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Two hierarchies of lattice soliton equations associated with a new discrete eigenvalue problem and Darboux transformation
Physics Letters A 338 (2005) 117–127 www.elsevier.com/locate/pla
Two hierarchies of lattice soliton equations associated with a new discrete eigenvalue problem and Darboux transformation Hong-Xiang Yang a,∗ , Xi-Xiang Xu b , Hai-Yong Ding c a Department of Computer Science and Technology, Taishan College, Taian 271021, China b College of Science, Shandong University of Science and Technology, Qingdao 266510, China c College of Information Science and Engineering, Shandong Agricultural University, Taian 271018, China
Received 5 November 2004; received in revised form 25 January 2005; accepted 11 February 2005
Communicated by A.P. Fordy
Abstract By considering a new discrete isospectral eigenvalue problem, two hierarchies of integrable positive and negative lattice models are derived. It is shown that they correspond to positive and negative power expansions of Lax operators with respect to the spectral parameter, respectively. And, each equation in the resulting hierarchies is integrable in Liouville sense. Moreover, a Darboux transformation is established for the typical equations by using gauge transformations of Lax pairs, from which the exact solutions are given. 2005 Elsevier B.V. All rights reserved. PACS: 02.30.Ik; 02.90.+p Keywords: Integrable lattice model; Positive and negative hierarchy; Darboux transformation; Soliton solutions
1. Introduction The role, is well known, played by nonlinear evolution equations of lattice versions in domain of nonlinear science. The investigation on integrable discrete systems and interrelated properties has always been one of the prominent events and has become the focus of common concern in the past few years. During the last decade, a considerable number of integrable lattice systems have been investigated successfully such as the Ablowitz–Ladik lattice [1], the Toda lattice [2], the Blaszak–Marciniak lattice [3,4], and so on [5–10]. The procedure of Lax pair * Corresponding author.
E-mail address: [email protected] (H.-X. Yang). 0375-9601/$ – see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2005.02.021
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is certainly a significant method to obtain the new Lax integrable lattice equations. So far, it is, however, still hard work to be carried out, mainly due to the difficulty in choosing a proper spectral problem. At the same time, quite a few systematic methods used for obtaining exact solutions of (1 + 1)-dimensional soliton equations are well established, such as the inverse scattering transform method [11,12], the bilinear transform method of Hirota [13], the Bäcklund [14] and Darboux transformation, Wronskian technique [15] and so forth. The Darboux transformation (DT), which is always used for proposing the explicit solutions of the soliton equations, was first introduced by Darboux in his study of Sturm–Liouville equation [16]. Extensive applications of the DT can be found in articles [17–21]. This Letter is devoted to introducing a new discrete isospectral problem 0 1 1 , Eψn (λ) = Un (u, λ)ψn (λ), Un (u, λ) = (1) rn λrn2 λrn + rn sn where the shift operator E is defined as Efn = f (n + 1) = fn+1 , E −1 fn = f (n − 1) = fn−1 , n ∈ Z, and u = (rn , sn )T is a 2-component potential function vector defined over R × Z and vanishes rapidly if |n| → ∞, ψn (λ) = (ψn1 (λ), ψn2 (λ))T , and, λ is the spectral parameter with λt = 0. By means of constructing a proper continuous time evolution equation d ψn (λ) = Γn{m} (u, λ)ψn (λ), dt
(2)
{m}
in which Γn (u, λ) is a suitable 2 × 2 matrix, and using the discrete zero-curvature equation d Un − EΓn{m} Un + Un Γn{m} = 0, (3) dt a pair of positive and negative hierarchies of integrable nonlinear lattice models are derived [9]. As the typical examples, the first two nonlinear lattice equations are given as follows, respectively, rn rnt = −rn rn−1 − rn sn , + rn+1 rn rn (4a) rn+1 snt = sn rn−1 − rn , rnt = − rsnn , (4b) rn snt = rnrn+1 sn+1 − rn−1 sn−1 . It is shown that these two hierarchies correspond to power expansions of Lax operators with respect to the spectral parameter, respectively, and each equation in resulting hierarchies is Liouville integrable. Further, by virtue of the gauge transformation, we find a DT for spectral problem (1). As an application, exact solutions of Eqs. (4) are given.
2. Integrable lattice models To get the associated lattice soliton hierarchies, we first solve the stationary discrete zero-curvature equation (EΓ1 )Un − Un Γ1 = 0,
(5)
where Γ1 =
∞ (m) −m an λ
cn λ−m (m)
m=0
bn λ−m−1 (m) −an λ−m (m)
.
H.-X. Yang et al. / Physics Letters A 338 (2005) 117–127
Then Eq. (5) leads to the initial relations (0) 1 (0) (0) an+1 + an(0) + bn+1 = 0, rn an+1 + an(0) + cn(0) = 0, rn and the recursion relations
119
(0) an+1 − an(0) = 0,
(6)
(m)
(m)
rn bn+1 =
cn , rn
1 (m+1) (m+1) (m) an+1 + an(m+1) + bn+1 + sn bn+1 = 0, rn (m+1) + an(m+1) + cn(m+1) + sn cn(m) = 0, rn an+1 1 (m) (m+1) (m) an+1 − an(m+1) = cn+1 − rn bn(m) + sn an(m) − an+1 . rn (0)
(0)
(7)
(0)
Now we choose an = −1/2, bn = 1/rn−1 , cn = rn and require selecting zero constants for the inverse operation (m) of the difference operator D in computing an , m 1. Under this requirement, recursion relations (7) uniquely (m) (m) (m) determine an , bn , cn , m 1 and the first few quantities are given by sn rn+1 rn 1 rn+1 rn rn (1) (1) (1) − , − rn sn , , bn+1 = − + cn = −rn + an = rn−1 rn rn rn−1 rn rn rn−1 an(2) = − Now, we set {m} Γ1
=
rn2 2 rn−1
−
rn rn−2
−
rn+1 rn sn−1 rn sn − − , rn−1 rn−1 rn−1
m (i) m−i+1 an λ (i)
i=0
cn λm−i+1
(i)
bn λm−i (i) −an λm−i+1
+
... .
(m+1)
an
0
0 (m+1)
−an
,
m 0.
