choice of imt~al condihon In analogy with the continuous case %e call thls slngulanly 'movable' The conjecture state^"^ that in int.. grable systems th~s smngularity musl dlsappear after a few lterdhons. Ths 1s what 1s meant by 'confmement' Also, memory of ml(lal condrtlons musl be recovered beyond singulanty We can present the method of implementmg singulanty confinement m the folluwlng way as Pan- lev6 analysls m the ARS method" (detalls m Appendix 1).

1) Find all posslble slngulanties and check that they are movable. ARS . Fmd all possible leadmg behaviours

2 Detemne when, at the earhest, the singulanhes can dlsappear. ARS Flnd the resonances

3 Check that fine cancellauons ensure lhat they actually disappear (gives constramis on the parameters) ARS Check compaublllty condiuons at resonances

For the purpose of dealing with d~fferential-drfference equatlons, neither Panlev6 method nor slngulanty confinemen1 1s enough to capture smgulames. But Ramam et a1 I)', 13' have shown how a nlce combinat~on af these luro methods will allow us to treat differential- difference equahons In fact, rt goes beyond m treatlng intep-dlfferenual equatlons as well. l%e bas~c idea 1s to consider the effect of a smgulanty 1n the conunuous vanable on the dis- citte evoluhon For Panlev6 property the smgulanty must be a pole, as well as the subsequent ones and In addition. l h ~ s must disappear afte~ a few Iterations (in the discrete variable). Thls idea 1s very fruitful in dealmg delay-&fferentlal equatlons As an appllcauon of this method a few delay-PainlevC equahons have also been obmned 17'

4 5 Pamlev4-smgulanty confinement analym for D M P

We illustrate the Painlev6 slngulanly confinement techmque on DAKP and study the smgular- it). structure of the solutions We mtroduce the following notations m our discussion t l= x.

350 K M T L . 1 AND S KANAGA VBL

DAKP equahon as

u,+2(l-n)u,-u, =g, t2(1-u)ux+!!,. (173)

Accordmg to singulantyconfmement analysis, we assume that a given r 1s regular and uslng the above equation. we should study the propagauon of s~npulantles that appear for u The l e h g behav~our around the f m stngulanty manlfold Kx, t) = 0 1s

To simphfy the calculaaons, we apply Kruskal's ansatz, i e we put Kx, t ) = x t dt) w~thout loss of generality In this situahon, # has a Taylor expansion and thus u be expressed In the Lament senes

where Q = 1. Umg the expanslous (174) and (175) III (173) and performing the usual Painlev6 analysis we find that ARS-resonances are j=-I, 2 and the compaubility condition for j= 2 a automattcally sausfied

Thr; is not enough to test lntegrablhty thmugh smgulanly confinement. For t h ~ s purpose we have to consider the fust and second iterations of the recursion (173) and pcriorm the usual Painlevd aoalys~s and check Ihe passing of the test The lteratlons of eqn (173) are

and

We apply the name of the slngulanty of the solution from the prevlous analysis to these above upsMed equahons while dong Pder.6 analys~s and notlce that DAW equation satisfies the slngular~ry continement cntenon and Painlev6 property, thus confmng the integrability of th~s equahon from slngulanty analysis point of vew Detals wlll be published elsewhere "' 5. A gauge equivalence of differential-difference Kadomtrev-Petviaqhvili equation

5.1 Introduction

In the previous sectlons we have studied the DAKP equation in wew of Sato theory We d e lived Lax pair, consewed quauhtles, generalized symmekes, Wronskm solutlons, rational soluhons and Lle pomt symmekes and tested the Pmlev6-smgulanty confinement pmperty,151, 152,1Sd, 15s In this sechon, we d~scuss a gauge eqwvalence of the DAKP equat~on.""

