- Email: [email protected]
New integrable lattice hierarchies
Physics Letters A 349 (2006) 439–445 www.elsevier.com/locate/pla
New integrable lattice hierarchies Andrew Pickering a , Zuo-Nong Zhu b,c,∗ a Area de Matemática Aplicada, ESCET, Universidad Rey Juan Carlos, c/ Tulipán s/n, 28933 Móstoles, Madrid, Spain b Departamento de Matemáticas, Universidad de Salamanca, Plaza de la Merced 1, 37008 Salamanca, Spain c Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200030, PR China
Received 10 June 2005; received in revised form 6 September 2005; accepted 15 September 2005 Available online 23 September 2005 Communicated by A.R. Bishop
Abstract In this Letter we give a new integrable four-field lattice hierarchy, associated to a new discrete spectral problem. We obtain our hierarchy as the compatibility condition of this spectral problem and an associated equation, constructed herein, for the time-evolution of eigenfunctions. We consider reductions of our hierarchy, which also of course admit discrete zero curvature representations, in detail. We find that our hierarchy includes many well-known integrable hierarchies as special cases, including the Toda lattice hierarchy, the modified Toda lattice hierarchy, the relativistic Toda lattice hierarchy, and the Volterra lattice hierarchy. We also obtain here a new integrable two-field lattice hierarchy, to which we give the name of Suris lattice hierarchy, since the first equation of this hierarchy has previously been given by Suris. The Hamiltonian structure of the Suris lattice hierarchy is obtained by means of a trace identity formula. 2005 Elsevier B.V. All rights reserved.
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 not only possess rich mathematical structures, but also have many applications in science, e.g., in mathematical physics, numerical analysis, statistical physics, and quantum physics. Perhaps the best known integrable lattice equation is the Toda lattice equation, which governs a system of unit masses connected by nonlinear springs whose restoring force is exponential. It has been shown that the Toda lattice possesses all the usual integrability properties, such as Lax pairs, Hamiltonian structures, infinitely many conservation laws, Bäcklund transformations, and soliton solutions [1–5]. Remarkably, the Toda lattice itself has been a source of new integrable lattices, with new integrable lattices related to the Toda lattice having been proposed, e.g., the modified Toda lattice [6], and the relativistic Toda lattice [7–10]. In its turn, the relativistic Toda lattice hierarchy has also been a source of new lattice * Corresponding author.
E-mail address: [email protected] (Z.-N. Zhu). 0375-9601/$ – see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2005.09.042
equations: Suris gave in [11] a new lattice equation related, via a highly non-trivial Bäcklund transformation, to the relativistic Toda lattice equation. The derivation of new integrable lattice hierarchies is of course an interesting and important topic. In this Letter, we consider the following discrete spectral problem: Eψ(n, t, λ) = U (u, λ)ψ(n, t, λ), λ + p(n, t) q(n, t) U (u, λ) = , λr(n, t) + s(n, t) θ
(1.1)
where the potential u = (p(n, t), q(n, t), r(n, t), s(n, t))T , θ = ±1, or 0, and the shift operator E is defined by Ef (n) = f (n + 1). To the best of our knowledge, the above discrete spectral problem has not before appeared in the literature. By constructing time-evolution equations for the eigenfunctions, and considering the associated discrete zero curvature condition, we obtain in Section 2 a new integrable lattice hierarchy. The approach used is by now well known; see, e.g., Refs. [12,13]. Reductions of our new hierarchy are discussed in detail in Section 3. We find that our integrable lattice hierarchy encompasses many well-known integrable lattice hierarchies. Amongst these are the above-mentioned Toda lattice, modi-
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fied Toda lattice, and relativistic Toda lattice hierarchies, as well as the Volterra lattice hierarchy. We also obtain the twofield lattice hierarchy derived by Merola, Ragnisco and Tu [14], the two-field lattice hierarchy of Ma and Xu [15], the threefield lattice hierarchy of Zhang et al. [16], as well as other integrable three-field lattice hierarchies. Within the context of two-field lattice hierarchies, perhaps our most important result is another new integrable lattice hierarchy obtained here, and which we name the Suris lattice hierarchy. It has as first member the above-mentioned lattice equation, related to the relativistic Toda lattice equation, obtained by Suris. We believe that the hierarchy corresponding to the Suris lattice equation has not previously been given. We also obtain, by means of a trace identity formula, the Hamiltonian structure of this hierarchy. Thus, we see that the above spectral problem provides a unifying approach to a wide variety of known and new lattice hierarchies. 2. Integrable lattice hierarchy related to the spectral problem (1.1) In this section, we construct the integrable lattice hierarchy related to the discrete spectral problem (1.1). In the course of this construction, and also later in the Letter, we will use at times, for reasons of simplicity, the following notation: f (1) = Ef (n),
(EΓ )Un − Un Γ = 0,
b −a
=
(2.1)
∞ am (n, t) m=0
cm (n, t)
(2.5)
θ Da + (λrn + sn )b − qn c(1) = 0,
(2.7)
(2.6)
where the difference operator D is defined by D = E − 1. It follows from Eqs. (2.4)–(2.6) that λ Da + θ (λrn + sn )b − θ qn c(1) = −pn Da + θ (λrn + sn )b − θ qn c(1) , if θ = ±1, and λ (λrn + sn )b − qn c(1) = −pn (λrn + sn )b − qn c(1) ,
if θ = 0.
So, Eq. (2.7) follows from Eqs. (2.4)–(2.6). Then, solving Eqs. (2.4)–(2.6), we have (1) (1) − rn bm+1 , am+1 = D −1 qn cm − pn Dam − sn bm (1) (1) + am − pn bm + θ bm , bm+1 = qn am (−1) (−1) cm+1 = rn−1 am+1 + am+1 + sn−1 am + am (−1) , − pn−1 cm + θ cm
(2.8)
and
fn = f (n, t).
The process of finding the integrable lattice hierarchy corresponding to (1.1) can be divided into two steps. The first step is to solve the stationary discrete zero curvature equation
where a Γ = c
λb = qn a (1) + a − pn b + θ b(1) , λc(1) = (λrn + sn ) a (1) + a − pn c(1) + θ c,
bm (n, t) −am (n, t)
b0 = 0,
(1) (1) c0 − rn a0 + a0 = 0,
(2.9)
where the operator D −1 is the formal inverse of D. We choose the constant a0 = 1/2, and so obtain a0 = 1/2, b0 = 0 and c0 = rn−1 , and then solve Eq. (2.8) through the following path: b1 → a1 → c1 → · · · → bm+1 → am+1 → cm+1 → · · ·
λ−m ,
to obtain
with a, b, c functions of the potential u. The solvability of Eq. (2.1) is crucial to obtaining an integrable lattice hierarchy. Then, we construct an equation for the time-evolution of the eigenfunctions dψn = V(m) (u, λ)ψn , dtm
(1)
Da0 + rn b0 = 0,
(2.2)
where m Γi λm−i + ∆m (u, λ), V(m) = λm Γ + + ∆m (u, λ) =
a1 = −qn rn−1 ,
b1 = q n ,
c1 = −rn−1 (pn−1 + qn rn−1 + qn−1 rn−2 ) + θ rn−2 + sn−1 , b2 = −qn (qn+1 rn + qn rn−1 + pn ) + θ qn+1 , a2 = qn pn−1 rn−1 + pn rn−1 − sn−1 + qn+1 rn rn−1 2 − θ (qn+1 rn−1 + qn rn−2 ), + qn−1 rn−1 rn−2 + qn rn−1 (−1) (−1) (−1) c2 = rn−1 a2 + a2 + sn−1 a1 + a1 − pn−1 c1 + θ c1 ,
We now realize these two steps for our choice of discrete spectral problem (1.1). The stationary discrete zero curvature equation (2.1) is equivalent to the following system:
and so on. The formal solvability of the system (2.8)–(2.9) is evident. However, one has to invert a difference operator in order to solve for am+1 . This then leads to the question of whether all am , bm and cm (m 0) are local. It turns out that they are so. Introduce a matrix V by Γ = V Un [12]: it follows from Eq. (2.1) that EΓ = Un V , and so D(trace(Γ m )) = trace((EΓ )m ) − trace(Γ m ) = trace((Un V )m ) − trace((V Un )m ) = 0. We thus obtain trace(Γ m ) = γm (t), a function of t only. From (2.8) we see that bm+1 and cm+1 are defined locally in terms of am , bm , am+1 and cm ,
λDa = qn c − pn Da − (λrn + sn )b(1) ,
bm+1 = f1 (am , bm ),
i=0
and where the matrix ∆m (u, λ) is an appropriate modification matrix, such that the integrable lattice hierarchy is derived from the following discrete zero curvature equation: U˙ n = EVn(m) Un − Un Vn(m) . (2.3)
(2.4)
cm+1 = f2 (am+1 , am , cm );
A. Pickering, Z.-N. Zhu / Physics Letters A 349 (2006) 439–445
now, from a 2 + bc = 12 trace(Γ 2 ) = γ2 (t), we add to these last the equation am+1 = −
m
ai am+1−i −
i=1
m+1
bi cm+1−i ,
m 0,
(2.10)
i=1
where we have used the facts that a0 = 1/2 and b0 = 0, and where we have taken γ2 (t) = 0 (this is in fact equivalent to setting to zero the arbitrary function of t that arises when, for each m 0, we solve the first of (2.8) for am+1 ). Then, since a0 , b0 , and c0 are local, we have that b1 is local; then a1 = −b1 c0 is local, and finally c1 is local. So, by mathematical induction, we conclude that am , bm and cm , for m 0, are local. It should also be noted that (2.10) gives us an alternative means of obtaining am+1 , m 0, i.e., algebraically rather than by solving a difference equation. Next we obtain an integrable lattice hierarchy from the discrete zero curvature equation (2.3), using an appropriate modification matrix ∆m . Assuming that Eqs. (2.8) and (2.9) hold, we obtain m m A B E λ Γ + Un − Un λ Γ + = (2.11) , C F where (1) − qn c m , A = pn Dam + sn bm (1) (1) B = qn am + am − pn bm + θ bm , (1) (1) C = pn cm − sn am + am − θ cm ,
F
(1) = qn cm
− sn bm − θ Dam .
