Integrable semi-discretizations of the Davey-Stewartson system and a (2+1) -dimensional Yajima-Oikawa system. II
aa r X i v : . [ n li n . S I] J u l Integrable semi-discretizations of theDavey–Stewartson system and a(2 + 1)-dimensional Ya jima–Oikawa system. II
Takayuki
Tsuchida
July 22, 2020
Abstract
This is a continuation of our previous paper arXiv:1904.07924,which is devoted to the construction of integrable semi-discretizationsof the Davey–Stewartson system and a (2 + 1)-dimensional Yajima–Oikawa system; in this series of papers, we refer to a discretization ofone of the two spatial variables as a semi-discretization. In this paper,we construct an integrable semi-discrete Davey–Stewartson system,which is essentially different from the semi-discrete Davey–Stewartsonsystem proposed in the previous paper arXiv:1904.07924. We first ob-tain integrable semi-discretizations of the two elementary flows thatcompose the Davey–Stewartson system by constructing their Lax-pair representations and show that these two elementary flows com-mute as in the continuous case. Then, we consider a linear combi-nation of the two elementary flows to obtain a new integrable semi-discretization of the Davey–Stewartson system. Using a linear trans-formation of the continuous independent variables, one of the two ele-mentary Davey–Stewartson flows can be identified with an integrablesemi-discretization of the (2 + 1)-dimensional Yajima–Oikawa systemproposed in https://link.aps.org/doi/10.1103/PhysRevE.91.062902 . ontents (2 + 1) -dimensional Yajima–Oikawa system 104 Concluding remarks 13References 14 Introduction
As a continuation of our previous paper [1], we consider the problem ofhow to discretize one of the two spatial variables in the Davey–Stewartsonsystem [2]. Both the Davey–Stewartson system [2] (also referred to as theBenney–Roskes system [3]) and the Calogero–Degasperis system [4] are inte-grable (2 + 1)-dimensional generalizations of the nonlinear Schr¨odinger equa-tion [5]. The integrability of the Davey–Stewartson system was establishedby Ablowitz and Haberman in 1975 [6] (also see [7–10]), who provided itsLax-pair representation [11]; the Lax-pair representation in 2 + 1 dimensionsis also referred to as the Manakov triad representation [12], particularly whenit is expressed in operator form.The Davey–Stewartson system [2] can be classified into three differenttypes [13]: the first type is ( i q t + q xx − q yy + ϕq = 0 , (1.1a) ϕ xx + ϕ yy = 2 σ h(cid:0) | q | (cid:1) xx − (cid:0) | q | (cid:1) yy i , (1.1b)where σ = 1 (focusing case) or σ = − ( i q t + q xx + q yy + ϕq = 0 , (1.2a) ϕ xx − ϕ yy = 2 h(cid:0) | q | (cid:1) xx + (cid:0) | q | (cid:1) yy i ; (1.2b)the third type is ( i q t + q xx − q yy + ϕq = 0 , (1.3a) ϕ xy = 2 h(cid:0) | q | (cid:1) xx − (cid:0) | q | (cid:1) yy i . (1.3b)Here, the subscripts t , x and y denote the partial differentiation with respectto these variables, q is a complex-valued function and ϕ is a real-valuedfunction. The first type (1.1) is outside the scope of this paper, but we stressthat only this type has two essentially different versions: the focusing case( σ = 1) and the defocusing case ( σ = − i q t + a ( q xx + 2 F q ) + b ( q yy + 2 Gq ) = 0 , (1.4a) F y = ( | q | ) x , (1.4b) G x = ( | q | ) y , (1.4c)by setting a = b = 1 and a = − b = 1, respectively. In (1.4), a and b are realconstants, and F (defined for the case a = 0) and G (defined for the case3 = 0) are nonlocal real-valued potentials, so the “constants” of integrationthat arise in solving (1.4b) for F and (1.4c) for G should also be real-valued.Clearly, the Davey–Stewartson system expressed in the form (1.4) is alinear combination of the two elementary flows:i q t + q xx + 2 F q = 0 , F y = ( | q | ) x , (1.5)and i q t + q yy + 2 Gq = 0 , G x = ( | q | ) y . (1.6)As described in our previous paper [1], the two elementary flows (1.5) and(1.6) commute [15, 16]; that is, the relation q t t = q t t holds true if andonly if the “constants” of integration appearing in F t and G t are chosenappropriately. By applying a linear change of the independent variables (see, e.g. , page 135 of [17]): e t = t + y, e x = x, e y = αy, (1.7)with a real constant α to (1.5), we obtain a (2 + 1)-dimensional generaliza-tion [18] of the Yajima–Oikawa system [19]:i q t + q xx + uq = 0 , u t + αu y = 2( | q | ) x . (1.8)Here, u := 2 F and we omit the tilde for brevity.The organization of this paper is as follows. In section 2, we provide anintegrable semi-discretization (discretization of one of the two spatial vari-ables, herein x ) of the elementary Davey–Stewartson flow (1.5) by present-ing its Lax-pair representation and show that it admits a straightforwardvector generalization. By changing the time part of the Lax-pair repre-sentation appropriately, we obtain an integrable semi-discretization of theelementary Davey–Stewartson flow (1.6). In section 3, we prove that thetwo elementary Davey–Stewartson flows in the semi-discrete case commuteunder a natural choice of the “constants” of integration. Thus, by taking alinear combination of the semi-discrete elementary Davey–Stewartson flows,we arrive at an integrable semi-discretization of the Davey–Stewartson sys-tem (1.4). In addition, using a linear transformation of the independentvariables like (1.7), we can convert the integrable semi-discretization of theelementary Davey–Stewartson flow (1.5) to an integrable semi-discretizationof the (2 + 1)-dimensional Yajima–Oikawa system (1.8), which essentially co-incides with the system recently proposed by G.-F. Yu and Z.-W. Xu [20].Note that its y -independent reduction, i.e., the (1 + 1)-dimensional discreteYajima–Oikawa system was studied in [21,22]. Concluding remarks are givenin section 4. 4 Integrable semi-discretizations of the twoelementary Davey–Stewartson flows
Inspired by the Lax-pair representation for the (1 + 1)-dimensional discreteYajima–Oikawa system (see Proposition 2.1 in [22]), we consider the followingsemi-discrete linear problem: ψ n,y = q n φ n + χ n r n , (2.1a) φ n +1 − φ n = 12 r n ( ψ n +1 + ψ n − ) , (2.1b) χ n +1 + χ n = 12 q n ( ψ n +1 + ψ n − ) , (2.1c)where n is a discrete spatial variable, y is a continuous spatial variable andthe subscript y denotes the differentiation by y .The linear wavefunction is composed of three components ψ n , φ n and χ n ;this is in contrast with the two-component wavefunction satisfying the spatiallinear problem for the continuous Davey–Stewartson system (1.4) [6, 14]: (cid:26) ψ y = qφ, (2.2a) φ x = − q ∗ ψ, (2.2b)where the asterisk denotes the complex conjugation.The dependent variables q n and r n in (2.1) are scalars, but it is possible toconsider a more general case of vector-valued variables q n and r n , which willbe touched upon at the end of subsection 2.2. Note that the first equation(2.1a) can be rewritten as ψ n,y = q n φ n +1 − χ n +1 r n , using the second and third equations (2.1b) and (2.1c). In view of the time part of the Lax-pair representation for the (1 + 1)-dimensional discrete Yajima–Oikawa system (see Proposition 2.1 in [22]),5e consider the following time evolution of the linear wavefunction: i ψ n,t = v n ( ψ n +1 + ψ n − ) − cψ n , (2.3a)i φ n,t = 12 v n r n − ( ψ n +1 + ψ n − ) − v n − r n ( ψ n + ψ n − ) , (2.3b)i χ n,t = 12 v n q n − ( ψ n +1 + ψ n − ) + 12 v n − q n ( ψ n + ψ n − ) − cχ n , (2.3c)where c is an arbitrary constant and v n is a scalar auxiliary function. Proposition 2.1.