One of the frenuent queshons asked m the theory of mtegrable systems is that of the rela- nonslup among vanous e~genvalue problems and of the assoc~ated systems. This qUeShOn has

slgwficanl implications, and to lnveshgdte it, gauge transforn&on has been inlioduced which connects one egenvalue problem to the other and subsequently one integrable equatlon to the othe, Such equivalence of integrable equauons ha5 been a subject of intensive researcl 17"'92

For example, a gauge equlvalence of the nonilnear Schro;dmger equahon and Heisenberg fer- romagnet equatlon was established by ~akshnianan,~~" and latcr Zabharov and ~ a k h t a j a n l ~ ~ showed a gauge equlvalence between the elgenvalue problems. It has been apphed by ~ u n d u ' " ~ " ~ to many ayqlems in both 1 + I and ?+ 1 dimensions It IS well known that three d~ffermt agenvalue problems, that is, Ablowltz-KaupXewelI-Segur (AKNS), Kaup-Newell (KN) and Wadatl-Konno-Ichlliana OVKI) are connected tluough gauge transformation R' In vlew of Salo theory, K~so denved the modified h~erarchics usmp gauge transfornation.'" There is a close connection among KP, modrficd KP and Harry-Dyn~ heraches which has heen eectablished through gauge transformahon An umfied approach to gauge Wansformation and rcapmcal links for a broad class of nonlmear evoluhon equations has also been mvestl-

I34 .'Sj

Mot~vated by these works we discuss a gauge equivalence of. the l~nca- elgenvalne problem of DAIW and dcr~re a d~lferennal-dtfkere~rcc cquauon rclated to D m through a gauge tans- f~nnatron."~ WE imd h e conserwJ quantlues and generalned spmnernes for t!xr system.

1 2 A g u q e equn~aience of'DAKP equation

We abrt unlh the pscado-rl~llrencc opcrator

!? = w; t w;K1 + w ; h 2 i

wheie the y s are functions o in , t,, t l , The formal Inverse of G la gmen by

v;i-' = 1,; + ";A-' t v p 2 + (17%

Usmg = W = I, we get

Rcd~ange the terns on the nght-hand side of the above expression (180) and compare the hke powers of A on both sides of (180) Thls results in an milnite number of equanons for v:s in

tennb of wjs, ?,,I = 1. 2,

Now we introduce a gauge transformabon for the L operdtor (48) defined m Section 2 by

P = )-'LO (182)

with

!t a possible to decompose in (183) as

Ixpandmg the nght-lland slde 01 equdllon (185), md comparing rt wlth eqn (183). we amve hat the e:s can be expressed in terms ot w;s for all I , J = 0, 1 , 2, We list the fmt few oi hem.

t the continuous case,'" the Be are definedby

BE = (L~)+

here O+ denotes the strictly posihve powers of a for the modified KP lue~mchy. Here also we pect the same and therefore define

DEFEREN11ALDJWERENCE KADOMTSBV-PEJ VIASHVLI EQUATION 353

(188)

where ()i involves s t n d y posluve powers o i h. and the generalued Lax equation is given by

k - = [ $ , L ] , k = 1 , 2 ,.... (189)

From (188) and (183), we get'

6 = urA

E2 = U'EU'AP -I-(U'EIL'-~~~ + ~ E U ; , +uZu;,

Umg the Lax equatlon (189) for k = 1 we have the followmg set of equahons

K= *%,; a, -- auk - U,E~A;I -u,u; + U'EU; -U;E-IU,

a,

and fork = 2 we have,

Solving the above aet of equations, we e v e at

Next we denve the conserved quanbties and generalized symmetries for this systcm

5.3. Conserved quantities

In Section 2, we have derived the conserved quanuues a n d generali~ed symmemes for D m equatlon (66) For this purpose we follow the procedure described m Maisukidiara et a1 'I7 Here, we &opt the same techmque and present the conserved quantiues and generalned symmetries of the equation (193) For thrs purpose, we fust consider the h e a r eigenvalue problem associated w~th the generalzed Lax equation (189)

K M ~ A M ~ L U M A N I H N D S KANAGA VPI

whue 4, = 0 We assume that = Ek - zk, and hence rewnte eqn (191) ,IS

It n noticed that i; coos~sts ol. terms mr-olvmg K', j=O, 1, 2. . Now we will express A:',

1 = 1,2, m terms of c-' For th~n purpose, first we frnd L? We assume that L-' rs of the form

Comparing the like pourers of A on both saldes of (197) wc get

Uang Leibnu mle (9) and (1961, we can denve the h~gher powers of c-J, = 1, 2, . We llsl some of tlm.

