On Generating Discrete Integrable Systems via Lie Algebras and Commutator

Commun. Theor. Phys. 65 (216 335 34 Vol. 65, No. 3, March 1, 216 On Generating Discrete Integrable Systems via Lie Algebras and Commutator Equations Yu-Feng Zhang ( ô 1 and Honwah Tam 2 1 College of Sciences, China University of Mining and Technology, Xuzhou 221116, China 2 Department of Computer Science, Hong Kong Baptist University, Hong Kong, China (Received November 4, 215 Abstract In the paper, we introduce the Lie algebras and the commutator equations to rewrite the Tu-d scheme for generating discrete integrable systems regularly. By the approach the various loop algebras of the Lie algebra A 1 are defined so that the well-known Toda hierarchy and a novel discrete integrable system are obtained, respectively. A reduction of the later hierarchy is just right the famous Ablowitz Ladik hierarchy. Finally, via two different enlarging Lie algebras of the Lie algebra A 1, we derive two resulting differential-difference integrable couplings of the Toda hierarchy, of course, they are all various discrete expanding integrable models of the Toda hierarchy. When the introduced spectral matrices are higher degrees, the way presented in the paper is more convenient to generate discrete integrable equations than the Tu-d scheme by using the software Maple. PACS numbers: 5.45.Yv, 2.3.Jr, 2.3.Ik Key words: discrte integrable system, Lie algebra, integrable coupling 1 Introduction In recent years, search for discrete integrable systems and their solutions, symmetries, Hamiltonian structures, Bäcklund transformations, conservation laws, and so on, has made rapidly developed. [1 16] A common approach for generating discrete integrable systems usually starts from the following discrete spectral problem Eψ = Uψ, (1 where ψ = (ψ 1,..., ψ N T is an N-vector and U = U(u, t, λ is an N N matrix which is dependent of a field vector u = (u 1,..., u p T, the time variable t and a spectral parameter λ, and Ef(n, t = f(n+1, t. To generate differential-difference integrable systems, a t-evolution part corresponding to Eq. (1 is introduced for some matrix V as follows ψ t = V ψ. (2 The compatibility condition of Eqs. (1 and (2 gives rise to a differential-difference equation U t = (EV U UV, (3 which is called a discrete zero-curvature equation. In terms of the scheme called the Tu-d scheme, [3] we should need to introduce a modified term for V if necessary, and denote by V (n = V + so that the discrete zerocurvature equation U tn = (EV (n U UV (n (4 leads to novel integrable discrete hierarchies. It is easy to see that Eq. (4 is the compatibility condition of the Lax pair Eψ = Uψ, ψ tn = V (n ψ. (5 Compared with the Tu scheme, [17] there is no commutator in the discrete zero-curvature equations (3 or (4. If we could construct a commutator appearing in Eq. (4, then we would follow the very familiar Tu scheme to generate discrete integrable systems, that is, we could imitate the well-known ideas of Tu scheme to investigate discrete integrable systems. An obvious difference between the Tu scheme and the Tu-d scheme reads that we could regularly construct the U and the V in the Lax pair (1 and (2 through Lie algebras. That is to say, set G to be a Lie algebra, and {e 1,..., e p } is a basis of G. Assume again G = G C([λ, λ 1 ], where C[λ, λ 1 ] represents the set of Laurent polynomials in λ. A basis of G is denoted by {e 1 (n,..., e p (n, n Z}. An element R G is called pseudo-regular if for ker ad R = {x x G, [x, R] = }, Im ad R = {x y G, x = [y, R]}, it holds that G = ker adr Im ad R, and ker ad R is commutative. In addition, we define gradations of G to be as follows Assume deg (X λ n = n, X G. U = R + u 1 e 1 + + u p e p, where u i (i = 1,...,p are component potential functions of the function u = (u 1,..., u p T. Denote α = deg(r, Supported by the National Natural Science Foundation of China under Grant No. 11371361, the Innovation Team of Jiangsu Province hosted by China University of Mining and Technology (214, and Hong Kong Research Grant Council under Grant No. HKBU22512, as well as the Natural Science Foundation of Shandong Province under Grant No. ZR213AL16 c 216 Chinese Physical Society and IOP Publishing Ltd http://www.iopscience.iop.org/ctp http://ctp.itp.ac.cn