Note that p˙ n ˙ Un = λ˙rn + s˙n
q˙n 0
(2.12)
∆m =
αm λem + hm
vm λδm + βm
(1) ∆22 = λ qn em − rn vm + θ Dδm + qn h(1) m − sn vm + θ Dβm . So the following conditions must be satisfied: (1) Dαm + rn vm − qn em = 0,
vm + qn δm = 0, (1) (1) + rn δm = 0, qn em − rn vm + θ Dδm = 0, (1) qn hm − sn vm + θ Dβm + F = 0. (1) em
Thus, in order to have a four-field lattice hierarchy, we must have δm = const (here 0 or 1), and qn rn−1 δm + α(t) −qn δm ∆m = (2.15) , hm − λrn−1 δm λδm + βm where α(t) is an arbitrary function of t , and where the functions hm and βm satisfy the equation: (1) qn cm (2.16) + sn δm + h(1) m − sn bm + θ D(βm − am ) = 0. In this way, we obtain the following new integrable lattice hierarchy: (pn )tm = pn Dam + (qn+1 rn − qn rn−1 )δm (1) (2.17) − qn+1 δm − qn (cm + hm ), + sn bm (qn )tm = bm+1 + qn qn+1 rn δm + pn δm − βm + α(t) − θ qn+1 δm , (1) − qn rn−1 δm − pn δm − α(t) (rn )tm = rn βm
(2.18)
+ (sn + θ rn−1 )δm + h(1) m , (1) (1) (sn )tm = sn βm − am − am − qn rn−1 δm − α(t) (1) + h(1) + pn cm m − θ (cm + hm ).
(2.19)
(2.20)
3. Reductions of the integrable lattice hierarchy (2.17)–(2.20)
,
and that F = 0 in general. We must therefore choose a modification matrix ∆m such that for some functions fn , gn , wn and zn , we have that fn gn . (EV(m) )Un − Un V(m) = (2.13) λzn + wn 0 Let
441
,
(2.14)
where αm , vm , em , hm , δm and βm are to be determined in terms of the field functions (pn , qn , rn , sn ). Then ∆11 ∆12 (E∆m )Un − Un ∆m = , ∆21 ∆22 where (1) (1) − qn em + pn Dαm + sn vm − qn hm , ∆11 = λ Dαm + rn vm (1) (1) ∆12 = −λ(vm + qn δm ) + qn αm − βm + θ vm − pn vm , (1) (1) + rn δm ∆21 = λ2 em (1) (1) (1) + λ h(1) m + pn em + rn βm − αm − θ em + sn δm (1) + pn h(1) m + sn βm − αm − θ hm ,
In this section, we discuss in detail reductions of our hierarchy (2.17)–(2.20). We find that it includes many well-known lattice hierarchies as special cases, including the Toda lattice hierarchy, the modified Toda lattice hierarchy, the relativistic Toda lattice hierarchy, and the Volterra lattice hierarchy. We obtain the two-field lattice hierarchy derived by Merola, Ragnisco and Tu. The lattice hierarchies of Ma and Xu, and of Zhang et al., are also included in our new lattice hierarchy. We also obtain here a new integrable lattice hierarchy, which we name the Suris lattice hierarchy since it has as first member the lattice equation related to the relativistic Toda lattice obtained by Suris. The derivation of this last hierarchy requires the use of a different modification matrix ∆m from (2.15) above; the other hierarchies mentioned here arise as direct reductions of the new hierarchy (2.17)–(2.20). Case 1. θ = 0, rn = 0, sn = −1. From Eq. (2.7), we know that F = 0. If we set δm = 0, then from Eq. (2.16) we have hm = 0. Taking α(t) = 0 and βm = −cm+1 ,
(3.1)
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(1) (1) + θ bm − epn bm e−qn + epn + γ (t). (qn )tm = am
then the lattice hierarchy (2.17)–(2.20) reduces to (pn )tm = pn Dam − Ebm − qn cm = −Dam+1 , (qn )tm = bm+1 + qn cm+1 ,
(3.2)
i.e., the Toda lattice hierarchy. The first two equations of this hierarchy are (pn )t1 = qn − qn+1 ,
(qn )t1 = qn (pn−1 − pn ),
(3.7)
For m = 0, θ = 1, γ (t) = − 12 , system (3.7) is the modified Toda lattice equation q˙n = epn ,
p˙ n = eqn+1 −qn − eqn −qn−1 .
(3.8)
Case 3. θ = 0, sn = 0, rn = −1.
(pn )t2 = qn+1 (pn + pn+1 ) − qn (pn + pn−1 ), 2 (qn )t2 = qn qn+1 − qn−1 + pn2 − pn−1 .
Taking δm = 0, hm = βm = −cm and α(t) = 0, we obtain the relativistic Toda lattice hierarchy:
If we set θ = 0, rn = 0, qn sn = ±1, and let δm = 1, hm = −sn−1 and βm = β(t), with β(t) an arbitrary function of t , then Eqs. (2.16) and (2.19) are identically satisfied. It can also be shown that the consistency condition d(qdtn sn ) = 0 holds identically. We thus obtain another Toda lattice hierarchy qn+1 qn (pn )tm = −Dam+1 ∓ ± , qn qn−1 (qn )tm = bm+1 + qn pn + γ (t) , (3.3)
(pn )tm = pn Dam ,
where γ (t) = α(t) − β(t) (of course, it is only this difference that is important). Let qn → eqn , sn → −e−qn ; then the Toda hierarchy (3.3) is rewritten as (pn )tm = −Dam+1 + eqn+1 −qn − eqn −qn−1 , (qn )tm = e−qn bm+1 + pn + γ (t).
(3.9)
For m = 1, (3.9) gives the relativistic Toda lattice equation, (pn )t1 = pn (qn+1 − qn ), (qn )t1 = qn (qn+1 − qn−1 − pn + pn−1 ).
(3.10)
Case 4. θ = 0, rn = α = const, sn = β = const. (−1)
β Taking δm = 0, α(t) = 0, hm = qn−1 bm − cm , βm = −hm /α, and noting Eq. (2.7), we see that Eqs. (2.16) and (2.19)–(2.20) are identically satisfied. It follows from Eqs. (2.17)–(2.18) that
βqn (−1) b , qn−1 m qn βqn (−1) b , (qn )tm = bm+1 − cm + α αqn−1 m (1) − (pn )tm = pn Dam + βbm
(3.4)
For m = 0, γ (t) = −1, (3.4) is the Toda lattice. For m = 1, the resulting equations are trivial. For m = 2, we obtain the following lattice equation: (pn )t2 = (pn+1 + pn )eqn+1 −qn − (pn + pn−1 )eqn −qn−1
(3.11)
which is the lattice hierarchy constructed by Ma and Xu. For m = 1, this gives the lattice equation (pn )t1 = αpn (qn − qn+1 ) + β(qn+1 − qn ),
+ eqn+1 −qn − eqn −qn−1 , (qn )t2 = eqn+1 −qn + eqn −qn−1 + pn2 + pn + γ (t).
