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Factorization of a hierarchy of the lattice soliton equations from a binary Bargmann symmetry constraint
Nonlinear Analysis 61 (2005) 1225 – 1240 www.elsevier.com/locate/na
Factorization of a hierarchy of the lattice soliton equations from a binary Bargmann symmetry constraint Xi-Xiang Xu∗ Department of Basic Courses, Shandong University of Science and Technology, Taian 271019, PR China Received 9 August 2003; accepted 24 January 2005
Abstract Staring from a discrete spectral problem, a hierarchy of the lattice soliton equations is derived. It is shown that each lattice equation in resulting hierarchy is Liouville integrable discrete Hamiltonian system. The binary nonlinearization of the Lax pairs and the adjoint Lax pairs of the resulting hierarchy is discussed. Each lattice soliton equation in the resulting hierarchy can be factored by an integrable symplectic map and a finite-dimensional integrable system in Liouville sense. Especially, factorization of a discrete Kdv equation is given. 䉷 2005 Elsevier Ltd. All rights reserved. MSC: 02.30.Ik Keywords: Lax pair; Lattice soliton equation; Discrete Hamiltonian system; Binary nonlinearization; Integrable symplectic map; Finite-dimensional integrable system
1. Introduction As it is well known, nonlinear integrable lattice equations have received considerable attention in recent years. Many nonlinear integrable lattice equations have been proposed and discussed, for instance the Ablowitz–Ladik lattice [1], the Toda lattice [15], and so on [10,14,19,21,22]. ∗ Tel.:/fax: +860 53 8622 6700.
E-mail address: [email protected]. 0362-546X/$ - see front matter 䉷 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2005.01.099
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An approach called the nonlinearization method of the Lax pairs of soliton equations has been developed to obtain finite-dimensional integrable Hamiltonian systems (FDIHSs) [4–6,23]. Under the constraints between the potentials and the eigenfunctions, the Lax pairs of soliton equations can be nonlinearized as finite-dimensional completely integrable systems in the Liouville sense. Recently, the binary nonlinearization method has been proposed by Ma et al. [7–9,12], which involves both the Lax pairs and the adjoint Lax pairs for soliton equations and thus the contents of research are much richer. Up to now, much work has been carried out in the study of the nonlinearization of the Lax pairs for continuous soliton equations [4–9,16–18,23] . However, there has been little discussion about the nonlinearization of the Lax pairs for lattice soliton equations. This paper is structured as follows. In Section 2, we would like to derive a hierarchy of lattice soliton equations from a discrete matrix spectral problem, a typical lattice equation in the resulting hierarchy turn into the Kdv equation in a continuous limit, and it may be transformed into the celebrated Volterra lattice [2,13]. Then, we establish their Hamiltonian structures by the aid of the discrete trace identity [15]. An infinite set of conserved functionals of the resulting hierarchy is given .The conserved functionals are in involution in pairs with respect to the corresponding Poisson bracket. In Section 3, the Lax pairs and the adjoint Lax pairs of the resulting hierarchy are binary nonlinearized by means of the binary Bargmann constraints. The spatial part of the Lax pairs and the adjoint Lax pairs is nonlinearized into a new integrable symplectic map [3,20]. Nonlinearization of the temporal parts of the Lax pairs and the adjoint Lax pairs generate a family of finite-dimensional Liouville integrable Hamiltonian systems. Therefore, each lattice soliton equation in the resulting hierarchy can be factored by an integrable symplectic map and a finite-dimensional integrable system in Liouville sense. The binary Bargmann constraint provides a Bäcklund transformation for each lattice soliton equation in the resulting hierarchy [11] and the integrable symplectic map and the finite-dimensional integrable system in Liouville sense. Finally, factorization of a discrete Kdv equation is given. 2. A hierarchy of the Liouville integrable lattice soliton equations Let us recall some fundamental conceptions. The shift operator E, the inverse of E and difference operator D are defined as follows: (Ef )(n) = f (n + 1), (E −1 f )(n) = f (n − 1), n ∈ Z, (Df )(n) = f (n + 1) − f (n) = (E − 1)f (n), n ∈ Z,
(1) (2)
and f (j ) = E j f, j ∈ Z. We assume that s = s(n, t) is a real function defined over Z × R, and s is required to be rapidly vanishing at infinity. is the spectral parameter and t = 0. A lattice equation st = K(s, Es, E −1 s, . . .)
(3)
is said to be a lattice soliton equation (or Lax integrable), if it can be rewritten as a compatibility condition Ut = (EV)U − UV
(4)
X.-X. Xu / Nonlinear Analysis 61 (2005) 1225 – 1240
1227
of a discrete spatial spectral problem E = U (s, )
(5)
and a corresponding continuous time evolution equations
t = V (s, ),
(6)
where U (s, ) and V (s, t) are of the some order square matrices. Eqs. (5) and (6) are said to be a Lax pair of the lattice soliton equation (3). Eq. (4) is called a discrete zero curvature representation of the lattice soliton equation (3). The Gateaux derivative and the variational derivative are defined by jH j H˜ J (s)[w] = (7) J (s + w) , E −m = j s js (m) =0 m∈Z
and the inner product is defined as f, g = f (n)g(n),
(8)
n∈Z
where f = f (n) and g = g(n) are required to be rapidly vanishing at the infinity, H˜ = ∗ ∗ n∈Z H (s(n)). Operator J is defined by f, J g = Jf , g , it is called the adjoint operator of J with respect to (8). If an operator J has the property J = −J ∗ , then J is called to be skew-symmetric. A linear operator J is called a Hamiltonian operator, if J is a skew-symmetric operator satisfying the Jacobi identity, i.e., J (s)[Jf ]g, h + Cycle(f, g, h) = 0. The associated Poisson bracket with a given Hamiltonian operator J is given by f g f g {f, g}J = ,J = J . s s s s
(9)
n∈Z
Note that it is natural that the Leibniz’rule holds for {f, g}J due to the usage of the variational derivative in the bracket. In virtue of [15], the discrete Hamiltonian system stm = J
H˜ m , m 0, s
is called the Liouville integrable if an infinite number of conserved functionals {H˜ m }∞ m=0 are in involution in pairs with respect to the Poisson bracket (9), i.e., {H˜ m , H˜ l }J = 0(mod D), m, l 0. Let us start the discrete spectral problem 1 0 −s E = U (s, ), U (s, ) = . (10) , = 1/s s 2
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In order to derive the corresponding hierarchy of lattice soliton equations, we proceed first to solve the stationary discrete zero curvature equation (E)U − U = 0.
(11)
We suppose that a solution of Eq. (11) is given by a b = , c −a then Eq. (11) is equivalent to b(1) + s 2 c = 0, b(1) − s 2 (a + a (1) ) = 0, c + (a + a (1) ) = 0, s 2 c(1) + b + (a (1) − a) = 0.
(12)
On setting a=
∞
am −2m ,
b=
∞
bm −2m+1 ,
c=
m=0
m=0
∞
cm −2m+1 ,
m=0
(12) yields equivalently (1)
b0 = 0,
c0 = 0,
(1)
(1)
a0 − a 0 = s 2 c 0 + b 0
and the recursion relation (1) + s 2 cm = 0, m 0, bm (1)
(1) ) = 0, m 0, bm+1 − s 2 (am + am (1)
(1)
(am+1 − am+1 ) + bm+1 + s 2 cm+1 = 0, m 0.