The compatibility conditions of the overdetermined linearsystems (2 . and (2 . for ψ n , φ n and χ n are equivalent to the followingsemi-discrete system in dimensions: i q n,t = v n ( q n +1 + q n − ) − cq n , (2.4a)i r n,t = − v n ( r n +1 + r n − ) + cr n , (2.4b) v n,y = 12 v n ( q n r n − + q n − r n − q n +1 r n − q n r n +1 ) . (2.4c)We can prove this proposition by a direct calculation. Specifically, using(2.1) and (2.3), the compatibility conditions can be rewritten as0 = i ψ n,yt − i ψ n,t y = (i q n,t − v n q n +1 − v n q n − + cq n ) φ n + χ n (i r n,t + v n r n +1 + v n r n − − cr n )+ (cid:20) − v n,y + 12 v n ( q n r n − + q n − r n − q n +1 r n − q n r n +1 ) (cid:21) ( ψ n +1 + ψ n − ) , (cid:20) r n ( ψ n +1 + ψ n − ) − φ n +1 + φ n (cid:21) t = 12 (i r n,t + v n r n +1 + v n r n − − cr n ) ( ψ n +1 + ψ n − ) , and 0 = i (cid:20) q n ( ψ n +1 + ψ n − ) − χ n +1 − χ n (cid:21) t = 12 (i q n,t − v n q n +1 − v n q n − + cq n ) ( ψ n +1 + ψ n − ) , which imply (2.4) and vice versa.If c ∈ R , we can impose the complex conjugation reduction: r n = − ∆q ∗ n , v ∗ n = v n ,
6n the system (2.4), where ∆ is an arbitrary real constant. In particular, ifwe set v n = 1 ∆ + F n , c = 2 ∆ , in (2.4) and impose the complex conjugation reduction, we obtain i q n,t = 1 ∆ ( q n +1 + q n − − q n ) + F n ( q n +1 + q n − ) , (2.5a) F n,y = 12 (cid:18) ∆ + ∆F n (cid:19) (cid:0) q n +1 q ∗ n + q n q ∗ n +1 − q n q ∗ n − − q n − q ∗ n (cid:1) , (2.5b)where q n ∈ C and F n ∈ R .By further setting q n ( y, t ) = q ( n∆, y, t ) , F n ( y, t ) = F ( n∆, y, t ) , the semi-discrete system (2.5) reduces in the continuous limit ∆ → ( i q t = q xx + 2 F q,F y = (cid:0) | q | (cid:1) x , (2.6)where x := n ∆. The system (2.6) can be identified with the elementaryDavey–Stewartson flow (1.5), up to a sign inversion of t . Thus, (2.5) canbe interpreted as an integrable semi-discretization of the elementary Davey–Stewartson flow (1.5).As in the continuous case [17, 23–25], the semi-discrete system (2.4) ad-mits a straightforward vector generalization: i q n,t = v n (cid:0) q n +1 + q n − (cid:1) − c q n , (2.7a)i r n,t = − v n ( r n +1 + r n − ) + c r n , (2.7b) v n,y = 12 v n (cid:0) h q n , r n − i + h q n − , r n i − h q n +1 , r n i − h q n , r n +1 i (cid:1) . (2.7c)Here, h · , · i stands for the standard scalar product. The Lax-pair represen-tation for (2.7) is given by the overdetermined linear systems: ψ n,y = h q n , φ n i + h χ n , r n i , φ n +1 − φ n = 12 r n ( ψ n +1 + ψ n − ) , χ n +1 + χ n = 12 q n ( ψ n +1 + ψ n − ) , (2.8)7nd i ψ n,t = v n ( ψ n +1 + ψ n − ) − cψ n , i φ n,t = 12 v n r n − ( ψ n +1 + ψ n − ) − v n − r n ( ψ n + ψ n − ) , i χ n,t = 12 v n q n − ( ψ n +1 + ψ n − ) + 12 v n − q n ( ψ n + ψ n − ) − c χ n . Note that the (1 + 1)-dimensional ( ∂ t = ∂ y ) reduction of the system (2.7)was studied in [21, 22]. To obtain an integrable semi-discretization of the elementary Davey–Stewartsonflow (1.6), we consider the following time evolution of the linear wavefunctioninvolving differentiation with respect to the continuous spatial variable y : i ψ n,t = − q n φ n,y + ( q n,y − q n q n − r n ) φ n + r n χ n,y − ( r n,y − q n r n r n − ) χ n , (2.9a)i φ n,t = − φ n,yy − h G n + ( q n − r n ) y i φ n + r n r n − χ n,y + J n χ n , (2.9b)i χ n,t = − q n q n − φ n,y + H n φ n + χ n,yy + h G n + ( q n r n − ) y i χ n , (2.9c)where G n , H n and J n are scalar auxiliary functions. Proposition 2.2.
