An integrable coupling family of Merola–Ragnisco–Tu lattice systems, its Hamiltonian structure and related nonisospectral integrable lattice family

Physics Letters A 374 (2010) 401–410

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Physics Letters A www.elsevier.com/locate/pla

An integrable coupling family of Merola–Ragnisco–Tu lattice systems, its Hamiltonian structure and related nonisospectral integrable lattice family Xi-Xiang Xu ∗ College of Science, Shandong University of Science and Technology, Qingdao, 266510, China

a r t i c l e

i n f o

a b s t r a c t

Article history: Received 10 September 2009 Accepted 4 November 2009 Available online 6 November 2009 Communicated by R. Wu

An integrable coupling family of Merola–Ragnisco–Tu lattice systems is derived from a four-by-four matrix spectral problem. The Hamiltonian structure of the resulting integrable coupling family is established by the discrete variational identity. Each lattice system in the resulting integrable coupling family is proved to be integrable discrete Hamiltonian system in Liouville sense. Ultimately, a nonisospectral integrable lattice family associated with the resulting integrable lattice family is constructed through discrete zero curvature representation. © 2009 Elsevier B.V. All rights reserved.

PACS: 02.30.Ik Keywords: Merola–Ragnisco–Tu lattice systems Integrable couplings Discrete variational identity Hamiltonian structure Nonisospectral integrable family

1. Introduction As is well known, the nonlinear integrable lattice systems are widely used in many fields of natural science in the last few years. Many nonlinear integrable lattice systems have been presented, both the isospectral case and nonisospectral case. And their integrable properties have been studied from different points of view [1–19]. Recently, the investigation on integrable couplings of soliton equations has attracted much attention [20–26]. It can be viewed as an approach to obtain new larger integrable lattice systems from a given integrable lattice systems. For a given integrable lattice system

v nt = K ( v n ),

(1)

we want to construct a new bigger triangular integrable system as follows:



vn zn





= t

K (vn) Z ( v n , zn )



(2)

. ∂ Z ( v ,z )

n n Here the function Z ( v n , zn ) should satisfy the non-triviality condition = 0, in which yn = v n , v n+1 , v n−1 , v n+2 , v n−2 , . . . . This ∂ yn statement means that the other lattice equations in the bigger system (2) involve the dependent variables of the original lattice system (1). Eq. (2) is call an integrable coupling of Eq. (1) [20–26]. A few methods to get integrable couplings of given integrable equations are presented by using perturbation methods [20,21], enlarging spectral problems [22,23], and semi-direct sums of Lie algebras [24–26], and so forth. Moreover, an important topic in the theory of integrable systems is to establish Hamiltonian structures of discrete integrable couplings. Usually, the Hamiltonian structures of integrable lattice systems may be established by the discrete trace identity [6]. But, it cannot be used to the case of discrete integrable couplings. In [26], the discrete trace identity has been generalized to discrete variational identity in the matrix Lie algebra which possesses a non-degenerate symmetric bilinear form. This work develops discrete trace identity. And the

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402

X.-X. Xu / Physics Letters A 374 (2010) 401–410

discrete variational identity may be used to establish Hamiltonian structure of discrete integrable couplings in case of semi-direct sums of Lie algebras. Furthermore, algebraic structure of discrete zero curvature equations of integrable lattice systems is established in [9,10], the key of the theory is to generating nonisospectral integrable lattice family from the discrete spectral problem associated with a given integrable lattice family. In [8], the discrete spectral problem



E ψn = U n ( v n , λ)ψn =

λ + rn sn rn sn

1



(3)

ψn ,

is introduced. Here v n = (rn , sn ) T and for a lattice function f n = f (n), the shift operator E and inverse of E are defined as follows

E −1 f n = f n −1 ,

E f n = f n +1 ,

n ∈ Z.

(4)

v n = (rn , sn ) , λ is the spectral parameter and λt = 0, namely, Eq. (3) is an isospectral problem. Staring from the matrix spectral problem (3), the Merola–Ragnisco–Tu lattice systems is derived. Furthermore, continuum limit, r-matrix structure, action-angle variables and a vector generalization of Merola–Ragnisco–Tu lattice systems are discussed in [8]. The typical member in the Merola–Ragnisco–Tu lattice systems is T



rnt = rn+1 − rn2 sn ,

(5)

snt = rn sn2 − sn−1 . In this Letter, we are going to introduce the four-by-four discrete matrix spectral problem

⎛ ⎜

λ + rn sn rn rn w n + sn un un sn 0 0

U n (χn , λ) = ⎜ ⎝





E ϕn = U n (χn , λ)ϕn ,

1 0 0

wn

λ + rn sn sn

⎞

0 ⎟ ⎟, rn ⎠ 1

(6)

in which χn = (rn , sn , un , w n ) T , rn = r (n, t ), sn = s(n, t ), un = u (n, t ), w n = w (n, t ) are real functions defined over Z × R and ϕn = (ϕn1 , ϕn2 , ϕn3 , ϕn4 )T is the eigenfunction vector. If we set