336 Communications in Theoretical Physics Vol. 65 ǫ i = deg(e i, i = 1,...,p. If α and ǫ i satisfy α >, α > ǫ i (i = 1,...,p, (6 then the stationary zero-curvature equation V x = [U, V ] (7 could have local solutions for the given spectral matrix U. Thus, under introducing the modified term of (λ n V + which is denoted by V (n = (λ n V + +, the continuous zero-curvature equation U t V (n x + [U, V (n ] = (8 generally could give rise to integrable hierarchies of evolution equations. Therefore, Eq. (6 is a guidance clue to construct U and V in Eqs. (6 and (7 so that Eq. (7 could have differential solutions for V, and Eq. (8 could acquire integrable ideal equations. In order to make the Tu-d scheme match the Tu scheme as possible, we rewrite the Tu-d scheme according to the Tu scheme. First of all, a proper Lie algebra G and its loop algebra G are introduced. Second, we apply the G to introduce U and V in Eqs. (1 and (2. Third, we solve a stationary zerocurvature equation similar to Eq. (7: ( V U = [U, V ], (9 where = E 1, [U, V ] = UV V U. Taking a modified term n for (V λ n +, denoted by V (n = (V λ n + + n, the discrte zero-curvature equation U tn = ( V (n U [U, V (n ] (1 could lead to novel differential-difference equations. Actually, Eq. (1 is a rewritten formula of Eq. (4. Finally, with the help of the discrete trace identity proposed by Tu, [3] we can derive the Hamiltonian structure of Eq. (1. In what follows, we shall apply the above version to some explicit applications by introducing two various loop algebras of a Lie algebra. 2 Generating Two Discrete Integrable Hierarchies The simplest basis of the Lie algebra A 1 reads ( ( 1 h 1 =, h 2 =, 1 ( ( 1 e =, f =, 1 with the commutative relations as follows [h 1, e] = e, [h 1, f] = f, [h 2, e] = e, [h 2, f] = f, [e, f] = h h 1 h 2. (11 It follows from Eq. (11 that [h, e] = 2e, [h, f] = 2f, h 1 h 1 = h 1, h 2 h 2 = h 2, h 1 h 2 = h 2 h 1 = ee = ff =, h 1 e = e, eh 1 =, h 1 f =, fh 1 = f, h 2 f = f, fh 2 =, h 2 e =, eh 2 = e, ef = h 1, fe = h 2. (12 We denote by G the above Lie algebra, that is, G = span{h 1, h 2, e, f} equipped with the commutative relations (11 and (12. 2.1 A Loop Algebra of the Lie Algebra G and the Toda Hierarchy A loop algebra of the Lie algebra G is the well-known form as follows G 1 = span{h 1 (n, h 2 (n, e(n, f(n}, along with degrees deg h i (n = deg e(n = deg f(n = n, i = 1, 2, where h i (n = h i λ n, e(n = eλ n, f(n = fλ n, i = 1, 2; n Z. We consider an isospectral problem by using G 1 Eϕ = Uϕ, U = h 2 (1 ph 2 (+e( vf(, ϕ t = Γϕ, Γ = n (a n h 1 ( n a n h 2 ( n+b n e( n+c n f( n. (13 A set of solutions to Eq. (9 for V is given by c n = vb (1 n, a(1 n +a n +b (1 n+1 pb(1 n =, va(1 n = c n+1 va n pc n, c (1 n a(1 n+1 +a n+1 +p(a (1 n a n = vb n, (14 where x (i n = E i x n, x = a, b, c; i = 1, 2,... Denote by Γ + = (a n (h 1 (m n h 2 (m n + b n e(m n + c n f(m n = λ m Γ Γ, n= then the stationary zero-curvature equation ( ΓU = [U, Γ] can be decomposed into the following form ( Γ + U [U, Γ + ] = ( Γ U + [U, Γ ]. (15 We observe that the left-hand side of (15 contains terms with degree more than, while the right-hand side contains terms with degree less than. Hence both sides of Eq. (15 contain only terms with degree being. Therefore, we have ( Γ + U [U, Γ + ] = ( b (1 m+1 + b m+1e( + (a (1 m+1 a m+1h 2 ( + c m+1 f(. Denoting by V (m = Γ + + m, m = b m+1 h 1 (, a direct calculation acquires ( V (m U [U, V (m ] = ( a m+1 h 2 ( + (c m+1 + vb m+1 f(. Thus, the discrete zero-curvature equation (1 permits

