Integrable semi-discretizations and full-discretization of the two dimensional Leznov lattice

Transcription

1 Integrable semi-discretizations and full-discretization of the two dimensional Leznov lattice Xing-Biao Hu and Guo-Fu Yu LSEC, Institute of Computational Mathematics and Scientific Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, P.O. Box 79, Beijing 00080, P.R. CHINA Department of Mathematics, Shanghai Jiao Tong University, Shanghai, 0040, P.R. CHINA Abstract In this paper, semi-discretizations and full-discretization of the Leznov lattice are investigated via Hirota s bilinear formalism. As a result, two integrable semi-discrete versions and one fully discrete version for the Leznov lattice are found. Bäcklund transformations, nonlinear superposition formulae and Lax pairs for these discrete versions are presented. Mathematics Subject Classification (000). 35Q58, 37K40 Keywords. Integrable system, Integrable discretization, Bäcklund transformation, Lax pair Introduction The nonlinear two-dimensional Leznov lattice given by [] ln θ(n) = θ(n + )p(n + ) θ(n)p(n) + θ(n )p(n ), x y () p(n) = θ(n + ) θ(n ), y () is a special case of the so-called UToda(m, m ) system with m =, m =. If we set the variable transformations a(n) = p(n + ), c(n) = θ(n + ), then () and () can be transformed into the form a y (n) = c(n + ) c(n ), (3) b y (n) = a(n )c(n ) a(n)c(n), (4) c x (n) = c(n)[b(n) b(n + )], (5) which is a two-dimensional generalization of the Blaszak-Marciniak lattice[, 3] In 985, Kupershmidt proposed the following integrable lattice [4]: a t (n) = c(n + ) c(n ), (6) b t (n) = a(n )c(n ) a(n)c(n), (7) c t (n) = c(n)[b(n) b(n + )]. (8) q i,t (n) = q i (n)[q 0 (n + i) q 0 (n)] + q i+ (n) q i+ (n ), 0 i N, (9) q N,t (n) = q N (n)[q 0 (n + N) q 0 (n)]. (0) Obviously, when N =, by the variable transformation q (n) = e u(n) u(n ), system (9),(0) is nothing but the Toda lattice Corresponding author u tt (n) = e u(n+) u(n) e u(n) u(n ). ()

2 When N =, by the Miura-like transformation q 0 (n) = b(n ), q (n) = a(n )c(n ), q (n) = c(n)c(n ). () the system (9)-(0) can be transformed into the three-field Blaszak-Marciniak lattice (6)-(8). From above observation we can see that two dimensional Toda lattice u ts (n) = e u(n+) u(n) e u(n) u(n ) (3) is different from the two-dimensional Leznov lattice since their one-dimensional versions correspond to different N values in the Kupershmidt lattice (9)-(0). For the two-dimensional Toda lattice, its integrable discretization form is the famous fully discrete KP equation (or Hirota-Miwa equation). It is quite natural and reasonable for us to consider integrable discrete versions for the two-dimensional Leznov lattice from the point of mathematical structure and substantial physical application. The purpose of this paper is to propose some discrete versions for the two-dimensional Leznov lattice. Until now, much attention has been paid to the problem of integrable discretizations of integrable systems. (See, e.g., [5]-[9] and references therein). It is highly nontrivial and of considerable interest to find integrable discretizations for integrable equations. Various approaches to the problem of integrable discretization are currently available. One of them is Hirota s approach [0]-[5]. Traditionally, Hirota s discretization of integrable equations is based on gauge invariance and soliton solutions. Here, we emphasize on discretizing integrable equations such that the resulting discrete bilinear equations have bilinear Bäcklund transformations. As a bonus, by doing so, we can usually derive Lax pairs for the resulting discrete equations. This method has been successfully applied to the discretization of + dimensional sinh-gordon equation [6]. It is well known that a variety of nonlinear integrable equations share many common features, among which are so called Bäcklund transformations(bts) and their associated nonlinear superposition formulae[7, 8, 9]. We can usually derive the nonlinear superposition formulae from the commutability of BTs. In the paper, we will show that the obtained semi-discretizations and full-discretization of the Leznov lattice do have such nice properties. The content of the paper is organized as follows. In section, an y-discrete version of the Leznov lattice is found. A BT and the corresponding nonlinear superposition formula and Lax pair are shown. In section 3, an x-discrete version of the Leznov lattice is found, its associated BT, superposition principle and Lax pair are given. Furthermore, a fully discrete version of the Leznov lattice is worked out in section 3. It turns out that the resulting fully discrete version of the Leznov lattice has a BT, nonlinear superposition formula and Lax pair. In section 4, conclusion and discussions are given. Finally we list some bilinear operator identities which are used in the paper in the appendix. Integrable semi-discrete version of the Leznov lattice in y-direction By the dependent variable transformation[0] system ()-() can be transformed into the form f(n + )f(n ) θ(n) = f(n), p(n) = D x D y f(n) f(n) f(n + )f(n ), D y(d x D y f(n) f(n)) (e f(n) f(n)) = sinh(d n )(e f(n) f(n)) f (n), (4) by introducing an auxiliary variable z and using (A3) from Appendix, we can decouple (4) into the bilinear form : (D y D z e + )f(n) f(n) = 0, (5) (D y D x D z e )f(n) f(n) = 0, (6) we remark that the technique of introducing auxiliary variables and then transforming multilinear equations into bilinear equations is typical in Hirota s bilinear formalism. The Hirota bilinear differential operator Dy m Dt k and the bilinear difference operator exp(δd n ) are defined by [], respectively, D m y D k t a b ( y ) m ( y t ) k t a(y, t)b(y, t ) y =y,t =t,