Yn = then





Un =

rn w n + sn un wn

Un 0˜

Yn Un

un 0

 ,

 ,

where 0˜ is the two-by-two zero matrix. Thus, Eq. (6) is an enlarging spectral problem of Eq. (3). This Letter is organized as follows. In Section 2, we will deduce an integrable coupling family of Merola–Ragnisco–Tu lattice systems from the isospectral problem (6). The first nonlinear lattice system in resulting integrable coupling family is

⎧ 2 ⎪ ⎪ rnt = rn+1 − rn sn , ⎪ ⎨ s = r s2 − s , nt n n n −1 ⎪ u = − r + rn2 sn + un+1 − 2rn sn un − rn2 w n , ⎪ nt n + 1 ⎪ ⎩ 2 w nt = −rn sn + sn−1 − w n−1 + 2rn sn w n + sn2 un .

(7)

In Section 3, we would like to establish the Hamiltonian structure of the resulting integrable coupling family by means of the discrete variational identity through a non-degenerate symmetric bilinear form, and a sequence of commuting conserved functionals for obtained discrete Hamiltonian systems is given. Then each obtained discrete Hamiltonian system is proved to be integrable in Liouville sense. In Section 4, we apply the method of generating nonisospectral integrable lattice family, which is proposed in [9,10], to the resulting integrable coupling family, a nonisospectral integrable lattice family associated with the resulting integrable coupling family is obtained by solving an initial nonisospectral discrete zero curvature equation and the corresponding characteristic operator equation and through discrete zero curvature equation. Finally, in Section 5, there will be some conclusions and remarks. 2. The integrable coupling family of Merola–Ragnisco–Tu lattice systems Let us here recall the construction of Merola–Ragnisco–Tu lattice systems [8]. Starting from the spectral problem (3), we choose the auxiliary spectral problem (m)

ψntm = V n ψn ,

(m)

Vn

=

m 



j =0

( j)

( j)

( 0)

1

( j)

bn

( j)

−an

an cn

( j) ( j)



λm− j , m  0,

( j)

where an , bn , cn , j  0, are uniquely determined by the following initial conditions and recursion relation:

an =



2

(m+1)

an+1

,

( 0)

b n = 0,

( 0)

c n = 0,

   − an(m+1) = −rn sn an(m+)1 − an(m) + rn cn(m) − sn bn(m+)1 ,

(8)

X.-X. Xu / Physics Letters A 374 (2010) 401–410

403

  = bn(m+)1 − rn sn bn(m) + rn an(m+)1 + an(m) ,  (m) (m+1) (m) (m) (m)  cn+1 = cn − rn sn cn+1 + sn an+1 + an . (m+1)

bn

(9)

It is easy to find that the compatible condition of Eqs. (3) and (8),

( E ψn )tm = E (ψntm ),

m  0,

is equivalent to the discrete zero curvature equations (m) 



U ntm = E V n

(m)

Un − Un Vn ,

m  0,

which give rise to the Merola–Ragnisco–Tu lattice systems



rn sn







(m+1)

bn

=

,

−cn(m++1 1)

tm

m  0.

(10)

The first nonlinear lattice system in the Merola–Ragnisco–Tu lattice systems (10) is



rnt = rn+1 − rn2 sn , snt = rn sn2 − sn−1 .

In what follows, we would like to derive an integrable coupling family of Merola–Ragnisco–Tu lattice systems. First of all, we consider the following set of four-by-four matrices

⎛

1 0 0 ⎜0 0 0 ⎜ ω1 = ⎝ 0 0 1 0 0 0

⎛

⎞

⎛

0 0⎟ ⎟, 0⎠ 0

0 1 0 ⎜0 0 0 ⎜ ω2 = ⎝ 0 0 0 0 0 0

⎞

⎛

⎞

⎛

⎞

0 ⎜1 ⎜ ω3 = ⎝ 0 0

0 0 0 0 0 0 0 1

0 0⎟ ⎟, 0⎠ 0

0 ⎜0 ⎜ ω4 = ⎝ 0 0

0 0 0 1 0 0⎟ ⎟, 0 0 0⎠ 0 0 1

0 0 0 1 ⎜0 0 0 0⎟ ⎟ ω6 = ⎜ ⎝ 0 0 0 0 ⎠, 0 0 0 0

0 ⎜0 ω7 = ⎜ ⎝0 0

0 0 0 1 0 0 0 0

0 0⎟ ⎟, 0⎠ 0

0 ⎜0 ω8 = ⎜ ⎝0 0

0 0 0 0

⎛

0 0 1 0 ⎜0 0 0 0⎟ ⎟ ω5 = ⎜ ⎝ 0 0 0 0 ⎠, 0 0 0 0

⎞

0 0⎟ ⎟, 1⎠ 0

⎞

⎛

⎞

⎛

⎞

0 0 0 0

0 1⎟ ⎟. 0⎠ 0

It is easy to see that

G = span{ω1 , ω2 , ω3 , ω4 , ω5 , ω6 , ω7 , ω8 },

G 1 = span{ω1 , ω2 , ω3 , ω4 },

G 2 = span{ω5 , ω6 , ω7 , ω8 }

˜ G˜ 1 , G˜ 2 [26]. For instance, the bases of the loop construct three Lie algebras. Then, we can obtain three corresponding loop algebras G, algebra G˜ are