(8)
The discrete zero-curvature equations {m} {m} Untm = EΓ1 Un − Un Γ1 ,
(9)
give rise to the following positive hierarchy of lattice soliton equations (m+1) , rntm = cn m 0. (m+1) sntm = sn (an+1 − an(m+1) ),
(10)
When m = 0, the resulting lattice system (10) reduces to Eq. (4a). And, the time part of the Lax pairs for Eq. (4a) is as follows 1 rn 1 − 2 λ + rn−1 rn−1 {1} Γ1 = (11) . rn 1 rn λ 2 λ − rn−1 By virtue of trace identity [2], the system (11) can be rewritten in Hamiltonian form (m+2) (m+2) an −an+1 ˜ rn δ Hm+1 rn , m 1, = J1 = J1 untm = (m+1) δu sn tm cn
(12)
rn
in which J1 =
0 −rn
rn rn E −1 r1n −
1 rn Ern
,
H˜ m =
rn an(m+1) + cn(m+1) − (k), (m + 1)rn
k∈Z
m 0.
(13)
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By means of direct calculation, noticing [2,8–10], it is easy to verify that J1 is a Hamiltonian operator. Hence, each equation in (10) or (12) is a discrete Hamiltonian equation. Set Φ11 Φ12 δ H˜ m−1 δ H˜ m . =Φ , Φ= (14) δu δu Φ21 Φ22 Ask for the help of recursion relations (7), we have 1 1 1 rn E −1 − Ern (1 + E)(1 − E)−1 rn , Φ11 = −sn − rn rn rn 1 1 1 Φ12 = − (1 + E) rn E −1 − Ern sn , rn rn rn Φ21 = −(1 + E)(1 − E)−1 rn , Φ22 = −sn , and
J1 Φ = M =
−rn (1 + E)(1 − E)−1 rn rn sn
−rn sn 0
.
From a direct verification, we get that the operator M is Hamiltonian [2,8–10], then (12) possesses twoHamiltonian structures, and, Φ = J1−1 M is a strong symmetry [22]. Moreover, the lattice soliton equations in (10) or the discrete Hamiltonian system in (12) are all discrete Liouville integrable systems [2,10]. Further, it is easy to verify that the operator Φ is invertible, and its inverse operator can be given by Ψ11 Ψ12 −1 , Ψ := Φ = (15) Ψ21 Ψ22 with 1 , sn 1 1 −1 1 rn E − Ern , Ψ12 = rn sn rn rn rn 1 Ψ21 = (1 + E)(1 − E)−1 , sn sn 1 1 1 −1 1 −1 1 rn E − Ern . Ψ22 = − − (1 + E)(1 − E) sn sn sn rn rn Ψ11 = −
Hereinbefore, we have proposed a hierarchy of lattice soliton equations, which can be easily found that they are derived from the discrete isospectral problem (1) and are so-called positive part corresponding to positive power expansion of Lax operators with respect to the spectral parameter. Now, we would like to briefly derive the negative hierarchy of lattice soliton equations related to the negative power expansion about the spectral parameter. To this end, we introduce the auxiliary spectral matrix ∞ (m) m (m) Bn λm−1 An λ . Γ2 = (m) (m) Cn λm −An λm m=0 Hence, the stationary discrete zero-curvature equation (EΓ2 )Un − Un Γ2 = 0,
H.-X. Yang et al. / Physics Letters A 338 (2005) 117–127
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implies that 1 A(0) n =− , 2
Bn(0) = Cn(0) = 0,
(16)
(m)
(m)
rn bn+1 =
cn , rn
1 (m) (m) (m+1) an+1 + an(m) + bn+1 + sn bn+1 = 0, rn (m) rn an+1 + an(m) + cn(m) + sn cn(m+1) = 0, (m+1) 1 (m+1) (m) sn an+1 − an(m+1) = cn+1 − rn bn(m+1) + an(m) − an+1 . rn And the first few quantities can be listed as follows A(1) n =
rn , rn−1 sn−1 sn
(1)
Bn+1 =
1 , rn sn
Cn(1) =
rn , sn
(m)
(17)
... .
Now, we set {−m}
Γ2
=
m (i) −m+i An λ
Cn λ−m+i (i)
i=0
Bn λ−m+i−1 (i) −An λ−m+i (i)
+
−An 0
0 (m) An
m 0.
,
(18)
Then, the discrete zero-curvature equations {−m} {−m} Untm = EΓ2 Un − Un Γ2 , lead to the following negative hierarchy of lattice soliton equations (m) rntm = −Cn , m 0, (m) (m) sntm = sn (An+1 − An ),
(19)
where the first nonlinear lattice system, when m = 1, is given by (4b) coming from negative lattice hierarchy (19). Similarly, the t-part of the Lax pairs for Eq. (4b) are shown as 1 − 2λ rn−1 s1n−1 λ {1} Γ2 = (20) . r n
sn
1 2λ
By using Eqs. (17), the discrete system (19) can be rewritten in the Hamiltonian form (m−1) (m−1) An −An+1 rn 0 δ H¯˜ m−1 rn m−1 = J2 = J2 utm = , m 1, = J2 Ψ 1 (m) δu sn tm Cn s n
rn
where H¯˜ m =
rn an(m) + cn(m) k∈Z
mrn
(k),
m 1,
H¯˜ 0 =
ln sn (k),
(21)
k∈Z
with J2 = −J1 and Ψ are defined by (15). Here the symbol “−” is used to make a distinction between H¯ and H only for convenience in discussion. Therefore, we arrive at a conclusion that J2 is a Hamiltonian operator, and N11 N12 , N = J2 Ψ = N21 N22
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with rn rn (1 + E)(1 − E)−1 , sn sn rn rn 1 −1 1 −1 1 N12 = + (1 + E)1 − E rn E , − sn sn sn rn rn 1 1 rn rn 1 N21 = − + Ern − rn E −1 (1 + E)(1 − E)−1 , sn rn rn sn sn 1 1 1 1 1 1 1 −1 −1 −1 1 −1 1 N22 = Ern − rn E (1 + E)(1 − E) − Ern + − Ern rn E rn E rn rn sn sn rn rn sn rn rn 1 1 1 − rn E −1 − Ern . rn rn sn
N11 = −
is a skew-symmetry operator. Hence, similarly, we obtain the following assertion that every equation in lattice soliton system (19) has the Liouville integrability [2,8,10] as well.