3 [ exp(δd n )a(n) b(n) exp δ( n ] n ) a(n)b(n ) n =n= a(n + δ)b(n δ). In this section, we consider an integrable semi-discretization of the Leznov lattice ()-(). First of all, we propose the following bilinear equations δ D z sinh(δd y )f f = (e +δdy e δdy )f f, (7) ( δ D x sinh(δd y ) D z e +δdy )f f = 0, (8) By some calculations, it is shown that in the continuum limit as δ 0, the system (7)-(8) is reduced to the Leznov lattice (5)-(6). Therefore the system (7)-(8) serves as a y-direction discrete version for (5)-(6). For simplicity we take δ = and rewrite variable y by m in the following discussion. In this case, the system (7)-(8) becomes By the dependent variable transformation D z e Dm f f = (e Dm+ e Dm )f f, (9) (D x e Dm D z e Dm+ )f f = 0. (0) u m,n = ln f m,n+ f m,n, v m,n = ln f m+,n f m,n, () the bilinear equations (9)-(0) can be transformed into the following nonlinear form u m+,n u m,n v m,n+ + v m,n = 0, () v m,n,x e um,n um+,n v m+,n,x e um+,n um+3,n +4e um+3,n+ um+,n 4e um+,n um,n = 0. (3) In fact, from () we can easily derive (). In order to obtain (3), we need to use bilinear operator identity (A3) from which it follows that sinh(d m)[d x e Dm f f] [e +Dm f f] = sinh(d n + D m )[e Dm+ f f] [e Dm f f] (4) where we have used (9) and (0). By some calculations, we can deduce (3) from (4). Since we are looking for integrable discretization of the Leznov lattice, we need to show integrability of (9)-(0). The justification to this is via Bäcklund transformation and Lax pair. We will show that (9)-(0) is integrable in the sense of having a Bäcklund transformation and Lax pair. Concerning bilinear equations (9)-(0), we have the following result: Proposition. A Bäcklund transformation for (9) and (0) is where λ, µ, γ and β are arbitrary constants. (D z λe γ)f(m, n) g(m, n) = 0, (5) (λe Dm + e Dm+ + µe Dm )f(m, n) g(m, n) = 0, (6) (D x + λe λγe + λd z e + β)f(m, n) g(m, n) = 0, (7) Proof. Let f(m, n) be a solution of Eqs. (9) and (0). If we can show that Eqs. (5)-(7) guarantee that the following two relations: P [D z e Dm (e Dm+ e Dm )]g(m, n) g(m, n) = 0, (8) P (D x e Dm D z e Dm+ )g(m, n) g(m, n) = 0, (9) hold, then Eqs. (5)-(7) form a Bäcklund transformation. 3

4 By use of (5)-(7), (A33), (A34), (A35), we have (e Dm f f)p = sinh(d m )(D z f g) (fg) 4 sinh( D n)(e +Dm f g) (e Dm f g) = λ sinh( D n )(edm f g) (e Dm f g) 4 sinh( D n = sinh( D n )(λedm f Dm+ g + e f g) (e Dm f g) = 0. Thus we have proved that (8) holds. Similarly )(e +D m f g) (e Dm f g) (e Dm f f)p = sinh(d m )(D x f g) (fg) (D z e +Dm f f)(e Dm g g) + (e Dm f f)(d z e Dm+ g g) +(e Dm+ f f)(d z e Dm g g) (D z e Dm f f)(e +Dm g g) (e Dm+ f f)(d z e Dm g g) + (D z e Dm f f)(e +Dm g g) = sinh(d m )(D x f g) (fg) 4D z cosh( D n)(e +Dm f g) (e Dm f g) 4(e +Dm f f)[(e +Dm e Dm )g g] + 4(e +Dm g g)[(e +Dm e Dm )f f] = sinh(d m )(D x f g) (fg) + λd z cosh( D n )(edm f g) (e Dm f g) +8 sinh( D n )(e +D m f g) (e Dm f g)] = sinh(d m )(D x f g) (fg) + λd z cosh( D n )(edm f g) (e Dm f g) 4 sinh( D n )(λe +D m f g) (e Dm f g)] = sinh(d m )(D x f g) (fg) 4λ sinh(d m )(fg) (e f g) +λ sinh(d m )[(D z f g) (e f g) + (D z e f g) (fg)] = sinh(d m )[(D x + λe λγe + λd z e )f g] (fg) = 0. Thus we have completed the proof of proposition. Using (5)-(7), we can easily obtain the following solution from the trivial solution f(m, n) = : g(m, n) = + exp(η), where η = pm + (ln (λ+)e p λ )n + [( λ + λ)e p λ + λ]z + [(λ + λ)e p λ λ λ (λ+)e p λ ]x and γ = λ, µ = λ, β = λ λ. Concerning the BT (5)-(7), we have the following result: Proposition. Let f 0 be a solution of the semi-discrete Leznov lattice (9)-(0). Suppose that f i (i =, ) is another solution of (9)-(0) which is related to f 0 under BT (5)-(7) with parameters (λ i, µ i, γ i, β i ) i.e., (λ i,µ i,γ i,β i) f 0 f i (i =, ), λ λ 0, f i 0 (i = 0,, ). Then f defined by e f 0 f = κ[λ e λ e ]f f (30) with κ being a non-zero constant, is a new solution which is related to f and f under the BT (5)-(7) with parameters (λ, µ, γ, β ), (λ, µ, γ, β ), respectively. As an application of the results, we can construct soliton solution of the semi-discrete Leznov lattice in y-direction. Choose, for example, f 0 =, κ = /(λ λ ). It can be easily verified that (λ, µ, γ, β ) + e η (λ, µ, γ, β ) (λ, µ, γ, β ) 3 + e η 3 F (λ, µ, γ, β ) where e ηi = s n i rm i epix+qiy+η0 i, pi = λ i (s i ) + λ i (s i ), q i = λ i λ i s i, r i = ( λi+/si λ i+ )/, β i = λ i λ i, γ i = λ i, µ i = λ i, 4