L-2 = q ; ~ - L q ; ~ - 2 t (-q;~-'q; + q ; ~ - z y ; + q ; ~ - l q ; +q;~-zq; )~-3+-

1-3 = q;~-'q;~-1q;A-3+... (199)

DIFFERENTIAL-DIFPERBNCE KADOMTSEV-PETVIAS~I EQUATION 355

using (198) and (199) we can present the values for A-), j = 1,2, . m iems of negative powers of L as

Now, using these lesults, we can wnte down eqn 1195) in the form

(201)

where ay's are functions of u', ois for all i. j= 0, 1. 2, and k = 1,2, On uslng L'y= h'y

in (201), we obtan

We denote o @ ) = x,"=,~?)>̂ ;J and hence eqn (202) becomes

Dlffemtiating eqn (203) wrth respect to the vanable t,, we wlll anive at the consewatlon laws

356 K M TAMlZfMANI AKD S KANAGA VEL

We the list first few of the u y ' s

From the above equdoons (207), we can exprcss u;,u;. . . , in term oi u' and we llsr the lint

few of u;s forj=0, 1, 2,

Now substituting the values of uA,u;, . In uj" (205), we ohtam the conserved densities of

the dfferenbal41iference equation (193) Wc 11st below some of them

In tha sechon, we denve the gemahzed symmetries of the eferential-d~fference equatron (193)."', li5 For thn purpose, we consider the linear agenvalue problem

and the adjolnt eigenwlue problem

Tr: =av:

where A 1s the spectral parameter and IS independent of n and f,". We follow the same procedure as In Sectlon 2 to compute the agenfunctions W. and y: They are given by

358

with

K M T M M M A N I AND S KMAGAVEL

For convenience, we denote u' as u, The equatms representing the hear elgenvalue problcm and 11s adjomt are gwen by

and

s = ~ h (216)

one can check that

(217)

1s connstent with (2141. The solution of eqn (217) 1s denved from

DWERENTAL-DIFFERENCE KADOMTSEV-PENIASHW EQUATION 359

We denote S = $ and &, = u Then, eqn (217) becomes

whch is nothug but the symmetry invariant equahon of (193). The soluhonS of eqn (220) are the generalized symmehies of eqn (1931. We first list a few generalized s y m m e ~ e s of (193):

5.5. 2-Reduc~on

In this section, we denve the 2-reduced gauge equivalent DAKP equation. For this, we consider

C =z2 (222) which will give the constraints

u'Eu; -u'u; +uPEu; +uh2 +u;E-~u' = 0

a'&; -uru; + u'Eu; + u;u; - u ; ~ " ~ ' + u ; ~ - 2 ~ ' + ~ ; ~ - 1 ~ ; +u;E-'u' = 0 (223)

Imposing these on the constaints in eqns (191) and (192) we fmay &ve at the reduced system

3MI K M TAMlWMANl AND S KANAGA VEL

Here u: =u' and eqn (224) is related to the Kac-van Mmbeke system4' by the thetransforma-

kon u: =log(%} The Pdey6-smgularity confinement andysis and Lie symmetry analysis

of (193) will be publ~sbed e lse~here . '~

Afknowledgements

In this paper we review some of the results obtmed recently with our collaborators Grammab- cos, Ramani and Tharmzhmi Tamizhmani.

We express our sueere thanks to B.Grammatms, A ram an^, Thamtzharasl Tarmzhmanr, Y. Oh@, J. Satsum* 1. Hietannlg 1 I C Nxmmo and W. Oevel for fnutful Qscusslons and encouragement at vanous stages KMT s supported by the Indo-French Project 1201-1 SK &s to thank the Cound for Sc~entlfic and Indushial Research (CSIR), Ind~a, for financial support

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DIFFERENTIAL-DIFFERENCE KADOmSEV-PETVIASHVILI EQUATION 363

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DIbTERENTIAL--DIFFERENCE KADOMTSEV-PETVIASHW EQUATION 369

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4ppendix I Painled adyais for PDEs