3. Darboux transformation and exact solutions In what follows, a DT for the typical equations of the resulting systems coming from positive and negative hierarchies will be set up. Further, the exact solutions of Eqs. (4) are explicitly given by the application of DT at the end of this part. 3.1. The Darboux transformation First, the eigenvalue problem and auxiliary problem of Eq. (4a) are as follows 0 1 1 Yn , Yn+1 = Un (u, λ)Yn = rn λrn2 λrn + rn sn 1 rn 1 − 2 λ + rn−1 r n−1 Ynt = Vn{1} (u, λ)Yn = Yn . rn 1 λrn 2 λ − rn−1
(22a)
(22b)
We assume that there is a gauge transformation Y˜n = Tn Yn ,
(23)
which can transform two spectral problems into Y˜n+1 = U˜ n Y˜n ,
Y˜nt = V˜n{1} Y˜n ,
with U˜ n = Tn+1 Un Tn−1 ,
V˜n{1} = Tnt + Tn Vn{1} Tn−1 ,
(24)
{1} {1} satisfying a fact that U˜ n , V˜n and Un , Vn have the same schedules, respectively. 1 2 T 1 2 T Let φn = (φn , φn ) , ψn = (ψn , ψn ) are two basic solutions of (22) and use (φn , ψn ) to define a 2 × 2 matrix Tn by Tn12 λ + Tn11 . Tn = (25) λTn21 λ + Tn22
H.-X. Yang et al. / Physics Letters A 338 (2005) 117–127
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In addition, we assume that λ1 and λ2 are two solutions of det Tn = 0. When λ = λi (i = 1, 2), we have 1 ψn1 + λi ψn1 Tn11 + ψn2 Tn12 φn + λi φn1 Tn11 + φn2 Tn12 ˜ , Yn = Tn Yn = λi φn1 Tn21 + φn2 + λi φn2 Tn22 λi ψn1 Tn21 + ψn2 + λi ψn2 Tn22 where two rows in matrix Y˜n are linear-correlated. Hence, there must be coefficients assumed γi (i = 1, 2) such that λi Tn21 + αi (n) λi + Tn22 = 0, λi + Tn11 + αi (n)Tn12 = 0, (26) with αi (n) =
φn2 (λi ) − γi ψn2 (λi ) , φn1 (λi ) − γi ψn1 (λi )
i = 1, 2.
(27)
A direct calculation for Eq. (26) gives us α2 (n)λ1 − α1 (n)λ2 , α1 (n) − α2 (n) α1 (n)α2 (n)(λ1 − λ2 ) , Tn21 = α1 (n)λ2 − α2 (n)λ1
λ2 − λ1 , α1 (n) − α2 (n) λ1 λ2 (α2 (n) − α1 (n)) Tn22 = , α1 (n)λ2 − α2 (n)λ1
Tn11 =
Tn12 =
(28)
where parameters λi and γi (λ1 = λ2 , γ1 = γ2 ) are suitably chosen such that all the denominators in (28), (27) are nonzero. Eqs. (22) and (27) present us αi (n + 1) = µi (n)/νi (n), with
i = 1, 2,
(29)
µi (n) = λi rn + αi (n)(λi + sn ), νi (n) =
αi (n) rn .
By using Eqs. (28) and (29), we have µ2 (n)ν1 (n)λ1 − µ1 (n)ν2 (n)λ2 , µ1 (n)ν2 (n) − µ2 (n)ν1 (n) µ1 (n)µ2 (n)(λ1 − λ2 ) 21 Tn+1 = , µ1 (n)ν2 (n)λ2 − µ2 (n)ν1 (n)λ1
11 = Tn+1
ν1 (n)ν2 (n)(λ2 − λ1 ) , µ1 (n)ν2 (n) − µ2 (n)ν1 (n) (µ2 (n)ν1 (n) − µ1 (n)ν2 (n))λ1 λ2 22 Tn+1 = . µ1 (n)ν2 (n)λ2 − µ2 (n)ν1 (n)λ1
12 Tn+1 =
Through direct and tedious calculations, from (28) and (30), we obtain the relations reading as 12 rn rn − Tn21 Tn+1 − Tn21 = 0.
(30)
(31)
Now, by using Eq. (28) we obtain det Tn = (λ − λ1 )(λ − λ2 ).
(32)
Hence, from all of above statements, we obtain following assertions. Proposition 1. The matrix U˜ n defined by (24) has the same form as Un , that is 1 0 r˜n , U˜ n = λ˜rn λ + s˜n in which the transformation formulae between old and new potentials are defined by r˜n = rn − Tn21 ,
s˜n = sn +
21 Tn+1
rn
22 − rn − Tn21 Tn12 + Tn+1 − Tn22 .
(33)
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The transformation (23) and (33): (Yn ; rn , sn ) → (Y˜n ; r˜n , s˜n ) is usually called a DT of eigenvalue problem (22a). And (33) is so-called a Bäcklund transformation (BT) between new and old potentials. Proof. Let Tn−1 = Tn∗ / det Tn and f11 (λ, n) f12 (λ, n) ∗ . Tn+1 Un Tn = f21 (λ, n) f22 (λ, n) It is easy to verify that f11 (λ, n), f12 (λ, n) or f21 (λ, n), f22 (λ, n) are second or cubic polynomials in λ, respectively. By virtue of (26) and (27), it can be verified that λi are roots of fij (λ, n) (i = 1, 2). Therefore, noticing (32), we have 0 0 p12 p11 ∗ , Tn+1 Un Tn = (det Tn )Pn = (det Tn ) 0 1 + p0 1 λp21 21 λp22 + p22 l (i, j = 1, 2; l = 0, 1) are independent of λ. At the same time, the above equation can be rewritten as where pij
Tn+1 Un = Pn Tn .
(34)
Equating the coefficients of λi (i = 0, 1, 2) in (34), noticing (31), we find 0 0 = p21 = 0, p11
1 p22 = 1,
1 = rn − Tn21 = r˜n , p21 0 = p12
12 rn Tn+1
Tn21
0 p22 = sn +
1 1 = , 21 r˜n rn − Tn
21 Tn+1
22 − rn − Tn21 Tn12 + Tn+1 − Tn22 = s˜n .
rn Thus we complete the proof.