No. 3 Communications in Theoretical Physics 337 the lattice hierarchy p tm = a m+1, v tm = c m+1 vb m+1 = vb (1 m+1 vb m+1, (16 which is completely consistent with that in Ref. [3], the well-known Toda hierarchy. Remark 1 It is easy to find that the pseduo-regular element in Eq. (13 is h 2 (1, whose degree reads degh 2 (1 = 1, which is more than other elements, satisfying the condition (6. In addition, Eq. (15 is similar to the decomposed equation in the Tu scheme ( λ n V +x + [U, (λ n V + ] = (λ n V x [U, (λ n V ]. (17 The above steps for computing the lattice hierarchy (16 are completely same with that by the Tu scheme, which hints that we could imitate all thoughts of Tu scheme to generate lattice hierarchies by introducing various Lie algebras and their resulting loop algebras. 2.2 Another Loop Algebra of the Lie Algebra G and a New Lattice Hierarchy as Another loop algebra of the Lie algebra G is defined G 2 = span{h 1 (n, h 2 (n, e(n, f(n}, where degh i (n = deg e(n = degf(n = 2n + p, i = 1, 2; p = 1,. An explicit loop algebra still denoted by G 2 satisfying the above requirements is given by G 2 = span{h i (n = h i λ 2n, e(n = eλ 2n 1, f(n = fλ 2n 1, i = 1, 2}, which possesses the following operating relations h i (mh i (n = h i (m + n, i = 1, 2, h i (mh j (n = e(me(n = f(mf(n =, i j, h 1 (me(n = e(m + n, e(mh 1 (n = h 1 (mf(n =, f(mh 1 (n = f(m + n = h 2 (mf(m, f(mh 2 (n = h 2 (me(n =, e(mh 2 (n = e(m + n, e(mf(n = h 1 (m + n 1, f(me(n = h 2 (m + n 1, [h 1 (m, e(n] = e(m + n, [h 1 (m, f(n] = f(m + n, [h 2 (m, e(n] = e(m + n, [h 2 (m, f(n] = f(m + n, from the above appearances we can derive that [h(m, e(n] = 2e(m + n, [h(m, f(n] = 2f(m + n, [e(m, f(n] = h(m + n 1 = h 1 (m + n 1 h 2 (m + n 1, mn Z. Obviously, the loop algebra G 2 is different from the previous G 1. In the following, we shall apply the loop algebra G 2 to investigate a discrete integrable hierarchy. Set U = h 1 (1 + qh 2 ( + re(1 + sf(1, Γ = n [a n (h 1 ( n h 2 ( n+b n e( n+c n f( n].(18 A direct calculation according to the stationary discrete equation ( ΓU [U, Γ] = (19 gives that a n+1 + sb (1 n = rc n, q a n = rc (1 n sb n, r(a (1 n+1 + a n+1 = b n+1 qb (1 n, c (1 n+1 s(a(1 n+1 + a n+1 = qc n. (2 The first equation in Eq. (2 can be derived from other three ones. In fact, we have q(a (1 n+1 + a n+1 = rc (1 n+1 sb n+1 = r[qc n + s(a (1 n+1 + a n+1] s[qb (1 n + r(a (1 n+1 + a n+1] = qrc n qsb (1 n. Denoting by Γ + = [a n h( n + b n e( n + c n f( n]λ 2m n= =λ 2m Γ Γ, then Eq. (19 can be decomposed into ( Γ + U [U, Γ + ] = ( Γ U + [U, Γ ]. (21 The degrees of the left-hand side of Eq. (21 are more than 1, while the right-hand side less than. Therefore, both sides should be 1,. Thus, Eq. (21 permits that ( Γ + U [U, Γ + ] = ( a m+1 h 1 ( + [b m+1 2ra m+1 ( a m+1 r]e( + [ c m+1 + 2sa m+1 + s( a m+1 c m+1 ]f(. We take a modified term m = δh 2 ( b m e( c m f(, and denote by V (m = Γ + + m, it can be computed that ( V (m U [U, V (m ] = (q δ rc (1 m + sb m h 2 ( + (b m rδe(1 + (s δ c (1 m + sδf(1. Therefore, Eq. (1 admits that q tm = q δ rc (1 m + sb m, r tm = b m rδ, s tm = sδ (1 c (1 m, (22 where δ is an arbitrary function with respect to m, t. Some reductions of Eq. (22 can be considered. Case 1 Taking q =, δ = a m, then Eq. (22 reduces to the famous Ablowtiz Ladik hierarchy r tm = b m ra m, s tm = sa m c (1 m. Case 2 Taking δ =, then Eq. (22 becomes q tm = q a m, r tm = b m, s tm = c (1 m, which is just the main result in Ref. [11].