5 F = + λ λ e p e η + λ e p λ e η + λ e p λ e p e η+η. λ λ λ λ λ λ In Fig. and Fig., we show the plotting of the -soliton solution u x and v x respectively, plotted with λ =.505, λ = 3.83, s = 0.8, s = 0.654, m = 4, z = 4. (a) (b) Figure : The -soliton solution: (a) u x (x = 0), (b) u x (a) (b) Figure : The -soliton solution: (a) v x (x = 0), (b) v x Starting from the bilinear BT (5)-(7), we can derive a Lax pair for the system ()-(3). Firstly, set ψ m,n = f m,n /g m,n, u m,n = ln g m,n+ g m,n, v m,n = ln g m+,n g m,n in (5)-(7). Then from the bilinear BT (5)-(7) and after some calculations, we can obtain the following Lax pair for ()-(3): λψ m+,n e um,n um+,n + ψ m+,n+ + µψ m,n = 0, (3) ψ m,n,x + λ ψ m,n e um,n um,n λψ m,n v m,n,x e um,n um+,n +λψ m,n e um+,n+ um,n + βψ m,n = 0. (3) 5

6 By differentiating eq. (3) with respect to x and by use of (3) to eliminate ψ m+,x, ψ m+,n+,x, ψ m,x, also by use of (3) to express ψ m,i by ψ m+,i and ψ m+,i+ (i = n, n, n), we can find that the coefficient of λ ψ m+,n is just (3). Together with the coefficient of λψ m+,n we can obtain (). So the compatibility condition of (3) and (3) yields the semi-discrete system ()-(3). 3 Integrable semi-discrete version of the Leznov lattice in x-direction Similar to section, we first propose the following bilinear equations: D y D z f f = (e )f f, (33) ɛ sinh(ɛd x)d y f f D z e +ɛdx f f = 0. (34) By some calculations, it is shown that in the continuum limit as ɛ 0, the system (33)-(34) is reduced to the Leznov lattice (5)-(6). Therefore the system (33)-(34) serves as a x-direction discrete version for (5)-(6). For simplicity we take ɛ = and rewrite variable x by k in the following discussion. In this case, the system (33)-(34) becomes Through the dependent variable transformation D y D z f f = (e )f f, (35) (D y e D k D z e +D k )f f = 0. (36) u n,k = ln f n+,k f n,k, v n,k = ln f n,k+ f n,k, (37) the bilinear equations (35)-(36) can be transformed into the following nonlinear form v n+,k v n,k + u n,k u n,k+ = 0, (38) e u n+,k+ u n,k e u n+,k+ u n,k + v n+,k,yy + v n+,k,y (u n,k,y u n+,k+,y ) = 0. (39) In fact, from (37) we can easily derive (38). In order to obtain (39), we need to use bilinear operator identity (A38) from which it follows that D y[d y e D k f f] [e +D k f f] = sinh(d n + D k )[e f f] f (40) where we have used (35) and (36). By some calculations, we can deduce (39) from (40). Concerning the bilinear equations (35)-(36), we have the following results: Proposition 3. A bilinear BT for equation (35) and (36) is (D y + λ e + µ)f(n, k) g(n, k) = 0, (4) (D z e λe + γe )f(n, k) g(n, k) = 0, (4) (λ e +D k D z e +D k γe +D k + βe D k )f(n, k) g(n, k) = 0. (43) where λ, µ, γ and β are arbitrary constants. Proof. Let f(n, k) be a solution of Eqs. (35) and (36). If we can show that Eqs. (4)-(43) guarantee that the following two relations: P [D y D z (e )]g(n, k) g(n, k) = 0, (44) P (D y e D k D z e +D k )g(n, k) g(n, k) = 0, (45) hold, then Eqs. (4)-(43) form a Bäcklund transformation. In analogy with the proof already given in [0], we know that P = 0 holds. Thus it suffices to show that P = 0. In this regard, by using (A33), (A39), 6