(qn )tm = bm+1 + qn cm .
(3.5)
(qn )t1 = αqn (qn−1 − qn+1 ) − qn (pn − pn−1 ).
(3.12)
Case 2. qn rn = θ = ±1, sn = 0.
It is very interesting to note that if we set pn = 0, β = 0 and α = −1, then the hierarchy (3.11) further reduces to the lattice hierarchy
Let δm = 1, hm = −cm , βm = am + β(t). Then Eqs. (2.16) and (2.20) are identically satisfied. We can also show that the consistency condition d(qdtn rn ) = 0 holds identically: in fact from Eqs. (2.18) and (2.19) we have
(qn )tm = bm+1 + qn cm = qn 1 + E −1 m−1 × qn+1 E 2 − qn E −1 + (E − 1)qn (E − 1)−1
d(qn rn ) = rn q˙n + qn r˙n dt (1) (1) (1) = 2θ am + θ rn bm − qn cm − pn rn bm , and it then follows from Eqs. (2.7) and (2.8) that d(qdtn rn ) = 0. Thus we see that the lattice hierarchy (2.17)–(2.20) reduces to the modified Toda hierarchy, qn qn+1 − , (pn )tm = pn Dam + θpn qn qn−1 (1) (1) + θ bm − pn bm + qn pn + γ (t) , (qn )tm = qn am (3.6) where γ (t) = α(t) − β(t). Letting qn → eqn , rn → θ e−qn , pn → epn , (3.6) is rewritten as (pn )tm = Dam + θ eqn+1 −qn − eqn −qn−1 ,
× (E − 1)qn ,
m 1,
(3.13)
which is just the Volterra lattice hierarchy. The first two equations of this hierarchy are q˙n = qn (qn+1 − qn−1 ), q˙n = qn qn+1 (qn+2 + qn+1 + qn ) − qn qn−1 (qn + qn−1 + qn−2 ).
(3.14) (3.15)
Case 5. rn = 0, pn = θ qn sn . In this part, we first show that a two-field lattice hierarchy of Merola, Ragnisco and Tu can be obtained from our hierarchy. Then we construct a new lattice hierarchy, which we name the Suris lattice hierarchy.
A. Pickering, Z.-N. Zhu / Physics Letters A 349 (2006) 439–445
We have rn = 0 and pn = θ qn sn . We also take δm = 1, hm = −sn−1 and βm = β(t); then the condition (2.16) and the d(qn sn ) n equation dp are satisfied, and our hierarchy (2.17)– dt = θ dt (2.20) reduces to qn+1 pn qn pn−1 − , (pn )tm = −Dam+1 − θ qn qn−1 (qn )tm = bm+1 + qn pn + γ (t) − θ qn+1 , (3.16)
qn gn+1 + θ hn+1 − sn fn − θ hn = 0.
443
(3.24)
The combinations (3.21) × qn − (3.22) × pn , and (3.23) × qn − (3.24) × pn , give pn (pn fn − qn en ) + qn (pn hn − qn gn ) = 0.
(3.25)
We now suppose that pn f n = q n en ,
pn hn = qn gn .
(3.26)
where γ (t) = α(t) − β(t). Under the transformation qn → θpn eqn , sn → e−qn , this lattice hierarchy is rewritten in the form (pn )tm = −Dam+1 − θ pn+1 eqn+1 −qn − pn eqn −qn−1 , 1 Dam+1 + θ bm+1 e−qn (qn )tm = pn + pn − θ eqn −qn−1 + γ (t). (3.17)
Substituting conditions (3.26) into Eqs. (3.21)–(3.24) gives the following two equations: sn qn+1 en+1 − pn en − qn gn = 0, pn + (3.27) pn+1 sn qn+1 gn+1 − sn en − θgn = 0, pn + (3.28) pn+1
Setting m = 0, θ = −1 and γ (t) = −1, Eq. (3.17) becomes the lattice equation:
a consequence of which is that
p˙ n = pn+1 e
qn+1 −qn
q˙n = pn + e
qn −qn−1
− pn e
qn −qn−1
,
.
(3.18)
The lattice hierarchy (3.17) is a two-field lattice hierarchy due to Merola, Ragnisco and Tu [14]. In [11], Suris proposed a new lattice equation, q¨n = −q˙n2 q˙n+1 eqn+1 −qn − q˙n−1 eqn −qn−1 , (3.19) which is related via a highly non-trivial Bäcklund transformation, also given in [11], to the relativistic Toda lattice equation. Our aim here is to construct a lattice hierarchy which has (3.19) as first member; to the best of our knowledge, such a lattice hierarchy is previously unknown. This can be done under the same reduction of our spectral problem (1.1), i.e., rn = 0 and pn = θ qn sn , but for a different choice of modification matrix ∆m . We set en fn α(t) 0 + ∆m = (3.20) λ−1 , 0 β(t) gn hn from which it follows that ∆˜ (E∆m )Un − Un ∆m = ˜ 11 ∆21
∆˜ 12 ∆˜ 22
,
where, with γ (t) = α(t) − β(t) (of course, given (3.20), it is only this difference that is important), ∆˜ 11 = Den + λ−1 (pn Den + sn fn+1 − qn gn ), ∆˜ 12 = −fn + γ (t)qn + λ−1 (qn en+1 + θfn+1 − pn fn − qn hn ), ∆˜ 21 = gn+1 − γ (t)sn + λ−1 (pn gn+1 + sn hn+1 − sn en − θgn ), ∆˜ 22 = λ−1 (qn gn+1 + θ hn+1 − sn fn − θ hn ). Thus, in order to obtain an integrable lattice hierarchy, the following equations must be satisfied: pn Den + sn fn+1 − qn gn = 0,
(3.21)
qn en+1 + θfn+1 − pn fn − qn hn = 0,
(3.22)
pn gn+1 + sn hn+1 − sn en − θgn = 0,
(3.23)
sn en+1 = pn gn+1 .
(3.29)
This last then allows us to obtain a solution of Eqs. (3.21)– (3.24) as pn qn−1 qn qn−1 en = , fn = , ∆ ∆ θpn θ qn , hn = , gn = (3.30) ∆ ∆ for a particular choice of integration constant, and where ∆ = pn qn−1 + θ qn . We thus obtain the lattice hierarchy (pn )tm = −Dam+1 + Den , (qn )tm = bm+1 − fn + γ (t)qn , (1)
(sn )tm = −cm+1 + gn+1 − γ (t)sn .
(3.31)
From Eqs. (2.7) and (2.8), we can prove that the condition d(qn sn ) dpn =θ dt dt is satisfied. Therefore, we have in fact constructed the following two-field lattice hierarchy: pn+1 qn pn qn−1 − , (pn )tm = −Dam+1 + pn+1 qn + θ qn+1 pn qn−1 + θ qn qn qn−1 + γ (t)qn . (qn )tm = bm+1 − (3.32) pn qn−1 + θ qn Let qn → eqn , sn → θpn e−qn ; then this lattice hierarchy is rewritten in the form: pn+1 pn (pn )tm = −Dam+1 + − , q −q n n+1 pn+1 + θ e pn + θ eqn −qn−1 1 + γ (t). (qn )tm = e−qn bm+1 − (3.33) pn + θ eqn −qn−1 For m = 0, θ = −1, and γ (t) = −1, Eq. (3.33) reduces to lattice equation eqn+1 −qn eqn −qn−1 − , q −q pn+1 − e n+1 n pn − eqn −qn−1 −1 q˙n = , pn − eqn −qn−1 p˙ n =
(3.34)
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which is equivalent to the Suris lattice equation (3.19). We have therefore successfully embedded the Suris lattice equation (3.19) in an integrable two-field lattice hierarchy.
