A new method to construct the integrable coupling system for discrete soliton equation with the Kronecker product

Physics Letters A 372 (2008) 3548–3554 www.elsevier.com/locate/pla

A new method to construct the integrable coupling system for discrete soliton equation with the Kronecker product Fajun Yu a,b,∗ , Li Li a a College of Mathematics and Systematic Science, Shenyang Normal University, Shenyang 110034, PR China b Department of Applied Mathematics, Dalian University of Technology, Dalian 116024, PR China

Received 12 November 2007; received in revised form 15 January 2008; accepted 7 February 2008 Available online 14 February 2008 Communicated by A.R. Bishop

Abstract It is shown that the Kronecker product can be applied to construct a new integrable coupling system of discrete soliton equation hierarchy in this Letter. A direct application to the generalized Toda lattice spectral problem leads to a novel integrable coupling system. It is also indicated that the study of integrable couplings by using of the Kronecker product is an efficient and straightforward method. © 2008 Elsevier B.V. All rights reserved. PACS: 02.30.Ik; 02.30.Jr Keywords: Kronecker product; Integrable coupling system; Discrete soliton equation hierarchy

1. Introduction In recent years there has been widespread interest in the study of integrable nonlinear lattice systems. It is well known that such lattice systems have many applications in science, e.g., in mathematical physics, numerical analysis, statistical physics, and quantum physics. Integrable coupling system is an interesting and important topic in soliton theory. The theory of integrable couplings brings other interesting results such as Lax pairs of block form and several spectral parameters [1–3,7], integrable constrained flows with higher multiplicity [4], local bi-Hamiltonian structures in higher dimensions [5] and hereditary recursion operators of higher order [1,6]. Integrable coupling system is presented by using Virasoro symmetry algebra [7]. In details, let ut = K(u) be a known integrable system, the following system  ut = K(u), vt = S(u, v),

(1)

(2)

is called an integrable coupling of the system (1), if vt = S(u, v) is also integrable and S(u, v) contains explicitly u or u-derivatives with respect to x. A few ways to construct integrable couplings are presented by using perturbations [8], enlarging spectral problems [9,10], and creating new loop algebras [11,12]. Profs. Ma, Xu and Zhang once obtained a discrete integrable couplings through the semi-direct * Corresponding author at: College of Mathematics and Systematic Science, Shenyang Normal University, Shenyang 110034, PR China. Tel.: +86 13478667302; fax: +86 411 81313446. E-mail address: [email protected] (F. Yu).

0375-9601/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2008.02.012

F. Yu, L. Li / Physics Letters A 372 (2008) 3548–3554

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sums of Lie algebras [13]. In Ref. [14], Ma proposed a beautiful method to generate Hamiltonian structures for integrable couplings associated with semi-direct sums of Lie algebras of the type. In [15] Ma and Guo considered the Lax representations and zerocurvature representations by the Kronecker product, however they did not obtain the integrable coupling system. We will show that the Kronecker product is an important and effective method to construct the discrete integrable couplings. In this Letter, we would like to present a relation between the Kronecker product and integrable couplings of discrete soliton equations. A feasible way to generate integrable couplings is presented through the Kronecker product. The general theory will be used to construct a novel class of integrable couplings of the nonlinear lattice soliton equation hierarchy. It will indicate that the study of integrable couplings using the Kronecker product is an important step towards constructing integrable systems. Example shows that the integrable couplings of a generalized Toda lattice hierarchy is presented. 2. Kronecker product Kronecker product, also known as a direct or a tensor product, is a concept having its origin on group theory and has important applications in particle physics. There are numerous applications of the Kronecker product in various fields including statistics, economics, control and matrix equation. Consider a matrix A = [aij ] of order (m × n) and a matrix B = [bij ] of order (r × s). The Kronecker product of the two matrices, denoted by A ⊗ B is defined as the partitioned matrix ⎞ ⎛ a11 B a12 B · · · a1n B ⎜ a21 B a22 B · · · a2n B ⎟ ⎜ . . (3) .. .. ⎟ ⎝ .. . . ⎠ am1 B

am2 B

· · · amn B

A ⊗ B is seen to be a matrix of order (mr × ns). It has mn blocks, the (ij )th block is the matrix aij B of order (r × s). We expect the Kronecker product to have the usual properties of a product. (I) If α is a scalar, then A ⊗ (αB) = α(A ⊗ B).