(13)
We choose the initial data a0 = −1,
b0 = 0
and require selecting zero constant for the inverse operation of the difference operator D in computing am , m 1. Thus, the recursion relation (13) uniquely determines am , bm , cm , m 1, and the first few quantities are given by a1 = −2(s (−1) )2 , a2 = −2(s
(−1) 4
b1 = −2(s (−1) )2 ,
) −2(ss
(−1) 2
) −2(s
c1 = 2, (−1) (−2) 2 s
) ,
b2 = −2(s (−1) )4 −2(s (−1) s (−2) )2 ,
c2 = 2s 2 + 2(s (−1) )2 , . . . . For any integer m 0, we choose m m ai 2m−2i bi 2m−2i+1 i=0 i=0 Vm = , m m 2m−2i+1 2m−2i ci − ai i=0
i=0
(14)
X.-X. Xu / Nonlinear Analysis 61 (2005) 1225 – 1240
then the following equation holds: 0 E(Vm )U − UVm = (1) −(am + am )/s
1229
(1) −s(am + am ) . 0
To present the associated hierarchy of lattice systems, we take an additional term cm+1 0 m = 2 cm+1 0 − 2 and define V (m) = Vm + m , m 0. We introduce the following continuous time evolution equations
tm = V (m) , m 0.
(15)
Then the compatibility conditions of Eqs. (10) and (15) gives rise to a hierarchy of discrete zero curvature equations Utm = (EV(m) )U − UV(m) , m 0.
(16)
It describes a hierarchy of lattice soliton equations stm = −s
(2)
(am − am ) , m 0. 2
(17)
When m = 0, Eqs. (17) become the following trivial system: st0 = 0. The first nonlinear lattice soliton equation in the family (17) is st1 = s((s (1) )2 − (s (−1) )2 ).
(18)
From [2], we know that the Eq. (18) turn into the Kdv equation in a continuous limit. Therefore, the Eq. (18) is called to be a discrete Kdv equation. Moreover, if we denote w = s 2 , t1 = 2, the Eq. (18) become w = w(w (1) − w (−1) ), this is the celebrated Volterra lattice [2]. To establish the Hamiltonian structures for (17), we define (a − b)/s sa V = U −1 = (c + a)/s sc and M, N ≡ Tr(MN ), where M and N are the some order square matrices. We have jU jU 0 0 0 −1 . = , = 0 1/s −1/s 2 −1/s 2 j js
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Hence
jU = c, V, j
a (1) jU c + a V, = −2 =2 . js s s
By the discrete trace identity [15], jU j jU , V, (n) = − V , j js j s n∈Z
with being a constant to be found. We have (1) 2a j − c(n) = . s j s
(19)
n∈Z
The substitution of a=
∞
ai −2m ,
b=
m=0
∞
bi −2m+1 ,
c=
m=0
∞
ci −2m+1
m=0
into (19), and comparing the coefficients of −2m−1 in Eqs. (19), we get (1) 2am cm+1 (n) = ( − 2m) . s s
(20)
n∈Z
When m = 0 in the Eq. (20), a direct calculation shows that = 0. So we obtain (1) cm+1 am (n) = , m > 0. s −4m s n∈Z
Set H˜ 0 = −2
(ln s)(n),
n∈Z
H˜ m =
cm+1 m∈Z
−4m
(n), m > 0,
(21)
we have (1) H˜ m am = , m 0. s s According to Eq. (13), we get
H˜ m−1 H˜ m = , s s where the recursion operator =
−1 (1 − E −1 )−1 (E −1 s 2 − s 2 E)(1 + E −1 )s. s
We introduce an operator J = s s/2, it is easy to verify that J is a Hamiltonian operator.
X.-X. Xu / Nonlinear Analysis 61 (2005) 1225 – 1240
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Now we can rewrite Eqs. (17) in the following forms: stm
(1) H˜ m−1 H˜ m am m −1 , m 0. =J = J = · · · = J =J 2s s s s
(22)
Therefore, each equation in (17) is a discrete Hamiltonian equation. Moreover, we have J =
−s (1 + E)(E −1 s 2 − s 2 E)(1 + E −1 )s. 2
Obviously, (J )∗ = −(J ). Proposition 1. {H˜ m }m 1 defined by (21) forms an infinite set of conserved functionals of the hierarchy (17), and H˜ m , m 1, are in involution in pairs with respect to the Poisson bracket (9). Proof. Based on (9), we have the following result ˜ ˜ H˜ m H˜ l m−1 H1 l−1 H1 ˜ ˜ Hm , Hl = ,J = , J J s s s s H˜ 1 H˜ 1 = m−1 , L ∗ J l−2 s s H˜ 1 H˜ 1 = m , J l−2 ={H˜ m+1 , H˜ l−1 }J = · · · = {H˜ m+l−1 , H˜ 1 }J . s s Similarly, we also get {H˜ l , H˜ m }J = {H˜ m+l−1 , H˜ 1 }J , thus {H˜ m , H˜ l }J = −{H˜ l , H˜ m }J . This implies {H˜ m , H˜ l }J = 0, m, l 1 and
˜m H (H˜ m )tl = Hm = , stl s n∈Z tl H˜ m H˜ l = , J = {H˜ m , H˜ l } = 0, m, l 1. s s
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˜ ∞ Therefore, {Km }∞ m=1 and {Hm }m=1 are infinitely commuting symmetries and infinitely conserved functionals of the lattice hierarchy (23). The proof is completed. From Proposition 1, we arrive at the following result. Theorem 1. Each lattice solution equations in (17) or the discrete Hamiltonian equation in (22) is a discrete Liouville integrable system. 3. Binary nonlinearization of the Lax pairs and the adjoint Lax pairs Let us consider the adjoint spectral problem of spectral problem (10) 1 E −1 = (E −1 U T (u, )) , = 2
(23)
and the continuous time evolution equations
tm = −VmT .
(24)
From (E −1 )tm = E −1 tm , we can find that E −1 UtTm = (E −1 U T )VmT − (E −1 VmT )(E −1 U T ).
(25)
It is easy to verify that Eq. (25) can be rewritten as Utm = (EVm )U − UVm . Therefore, the lattice soliton equations (17) have another kind of discrete zero curvature representation (25). The integrality condition (E −1 )tm = E −1 tm of overdetermined systems (23) and (24) is still the mth lattice soliton equation (17). The Eqs. (23) and (24) are called the adjoint Lax pairs of (17). The adjoint Lax pairs (23) and (24) can help us determine the variational derivative of the spectral parameter with respect to the potential s [11]. Let 1 , 2 , . . . , N be N distinct eigenvalues of spectral problem (10). We have 1j E 1j = U (u, j ) , E 2j −12j E 1j 1j −1 T = (E U (u, j )) , 1j N, (26) 2j E −1 2j 1j 1j , = Vm (u, j ) 2j tm 2j 1j 1j , 1j N, = −VmT (u, j ) (27) 2j tm 2j (E 1j , E 2j ) = (1j , 2j )U (u, j )T , 1 j N ,
(28)
(E 1j , E 2j ) = ( 1j , 2j )U (u, )−1 , 1 j N .