7 (A40), we have P [e D k f f] = sinh(d k )(D y f g) (fg) 4 sinh( D n )(D ze +D k f g) (e D k f g) +4 sinh( D n + D k)(e f g) (D z e f g) = λ sinh(d k )(e f g) (fg) 4 sinh( D n )(D ze +D k f g) (e D k f g) 4γ sinh( D n + D k)(e f g) (e f g) = λ sinh( D n )(e +D k f g) (e D k f g) 4 sinh( D n )(D ze +D k f g) (e D k f g) 4γ sinh( D n )(e +D k f g) (e D k f g) = sinh( D n )[(λ e +D k D z e +D k γe +D k )f g] (e D k f g) = 0. Thus we have completed the proof of Proposition 3. Using (4)-(43), we can easily obtain the following solution from the trivial solution f(m, n) = : g(m, n) = + exp(η), where η = pn + ( ln λ λe p λ λ )k + (λ e p λ )y + (λ λe p )z and γ = λ, µ = λ, β = λ λ. Concerning the BT (4)-(43), we have the following result: Proposition 4. Let f 0 be a solution of the semi-discrete Leznov lattice (35)-(36). Suppose that f i (i =, ) is another solution of (35)-(36) which is related to f 0 under BT (4)-(43) with parameters (λ i, µ i, γ i, β i ) i.e., (λ i,µ i,γ i,β i) f 0 f i (i =, ), λ λ 0, f i 0 (i = 0,, ). Then f defined by e f 0 f = c [λ e λ e ]f f (c is a nonzero constant), (46) is a new solution which is related to f and f under the BT (4)-(43) with parameters (λ, µ, γ, β ), (λ, µ, γ, β ), respectively. Starting from the bilinear BT (4)-(43), we can derive a Lax pair for the system (38)-(39). Firstly, set ψ n,k = f n,k /g n,k, u n,k = ln g n+,k g n,k, v n,k = ln g n,k+ g n,k in (4)-(43). Then from the bilinear BT (4)-(43) and after some calculations, we can obtain the following Lax pair for (38)-(39): λ ψ n,k+ e u n,k u n,k+ λψ n+,k+ v n+,k,y ψ n+,k+ e u n,k u n+,k+ + βψ n,k = 0, (47) ψ n,k,y + λ ψ n,k e u n,k u n,k + µψ n,k = 0. (48) Similar to the deduction of compatibility in the above section, by differentiating eq. (47) with y and substitute (48) into the differential expression to eliminate ψ k+,y, ψ n+,k+,y, ψ n+,k+,y, ψ k,y, it can be shown that the compatibility condition of (47)-(48) generates the system (38)-(39). 4 Integrable fully-discrete version of the Leznov lattice First of all, we propose the following bilinear equations δ sinh(δd y)d z f f = (e +δdy e δdy )f f, (49) [ δɛ sinh(δd y) sinh(ɛd x ) D z e +δdy+ɛdx ]f f = 0. (50) In the continuum limit as ɛ 0 and δ 0, the system (49)-(50) is reduced to the Leznov lattice (5)- (6). Therefore the system (49)-(50) serves as a full-discrete version for (5)-(6). For simplicity we take 7

8 ɛ =, δ = and rewrite variable y by m and x by k in the following discussion. In this case the system (49)-(50) becomes Through the independent variable transformation eqs. (5) and (5) can be transformed into the nonlinear form D z e Dm f f = (e +Dm e Dm )f f, (5) [sinh(d m ) sinh(d k ) D z e +Dm+D k ]f f = 0. (5) u m,n,k = ln f m,n+,k, v m,n,k = ln f m,n,k+, (53) f m,n,k f m,n,k e u m,n,k+ u m+,n,k+ +v m,n,k v m+,n,k e u m,n,k u m+,n,k+ +e u m 3,n,k u m,n,k+ e u m 3,n,k+ u m,n,k+ +v m 3,n,k v m,n,k +8e u m+,n+,k+ u m,n,k+ 8e u m,n,k u m 3,n,k = 0, (54) u m,n,k+ u m,n,k + v m,n,k v m,n+,k = 0. (55) In fact, from (53) we can easily derive (55). In order to obtain (54), we need to use bilinear operator identity (A4) from which it follows that sinh(d m)[sinh(d m ) sinh(d k )f f] [e Dm++D k f f] = sinh(d m + D n + D k )(e Dm+ f f) (e Dm f f) where we have used (5) and (5). By some calculations, we can deduce (54) from (56). Concerning the bilinear equations (5)-(5), we have the following results: Proposition 5. A Bäcklund transformation for the fully discrete Leznov lattice (5)-(5) is (D z e λe γe )f g = 0, (57) (λe Dm + e Dm+ + µe Dm )f g = 0, (58) (D z e D k D m γe D k D m + µ edm+ D k + βe +D k+d m )f g = 0. (59) where λ, µ, γ and β are arbitrary constants. Proof. Let f(n) be a solution of Eqs. (5) and (5). What we need to prove is that the function g satisfying (57)-(59) is another solution of Eqs. (5) and (5), i.e., (56) P [D z e Dm (e +Dm e Dm )]g g = 0, (60) P [sinh(d m ) sinh(d k ) D z e +Dm+D k ]g g = 0. (6) In the section, we have proved that (60) holds. Thus it suffices to show that P = 0. In this regards, by (A4)-(A43), we have P [e Dm+D k f f] = [(sinh(d m ) sinh(d k ) D z e +Dm+D k )f f][e Dm+D k g g] [(sinh(d m ) sinh(d k ) D z e +Dm+D k )g g][e Dm+D k f f] = sinh(d k )(e Dm f g) (e Dm f g) 4 sinh( D n + D k + D m )(D z e f g) (e f g) +4 sinh( D n )(e +D k +D m f g) (D z e D k D m f g) = µ sinh(d k)(e Dm f g) (e Dm+ f g) 4γ sinh( D n + D k + D m )(e f g) (e f g) +4 sinh( D n )(e +D k +D m f g) (D z e D k D m f g) = µ sinh(d n )(e +D m+d k Dm+ f g) (e D k f g) 4γ sinh( D n +4 sinh( D n )(e +D k +D m f g) (D z e D k D m f g) )(e +D k +D m f g) (e D k D m f g) = sinh( D n )(e +D m+d k f g) [(µ Dm+ e D k γe D k D m + D z e D k D m )f g] = 0. Thus we have completed the proof of Proposition 5. 8