ωi (m) = λm ωi , i = 1, 2, . . . , 8, m ∈ Z , and the commutation operation in the G˜ is as follows





ωi (m), ω j (n) = [ωi , ω j ]λm+n , i = 1, 2, . . . , 8, n, m ∈ Z .

It is easy to verify that

G = G 1 ⊕ G 2,

[G 1 , G 2 ] ≡ G 1 G 2 − G 2 G 1 ⊆ G 2 .

Then we know that G˜ is a semi-direct sum of G˜ 1 and G˜ 2 . Let us consider the four-by-four matrix spectral problem (6), namely, 

E ϕn = U n (χn , λ)ϕn , 

where U n (χn , λ) may be also expressed as 

U n (χn , λ) = ω1 (1) + rn sn ω1 (0) + rn ω2 (0) + sn ω3 (0) + ω4 (0) + (rn w n + sn un )ω5 (0) + un ω6 (0) + w n ω7 (0), 

˜ To get the where rn , sn , un , w n are defined as before, λ is the spectral parameter and λt = 0. Evidently, the matrix U n belong to G. integrable coupling family of Merola–Ragnisco–Tu lattice systems (10), we first solve the stationary discrete zero curvature equation 







( E Πn )U n − U n Πn = Πn+1 U n − U n Πn = 0 with

⎛ Πn =

(m)

an

(m)

(m)

bn

en

(m)

fn

(11)

⎞

∞ ⎜ (m) (m) (m) (m) ⎟  −an gn −en ⎟ −m ⎜ cn . ⎜ (m) (m) ⎟ λ ⎝ 0 ⎠ 0 an bn m =0 (m) (m) 0 0 cn −an

Eq. (11) implies the initial conditions:



( 0)

( 0) 

an+1 − an

= 0,

( 0)

b n = 0,

( 0)

c n +1 = 0,



( 0)

( 0) 

e n +1 − e n

= 0,

( 0)

fn

= 0,

( 0)

g n +1 = 0,

404

X.-X. Xu / Physics Letters A 374 (2010) 401–410

and, for m  0, the recursion relation:

  − an(m+1) = rn sn an(m) − an(m+)1 + rn cn(m) − sn bn(m+)1 ,   (m+1) bn = bn(m+)1 + rn an(m) + an(m+)1 − rn sn bn(m) ,  (m) (m+1) (m) (m)  (m) cn+1 = cn + sn an + an+1 − rn sn cn+1 ,    (m) (m+1) (m+1) m) (m) (m) (m)  e n +1 − e n = rn sn en(m) − en(m+)1 + rn gn(m) − sn f n(+ 1 + u n cn − w n bn+1 + (sn u n + rn w n ) an − an+1 ,     (m+1) m) (m) fn = f n(+ + rn en(m+)1 + en(m) + un an(m+)1 + an(m) − (rn w n + sn un )bn(m) , 1 − rn sn f n  (m)   (m+1) (m) (m) (m)  + w n an(m+)1 + an(m) − (rn w n + sn un )cn(m+)1 . gn+1 = gn − rn sn gn+1 + sn en+1 + en (m+1)

an+1

(12)

We first research the locality of solution of Eqs. (12). Lemma 1. If Πn solves Eq. (11), then ( E (Πn2 )) − Πn2 = 0. 

Proof. As U n is invertible, using Eq. (11), we easily get −1



( E Πn ) = U n Πn U n . Then, we have

 

E Πn2



−1



= U n Πn2 U n .

Therefore

 

E Πn2



−1



− Πn2 = U n Πn2 U n − Πn2 .

It is easy to verify that 



U n Πn2 = Πn2 U n ,

Thus, we obtain









E Πn2 − Πn2 = D Πn2 = 0. The proof is completed.

2

Lemma 2. If the initial values in Eq. (11) are chosen as

1

( 0)

an = (m)

2

(m)

1 ( 0) en = − , 2

( 0)

c n = 0,

, (m)

(m)

(m)

then an , bn , cn , en , f n variables rn , sn , un and w n .