338 Communications in Theoretical Physics Vol. 65 Case 3 Taking δ = a (1 m + a m, then Eq. (22 can reduce to q tm = 2qa m, r tm = qb (1 m 1, s tm = c (1 m + s(a (1 m + a (2 m. (23 Again set q =, Eq. (23 reduces to a new lattice hierarchy: s tm = c (1 m + s(a(1 m + a(2 m. As similar to the case where the Hamiltonian structure of the Toda hierarchy (16 was derived from the discrete trace identity in Ref. [3], the Hamiltonian structures of Eqs. (22 and (23 could be investigated by the discrete trace identity, here we do not further discuss them. 3 Linear and Nonlinear Discrete Integrable Models of Toda Hierarchy As we know that some continuous expanding integrable models of the known integrable systems, such as the AKNS system, the KN system,the KdV system, and so on, were obtained by enlarging the Lie algebra A 1, e.g. see Refs. [18 2]. In what follows, we want to extend the approach to the case of discrete integrable hierarchies. That is, we extend the Tu scheme for generating continuous expanding integrable models to the case by introducing commutators, as presented above, so that a great number of discrete expanding integrable systems could be readily generated just like generating expansion integrable models of continuous integrable systems. In the section, we only investigate the linear and nonlinear discrete expanding integrable models of the Toda hierarchy so that our method will be illustrated. 3.1 A Linear Discrete Integrable Coupling Set Ref. [18] 1 h 1 =, h 2 = 1, 1 e =, f = 1, 1 g 1 =, g 2 = 1, and denote L by L = span{h 1, h 2, e, f, g 1, g 2 }. Assume L 1 = span{h 1, h 2, e, f}, L 2 = span{g 1, g 2 }, then it is easy to see that L = L 1 L 2, L 1 = G, [L1, L 2 ] L 2, [L 1, L 1 ] L 1, [L 2, L 2 ] = {}. Obviously, the linear space L is an enlarging Lie algebra of the Lie algebra G presented before. Since the Lie subalgebra L 1 is isomorphic to the Lie algebra G, then of course have the common operation relations. Therefore, we only consider the operating relations among {g 1, g 2 } with {h 1, h 2, e, f}. It is easy to compute by Maple that g 1 g 1 = g 2 g 2 = g 1 g 2 = g 2 g 1 =, h 1 g 1 = g 1, g 1 h 1 =, h 2 g 1 = g 1 h 2 =, eg 1 = g 1 e =, fg 1 = g 2, g 1 f = h 1 g 2 = g 2 h 1 =, h 2 g 2 = g 2, g 2 h 2 =, eg 2 = g 1, g 2 e =, fg 2 = g 2 f =, [h 2, g 1 ] =, [h 2, g 2 ] = g 2, [e, g 1 ] =, [f, g 1 ] = g 2, [e, g 2 ] = g 1, [f, g 2 ] =, [h 1, g 1 ] = g 1, [h 1, g 2 ] =. (24 A loop algebra of the Lie algebra L is defined as L = span{h 1 (n, h 2 (n, e(n, f(n, g 1 (n, g 2 (n}, where degx(n = n, X L. Applying the loop algebra L introduces the following isospectral problems Eψ = Uψ, U = h 2 (1 ph 2 ( + e( vf( + qg 1 ( + rg 2 (,(25 ψ t = Γψ, Γ = n [a n (h 1 ( n h 2 ( n + b n e( n + c n f( n + d n g 1 ( n + e n g 2 ( n]. (26 The stationary discrete zero-curvature equation (9 permits the following equatiosn by using Eqs. (11 (12 and (24 (26: c n = vb (1 n, a(1 n + a n + b (1 n+1 pb(1 n =, va (1 n = c n+1 va n pc n, c (1 n a(1 n+1 + a n+1 + p a n = vb n, a (1 n q + b(1 n r = e n, e n+1 = a (1 n r c(1 n q vd n + pe n. (27 The first four equations in Eq. (27 are the same with Eq. (14. Set Γ + = [a n (h 1 ( n h 2 ( n + b n e( n + c n f( n n= + d n g 1 ( n + e n g 2 ( n]λ 2m = λ 2m Γ Γ, similar to the previous discussions, we obtain that ( Γ + U [U, Γ + ] = b (1 m+1 e( + ( a m+1 h 2 ( + c m+1 f( + e m+1 g 2 (. Taking V (m = Γ + + m, m = b m+1 h 1 (, a direct computation yields that ( V (m U [U, V (m ] = ( a m+1 h 2 ( + (c m+1 + vb m+1 f( + b (1 m+1 g 1( + e m+1 g 2 (. Thus, the discrete zero-curvature equation (1 permits the integrable hierarchy p tm = a m+1, v tm = vb (1 m+1 vb m+1, q tm = qb (1 m+1, r t m = e m+1. (28 When we take q = r =, Eq. (28 just reduces to the well-known Toda hierarchy (16. Therefore, Eq. (28 is an integrable coupling of the Toda hierarchy, of course, also

No. 3 Communications in Theoretical Physics 339 a discrete integrable expanding model of the Toda hierarchy. In what follows, we deduce a discrete integrable coupling of the Toda equation. For the sake, we take b = c =, a = 1 2, b 1 = 1, d n =, n =, 1, 2,... According to Eq. (27, we have a 1 =, a 2 = v, b 2 = p ( 1, a 3 = v(p + p ( 1, b 3 = v ( 1 v (p ( 1 2, e = 1 2 q, e 1 = 1 (q + r, 2 c 1 = v, e 2 = v (1 r v (1 q + 1 2 p(q + r, c 2 = vp, e 3 = v (1 r v (1 p (1 q + pv (1 r pv (1 q + 1 2 p2 (q + r,... Therefore, we get a discrete integrable coupling of the Toda equation as follows p t2 = [v(p + p ( 1 ], v t2 = v [v ( 1 + v + (p ( 1 2 ], q t2 = q[v ( 1 + v + (p ( 1 2 ], r t2 = (p + 1v (1 r ( pv (1 q + 1 2 p2 (q + r. (29 It is easy to see that Eq. (29 is a linear discrete integrable coupling with respect to the new variables q and r. Therefore, Eq. (28 is known as a linear discrete integrable coupling of the Toda hierarchy. 3.2 A Nonlinear Discrete Integrable Coupling To search for nonlinear discrete integrable coupling of the Toda hierarchy, we should enlarge the Lie algebra G to a Lie subalgebra of the Lie algebra A 3. Therefore, we set ( ( h1 h2 H 1 =, H 2 =, h 1 ( e E = e ( h1 T 1 = h 1 ( e T 3 = e, F =, T 2 =, T 4 = where h 1, h 2, e, f A 1. We denote h 2 ( f, f ( h2, h 2 ( f f, Q = span{h 1, H 2, E, F, T 1, T 2, T 3, T 4, T 5 }, Q = Q 1 Q 2, here Q 1 = span{h 1, H 2, E, F }, Q 2 = span{t 1, T 2, T 3, T 4 }. If equipped with an operation for Q 1 as follows [X, Y ] = XY Y X, X, Y Q 1, then Q 1 is a Lie algebra which is isomorphic to the previous Lie algebra G. Hence, Q 1 and G possess the same commutative relations. It is easy to