On a (2+1)-dimensional generalization of the Ablowitz-Ladik lattice and a discrete Davey-Stewartson system

On a (2 + 1)-dimensional gen eralization of the Ablo witz–Ladik lattice and a discrete Da v ey–Stew artson system T ak a yuk i Tsuc hid a ∗ Okayama Institute for Quantum Physics, Kyoyama 1-9-1, Okayama 700-0015, Jap an Aristo phanes Dimakis † Dep artment of Financial and Managem ent Engi n e ering University of the A e ge an, 41 Kountourioti Str., GR-82100 Chios, Gr e e c e Marc h 2 , 202 2 Abstract W e prop ose a natural (2 + 1)-dimensional generalization of the Ablo witz–Ladik lattice that is an integrable space discretization of the cub ic nonlin ear Schr¨ odinger (NLS) system in 1 + 1 dimensions. By further requiring rotational symmetry of ord er 2 in the t wo -dimen s ional lattice , w e iden tify an app ropriate c hange of d ep enden t v ariables, whic h translates the (2 + 1)-dimensional Ab lo witz–Ladik lat tice in to a suitable space discretiz ation of the Da ve y–Stewartson system. T he space-discrete Da ve y–Stewartson sy s tem has a Lax pair and allo ws the complex conjugation reduction b et w een t wo dep endent v ariables as in the con tin u ou s case. Moreo ver, it is ideally symmetric with resp ect to space reflections. Using the Hirota b ilinear metho d, w e co n struct some exact solutions su c h as multidromion solutions. P A CS: 02.30 .Ik, 05.45.Yv MSC: 37K1 0 , 3 7K15, 3 7K60 ∗ E-mail: tsuchida at poisson.ms.u- to kyo.ac.jp † E-mail: dimakis at aegea n.gr 1 Con tents 1 In tro duction 3 2 Deriv ation based on Lax pairs 5 2.1 (2 + 1)- dimensional Ablo witz–Ladik lattice . . . . . . . . . . . 5 2.2 Con tin uum limit . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.3 Noncomm utativ e extension . . . . . . . . . . . . . . . . . . . . 8 2.4 Appropriate c hang e of dep enden t v a r iables . . . . . . . . . . . 9 2.5 Space-discrete Da v ey–Stew artson system . . . . . . . . . . . . 11 3 Solutions b y the Hirota method 15 3.1 Decomp osition in to four commutativ e flo ws . . . . . . . . . . . 15 3.2 Bilinearization . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.3 General solution fo rm ulas . . . . . . . . . . . . . . . . . . . . 18 3.4 Solitons and dro mio ns . . . . . . . . . . . . . . . . . . . . . . 20 4 Concluding remarks 26 References 27 2 1 In tro duction More than 40 y ears hav e passed since the Kortew eg–de V ries (KdV) equation w as solv ed by Gardner et al. [1] using the in ve rse scattering metho d based o n its Lax pair [2]. The n umber of kno wn in tegrable systems follow ing the KdV equation, pa r ticularly partial differen tial equations (PDEs) in 1 + 1 space- time dime nsions, has increased enormously , and v ar ious tec hniques to study them ha v e b een dev elop ed. Recen tly the cente r of researc hers’ in terest has shifted from con tinuous PDEs to differen tial-difference or partial differenc e equations wherein at least one of the indep enden t v ariables tak es discrete v alues. A ma jo r problem in this t r end is ho w to find a suitable difference analogue of a giv en differen tial equation. The suitable discretization of an in tegrable contin uous system is generally required t o retain the integrabil- it y [3], but that is not sufficien t if the original contin uous system has some essen tial inte rnal symme tries. This b ecomes conspicuous if w e consider in- tegrable discretizations of nonlinear Sc hr¨ odinger (NLS)- t yp e system s, i.e., t w o- comp onen t s ystems of second order tha t a llo w the complex conjug a tion reduction b et wee n the t w o dep enden t v ariables. As a pro tot ypical example, w e discuss the cubic NLS system in 1 + 1 dimensions [4 ], i q t + q xx − 2 q 2 r = 0 , (1.1a) i r t − r xx + 2 r 2 q = 0 . (1.1b) Note t ha t the r eduction r = σ q ∗ with a r eal constan t σ simplifies the t w o- comp onen t system (1.1) to t he scalar NLS equation [5, 6]. In addition, (1.1) is in v ariant unde r the space reflection x → − x a s w ell as the time reflection t → − t with q ↔ r . The suitable and elegan t space discretization of the NLS system (1.1) was prop o sed b y Ablo witz and Ladik [7 ] in the form i q n,t + ( q n +1 + q n − 1 − 2 q n ) − q n r n ( q n +1 + q n − 1 ) = 0 , (1.2a) i r n,t − ( r n +1 + r n − 1 − 2 r n ) + r n q n ( r n +1 + r n − 1 ) = 0 . (1.2b) Indeed, system (1.2) is integrable and, with a rescaling of v ariables, reduce s to (1.1 ) in the con tin uous space limit. Moreov er, (1.2 ) allo ws t he complex conjugation reduction b etw een q n and r n and p ossesses the same inv ariance prop erties with resp ect to the space/time reflection as the con tinuous system (1.1). Although 35 years hav e alr eady passed since their w ork, the Ablowitz – Ladik discretization (1.2) is still a rare example o f success. Indeed, eve n now, only a small num b er o f suitable space discretizations of integrable NLS-ty p e systems are kno wn (see, e.g. , [8 ]); they are all (1 + 1)-dimensional systems with only one discrete spatial v ariable. The problem of ho w to discretize the 3 con tin uous time v ariable in suc h systems is an in teresting topic [9], but w e do not discuss it in this pap er. The main ob jectiv e o f this pap er is to pro vide the first example of a suitable discretization o f an NLS-type sy stem in 2 + 1 dimensions. In par- ticular, w e consider the discretization of b oth spatial v aria bles in a (2 + 1)- dimensional NLS system know n as the Da v ey–Stew artson system [10] (also see [11]). Note that the Da v ey–Stew artson system is integrable [12 – 16] and app ears to b e the only gen uinely (2 + 1)-dimensional generalization of the NLS system (1.1) (cf. the Calogero–D egasp eris system [17]). Moreov er, ev en if w e include other t yp es of in tegra ble systems, the list of kno wn systems with t w o discrete and one con tin uous indep enden t v ariables is still v ery short. Th us, it is a highly nontrivial and ch allenging task to obtain the suitable space discretization of the Dav ey–Stew artson sys tem. T o solve this