The compatibility conditions of the overdetermined linearsystems (2 . and (2 . for ψ n , φ n and χ n are equivalent to the followingsemi-discrete system in dimensions: i q n,t = q n,yy + G n q n − H n r n − q n,y q n − r n , i r n,t = − r n,yy − J n q n − G n r n + q n r n − r n,y ,G n +1 − G n = − ( q n +1 r n + q n r n +1 ) y + 12 q n r n ( q n +1 r n +1 − q n − r n − ) ,H n +1 + H n = − q n,y ( q n +1 + q n − ) − q n ( q n +1 r n +1 − q n − r n − ) ,J n +1 + J n = r n,y ( r n +1 + r n − ) + 12 r n ( q n +1 r n +1 − q n − r n − ) . (2.10)8his proposition can be proved by a straightforward calculation. Indeed,using (2.1) and (2.9), we can rewrite the compatibility conditions as0 = i ( ψ n,yt − ψ n,t y )= (i q n,t − q n,yy − G n q n + H n r n + q n,y q n − r n ) φ n + (i r n,t + r n,yy + J n q n + G n r n − q n r n − r n,y ) χ n , (cid:20) r n ( ψ n +1 + ψ n − ) + φ n − φ n +1 (cid:21) t = 12 h i r n,t + r n,yy − J n +1 q n + G n +1 r n + ( q n +1 r n ) y r n + ( q n r n +1 r n ) y i ( ψ n +1 + ψ n − )+ (cid:20) G n +1 − G n + ( q n +1 r n + q n r n +1 ) y − q n r n ( q n +1 r n +1 − q n − r n − ) (cid:21) φ n + (cid:20) J n +1 + J n − r n,y ( r n +1 + r n − ) − r n ( q n +1 r n +1 − q n − r n − ) (cid:21) χ n , and0 = i (cid:20) q n ( ψ n +1 + ψ n − ) − χ n − χ n +1 (cid:21) t = 12 h i q n,t − q n,yy − G n +1 q n − H n +1 r n − ( q n r n +1 ) y q n − ( q n +1 q n r n ) y i ( ψ n +1 + ψ n − ) − (cid:20) H n +1 + H n + q n,y ( q n +1 + q n − ) + 12 q n ( q n +1 r n +1 − q n − r n − ) (cid:21) φ n + (cid:20) G n +1 − G n + ( q n +1 r n + q n r n +1 ) y − q n r n ( q n +1 r n +1 − q n − r n − ) (cid:21) χ n , which are, as a whole, equivalent to (2.10).We can impose the complex conjugation reduction r n = − ∆q ∗ n , G ∗ n = G n and J n = − ∆ H ∗ n on (2.10) to obtain i q n,t = q n,yy + G n q n + ∆H n q ∗ n + ∆q n,y q n − q ∗ n ,G n +1 − G n = ∆ (cid:0) q n +1 q ∗ n + q n q ∗ n +1 (cid:1) y + 12 ∆ | q n | (cid:0) | q n +1 | − | q n − | (cid:1) ,H n +1 + H n = − q n,y ( q n +1 + q n − ) + 12 ∆q n (cid:0) | q n +1 | − | q n − | (cid:1) , (2.11)where ∆ is an arbitrary real constant. If we interpret ∆ as a lattice param-eter, set x := n ∆ and consider the continuous limit ∆ →
0, (2.11) reducesto the elementary Davey–Stewartson flow (1.6), up to time reversal and aminor change of notation. 9n subsection 2.2, we showed that the semi-discrete system (2.4) admitsthe vector generalization (2.7). Analogously, we can construct a vector gen-eralization of the semi-discrete system (2.10), which is associated with thelinear problem (2.8). However, the equations of motion for this vector gen-eralization are highly nonlocal and complicated, so we do not present themhere. (2 + 1) -dimensionalYa jima–Oikawa system
Let us first demonstrate that in the generic case the semi-discrete flow (2.4)and the semi-discrete flow (2.10) commute. Using (2.4) and (2.10), we obtain2i (log v n ) yt = i ( q n r n − + q n − r n − q n +1 r n − q n r n +1 ) t = (cid:20) q n,y r n − − q n r n − ,y + q n − ,y r n − q n − r n,y −
12 ( q n r n − ) + 12 ( q n − r n ) (cid:21) y − [ n → n + 1] y , which impliesi (log v n ) t = i v n,t v n = (cid:20)
12 ( q n,y r n − − q n r n − ,y + q n − ,y r n − q n − r n,y ) −
14 ( q n r n − ) + 14 ( q n − r n ) (cid:21) − [ n → n + 1] + f n ( t , t ) , (3.1)where f n ( t , t ) is a y -independent function. Moreover, using (2.4) and (2.10),we also obtaini G n,t = 12 v n [ − ( q n r n − − q n +1 r n − q n r n +1 ) q n +1 r n − + ( q n − r n − q n +1 r n − q n r n +1 ) q n − r n +1 + q n − r n − ( q n − r n − q n r n − )] − v n ( q n +1 r n − − q n − r n +1 ) y + 12 v n − q n r n ( q n r n − + q n − r n − − q n − r n − q n − r n − ) + g ( y, t , t ) , (3.2)10 H n,t = − cH n − v n − q n,y ( q n + q n − ) − v n q n − ,y ( q n +1 + q n − ) − v n q n − ( q n +1 r n + q n − r n − q n r n +1 − q n r n − )+ 12 v n − q n ( q n r n − + q n − r n − − q n − r n − q n − r n − ) + ( − n h ( y, t , t ) , (3.3)andi J n,t = 2 cJ n − v n r n − ,y ( r n +1 + r n − ) − v n − r n,y ( r n + r n − )+ 12 v n r n − ( q n +1 r n + q n − r n − q n r n +1 − q n r n − ) − v n − r n ( q n r n − + q n − r n − − q n − r n − q n − r n − ) + ( − n j ( y, t , t ) , (3.4)where g ( y, t , t ), h ( y, t , t ) and j ( y, t , t ) are n -independent functions.By a straightforward calculation, we can prove the following proposition. Proposition 3.1.