9 Using the BT (57)-(59), we can obtain the following solution from the trivial solution f(m, n, k) = : g(m, n, k) = + exp(η) where η = λ+e q (ln λ+ )m + qn + (ln λ +4λ λ e q +4λe q )k + (λ λeq )z and µ = λ, β = λ + 4+λ, γ = λ. Proposition 6. Let f 0 be a solution of the fully discrete Leznov lattice (5)-(5). Suppose that f i (i =, ) is another solution of (5)-(5) which is related to f 0 under BT (57)-(59) with parameters (λ i, µ i, γ i, β i ) i.e., (λ i,µ i,γ i,β i) f 0 f i (i =, ), λ λ 0, f i 0 (i = 0,, ). Then f defined by (λ e λ e )f f = ke f 0 f (k is a nonzero constant), (6) is a new solution which is related to f and f under the BT (57)-(59) with parameters (λ, µ, γ, β ), (λ, µ, γ, β ), respectively. In the following, we derive a Lax pair for the fully discrete Leznov equation (5)-(5). Let Then from eqs. (57)-(59) we can obtain ψ = f g, u = ln g n+ g, v = ln g k+ g. In the above, we have denoted λ ψ n,k (λc m,n,k + λ 4 A m,n,k)ψ k +( B m,n,k A m,n,kc m,n,k )ψ n+,k + µβψ n+,k+ = 0, (63) λψ m+ + µψ m + C m,n,k ψ m+,n+ = 0. (64) A m,n,k e u m,n,k +v m,n,k u m+,n,k v m+,n+,k e u m,n,k u m+,n,k+, B m,n,k e v m,n,k v m+,n+,k, C m,n,k e u m+,n,k u m,n,k, for simplicity. In order to derive Lax pair for (54)-(55), we first shift k to k + in eq.(64), then substitute those ψ with index k + by k, m by m + by use of (63) and (64). Through some calculations we can see that the compatibility condition of (63)-(64) generates eqs. (54)-(55). So they constitute the Lax pair for (54)-(55). 5 Conclusion and discussions In this paper,we have presented two semi-discrete integrable versions and one fully discrete integrable version for the Leznov lattice. Corresponding BTs, nonlinear superposition formulae and Lax pairs are derived. Based on these obtained results, it is natural to further consider some other integrable properties such as determinant structures of N-soliton solutions for these semi-discrete and fully discrete integrable versions of the Leznov lattice. Besides, it would be interesting to work out their pfaffian versions for these semidiscrete and fully discrete Leznov lattice. Finally it is noted that in the paper the semi-discrete and fully discrete forms are always compared with bilinear equations (5)-(6). It is natural to inquire what are discrete analogues of ()-() and what are the discretizations of p and θ. In order to answer these, let us first consider y-directional discrete version of the Leznov lattice (7)-(8). In this case, we have the following equation, by use of (A3) with D m = δd y, δ sinh(δd y)[d x sinh(δd y )f f] [e +δdy f f] = sinh(d n + δd y )[e δdy+ f f] [e δdy f f]. (65) By the dependent variable transformation θ(n, x, y) = f(n +, x, y + δ)f(n, x, y δ), p(n, x, y) = D x sinh(δd y )f f f(n, x, y + δ)f(n, x, y δ) δ f(n +, x, y + δ)f(n, x, y δ) (66) 9