problem, w e first prop ose a nat ura l (2 + 1 ) - dimensional generalization of the Ablo witz– Ladik lattice (cf. (1.2)) b y constructing its Lax pair. This (2 + 1 )-dimensional Ablo witz–Ladik lattice certainly reduces to the Dav ey–Stew artson syste m in the con tin uous space limit. A relev an t Lax pair as w ell as t he resulting sy s- tem w as previously studied b y o ther authors [18] ( also see [19]), but the t ime part of our Lax pair is essen tially more general than the previously kno wn one [18]. As a result, the time ev olutio n of our system is a linear com- bination of four elemen tary time ev olutions, t w o of whic h w ere previously unkno wn. Mor eov er, it can b e show n that the four time evolutions are m u- tually comm uta t iv e. Thus, the (2 + 1)-dimensional Ablo witz–Ladik lattice is general enough a nd app ears to b e promising. Ho wev er, it do es not allo w the complex conjugatio n reduction directly and thus is not a suitable space discretization of the Dav ey–Stew artson sys tem in its presen t fo rm. T o fix this shortcoming, we only ha ve to consider a certain nonlo cal transformation of dep enden t v ariables, whic h symmetrize s the equations of motion. Th us, w e obtain the suitable space discretization o f the Dav ey–Stew artson sys tem that indeed allows the complex conjugation reduction betw een the new v ari- ables after the tr ansformation. In addition, the in v ariance prop erties of the con tin uous Da v ey–Stew artson system with respect to space/time reflections turn out to b e prop erly incorp orated in our space-discrete Dav ey–Stew artson system. This pa p er is organized as follo ws. In section 2, w e prop ose a (2 + 1)- dimensional ve rsion of the Ablo witz–Ladik lattice b y considering an appropri- ate generalization of the Lax pair for the original Ablowitz–Ladik lattice. T o unco v er ho w the complex conjugation r eduction can b e imposed as an NLS- t yp e system, w e consider a nonlocal c hange of dependen t v ariables; it can turn the (2 + 1)-dimensional Ablo witz–Ladik lattice in to the su ita ble space discretization of the Dav ey–Stew artson system that indeed allows the com- 4 plex conjug ation reduc tion b etw een the new dependen t v aria bles. In sec tion 3, we elucidate how the general time ev olution considered can b e decomp osed in to four elemen tary time ev olutio ns corresp onding to the four directions on the tw o- dimensional lattice. On the basis of this decomp osition and using the Hirota bilinear metho d [20], we construct some exact solutions of the (2 + 1)- dimensional Ablowitz –L adik lattice and the space-discrete D a v ey–Stew artson system. In particular, m ultidromion solutions ar e presen ted explicitly . The last sec tion, section 4, is dev oted to concluding remarks . 2 Deriv atio n based on Lax pairs 2.1 (2 + 1) -dimensional Ablo witz–Ladik latti ce As a generalization of the La x pair in tro duced by Ablo witz and Ladik [7], w e consider the following linear system on the t w o- dimensional lattice: ∆ + n ψ n,m := ψ n +1 ,m − ψ n,m = q n,m φ n,m , (2.1a) ∆ + m φ n,m := φ n,m +1 − φ n,m = r n,m ψ n,m , (2.1b) ∂ ψ n,m ∂ t = aψ n,m +1 + A n,m ψ n,m + C n,m ψ n,m − 1 + bq n − 1 ,m φ n,m − q n,m D n,m φ n − 1 ,m , (2.1c) ∂ φ n,m ∂ t = bφ n +1 ,m + B n,m φ n,m + D n,m φ n − 1 ,m + ar n,m − 1 ψ n,m − r n,m C n,m ψ n,m − 1 . (2.1d) Here, ∆ + n and ∆ + m denote the for ward difference op erat ors in eac h spatial direction, and the parameters a and b are arbitra r y constan ts. The time dep endence of the functions is usually suppresse d. The compatibilit y condi- tions ∂ t ∆ + n ψ n,m = ∆ + n ∂ t ψ n,m and ∂ t ∆ + m φ n,m = ∆ + m ∂ t φ n,m for the linear sys- 5 tem (2.1) pro vide the ( 2 + 1)-dimensional Ablow itz–Ladik lattice, ∂ q n,m ∂ t + q n +1 ,m D n +1 ,m + bq n − 1 ,m − aq n,m +1 − C n +1 ,m q n,m − 1 + q n,m B n,m − A n +1 ,m q n,m = 0 , (2.2a) ∂ r n,m ∂ t + r n,m +1 C n,m +1 + ar n,m − 1 − br n +1 ,m − D n,m +1 r n − 1 ,m + r n,m A n,m − B n,m +1 r n,m = 0 , (2.2b) A n +1 ,m − A n,m = − a ( q n,m +1 r n,m − q n,m r n,m − 1 ) , (2.2c) B n,m +1 − B n,m = − b ( r n +1 ,m q n,m − r n,m q n − 1 ,m ) , (2.2d) (1 − q n,m r n,m ) C n,m = C n +1 ,m (1 − q n,m − 1 r n,m − 1 ) , (2.2e) (1 − r n,m q n,m ) D n,m = D n,m +1 (1 − r n − 1 ,m q n − 1 ,m ) . (2.2f ) Indeed, if all the functions depend on n and m only through n + m , (2.2) reduces t o the original Ablo witz–Ladik lattice [7]; the latter con tains the in tegrable discrete NLS system (1 .2) as a sp ecial case. T o restore the (1 + 1)- dimensional Lax pair in volvin g the sp ectral parameter z , w e set ψ n,m = z m − n ψ ′ n,m and φ n,m = z m − n − 1 φ ′ n,m , rewrite the linear problem (2.1 ) in terms of ψ ′ n,m and φ ′ n,m , and then consider t he dime nsional reduc tio n. Under appropriate bo undar y conditions at spatial infinit y , w e can use (2.2c)–(2.2f) recursiv ely to express the auxiliary fields A n,m , B n,m , C n,m , and D n,m globally in terms o f q n,m and r n,m . Th us, they can be considered as the defining relations for the auxiliary fields, and the (2 + 1)-dimensional Ablo witz–Ladik lat t ice (2.2) has in trinsically nonlo cal nonlinearit y . Note that relations (2.2c)–(2.2f) already app eared in the literature on an in tegrable time discretization o f the (1 + 1) - dimensional Ablo witz–Ladik lattice [3, 21, 22]. Inciden t a lly , in the stationary case of ∂ t q n,m = ∂ t r n,m = 0, (2.2) reduc es to a non trivial system of partia l difference equations in 1 + 1 dimensions. Using the simple transformation q n,m = q ′ n,m e γ t , r n,m = r ′ n,m e − γ t , where γ is a cons tant, and omitting the prime, we can in tro duce the terms + γ q n,m and − γ r n,m in (2.2a) and (2.2b), res p ectiv ely . Clearly , thes e terms can be absorb ed b y constan t shifts of A n,m and B n,m . 