(29)
X.-X. Xu / Nonlinear Analysis 61 (2005) 1225 – 1240
From [11], it is known that j 2j (1) 1j (1) j 2j 2j (1) − 2j 1j − = j − s2 s s2 1j 1j − 2j 2j + 2j 2j 1j = j − , s
1233
(30)
where j /s is a variational derivative for eigenvalue j , j , 1 j N , are constants and i , i , i = 1, 2 are required to be rapidly vanishing at the infinity, Eq. (30) will play a central role in the binary nonlinearization method. We can deduce that 1j 1j − 2j 2j 2j 2j 1j − − s s − 2j 2j 1j 1j 1j 2j 2j = 2j − − . s s The following notations are used:
i = (i1 , i2 , . . . , iN ),
i = ( i1 , i2 , . . . , iN ), i = 1, 2,
= diag(1 , 2 , . . . , N ). And the inner product in R N is denoted by . , . . Consider the discrete constraint J
N j H 1 , =J s s
(31)
j =1
where j =1, j =1, 2, . . . , N. Because of tm =0, J (j /s), 1 j N , are all symmetries of stm = J (H˜ m /s), m 0. Thus (34) (or (35)) is a symmetry constraint. From (31), we have (1) H1 a1 1 , 1 − 2 , 2 2 , 1 = = +2 , s s s s
the symmetry constraint (31) yields s = 21 1 , 1 − 2 , 2 + 2 2 , 1 .
(32)
Because it is possible to solve (31) for s, and s depend on i , i , i = 1, 2, the Eq. (31) or (32) is called a binary Bargmann symmetry constraint. Under constraint (32), Put the N copies of (26) and (29) in vector form, we obtain a discrete Bargmann system E 1j = − 21 1 , 1 − 2 , 2 + 2 2 , 1 2j , 1 j N , E 2j = − √
2(1j + j 2j ) 1 , 1 − 2 , 2 + 2 2 , 1
, 1j N,
1234
X.-X. Xu / Nonlinear Analysis 61 (2005) 1225 – 1240
E 1j = √ E 2j =
1 2
2(j 1j − 2j ) 1 , 1 − 2 , 2 + 2 2 , 1
, 1j N,
1 , 1 − 2 , 2 + 2 2 , 1 1j , 1 j N .
(33)
Setting fi = fi ( 1 , 2 , 1 , 2 ),
gi = gi ( 1 , 2 , 1 , 2 ), 1 i 2N .
j 1j − 2j fi = √ , 1j N, 2 1 , 1 − 2 , 2 + 2 2 , 1 fN+j = 2 1 , 1 − 2 , 2 + 2 2 , 1 1j , 1 j N . gj = − 21 1 , 1 − 2 , 2 + 2 2 , 1 2j , 1 j N , gN+j = − √
2(1j + j 2j ) 1 , 1 − 2 , 2 + 2 2 , 1
.
4N We 2 define Poisson bracket for any pair of functions f and g in sympletic space (R , i=1 di d i ) as follows:
{f, g} =
N 2
i=1 j =1
jf jg jf jg − j ij jij jij j ij
2 jf jg jf jg = − , , , ji j i j i ji
(34)
i=1
which is a skew-symmetric bilinear, and satisfies the Jacobi identity. In particular, f and g are called involutive if {f, g} = 0. Proposition 2. Eq. (33) determines a symplectic map H : [17,18] H (1 , 2 , 1 , 2 ) = (E 1 , E 2 , E 1 , E 2 ). Proof. Through laborious, but direct computation, we get {fi , fj } = {gi , gj } = 0, This completes the proof.
{fi , gj } = ij , 1 i, j 2N .
(35)
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1235
We can solve Eqs. (13) as follows: b˜0 = 0,
a˜ 0 = −1, a˜ m =
c˜0 = 0,
1 , 1 − 2 , 2 , b˜1 = 2
c˜1 = 2,
2m−2 1 , 1 − 2m−2 2 , 2 , m 1, 2
b˜m = 2m−3 1, 2 , m 2, c˜m = 2m−3 2 , 1 , m 2.
(36)
Set F˜0 = a˜ 02 = 1, F˜m =
m
a˜ i a˜ m−i +
i=0
m
b˜i c˜m−i+1 = −( 2m−2 1 , 1 − 2m−2 2 , 2 )
i=1
1 + 2 2m−3 1 , 2 + ( 1 , 1 − 2 , 2 ) 2m−3 2 , 1 2 +
m−2
2i−1 1 , 2 2m−2i−3 2 , 1 +
i=1
−
2i−2
m−1 1 ( 2i−2 1 , 1 4 i=1
2 2 )(
2m−2i−2
1 , 1 −
2m−2i−2
2 2 ), m 1.
(37)
Proposition 3. DF˜m = 0, m 0,
(38)
i.e., F˜m , m 0, constitutes a hierarchy of integrals of motion for symplectic map H (35) (or the discrete flow (33)). Proof. Because expressions (36) are the solutions of Eqs. (13), we have a˜ =
∞
a˜ m −2m ,
m=0
b˜ =
∞ m=0
b˜m −2m+1 ,
c˜ =
∞
c˜m −2m+1
m=0
is a solution set of Eqs. (12), thus D(a˜ 2 + b˜ c) ˜ = 0, this implies DF˜m = 0, m 0. This completes the proof.
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X.-X. Xu / Nonlinear Analysis 61 (2005) 1225 – 1240
In the following, we would like to discuss the Liouville integrability on the nonlinearized temporal parts of the Lax pairs and adjoint Lax pairs. Under the control of Eq. (32), the temporal parts of the Lax pairs and the adjoint Lax pairs by substituting (36) into (27) become
1j 2j 1j 2j
tm
=V
=V
(m)
tm
1j , j = 1, 2, . . . , N, B
(m)
2j
1j , j = 1, 2, . . . , N. B
(39)
2j
Here the subscript B means substitution of (32) into the expression. Through a direct calculation, we find that Eqs. (39) may be rewritten as
it m =
jF˜m+1 , ji
it m = −
jF˜m+1 , i = 1, 2. j i
(40)
Proposition 4. F˜m , m 0, are in involution in pairs with respect to the Poisson bracket (34), and constitutes a hierarchy of involutive integrals of motion for Eqs. (40). Proof. Through a direct calculation, we can get
˜ tm = [V˜ (m) , ˜ ], where
a˜ ˜ = c˜
b˜ −a˜
∞ a˜ m −2m m=0 = ∞ c˜m −2m+1 m=0
∞
b˜m −2m+1
m=0 ∞
−
m=0
a˜ m −2m
.
This yields 2
d d ˜ 2 = d [V˜ (m) , ˜ 2 ] = 0, (a˜ 2 + b˜ c) ˜ = Tr dtm dtm dtm
i.e., d ˜ Fl = 21 {F˜l , F˜m } = 0, l, m 1. dtm This elucidates that F˜m , m 1, constitutes a hierarchy of involutive integrals of motion for Eqs. (40). F˜m , m 0, are in involution in pairs with respect to the Poisson bracket (34). This completes the proof.
X.-X. Xu / Nonlinear Analysis 61 (2005) 1225 – 1240
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Set F¯j = 1j 1j + 2j 2j , 1 j N. It is easy to calculate that d ¯ Fj = 0, 1 j N, m 0, dtm {F˜m , F¯j } = 0, m 0, 1 j N, {F¯i , F¯j } = 0, 1 i, j N .