( 0)

g n = 0,

(m)

, gn , m  1, which are solved by Eqs. (12), are all local, and they are just difference polynomials in the dependent (m+1)

(m+1)

(m)

(m)

and cn can be determined locally by an , bn and Proof. First, from second and third equations in Eqs. (12), we obtain that bn (m) (m+1) (m+1) and gn can be determined locally by cn , m  0. Secondly, according to fifth and sixth equations in Eqs. (12), we find that f n (m) (m) (m) (m) (m) (m) (m+1) (m+1) an , bn , cn , en , f n , gn , m  0. Finally, to get an and en from first and fourth equations in Eqs. (12), we are going to apply (m+1) (m+1) and en may be obtained operator D −1 = ( E − 1)−1 to solve the corresponding difference equations. Below, we shall show that an through an algebraic method rather than by solving the difference equations. A direct calculation shows

⎛

an2 + bn cn ⎜ 0 Πn2 = ⎜ ⎝ 0 0

2an en + cn f n + bn gn 0 an2 + bn cn 0

0 an2 + bn cn 0 0

0

⎞

2an en + cn f n + bn gn ⎟ ⎟. ⎠ 0 an2 + bn cn

Based on Lemma 1, we can obtain that

an2 + bn cn = ξ1 (t ),

2an en + cn f n + bn gn = ξ2 (t ) (m+1)

with ξ1 (t ) and ξ2 (t ) are arbitrary functions of time variable t only. We then obtain two recursion relations for an (m+1)

an

=

m 

( j ) (m− j +1)

an an

+

j =1

(m+1)

en

=2

m  j =1

m 

( j ) (m− j +1)

bn c n

− η1 (t ),

m  1,

i =1

( j ) (m− j +1)

an en

+

m  i =1

( j ) (m− j +1)

cn f n

+

m  j =1

( j ) (m− j +1)

bn g n

− an(m+1) − η2 (t ),

m  1.

(m+1)

and en

:

X.-X. Xu / Physics Letters A 374 (2010) 401–410

(m+1)

(m+1)

(m+1)

We choose ξ1 (t ) = ξ2 (t ) = 0. Then, we can obtain that an , bn , cn difference polynomials in the four dependent variables rn , sn , un and w n . The proof is completed. 2 (m)

(m)

(m)

(m)

(m)

Therefore, by Lemma 2, an , bn , cn , en , f n (1 )

(1 )

an = 0, (2 )

(2 )

(m)

Vn

⎛

(m)

an

(m)

(m)

bn

= −rn + un ,

(2 )

, m  0 are all local, and they are just

(1 )

gn = −sn−1 + w n−1 ,

(2 )

en = rn sn−1 − rn w n−1 − un sn−1 ,

(2 )

gn = −sn−2 + w n−2 + rn−1 sn2−1 − 2rn−1 sn−1 w n−1 − sn2−1 un−1 ,

fn

⎟ m ⎜ (m)  ⎜ cn −an(m) gn(m) −en(m) ⎟ m−i ⎜ ⎟λ . ⎜ (m) (m) ⎟ 0 an bn ⎠ i =0 ⎝ 0

=

0

(m)

0

....

⎞

(m)

en

(1 )

fn

cn = sn−2 − rn−1 sn2−1 ,

= −rn+1 + un+1 + rn2 sn − 2rn sn un − rn2 w n ,

Now we define

(m+1)

, gn

(m)

e n = 0,

(2 )

(m+1)

, fn

, gn , m  1, in Eqs. (12) can be uniquely determined. In particular, we have

(1 )

cn = sn−1 ,

bn = rn+1 − rn2 sn ,

an = −rn sn−1 , fn

(1 )

bn = rn ,

(m+1)

, en

405

(13)

(m)

−an

cn

If we set the time evolution of the eigenfunction of the isospectral problem (6) obey the differential equation

ϕntm = V n(m) ϕn ,

(14)

then the compatibility conditions between Eqs. (6) and (14) are (m) 





U ntm = E V n



(m)

Un − Un Vn ,

m  0,

(15)

which give rise to the following family of integrable lattice systems

⎛

rn

⎛

⎞

⎜ sn ⎟ ⎜ ⎟ ⎝ un ⎠ wn

tm

(m+1)

bn

⎞

⎟ ⎜ ⎜ −c (m+1) ⎟ ⎜ n +1 ⎟ = ⎜ (m+1) ⎟ , ⎟ ⎜ fn ⎠ ⎝ (m+1) − g n +1

m  0.

(16)

It is easy to find that the first two components in Eqs. (16) constitute Eqs. (10), therefore Eqs. (16) is an integrable coupling family of the Merola–Ragnisco–Tu lattice systems (10). When m = 1, Eqs. (16) is

⎛

⎛

⎞ rn+1 − rn2 sn rn ⎜ ⎜ sn ⎟ rn sn2 − sn−1 ⎜ ⎜ ⎟ ⎝ un ⎠ = ⎜ 2 ⎝ −rn+1 + rn sn + un+1 − 2rn sn un − rn2 w n wn t 1 −rn sn2 + sn−1 − w n−1 + 2rn sn w n + sn2 un

⎞

⎟ ⎟ ⎟. ⎠

(17)