2.2 Con tin uu m limit By c ho osing the parameters appropriately and taking the con tinuous space limit, w e can reduce system (2.2) to the con t in uous Dav ey–Stew artson sys- 6 tem. T o se e this, w e first shift the auxiliary fields as A n,m = b − a + b A n,m , B n,m = a − b + b B n,m , C n,m = a + b C n,m , D n,m = b + b D n,m , and rewrite ( 2 .2) as ∂ q n,m ∂ t + b ( q n +1 ,m + q n − 1 ,m − 2 q n,m ) − a ( q n,m +1 + q n,m − 1 − 2 q n,m ) + q n +1 ,m b D n +1 ,m − b C n +1 ,m q n,m − 1 + q n,m b B n,m − b A n +1 ,m q n,m = 0 , (2.3a) ∂ r n,m ∂ t + a ( r n,m +1 + r n,m − 1 − 2 r n,m ) − b ( r n +1 ,m + r n − 1 ,m − 2 r n,m ) + r n,m +1 b C n,m +1 − b D n,m +1 r n − 1 ,m + r n,m b A n,m − b B n,m +1 r n,m = 0 , (2.3b) b A n +1 ,m − b A n,m = − a ( q n,m +1 r n,m − q n,m r n,m − 1 ) , (2.3c) b B n,m +1 − b B n,m = − b ( r n +1 ,m q n,m − r n,m q n − 1 ,m ) , (2.3d) b C n +1 ,m − b C n,m = − a ( q n,m r n,m − q n,m − 1 r n,m − 1 ) + b C n +1 ,m q n,m − 1 r n,m − 1 − q n,m r n,m b C n,m , (2.3e) b D n,m +1 − b D n,m = − b ( r n,m q n,m − r n − 1 ,m q n − 1 ,m ) + b D n,m +1 r n − 1 ,m q n − 1 ,m − r n,m q n,m b D n,m . (2.3f ) Subsequen tly , w e rescale the v ariables and pa rameters as q n,m = ∆x · q ( x, y ) , r n,m = ∆y · r ( x, y ) , x := n ∆x, y := m ∆y , b A n,m = A ( x, y ) , b B n,m = B ( x, y ) , b C n,m = C ( x, y ) , b D n,m = D ( x, y ) , wherein the time dep endence is s uppressed and a = α ( ∆y ) 2 , b = β ( ∆x ) 2 . Th us, in t he con tinuum limit ∆x, ∆y → 0, (2.3) reduces to the con tinuous Da vey –Stewartson sys tem [1 3, 14, 16, 23, 24], q t + β q xx − α q y y − ( A + C ) q + q ( B + D ) = 0 , (2.4a) r t + α r y y − β r xx + r ( A + C ) − ( B + D ) r = 0 , (2.4b) A x = C x = − α ( q r ) y , (2.4c) B y = D y = − β ( r q ) x . (2.4d) 7 Note that the Da v ey–Stew artson sys tem (2.4) is a linear com bination of the t w o comm uting flows corresp o nding to α = 0, β 6 = 0 and α 6 = 0, β = 0 [25, 26] (also see [24]). In subsection 3.1, we presen t its discrete a nalogue, that is, the (2 + 1)-dimensional Ablo witz–Ladik lattice ( 2.2) is a linear combination of four comm uting flo ws. This is a quite natural result b ecause (i) eac h of the tw o Dav ey–Stew artson flows pro vides an asymmetric (2 + 1)-dimensional generalization of the NLS sy stem and (ii) the Ablo witz–Ladik discretization of t he NLS system is actually a sum of tw o elemen tary flo ws (a nd one trivial flo w) in the same hierarc h y [3, 7, 27 – 29]. 2.3 Noncomm utativ e extension Actually , the (2 + 1)-dimensional Ablo witz–Ladik la t t ice (2.2) is integrable in the general case where the dep enden t v ariables tak e their v alues in matrices, as long as the op erations suc h a s addition and multiplication mak e sense. In tha t case, “1” in (2.2e) and (2.2 f) should b e in terpreted as the identit y matrix. W e can further generalize it to a v a r ia ble-co efficien t system wherein the parameters a and b b ecome arbitrary matrix-v alued functions of one s patia l v ariable as a m := a ( m ) and b n := b ( n ). T o obtain suc h a n extens ion, we consider the following g eneralization of the linear system (2.1): ψ n +1 ,m = z ψ n,m + q n,m φ n,m , (2.5a) φ n,m +1 = z − 1 φ n,m + r n,m ψ n,m , (2.5b) ∂ ψ n,m ∂ t = z a m ψ n,m +1 + A n,m ψ n,m + z − 1 C n,m ψ n,m − 1 + z − 1 q n − 1 ,m b n − 1 φ n,m − q n,m D n,m φ n − 1 ,m , (2.5c) ∂ φ n,m ∂ t = z − 1 b n φ n +1 ,m + B n,m φ n,m + z D n,m φ n − 1 ,m + z r n,m − 1 a m − 1 ψ n,m − r n,m C n,m ψ n,m − 1 . (2.5d) Note that the “sp ectral parameter” z is nonessen tial in t he (2 + 1)-dimensional case and can b e fixed at 1 as des crib ed in subsection 2.1. The compatibilit y conditions for the linear system (2.5) indeed pro vide the noncommutativ e 8 system with site-dep enden t coefficien ts, ∂ q n,m ∂ 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 + q n,m B n,m − A n +1 ,m q n,m = 0 , (2.6a) ∂ r n,m ∂ 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 + r n,m A n,m − B n,m +1 r n,m = 0 , (2.6b) 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 , (2.6c) 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 , (2.6d) ( I − q n,m r n,m ) C n,m = C n +1 ,m ( I − q n,m − 1 r n,m − 1 ) , (2.6e) ( I − r n,m q n,m ) D n,m = D n,m +1 ( I − r n − 1 ,m q n − 1 ,m ) . (2.6f ) If q n,m and r n,m are rectangular matrices, the iden tity ma t rix I in (2.6e) and that in (2.6f) ha ve unequal sizes. In the comm utative case of the parameters, the site-dependen t nature o f ( 2 .6) is nonessen tial if b o th Q ∞ m = −∞ a m and Q ∞ n = −∞ b n tak e nonzero finite v alues. Indeed, if w e c hange the v ariables as q n,m = m − 1 Y j = −∞ a j ! − 1 n − 1 Y k = −∞ b k ! e q n,m , r n,m = m − 1 Y j = −∞ a j ! n − 1 Y k = −∞ b k ! − 1 e r n,m , C n,m = a − 1 m − 1 e C n,m , D n,m = b − 1 n − 1 e D n,m , the site-dep enden t parameters can b e normalized to 1. W e can also obtain a similar result in the noncomm uta t ive case. 2.4 Appropriate c hange of dep enden t v ariables F o r simplicit y , in the follow ing discussion, we consider only the comm utativ e and constan t-co efficie nt case wherein the parameters a a nd b are constan ts and all the quan tities are scalar. Th us, the low est-order conserv ation law for (2.2) is giv en by ∂ log(1 − q n,m r n,m ) ∂ t = ∆ + n  − bq 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  + ∆ + m  − aq 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  . (2.7) The existen ce of an ultra lo cal conserv ed density log (1 − q n,m r n,m ) implies that a nonlo cal transformation inv olving infinite pro ducts of (1 − q n,m r n,m ) δ 9 with δ 6 = 0 could b e applied (cf. [30]); this is indeed the case as w e will see b elo w. The (2 + 1) - dimensional Ablowitz–Ladik lattice (2.2) is in v ariant under a space reflection ( n, m ) → ( − m, − n ) with a minor redefinition of the pa- rameters and the auxiliary fields. Ho w ev er, (2.2) do es not allow the complex conjugation reduction b etw een q n,m and r n,m in t he lo cal f o rm. Therefore, w e need t o iden tify new “conjugate” v aria bles instead of q n,m and r n,m and rewrite (2.2) in a more symmetric form using the new v ariables. F o r this purp ose, w e conside r a gauge t r a nsformation so that the spatial part of the Lax