DF¯j = 0,
(41)
Proposition 5. F˜m+1 , 1 m N, F¯j , 1 j N , are functionally independent over some region of R 4N . Proof. A straightforward computation gives m
m
i=0
i=0
c˜m+1 jF˜m+1 c˜i 2m−2i+1 2 + 1 , = a˜ i 2m−2i 1 + 2 j 1
(42)
m m c˜m+1 jF˜m+1 =− ai 2m−2i 2 + 2 . b˜i 2m−2i+1 1 − j 2 2
(43)
i=0
i=1
Thus, we have jF˜m+1 j 1
= −2 2m 1 ,
1 1 = 2 =0, 2 =− 2 1
jF˜m+1 j 2
= 2m 2 , m 0.
(44)
1 1 = 2 =0, 2 =− 2 1
Assume that the result on the functional independence there exist 2N real is not2 true. Then N 2 = 0, such that numbers 1 , 2 , . . . , N , 1 , 2 , . . . , N , satisfying N
+ i=1 j j =0 j N i=1
i F¯i +
N j =1
j F˜j = 0.
A direct calculation gives rise to
jF¯j = ij j l , j, l = 1, 2, . . . , N. jil
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X.-X. Xu / Nonlinear Analysis 61 (2005) 1225 – 1240
We can compute
jF¯1 j 1 det jF¯1 j 2
··· ···
11 .. . 0 = det 21 . .. 0
jF¯N j 1 jF¯N
jF˜2 j 1 jF˜2
j 2
j 2
··· .. . ··· ··· .. . ···
0 .. .
1N 0 .. .
2N
··· ···
jF˜N+1 j 1 jF˜N+1
−221 11 .. .
−22N 1N 21 21 .. .
2N 2N
j 2 ··· .. . ··· ··· .. . ···
−22N 1 11 .. . −22N 1N N 2N 1 21 .. . 2N N 2N
1 21 N N 2 N = (−1)N 23 3j 1 2 21j ··· j =1 j =1 1 2N
· · · 2N−2 1 2N−2 · · · 2 . ··· · · · 2N−2 N
By observing that the Vandermode determinant V (21 , 22 , . . . , 2N ) = 0, this means that all
i , i must be zero. Therefore the functions F¯j , F˜j +1 , j = 1, 2, . . . , N, can be functionally independent at least on certain region of R 4N . This completes the proof. Summing up, according to the Propositions 3–5, we have the following theorem. Theorem 2. Symplectic map (35) is integrable and nonlinearized temporal parts of the Lax pairs and the adjoint Lax pairs (40) are all finite-dimensional integrable systems in the Liouville sense. Moreover, Propositions 5 and 6 provide us with an involutive and independent system of polynomials, which are completely new as far as we know. Let ( 1 (n, tm ), 2 (n, tm ), 1 (n, tm ), 2 (n, tm )) be a solution of the discrete flow (34) and the finite-dimensional completely integrable systems (40). Theorem 3. The function s(n, tm ) = 21 1 (n, tm ), 1 (n, tm − 2 (n, tm ), 2 (n, tm ) +2 2 (n, tm ), 1 (n, tm ) (45) solves the lattice soliton equation (17) or Hamiltonian system (22).
X.-X. Xu / Nonlinear Analysis 61 (2005) 1225 – 1240
1239
Proof. Making use of Eqs. (36) and (40), we have
js(n, tm ) 2a˜ m+1 − a˜ 1 c˜m+1 + c˜m+2 − 21 c˜2 c˜m+1 + 21 a˜ 1 c˜m+1 − b˜m+1 = jtm 2s =
(1)
(2)
(a˜ m − a˜ m ) s(c˜m − c˜m ) = −s . 2 2
Thus the proof is completed.
According to Theorem 3, construction of solutions (45) to the hierarchy (17) is split into finding i (n, tm ) i (n, tm ), i = 1, 2, to the integrable symplectic map (34) (or discrete flow (35)) and the finite-dimensional complete integrable systems (40). In fact, in view of s(n, tm ) = 21 1 (n, tm ), 1 (n, tm − 2 (n, tm ), 2 (n, tm ) +2 2 (n, tm ), 1 (n, tm ) is a Bäcklund transformation between the lattice soliton equation (17) and the integrable symplectic map (35) (or discrete flow (34)) and the finite-dimensional completely integrable systems (40). As a result of Theorem 3, we have the following conclusion. Theorem 4. Let ( 1 (n, t1 ), 2 (n, t1 ), 1 (n, t1 ), 2 (n, t1 )) be a solution of the discrete flow (34) and following continuous time evolution equation
it 1 =
jF˜2 , ji
it 1 = −
jF˜2 , i = 1, 2. j i
Then s(n, t1 ) =
1 2
1 (n, t1 ), 1 (n, t1 − 2 (n, t1 ), 2 (n, t1 ) + 2 2 (n, t1 ), 1 (n, t1 )
is a solution of the discrete Kdv equation st1 = s((s (1) )2 − (s (−1) )2 ). References [1] [2] [3] [4] [5]
M. Ablowitz, J. Ladik, Nonlinear differential-deference equation, J. Math. Phys. 16 (1975) 598–603. O.I. Bogoyavlensky, Integrable discretizations of the Kdv equation, Phys. Lett. A 134 (1988) 34–38. M. Bruschi, O. Ragnisco, P.M. Santini, G.Z. Tu, Integrable sympiectic maps, Physica D 49 (1991) 273–294. C.W. Cao, Nonlinearization of the Lax system for AKNS hierarchy, Sci. China Ser. A 33 (1990) 528–536. C.W. Cao, X.G. Geng, Classical integrable systems generated through nonlinearization of eigenvalue problems in nonlinear physics, in: C.H. Gu,Y.S. Li, G.Z. Tu (Eds.), Nonlinear Physics, Proceedings, Shanghai, 1989, Springer, Berlin, 1990, pp. 68–76. [6] W.C. Cao, X.G. Geng, Y.T. Wu, From the special 2+1 Toda lattice to the Kadomtsev–Petvashvili equation, J. Phys. A: Math. Gen. 32 (1999) 8059–8078. [7] Y.S. Li, W.X. Ma, Binary nonlinearization of AKNS spectral problem under higher-order symmetry constraints, Chaos Soliton Fract. 11 (2000) 697–710.