If we take sn = 0, hm = −cm , α(t) = 0 and βm = am , then we obtain the three-field lattice hierarchy: (pn )tm = pn Dam + (qn+1 rn − qn rn−1 )δm ,
Remark. The discrete linear problem associated with the Suris lattice equation (3.19) is λ + pn eqn ψn , Eψn = −pn e−qn −1 dψn λ−1 pn eqn−1 eqn +qn−1 = ψn , −pn −eqn dt pn eqn−1 − eqn
(qn )tm = bm+1 + qn (qn+1 rn δm + pn δm − am ) − θ qn+1 δm , (1) (rn )tm = rn am − qn rn−1 δm − pn δm
where we have set α(t) = −1/2 and β(t) = 1/2. This Lax representation is to be compared with that given by Suris in [11] for the lattice equation (3.19). We now return to our hierarchy (2.17)–(2.20). We have so far discussed two-field reductions of this hierarchy. Indeed, we have seen that this hierarchy contains many well-known twofield hierarchies, as well as a new integrable two-field lattice hierarchy. That is, the importance and interest of the hierarchy (2.17)–(2.20) has already been demonstrated. However, there remains the question of whether our hierarchy contains interesting reductions other than those given above. In particular, it remains to consider three-field reductions of our hierarchy (2.17)–(2.20). Case 6. rn = 0. From (2.7), we see that F = 0. We now let ∆m = O, and so obtain the lattice hierarchy given in [16], (pn )tm = −Dam+1 ,
(qn )tm = bm+1 ,
(1)
(sn )tm = −cm+1 . If we let α(t) ∆m = −sn−1
(3.35)
−qn λ + β(t)
,
(3.36)
then the hierarchy (2.17)–(2.20) reduces to (pn )tm = −Dam+1 − sn qn+1 + qn sn−1 , (qn )tm = bm+1 + qn pn + γ (t) − θ qn+1 , (1) (sn )tm = −cm+1 − sn pn + γ (t) + θ sn−1 ,
(3.37)
where γ (t) = α(t) − β(t). Case 7. If we take qn rn = θ , δm = 0, hm = 0, α(t) = 0 and βm = d(qn rn ) −D −1 ( bm+1 = 0, and thus we obtain qn ), it follows that dt the following three-field lattice hierarchy: (1) − qn c m , (pn )tm = pn Dam + sn bm
(qn )tm = bm+1 − qn βm , (1) (1) (1) (sn )tm = sn βm − am − am + pn cm − θ cm .
(3.38)
(1) . + θ rn−1 δm − cm
(3.39)
Thus our hierarchy (2.17)–(2.20) also contains a number of interesting three-field reductions. 4. Hamiltonian structure of the Suris lattice hierarchy (3.32) In this section, we construct the Hamiltonian structure for the Suris lattice hierarchy (3.32). This we do by making use of the following trace identity [13]: δ ∂Un δ , trace W δpn δqn ∂λ ∂Un ∂Un −µ ∂ µ = λ λ trace W , trace W , (4.1) ∂λ ∂pn ∂qn where W = Γ Un−1 . We seek a Hamiltonian operator J and Hamiltonian Hm , such that the Suris lattice hierarchy (3.32) can be written in the form δ δ T pn ut m = (4.2) =J , Hm+1 , qn t δpn δqn m where
δf ∂f −n = E . (n) δui ∂ui n∈Z
(4.3)
We have
θ θ a − bsn −aqn + λb + bpn , W = Γ Un−1 = λ θ c + asn −cqn − λa − apn ∂Un = λ−1 a − bpn qn−1 , trace W ∂λ ∂Un trace W = bqn−1 , ∂pn ∂Un = λ−1 c + 2apn qn−1 − λbpn qn−2 − bpn2 qn−2 . trace W ∂qn Note that δ δ , am − bm pn qn−1 δpn δqn = (µ − m) bm qn−1 , cm−1 + 2am−1 pn qn−1 − bm pn qn−2 − bm−1 pn2 qn−2 . (4.4) Let m = 1; it follows that µ = 0 and δ δ , hm+1 δpn δqn = bm+1 qn−1 , cm + 2am pn qn−1 − bm pn2 qn−2 − bm+1 pn qn−2 = qn−1 (bm+1 , Dam+1 ),
(4.5)
A. Pickering, Z.-N. Zhu / Physics Letters A 349 (2006) 439–445
where hm = bm pn qn−1 − am /m.
(4.6)
We thus obtain that the Suris hierarchy (3.32) can be written
pn+1 qn pn qn−1 δ δ T pn+1 qn +θqn+1 − pn qn−1 +θqn , hm+1 + , utm = J n qn−1 γ (t)qn − pn qqn−1 δpn δqn +θqn (4.7) where 0 J= qn
−qn 0
.
(4.8)
Now let f = ln qn − ln(pn qn−1 + θ qn ) + γ (t)pn . Then
δf J
δpn δf δqn
=
pn+1 qn pn+1 qn +θqn+1
γ (t)qn −
pn qn−1 pn qn−1 +θqn qn qn−1 pn qn−1 +θqn
−
of great interest is the embedding of the Suris lattice equation (3.19) within a corresponding integrable hierarchy. We have also given the Hamiltonian structure for this new Suris lattice hierarchy. Here it is worth remarking that the derivation of new integrable lattice hierarchies is of course always an interesting and important topic. Some problems deserve further investigation. For example, given that there exists a Bäcklund transformation between the Suris lattice equation and the relativistic Toda lattice equation [11], what is the relationship between the Suris lattice hierarchy given here and the relativistic Toda lattice hierarchy? It seems natural to expect some kind of relationship, perhaps an extension of Suris’ Bäcklund transformation, between these hierarchies. Another open question is that of whether our new integrable three- and four-field lattice hierarchies can be put in Hamiltonian form. Acknowledgements
.
We thus obtain that the Suris lattice hierarchy (3.32) can indeed be written in the form (4.2), with Hamiltonian operator J being as given by (4.8), and the Hamiltonian Hm+1 being Hm+1 = hm+1 + ln qn − ln(pn qn−1 + θ qn ) + γ (t)pn .
445
(4.9)
The question of Hamiltonian structures is of course intimately related to that of a geometric understanding of integrability. One of the best known exponents of this point of view is Magri, who argues an interpretation of the inverse scattering transform not so much as a nonlinear analogue of Fourier analysis but rather as an extension of Jacobi’s method of separation of variables. This geometric point of view would also allow an alternative explanation of our various reductions of the discrete spectral problem (1.1). 5. Conclusions In this Letter, we have given a new discrete spectral problem, and a new hierarchy of integrable four-field lattice equations. We have considered reductions, which also of course admit zero curvature representations, in detail. Our new integrable lattice hierarchy has been found to include as special cases many well-known integrable lattice hierarchies, including the Toda, modified Toda, relativistic Toda, and Volterra lattice hierarchies. In addition, the two-field lattice hierarchy of Merola, Ragnisco and Tu, the two-field lattice hierarchy of Ma and Xu, and the three-field lattice hierarchy of Zhang et al., are also included in our hierarchy, as are other integrable three-field lattice hierarchies. Within the context of two-field lattice hierarchies,
The work of A.P. is supported in part by the DGESYC under contract BFM2002-02609, that of A.P. and Z.-N.Z. by the Junta de Castilla y León under contract SA011/04, and that of Z.-N.Z. by the National Natural Science Foundation of China under grant No. 10471092. Z.-N.Z. thanks the Ministry of Education and Science of Spain for financial support under the programme “Ayudas para estancias de profesores, investigadores, doctores y tecnólogos extranjeros en España”. He also thanks Professor Muñoz Porras of the Department of Mathematics, University of Salamanca, for his kind hospitality. We are also very grateful to the referees for their useful remarks on the original manuscript. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]
M. Toda, J. Phys. Soc. Jpn. 22 (1967) 431. M. Toda, Prog. Theor. Phys. Suppl. 45 (1970) 174. M. Toda, Theory of Nonlinear Lattice, Springer, Berlin, 1981. H. Flaschka, Phys. Rev. B 9 (1974) 1924. H. Flaschka, Prog. Theor. Phys. 51 (1974) 703. Y.B. Suris, J. Phys. A: Math. Gen. 30 (1997) 2235. M. Bruschi, O. Ragnisco, Phys. Lett. A 129 (1988) 21. M. Bruschi, O. Ragnisco, Phys. Lett. A 134 (1989) 365. M. Bruschi, O. Ragnisco, Inverse Problems 5 (1989) 389. S.N.M. Ruijsenaars, Commun. Math. Phys. 133 (1990) 217. Y.B. Suris, J. Phys. A: Math. Gen. 30 (1997) 1745. O. Ragnisco, P.M. Santini, Inverse Problems 6 (1990) 441. G.Z. Tu, J. Phys. A: Math. Gen. 23 (1990) 3903. I. Merola, O. Ragnisco, G.-Z. Tu, Inverse Problems 10 (1994) 1315. W.X. Ma, X.X. Xu, J. Phys. A: Math. Gen. 37 (2004) 1323. H.W. Zhang, G.Z. Tu, W. Oevel, B. Fuchssteiner, J. Math. Phys. 32 (1991) 1908.