represen tatio n obtains in v ar ia nce with resp ect to the com bined space reflection ( n, m ) → ( − n, − m ) or, equiv alen tly , a 180 degree rotation around the origin. Th us, w e apply the g a uge tra nsformation ψ n,m = X n,m Ψ n,m , φ n,m = Y n,m Φ n,m (2.8) to (2.1a) and (2.1b) and change the dependent v ariables as u n,m = Y n,m X n,m q n,m , v n,m = X n,m Y n,m r n,m . (2.9) Here, X n,m and Y n,m are define d as X n,m := 1 h m n − 1 Y j = −∞ p 1 − q j,m r j,m , Y n,m := 1 l n m − 1 Y k = −∞ p 1 − q n,k r n,k . (2.10) The norming functions h m ( t ) and l n ( t ) are introduced to realize the complex conjugation reduction b et we en u n,m and v n,m ; they will b e dete rmined later. One can also use  Q + ∞ k = m p 1 − q n,k r n,k  − 1 instead of Q m − 1 k = −∞ p 1 − q n,k r n,k to main tain the in v ar ia nce under the space reflection ( n, m ) → ( − m, − n ). This mo dificatio n causes no ess ential difference in the following discussion, so the tra nsformed system can become ideally symmetric with resp ect to space reflections. Here and hereafter, w e assume that | q n,m r n,m | ≪ 1 so that p 1 − q n,m r n,m and its in ve rse as w ell as their infinite pro ducts as consid- ered ab ov e are uniquely and w ell defined. F o r example, w e consider that q n,m = O ( ∆x ) and r n,m = O ( ∆y ) (cf. (2.1a) a nd (2.1b)), and p 1 − q n,m r n,m is defined as the Maclaurin series in q n,m r n,m = O ( ∆x ∆y ). Note that u n,m v n,m = q n,m r n,m , so that the in ve rse t r ansformation of (2.9) can b e obtained immediately . Th us, the spatial part of the Lax represen ta t ion acquires the form  Ψ n +1 ,m Φ n,m +1  = 1 p 1 − u n,m v n,m  1 u n,m v n,m 1   Ψ n,m Φ n,m  . (2.11) 10 V ery recen tly , D. Zakharov has considered essen tially the same scattering problem in [31, 32]. How ev er, this is an acciden tal coincidence, b ecause the first author arriv ed at this Lax represe ntation as w ell as its generalization implied in subsection 2.5 inde p enden tly b efore the papers [31, 32] app eared. The in v ariance under the com bined space reflection ( n, m ) → ( − n, − m ) can b e easily s een if w e shift the indices of the linear w a ve function b y 1 / 2, i.e., " b Ψ n + 1 2 ,m b Φ n,m + 1 2 # = 1 p 1 − u n,m v n,m  1 u n,m v n,m 1  " b Ψ n − 1 2 ,m b Φ n,m − 1 2 # , where Ψ n,m =: b Ψ n − 1 2 ,m and Φ n,m =: b Φ n,m − 1 2 . Indeed, b ecause the determinan t of the spatial Lax matrix ab o v e is unit y , w e obtain " b Ψ n − 1 2 ,m − b Φ n,m − 1 2 # = 1 p 1 − u n,m v n,m  1 u n,m v n,m 1  " b Ψ n + 1 2 ,m − b Φ n,m + 1 2 # . It should b e p ossible to apply the tr a nsformation (2.9) with (2.10) di- rectly to the (2 + 1)-dimensional Ablow itz–Ladik lattice (2.2) and deriv e the transformed equations of motion with the aid of the conse rv ation law (2.7). Ho w ev er, the nonlo cal nature of (2.2 ) mak es such a computation rather com- plicated and difficult. Th us, as an alternative , w e apply the gauge transfor- mation ( 2 .8) with (2.9) and (2.10) to the time part of the Lax represen tation, (2.1c) and (2.1d), and determine the time ev olution of the gauge-tr a nsformed w a ve function, ∂ t Ψ n,m and ∂ t Φ n,m . Its compatibility with the scattering prob- lem (2.11) can pro vide the transformed equations of motion. 2.5 Space-discrete Da v ey–Stew artson system Before applying the t r a nsformation desc rib ed in subse ction 2 .4, we fix the b oundary conditions for the (2 + 1)- dimensional Ablow itz–Ladik lattice (2.2) as lim n →−∞ ( q n,m , r n,m ) = lim m →−∞ ( q n,m , r n,m ) = 0 , (2.12a) lim n →−∞ C n,m = c  h m − 1 h m  2 , lim m →−∞ D n,m = d  l n − 1 l n  2 . (2.12b) Ho w ev er, w e do not fix lim n →−∞ A n,m and lim m →−∞ B n,m in order to obtain in teresting solutions suc h as dromion solutions; they can also dep end on the remaining spatial v ariable and time t . In (2.12a), the dynamical v ariables q n,m and r n,m are a ssumed to approach zero sufficien t ly ra pidly . In (2.12b), 11 c and d are constan ts. The defining relations (2.2e) and (2.2 f) enable C n,m and D n,m to be express ed g lo bally as C n,m = c  X n,m X n,m − 1  2 , D n,m = d  Y n,m Y n − 1 ,m  2 . Th us, the gauge transformatio n (2.8) with (2.9) and (2.10) c hanges (2.1c) and (2.1d) to ∂ Ψ n,m ∂ t = a X n,m +1 X n,m Ψ n,m +1 + e A n,m Ψ n,m + c X n,m X n,m − 1 Ψ n,m − 1 + b Y n,m Y n − 1 ,m +1 u n − 1 ,m Φ n,m − d Y n,m Y n − 1 ,m u n,m Φ n − 1 ,m and ∂ Φ n,m ∂ t = b Y n +1 ,m Y n,m Φ n +1 ,m + e B n,m Φ n,m + d Y n,m Y n − 1 ,m Φ n − 1 ,m + a X n,m X n +1 ,m − 1 v n,m − 1 Ψ n,m − c X n,m X n,m − 1 v n,m Ψ n,m − 1 , where e A n,m := A n,m − ∂ t X n,m X n,m , e B n,m := B n,m − ∂ t Y n,m Y n,m . (2.13) Recalling that q n,m r n,m = u n,m v n,m , the ab ov e relatio ns com bined with (2.11) comprise the Lax represen ta t io n for the transformed system. T o expres s it in a concise form, w e in tro duce the quan tities w n,m := 1 p 1 − q n,m r n,m = 1 p 1 − u n,m v n,m , (2.14a) f n,m := X n,m +1 X n,m = h m Q n − 1 j = −∞ p 1 − u j,m +1 v j,m +1 h m +1 Q n − 1 j = −∞ p 1 − u j,m v j,m , (2.14b) g n,m := Y n +1 ,m Y n,m = l n Q m − 1 k = −∞ p 1 − u n +1 ,k v n +1 ,k l n +1 Q m − 1 k = −∞ p 1 − u n,k v n,k . (2.14c) 12 Th us, w e obtain the Lax represe ntation in the form, Ψ n +1 ,m = w n,m Ψ n,m + w n,m u n,m Φ n,m , (2.15a) Φ n,m +1 = w n,m Φ n,m + w n,m v n,m Ψ n,m , (2.15b) ∂ Ψ n,m ∂ t = af n,m Ψ n,m +1 + e A n,m Ψ n,m + cf n,m − 1 Ψ n,m − 1 + bw n − 1 ,m g n − 1 ,m u n − 1 ,m Φ n,m − dg n − 1 ,m u n,m Φ n − 1 ,m , (2.15c) ∂ Φ n,m ∂ t = bg n,m Φ n +1 ,m + e B n,m Φ n,m + d g n − 1 ,m Φ n − 1 ,m + aw n,m − 1 f n,m − 1 v n,m − 1 Ψ n,m − cf n,m − 1 v n,m Ψ n,m − 1 . (2.15d) The corresponding b oundary conditions are giv en b y lim n →−∞ ( u n,m , v n,m ) = lim m →−∞ ( u n,m , v n,m ) = 0 , lim n →−∞ f n,m = h m h m +1 , lim m →−∞ g n,m = l n l n +1 . Actually , w e can generalize (2.15) to a more general form wherein the spatial part is giv en b y Ψ n +1 ,m = w n,m Ψ n,m + q n,m Φ n,m , Φ n,m +1 = s n,m Φ n,m + r n,m Ψ n,m , with four indep endent