10 From (65) and (66), we can easily derive y-directional discretization of the Leznov lattice ()-() (p(n, x, y + δ) p(n, x, y δ)) = θ(n +, x, y + δ) θ(n, x, y δ) δ x δ sinh(δ y) ln θ(n, x, y) = θ(n +, x, y + δ)p(n +, x, y + δ) θ(n, x, y + δ)p(n, x, y + δ) θ(n, x, y δ)p(n, x, y δ) + θ(n, x, y δ)p(n, x, y δ). We now turn to consider x-directional discrete version of the Leznov lattice (33)-(34). In this case, by the following dependent variable transformation f(n +, x, y)f(n, x, y) θ(n, x, y) =, f(n, x, y)f(n, x, y) p(n, x, y) = D y sinh(ɛd x )f f f(n +, x, y), q(n, x, y) = ɛ f(n +, x + ɛ, y)f(n, x ɛ, y) f(n, x, y) and by use of (A38) with D n = ɛd x, we have from (33)-(34) the following x-directional discrete version of the Leznov lattice ()-() p y (n, x, y) = θ(n +, x + ɛ, y) θ(n, x ɛ, y) (67) q(n, x, y) θ(n, x, y) = q(n, x, y) (68) y ɛ sinh(ɛ q(n +, x + ɛ, y) x) ln θ(n, x, y) = p(n +, x, y) q(n, x ɛ, y) q(n, x + ɛ, y), x + ɛ, y) p(n, x, y) + p(n, x, y)q(n q(n, x ɛ, y) q(n, x ɛ, y). (69) Similarly we can also further consider fully discrete version of the Leznov lattice ()-() by the dependent variable transformation f(n +, x, y + δ)f(n, x, y δ) θ(n, x, y) = f(n, x, y + δ)f(n, x, y δ) f(n +, x + ɛ, y)f(n, x ɛ, y) r ɛ (n, x, y) = f(n +, x ɛ, y)f(n, x + ɛ, y), p(n, x, y) = ɛδ and by use of (A4) with D k = ɛd x and D m = δd y. In fact, from (49)-(50), we have sinh(ɛd x ) sinh(δd y )f f f(n +, x + ɛ, y + δ)f(n, x ɛ, y δ) [p(n, x, y + δ) p(n, x, y δ)] = θ(n +, x + ɛ, y + δ) θ(n, x ɛ, y δ) δ 4ɛδ[θ(n, x ɛ, y)p(n, x, y) θ(n +, x + ɛ, y)p(n +, x, y)] = r ɛ (n, x, y + δ) r ɛ (n, x, y δ) ɛ δ sinh(ɛ x) sinh(δ y ) ln θ(n, x, y) = 4ɛδ [ln r ɛ(n, x, y + δ) ln r ɛ (n, x, y) ln r ɛ (n, x, y) + ln r ɛ (n, x, y δ)]. It can be easily verified that when δ 0, the above system becomes (67)-(69). Acknowledgements The authors would like to thank the referee for valuable comments. This work was supported by the National Natural Science Foundation of China (grant no ), the knowledge innovation program of the Institute of Computational Math., AMSS and State Key Laboratory of Scientific and Engineering Computing, Chinese Academy of Sciences. Appendix A. Proofs of propositions, 4, 6 A. Proof of Proposition : 0

11 Analogous to the deduction in [] and [3], we can show that D z f f + (γ γ )f f κ e f 0 f = 0, (A) (D z λ e γ )f f = 0, (A) (D z λ e γ )f f = 0. (A3) (λ e Dm + e Dm+ + µ e Dm )f f = 0, (A4) (λ e Dm + e Dm+ + µ e Dm )f f = 0, (A5) Thus, in order to prove Proposition, it suffices to show that (D x + λ e λ γ e + λ D z e + β )f f = 0, (A6) (D x + λ e λ γ e + λ D z e + β )f f = 0. (A7) Since f and f are two solutions of (9), we have, by using (30), (A34)-(A37) and f 0 (λ i,µ i,γ i,β i) f i (i =, ), that Dm 0 = [(λ e Dm+ + e + µ e Dm Dm )f 0 f ][e f f 0 ] Dm [(λ e Dm+ + e + µ e Dm Dm )f 0 f ][e f f 0 ] = e Dm (e f0 f 0 ) [(λ e λ e Dm )f f ] + (e Dm+ f0 f 0 )[(µ e Dm µ e )f f ] Dm = (e f0 f 0 )[ edm Dm+ f0 f + µ e Dm f f µ e f f ], κ so we have Similarly, κ edm Dm f0 f = (µ e Dm+ µ e )f f. (A8) Dm 0 = [(e + Dm+ e + µ e D m Dm+ )f 0 f ][e f f 0 ] λ λ = Dm [(e + Dm+ e + µ e D m Dm+ )f 0 f ][e f f 0 ] λ λ e Dm (e f0 f 0 ) [(λ e λ e )f f ] + e (e D m f 0 f 0 ) [( µ e Dm µ e Dm )f f ] λ λ λ λ = e (e D m f 0 f 0 ) [ λ λ κ edm f 0 f + µ λ e Dm f f µ λ e Dm f f ], which means that κ edm f 0 f = (µ λ e Dm µ λ e Dm )f f. (A9)