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[8] W.X. Ma, Binary nonlinearization for the Dirac systems, Chinese Ann. Math. Ser. B 18 (1997) 79–88. [9] W.X. Ma, Q. Ding, W.G. Zhang, B.Q. Lu, Binary non-linearization of Lax pairs of Kaup–Newell solito hierarchy, IL Nuovo Cimento B 111 (1996) 1135–1149. [10] W.X. Ma, B. Fuchssteiner, Algebraic structures of discrete zero curvature equations and master symmetries of discrete evolution equations, J. Math. Phys. 40 (1999) 2400–2418. [11] W.X. Ma, X.G. Geng, Bäcklund transformations of soliton systems from symmetry constraints, in: Proceedings of the AARMS-CRM Workshop on Bäcklund & Darboux Transformations: The Geometry of Soliton Theory, Halifax, Canada, 1999. [12] W.X. Ma, W. Strmpp, An explicit symmetry constraint for the Lax pairs and the adjoint Lax pairs of AKNS systems, Phys. Lett. A 185 (1994) 277–286. [13] K. Narita, Soliton solutions for the coupled discrete Kdv equations under nonvanishing boundary conditions at infinity, J. Phys. Soc. Jpn. 71 (2002) 2401–2405. [14] W. Oevel, H. Zhang, B. Fuchssteiner, Mastersymmetries and multi-Hamiltonian formulations for some integrable lattice systems, Prog. Theor. Phys. 81 (1989) 294–308. [15] G.Z. Tu, A trace identity and its applications to the theory of discrete integrable systems, J. Phys. A: Math. Gen. 23 (1990) 3903–3922. [16] X.X. Xu, A generalized Wadati–Konno–Ichikawa hierarchy and new finite-dimensional integrable systems, Phys. Lett. A 301 (2002) 250–262. [17] X.X. Xu, New finite-dimensional integrable system by binary nonlinearization of WKI equations, Chinese Phys. Lett. 12 (1995) 513–516. [18] X.X. Xu, H.H. Dong, A hierarchy of integrable nonlinear lattice equations and new integrable symplectic map, Commun. Theor. Phys. (Beijing, China) 38 (2002) 523–528. [19] Y.T. Wu, X.G. Geng, A new integrable symplectic map associated with lattice soliton equations, J. Math. Phys. 37 (1996) 2338–2345. [20] Y.B. Zeng, Y.S. Li, New symplectic maps: integrability and Lax representation, Chinese Ann. Math. 18B (1997) 457–466. [21] D.J. Zhang, D.Y. Chen, Hamiltonian structure of discrete soliton systems, J. Phys. A: Gen. Math. 35 (2002) 7225–7241. [22] H.W. Zhang, G.Z. Tu, W. Oevel, B. Fuchssteiner, Symmetries, conserved quantities, and hierarchies for some lattice systems with soliton structure, J. Math. Phys. 32 (1991) 1908–1918. [23] R.G. Zhou, W.X. Ma, Classical r-matrix structures of integrable mapping related to the Volterra lattice, Phys. Lett. A 269 (2000) 103–111.
Factorization of a hierarchy of the lattice soliton equations from a binary Bargmann symmetry constraint Xi-Xiang Xu∗ Department of Basic Courses, Shandong University of Science and Technology, Taian 271019, PR China Received 9 August 2003; accepted 24 January 2005
Abstract Staring from a discrete spectral problem, a hierarchy of the lattice soliton equations is derived. It is shown that each lattice equation in resulting hierarchy is Liouville integrable discrete Hamiltonian system. The binary nonlinearization of the Lax pairs and the adjoint Lax pairs of the resulting hierarchy is discussed. Each lattice soliton equation in the resulting hierarchy can be factored by an integrable symplectic map and a finite-dimensional integrable system in Liouville sense. Especially, factorization of a discrete Kdv equation is given. 䉷 2005 Elsevier Ltd. All rights reserved. MSC: 02.30.Ik Keywords: Lax pair; Lattice soliton equation; Discrete Hamiltonian system; Binary nonlinearization; Integrable symplectic map; Finite-dimensional integrable system
1. Introduction As it is well known, nonlinear integrable lattice equations have received considerable attention in recent years. Many nonlinear integrable lattice equations have been proposed and discussed, for instance the Ablowitz–Ladik lattice [1], the Toda lattice [15], and so on [10,14,19,21,22]. ∗ Tel.:/fax: +860 53 8622 6700.
E-mail address: [email protected]. 0362-546X/$ - see front matter 䉷 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2005.01.099
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An approach called the nonlinearization method of the Lax pairs of soliton equations has been developed to obtain finite-dimensional integrable Hamiltonian systems (FDIHSs) [4–6,23]. Under the constraints between the potentials and the eigenfunctions, the Lax pairs of soliton equations can be nonlinearized as finite-dimensional completely integrable systems in the Liouville sense. Recently, the binary nonlinearization method has been proposed by Ma et al. [7–9,12], which involves both the Lax pairs and the adjoint Lax pairs for soliton equations and thus the contents of research are much richer. Up to now, much work has been carried out in the study of the nonlinearization of the Lax pairs for continuous soliton equations [4–9,16–18,23] . However, there has been little discussion about the nonlinearization of the Lax pairs for lattice soliton equations. This paper is structured as follows. In Section 2, we would like to derive a hierarchy of lattice soliton equations from a discrete matrix spectral problem, a typical lattice equation in the resulting hierarchy turn into the Kdv equation in a continuous limit, and it may be transformed into the celebrated Volterra lattice [2,13]. Then, we establish their Hamiltonian structures by the aid of the discrete trace identity [15]. An infinite set of conserved functionals of the resulting hierarchy is given .The conserved functionals are in involution in pairs with respect to the corresponding Poisson bracket. In Section 3, the Lax pairs and the adjoint Lax pairs of the resulting hierarchy are binary nonlinearized by means of the binary Bargmann constraints. The spatial part of the Lax pairs and the adjoint Lax pairs is nonlinearized into a new integrable symplectic map [3,20]. Nonlinearization of the temporal parts of the Lax pairs and the adjoint Lax pairs generate a family of finite-dimensional Liouville integrable Hamiltonian systems. Therefore, each lattice soliton equation in the resulting hierarchy can be factored by an integrable symplectic map and a finite-dimensional integrable system in Liouville sense. The binary Bargmann constraint provides a Bäcklund transformation for each lattice soliton equation in the resulting hierarchy [11] and the integrable symplectic map and the finite-dimensional integrable system in Liouville sense. Finally, factorization of a discrete Kdv equation is given. 2. A hierarchy of the Liouville integrable lattice soliton equations Let us recall some fundamental conceptions. The shift operator E, the inverse of E and difference operator D are defined as follows: (Ef )(n) = f (n + 1), (E −1 f )(n) = f (n − 1), n ∈ Z, (Df )(n) = f (n + 1) − f (n) = (E − 1)f (n), n ∈ Z,
(1) (2)
and f (j ) = E j f, j ∈ Z. We assume that s = s(n, t) is a real function defined over Z × R, and s is required to be rapidly vanishing at infinity. is the spectral parameter and t = 0. A lattice equation st = K(s, Es, E −1 s, . . .)