functions w n,m , s n,m , q n,m , and r n,m . Th us, it is p o s- sible to start with this general La x represe ntation and then consider the reduction. How ev er, w e skip suc h a discussion to main tain an easy-to-read flo w of the pap er. The compatibility conditions for the linear system (2.15) with w n,m = (1 − u n,m v n,m ) − 1 2 pro vide the time ev olution equations f o r u n,m and v n,m . They can b e written in a natural compact form using new auxiliary fields α n,m and β n,m defined as e A n,m =: 1 2 w n − 1 ,m g n − 1 ,m ( bu n − 1 ,m v n,m − du n,m v n − 1 ,m ) − 1 2 α n,m , (2.16a) e B n,m =: 1 2 w n,m − 1 f n,m − 1 ( au n,m v n,m − 1 − cu n,m − 1 v n,m ) − 1 2 β n,m . (2.16b) 13 Th us, w e finally arriv e at the desired system, ∂ u n,m ∂ t + (1 − u n,m v n,m ) ( dw n,m g n,m u n +1 ,m + bw n − 1 ,m g n − 1 ,m u n − 1 ,m − aw n,m f n,m u n,m +1 − cw n,m − 1 f n,m − 1 u n,m − 1 ) + 1 2 u n,m [ α n,m − w n,m − 1 f n,m − 1 ( au n,m v n,m − 1 + cu n,m − 1 v n,m ) − β n,m + w n − 1 ,m g n − 1 ,m ( bu n − 1 ,m v n,m + d u n,m v n − 1 ,m )] = 0 , (2.17a) ∂ v n,m ∂ t + (1 − u n,m v n,m ) ( cw n,m f n,m v n,m +1 + aw n,m − 1 f n,m − 1 v n,m − 1 − bw n,m g n,m v n +1 ,m − dw n − 1 ,m g n − 1 ,m v n − 1 ,m ) − 1 2 v n,m [ α n,m − w n,m − 1 f n,m − 1 ( au n,m v n,m − 1 + cu n,m − 1 v n,m ) − β n,m + w n − 1 ,m g n − 1 ,m ( bu n − 1 ,m v n,m + d u n,m v n − 1 ,m )] = 0 , (2.17b) w n,m f n,m = w n,m +1 f n +1 ,m if ( a, c ) 6 = 0 , (2.17c) w n,m g n,m = w n +1 ,m g n,m +1 if ( b, d ) 6 = 0 , (2.17d) ∆ + n α n,m = ∆ + m [ w n,m − 1 f n,m − 1 ( au n,m v n,m − 1 + cu n,m − 1 v n,m )] , (2.17e) ∆ + m β n,m = ∆ + n [ w n − 1 ,m g n − 1 ,m ( bu n − 1 ,m v n,m + d u n,m v n − 1 ,m )] . (2.17f ) Here, a , b , c , and d are constan ts and w n,m = (1 − u n,m v n,m ) − 1 2 . In the same w a y as (2.2), (2.17) also admits a dimensional reduction to the Ablowitz– Ladik lattice [7]. Using (2.17 c)–(2.17f), w e can rewrite (2.17a) and (2.17b) in a more symmetric form with respect to space reflections. When c = − a ∗ and d = − b ∗ , the (2 + 1)-dimensional system (2.17) allo ws the complex con- jugation reduction v n,m = σ u ∗ n,m with a real constan t σ ; in this reduction, the auxiliary fields f n,m and g n,m b ecome real-v alued, while t he auxiliary fields α n,m and β n,m b ecome purely imaginary . In particular, (2.17) with purely imaginary a , b , c (= − a ∗ ), and d (= − b ∗ ) provides the suitable space discretization of the Dav ey–Stew artson system (cf. (2.4)). Similarly to the con tinuous case (cf. [33 – 35 ]) , when c = − a and d = − b , w e can consider the reduction of v n,m = σ u n,m and α n,m = β n,m = 0 to obtain 14 a (2 + 1)-dimensional analogue o f the mo dified V olterra lattice [36], ∂ u n,m ∂ t =  1 − σ u 2 n,m  ( bw n,m g n,m u n +1 ,m − bw n − 1 ,m g n − 1 ,m u n − 1 ,m + aw n,m f n,m u n,m +1 − aw n,m − 1 f n,m − 1 u n,m − 1 ) , (2.18a) w n,m f n,m = w n,m +1 f n +1 ,m if a 6 = 0 , (2.18b) w n,m g n,m = w n +1 ,m g n,m +1 if b 6 = 0 . (2.18c) Here, w n,m =  1 − σ u 2 n,m  − 1 2 . It w ould b e in teresting to lo ok for a relationship b et we en (2.1 8) and the discrete mo dified Nizhnik–V eselo v–Novik ov hierarc h y in [31] (also see [37 – 39]). 3 Solutio n s b y the Hirota meth o d In this section, w e discuss how to construct exact solutions of the dis crete Da vey –Stewartson system (2.17) using the Hirota bilinear metho d [20]. Be- cause of the complexit y and irrationality of the equations of motion, it w ould b e to o hard to solv e ( 2 .17) directly , so w e ta k e an alternativ e approach. First, w e bilinearize the (2 + 1)- dimensional Ablowitz–Ladik lat t ice (2.2). Sub- sequen tly , w e consider the effect of the nonlo cal transformation (2 .9) with (2.10), (2.1 3), (2.14) , and (2.16) in the bilinear formalism. The infinite pro d- ucts appearing in the nonlo cal transformatio n can ess entially be ex pressed lo cally in terms of a “tau function”. Thus , w e can obta in exact solutions of (2.2) and (2.1 7) conc urrently from the same set of bilinear equations. 3.1 Decomp osition in to four comm utativ e flo ws Before applying the Hirota bilinear metho d, w e demonstrate that the (2 + 1)- dimensional Ablo witz–Ladik lattice (2.2) can be decomp osed in to the four elemen ta ry flows . In view of (2 .2 c), (2.2d), and (2.12b), w e rescale the aux- iliary fields as A n,m =: aA (0) n,m , B n,m =: bB (0) n,m , C n,m =: c C (0) n,m , D n,m =: d D (0) n,m . (3.1) The corresponding b oundary conditions are lim n →−∞ ( q n,m , r n,m ) = lim m →−∞ ( q n,m , r n,m ) = 0 , lim n →−∞ C (0) n,m =  h m − 1 h m  2 , lim m →−∞ D (0) n,m =  l n − 1 l n  2 . 15 Th us, conside ring the simplest cases where only one of t he parameters a , b , c , and d do es not v a nish, w e obtain the f o ur elemen tar y sy stems: • a -syste m ∂ t a q n,m = q n,m +1 + A (0) n +1 ,m q n,m , (3.2a) ∂ t a r n,m = − r n,m − 1 − r n,m A (0) n,m , (3.2b) A (0) n +1 ,m − A (0) n,m = − ( q n,m +1 r n,m − q n,m r n,m − 1 ) , (3.2c) • b -sys tem ∂ t b q n,m = − q n − 1 ,m − q n,m B (0) n,m , (3.3a) ∂ t b r n,m = r n +1 ,m + B (0) n,m +1 r n,m , (3.3b) B (0) n,m +1 − B (0) n,m = − ( r n +1 ,m q n,m − r n,m q n − 1 ,m ) , ( 3 .3c) • c -system ∂ t c q n,m = C (0) n +1 ,m q n,m − 1 , (3.4a) ∂ t c r n,m = − r n,m +1 C (0) n,m +1 , (3.4b) (1 − q n,m r n,m ) C (0) n,m = C (0) n +1 ,m (1 − q n,m − 1 r n,m − 1 ) , (3.4c) • d -system ∂ t d q n,m = − q n +1 ,m D (0) n +1 ,m , ( 3 .5a) ∂ t d r n,m = D (0) n,m +1 r n − 1 ,m , (3.5b) (1 − r n,m q n,m ) D (0) n,m = D (0) n,m +1 (1 − r n − 1 ,m q n − 1 ,m ) . (3.5c) Clearly , the time ev olution in (2.2) is a linear com binatio n of these four time ev olutions, that is, ∂ t = a∂ t a + b∂ t b + c∂ t c + d ∂ t d . In fact, they a re m utually comm utativ e, so the ab o v e four systems b elong to the same in tegra ble hier- arc h y as the original system (2 .2). T o chec k the commu ta tivit y conditions ∂ t α ∂ t β q n,m = ∂ t β ∂ t α q n,m and ∂ t α ∂ t β r n,m = ∂ t β ∂ t α r n,m for { α, β } ⊂ { a, b, c, d } , w e need to k now ho w to expre ss time deriv atives of the auxiliary fields. Us- ing (3.2)–(3.5), w e can obtain all necessary expressions in the lo cal forms, e.g. , ∂ t b A (0) n,m = − ( q n − 1 ,m +1 r n,m − q n − 1 ,m r n,m − 1 ) , ∂ t c A (0) n,m = −  C (0) n,m +1 − C (0) n,m  , ∂ t d A (0) n,m = q n,m +1 D (0) n,m +1 r n − 1 ,m − q n,m D (0) n,m r n − 1 ,m − 1 , etc . 