(4)

(II) The product is distributive with respect to addition, that is (A + B) ⊗ C = A ⊗ C + B ⊗ C,

(5)

A ⊗ (B + C) = A ⊗ B + A ⊗ C.

(6)

(III) The product is associative A ⊗ (B ⊗ C) = (A ⊗ B) ⊗ C.

(7)

(VI) The ‘Mixed Product Rule’ (A ⊗ B)(C ⊗ D) = AC ⊗ BD.

(8)

It is an important result in our constructing integrable coupling system. (VII) Given A(m × m), B(n × n) and subject to the existence of the various inverses, (A ⊗ B)−1 = A−1 ⊗ B −1 .

(9)

(VIII) Given the two matrices A and B of order n × n and m × m, respectively, |A ⊗ B| = |A|m |B|n .

(10)

3. A new integrable coupling system of discrete soliton equation with the Kronecker product Let IN denote the unit matrix of order N , N ∈ Z. For two matrices A = (aij )pq , B = (bkl )rs , the Kronecker product A ⊗ B is defined by A ⊗ B = (aij B)(pr)×(qs) , or equivalently

(11)

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F. Yu, L. Li / Physics Letters A 372 (2008) 3548–3554

(A ⊗ B)ij,kl = aik bj l , 





aip cpk bj q dql (A ⊗ B)(C ⊗ D) ij,kl = (AC ⊗ BD)ij,kl = p

(12)

q

provided that the matrices AC and BD make sense. The relation will be used to construct a new integrable coupling system of soliton equation hierarchy. Assume that an integrable model has two discrete integrable model with zero curvature representation U1t − (EV1 )U1 + U1 V1 = 0, U2t − (EV2 )U2 + U2 V2 = 0,

(13)

where U1 , V1 and U2 , V2 are M × M and N × N matrices, respectively. Prof. Ma defined U3 and V3 [15] U3 = U1 ⊗ U2 ,

V3 = V1 ⊗ IN + IM ⊗ V2 .

(14)

Then the same integrable model has another discrete zero curvature representation U3t − (EV3 )U3 + U3 V3 = 0.

(15)

However, Eq. (15) is not suit for the definition of integrable couplings. We will give a new pair U3 and V3 by Kronecker product, which can produce the integrable coupling system. Let U3 and V3 , the new Lax pair form be presented as follows U3 = IN ⊗ U1 + TN ⊗ U2 ,

V3 = IN ⊗ V1 + TN ⊗ V2 ,

(16)

where TN is a matrix of N × N , the elements are zero besides the top right corner. Through the computation, we get (EV3 )U3 − U3 V3 = IN2 ⊗ (EV1 )U1 − IN2 ⊗ U1 V1 + IN TN ⊗ (EV1 )U2 − IN TN ⊗ U1 V2 + TN IN ⊗ (EV2 )U1 − IN TN ⊗ U2 V1 + TN2 ⊗ (EV2 )U2 − TN2 ⊗ U2 V2 ,

(17)

or (EV3 )U3 − U3 V3 = IN ⊗ (EV1 )U1 − IN ⊗ U1 V1 + TN ⊗ (EV1 )U2 − TN ⊗ U1 V2 + TN ⊗ (EV2 )U1 − TN ⊗ U2 V1 + TN2 ⊗ (EV2 )U2 − TN2 ⊗ U2 V2 , where IN and TN are the same ranks and TN2 = 0. We take a pair of enlarged matrix spectral (16) and the discrete stationary zero curvature equation (EV3 )U3 − U3 V3 = 0

(18)

into the discrete zero curvature equation (15), we get the new discrete zero curvature equations  U1t − (EV1 )U1 + U1 V1 = 0, U2t − (EV2 )U1 + U1 V2 − (EV1 )U2 + U2 V1 = 0.