2
In the way similar to the proof of Proposition 1, we can then verify the following fact. {1}
Proposition 2. Under the transformation (33), the matrix V˜n 1 r˜n 1 − 2 λ + r˜n−1 r ˜ n−1 V˜n{1} = . r˜n 1 λ˜rn 2 λ − r˜n−1
{1}
defined by (24) has the same form as Vn , that is
The fact of equivalence between differential-difference equation (4a) with the discrete zero-curvature equation {1} {1} Unt − Vn+1 Un + Un Vn = 0, noticing Propositions 1 and 2, yield following proposition. Proposition 3. Every solution (rn , sn ) of (4a) is mapped into a new solution (˜rn , s˜n ) under the BT (33). Similarly, noticing a fact that Eqs. (4a) and (4b) are based on the same eigenvalue problem (1), we can construct a Darboux matrix similar to (25) for Eq. (4b). So we arrive at another assertion for Eq. (4b). Proposition 4. Every solution (rn , sn ) of (4b) is mapped into a new solution (˜rn , s˜n ) under the DT ¯ 11 T¯ 12 n ˜n = λ + Tn Y Yn , λT¯n21 λ + T¯n22 r˜n = rn − T¯n21 , T¯ 21 s˜n = sn + n+1 − (rn − T¯n21 )T¯n12 + T¯ 22 − T¯n22 . rn
n+1
(35)
H.-X. Yang et al. / Physics Letters A 338 (2005) 117–127
125
Here the symbol “−” is same as that in (21) to make a distinction between letters in context only for the sake of convenience in discussion. Actually, the essential quantities needed for the establishment of DT for Eq. (35) (such 12 , µ ¯ i (n), ν¯ i (n), etc.) are similar in form to those appearing in relations (28), (29) and (30) apart from a as T¯n11 , T¯n+1 little shaped change of λi , αi (n) → λ¯ i , α¯ i (n), respectively. 3.2. Exact solutions As below, some exact solutions will be given by using DT. Substituting the trivial solution rn = 1, sn = −2 of (4a) into (22), then we have 1 1 −2λ + 1 0 1 Yn , Yn , Ynt = Yn+1 = 1 λ λ−2 λ 2λ − 1 from which two real basic solutions of Eq. (4a) are presented as √ √ 2 2 λ +4 λ − 2 + λ2 + 4 n √ , t exp φn = 2 2 λ − 2 + λ2 + 4 √ √ 2 2 λ +4 λ − 2 − λ2 + 4 n √ ψn = , t exp − 2 2 λ − 2 − λ2 + 4 where λ ∈ R − {0}. From (27) we have ξi2n exp(t λ2i + 4 )(λi − 2 + λ2i + 4 ) − γi (λi − 2 − λ2i + 4 ) αi (n) = , 2ξi2n exp(t λ2i + 4 ) − 2γi λi −2+ λ2i +4 √ , 2i |λi |
i = 1, 2, in which i2 = −1. with ξi = So the new solutions of Eq. (4a) by applying BT (33) can be shown as r˜n = 1 +
α1 (n)α2 (n)(λ1 − λ2 ) , λ1 α2 (n) − λ2 α1 (n)
λ1 λ2 (α1 (n) − α2 (n))2 − α1 (n)α2 (n)(λ1 − λ2 )2 (α1 (n) − α2 (n))(λ2 α1 (n) − λ1 α2 (n)) η1 η2 (λ1 − λ2 ) + (η2 α1 (n) − η1 α2 (n))λ1 λ2 λ1 − λ2 , + + η1 α2 (n)λ2 − η2 α1 (n)λ1 α1 (n) − α2 (n)
s˜n = −2 +
with ηi = λi + αi (n)(λi − 2), i = 1, 2. Similarly, from the seed solutions rn = exp(−t), sn = 1, by virtue of DT (35), we can obtain the new solutions of Eq. (4b) r˜n = exp(−t) +
α¯ 1 (n)α¯ 2 (n)(λ¯ 1 − λ¯ 2 ) , λ¯ 1 α¯ 2 (n) − λ¯ 2 α¯ 1 (n)
λ¯ 1 λ¯ 2 (α¯ 1 (n) − α¯ 2 (n))2 − α¯ 1 (n)α¯ 2 (n)(λ¯ 1 − λ¯ 2 )2 (α¯ 1 (n) − α¯ 2 (n))(λ¯ 2 α¯ 1 (n) − λ¯ 1 α¯ 2 (n)) λ¯ 1 − λ¯ 2 η¯ 1 η¯ 2 (λ¯ 1 − λ¯ 2 ) + (η¯ 2 α¯ 1 (n) − η¯ 1 α¯ 2 (n))λ¯ 1 λ¯ 2 + + exp(−t) , α¯ 1 (n) − α¯ 2 (n) η¯ 1 α¯ 2 (n)λ¯ 2 − η¯ 2 α¯ 1 (n)λ¯ 1
s˜n = 1 +
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H.-X. Yang et al. / Physics Letters A 338 (2005) 117–127
with η¯ i = exp(−t)λ¯ i + α¯ i (n)(λ¯ i + 1), i = 1, 2. Where ¯
α¯ i (n) = with ξ¯i =
(ξ¯i )2n exp( λζ¯ i t)(1 + λ¯ i + ζ¯i ) − γ¯i (1 + λ¯ i − ζ¯i )
1+√ λ¯ i +ζ¯i ¯ , ζi 2i |λ¯ i |
i
¯
¯
2(ξ¯i )2n exp( λiλ¯+ζi t) − 2γ¯i =
,
i
(λ¯ i + 1)2 + 4λ¯ i , i = 1, 2 and i2 = −1.
The plots of solutions r˜ , s˜ , for Eq. (4a) by using DT (23) and (33) are given in Fig. 1 with the parameters are chosen as λ1 = −2, λ2 = −0.01, γ1 = 3, γ2 = −1. Further, if the resulting solutions are taken as the new starting
Fig. 1. The soliton solutions with λ1 = −2, λ2 = −0.01, γ1 = 3, γ2 = −1.
H.-X. Yang et al. / Physics Letters A 338 (2005) 117–127
127
point, we can make the DT once again and engender another set of new explicit solutions. This process can be done continuously and the multi-soliton solutions result usually.
4. Conclusions and remarks In this Letter, based on a new discrete isospectral problem, two hierarchies of nonlinear integrable lattice equations are derived. It is shown that every equation in the resulting models is integrable in Liouville sense. It is also shown that these two hierarchies correspond to positive and negative power expansions concerning spectral parameter, respectively. In addition, with the help of the gauge transformations of Lax pairs, a Darboux transformation is established for the first nonlinear equations of resulting hierarchies, from which the exact solutions result. Searching for new integrable discrete systems is still a significant but hard task in soliton theory. The procedure provided in this study, to some extent, presents a design and implementation for new soliton hierarchies of lattice versions. We can assert that there should be some other spectral problems fitting for the present studies even in higher order. Furthermore, from (13) and (14), we have concluded that both operators −r(1 + E)(1 − E)−1 r −rs 0 r , , M = J1 = rs 0 −r rE −1 1r − 1r Er are Hamiltonian, which prompts us with a bold conjecture of the bi-Hamiltonian structure for system (12). This conjecture, together with other properties of resulting hierarchies, would be fulfilled in other papers.