16 Here, w e assumed that all “in t egr a tion constan ts” etc. can be set equal t o zero. With these lo cal express ions, w e can c hec k the comm utat ivit y of the four flo ws b y direct computatio ns. Note that lim n →−∞ ∂ t c A (0) n,m , lim m →−∞ ∂ t d B (0) n,m , lim n →−∞ ∂ t a C (0) n,m , and lim m →−∞ ∂ t b D (0) n,m do no t v anish in general. Th us, the t a -flo w can change the b oundary v alue of the auxiliary field in the t c -flo w and vice v ersa; the same applies for the t b -flo w and t d -flo w. 3.2 Bilinearization Because the four systems (3.2)–(3.5) ar e compatible, in the sense that their flo ws mutually comm ute, w e will consider here their common solution de- noted as q n,m ( t a , t b , t c , t d ), r n,m ( t a , t b , t c , t d ), etc. Here, t a , t b , t c , and t d are indep enden t argumen ts. Thus , t he solution of the original system (2.2) is obtained b y setting t a = at, t b = bt, t c = ct, t d = dt, (3.6) whic h indeed implie s the relation ∂ t = a∂ t a + b∂ t b + c∂ t c + d ∂ t d . W e assume a solution expres sible in the form, q n,m = G n,m F n +1 ,m , r n,m = H n,m F n,m +1 , (3.7a) A (0) n,m = ∂ t a log  F n,m +1 F n,m  = ∆ + m  ∂ t a F n,m F n,m  , (3.7b) B (0) n,m = ∂ t b log  F n +1 ,m F n,m  = ∆ + n  ∂ t b F n,m F n,m  , (3.7c) C (0) n,m = F n,m +1 F n,m − 1 ( F n,m ) 2 , D (0) n,m = F n +1 ,m F n − 1 ,m ( F n,m ) 2 , (3.7d) and bilinearize the four system s (3.2)–(3.5) in terms of the “tau f unctions” F n,m , G n,m , and H n,m as fo llows: • a -syste m F n +1 ,m +1 ∂ t a G n,m − G n,m ∂ t a F n +1 ,m +1 = F n +1 ,m G n,m +1 , ( 3.8a) F n,m ∂ t a H n,m − H n,m ∂ t a F n,m = − F n,m +1 H n,m − 1 , (3.8b) F n,m ∂ t a F n +1 ,m − F n +1 ,m ∂ t a F n,m = − G n,m H n,m − 1 , (3.8c) • b -sys tem F n,m ∂ t b G n,m − G n,m ∂ t b F n,m = − F n +1 ,m G n − 1 ,m , (3.9a) F n +1 ,m +1 ∂ t b H n,m − H n,m ∂ t b F n +1 ,m +1 = F n,m +1 H n +1 ,m , (3.9b) F n,m ∂ t b F n,m +1 − F n,m +1 ∂ t b F n,m = − G n − 1 ,m H n,m , (3.9c) 17 • c -system F n +1 ,m ∂ t c G n,m − G n,m ∂ t c F n +1 ,m = F n +1 ,m +1 G n,m − 1 , (3.10a) F n,m +1 ∂ t c H n,m − H n,m ∂ t c F n,m +1 = − F n,m H n,m +1 , (3.10b) F n +1 ,m F n,m +1 − F n +1 ,m +1 F n,m = G n,m H n,m , (3.10c) • d -system F n +1 ,m ∂ t d G n,m − G n,m ∂ t d F n +1 ,m = − F n,m G n +1 ,m , (3.11a) F n,m +1 ∂ t d H n,m − H n,m ∂ t d F n,m +1 = F n +1 ,m +1 H n − 1 ,m , (3.11b) F n +1 ,m F n,m +1 − F n +1 ,m +1 F n,m = G n,m H n,m . (3.11c) T o b e precise, eac h tr iplet of bilinear equations gives a sufficien t condition for the corresp onding o r ig inal system. Note that for the c -system and d - system, the bilinear forms as w ell as some exact solutions w ere studied in [18]. 3.3 General solution form ulas Once a solution of t he bilinear equations (3.8 ) –(3.11) is obtained, formula (3.7) with (3 .1) and (3.6) provide s the solution o f the (2 + 1)-dimensional Ablo witz–Ladik lattice (2.2). W e assum e that it satisfies the boundary con- ditions (2.12). Th us, b y applying the nonlo cal transformation (2.9) with (2.10), (2.13), (2 .14), and (2.16), we can also obtain the solution of the dis- crete Dav ey–Stew artson system (2 .1 7). T o ev aluate t he effect of this nonlo cal transformation, w e use (3 .7 a) and (3.10 c) (o r (3.11c)) to r ewrite the infinite pro ducts as n − 1 Y j = −∞ p 1 − q j,m r j,m = n − 1 Y j = −∞ s F j +1 ,m +1 F j,m F j +1 ,m F j,m +1 = s F n,m +1 F n,m lim j →−∞ F j,m F j,m +1 , m − 1 Y k = −∞ p 1 − q n,k r n,k = m − 1 Y k = −∞ s F n +1 ,k +1 F n,k F n +1 ,k F n,k +1 = s F n +1 ,m F n,m lim k →−∞ F n,k F n +1 ,k . Because w e assumed | q n,m r n,m | ≪ 1 , the v alue o f ( F n +1 ,m +1 F n,m ) / ( F n +1 ,m F n,m +1 ) is alwa ys restricted to the neigh b orho o d of 1. F or simplic ity , in considering the solution of (2.17), we also assum e that F n,m is positive (or, at least, | arg F n,m | is sufficien tly small); the p ositivit y conditio n F n,m > 0 can fully justify the use of the form ulas for the square ro ot, suc h as √ X 2 = X and p X/ Y = √ X / √ Y . W e set the norming functions h m and l n in (2.10) as h m = s lim j →−∞ F j,m F j,m +1 , l n = s lim k →−∞ F n,k F n +1 ,k . 18 Th us, w e obtain X n,m = s F n,m +1 F n,m , Y n,m = s F n +1 ,m F n,m . (3.12) After all, w e can express the transformation from (2.2) to (2.17) lo cally in terms of the “ tau function” F n,m . Combining (2.9), (2.13), (2.14), (3.1), (3.7), and (3.12), we arriv e at general solution form ulas for the discrete D a v ey– Stew artson system (2.17) in the form, u n,m = G n,m p F n +1 ,m F n,m +1 , v n,m = H n,m p F n +1 ,m F n,m +1 , (3.13a ) e A n,m = 1 2 ( a∂ t a − b∂ t b − c∂ t c − d∂ t d ) log  F n,m +1 F n,m  , ( 3.13b) e B n,m = 1 2 ( b∂ t b − a∂ t a − c∂ t c − d∂ t d ) log  F n +1 ,m F n,m  , (3.13c) f n,m = p F n,m +2 F n,m F n,m +1 , g n,m = p F n +2 ,m F n,m F n +1 ,m . (3.13d) Here, the time v ariables are set as in (3.6) and the auxiliary fields α n,m and β n,m are determine d from e A n,m and e B n,m through (2.16). Using the bilinear equations (3.8c) and (3.9c) and noting that (2.17e) and (2.1 7f) are iden tities in a , b , c , and d , w e obtain compact expressions for α n,m and β n,m , α n,m = ( − a∂ t a + c∂ t c ) log  F n,m +1 F n,m  , (3.14a) β n,m = ( − b∂ t b + d ∂ t d ) log  F n +1 ,m F n,m  , (3.14b) and new bilinear equations, F n,m ∂ t c F n +1 ,m − F n +1 ,m ∂ t c F n,m = G n,m − 1 H n,m , (3.15) F n,m ∂ t d F n,m +1 − F n,m +1 ∂ t d F n,m = G n,m H n − 1 ,m . (3.16) Note that (3.15) and (3 .16) fill in the piece missing in (3.8)–(3.11). In t he next subsection, w e construct common solutions to all these bilinear equations. As described b elo w ( 2 .17), when c = − a ∗ and d = − b ∗ , w e can imp ose the reduction v n,m = σ u ∗ n,m with a real constan t σ . This reduction can b e realized b y requiring that F n,m > 0 and H n,m = σ G ∗ n,m . 