12 Since f and f are two solutions of (0), we have, by use of (30),(A8), (A9),that 0 = [(D x e Dm D z e +Dm )f f ]e Dm f f [(D x e Dm D z e +Dm )f f ]e Dm f f = D x (e Dm f f ) (e Dm f f ) 4D z cosh( D n )(edm+ Dm f f ) (e f f ) = = = = +8 sinh( D n )(edm+ Dm f f ) (e f f ) D x [(µ λ e Dm µ λ e Dm )f f ] (e Dm f f ) µ λ 4 D z cosh( D n µ )[(µ e Dm+ Dm µ e Dm )f f ] (e f f ) + 8 sinh( D n µ )[(µ e Dm+ Dm µ e Dm )f f ] (e f f ) µ λ k D x(e Dm f 0 f ) (e Dm f f ) + 4 D z cosh( D n κµ )(edm+ Dm f0 f ) (e f f ) 8 sinh( D n kµ )(edm+ Dm f0 f ) (e f f ) µ λ κ D x(e Dm f 0 f ) (e Dm f f ) + 4 e Dm [(D z f 0 f ) (e f f ) + (f 0 f ) ( D z e f f ) κµ +(D z e f 0 f ) (f f ) (e f 0 f ) (D z f f )] 8 sinh( D n κµ )(edm+ f0 f ) (e e Dm (f 0 f ) [(γ e D z e e D x β )f f ] κµ λ λ Dm which implies that (A6) holds. Similarly, we can show that (A7) also holds. Therefore, we have completed the proof of Proposition. A. Proof of Proposition 4: Analogous to the deduction in [] and [3], we can show that D y f f + (µ µ )f f + c e f 0 f = 0, λ λ (A0) (D y + λ + µ e )f f = 0, (A) (D y + λ + µ e )f f = 0. (A) (D z e λ e + γ e )f f = 0, (A3) (D z e λ e + γ e )f f = 0. (A4) Thus, in order to prove Proposition 4, it suffices to show that (λ e +D k D z e +D k γ e +D k + β e D k )f f = 0, (A5) (λ e +D k D z e +D k γ e +D k + β e D k )f f = 0. (A6) f f ) From [(D z e λ e + γ e )f0 f ]f (n λ ) (A7) [(D z e λ e + γ e )f0 f ]f (n ) = 0, (A8) λ we can deduce that [4] [λ D z e (/) + λ D z e (/) λ γ e (/) + λ γ e (/) ]f f cd z e (/) f 0 f = 0.(A9) (λ i,µ i,γ i,β i) Since f and f are two solutions of (35)-(36), we have, by using (46), (A0), (A5), (A9), and f 0

13 f i (i =, ), that 0 = [(D y e D k D z e +D k )f f ]e D k f f [(D y e D k D z e +D k )f f ]e D k f f = sinh(d k )(D y f f ) (f f ) D z cosh( D n + D k)(e f f ) (e f f ) sinh( D n + D k)[(d z e f f ) (e f f ) (e f f ) (D z e f f )] = sinh(d k )(D y f f ) (f f ) + λ D z cosh( D n + D k)[(λ e λ e )f f ] (e f f ) λ sinh( D n + D k)[(λ D z e + λ D z e f f )] (e f f ) +(D z e f f ) [(λ e λ e )f f ] = c λ λ sinh(d k )(e f 0 f ) (f f ) + cd z cosh( D n + D k)(e f f ) (e f0 f ) 4cγ λ sinh( D n + D k)(e f0 f ) (e f f ) c sinh( D n λ + D k)[(d z e f0 f ) (e f f ) + (e f0 f ) (D z e f f )] = c e [λ λ (ed k f0 f ) (e D k f f ) λ (e D k f0 f ) (e D k f f ) +(e D k f0 f ) (D z e D k+ f f ) (D z e D k+ f0 f ) (e D k f f ) γ (e D k+ f0 f ) (e D k f f ) + γ (e D k f0 f ) (e D k+ f f )] = c e [e D k λ f0 f ] [( λ e +D k + D z e +D k + γ e +D k β e D k )f f ], which implies that (A5) holds. Similarly we can prove (A6) also holds. Therefore we have completed the proof of the propositions 4. A.3 Proof of Proposition 6: we can show that D z f f + (γ γ )f f kf 0 f = 0, Thus, in order to prove Proposition 6, it suffices to show that (A0) (µ λ e Dm µ λ e Dm )f f = ke Dm+ f 0 f (A) (D z e λ e γ e )f f = 0, (A) (D z e λ e γ e )f f = 0. (A3) (λ e Dm + e Dm+ + µ e Dm )f f = 0, (A4) (λ e Dm + e Dm+ + µ e Dm )f f = 0, (A5) From (D z e D k D m γ e D k D m + e Dm+ D k + β e +D k+d m )f f = 0, (A6) µ (D z e D k D m γ e D k D m + e Dm+ D k + β e +D k+d m )f f = 0. (A7) µ λ [(D z e λ e γ e )f0 f ]f (n + ) (A8) [(D z e λ e γ e )f0 f ]f (n + ) = 0, (A9) λ we can deduce that [4] kd z e f0 f + (λ D z e + λ D z e )f f + (λ γ e λ γ e )f f = 0. (A30) 3