New integrable lattice hierarchies Andrew Pickering a , Zuo-Nong Zhu b,c,∗ a Area de Matemática Aplicada, ESCET, Universidad Rey Juan Carlos, c/ Tulipán s/n, 28933 Móstoles, Madrid, Spain b Departamento de Matemáticas, Universidad de Salamanca, Plaza de la Merced 1, 37008 Salamanca, Spain c Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200030, PR China
Received 10 June 2005; received in revised form 6 September 2005; accepted 15 September 2005 Available online 23 September 2005 Communicated by A.R. Bishop
Abstract In this Letter we give a new integrable four-field lattice hierarchy, associated to a new discrete spectral problem. We obtain our hierarchy as the compatibility condition of this spectral problem and an associated equation, constructed herein, for the time-evolution of eigenfunctions. We consider reductions of our hierarchy, which also of course admit discrete zero curvature representations, in detail. We find that our hierarchy includes many well-known integrable hierarchies as special cases, including the Toda lattice hierarchy, the modified Toda lattice hierarchy, the relativistic Toda lattice hierarchy, and the Volterra lattice hierarchy. We also obtain here a new integrable two-field lattice hierarchy, to which we give the name of Suris lattice hierarchy, since the first equation of this hierarchy has previously been given by Suris. The Hamiltonian structure of the Suris lattice hierarchy is obtained by means of a trace identity formula. 2005 Elsevier B.V. All rights reserved.
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 not only possess rich mathematical structures, but also have many applications in science, e.g., in mathematical physics, numerical analysis, statistical physics, and quantum physics. Perhaps the best known integrable lattice equation is the Toda lattice equation, which governs a system of unit masses connected by nonlinear springs whose restoring force is exponential. It has been shown that the Toda lattice possesses all the usual integrability properties, such as Lax pairs, Hamiltonian structures, infinitely many conservation laws, Bäcklund transformations, and soliton solutions [1–5]. Remarkably, the Toda lattice itself has been a source of new integrable lattices, with new integrable lattices related to the Toda lattice having been proposed, e.g., the modified Toda lattice [6], and the relativistic Toda lattice [7–10]. In its turn, the relativistic Toda lattice hierarchy has also been a source of new lattice * Corresponding author.
E-mail address: [email protected] (Z.-N. Zhu). 0375-9601/$ – see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2005.09.042
equations: Suris gave in [11] a new lattice equation related, via a highly non-trivial Bäcklund transformation, to the relativistic Toda lattice equation. The derivation of new integrable lattice hierarchies is of course an interesting and important topic. In this Letter, we consider the following discrete spectral problem: Eψ(n, t, λ) = U (u, λ)ψ(n, t, λ), λ + p(n, t) q(n, t) U (u, λ) = , λr(n, t) + s(n, t) θ
(1.1)
where the potential u = (p(n, t), q(n, t), r(n, t), s(n, t))T , θ = ±1, or 0, and the shift operator E is defined by Ef (n) = f (n + 1). To the best of our knowledge, the above discrete spectral problem has not before appeared in the literature. By constructing time-evolution equations for the eigenfunctions, and considering the associated discrete zero curvature condition, we obtain in Section 2 a new integrable lattice hierarchy. The approach used is by now well known; see, e.g., Refs. [12,13]. Reductions of our new hierarchy are discussed in detail in Section 3. We find that our integrable lattice hierarchy encompasses many well-known integrable lattice hierarchies. Amongst these are the above-mentioned Toda lattice, modi-
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fied Toda lattice, and relativistic Toda lattice hierarchies, as well as the Volterra lattice hierarchy. We also obtain the twofield lattice hierarchy derived by Merola, Ragnisco and Tu [14], the two-field lattice hierarchy of Ma and Xu [15], the threefield lattice hierarchy of Zhang et al. [16], as well as other integrable three-field lattice hierarchies. Within the context of two-field lattice hierarchies, perhaps our most important result is another new integrable lattice hierarchy obtained here, and which we name the Suris lattice hierarchy. It has as first member the above-mentioned lattice equation, related to the relativistic Toda lattice equation, obtained by Suris. We believe that the hierarchy corresponding to the Suris lattice equation has not previously been given. We also obtain, by means of a trace identity formula, the Hamiltonian structure of this hierarchy. Thus, we see that the above spectral problem provides a unifying approach to a wide variety of known and new lattice hierarchies. 2. Integrable lattice hierarchy related to the spectral problem (1.1) In this section, we construct the integrable lattice hierarchy related to the discrete spectral problem (1.1). In the course of this construction, and also later in the Letter, we will use at times, for reasons of simplicity, the following notation: f (1) = Ef (n),
(EΓ )Un − Un Γ = 0,
b −a
=
(2.1)
∞ am (n, t) m=0
cm (n, t)
(2.5)
θ Da + (λrn + sn )b − qn c(1) = 0,
(2.7)
(2.6)
where the difference operator D is defined by D = E − 1. It follows from Eqs. (2.4)–(2.6) that λ Da + θ (λrn + sn )b − θ qn c(1) = −pn Da + θ (λrn + sn )b − θ qn c(1) , if θ = ±1, and λ (λrn + sn )b − qn c(1) = −pn (λrn + sn )b − qn c(1) ,
if θ = 0.
So, Eq. (2.7) follows from Eqs. (2.4)–(2.6). Then, solving Eqs. (2.4)–(2.6), we have (1) (1) − rn bm+1 , am+1 = D −1 qn cm − pn Dam − sn bm (1) (1) + am − pn bm + θ bm , bm+1 = qn am (−1) (−1) cm+1 = rn−1 am+1 + am+1 + sn−1 am + am (−1) , − pn−1 cm + θ cm
(2.8)
and
fn = f (n, t).