Acknowledgements The authors would like to express their sincere thanks to referees for their helpful suggestions.
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Two hierarchies of lattice soliton equations associated with a new discrete eigenvalue problem and Darboux transformation Hong-Xiang Yang a,∗ , Xi-Xiang Xu b , Hai-Yong Ding c a Department of Computer Science and Technology, Taishan College, Taian 271021, China b College of Science, Shandong University of Science and Technology, Qingdao 266510, China c College of Information Science and Engineering, Shandong Agricultural University, Taian 271018, China
Received 5 November 2004; received in revised form 25 January 2005; accepted 11 February 2005
Communicated by A.P. Fordy
Abstract By considering a new discrete isospectral eigenvalue problem, two hierarchies of integrable positive and negative lattice models are derived. It is shown that they correspond to positive and negative power expansions of Lax operators with respect to the spectral parameter, respectively. And, each equation in the resulting hierarchies is integrable in Liouville sense. Moreover, a Darboux transformation is established for the typical equations by using gauge transformations of Lax pairs, from which the exact solutions are given. 2005 Elsevier B.V. All rights reserved. PACS: 02.30.Ik; 02.90.+p Keywords: Integrable lattice model; Positive and negative hierarchy; Darboux transformation; Soliton solutions
1. Introduction The role, is well known, played by nonlinear evolution equations of lattice versions in domain of nonlinear science. The investigation on integrable discrete systems and interrelated properties has always been one of the prominent events and has become the focus of common concern in the past few years. During the last decade, a considerable number of integrable lattice systems have been investigated successfully such as the Ablowitz–Ladik lattice [1], the Toda lattice [2], the Blaszak–Marciniak lattice [3,4], and so on [5–10]. The procedure of Lax pair * Corresponding author.
E-mail address: [email protected] (H.-X. Yang). 0375-9601/$ – see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2005.02.021
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is certainly a significant method to obtain the new Lax integrable lattice equations. So far, it is, however, still hard work to be carried out, mainly due to the difficulty in choosing a proper spectral problem. At the same time, quite a few systematic methods used for obtaining exact solutions of (1 + 1)-dimensional soliton equations are well established, such as the inverse scattering transform method [11,12], the bilinear transform method of Hirota [13], the Bäcklund [14] and Darboux transformation, Wronskian technique [15] and so forth. The Darboux transformation (DT), which is always used for proposing the explicit solutions of the soliton equations, was first introduced by Darboux in his study of Sturm–Liouville equation [16]. Extensive applications of the DT can be found in articles [17–21]. This Letter is devoted to introducing a new discrete isospectral problem 0 1 1 , Eψn (λ) = Un (u, λ)ψn (λ), Un (u, λ) = (1) rn λrn2 λrn + rn sn where the shift operator E is defined as Efn = f (n + 1) = fn+1 , E −1 fn = f (n − 1) = fn−1 , n ∈ Z, and u = (rn , sn )T is a 2-component potential function vector defined over R × Z and vanishes rapidly if |n| → ∞, ψn (λ) = (ψn1 (λ), ψn2 (λ))T , and, λ is the spectral parameter with λt = 0. By means of constructing a proper continuous time evolution equation d ψn (λ) = Γn{m} (u, λ)ψn (λ), dt
(2)
{m}
in which Γn (u, λ) is a suitable 2 × 2 matrix, and using the discrete zero-curvature equation d Un − EΓn{m} Un + Un Γn{m} = 0, (3) dt a pair of positive and negative hierarchies of integrable nonlinear lattice models are derived [9]. As the typical examples, the first two nonlinear lattice equations are given as follows, respectively, rn rnt = −rn rn−1 − rn sn , + rn+1 rn rn (4a) rn+1 snt = sn rn−1 − rn , rnt = − rsnn , (4b) rn snt = rnrn+1 sn+1 − rn−1 sn−1 . It is shown that these two hierarchies correspond to power expansions of Lax operators with respect to the spectral parameter, respectively, and each equation in resulting hierarchies is Liouville integrable. Further, by virtue of the gauge transformation, we find a DT for spectral problem (1). As an application, exact solutions of Eqs. (4) are given.
2. Integrable lattice models To get the associated lattice soliton hierarchies, we first solve the stationary discrete zero-curvature equation (EΓ1 )Un − Un Γ1 = 0,
(5)
where Γ1 =
∞ (m) −m an λ
cn λ−m (m)
m=0
bn λ−m−1 (m) −an λ−m (m)
.
H.-X. Yang et al. / Physics Letters A 338 (2005) 117–127
Then Eq. (5) leads to the initial relations (0) 1 (0) (0) an+1 + an(0) + bn+1 = 0, rn an+1 + an(0) + cn(0) = 0, rn and the recursion relations
119
(0) an+1 − an(0) = 0,
(6)
(m)
(m)
rn bn+1 =
cn , rn
1 (m+1) (m+1) (m) an+1 + an(m+1) + bn+1 + sn bn+1 = 0, rn (m+1) + an(m+1) + cn(m+1) + sn cn(m) = 0, rn an+1 1 (m) (m+1) (m) an+1 − an(m+1) = cn+1 − rn bn(m) + sn an(m) − an+1 . rn (0)
(0)
(7)
(0)
Now we choose an = −1/2, bn = 1/rn−1 , cn = rn and require selecting zero constants for the inverse operation (m) of the difference operator D in computing an , m 1. Under this requirement, recursion relations (7) uniquely (m) (m) (m) determine an , bn , cn , m 1 and the first few quantities are given by sn rn+1 rn 1 rn+1 rn rn (1) (1) (1) − , − rn sn , , bn+1 = − + cn = −rn + an = rn−1 rn rn rn−1 rn rn rn−1 an(2) = − Now, we set {m} Γ1
=
rn2 2 rn−1
−
rn rn−2
−
rn+1 rn sn−1 rn sn − − , rn−1 rn−1 rn−1
m (i) m−i+1 an λ (i)
i=0
cn λm−i+1
(i)
bn λm−i (i) −an λm−i+1
+
... .
(m+1)
an
0
0 (m+1)
−an
,
m 0.