compute by Maple that T 1 T 1 = T 1, T 2 T 2 = T 2, T 1 T 2 = T 2 T 1 = T 3 T 3 = T 4 T 4 =, T 3 T 4 = T 1, T 4 T 3 = T 2, T 1 T 3 = T 3, T 3 T 1 = T 1 T 4 =, T 4 T 1 = T 4, T 2 T 3 =, T 3 T 2 = T 3, T 2 T 4 = T 4, T 4 T 2 =, H 1 T 1 = T 1, T 1 H 1 = T 1, H 1 T 2 = T 2 H 1 =, H 1 T 3 = T 3, T 3 H 1 =, H 1 T 4 =, T 4 H 1 = T 4, H 2 T 1 = T 1 H 2 =, H 2 T 2 = T 2 H 2 = T 2, H 2 T 3 =, T 3 H 2 = T 3, H 2 T 4 = T 4, T 4 H 2 =, ET 1 =, T 1 E = T 3, ET 2 = T 3, T 2 E =, ET 3 = T 3 E =, ET 4 = T 1, T 4 E = T 2, FT 1 = T 4, T 1 F =, FT 2 =, T 2 F = T 4, FT 3 = T 2, T 3 F = T 1, FT 4 = T 4 F =, [T 4, H 1 ] = T 4, [T 4, T 1 ] = T 4, [T 4, T 2 ] = T 4, [T 3, T 4 ] = T 1 T 2, [T 3, T 1 ] = T 3, [T 3, T 2 ] = T 3, [H 2, T 3 ] = T 3, [H 2, T 4 ] = T 4, [E, T 3 ] =, [E, T 4 ] = T 1 T 2, [F, T 3 ] = T 2 T 1, [F, T 4 ] =, [H 1, T 3 ] = T 3, [H 1, T 4 ] = T 4, [T 3, T 4 ] = T 1 T 2, [F, T 3 ] = T 2 T 1, [E, T 1 ] = T 3, [E, T 2 ] = T 3, [F, T 1 ] = T 4, [F, T 2 ] = T 4,... From the above computations, we find that [Q 2, Q 2 ] Q 1, [Q 1, Q 2 ] Q 2, which is apparently different from that of L 1 and L 2 presented before. Actually, the Lie algebra L is a Lie algebra of a homogeneous space of a Lie group, while the Q is a Lie algebra of a symmetric space of a Lie group. A corresponding loop algebra of the Lie algebra Q is defined as Q = span{h 1 (n, H 2 (n, E(n, F(n, T 1 (n, T 2 (n, T 3 (n, T 4 (n}, where degx(n = n, X Q. In what follows, we consider the isospectral problems by using the loop algebra Q: Eϕ = Uϕ, U = (λ ph 2 ( + E( vf( + ut 3 ( + st 4 ( + wt 2 (, (3 ψ t =Γψ, Γ= n [a n (H 1 ( n H 2 ( n + b n E( n + c n F( n + d n T 3 ( n + g n T 4 ( n + q n (T 1 ( n T 2 ( n], (31 where the symbol E in the first equation of Eq. (3 stands for forward operator Ef(n = f(n + 1, which is different from the E in U and Γ in Eqs. (3 and (31. The compatibility condition of Eqs. (3 and (31 can be written as the form (1 along with a commutator operation, the corresponding stationary discrete zero-curvature equation (9 directly leads to c n = vb (1 n, a (1 n + a n + b (1 n+1 pb(1 n =, va (1 n = c n+1 va n pc n,

34 Communications in Theoretical Physics Vol. 65 c (1 n sb (1 n a(1 n+1 + a n+1 + p a n = vb n, uc(1 n vd(1 n g(1 n = ug n 2sd n, d (1 n+1 + u(a(1 n + a n pd (1 n + q n (1 (1 + u +g n (1 + u + w( b n + d n + b n + d n =, g (1 n+1 + s(a(1 n + a n = pg n (1 + (v sq(1 n +(v sq n w(c n + g n, q n+1 + p( q n + u( g n w (a n + q n =. (32 Similar to the previous discussions, we denote by Γ + = [a n (H 1 (m n H 2 (m n + b n E(m n n= + c n F(m n + d n T 3 (m n + g n T 4 (m n + q n (T 1 (m n T 2 (m n] = λ m Γ Γ, after calculations one infers that ( Γ + U [U, Γ + ] = (b m+1 b (1 m+1 E( + ( a m+1 H 2 ( + c m+1 F( d (1 n+1 T 3( + g (1 m+1 T 4( + ( q