ARS" proposed an algorithm to maiyse the P a n l e d propefirty of ODES Tbls has been ax- tendcd hy WTC" The Yanlevt analysis for PDEs due to WTC can be slated as fullows Let us conader the ebolunon equatrrm of the form

where K h ) e some nonlroear functlon of u and rts denvatlves of order "i m the complex do- mam Wc say chat an NPDE posccases he genaralned PanlevC propcrtyY'." of the following two condiuons are satosfred

(a) The soluuons of the NPDE(A 1) must he 'slngle-valued' about the 'non-charactcnsnc' movable angnlarlty 1ixifol6. More precselg, rf the sungular~cy manifold 1s detenn~ned by

where b= Ks, t), u,= lr,(r, t ) , uo t 0 are analytrc funcuons of (x, 1 ) rn a ne~ghbourhood of the mamfdd and a ~ s a negatlw integer

(b) Poen by Cauchy-Kwdle~skaya a theorem, the solution (A 31 5hoald contm N m l W q imcuoni, one of them beurg h e functloo 4 and othm comlng lrom ihe u,s The algo-

DIFFERENTlhlrDIPFEXENCE KADOMTSEV-PETVIASHVLI EQUATION 371

nthmlc procedure to test the glven nonlinear evolut~on equauon for its genedzed Pm- lev6 property consists essentially of three steps. We shall descnbe each of these steps below

Leadmg order unalysls

The analysis starts wlth the detemi~nat~on of the possrblc values of a and uo In the expansion

(A 3) For each values of a. lhe homogeneous terms with the highest degree may balance each other The terms that balance each other are called leadmg terms Then aU the as must be negahve intcgers by (a)

For each cho~ce of the Ct, an algebriuc equahon lor the uo III (A 3) is usually obtained by re- quuing that (he cocffic~ent, say A of the dom~nanl term ~6~ should vmsh. v h r e d 1s the lughest degree It 1s arb~trary, A should ldent~cally vantsh.

After ~den!lfying all the poss~ble branches in the solul~on (A 31, our next aim is to find the resonances When the cocftic~ent u, of the temi @'" m the express~on (A 3) 1s arbdnry, then we say that the resonance occurs at j ln the above senes. In order to find the resonance values, we subshtute

~=u0$"+u,@"" (A 4)

m eqn (A I), retaning only the most dominant terms, and exrracting the coefficient

& I ) = Q(,)u, of the [ e m Jic-" Then Q0) =0 1s called the resonance equauon, in urbch -1

IS always a root, which corresponda to the srbmaq nature OK $. Subsumtmg the values of u, (oblained earl~cr m Ihe leadmg order analys~s) m the resonance equation, one can find the re- niarning roots of Q(f)

Hanng obtatned the resonance values, we have to show that necessary arb~trary functions enst at these resonance values ID the senes without the introduchon of any movable cnucal man- fold. Let r, be the highest of the allowed resonance values Then we subsUNte

In the ong~nal equation (A I ) and for] = 0, 1. 2, , r, requlres

QOu, + R, = 0, (A 6)

where the lefl-hand side oieqn (A 6) IS the cwlfiment of fliCtYand R, is a polynomal ln the partla1 derivatives of $and y s (k = 0, 1, , j - I). Srnce Q(]] = 0. R, slrould rdenlically vamsb for any resonance, and tn whrch case u, is arbrkary Supposc if 11 ,a not so, we have to inbo- duce loganthm~c terms of the form a, + b, log $ rn the senes But due to this add~tion, the l~gaithrmc smngulanues will appear in the solution manrfold. Thus, the cond~tton R, = 0 en- s m s that the soluuon IS f~-ee from movable critical ma~~~lblds .

312 K M TAMZHMANI AND S KANAOA VEL

Note: We have noticed rather late In production stage that eqns (4) and (5) do not figure ~n the paper of & M. Tamizhmani and S Kanaga VeI. 1998, 78, 311-372 This o~nission bas proba- bly accurred during revision

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Differential-difference Kadomtsev-Petviashvili equation