(8)
The discrete zero-curvature equations {m} {m} Untm = EΓ1 Un − Un Γ1 ,
(9)
give rise to the following positive hierarchy of lattice soliton equations (m+1) , rntm = cn m 0. (m+1) sntm = sn (an+1 − an(m+1) ),
(10)
When m = 0, the resulting lattice system (10) reduces to Eq. (4a). And, the time part of the Lax pairs for Eq. (4a) is as follows 1 rn 1 − 2 λ + rn−1 rn−1 {1} Γ1 = (11) . rn 1 rn λ 2 λ − rn−1 By virtue of trace identity [2], the system (11) can be rewritten in Hamiltonian form (m+2) (m+2) an −an+1 ˜ rn δ Hm+1 rn , m 1, = J1 = J1 untm = (m+1) δu sn tm cn
(12)
rn
in which J1 =
0 −rn
rn rn E −1 r1n −
1 rn Ern
,
H˜ m =
rn an(m+1) + cn(m+1) − (k), (m + 1)rn
k∈Z
m 0.
(13)
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By means of direct calculation, noticing [2,8–10], it is easy to verify that J1 is a Hamiltonian operator. Hence, each equation in (10) or (12) is a discrete Hamiltonian equation. Set Φ11 Φ12 δ H˜ m−1 δ H˜ m . =Φ , Φ= (14) δu δu Φ21 Φ22 Ask for the help of recursion relations (7), we have 1 1 1 rn E −1 − Ern (1 + E)(1 − E)−1 rn , Φ11 = −sn − rn rn rn 1 1 1 Φ12 = − (1 + E) rn E −1 − Ern sn , rn rn rn Φ21 = −(1 + E)(1 − E)−1 rn , Φ22 = −sn , and
J1 Φ = M =
−rn (1 + E)(1 − E)−1 rn rn sn
−rn sn 0
.
From a direct verification, we get that the operator M is Hamiltonian [2,8–10], then (12) possesses twoHamiltonian structures, and, Φ = J1−1 M is a strong symmetry [22]. Moreover, the lattice soliton equations in (10) or the discrete Hamiltonian system in (12) are all discrete Liouville integrable systems [2,10]. Further, it is easy to verify that the operator Φ is invertible, and its inverse operator can be given by Ψ11 Ψ12 −1 , Ψ := Φ = (15) Ψ21 Ψ22 with 1 , sn 1 1 −1 1 rn E − Ern , Ψ12 = rn sn rn rn rn 1 Ψ21 = (1 + E)(1 − E)−1 , sn sn 1 1 1 −1 1 −1 1 rn E − Ern . Ψ22 = − − (1 + E)(1 − E) sn sn sn rn rn Ψ11 = −
Hereinbefore, we have proposed a hierarchy of lattice soliton equations, which can be easily found that they are derived from the discrete isospectral problem (1) and are so-called positive part corresponding to positive power expansion of Lax operators with respect to the spectral parameter. Now, we would like to briefly derive the negative hierarchy of lattice soliton equations related to the negative power expansion about the spectral parameter. To this end, we introduce the auxiliary spectral matrix ∞ (m) m (m) Bn λm−1 An λ . Γ2 = (m) (m) Cn λm −An λm m=0 Hence, the stationary discrete zero-curvature equation (EΓ2 )Un − Un Γ2 = 0,
H.-X. Yang et al. / Physics Letters A 338 (2005) 117–127
121
implies that 1 A(0) n =− , 2
Bn(0) = Cn(0) = 0,
(16)
(m)
(m)
rn bn+1 =
cn , rn
1 (m) (m) (m+1) an+1 + an(m) + bn+1 + sn bn+1 = 0, rn (m) rn an+1 + an(m) + cn(m) + sn cn(m+1) = 0, (m+1) 1 (m+1) (m) sn an+1 − an(m+1) = cn+1 − rn bn(m+1) + an(m) − an+1 . rn And the first few quantities can be listed as follows A(1) n =
rn , rn−1 sn−1 sn
(1)
Bn+1 =
1 , rn sn
Cn(1) =
rn , sn
(m)
(17)
... .
Now, we set {−m}
Γ2
=
m (i) −m+i An λ
Cn λ−m+i (i)
i=0
Bn λ−m+i−1 (i) −An λ−m+i (i)
+
−An 0
0 (m) An
m 0.
,
(18)
Then, the discrete zero-curvature equations {−m} {−m} Untm = EΓ2 Un − Un Γ2 , lead to the following negative hierarchy of lattice soliton equations (m) rntm = −Cn , m 0, (m) (m) sntm = sn (An+1 − An ),
(19)
where the first nonlinear lattice system, when m = 1, is given by (4b) coming from negative lattice hierarchy (19). Similarly, the t-part of the Lax pairs for Eq. (4b) are shown as 1 − 2λ rn−1 s1n−1 λ {1} Γ2 = (20) . r n
sn
1 2λ
By using Eqs. (17), the discrete system (19) can be rewritten in the Hamiltonian form (m−1) (m−1) An −An+1 rn 0 δ H¯˜ m−1 rn m−1 = J2 = J2 utm = , m 1, = J2 Ψ 1 (m) δu sn tm Cn s n
rn
where H¯˜ m =
rn an(m) + cn(m) k∈Z
mrn
(k),
m 1,
H¯˜ 0 =
ln sn (k),
(21)
k∈Z
with J2 = −J1 and Ψ are defined by (15). Here the symbol “−” is used to make a distinction between H¯ and H only for convenience in discussion. Therefore, we arrive at a conclusion that J2 is a Hamiltonian operator, and N11 N12 , N = J2 Ψ = N21 N22
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with rn rn (1 + E)(1 − E)−1 , sn sn rn rn 1 −1 1 −1 1 N12 = + (1 + E)1 − E rn E , − sn sn sn rn rn 1 1 rn rn 1 N21 = − + Ern − rn E −1 (1 + E)(1 − E)−1 , sn rn rn sn sn 1 1 1 1 1 1 1 −1 −1 −1 1 −1 1 N22 = Ern − rn E (1 + E)(1 − E) − Ern + − Ern rn E rn E rn rn sn sn rn rn sn rn rn 1 1 1 − rn E −1 − Ern . rn rn sn
N11 = −
is a skew-symmetry operator. Hence, similarly, we obtain the following assertion that every equation in lattice soliton system (19) has the Liouville integrability [2,8,10] as well.