(3)
is said to be a lattice soliton equation (or Lax integrable), if it can be rewritten as a compatibility condition Ut = (EV)U − UV
(4)
X.-X. Xu / Nonlinear Analysis 61 (2005) 1225 – 1240
1227
of a discrete spatial spectral problem E = U (s, )
(5)
and a corresponding continuous time evolution equations
t = V (s, ),
(6)
where U (s, ) and V (s, t) are of the some order square matrices. Eqs. (5) and (6) are said to be a Lax pair of the lattice soliton equation (3). Eq. (4) is called a discrete zero curvature representation of the lattice soliton equation (3). The Gateaux derivative and the variational derivative are defined by jH j H˜ J (s)[w] = (7) J (s + w) , E −m = j s js (m) =0 m∈Z
and the inner product is defined as f, g = f (n)g(n),
(8)
n∈Z
where f = f (n) and g = g(n) are required to be rapidly vanishing at the infinity, H˜ = ∗ ∗ n∈Z H (s(n)). Operator J is defined by f, J g = Jf , g , it is called the adjoint operator of J with respect to (8). If an operator J has the property J = −J ∗ , then J is called to be skew-symmetric. A linear operator J is called a Hamiltonian operator, if J is a skew-symmetric operator satisfying the Jacobi identity, i.e., J (s)[Jf ]g, h + Cycle(f, g, h) = 0. The associated Poisson bracket with a given Hamiltonian operator J is given by f g f g {f, g}J = ,J = J . s s s s
(9)
n∈Z
Note that it is natural that the Leibniz’rule holds for {f, g}J due to the usage of the variational derivative in the bracket. In virtue of [15], the discrete Hamiltonian system stm = J
H˜ m , m 0, s
is called the Liouville integrable if an infinite number of conserved functionals {H˜ m }∞ m=0 are in involution in pairs with respect to the Poisson bracket (9), i.e., {H˜ m , H˜ l }J = 0(mod D), m, l 0. Let us start the discrete spectral problem 1 0 −s E = U (s, ), U (s, ) = . (10) , = 1/s s 2
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In order to derive the corresponding hierarchy of lattice soliton equations, we proceed first to solve the stationary discrete zero curvature equation (E)U − U = 0.
(11)
We suppose that a solution of Eq. (11) is given by a b = , c −a then Eq. (11) is equivalent to b(1) + s 2 c = 0, b(1) − s 2 (a + a (1) ) = 0, c + (a + a (1) ) = 0, s 2 c(1) + b + (a (1) − a) = 0.
(12)
On setting a=
∞
am −2m ,
b=
∞
bm −2m+1 ,
c=
m=0
m=0
∞
cm −2m+1 ,
m=0
(12) yields equivalently (1)
b0 = 0,
c0 = 0,
(1)
(1)
a0 − a 0 = s 2 c 0 + b 0
and the recursion relation (1) + s 2 cm = 0, m 0, bm (1)
(1) ) = 0, m 0, bm+1 − s 2 (am + am (1)
(1)
(am+1 − am+1 ) + bm+1 + s 2 cm+1 = 0, m 0.
(13)
We choose the initial data a0 = −1,
b0 = 0
and require selecting zero constant for the inverse operation of the difference operator D in computing am , m 1. Thus, the recursion relation (13) uniquely determines am , bm , cm , m 1, and the first few quantities are given by a1 = −2(s (−1) )2 , a2 = −2(s
(−1) 4
b1 = −2(s (−1) )2 ,
) −2(ss
(−1) 2
) −2(s
c1 = 2, (−1) (−2) 2 s
) ,
b2 = −2(s (−1) )4 −2(s (−1) s (−2) )2 ,
c2 = 2s 2 + 2(s (−1) )2 , . . . . For any integer m 0, we choose m m ai 2m−2i bi 2m−2i+1 i=0 i=0 Vm = , m m 2m−2i+1 2m−2i ci − ai i=0
i=0
(14)
X.-X. Xu / Nonlinear Analysis 61 (2005) 1225 – 1240
then the following equation holds: 0 E(Vm )U − UVm = (1) −(am + am )/s
1229
(1) −s(am + am ) . 0
To present the associated hierarchy of lattice systems, we take an additional term cm+1 0 m = 2 cm+1 0 − 2 and define V (m) = Vm + m , m 0. We introduce the following continuous time evolution equations
tm = V (m) , m 0.
(15)
Then the compatibility conditions of Eqs. (10) and (15) gives rise to a hierarchy of discrete zero curvature equations Utm = (EV(m) )U − UV(m) , m 0.
(16)
It describes a hierarchy of lattice soliton equations stm = −s
(2)
(am − am ) , m 0. 2
(17)
When m = 0, Eqs. (17) become the following trivial system: st0 = 0. The first nonlinear lattice soliton equation in the family (17) is st1 = s((s (1) )2 − (s (−1) )2 ).
(18)
From [2], we know that the Eq. (18) turn into the Kdv equation in a continuous limit. Therefore, the Eq. (18) is called to be a discrete Kdv equation. Moreover, if we denote w = s 2 , t1 = 2, the Eq. (18) become w = w(w (1) − w (−1) ), this is the celebrated Volterra lattice [2]. To establish the Hamiltonian structures for (17), we define (a − b)/s sa V = U −1 = (c + a)/s sc and M, N ≡ Tr(MN ), where M and N are the some order square matrices. We have jU jU 0 0 0 −1 . = , = 0 1/s −1/s 2 −1/s 2 j js
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X.-X. Xu / Nonlinear Analysis 61 (2005) 1225 – 1240
Hence
jU = c, V, j
a (1) jU c + a V, = −2 =2 . js s s
By the discrete trace identity [15], jU j jU , V, (n) = − V , j js j s n∈Z
with being a constant to be found. We have (1) 2a j − c(n) = . s j s
(19)
n∈Z
The substitution of a=
∞
ai −2m ,
b=
m=0
∞
bi −2m+1 ,
c=
m=0
∞
ci −2m+1
m=0
into (19), and comparing the coefficients of −2m−1 in Eqs. (19), we get (1) 2am cm+1 (n) = ( − 2m) . s s
(20)
n∈Z
When m = 0 in the Eq. (20), a direct calculation shows that = 0. So we obtain (1) cm+1 am (n) = , m > 0. s −4m s n∈Z
Set H˜ 0 = −2
(ln s)(n),
n∈Z
H˜ m =
cm+1 m∈Z
−4m
(n), m > 0,
(21)
we have (1) H˜ m am = , m 0. s s According to Eq. (13), we get
H˜ m−1 H˜ m = , s s where the recursion operator =
−1 (1 − E −1 )−1 (E −1 s 2 − s 2 E)(1 + E −1 )s. s
We introduce an operator J = s s/2, it is easy to verify that J is a Hamiltonian operator.
X.-X. Xu / Nonlinear Analysis 61 (2005) 1225 – 1240
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Now we can rewrite Eqs. (17) in the following forms: stm
(1) H˜ m−1 H˜ m am m −1 , m 0. =J = J = · · · = J =J 2s s s s
(22)
Therefore, each equation in (17) is a discrete Hamiltonian equation. Moreover, we have J =
−s (1 + E)(E −1 s 2 − s 2 E)(1 + E −1 )s. 2
Obviously, (J )∗ = −(J ). Proposition 1. {H˜ m }m 1 defined by (21) forms an infinite set of conserved functionals of the hierarchy (17), and H˜ m , m 1, are in involution in pairs with respect to the Poisson bracket (9). Proof. Based on (9), we have the following result ˜ ˜ H˜ m H˜ l m−1 H1 l−1 H1 ˜ ˜ Hm , Hl = ,J = , J J s s s s H˜ 1 H˜ 1 = m−1 , L ∗ J l−2 s s H˜ 1 H˜ 1 = m , J l−2 ={H˜ m+1 , H˜ l−1 }J = · · · = {H˜ m+l−1 , H˜ 1 }J . s s Similarly, we also get {H˜ l , H˜ m }J = {H˜ m+l−1 , H˜ 1 }J , thus {H˜ m , H˜ l }J = −{H˜ l , H˜ m }J . This implies {H˜ m , H˜ l }J = 0, m, l 1 and
˜m H (H˜ m )tl = Hm = , stl s n∈Z tl H˜ m H˜ l = , J = {H˜ m , H˜ l } = 0, m, l 1. s s
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X.-X. Xu / Nonlinear Analysis 61 (2005) 1225 – 1240
˜ ∞ Therefore, {Km }∞ m=1 and {Hm }m=1 are infinitely commuting symmetries and infinitely conserved functionals of the lattice hierarchy (23). The proof is completed. From Proposition 1, we arrive at the following result. Theorem 1. Each lattice solution equations in (17) or the discrete Hamiltonian equation in (22) is a discrete Liouville integrable system. 3. Binary nonlinearization of the Lax pairs and the adjoint Lax pairs Let us consider the adjoint spectral problem of spectral problem (10) 1 E −1 = (E −1 U T (u, )) , = 2
(23)
and the continuous time evolution equations
tm = −VmT .