Obviously, this is an integrable coupling of Merola–Ragnisco–Tu lattice (5). If we set the Merola–Ragnisco–Tu lattice (5) in following form

 v nt =

rn sn



 = K (vn) = t

rn+1 − rn2 sn rn sn2 − sn−1 )



,

then the integrable coupling (17) may be represented as



v nt = K ( v n ),

(− zn )t = K ( v n ) + K ( v n )[(− zn )] in which v n = (rn , sn ) T , zn = (un , w n ) T , K ( v n )[ zn ] denotes the Gateaux derivative of K ( v n ) with respect to v n in a direction zn , namely, K ( v n )[ zn ] = ∂∂ε K ( v n + ε zn )|ε=0 . Therefore, in view of perturbation theory, the integrable coupling (17) is a first-order perturbation system [20,21] of the Merola–Ragnisco–Tu lattice (5). 3. Hamiltonian structure and Liouville integrability Next, we define a map

η : G˜ → R 8 ,

A → (a1 , a2 , . . . , a8 ) T ,

⎛

a1 ⎜ a3 ⎜ A=⎝ 0 0

Then

η( A ) = a = (a1 , a2 , a3 , a4 , a5 , a6 , a7 , a8 )T ∈ R 8 .

a2 a4 0 0

a5 a7 a1 a3

⎞

a6 a8 ⎟ ⎟ , A ∈ G˜ . a2 ⎠ a4

(18)

406

X.-X. Xu / Physics Letters A 374 (2010) 401–410

The commutator [·,·] on R 8 can be denoted by

[a, b]TR 8 = a T R (b), where

⎛

0 ⎜ b3 ⎜ ⎜ −b2 ⎜ ⎜ 0 R (b) = ⎜ ⎜ 0 ⎜ ⎜ 0 ⎝ 0 0

a, b ∈ R 8 ,

(19)

−b3

b2

b4 − b1 0 −b2 0 0 0 0

0 b1 − b4 b3 0 0 0 0

We choose the matrix

⎛

0 ⎜0 ⎜ ⎜0 ⎜ ⎜0 Q =⎜ ⎜1 ⎜ ⎜0 ⎝0 0

0 0 0 0 0 0 1 0

0 0 0 0 0 1 0 0

0 −b3 b2 0 0 0 0 0

0 b7 −b6 0 0 b3 −b2 0

−b7

b6

b8 − b5 0 −b6 b2 b4 − b1 0 −b2

0 b5 − b8 b7 −b3 0 b1 − b4 b3

⎞

0 −b7 ⎟ ⎟ b6 ⎟ ⎟ 0 ⎟ ⎟. 0 ⎟ ⎟ −b3 ⎟ b2 ⎠ 0

⎞

0 0 0 0 0 0 0 1

1 0 0 0 0 0 1 0⎟ ⎟ 0 1 0 0⎟ ⎟ 0 0 0 1⎟ ⎟. 0 0 0 0⎟ ⎟ 0 0 0 0⎟ 0 0 0 0⎠ 0 0 0 0

(20)

It is easy to verify that Q meets

QT = Q, and



T

Q R (b)

for all b ∈ R 8 .

= − R (b) Q ,

Following [26], a non-degenerate symmetric bilinear form on R 8 may be determined by

a, b R 8 = a T Q b.

(21)

And then, we can obtain following non-degenerate bilinear form on G˜

  A , B G˜ = η( A ), η( B ) R 8 = (a1 , a2 , a3 , a4 , a5 , a6 , a7 , a8 ) Q (b1 , b2 , b3 , b4 , b5 , b6 , b7 , b8 )T .

(22)

In what follows, we shall establish the Hamiltonian structure for the integrable coupling family (16) by means of the discrete variational identity [26]

      δ  ¯ ∂Un ∂ γ ¯ ∂Un V n, = λ−γ , λ V n, δ χn ∂λ G˜ ∂λ ∂ χn G˜

(23)

n∈ Z

 −1 ˜n −m ( ∂ H n ). where V¯ n = Πn U n , the variational derivative is defined by δδ H χn = m∈ Z E ∂ χn+m Through a direct computation, we obtain

 



∂ Un V¯ n , ∂λ 

V¯ n ,

∂ Un ∂ rn

 

G˜

G˜

=

en − sn f n − w n bn

λ

= g n +1 ,



,



V¯ n ,

∂ Un ∂ sn



 G˜

= fn,

Eqs. (23) and (24) imply

⎛

g n +1



V¯ n ,

∂Un ∂ un



 G˜

= c n +1 ,



V¯ n ,

∂ Un ∂ wn

 G˜

= bn .