19 3.4 Solitons and dromions The set o f bilinear equations (3.8)–(3.11) together with (3.15) a nd (3.16) is not ideally symmetric in its presen t form. In particular, it is not clear wh y reductions s uch as F ∗ n,m = F n,m and H n,m = σ G ∗ n,m are allow ed. T o restore the symmetry , w e need only to rewrite (3.8a), (3 .8b), (3.9a), and (3.9b) using (3.8c), (3.9 c), and (3.10c) (or (3.11c)). F or example , using (3.8c) and then (3.10c), (3.8a) can b e rewritten as F n,m +1 ∂ t a G n,m − G n,m ∂ t a F n,m +1 = F n,m G n,m +1 . Th us, the full set o f bilinear equations can b e reform ulated in the symmetric form, F n +1 ,m F n,m +1 − F n +1 ,m +1 F n,m = G n,m H n,m , (3.17) F n,m +1 ∂ t a G n,m − G n,m ∂ t a F n,m +1 = F n,m G n,m +1 , (3.18a) F n +1 ,m ∂ t a H n,m − H n,m ∂ t a F n +1 ,m = − F n +1 ,m +1 H n,m − 1 , (3.18b) F n,m ∂ t a F n +1 ,m − F n +1 ,m ∂ t a F n,m = − G n,m H n,m − 1 , (3.18c) F n,m +1 ∂ t b G n,m − G n,m ∂ t b F n,m +1 = − F n +1 ,m +1 G n − 1 ,m , (3.19a) F n +1 ,m ∂ t b H n,m − H n,m ∂ t b F n +1 ,m = F n,m H n +1 ,m , (3.19b) F n,m ∂ t b F n,m +1 − F n,m +1 ∂ t b F n,m = − G n − 1 ,m H n,m , ( 3 .19c) F n +1 ,m ∂ t c G n,m − G n,m ∂ t c F n +1 ,m = F n +1 ,m +1 G n,m − 1 , (3.20a ) F n,m +1 ∂ t c H n,m − H n,m ∂ t c F n,m +1 = − F n,m H n,m +1 , (3.20b) F n,m ∂ t c F n +1 ,m − F n +1 ,m ∂ t c F n,m = G n,m − 1 H n,m , (3.20c) F n +1 ,m ∂ t d G n,m − G n,m ∂ t d F n +1 ,m = − F n,m G n +1 ,m , (3.21a) F n,m +1 ∂ t d H n,m − H n,m ∂ t d F n,m +1 = F n +1 ,m +1 H n − 1 ,m , (3.21b) F n,m ∂ t d F n,m +1 − F n,m +1 ∂ t d F n,m = G n,m H n − 1 ,m . (3.21c) It is no w clear that the t a -flo w and t b -flo w can b e identified with the t c - flo w a nd t d -flo w, resp ectiv ely , thro ugh the complex conjugatio n reduction. Moreo v er, ( 3 .18) and (3.19) corresp o nd to eac h other by the in terc hange of n and m , up to a redefinition of the v a riables. With t hese symmetries in mind, w e can considerably reduce the task of c onstructing explicit s olutio ns to the ab ov e 13 bilinear equations. In the same wa y as in the contin uous case (see, e.g. , [40]), w e can construct the o ne-soliton solution and a tw o-solito n solution straightforw ardly . The 20 one-soliton solution is giv en b y F n,m = 1 − g g (1 − pp )(1 − q q ) ( p p ) n ( q q ) m e ω + ω , G n,m = g p n q m e ω , H n,m = g p n q m e ω , where ω := q t a − p − 1 t b + q − 1 t c − p t d , ω := − q − 1 t a + p t b − q t c + p − 1 t d , a nd g , p , q , etc. are nonzero constan ts. The constan t q should not be confused with q n,m . A ctually , w e can shift t a , t b , t c , and t d in (3.6) b y ar bit r a ry constan ts, but t his freedom can b e absorb ed b y rescaling g and g . When c = − a ∗ and d = − b ∗ (cf. (3.6)), w e set p = p ∗ , q = q ∗ , and g = σ g ∗ with σ (1 − | p | 2 )(1 − | q | 2 ) < 0. Th us, F n,m > 0 and H n,m = σ G ∗ n,m , so the complex conjugation reduction is realized. T o sa v e space, w e omit a rat her lengthy expression for a tw o-soliton solu- tion. These soliton solutions are direct (2 + 1)-dimensional analogues of t he soliton solutions of the (1 + 1)-dimensional systems suc h as (1.2). With an appropriate c hoice of the parameters, they represe nt s tra igh t line solitons in the ph ysical v a riables (cf. (3.7a) or (3.13a)) and th us are not lo calized. In the following, w e obtain more interes ting solutions, that is, dromion solutions; dromions [41] are sp at ia lly lo calized “solitons” that deca y exp o- nen tially in all dire ctions [35] and can exhibit non trivial in teraction proper- ties [40, 42]. More details as w ell as a n extensiv e list of references can b e found in the review article [43]. In analogy with the contin uous case [40], the one-dromion solution is obtained as F n,m = 1 + α 11 1 − pp ( p p ) n e ω 1 + ω 1 + α 22 1 − q q ( q q ) m e ω 2 + ω 2 + α 11 α 22 − α 12 α 21 (1 − pp )(1 − q q ) ( p p ) n ( q q ) m e ω 1 + ω 2 + ω 1 + ω 2 , G n,m = α 12 p n q m e ω 1 + ω 2 , H n,m = α 21 p n q m e ω 1 + ω 2 , where ω 1 := − p − 1 t b − p t d , ω 1 := p t b + p − 1 t d , ω 2 := q t a + q − 1 t c , ω 2 := − q − 1 t a − q t c . When c = − a ∗ and d = − b ∗ (cf. (3.6 )), w e set p = p ∗ , q = q ∗ , and α 21 = σ α ∗ 12 so t hat H n,m = σ G ∗ n,m . Moreov er, if the constant co efficien ts of the three terms in F n,m are p o sitiv e, then F n,m > 0, so the complex conjugation re- duction is realized. Note that the ab ov e F n,m can be written in a 2 × 2 determinan t form, F n,m = det  I +  α 11 α 21 α 12 α 22   p n e ω 1 q m e ω 2   1 1 − pp 1 1 − q q   p n e ω 1 q m e ω 2  . 21 F o llowing the Gilson–Nimmo approac h [42] in the con tinuous case, we construct the m ultidromion solution called the ( M , N )-dro mion solution. The one-dromion solutio n corresponds to the simplest case of M = N = 1. Hereinafter, w e suppress the subscripts of the functions represen ting their dep endence o n the spatial v ariables n and m . When t hey are shifted, w e express it using the shift op erators ( S n Z ) n,m := Z n +1 ,m , ( S m Z ) n,m := Z n,m +1 . W e will consider v arious ( M + N ) × ( M + N ) matrices; they all ha ve the same shape as 2 × 2 block matrices, so op erations can b e p erformed blo c k- wise. F or simplicit y , off-diagonal zeros in the blo c k diagonal matr ices are omitted. W e in tro duce t wo diagonal matrices as Ξ :=  Ξ 1 Ξ 2  , Ξ 1 := diag ( ϕ 1 , . . . , ϕ M ) , Ξ 2 := diag ( χ 1 , . . . , χ N ) , Ξ := Ξ 1 Ξ 2 ! , Ξ 1 := diag ( ϕ 1 , . . . , ϕ M ) , Ξ 2 := diag ( χ 1 , . . . , χ N ) , where ϕ i ( n ) := p n i e − p − 1 i t b − p i t d , χ i ( m ) := q m i e − q − 1 i t a − q i t c , ϕ i ( n ) := p n i e p i t b + p − 1 i t d , χ i ( m ) := q m i e q i t a + q − 1 i t c . W e also in tro duce R :=  P Q  , P := diag ( p 1 , . . . , p M ) , Q := diag ( q 1 , . . . , q N ) , R := P Q ! , P := diag( p 1 , . . . , p M ) , Q := diag ( q 1 , . . . , q N ) . 