The process of finding the integrable lattice hierarchy corresponding to (1.1) can be divided into two steps. The first step is to solve the stationary discrete zero curvature equation
where a Γ = c
λb = qn a (1) + a − pn b + θ b(1) , λc(1) = (λrn + sn ) a (1) + a − pn c(1) + θ c,
bm (n, t) −am (n, t)
b0 = 0,
(1) (1) c0 − rn a0 + a0 = 0,
(2.9)
where the operator D −1 is the formal inverse of D. We choose the constant a0 = 1/2, and so obtain a0 = 1/2, b0 = 0 and c0 = rn−1 , and then solve Eq. (2.8) through the following path: b1 → a1 → c1 → · · · → bm+1 → am+1 → cm+1 → · · ·
λ−m ,
to obtain
with a, b, c functions of the potential u. The solvability of Eq. (2.1) is crucial to obtaining an integrable lattice hierarchy. Then, we construct an equation for the time-evolution of the eigenfunctions dψn = V(m) (u, λ)ψn , dtm
(1)
Da0 + rn b0 = 0,
(2.2)
where m Γi λm−i + ∆m (u, λ), V(m) = λm Γ + + ∆m (u, λ) =
a1 = −qn rn−1 ,
b1 = q n ,
c1 = −rn−1 (pn−1 + qn rn−1 + qn−1 rn−2 ) + θ rn−2 + sn−1 , b2 = −qn (qn+1 rn + qn rn−1 + pn ) + θ qn+1 , a2 = qn pn−1 rn−1 + pn rn−1 − sn−1 + qn+1 rn rn−1 2 − θ (qn+1 rn−1 + qn rn−2 ), + qn−1 rn−1 rn−2 + qn rn−1 (−1) (−1) (−1) c2 = rn−1 a2 + a2 + sn−1 a1 + a1 − pn−1 c1 + θ c1 ,
We now realize these two steps for our choice of discrete spectral problem (1.1). The stationary discrete zero curvature equation (2.1) is equivalent to the following system:
and so on. The formal solvability of the system (2.8)–(2.9) is evident. However, one has to invert a difference operator in order to solve for am+1 . This then leads to the question of whether all am , bm and cm (m 0) are local. It turns out that they are so. Introduce a matrix V by Γ = V Un [12]: it follows from Eq. (2.1) that EΓ = Un V , and so D(trace(Γ m )) = trace((EΓ )m ) − trace(Γ m ) = trace((Un V )m ) − trace((V Un )m ) = 0. We thus obtain trace(Γ m ) = γm (t), a function of t only. From (2.8) we see that bm+1 and cm+1 are defined locally in terms of am , bm , am+1 and cm ,
λDa = qn c − pn Da − (λrn + sn )b(1) ,
bm+1 = f1 (am , bm ),
i=0
and where the matrix ∆m (u, λ) is an appropriate modification matrix, such that the integrable lattice hierarchy is derived from the following discrete zero curvature equation: U˙ n = EVn(m) Un − Un Vn(m) . (2.3)
(2.4)
cm+1 = f2 (am+1 , am , cm );
A. Pickering, Z.-N. Zhu / Physics Letters A 349 (2006) 439–445
now, from a 2 + bc = 12 trace(Γ 2 ) = γ2 (t), we add to these last the equation am+1 = −
m
ai am+1−i −
i=1
m+1
bi cm+1−i ,
m 0,
(2.10)
i=1
where we have used the facts that a0 = 1/2 and b0 = 0, and where we have taken γ2 (t) = 0 (this is in fact equivalent to setting to zero the arbitrary function of t that arises when, for each m 0, we solve the first of (2.8) for am+1 ). Then, since a0 , b0 , and c0 are local, we have that b1 is local; then a1 = −b1 c0 is local, and finally c1 is local. So, by mathematical induction, we conclude that am , bm and cm , for m 0, are local. It should also be noted that (2.10) gives us an alternative means of obtaining am+1 , m 0, i.e., algebraically rather than by solving a difference equation. Next we obtain an integrable lattice hierarchy from the discrete zero curvature equation (2.3), using an appropriate modification matrix ∆m . Assuming that Eqs. (2.8) and (2.9) hold, we obtain m m A B E λ Γ + Un − Un λ Γ + = (2.11) , C F where (1) − qn c m , A = pn Dam + sn bm (1) (1) B = qn am + am − pn bm + θ bm , (1) (1) C = pn cm − sn am + am − θ cm ,
F
(1) = qn cm
− sn bm − θ Dam .
Note that p˙ n ˙ Un = λ˙rn + s˙n
q˙n 0
(2.12)
∆m =
αm λem + hm
vm λδm + βm
(1) ∆22 = λ qn em − rn vm + θ Dδm + qn h(1) m − sn vm + θ Dβm . So the following conditions must be satisfied: (1) Dαm + rn vm − qn em = 0,
vm + qn δm = 0, (1) (1) + rn δm = 0, qn em − rn vm + θ Dδm = 0, (1) qn hm − sn vm + θ Dβm + F = 0. (1) em
Thus, in order to have a four-field lattice hierarchy, we must have δm = const (here 0 or 1), and qn rn−1 δm + α(t) −qn δm ∆m = (2.15) , hm − λrn−1 δm λδm + βm where α(t) is an arbitrary function of t , and where the functions hm and βm satisfy the equation: (1) qn cm (2.16) + sn δm + h(1) m − sn bm + θ D(βm − am ) = 0. In this way, we obtain the following new integrable lattice hierarchy: (pn )tm = pn Dam + (qn+1 rn − qn rn−1 )δm (1) (2.17) − qn+1 δm − qn (cm + hm ), + sn bm (qn )tm = bm+1 + qn qn+1 rn δm + pn δm − βm + α(t) − θ qn+1 δm , (1) − qn rn−1 δm − pn δm − α(t) (rn )tm = rn βm
(2.18)
+ (sn + θ rn−1 )δm + h(1) m , (1) (1) (sn )tm = sn βm − am − am − qn rn−1 δm − α(t) (1) + h(1) + pn cm m − θ (cm + hm ).
(2.19)
(2.20)
3. Reductions of the integrable lattice hierarchy (2.17)–(2.20)
,
and that F = 0 in general. We must therefore choose a modification matrix ∆m such that for some functions fn , gn , wn and zn , we have that fn gn . (EV(m) )Un − Un V(m) = (2.13) λzn + wn 0 Let
441
,
(2.14)
where αm , vm , em , hm , δm and βm are to be determined in terms of the field functions (pn , qn , rn , sn ). Then ∆11 ∆12 (E∆m )Un − Un ∆m = , ∆21 ∆22 where (1) (1) − qn em + pn Dαm + sn vm − qn hm , ∆11 = λ Dαm + rn vm (1) (1) ∆12 = −λ(vm + qn δm ) + qn αm − βm + θ vm − pn vm , (1) (1) + rn δm ∆21 = λ2 em (1) (1) (1) + λ h(1) m + pn em + rn βm − αm − θ em + sn δm (1) + pn h(1) m + sn βm − αm − θ hm ,
In this section, we discuss in detail reductions of our hierarchy (2.17)–(2.20). We find that it includes many well-known lattice hierarchies as special cases, including the Toda lattice hierarchy, the modified Toda lattice hierarchy, the relativistic Toda lattice hierarchy, and the Volterra lattice hierarchy. We obtain the two-field lattice hierarchy derived by Merola, Ragnisco and Tu. The lattice hierarchies of Ma and Xu, and of Zhang et al., are also included in our new lattice hierarchy. We also obtain here a new integrable lattice hierarchy, which we name the Suris lattice hierarchy since it has as first member the lattice equation related to the relativistic Toda lattice obtained by Suris. The derivation of this last hierarchy requires the use of a different modification matrix ∆m from (2.15) above; the other hierarchies mentioned here arise as direct reductions of the new hierarchy (2.17)–(2.20). Case 1. θ = 0, rn = 0, sn = −1. From Eq. (2.7), we know that F = 0. If we set δm = 0, then from Eq. (2.16) we have hm = 0. Taking α(t) = 0 and βm = −cm+1 ,
(3.1)
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(1) (1) + θ bm − epn bm e−qn + epn + γ (t). (qn )tm = am
then the lattice hierarchy (2.17)–(2.20) reduces to (pn )tm = pn Dam − Ebm − qn cm = −Dam+1 , (qn )tm = bm+1 + qn cm+1 ,
(3.2)
i.e., the Toda lattice hierarchy. The first two equations of this hierarchy are (pn )t1 = qn − qn+1 ,
(qn )t1 = qn (pn−1 − pn ),
(3.7)
For m = 0, θ = 1, γ (t) = − 12 , system (3.7) is the modified Toda lattice equation q˙n = epn ,
p˙ n = eqn+1 −qn − eqn −qn−1 .
(3.8)
Case 3. θ = 0, sn = 0, rn = −1.
(pn )t2 = qn+1 (pn + pn+1 ) − qn (pn + pn−1 ), 2 (qn )t2 = qn qn+1 − qn−1 + pn2 − pn−1 .
Taking δm = 0, hm = βm = −cm and α(t) = 0, we obtain the relativistic Toda lattice hierarchy:
If we set θ = 0, rn = 0, qn sn = ±1, and let δm = 1, hm = −sn−1 and βm = β(t), with β(t) an arbitrary function of t , then Eqs. (2.16) and (2.19) are identically satisfied. It can also be shown that the consistency condition d(qdtn sn ) = 0 holds identically. We thus obtain another Toda lattice hierarchy qn+1 qn (pn )tm = −Dam+1 ∓ ± , qn qn−1 (qn )tm = bm+1 + qn pn + γ (t) , (3.3)
(pn )tm = pn Dam ,
where γ (t) = α(t) − β(t) (of course, it is only this difference that is important). Let qn → eqn , sn → −e−qn ; then the Toda hierarchy (3.3) is rewritten as (pn )tm = −Dam+1 + eqn+1 −qn − eqn −qn−1 , (qn )tm = e−qn bm+1 + pn + γ (t).
(3.9)
For m = 1, (3.9) gives the relativistic Toda lattice equation, (pn )t1 = pn (qn+1 − qn ), (qn )t1 = qn (qn+1 − qn−1 − pn + pn−1 ).