(19)

The second equation here exactly presents Eq. (2). And the second equation provides a coupling system for the first equation in Eq. (19). To summarize, the Kronecker product provides a great choice of candidates of discrete integrable couplings for Eq. (19). Next let us shed light on the above general idea of constructing coupling systems by a particular class of the Kronecker product. Consider the following two kinds of Kronecker product: Example 1. The case of 2 × 2 Kronecker product: Let U and V have the forms 



0 1 1 0 ⊗ U4 , U= ⊗ U1 + 0 0 0 1

  0 1 1 0 ⊗ V4 , ⊗ V1 + V= 0 0 0 1 we get a new pair of U and V ,

  V1 U1 U4 , V= U= 0 U1 0

V4 V1

(20) (21)

.

Therefore, the corresponding enlarged zero curvature equation is equivalent to  U1t − (EV1 )U1 + U1 V1 = 0, U4t − (EV4 )U1 + U1 V4 − (EV1 )U4 + U4 V1 = 0.

(22)

(23)

F. Yu, L. Li / Physics Letters A 372 (2008) 3548–3554

Example 2. The case of 4 × 4 Kronecker product: Let U and V have the forms ⎛ ⎞ 1 0 0 0 

 0 1 1 ⎜0 1 0 0⎟ U =⎝ ⊗ ⎠ ⊗ U1 + 0 0 1 0 0 0 0 0 0 0 1 ⎛ ⎞ 1 0 0 0

  1 0 1 ⎜0 1 0 0⎟ ⊗ V =⎝ ⎠ ⊗ V1 + 0 0 0 0 0 1 0 0 0 0 1

0 1

0 1

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⊗ U5 ,

(24)

⊗ V5 ,

(25)

from the zero curvature equation, we get  U1t − (EV1 )U1 + U1 V1 = 0, U5t − (EV5 )U1 + U1 V5 − (EV1 )U5 + U5 V1 = 0.

(26)

If IN and TN are the same rank unit matrices in (16), we will have the new discrete zero curvature equations  U1t − (EV1 )U1 + U1 V1 = 0, U2t − (EV2 )U1 + U1 V2 − (EV1 )U2 + U2 V1 − (EV2 )U2 + U2 V2 = 0.

(27)

So Eqs. (19) and (27) provide two kinds of coupling systems for Eq. (4) though the Kronecker product. 4. Integrable coupling system of the generalized Toda lattice equation hierarchy By using of the theory on Lie algebra, some discrete integrable coupling systems of the known equation hierarchies have been obtained, such as the Toda hierarchy, the modified KdV lattice equation hierarchy, the cubic Volterra lattice equation hierarchy [13, 16–20]. In this Letter, in order to obtain a new integrable coupling system, we illustrate a new approach by the Kronecker product to nonlinear soliton equation hierarchy. A new Lax pair, U3 and V3 is presented as follows U3 = U1 ⊗ IN + TN ⊗ U2 ,

V3 = V1 ⊗ IN + TN ⊗ V2 ,

where TN is an upper triangle matrix, U1 , V1 , U2 , V2 are matrices of 2 × 2, IN = equation, we have  U1t − (EV1 )U1 + U1 V1 = 0, U2t − (EV2 )U1 + U1 V2 − (EV1 )U2 + U2 V1 = 0. Consider the following isospectral problem: 

0 −rn Eψn = U1 ψn , U1 = , qn λ + qn rn

 V1 =

an cn

bn −an

1 0

01

, TN =

0 1

00

(28) . From the zero curvature

(29)

.

(30)

From the stationary discrete zero curvature equation (EV1 )U1 − U1 V1 = 0,

(31)

it gives rise to qn bn+1 + rn cn = 0,

(32a)

−rn (an + an+1 ) + qn rn bn+1 + bn+1 λ = 0,

(32b)

qn (an + an+1 ) + qn rn cn + cn λ = 0,

(32c)

−rn cn+1 − qn bn + qn rn (an − an+1 ) + (an − an+1 )λ = 0,

(32d)

where Eq. (32a) can obtain through computing qn (32b) + rn (32c). Setting a m λ−m , b= bm λ−m , c= cm λ−m , a= m0

m0

m0

(33)