22 Then, the following relatio ns hold: S n Ξ = Ξ  P I  , S m Ξ = Ξ  I Q  , S n Ξ =  P I  Ξ , S m Ξ =  I Q  Ξ , ∂ t a Ξ = − Ξ  O Q − 1  , ∂ t b Ξ = − Ξ  P − 1 O  , ∂ t c Ξ = − Ξ  O Q  , ∂ t d Ξ = − Ξ  P O  , ∂ t a Ξ =  O Q  Ξ , ∂ t b Ξ =  P O  Ξ , ∂ t c Ξ =  O Q − 1  Ξ , ∂ t d Ξ =  P − 1 O  Ξ . Note that the order of tw o diagonal mat r ices on the right-hand side can be c hanged b ecause they comm ute. Moreo v er, we in tro duce M × M matrices K a nd E M as ( K ) ij := 1 1 − p i p j , ( E M ) ij := 1 , 1 ≤ i, j ≤ M , and N × N matrices L and E N as ( L ) k l := 1 1 − q k q l , ( E N ) k l := 1 , 1 ≤ k , l ≤ N . They satisfy the relations, K − P K P = E M , L − QLQ = E N . W e define ( M + N )-comp onen t column v ectors as e M := (1 , . . . , 1 , 0 , . . . , 0 ) T , e N := (0 , . . . , 0 , 1 , . . . , 1) T , and l := Ξ e M = ( ϕ 1 , . . . , ϕ M , 0 , . . . , 0) T , m := Ξ e N = (0 , . . . , 0 , χ 1 , . . . , χ N ) T , l := Ξ e M = ( ϕ 1 , . . . , ϕ M , 0 , . . . , 0) T , m := Ξ e N = (0 , . . . , 0 , χ 1 , . . . , χ N ) T . 23 Then, w e can easily show the follow ing relations: S n l = R l , S m l = l , S n m = m , S m m = R m , S n l = R l , S m l = l , S n m = m , S m m = R m . W e set t he “tau function” F n,m as F = det F . Here, the ( M + N ) × ( M + N ) matrix F is defined as F := I + A Ξ  K L  Ξ , where A is a constan t ( M + N ) × ( M + N ) matrix. W e also set the o ther “tau functions” G n,m and H n,m as G =  m T F − 1 A l  F , H =  l T F − 1 A m  F . Note that the one-dromion solution is repro duced by setting M = N = 1, up to a minor redefinition of the parameters. Let us c hec k that these “tau functions” indeed satisfy the only bilinear equation without time deriv ative s, (3.17). With the aid of the previous relations, w e ha ve S n F − F = A Ξ  P I   K L   P I  Ξ − A Ξ  K L  Ξ = −A Ξ  K − P K P O  Ξ = −A Ξ e M e T M Ξ = −A l l T . Th us, w e obtain S n F = det  F − A l l T  = det  I − F − 1 A l l T  F =  1 − l T F − 1 A l  F . Similarly , w e obtain S m F =  1 − m T F − 1 A m  F . 24 T o compute S n S m F , w e still need to know S n F − 1 . This can b e express ed as S n F − 1 = ( S n F ) − 1 =  F − A l l T  − 1 = F − 1 + F − 1 A l l T F − 1 1 − l T F − 1 A l . Here, w e used the so-called Sherman–Morrison f orm ula; recall that A l is a column v ector and l T is a ro w v ector. Com bining the abov e results, w e obtain ( S n S m F ) F =  1 − m T  S n F − 1  A m  ( S n F ) F = ( S n F )  1 − m T F − 1 A m  F −  m T F − 1 A l  F  l T F − 1 A m  F = ( S n F ) ( S m F ) − GH . This completes the pro of of (3.1 7). It is a direct but length y calculation to che ck the remaining 12 equations (3.18a)–(3 .2 1c) in volvin g time deriv a t iv es. F or example, in order to c hec k (3.18b), w e need the follo wing in termediate formulas: ∂ t a F = −  m T F − 1 A R − 1 m  F , ∂ t a H =  l T F − 1 A R − 1 m   m T F − 1 A m − 1  F −  l T F − 1 A m   m T F − 1 A R − 1 m  F , S − 1 m H =  l T F − 1 A R − 1 m  F . T o obtain the first formu la, w e first c ompute ∂ t a F as ∂ t a F = −A Ξ  O Q − 1   K L  Ξ + A Ξ  K L   O Q  Ξ = −A Ξ  O Q − 1 E N  Ξ = −A R − 1 Ξ e N e T N Ξ = −A R − 1 m m T . Then, w e m ultiply it b y F − 1 and tak e the trace. 25 4 Conclud ing remarks In this pap er, w e hav e studied a suitable space discretization of the Dav ey– Stew artson system. The Dav ey–Stew artson system is an in tegra ble NLS system in 2 + 1 dimensions, w hich in v olve s tw o spatial v ariables on an equal fo oting and allo ws the complex conjugation reduction b et w een the dep enden t v ariables. W e start ed with a na tural (2 + 1)- dimensional generalization of the Ablo witz–Ladik lattice and then considered a nonlo cal c hange of dep enden t v ariables to symmetrize the equations of motion. Consequen tly , w e obtained the space-discrete Dav ey–Stew art son system inheriting most of the imp ortant prop erties of the contin uo us syste m; in particular, it is integrable and a llo ws the complex conjugation reduction. The price to pa y is the irrationalit y of the equations of motion and their high degree of nonlo calit y , whic h are not seen in the con tinuous case. Through a simple reduction, w e reduced the degree of nonlo cality and obtained a disc rete mo dified KdV-type sys tem in t w o spatial dimensions, namely , the (2 + 1)- dimensional mo dified V olterra lattice (2.18). The (2 + 1 ) - dimensional Ablowitz–Ladik lattice, as w ell as the space- discrete D a ve y–Stew art son system, is a sup erp osition of four elemen tary flo ws that are mutually comm utative. Na turally , the num b er of elemen tary flo ws is equal to the n um b er of directions on the square lattice. No te also that b oth the (1 + 1)-dimensional Ablowitz–Ladik lattice and the con tin uous Da vey –Stewartson sys tem can b e written as a sum o f t wo comm uting flows . W e conjecture that the (2 + 1 ) - dimensional Ablo witz–Ladik lattice and the space-discrete Dav ey–Stew art son system possess four infinite sets of higher symmetries . As in the original Ablo witz–Ladik lattice [44, 45 ], eac h set of symmetries could b e generated from a s ingle dis crete-time system using the Maclaurin expansion in the step-siz e parameter. It w ould b e in teresting to pro vide a more precise description within the framew or k of the Sato theory , e.g. , the discrete t wo-component KP hierarc h y (cf. [19]). Using the Hirota bilinear metho d, w e ha v e constructed exact solutions suc h a s the multidromion solutio ns of the (2 + 1)-dimensional Ablowitz– Ladik lattice and the space - discrete Da v ey–Stew artson sys tem concurren tly . Their solutions can be obtained fro m the same set of bilinear equations, al- though their bilinearizing transformatio ns a re rather differen t (cf. (3.7a) and (3.13a)). N ot e that ( 3 .13a) reflects the irr ationality o f the space-discrete Da vey –Stewartson system that can, ho w ev er, allo w the complex conjuga t ion reduction. 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