(3.10)
Case 4. θ = 0, rn = α = const, sn = β = const. (−1)
β Taking δm = 0, α(t) = 0, hm = qn−1 bm − cm , βm = −hm /α, and noting Eq. (2.7), we see that Eqs. (2.16) and (2.19)–(2.20) are identically satisfied. It follows from Eqs. (2.17)–(2.18) that
βqn (−1) b , qn−1 m qn βqn (−1) b , (qn )tm = bm+1 − cm + α αqn−1 m (1) − (pn )tm = pn Dam + βbm
(3.4)
For m = 0, γ (t) = −1, (3.4) is the Toda lattice. For m = 1, the resulting equations are trivial. For m = 2, we obtain the following lattice equation: (pn )t2 = (pn+1 + pn )eqn+1 −qn − (pn + pn−1 )eqn −qn−1
(3.11)
which is the lattice hierarchy constructed by Ma and Xu. For m = 1, this gives the lattice equation (pn )t1 = αpn (qn − qn+1 ) + β(qn+1 − qn ),
+ eqn+1 −qn − eqn −qn−1 , (qn )t2 = eqn+1 −qn + eqn −qn−1 + pn2 + pn + γ (t).
(qn )tm = bm+1 + qn cm .
(3.5)
(qn )t1 = αqn (qn−1 − qn+1 ) − qn (pn − pn−1 ).
(3.12)
Case 2. qn rn = θ = ±1, sn = 0.
It is very interesting to note that if we set pn = 0, β = 0 and α = −1, then the hierarchy (3.11) further reduces to the lattice hierarchy
Let δm = 1, hm = −cm , βm = am + β(t). Then Eqs. (2.16) and (2.20) are identically satisfied. We can also show that the consistency condition d(qdtn rn ) = 0 holds identically: in fact from Eqs. (2.18) and (2.19) we have
(qn )tm = bm+1 + qn cm = qn 1 + E −1 m−1 × qn+1 E 2 − qn E −1 + (E − 1)qn (E − 1)−1
d(qn rn ) = rn q˙n + qn r˙n dt (1) (1) (1) = 2θ am + θ rn bm − qn cm − pn rn bm , and it then follows from Eqs. (2.7) and (2.8) that d(qdtn rn ) = 0. Thus we see that the lattice hierarchy (2.17)–(2.20) reduces to the modified Toda hierarchy, qn qn+1 − , (pn )tm = pn Dam + θpn qn qn−1 (1) (1) + θ bm − pn bm + qn pn + γ (t) , (qn )tm = qn am (3.6) where γ (t) = α(t) − β(t). Letting qn → eqn , rn → θ e−qn , pn → epn , (3.6) is rewritten as (pn )tm = Dam + θ eqn+1 −qn − eqn −qn−1 ,
× (E − 1)qn ,
m 1,
(3.13)
which is just the Volterra lattice hierarchy. The first two equations of this hierarchy are q˙n = qn (qn+1 − qn−1 ), q˙n = qn qn+1 (qn+2 + qn+1 + qn ) − qn qn−1 (qn + qn−1 + qn−2 ).
(3.14) (3.15)
Case 5. rn = 0, pn = θ qn sn . In this part, we first show that a two-field lattice hierarchy of Merola, Ragnisco and Tu can be obtained from our hierarchy. Then we construct a new lattice hierarchy, which we name the Suris lattice hierarchy.
A. Pickering, Z.-N. Zhu / Physics Letters A 349 (2006) 439–445
We have rn = 0 and pn = θ qn sn . We also take δm = 1, hm = −sn−1 and βm = β(t); then the condition (2.16) and the d(qn sn ) n equation dp are satisfied, and our hierarchy (2.17)– dt = θ dt (2.20) reduces to qn+1 pn qn pn−1 − , (pn )tm = −Dam+1 − θ qn qn−1 (qn )tm = bm+1 + qn pn + γ (t) − θ qn+1 , (3.16)
qn gn+1 + θ hn+1 − sn fn − θ hn = 0.
443
(3.24)
The combinations (3.21) × qn − (3.22) × pn , and (3.23) × qn − (3.24) × pn , give pn (pn fn − qn en ) + qn (pn hn − qn gn ) = 0.
(3.25)
We now suppose that pn f n = q n en ,
pn hn = qn gn .
(3.26)
where γ (t) = α(t) − β(t). Under the transformation qn → θpn eqn , sn → e−qn , this lattice hierarchy is rewritten in the form (pn )tm = −Dam+1 − θ pn+1 eqn+1 −qn − pn eqn −qn−1 , 1 Dam+1 + θ bm+1 e−qn (qn )tm = pn + pn − θ eqn −qn−1 + γ (t). (3.17)
Substituting conditions (3.26) into Eqs. (3.21)–(3.24) gives the following two equations: sn qn+1 en+1 − pn en − qn gn = 0, pn + (3.27) pn+1 sn qn+1 gn+1 − sn en − θgn = 0, pn + (3.28) pn+1
Setting m = 0, θ = −1 and γ (t) = −1, Eq. (3.17) becomes the lattice equation:
a consequence of which is that
p˙ n = pn+1 e
qn+1 −qn
q˙n = pn + e
qn −qn−1
− pn e
qn −qn−1
,
.
(3.18)
The lattice hierarchy (3.17) is a two-field lattice hierarchy due to Merola, Ragnisco and Tu [14]. In [11], Suris proposed a new lattice equation, q¨n = −q˙n2 q˙n+1 eqn+1 −qn − q˙n−1 eqn −qn−1 , (3.19) which is related via a highly non-trivial Bäcklund transformation, also given in [11], to the relativistic Toda lattice equation. Our aim here is to construct a lattice hierarchy which has (3.19) as first member; to the best of our knowledge, such a lattice hierarchy is previously unknown. This can be done under the same reduction of our spectral problem (1.1), i.e., rn = 0 and pn = θ qn sn , but for a different choice of modification matrix ∆m . We set en fn α(t) 0 + ∆m = (3.20) λ−1 , 0 β(t) gn hn from which it follows that ∆˜ (E∆m )Un − Un ∆m = ˜ 11 ∆21
∆˜ 12 ∆˜ 22
,
where, with γ (t) = α(t) − β(t) (of course, given (3.20), it is only this difference that is important), ∆˜ 11 = Den + λ−1 (pn Den + sn fn+1 − qn gn ), ∆˜ 12 = −fn + γ (t)qn + λ−1 (qn en+1 + θfn+1 − pn fn − qn hn ), ∆˜ 21 = gn+1 − γ (t)sn + λ−1 (pn gn+1 + sn hn+1 − sn en − θgn ), ∆˜ 22 = λ−1 (qn gn+1 + θ hn+1 − sn fn − θ hn ). Thus, in order to obtain an integrable lattice hierarchy, the following equations must be satisfied: pn Den + sn fn+1 − qn gn = 0,
(3.21)
qn en+1 + θfn+1 − pn fn − qn hn = 0,
(3.22)
pn gn+1 + sn hn+1 − sn en − θgn = 0,
(3.23)
sn en+1 = pn gn+1 .
(3.29)
This last then allows us to obtain a solution of Eqs. (3.21)– (3.24) as pn qn−1 qn qn−1 en = , fn = , ∆ ∆ θpn θ qn , hn = , gn = (3.30) ∆ ∆ for a particular choice of integration constant, and where ∆ = pn qn−1 + θ qn . We thus obtain the lattice hierarchy (pn )tm = −Dam+1 + Den , (qn )tm = bm+1 − fn + γ (t)qn , (1)
(sn )tm = −cm+1 + gn+1 − γ (t)sn .
(3.31)
From Eqs. (2.7) and (2.8), we can prove that the condition d(qn sn ) dpn =θ dt dt is satisfied. Therefore, we have in fact constructed the following two-field lattice hierarchy: pn+1 qn pn qn−1 − , (pn )tm = −Dam+1 + pn+1 qn + θ qn+1 pn qn−1 + θ qn qn qn−1 + γ (t)qn . (qn )tm = bm+1 − (3.32) pn qn−1 + θ qn Let qn → eqn , sn → θpn e−qn ; then this lattice hierarchy is rewritten in the form: pn+1 pn (pn )tm = −Dam+1 + − , q −q n n+1 pn+1 + θ e pn + θ eqn −qn−1 1 + γ (t). (qn )tm = e−qn bm+1 − (3.33) pn + θ eqn −qn−1 For m = 0, θ = −1, and γ (t) = −1, Eq. (3.33) reduces to lattice equation eqn+1 −qn eqn −qn−1 − , q −q pn+1 − e n+1 n pn − eqn −qn−1 −1 q˙n = , pn − eqn −qn−1 p˙ n =
(3.34)
444
A. Pickering, Z.-N. Zhu / Physics Letters A 349 (2006) 439–445
which is equivalent to the Suris lattice equation (3.19). We have therefore successfully embedded the Suris lattice equation (3.19) in an integrable two-field lattice hierarchy.
