Physics Letters A 374 (2010) 1922–1926

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Physics Letters A

www.elsevier.com/locate/pla

An integrable lattice equation related to the nKN system

Xiao Yang a,∗, Junmin Wang b

a Department of Mathematics, Zhengzhou University, Zhengzhou, Henan 450002, PR Chinab School of Information, Henan University of Finance and Economics, Zhengzhou, Henan 450002, PR China

a r t i c l e i n f o a b s t r a c t

Article history:Received 20 October 2009Accepted 26 February 2010Available online 6 March 2010Communicated by A.P. Fordy

Keywords:Lattice equationLax pairIntegrable symplectic mapAlgebro-geometric solution

The Lax pair of a lattice equation is given by a discrete spectral problem and the negative Kaup–Newellspectral problem. Based on the Lax nonlinearization technique, they are transformed into a symplecticmap and a Hamiltonian system, which are integrable in the Liouville sense and are straightened out in theJacobi coordinates. An algebraic–geometric solution of the lattice equation is obtained by the Riemann–Jacobi inversion. Explicit solutions of the associated lattice hierarchy are given.

© 2010 Elsevier B.V. All rights reserved.

1. Introduction

Since the continuous systems have been well developed tosome degree, people turn to care about the discrete ones. In thestudy of the integrable models containing discrete variables, to ob-tain integrable symplectic maps is essential. A symplectic map isa map that preserves a symplectic form [1,2]. It arises naturallyin physical systems and is closed related to Hamiltonian systems.There are many applications of symplectic maps including acceler-ate chemical, condensed matter, plasma and fluid physics [1–8].

In this Letter, a lattice hierarchy associated with a discrete spec-tral problem is studied, with special emphasis on the lattice equa-tion{

an = a2nbn+1 − an−1,

bn = bn+1 − b2nan−1,

(1.1)

where n corresponds to the symplectic map S (S-flow), · denotes∂τ−1 which corresponds to the negative Kaup–Newell (nKN) system(H−1-flow). As the importance of the KN hierarchy in physics andapplied mathematics [9,10], the works on the positive KN flowsare plenty [9–13], but on the negative are rare [14–16]. In [16],the negative systems {H−k} were obtained, among which H−1 wasspecial studied, and named nKN system:

H−1 = 1

2

⟨A−2 p,q

⟩ − 1

2

〈A−1 p, p〉〈A−1q,q〉2〈p,q〉 − 1

,

∂

(p

q

)= I∇H =

(−∂ H−1/∂q

∂ H−1/∂ p

). (1.2)

* Corresponding author.E-mail address: yx@zzu.edu.cn (X. Yang).

0375-9601/$ – see front matter © 2010 Elsevier B.V. All rights reserved.doi:10.1016/j.physleta.2010.02.080

In the present Letter, assisted with the results in [16], explicit solu-tions of the lattice hierarchy are calculated, an algebraic–geometricsolution of Eq. (1.1) is obtained.

2. The Lax pair and the symplectic map S

We start from the discrete spectral problem:

Eχ = Uχ, U =(

λ(1 + ab) a

b λ−1

), (2.1)

where a,b are potentials, satisfy 1 + ab �= 0, λ is a constant spec-tral parameter, E is the shift operator with regard to the dis-crete variable n: E f (n) = f (n + 1), E− f (n) = f (n − 1), � f (n) =f (n + 1) − f (n).

Let γ be an arbitrary smooth function, define a linear map:

σ : γ �→ V = σλ(γ ) =( −γ 3 λ−1γ 1

λ−1 E−γ 2 γ 3

).

Then we have the discrete commutative relation (fundamentalidentity):

(E V )U − U V = σλ

{(K − λ−2 J

)γ

}, (2.2)

with

σλ(ξ) �(

λ{bξ1 + aξ2 − (1 + ab)ξ3} ξ1

ξ2 λ−1ξ3

),

K =⎛⎝−(1 + ab) 0 −a(E + 1)

0 1 + ab b(E + 1)

⎞⎠ ,

−b a �

X. Yang, J. Wang / Physics Letters A 374 (2010) 1922–1926 1923

J =(−E 0 0

0 E− 00 0 0

).