(24)

⎞

δ  (en − sn f n − w n bn ) ∂ γ⎜ f ⎟ = λ−γ λ ⎜ n ⎟. δ χn λ ∂λ ⎝ cn+1 ⎠ n∈ Z

Substituting expansions an =

∞

∞

(m) −m , m=0 an λ

bn =

bn ∞

(m) −m , m=0 bn λ −m−1

(m) −m into (25), and equating the coefficient of m=0 gn λ

⎛

δ  δ χn

(m)

en

(m)

− sn f n

(m) 

− w n bn

(25)

(m)

g n +1

⎜ (m) ⎜ f ⎜ n = (γ − m) ⎜ (m) ⎜c ⎝ n +1 (m)

bn

λ

⎞ ⎟ ⎟ ⎟ ⎟. ⎟ ⎠

cn =

∞

(m) −m , m=0 cn λ

en =

∞

(m) −m , m=0 en λ

fn =

∞

m=0

(m) −m λ , gn

fn

=

, we have

(26)

X.-X. Xu / Physics Letters A 374 (2010) 401–410

407

γ is a constant. Taking m = 0 yields the constant γ = 0. We therefore conclude that ⎞ ⎛ (m)

In Eq. (24),

g

1 ⎟ ⎜ n(+ ⎜ f m) ⎟ ⎜ n ⎟ = ⎜ (m) ⎟ , ⎟ ⎜c ⎝ n +1 ⎠

δ H˜ n δ χn

(m)

m > 0,

(27)

(m)

bn where

˜n H

(m)

=



(m)

(m)

Hn ,

Hn

(m)

=−

en

(m)

− sn f n

(m)

− w n bn

m

n∈ Z

, m  1.

Ask for help the recursion relations (12), we obtain the following recursion structure

δ H˜ n(m) δ H˜ n(m+1) =Ψ , δ χn δ χn

(28)

where

⎛ ⎜ ⎜ Ψ =⎜ ⎝

E −1 − rn sn − sn Θ rn

sn Θ sn

Ψ13

sn Θ w n + w n Θ sn

−rn Θ rn

E − rn sn + rn Θ sn

−rn Θ un − un Θ rn

Ψ24

0

0

E −1 − rn sn − sn Θ rn

sn Θ sn

0

0

−rn Θ rn

E − rn sn + rn Θ sn

⎞ ⎟ ⎟ ⎟, ⎠

in which

Θ = (1 + E )(1 − E )−1 ,

Ψ13 = −sn Θ un − w n Θ rn − rn w n − sn un ,

Ψ24 = un Θ sn + rn Θ w n − rn w n − sn un .

Make use of Eqs. (27), we find that the family (16) has following Hamiltonian structure:

⎛

χntm

rn

⎛

⎞ (m+1)

δ H˜ n =M δ χn

⎜ sn ⎟ ⎟ =⎜ ⎝ un ⎠ wn

( 0)

g n +1

⎜ ( 0) (m) ⎜ f δ H˜ n ⎜ n =N = M Ψ m +1 ⎜ ( 0) ⎜c δ χn ⎝ n +1 ( 0)

tm

⎞ ⎟ ⎟ ⎟ ⎟, ⎟ ⎠

m  0,

(29)

bn

where the discrete Hamiltonian operator

⎛

0 ⎜ 0 M =⎜ ⎝ 0 −1

⎞

0 0 1 0 −1 0 ⎟ ⎟ 1 0 0⎠ 0 0 0

and it is not difficult to verify that

⎛

0

0

−rn Θ rn

E − rn sn + rn Θ sn

0

0

− E −1 + rn sn + sn Θ rn

−sn Θ sn

−rn Θ rn

E − rn sn + rn Θ sn

−rn Θ un − un Θ rn

N 34

− E −1 + rn sn + sn Θ rn

−sn Θ sn

N 43

−sn Θ w n − w n Θ sn

⎜ ⎜ ⎝

N = MΨ = ⎜

⎞ ⎟ ⎟ ⎟, ⎠

where

N 34 = un Θ sn + rn Θ w n − rn w n − sn un ,

N 43 = sn Θ un + w n Θ rn + rn w n + sn un .

In particular, Eq. (17) has Hamiltonian structure

⎛

χnt1

⎞

rn ˜ (2 ) ⎜ sn ⎟ ⎟ = M δ Hn . ⎜ =⎝ un ⎠ δ χn wn t 1

Here

˜n =− H (2 )

 1  rn sn−1 − rn w n−1 − un sn−1 + rn+1 sn − sn un+1 − rn+1 w n − rn2 sn2 + 2rn sn2 un + 2rn2 sn w n . 2 n∈ Z

(30)

408

X.-X. Xu / Physics Letters A 374 (2010) 401–410

Moreover, it is easy to verify that Eqs. (29) may be rewritten as following form

⎛

χntm

⎛

⎞

rn

⎜ sn ⎟ ⎟ =⎜ ⎝ un ⎠ wn

(m)