Equations (2 . , (2 . and (3 . – (3 . imply the commu-tativity of ∂ t and ∂ t , i.e., q n,t t = q n,t t and r n,t t = r n,t t , if and only if the “constants” of integration f n ( t , t ) , g ( y, t , t ) , h ( y, t , t ) and j ( y, t , t ) all vanish identically. Note that it is possible to decompose the semi-discrete flow (2.4) intomore fundamental flows by extracting the trivial zeroth flow from (2.4): (cid:26) q n,t = − q n , (3.5a) r n,t = r n . (3.5b)In the generic case, the zeroth flow (3.5) commutes with the semi-discreteflow (2.4) (for any value of c , say, c = 0) and the semi-discrete flow (2.10);that is, q n,t t = q n,t t and r n,t t = r n,t t , and q n,t t = q n,t t and r n,t t = r n,t t , if the corresponding “constants” of integration vanish.In view of the commutativity of the semi-discrete flow (2.4) and the semi-discrete flow (2.10), we can naturally consider a linear combination of thetwo flows: ∂ t := a∂ t + b∂ t , ac → β . Thus, the time evolution of the linearwavefunction can be written as i ψ n,t = av n ( ψ n +1 + ψ n − ) − βψ n + b [ − q n φ n,y + ( q n,y − q n q n − r n ) φ n + r n χ n,y − ( r n,y − q n r n r n − ) χ n ] , (3.6a)i φ n,t = a (cid:20) v n r n − ( ψ n +1 + ψ n − ) − v n − r n ( ψ n + ψ n − ) (cid:21) + b n − φ n,yy − h G n + ( q n − r n ) y i φ n + r n r n − χ n,y + J n χ n o , (3.6b)i χ n,t = a (cid:20) v n q n − ( ψ n +1 + ψ n − ) + 12 v n − q n ( ψ n + ψ n − ) (cid:21) − βχ n + b n − q n q n − φ n,y + H n φ n + χ n,yy + h G n + ( q n r n − ) y i χ n o , (3.6c)where a , β and b are constants (or, more generally, arbitrary functions of thetime variable t [4]) and v n , G n , H n and J n are auxiliary functions.By a straightforward calculation, we can prove that the following propo-sition holds true. Proposition 3.2.