If we take sn = 0, hm = −cm , α(t) = 0 and βm = am , then we obtain the three-field lattice hierarchy: (pn )tm = pn Dam + (qn+1 rn − qn rn−1 )δm ,
Remark. The discrete linear problem associated with the Suris lattice equation (3.19) is λ + pn eqn ψn , Eψn = −pn e−qn −1 dψn λ−1 pn eqn−1 eqn +qn−1 = ψn , −pn −eqn dt pn eqn−1 − eqn
(qn )tm = bm+1 + qn (qn+1 rn δm + pn δm − am ) − θ qn+1 δm , (1) (rn )tm = rn am − qn rn−1 δm − pn δm
where we have set α(t) = −1/2 and β(t) = 1/2. This Lax representation is to be compared with that given by Suris in [11] for the lattice equation (3.19). We now return to our hierarchy (2.17)–(2.20). We have so far discussed two-field reductions of this hierarchy. Indeed, we have seen that this hierarchy contains many well-known twofield hierarchies, as well as a new integrable two-field lattice hierarchy. That is, the importance and interest of the hierarchy (2.17)–(2.20) has already been demonstrated. However, there remains the question of whether our hierarchy contains interesting reductions other than those given above. In particular, it remains to consider three-field reductions of our hierarchy (2.17)–(2.20). Case 6. rn = 0. From (2.7), we see that F = 0. We now let ∆m = O, and so obtain the lattice hierarchy given in [16], (pn )tm = −Dam+1 ,
(qn )tm = bm+1 ,
(1)
(sn )tm = −cm+1 . If we let α(t) ∆m = −sn−1
(3.35)
−qn λ + β(t)
,
(3.36)
then the hierarchy (2.17)–(2.20) reduces to (pn )tm = −Dam+1 − sn qn+1 + qn sn−1 , (qn )tm = bm+1 + qn pn + γ (t) − θ qn+1 , (1) (sn )tm = −cm+1 − sn pn + γ (t) + θ sn−1 ,
(3.37)
where γ (t) = α(t) − β(t). Case 7. If we take qn rn = θ , δm = 0, hm = 0, α(t) = 0 and βm = d(qn rn ) −D −1 ( bm+1 = 0, and thus we obtain qn ), it follows that dt the following three-field lattice hierarchy: (1) − qn c m , (pn )tm = pn Dam + sn bm
(qn )tm = bm+1 − qn βm , (1) (1) (1) (sn )tm = sn βm − am − am + pn cm − θ cm .
(3.38)
(1) . + θ rn−1 δm − cm
(3.39)
Thus our hierarchy (2.17)–(2.20) also contains a number of interesting three-field reductions. 4. Hamiltonian structure of the Suris lattice hierarchy (3.32) In this section, we construct the Hamiltonian structure for the Suris lattice hierarchy (3.32). This we do by making use of the following trace identity [13]: δ ∂Un δ , trace W δpn δqn ∂λ ∂Un ∂Un −µ ∂ µ = λ λ trace W , trace W , (4.1) ∂λ ∂pn ∂qn where W = Γ Un−1 . We seek a Hamiltonian operator J and Hamiltonian Hm , such that the Suris lattice hierarchy (3.32) can be written in the form δ δ T pn ut m = (4.2) =J , Hm+1 , qn t δpn δqn m where
δf ∂f −n = E . (n) δui ∂ui n∈Z
(4.3)
We have
θ θ a − bsn −aqn + λb + bpn , W = Γ Un−1 = λ θ c + asn −cqn − λa − apn ∂Un = λ−1 a − bpn qn−1 , trace W ∂λ ∂Un trace W = bqn−1 , ∂pn ∂Un = λ−1 c + 2apn qn−1 − λbpn qn−2 − bpn2 qn−2 . trace W ∂qn Note that δ δ , am − bm pn qn−1 δpn δqn = (µ − m) bm qn−1 , cm−1 + 2am−1 pn qn−1 − bm pn qn−2 − bm−1 pn2 qn−2 . (4.4) Let m = 1; it follows that µ = 0 and δ δ , hm+1 δpn δqn = bm+1 qn−1 , cm + 2am pn qn−1 − bm pn2 qn−2 − bm+1 pn qn−2 = qn−1 (bm+1 , Dam+1 ),
(4.5)
A. Pickering, Z.-N. Zhu / Physics Letters A 349 (2006) 439–445
where hm = bm pn qn−1 − am /m.
(4.6)
We thus obtain that the Suris hierarchy (3.32) can be written
pn+1 qn pn qn−1 δ δ T pn+1 qn +θqn+1 − pn qn−1 +θqn , hm+1 + , utm = J n qn−1 γ (t)qn − pn qqn−1 δpn δqn +θqn (4.7) where 0 J= qn
−qn 0
.
(4.8)
Now let f = ln qn − ln(pn qn−1 + θ qn ) + γ (t)pn . Then
δf J
δpn δf δqn
=
pn+1 qn pn+1 qn +θqn+1
γ (t)qn −
pn qn−1 pn qn−1 +θqn qn qn−1 pn qn−1 +θqn
−
of great interest is the embedding of the Suris lattice equation (3.19) within a corresponding integrable hierarchy. We have also given the Hamiltonian structure for this new Suris lattice hierarchy. Here it is worth remarking that the derivation of new integrable lattice hierarchies is of course always an interesting and important topic. Some problems deserve further investigation. For example, given that there exists a Bäcklund transformation between the Suris lattice equation and the relativistic Toda lattice equation [11], what is the relationship between the Suris lattice hierarchy given here and the relativistic Toda lattice hierarchy? It seems natural to expect some kind of relationship, perhaps an extension of Suris’ Bäcklund transformation, between these hierarchies. Another open question is that of whether our new integrable three- and four-field lattice hierarchies can be put in Hamiltonian form. Acknowledgements
.
We thus obtain that the Suris lattice hierarchy (3.32) can indeed be written in the form (4.2), with Hamiltonian operator J being as given by (4.8), and the Hamiltonian Hm+1 being Hm+1 = hm+1 + ln qn − ln(pn qn−1 + θ qn ) + γ (t)pn .
445
(4.9)
The question of Hamiltonian structures is of course intimately related to that of a geometric understanding of integrability. One of the best known exponents of this point of view is Magri, who argues an interpretation of the inverse scattering transform not so much as a nonlinear analogue of Fourier analysis but rather as an extension of Jacobi’s method of separation of variables. This geometric point of view would also allow an alternative explanation of our various reductions of the discrete spectral problem (1.1). 5. Conclusions In this Letter, we have given a new discrete spectral problem, and a new hierarchy of integrable four-field lattice equations. We have considered reductions, which also of course admit zero curvature representations, in detail. Our new integrable lattice hierarchy has been found to include as special cases many well-known integrable lattice hierarchies, including the Toda, modified Toda, relativistic Toda, and Volterra lattice hierarchies. In addition, the two-field lattice hierarchy of Merola, Ragnisco and Tu, the two-field lattice hierarchy of Ma and Xu, and the three-field lattice hierarchy of Zhang et al., are also included in our hierarchy, as are other integrable three-field lattice hierarchies. Within the context of two-field lattice hierarchies,
The work of A.P. is supported in part by the DGESYC under contract BFM2002-02609, that of A.P. and Z.-N.Z. by the Junta de Castilla y León under contract SA011/04, and that of Z.-N.Z. by the National Natural Science Foundation of China under grant No. 10471092. Z.-N.Z. thanks the Ministry of Education and Science of Spain for financial support under the programme “Ayudas para estancias de profesores, investigadores, doctores y tecnólogos extranjeros en España”. He also thanks Professor Muñoz Porras of the Department of Mathematics, University of Salamanca, for his kind hospitality. We are also very grateful to the referees for their useful remarks on the original manuscript. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]
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