(24)
From (E −1 )tm = E −1 tm , we can find that E −1 UtTm = (E −1 U T )VmT − (E −1 VmT )(E −1 U T ).
(25)
It is easy to verify that Eq. (25) can be rewritten as Utm = (EVm )U − UVm . Therefore, the lattice soliton equations (17) have another kind of discrete zero curvature representation (25). The integrality condition (E −1 )tm = E −1 tm of overdetermined systems (23) and (24) is still the mth lattice soliton equation (17). The Eqs. (23) and (24) are called the adjoint Lax pairs of (17). The adjoint Lax pairs (23) and (24) can help us determine the variational derivative of the spectral parameter with respect to the potential s [11]. Let 1 , 2 , . . . , N be N distinct eigenvalues of spectral problem (10). We have 1j E 1j = U (u, j ) , E 2j −12j E 1j 1j −1 T = (E U (u, j )) , 1j N, (26) 2j E −1 2j 1j 1j , = Vm (u, j ) 2j tm 2j 1j 1j , 1j N, = −VmT (u, j ) (27) 2j tm 2j (E 1j , E 2j ) = (1j , 2j )U (u, j )T , 1 j N ,
(28)
(E 1j , E 2j ) = ( 1j , 2j )U (u, )−1 , 1 j N .
(29)
X.-X. Xu / Nonlinear Analysis 61 (2005) 1225 – 1240
From [11], it is known that j 2j (1) 1j (1) j 2j 2j (1) − 2j 1j − = j − s2 s s2 1j 1j − 2j 2j + 2j 2j 1j = j − , s
1233
(30)
where j /s is a variational derivative for eigenvalue j , j , 1 j N , are constants and i , i , i = 1, 2 are required to be rapidly vanishing at the infinity, Eq. (30) will play a central role in the binary nonlinearization method. We can deduce that 1j 1j − 2j 2j 2j 2j 1j − − s s − 2j 2j 1j 1j 1j 2j 2j = 2j − − . s s The following notations are used:
i = (i1 , i2 , . . . , iN ),
i = ( i1 , i2 , . . . , iN ), i = 1, 2,
= diag(1 , 2 , . . . , N ). And the inner product in R N is denoted by . , . . Consider the discrete constraint J
N j H 1 , =J s s
(31)
j =1
where j =1, j =1, 2, . . . , N. Because of tm =0, J (j /s), 1 j N , are all symmetries of stm = J (H˜ m /s), m 0. Thus (34) (or (35)) is a symmetry constraint. From (31), we have (1) H1 a1 1 , 1 − 2 , 2 2 , 1 = = +2 , s s s s
the symmetry constraint (31) yields s = 21 1 , 1 − 2 , 2 + 2 2 , 1 .
(32)
Because it is possible to solve (31) for s, and s depend on i , i , i = 1, 2, the Eq. (31) or (32) is called a binary Bargmann symmetry constraint. Under constraint (32), Put the N copies of (26) and (29) in vector form, we obtain a discrete Bargmann system E 1j = − 21 1 , 1 − 2 , 2 + 2 2 , 1 2j , 1 j N , E 2j = − √
2(1j + j 2j ) 1 , 1 − 2 , 2 + 2 2 , 1
, 1j N,
1234
X.-X. Xu / Nonlinear Analysis 61 (2005) 1225 – 1240
E 1j = √ E 2j =
1 2
2(j 1j − 2j ) 1 , 1 − 2 , 2 + 2 2 , 1
, 1j N,
1 , 1 − 2 , 2 + 2 2 , 1 1j , 1 j N .
(33)
Setting fi = fi ( 1 , 2 , 1 , 2 ),
gi = gi ( 1 , 2 , 1 , 2 ), 1 i 2N .
j 1j − 2j fi = √ , 1j N, 2 1 , 1 − 2 , 2 + 2 2 , 1 fN+j = 2 1 , 1 − 2 , 2 + 2 2 , 1 1j , 1 j N . gj = − 21 1 , 1 − 2 , 2 + 2 2 , 1 2j , 1 j N , gN+j = − √
2(1j + j 2j ) 1 , 1 − 2 , 2 + 2 2 , 1
.
4N We 2 define Poisson bracket for any pair of functions f and g in sympletic space (R , i=1 di d i ) as follows:
{f, g} =
N 2
i=1 j =1
jf jg jf jg − j ij jij jij j ij
2 jf jg jf jg = − , , , ji j i j i ji
(34)
i=1
which is a skew-symmetric bilinear, and satisfies the Jacobi identity. In particular, f and g are called involutive if {f, g} = 0. Proposition 2. Eq. (33) determines a symplectic map H : [17,18] H (1 , 2 , 1 , 2 ) = (E 1 , E 2 , E 1 , E 2 ). Proof. Through laborious, but direct computation, we get {fi , fj } = {gi , gj } = 0, This completes the proof.
{fi , gj } = ij , 1 i, j 2N .
(35)
X.-X. Xu / Nonlinear Analysis 61 (2005) 1225 – 1240
1235
We can solve Eqs. (13) as follows: b˜0 = 0,
a˜ 0 = −1, a˜ m =
c˜0 = 0,
1 , 1 − 2 , 2 , b˜1 = 2
c˜1 = 2,
2m−2 1 , 1 − 2m−2 2 , 2 , m 1, 2
b˜m = 2m−3 1, 2 , m 2, c˜m = 2m−3 2 , 1 , m 2.
(36)
Set F˜0 = a˜ 02 = 1, F˜m =
m
a˜ i a˜ m−i +
i=0
m
b˜i c˜m−i+1 = −( 2m−2 1 , 1 − 2m−2 2 , 2 )
i=1
1 + 2 2m−3 1 , 2 + ( 1 , 1 − 2 , 2 ) 2m−3 2 , 1 2 +
m−2
2i−1 1 , 2 2m−2i−3 2 , 1 +
i=1
−
2i−2
m−1 1 ( 2i−2 1 , 1 4 i=1
2 2 )(
2m−2i−2
1 , 1 −
2m−2i−2
2 2 ), m 1.
(37)
Proposition 3. DF˜m = 0, m 0,
(38)
i.e., F˜m , m 0, constitutes a hierarchy of integrals of motion for symplectic map H (35) (or the discrete flow (33)). Proof. Because expressions (36) are the solutions of Eqs. (13), we have a˜ =
∞
a˜ m −2m ,
m=0
b˜ =
∞ m=0
b˜m −2m+1 ,
c˜ =
∞
c˜m −2m+1
m=0
is a solution set of Eqs. (12), thus D(a˜ 2 + b˜ c) ˜ = 0, this implies DF˜m = 0, m 0. This completes the proof.