The compatibility conditions of the overdetermined linearsystems (2 . and (3 . for ψ n , φ n and χ n are equivalent to the followingsemi-discrete system in dimensions: i q n,t = av n ( q n +1 + q n − ) − β q n + b ( q n,yy + G n q n − H n r n − q n,y q n − r n ) , i r n,t = − av n ( r n +1 + r n − ) + β r n + b ( − r n,yy − J n q n − G n r n + q n r n − r n,y ) ,v n,y = 12 v n ( q n r n − + q n − r n − q n +1 r n − q n r n +1 ) if a = 0 ,G n +1 − G n = − ( q n +1 r n + q n r n +1 ) y + 12 q n r n ( q n +1 r n +1 − q n − r n − ) if b = 0 ,H n +1 + H n = − q n,y ( q n +1 + q n − ) − q n ( q n +1 r n +1 − q n − r n − ) if b = 0 ,J n +1 + J n = r n,y ( r n +1 + r n − ) + 12 r n ( q n +1 r n +1 − q n − r n − ) if b = 0 . (3.7)If a, β, b ∈ R , we can impose the complex conjugation reduction: r n = − ∆q ∗ n , v ∗ n = v n , G ∗ n = G n , J n = − ∆ H ∗ n , on the system (3.7), where ∆ is an arbitrary real constant. By setting v n = 1 ∆ + F n , β = 2 ∆ a, i q n,t = a (cid:20) ∆ ( q n +1 + q n − − q n ) + F n ( q n +1 + q n − ) (cid:21) + b ( q n,yy + G n q n + ∆H n q ∗ n + ∆q n,y q n − q ∗ n ) ,F n,y = 12 (cid:18) ∆ + ∆F n (cid:19) (cid:0) q n +1 q ∗ n + q n q ∗ n +1 − q n q ∗ n − − q n − q ∗ n (cid:1) if a = 0 ,G n +1 − G n = ∆ (cid:0) q n +1 q ∗ n + q n q ∗ n +1 (cid:1) y + 12 ∆ | q n | (cid:0) | q n +1 | − | q n − | (cid:1) if b = 0 ,H n +1 + H n = − q n,y ( q n +1 + q n − ) + 12 ∆q n (cid:0) | q n +1 | − | q n − | (cid:1) if b = 0 , (3.8)where a and b are real constants, q n , H n ∈ C and F n , G n ∈ R . The semi-discrete system (3.8) with a sign inversion of t and a minor change of nota-tion reduces in the continuous limit ∆ → x := n ∆. Thus, (3.8) can be regarded as an integrable semi-discretization of the Davey–Stewartson system (1.4).If we consider a linear change of the independent variables: e t = t + y, e y = αy, (3.9)where α is an arbitrary real constant, the semi-discrete elementary Davey–Stewartson flow (2.5) is transformed to i q n,t = 1 ∆ ( q n +1 + q n − − q n ) + F n ( q n +1 + q n − ) ,F n,t + αF n,y = 12 (cid:18) ∆ + ∆F n (cid:19) (cid:0) q n +1 q ∗ n + q n q ∗ n +1 − q n q ∗ n − − q n − q ∗ n (cid:1) . (3.10)Here, we omit the tilde of the continuous independent variables for brevity.The system (3.10) can be interpreted as an integrable semi-discretizationof the (2 + 1)-dimensional Yajima–Oikawa system (1.8), up to a rescalingof variables; note that (3.10) essentially coincides with the system recentlyproposed by G.-F. Yu and Z.-W. Xu [20]. If we discard the dependence on y , (3.10) reduces to the (1 + 1)-dimensional discrete Yajima–Oikawa systemstudied in [21, 22]. As a continuation of our previous paper [1], we studied the problem of howto discretize the spatial variable x in the Davey–Stewartson system (1.4) and13he (2 + 1)-dimensional Yajima–Oikawa system (1.8). To guarantee the in-tegrability of the semi-discretization, we start with the linear problem (2.1),wherein we can impose the complex conjugation reduction r n = − ∆q ∗ n with areal constant ∆ . By associating (2.1) with an appropriate time-evolutionarysystem of the linear wavefunction and computing the compatibility condi-tions, we obtain (2.5) (resp. (2.11)) as an integrable semi-discretization ofthe elementary Davey–Stewartson flow (1.5) (resp. (1.6)). Note that (2.5)(or, more precisely, its original form (2.4)) admits the simple vector gener-alization (2.7). It is shown that the two elementary flows (2.5) and (2.11)(or, more generally, (2.4) and (2.10)) commute under a natural choice ofthe “constants” of integration. Thus, we can take a linear combination ofthem to obtain (3.8), which provides an integrable semi-discretization of theDavey–Stewartson system (1.4). By changing the independent variables as in(3.9), we convert the semi-discrete elementary Davey–Stewartson flow (2.5)to the system (3.10), where the tilde of the continuous independent vari-ables is omitted. The system (3.10) gives an integrable semi-discretizationof the (2 + 1)-dimensional Yajima–Oikawa system (1.8), which is essentiallyequivalent to the semi-discrete system recently proposed by G.-F. Yu andZ.-W. Xu [20].We finally remark that another integrable discretization of the Davey–Stewartson system (1.4) can be found in [26] (also see some preceding resultsin [27, 28]), wherein both spatial variables x and y are discretized. References [1] T. Tsuchida:
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