3. Darboux transformation and exact solutions In what follows, a DT for the typical equations of the resulting systems coming from positive and negative hierarchies will be set up. Further, the exact solutions of Eqs. (4) are explicitly given by the application of DT at the end of this part. 3.1. The Darboux transformation First, the eigenvalue problem and auxiliary problem of Eq. (4a) are as follows 0 1 1 Yn , Yn+1 = Un (u, λ)Yn = rn λrn2 λrn + rn sn 1 rn 1 − 2 λ + rn−1 r n−1 Ynt = Vn{1} (u, λ)Yn = Yn . rn 1 λrn 2 λ − rn−1
(22a)
(22b)
We assume that there is a gauge transformation Y˜n = Tn Yn ,
(23)
which can transform two spectral problems into Y˜n+1 = U˜ n Y˜n ,
Y˜nt = V˜n{1} Y˜n ,
with U˜ n = Tn+1 Un Tn−1 ,
V˜n{1} = Tnt + Tn Vn{1} Tn−1 ,
(24)
{1} {1} satisfying a fact that U˜ n , V˜n and Un , Vn have the same schedules, respectively. 1 2 T 1 2 T Let φn = (φn , φn ) , ψn = (ψn , ψn ) are two basic solutions of (22) and use (φn , ψn ) to define a 2 × 2 matrix Tn by Tn12 λ + Tn11 . Tn = (25) λTn21 λ + Tn22
H.-X. Yang et al. / Physics Letters A 338 (2005) 117–127
123
In addition, we assume that λ1 and λ2 are two solutions of det Tn = 0. When λ = λi (i = 1, 2), we have 1 ψn1 + λi ψn1 Tn11 + ψn2 Tn12 φn + λi φn1 Tn11 + φn2 Tn12 ˜ , Yn = Tn Yn = λi φn1 Tn21 + φn2 + λi φn2 Tn22 λi ψn1 Tn21 + ψn2 + λi ψn2 Tn22 where two rows in matrix Y˜n are linear-correlated. Hence, there must be coefficients assumed γi (i = 1, 2) such that λi Tn21 + αi (n) λi + Tn22 = 0, λi + Tn11 + αi (n)Tn12 = 0, (26) with αi (n) =
φn2 (λi ) − γi ψn2 (λi ) , φn1 (λi ) − γi ψn1 (λi )
i = 1, 2.
(27)
A direct calculation for Eq. (26) gives us α2 (n)λ1 − α1 (n)λ2 , α1 (n) − α2 (n) α1 (n)α2 (n)(λ1 − λ2 ) , Tn21 = α1 (n)λ2 − α2 (n)λ1
λ2 − λ1 , α1 (n) − α2 (n) λ1 λ2 (α2 (n) − α1 (n)) Tn22 = , α1 (n)λ2 − α2 (n)λ1
Tn11 =
Tn12 =
(28)
where parameters λi and γi (λ1 = λ2 , γ1 = γ2 ) are suitably chosen such that all the denominators in (28), (27) are nonzero. Eqs. (22) and (27) present us αi (n + 1) = µi (n)/νi (n), with
i = 1, 2,
(29)
µi (n) = λi rn + αi (n)(λi + sn ), νi (n) =
αi (n) rn .
By using Eqs. (28) and (29), we have µ2 (n)ν1 (n)λ1 − µ1 (n)ν2 (n)λ2 , µ1 (n)ν2 (n) − µ2 (n)ν1 (n) µ1 (n)µ2 (n)(λ1 − λ2 ) 21 Tn+1 = , µ1 (n)ν2 (n)λ2 − µ2 (n)ν1 (n)λ1
11 = Tn+1
ν1 (n)ν2 (n)(λ2 − λ1 ) , µ1 (n)ν2 (n) − µ2 (n)ν1 (n) (µ2 (n)ν1 (n) − µ1 (n)ν2 (n))λ1 λ2 22 Tn+1 = . µ1 (n)ν2 (n)λ2 − µ2 (n)ν1 (n)λ1
12 Tn+1 =
Through direct and tedious calculations, from (28) and (30), we obtain the relations reading as 12 rn rn − Tn21 Tn+1 − Tn21 = 0.
(30)
(31)
Now, by using Eq. (28) we obtain det Tn = (λ − λ1 )(λ − λ2 ).
(32)
Hence, from all of above statements, we obtain following assertions. Proposition 1. The matrix U˜ n defined by (24) has the same form as Un , that is 1 0 r˜n , U˜ n = λ˜rn λ + s˜n in which the transformation formulae between old and new potentials are defined by r˜n = rn − Tn21 ,
s˜n = sn +
21 Tn+1
rn
22 − rn − Tn21 Tn12 + Tn+1 − Tn22 .
(33)
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The transformation (23) and (33): (Yn ; rn , sn ) → (Y˜n ; r˜n , s˜n ) is usually called a DT of eigenvalue problem (22a). And (33) is so-called a Bäcklund transformation (BT) between new and old potentials. Proof. Let Tn−1 = Tn∗ / det Tn and f11 (λ, n) f12 (λ, n) ∗ . Tn+1 Un Tn = f21 (λ, n) f22 (λ, n) It is easy to verify that f11 (λ, n), f12 (λ, n) or f21 (λ, n), f22 (λ, n) are second or cubic polynomials in λ, respectively. By virtue of (26) and (27), it can be verified that λi are roots of fij (λ, n) (i = 1, 2). Therefore, noticing (32), we have 0 0 p12 p11 ∗ , Tn+1 Un Tn = (det Tn )Pn = (det Tn ) 0 1 + p0 1 λp21 21 λp22 + p22 l (i, j = 1, 2; l = 0, 1) are independent of λ. At the same time, the above equation can be rewritten as where pij
Tn+1 Un = Pn Tn .
(34)
Equating the coefficients of λi (i = 0, 1, 2) in (34), noticing (31), we find 0 0 = p21 = 0, p11
1 p22 = 1,
1 = rn − Tn21 = r˜n , p21 0 = p12
12 rn Tn+1
Tn21
0 p22 = sn +
1 1 = , 21 r˜n rn − Tn
21 Tn+1
22 − rn − Tn21 Tn12 + Tn+1 − Tn22 = s˜n .
rn Thus we complete the proof.