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X.-X. Xu / Nonlinear Analysis 61 (2005) 1225 – 1240
In the following, we would like to discuss the Liouville integrability on the nonlinearized temporal parts of the Lax pairs and adjoint Lax pairs. Under the control of Eq. (32), the temporal parts of the Lax pairs and the adjoint Lax pairs by substituting (36) into (27) become
1j 2j 1j 2j
tm
=V
=V
(m)
tm
1j , j = 1, 2, . . . , N, B
(m)
2j
1j , j = 1, 2, . . . , N. B
(39)
2j
Here the subscript B means substitution of (32) into the expression. Through a direct calculation, we find that Eqs. (39) may be rewritten as
it m =
jF˜m+1 , ji
it m = −
jF˜m+1 , i = 1, 2. j i
(40)
Proposition 4. F˜m , m 0, are in involution in pairs with respect to the Poisson bracket (34), and constitutes a hierarchy of involutive integrals of motion for Eqs. (40). Proof. Through a direct calculation, we can get
˜ tm = [V˜ (m) , ˜ ], where
a˜ ˜ = c˜
b˜ −a˜
∞ a˜ m −2m m=0 = ∞ c˜m −2m+1 m=0
∞
b˜m −2m+1
m=0 ∞
−
m=0
a˜ m −2m
.
This yields 2
d d ˜ 2 = d [V˜ (m) , ˜ 2 ] = 0, (a˜ 2 + b˜ c) ˜ = Tr dtm dtm dtm
i.e., d ˜ Fl = 21 {F˜l , F˜m } = 0, l, m 1. dtm This elucidates that F˜m , m 1, constitutes a hierarchy of involutive integrals of motion for Eqs. (40). F˜m , m 0, are in involution in pairs with respect to the Poisson bracket (34). This completes the proof.
X.-X. Xu / Nonlinear Analysis 61 (2005) 1225 – 1240
1237
Set F¯j = 1j 1j + 2j 2j , 1 j N. It is easy to calculate that d ¯ Fj = 0, 1 j N, m 0, dtm {F˜m , F¯j } = 0, m 0, 1 j N, {F¯i , F¯j } = 0, 1 i, j N .
DF¯j = 0,
(41)
Proposition 5. F˜m+1 , 1 m N, F¯j , 1 j N , are functionally independent over some region of R 4N . Proof. A straightforward computation gives m
m
i=0
i=0
c˜m+1 jF˜m+1 c˜i 2m−2i+1 2 + 1 , = a˜ i 2m−2i 1 + 2 j 1
(42)
m m c˜m+1 jF˜m+1 =− ai 2m−2i 2 + 2 . b˜i 2m−2i+1 1 − j 2 2
(43)
i=0
i=1
Thus, we have jF˜m+1 j 1
= −2 2m 1 ,
1 1 = 2 =0, 2 =− 2 1
jF˜m+1 j 2
= 2m 2 , m 0.
(44)
1 1 = 2 =0, 2 =− 2 1
Assume that the result on the functional independence there exist 2N real is not2 true. Then N 2 = 0, such that numbers 1 , 2 , . . . , N , 1 , 2 , . . . , N , satisfying N
+ i=1 j j =0 j N i=1
i F¯i +
N j =1
j F˜j = 0.
A direct calculation gives rise to
jF¯j = ij j l , j, l = 1, 2, . . . , N. jil
1238
X.-X. Xu / Nonlinear Analysis 61 (2005) 1225 – 1240
We can compute
jF¯1 j 1 det jF¯1 j 2
··· ···
11 .. . 0 = det 21 . .. 0
jF¯N j 1 jF¯N
jF˜2 j 1 jF˜2
j 2
j 2
··· .. . ··· ··· .. . ···
0 .. .
1N 0 .. .
2N
··· ···
jF˜N+1 j 1 jF˜N+1
−221 11 .. .
−22N 1N 21 21 .. .
2N 2N
j 2 ··· .. . ··· ··· .. . ···
−22N 1 11 .. . −22N 1N N 2N 1 21 .. . 2N N 2N
1 21 N N 2 N = (−1)N 23 3j 1 2 21j ··· j =1 j =1 1 2N
· · · 2N−2 1 2N−2 · · · 2 . ··· · · · 2N−2 N
By observing that the Vandermode determinant V (21 , 22 , . . . , 2N ) = 0, this means that all
i , i must be zero. Therefore the functions F¯j , F˜j +1 , j = 1, 2, . . . , N, can be functionally independent at least on certain region of R 4N . This completes the proof. Summing up, according to the Propositions 3–5, we have the following theorem. Theorem 2. Symplectic map (35) is integrable and nonlinearized temporal parts of the Lax pairs and the adjoint Lax pairs (40) are all finite-dimensional integrable systems in the Liouville sense. Moreover, Propositions 5 and 6 provide us with an involutive and independent system of polynomials, which are completely new as far as we know. Let ( 1 (n, tm ), 2 (n, tm ), 1 (n, tm ), 2 (n, tm )) be a solution of the discrete flow (34) and the finite-dimensional completely integrable systems (40). Theorem 3. The function s(n, tm ) = 21 1 (n, tm ), 1 (n, tm − 2 (n, tm ), 2 (n, tm ) +2 2 (n, tm ), 1 (n, tm ) (45) solves the lattice soliton equation (17) or Hamiltonian system (22).
X.-X. Xu / Nonlinear Analysis 61 (2005) 1225 – 1240
1239
Proof. Making use of Eqs. (36) and (40), we have
js(n, tm ) 2a˜ m+1 − a˜ 1 c˜m+1 + c˜m+2 − 21 c˜2 c˜m+1 + 21 a˜ 1 c˜m+1 − b˜m+1 = jtm 2s =
(1)
(2)
(a˜ m − a˜ m ) s(c˜m − c˜m ) = −s . 2 2
Thus the proof is completed.
According to Theorem 3, construction of solutions (45) to the hierarchy (17) is split into finding i (n, tm ) i (n, tm ), i = 1, 2, to the integrable symplectic map (34) (or discrete flow (35)) and the finite-dimensional complete integrable systems (40). In fact, in view of s(n, tm ) = 21 1 (n, tm ), 1 (n, tm − 2 (n, tm ), 2 (n, tm ) +2 2 (n, tm ), 1 (n, tm ) is a Bäcklund transformation between the lattice soliton equation (17) and the integrable symplectic map (35) (or discrete flow (34)) and the finite-dimensional completely integrable systems (40). As a result of Theorem 3, we have the following conclusion. Theorem 4. Let ( 1 (n, t1 ), 2 (n, t1 ), 1 (n, t1 ), 2 (n, t1 )) be a solution of the discrete flow (34) and following continuous time evolution equation
it 1 =
jF˜2 , ji
it 1 = −
jF˜2 , i = 1, 2. j i
Then s(n, t1 ) =
1 2
1 (n, t1 ), 1 (n, t1 − 2 (n, t1 ), 2 (n, t1 ) + 2 2 (n, t1 ), 1 (n, t1 )
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