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F. Yu, L. Li / Physics Letters A 372 (2008) 3548–3554

by using of the stationary discrete zero curvature equation (31), we obtain ⎧ (m) (m) qn bn+1 + rn cn = 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ b(m+1) − rn (an(m) + a (m) ) + qn rn b(m) = 0, n+1

n+1

n+1

(34)

(m+1) (m) (m) (m) ⎪ ⎪ + qn (an + an+1 ) + qn rn cn = 0, cn ⎪ ⎪ ⎪ ⎩ (m+1) (m+1) (m) (m) (m) (m) − an+1 ) − rn cn+1 − qn bn + qn rn (an − an+1 ) = 0. (an (0)

Taking an = − 12 , we have ⎧ (0) (0) cn = 0, ⎪ ⎨ bn = 0, (1) (1) (0) (0) an+1 − an = −rn cn+1 − qn bn , ⎪ ⎩ (1) (1) bn = −rn−1 , an = −rn−1 qn ,

(35) (1) cn

= qn .

Letting (m) V1n

=

m  (i) m−i an λ

(i)

bn λm−i −an(i) λm−i

cn(i) λm−i

i=1

we get

(m)

EV1n U

Considering (m) n



=

 (m) − U V1n

(m)

−an 0

=

0 , 0

EVn(m) U − U Vn(m) =

m  0,

,



(m+1)

−bn+1

0 (m+1)

(m+1)

cn

an+1

(36)

.

(m+1)

− an

(m)

Vn(m) = V1n + n ,



(m+1)

−bn+1

0 (m+1)

cn

(37)

(m)

+ qn a n

(m+1)

an+1

(m)

+ rn an+1 (m+1)

− an

 ,

m  0.

By using of Tu method, the discrete zero equation (13) admits the following the nonlinear lattice soliton equation hierarchy:  (m+1) (m) (m) (m) rn,tm = bn+1 − rn an+1 = −qn rn bn+1 + rn an , (m+1)

qn,tm = cn

(m)

+ qn a n

(m)

= −qn rn cn

(m)

− qn an+1 .

When n = 1, the first nonlinear lattice soliton equation in the hierarchy (39) is  rnt = qn rn (rn−1 − rn ), qnt = qn rn (qn+1 − qn ), which is a new nonlinear lattice soliton equation. It is the generalized Toda lattice equation. Set 

0 v2 U2 (v) = , v3 v1

(38)

(39)

(40)

(41)

where vi , 1  i  3, are some new dependent variables and v = (v1 , v2 , v3 )T , u¯ = (q, r, v1 , v2 , v3 )T . To obtain the corresponding enlarged stationary discrete zero curvature equation (EV3 )U3 − U3 V3 = 0, i.e., (EV2 )U1 − U1 V2 + (EV1 )U2 − U2 V1 = 0, take

 V2 (u, ¯ λ) =

and e=

m0

e g

em λ−m ,

(42)

f , −e f=

(43) m0

fm λ−m ,

g=

m0

gm λ−m .

F. Yu, L. Li / Physics Letters A 372 (2008) 3548–3554

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Obviously, we have (EV2 )U1 − U1 V2  =

(m)

(m)

qn fn+1 + rn gn

(m)

(m)

(m+1)

−qn (en+1 + en ) − gn 

(EV1 )U2 − U2 V1 =

(m)

− qn rn gn

(m)

(m)

(m)

(m)

(m)

− en+1

−rn gn+1 − qn fn

(m)

(m)

(m+1)

(m)

v3n bn+1 − v2n cn −v1n cn

(m)

(m+1)

−rn (en+1 + en ) + fn

(m+1)

+ en

(m)

(m)

(m)

(m)

(m)

(m)

,

(44a)



(m)

−v1n (an+1 − an ) − v3n bn

(m)

− qn rn (en+1 − en )

v2n (an+1 + an ) + v1n bn+1

− v3n (an+1 + an )



(m)

+ qn rn fn

(m)