2
In the way similar to the proof of Proposition 1, we can then verify the following fact. {1}
Proposition 2. Under the transformation (33), the matrix V˜n 1 r˜n 1 − 2 λ + r˜n−1 r ˜ n−1 V˜n{1} = . r˜n 1 λ˜rn 2 λ − r˜n−1
{1}
defined by (24) has the same form as Vn , that is
The fact of equivalence between differential-difference equation (4a) with the discrete zero-curvature equation {1} {1} Unt − Vn+1 Un + Un Vn = 0, noticing Propositions 1 and 2, yield following proposition. Proposition 3. Every solution (rn , sn ) of (4a) is mapped into a new solution (˜rn , s˜n ) under the BT (33). Similarly, noticing a fact that Eqs. (4a) and (4b) are based on the same eigenvalue problem (1), we can construct a Darboux matrix similar to (25) for Eq. (4b). So we arrive at another assertion for Eq. (4b). Proposition 4. Every solution (rn , sn ) of (4b) is mapped into a new solution (˜rn , s˜n ) under the DT ¯ 11 T¯ 12 n ˜n = λ + Tn Y Yn , λT¯n21 λ + T¯n22 r˜n = rn − T¯n21 , T¯ 21 s˜n = sn + n+1 − (rn − T¯n21 )T¯n12 + T¯ 22 − T¯n22 . rn
n+1
(35)
H.-X. Yang et al. / Physics Letters A 338 (2005) 117–127
125
Here the symbol “−” is same as that in (21) to make a distinction between letters in context only for the sake of convenience in discussion. Actually, the essential quantities needed for the establishment of DT for Eq. (35) (such 12 , µ ¯ i (n), ν¯ i (n), etc.) are similar in form to those appearing in relations (28), (29) and (30) apart from a as T¯n11 , T¯n+1 little shaped change of λi , αi (n) → λ¯ i , α¯ i (n), respectively. 3.2. Exact solutions As below, some exact solutions will be given by using DT. Substituting the trivial solution rn = 1, sn = −2 of (4a) into (22), then we have 1 1 −2λ + 1 0 1 Yn , Yn , Ynt = Yn+1 = 1 λ λ−2 λ 2λ − 1 from which two real basic solutions of Eq. (4a) are presented as √ √ 2 2 λ +4 λ − 2 + λ2 + 4 n √ , t exp φn = 2 2 λ − 2 + λ2 + 4 √ √ 2 2 λ +4 λ − 2 − λ2 + 4 n √ ψn = , t exp − 2 2 λ − 2 − λ2 + 4 where λ ∈ R − {0}. From (27) we have ξi2n exp(t λ2i + 4 )(λi − 2 + λ2i + 4 ) − γi (λi − 2 − λ2i + 4 ) αi (n) = , 2ξi2n exp(t λ2i + 4 ) − 2γi λi −2+ λ2i +4 √ , 2i |λi |
i = 1, 2, in which i2 = −1. with ξi = So the new solutions of Eq. (4a) by applying BT (33) can be shown as r˜n = 1 +
α1 (n)α2 (n)(λ1 − λ2 ) , λ1 α2 (n) − λ2 α1 (n)
λ1 λ2 (α1 (n) − α2 (n))2 − α1 (n)α2 (n)(λ1 − λ2 )2 (α1 (n) − α2 (n))(λ2 α1 (n) − λ1 α2 (n)) η1 η2 (λ1 − λ2 ) + (η2 α1 (n) − η1 α2 (n))λ1 λ2 λ1 − λ2 , + + η1 α2 (n)λ2 − η2 α1 (n)λ1 α1 (n) − α2 (n)
s˜n = −2 +
with ηi = λi + αi (n)(λi − 2), i = 1, 2. Similarly, from the seed solutions rn = exp(−t), sn = 1, by virtue of DT (35), we can obtain the new solutions of Eq. (4b) r˜n = exp(−t) +
α¯ 1 (n)α¯ 2 (n)(λ¯ 1 − λ¯ 2 ) , λ¯ 1 α¯ 2 (n) − λ¯ 2 α¯ 1 (n)
λ¯ 1 λ¯ 2 (α¯ 1 (n) − α¯ 2 (n))2 − α¯ 1 (n)α¯ 2 (n)(λ¯ 1 − λ¯ 2 )2 (α¯ 1 (n) − α¯ 2 (n))(λ¯ 2 α¯ 1 (n) − λ¯ 1 α¯ 2 (n)) λ¯ 1 − λ¯ 2 η¯ 1 η¯ 2 (λ¯ 1 − λ¯ 2 ) + (η¯ 2 α¯ 1 (n) − η¯ 1 α¯ 2 (n))λ¯ 1 λ¯ 2 + + exp(−t) , α¯ 1 (n) − α¯ 2 (n) η¯ 1 α¯ 2 (n)λ¯ 2 − η¯ 2 α¯ 1 (n)λ¯ 1
s˜n = 1 +
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H.-X. Yang et al. / Physics Letters A 338 (2005) 117–127
with η¯ i = exp(−t)λ¯ i + α¯ i (n)(λ¯ i + 1), i = 1, 2. Where ¯
α¯ i (n) = with ξ¯i =
(ξ¯i )2n exp( λζ¯ i t)(1 + λ¯ i + ζ¯i ) − γ¯i (1 + λ¯ i − ζ¯i )
1+√ λ¯ i +ζ¯i ¯ , ζi 2i |λ¯ i |
i
¯
¯
2(ξ¯i )2n exp( λiλ¯+ζi t) − 2γ¯i =
,
i
(λ¯ i + 1)2 + 4λ¯ i , i = 1, 2 and i2 = −1.
The plots of solutions r˜ , s˜ , for Eq. (4a) by using DT (23) and (33) are given in Fig. 1 with the parameters are chosen as λ1 = −2, λ2 = −0.01, γ1 = 3, γ2 = −1. Further, if the resulting solutions are taken as the new starting
Fig. 1. The soliton solutions with λ1 = −2, λ2 = −0.01, γ1 = 3, γ2 = −1.
H.-X. Yang et al. / Physics Letters A 338 (2005) 117–127
127
point, we can make the DT once again and engender another set of new explicit solutions. This process can be done continuously and the multi-soliton solutions result usually.
4. Conclusions and remarks In this Letter, based on a new discrete isospectral problem, two hierarchies of nonlinear integrable lattice equations are derived. It is shown that every equation in the resulting models is integrable in Liouville sense. It is also shown that these two hierarchies correspond to positive and negative power expansions concerning spectral parameter, respectively. In addition, with the help of the gauge transformations of Lax pairs, a Darboux transformation is established for the first nonlinear equations of resulting hierarchies, from which the exact solutions result. Searching for new integrable discrete systems is still a significant but hard task in soliton theory. The procedure provided in this study, to some extent, presents a design and implementation for new soliton hierarchies of lattice versions. We can assert that there should be some other spectral problems fitting for the present studies even in higher order. Furthermore, from (13) and (14), we have concluded that both operators −r(1 + E)(1 − E)−1 r −rs 0 r , , M = J1 = rs 0 −r rE −1 1r − 1r Er are Hamiltonian, which prompts us with a bold conjecture of the bi-Hamiltonian structure for system (12). This conjecture, together with other properties of resulting hierarchies, would be fulfilled in other papers.
Acknowledgements The authors would like to express their sincere thanks to referees for their helpful suggestions.
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