= Kn

=Φ

m +1

( 0)

g n +1

⎜ ( 0) ⎜ f ⎜ n M ⎜ ( 0) ⎜c ⎝ n +1 ( 0)

tm

⎞

⎟ ⎟ ⎟ ⎟, ⎟ ⎠

m  0,

bn

In which the recursion operator

⎛

E − rn sn + rn Θ sn

rn Θ rn

0

0

−sn Θ sn

E −1 − rn sn − sn Θ rn

0

0

Φ31

u n Θ rn + rn Θ u n

E − rn sn + rn Θ sn

rn Θ rn

− w n Θ sn − sn Θ w n

Φ42

−sn Θ sn

E −1 − rn sn − sn Θ rn

⎜ ⎜ Φ =Ψ∗ =⎜ ⎝

⎞ ⎟ ⎟ ⎟, ⎠

here Φ31 = un Θ sn + rn Θ w n − rn w n − sn un , Φ42 = −sn Θ un − w n Θ rn − rn w n − sn un . Based on a given Hamiltonian operator M, we can define a corresponding Poisson bracket

 { f n , gn } M =

   δ gn δ gn δ fn δ fn = ,M ,M . δ χn δ χn δ χn δ χn R 4

(31)

n∈ Z

Then, we can find that N ∗ = − N, namely, Ψ ∗ M = M Ψ . It follows from the general theory on the Liouville integrability of zero curvature equations [6,16,17] that



(l) 

˜ n , H˜ n H (m)

M

=

 ˜ (m) (l)  δ H˜ n δ Hn = 0, ,M δ χn δ χn

m , l > 0.

Therefore, for the mth integrable coupling (16), we have



(m) 

˜n H

tl

=

  ˜ (m)  ˜ (m) (l)    δ H˜ n δ Hn δ Hn = H˜ n(m) , H˜ n(l) M = 0, , untl = ,M δ χn δ χn δ χn

m , l  1.

(32)

Based on above discussion, we can obtain the following lemma and theorem. Lemma 3. Each lattice system in the family (16) or the discrete Hamiltonian system (29) has infinitely many common conserved functionals (m) ∞ { H˜ n }m =0 . Theorem 1. The discrete integrable couplings (16) of Merola–Ragnisco–Tu lattice systems are all discrete Liouville integrable Hamiltonian systems. In particular, Eq. (17), namely,

⎧ rnt1 = rn+1 − rn2 sn , ⎪ ⎪ ⎪ ⎨ s = r s2 − s , nt 1 n n n −1 ⎪ unt1 = −rn+1 + rn2 sn + un+1 − 2rn sn un − rn2 w n , ⎪ ⎪ ⎩ w nt1 = −rn sn2 + sn−1 − w n−1 + 2rn sn w n + sn2 un is a discrete integrable system in Liouville sense. 4. The related nonisospectral integrable lattice family In Section 2, we have discussed the spectral problem (6) in the isospectral case, namely, ddtλ = 0. In this section, we shall investigate the spectral problem (6) in the nonisospectral case. Now let us suppose that ddtλ = f (λ) = λm+1 in the spectral problem (6), we are going to derive the nonisospectral integrable family associated with the integrable coupling family (16). Based on the approach proposed in [9,10], we first consider the nonisospectral initial discrete zero curvature equation





( 0)  

E Wn



( 0)

Un − Un Wn

 ∂ Un   = U n ρn(0) + λ . ∂λ

In this equation, the corresponding quantities easily are solved as follows:

⎛ ( 0)

Wn

n

0 − 12

0

⎞

⎜0 0 ⎜ =⎜ ⎝0 0

n

⎟ ⎟ ⎟ 0⎠

0 0

0

0

0

1 2

X.-X. Xu / Physics Letters A 374 (2010) 401–410

and

⎛

409

⎞

(n + 1)rn

⎜ ⎟ −nsn ⎜ ⎟ ⎟. ⎝ −rn + (n + 1)un ⎠

ρn(0) = ⎜

sn − nw n Next, we find a solution of the following operator equation











E Ω( X ) U n − U n Ω( X ) = U n [Φ X ] − λU n [ X ],



X = X n1 , X n2 , X n3 , X n4

T

.

(33)

This equation is called the characteristic operator equation in [9,10]. In Eq. (33), we choose the undetermined operator Ω = Ω( X ) as follows

⎛

Ωn11 ( X ) Ωn12 ( X ) Ωn13 ( X ) Ωn14 ( X )

⎞

⎜ Ω 21 ( X ) Ω 22 ( X ) Ω 23 ( X ) Ω 24 ( X ) ⎟ ⎜ ⎟ n n n Ω( X ) = ⎜ n ⎟. 11 12 ⎝ 0 0 Ωn ( X ) Ωn ( X ) ⎠ 0

Ωn21 ( X ) Ωn22 ( X )

0

Through a direct calculation, we can obtain that

  Ωn11 ( X ) = (1 − E )−1 rn Xn2 + sn Xn1 ,

Ωn12 ( X ) = Xn1 , Ωn21 ( X ) = − E −1 Xn2 ,   Ωn13 ( X ) = (1 − E )−1 rn Xn4 + sn Xn3 + un Xn2 + w n Xn1 ,

Ωn22 ( X ) = −Ωn11 ( X ),

Ωn23 ( X ) = − E −1 Xn4 ,

Ωn14 ( X ) = Xn3 ,

Ωn24 ( X ) = −Ωn13 ( X ).