+ v2n cn+1

,

(44b)

from Eq. (32), the enlarged stationary discrete zero curvature equation (30) is given as follows ⎧ (m) (m) (m) (m) qn fn+1 + rn gn + v3n bn+1 − v2n cn = 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ v2n (a (m) + an(m) ) + v1n b(m) − rn (e(m) + en(m) ) + fn(m+1) + qn rn fn(m) = 0, n+1 n+1 n+1 (m) (m) (m) (m+1) (m) (m) (m) ⎪ ⎪ − qn rn gn − v3n (an+1 + an ) = 0, −v1n cn − qn (en+1 + en ) − gn ⎪ ⎪ ⎪ ⎩ (m) (m) (m+1) (m+1) (m) (m) (m) (m) (m) (m) − qn rn (en+1 − en ) − v1n (an+1 − an ) − v3n bn + v2n cn+1 = 0. −rn gn+1 − qn fn − en+1 + en (0)

(0)

(0)

(i)

(i)

(45)

(i)

If set fn = gn = en = 0, we see that all sets of functions en , fn and gn are uniquely determined. In particular, the first few sets are: ⎧ (1) (1) (1) ⎪ gn = v3n , en = 0, fn = v2n , ⎪ ⎪ ⎪ ⎨ f (2) = r v − q r v + r q v + q r v , n 1n n n 2n n n+1 2n n n−1 2n n (46) (2) ⎪ ⎪ gn = −qn v1n − qn rn v3n + rn qn+1 v3n + qn rn−1 v3n , ⎪ ⎪ ⎩ (2) en = qn v2n − rn−1 v3n − (E − 1)−1 v1n (qn rn−1 ). Based on Eq. (29), it gives rise to ⎛

⎞

⎛

(m+1)

−bn+1

(m)

+ rn an+1

⎞

rn ⎟ ⎜ (m+1) ⎜ cn + qn an(m) ⎟ ⎜ qn ⎟ ⎟ ⎜ (m+1) (m+1) ⎟ ⎜ ⎟ ⎜ U¯ tn = ⎜ v1n ⎟ = ⎜ −en+1 + en ⎟. ⎟ ⎜ ⎝ ⎠ v2n ⎜ f (m+1) + v a (m) ⎟ 2n n+1 ⎠ ⎝ n v3n t −gn(m+1) − v3n an(m)

(47)

By using of Eqs. (46) and (47), the first two systems in (47) are obtained, when m = 0, ⎧ rn,t0 = 12 rn , ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ ⎨ qn,t0 = 2 qn , v1n,t0 = 0, ⎪ ⎪ ⎪ ⎪ v2n,t0 = 12 v2n , ⎪ ⎪ ⎩ v3n,t0 = − 12 v3n ;

(48)

when m = 1, ⎧ rn,t = qn rn (rn−1 − rn ), ⎪ ⎪ 1 ⎪ ⎨ qn,t1 = qn rn (qn+1 − qn ), v1n,t1 = v3n rn−1 − v3n+1 rn − v2n (qn − qn+1 ) + v1n (qn rn−1 + rn qn+1 ), ⎪ ⎪ ⎪ ⎩ v2n,t1 = rn v1n − qn v2n (rn − rn−1 ), v3n,t1 = qn v1n + rn v3n (qn − qn+1 ).

(49)

We obtain the discrete integrable coupling system of a generalized Toda lattice soliton equation (49). When v1n = v2n = v3n = 0, we obtain the generalized Toda lattice soliton equations. The integrable couplings of generalized Toda lattice equation hierarchy is constructed by using the Kronecker product. So the Kronecker product is an efficient and straightforward method to construct the integrable coupling system of discrete soliton hierarchy.

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F. Yu, L. Li / Physics Letters A 372 (2008) 3548–3554

5. Conclusions In this Letter we present the integrable coupling system of generalized Toda lattice equation hierarchy by using the Kronecker product. The Kronecker product is a powerful tool to construct the discrete integrable coupling system of soliton hierarchy. The key idea in our construction is to establish a relation between the Kronecker product and the Lax pairs of soliton equations. The general idea in our analysis could also be applied to other types of soliton equations. In Ref. [14], Ma and Chen presented an effective method to construct Hamiltonian integrable coupling structures with five dependent variables for the AKNS hierarchy. We will further consider how to construct Hamiltonian integrable coupling structures of the discrete soliton equation hierarchy in the future. Acknowledgement This work was supported by the National Key Basic Research Development of China (2004CB318000). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]

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