m+1 T 2 (. Taking V (m = Γ + + b m+1 H 1 (, we can compute that ( V (m U [U, V (m ] = ( a m+1 H 2 ( + (c m+1 + vb m+1 F( + ( q m+1 T 2 ( + ( d (1 m+1 ub m+1t 3 ( + (g (1 m+1 + sb m+1t 4 (. Hence, the discrete zero-curvature equation (1 admits p tm = a m+1, v tm = vb (1 m+1 vb m+1 = v b m+1, u tm = d (1 m+1 ub m+1, s tm = g (1 m+1 + sb m+1, w tm = g m+1. (33 When u = s = w =, Eq. (33 reduces to the Toda hierarchy. Hence, it is a discrete expanding integrable hierarchy of the Toda hierarchy, of course, a kind of discrete integrable coupling of the Toda hierarchy. In order to further recognize Eq. (33, we consider its a simple reduction when we take m = 1. Given the initial values b = c = d = g =, a = 1/2, b 1 = 1, c 1 = v, Eq. (33 reduces to p t1 = v (1 + v, v t1 = vp + vp ( 1, u t1 = pu ( 1 2uw + wu ( 1 w + up ( 1, s t1 = ps + ws ( 1 wv sp ( 1, w t1 = us ( 1 su, which is a set of nonlinear differential-difference equations with respect to the new variables u, s, w. Therefore, we conclude that Eq. (33 is a nonlinear discrete integrable coupling of the Toda hierarchy, which is completely different from the discrete integrable coupling (28. Remark 2 With the help of EQs. (9 and (1, generating discrete integrable hierarchies seems more complicated than the Tu-d scheme. However, we find that when the spectral matrices U and V in (1 and (2 are higher degrees, it is more convenient to deduce discrete integrable systems than the Tu-d scheme with the aid of software Maple. In addition, we do not again discuss the Hamiltonian structure of Eq. (33 in the paper. References [1] M.J. Ablowitz and J.F. Ladik, J. Math. Phys. 16 (1975 598. [2] I. Merola, O. Ragnisco, and G.Z. Tu, Inverse Problems 1 (1994 1315. [3] G.Z. Tu, J. Phys. A 23 (199 393. [4] M. Bruschi, O. Ragnisco, P.M. Sanitini, and G.Z. Tu, Physica D 4 (1991 273. [5] H.W. Zhang, G.Z. Tu, W. Oevel, and B. Fuchssteiner, J. Math. Phys. 32 (1991 198. [6] C.W. Cao, X.G. Geng, and Y.T. Wu, J. Phys. A 32 (1999 859. [7] C.W. Cao and X.X. Xu, Commun. Theor. Phys. 58 (212 469. [8] C.W. Cao and X. Yang, J. Phys. A 41 (28 2523 (19pp. [9] X.G. Geng, H.H. Dai, and J.Y. Zhu, Stud. Appl. Math. 118 (27 281. [1] X.G. Geng, J. Math. Phys. 44 (23 4573. [11] E.G. Fan and Z.H. Yang, Int. J. Theor. Phys. 48 (29 1. [12] Z.N. Zhu and H.W. Tam, J. Phys. A 37 (24 3175. [13] D.J. Zhang and D.Y. Chen, J. Phys. A 35 (22 7225. [14] Y.B. Suris, J. Phys. A 3 (1997 2235. [15] W.X. Ma and B. Fuchsteiner, J. Math. Phys. 4 (1999 24. [16] Z.J. Qiao, Chin. Sci. Bull. 44 (1999 114. [17] G.Z. Tu, J. Math. Phys. 3 (1989 33. [18] Y.F. Zhang and H.Q. Zhang, J. Math. Phys. 43 (22 466. [19] W.X. Ma and M. Chen, J. Phys. A 39 (26 1787. [2] W.X. Ma, J.H. Meng, and H.Q. Zhang, Global J. Math. Sciences 1 (212 1.