We set

ρn(m) = Φ m ρn(0) , m  1, and (m)

Wn

( 0)

= λm W n +

m 

 ( j −1 )  , λm− j Ω ρn

m  1.

j =1

Then, we have



(m) 



(m)

Un − Un Wn

E Wn

m       ( j −1 )    ( j −1)    = λm E W n(0) U n − U n W n(0) + λm− j E Ω ρn U n − U n Ω ρn





 ∂ Un   = λm U n ρn(0) + λ ∂λ 

 

( 0) 

= λm U n ρn



+λ

∂ Un ∂λ

j =1

 +

m 

  ( j −1)    ( j −1)  − λU n ρn λm− j U n Φ ρn

j =1

 +

m 

  ( j )    ( j −1)  λm− j U n ρn − λU n ρn

j =1



 ∂ Un   = U n ρn(m) + λm+1 , ∂λ

m  1.

Therefore, from the nonisospectral (λt = λm+1 ) zero curvature representations 



(m) 

U ntm = E W n



(m)

Un − Un Wn ,

m  0,

(34)

we can deduce an integrable family of nonisospectral discrete evolution equations

χntm = ρn(m) = Φ m ρn(0) , m  0.

(35)

The first equation in Eqs. (35) is

⎧ rnt0 = (n + 1)rn , ⎪ ⎪ ⎪ ⎨s nt 01 = −nsn , ⎪ u ⎪ nt0 = −rn + (n + 1)un , ⎪ ⎩ w nt0 = sn − nw n .

(36)

(0)

Eq. (6) and ϕnt0 = W n ϕn (λt = λ) form the Lax pair of Eq. (36). When m = 1, Eqs. (35) become

⎧ rnt1 = (n + 2)rn+1 − (n + 1)rn2 sn + rn Θ rn sn , ⎪ ⎪ ⎪ ⎨ s = nr s2 − (n − 1)s nt 1 n n n−1 − sn Θ rn sn , 2 ⎪ u = − r + r s + (n + 2)un+1 − 2(n + 1)rn sn un − (n + 1)rn2 w n + un Θ rn sn + rn Θ(rn w n + sn un ), ⎪ nt1 n +1 n n ⎪ ⎩ 2 w nt1 = −rn sn + sn−1 − (n − 1) w n−1 + 2nrn sn w n + nsn2 un − w n Θ rn sn − sn Θ(rn w n + sn un ).

(37)

410

X.-X. Xu / Physics Letters A 374 (2010) 401–410

Based on Eqs. (34) or through a direct calculation, we can obtain that the nonisospectral discrete evolution equation (37) possess nonisospectral (λt = λ2 ) discrete zero curvature representation



(1 ) 

U nt1 = E W n Here

⎛ (1 )

Wn

⎜ ⎜ =⎜ ⎜ ⎝

(1 )

Un − Un Wn .

nλ + (1 − E )−1 rn sn

(n + 1)rn

− λ2 + (1 − E )−1 (rn w n + sn un )

(n − 1)sn−1

−(1 − E )−1 rn sn

0

0

nλ + (1 − E )−1 rn sn

0

0

(n − 1)sn−1

−sn−1 + (n − 1) w n−1

⎞

−rn + (n + 1)un

λ 2

⎟ − (1 − E )−1 (rn w n + sn un ) ⎟ ⎟. ⎟ (n + 1)rn ⎠ − 1 −(1 − E ) rn sn (0)

We can find that the nonisospectral integrable lattice family (34) is non-local except the first one of Eqs. (36) since W n

is local and the

(l)

other nonisospectral Lax operators W n , l  1, are non-local. 5. Conclusions and remarks By introducing a proper four-by-four isospectral problem, an integrable coupling family of Merola–Ragnisco–Tu lattice systems have been derived through the isospectral discrete zero curvature equations. The Hamiltonian structure for the resulting integrable coupling family is established by discrete variational identity, which is derived from a non-degenerate symmetric bilinear form. A sequence of commuting conserved functionals for obtained discrete Hamiltonian systems is presented, the Liouville integrability of the resulting discrete Hamiltonian equations is demonstrated. Finally, a nonisospectral integrable lattice family associated with the resulting integrable coupling family is constructed through the nonisospectral discrete zero curvature equation. Moreover, like the case in [9,10], we can further investigate the vector field algebra of integrable coupling family (16) and the nonisospectral integrable lattice family (34), and master symmetries the isospectral integrable coupling family (18). In addition, many other interesting integrability problems deserve further investigation for the integrable coupling family (16) and the nonisospectral integrable lattice family (34). For instance, Darboux transformation, nonlinearization of Lax pairs, conservation laws and so on. Acknowledgements This work was supported by the Science and Technology Plan Projects of the Educational Department of Shandong Province of China (Grant No. J08LI08). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26]

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