14 (λ i,µ i,γ i,β i) Since f and f are two solutions of (57)-(59), we have, by using (6), (A), (A30), (59) and f 0 f i (i =, ), that 0 = { [sinh(d m ) sinh(d k ) D z e +Dm+D k ]f f } (e D m D k f f ) { [sinh(d m ) sinh(d k ) D z e +Dm+D k ]f f } (e D m D k f f ) = sinh(d k )(e Dm f f ) (e Dm f f ) D z cosh( D n + D m + D k )(e f f ) (e f f ) sinh( D n + D m + D k )[(D z e f f ) (e f f ) + (D z e f f ) (e f f )] = µ λ sinh(d k )(e Dm f f ) (µ λ e Dm µ λ e Dm f f ) λ D z cosh( D n + D m + D k )[(λ e λ e )f f ] (e f f ) sinh( D n λ + D m + D k ){[(λ D z e + λ D z e )f f ] (e f f ) +(D z e f f ) [(λ e λ e )f f ]} = k e { µ (e +Dm+D k f f ) (e D k+d m+ f0 f ) + µ λ (ed k+d m+ f f ) (e D k+d m+ f0 f ) γ (e Dm D k f f ) (e Dm+D k+ f0 f ) + γ (e Dm+D k+ f f ) (e Dm D k f 0 f ) +(D z e Dm D k f f ) (e +D k+d m f 0 f ) (e +D k+d m f f ) (D z e Dm D k f 0 f )} = k λ e {[(µ e D k+d m+ γ e Dm D k + D z e Dm D k + β e +Dm+D k )f f ] (e +Dm+D k f 0 f )}, which implies that (A6) holds. Similarly we can prove (A7) also holds. Therefore we have completed the proof of the Propositions 6. Appendix B. Hirota bilinear operator identities. The following bilinear operator identities hold for arbitrary functions a, b, c, and d. D y (D z e a a) (e a a) = sinh(d n )(D y D z a a) a. sinh(d m )(D z e Dm+ a a) (e Dm+ a a) = sinh(d n + D m )(D z e Dm a a) (e Dm a a). (A3) (A3) [D x e Dm a a][e Dm b b] (e Dm a a)(d x e Dm b b) = sinh( D m)(d x a b) ab. (A33) [e +Dm a a][e Dm b b] [e +Dm b b][e Dm a a] = sinh( D n)(e +Dm a b) (e Dm a b). (A34) sinh( D m)a a = 0. (A35) [e Dm a b][e Dm c a] = e Dm (e a a) (e b c) (A36) [D z e +Dm a a][e Dm b b] [D z e +Dm a a][e Dm b b] = D z cosh( D n)(e +Dm a b) (e Dm a b) +[e +Dm a a][d z e Dm b b] [D z e Dm a a][e Dm+ b b] (A37) sinh(d n + D k )(D y D z a a) a = D y (D z e +D k a a) (e +D k a a). (A38) 4

15 D z cosh( D n + D k )(e a b) (e a b) = sinh( D n)[(d z e +D k a b) (e D k a b) (e +D k a b) (D z e D k a b)] (A39) [D z e +D k a a][e D k b b] [D z e +D k b b][e D k a a] = D z cosh( D n + D k )(e a b) (e a b) (A40) + sinh( D n + D k )[(D z e a b) (e a b) (e a b) (D z e a b)] sinh(d m + D n + D k )(D z e Dm a a) (e Dm a a) = sinh(d m )(D z e Dm++D k a a) (e Dm++D k a a). (A4) [e Dm D k a a][e Dm+D k b b] [e Dm D k b b][e Dm+D k a a] = sinh(d k )(e Dm a b) (e Dm a b) (A4) [D z e +Dm+D k a a][e Dm+D k b b] [D z e +Dm+D k b b][e Dm+D k a a] = sinh( D n + D k + D m )(D z e a b) (e a b) (A43) sinh( D n)(e +D k+d m a b) (D z e D k D m a b) References [] Leznov A N.: Graded Lie algebras, representation theory, integrable mappings and integrable systems, Theor. Math. Phys. () -8 (000) [] M.Blaszak,K.Marciniak,J.Math.Phys.35 (994)466 [3] X.B.Hu,Z.N.Zhu,J.Math.Phys.39 (998) 4766 [4] B.A.Kupershmidt,Discrete Lax equations and differential-difference calculus, Asterisque (3),, (985). [5] Levi D, Vinet L, and Winternitz P. (ed.): Symmetries and Integrability of Difference Equations, CRM Proceedings & Lecture Notes, Vol. 9, American Mathematical Society, (996) [6] Clarkson P A and Nijhoff F W. (ed.): Symmetries and Integrability of Difference Equations, London Mathematical Society Lecture Note Series 55, Cambridge, New York, Cambridge University Press, (999) [7] Levi D and Ragnisco O.(ed.): SIDE III Symmetries and Integrability of Difference Equations, CRM Proceedings & Lecture Notes, Vol. 5, American Mathematical Society, (000) [8] Hietarinta J, Nijhoff F W, and Satsuma J.(ed.): Symmetries and integrability of difference equations, Institute of Physics Publishing, Bristol, (00) [9] Suris Y B.: The problem of integrable discretization: Hamiltonian approach. Progress in Mathematics, 9. Birkhäuser Verlag, Basel. (003) [0] Hirota R.: Nonlinear partial difference equations. I. A difference analogue of the Korteweg-de Vries equation. J. Phys. Soc. Japan (977) [] Hirota R.: Nonlinear partial difference equations. II. Discrete-time Toda equation. J. Phys. Soc. Japan (977) [] Hirota R.: Nonlinear partial difference equations. III. Discrete sine-gordon equation. J. Phys. Soc. Japan (977) [3] Hirota R.: Nonlinear partial difference equations. IV. Bäcklund transformations for the discrete-time Toda equation. J. Phys. Soc. Japan (978) [4] Hirota R.: Discrete analogue of a generalized Toda equation. J. Phys. Soc. Japan (98) 5

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