Define the Lenart sequence {g− j} recursively by

K g− j = J g− j−1, J g0 = 0, j = 0,1,2, . . . ,

g0 =⎛⎝ 0

012

⎞⎠ , g−1 =⎛⎝ E−a

Eb

−bE−a

⎞⎠ ,

g−2 =⎛⎝ E−2a − bE−a2

E2b − aEb2

b2 E−a2 − E−a − Eb − bE2a

⎞⎠ . . . .

Let γ− j = ∑ js=0 g−sλ

2s−2 j , V− j = σλ(γ− j), substitute V− j into thezero curvature equation Uτ j − (E V )U + U V = 0, we have theisospectral class of the discrete spectral problem (2.1):

d

dτ j

(a

b

)= P J g− j−1 � X j, P : {α,β,γ }T → {α,β}T , (2.3)

which implies a lattice hierarchy{an,τ0 = −an,

bn,τ0 = bn,{an,τ−1 = a2

nbn+1 − an−1,

bn,τ−1 = bn+1 − an−1b2n,⎧⎪⎪⎨⎪⎪⎩

an,τ−2 = −an−2 + a2n−1bn + a2

nbn+2 − a3nb2

n+1

+ 2anan−1bn+1,

bn,τ−2 = bn+2 − anb2n+1 − an−2b2

n + a2n−1b3

n

− 2an−1bnbn+1,

. . . . (2.4)

Note τ−1 by ·, We have Eq. (1.1) and it’s Lax pair

∂τ−1χ = V−1χ, V−1 =(− 1

2 λ−2 + bE−a λ−1 E−a

λ−1b 12 λ−2 − bE−a

);

Eχ = Uχ, U =(

λ(1 + ab) a

b λ−1

). (2.5)

Note 〈ξ,η〉 = ∑Nj=1 ξ jη j , A = diag(α1, . . . ,αN ), define the

Bargmann map f S :(a

b

)= f S(p,q) =

( E〈A−1 p,p〉2〈p,q〉−1

− 〈A−1q,q〉2〈p,q〉−1

). (2.6)

Under the Bargmann map f S , Eq. (2.5) is equal to Eq. (1.2). It be-longs to the nKN system.

A direct calculation gives

〈Ap, p〉 = − a

1 + ab, E〈Aq,q〉 = b

1 + ab, (2.7)

which is important in the following discuss.Consider the map S : R

2 → R2,

E

(p

q

)=

(p

q

)= S

(p

q

)=

(A(1 + ab)p + aq

bp + A−1q

)(a,b)= f S (p,q)

. (2.8)

Exactly, the jth component of Eq. (2.8) is the discrete spectralproblem (2.1) with λ = α j .

Proposition 2.1. Under the Bargmann map f S , E〈p,q〉 = 〈p, q〉 =〈p,q〉.

Proof. By Eqs. (2.7) and (2.8), we have

E〈p,q〉 = b〈Ap, p〉 + aE〈Aq,q〉 + 〈p,q〉 = 〈p,q〉. �Proposition 2.2. S is symplectic: S∗(dp ∧ dq) = dp ∧ dq.

Proof. Also by Eqs. (2.7) and (2.8), we have

dp ∧ dp = dp ∧ dq + 1

2d〈Ap, p〉 ∧ db + 1

2da ∧ d〈Aq,q〉

= dp ∧ dq. �3. The integrable Hamilton systems and decomposition of thelattice hierarchy

In this section, we’ll give the Hamilton systems and prove theLiouville integrability, then express the solutions of the lattice hi-erarchy (2.4) in (p,q) variables through decomposition.

Under the Bargmann map f S , direct calculations give

Proposition 3.1. Suppose (p,q) solves Eq. (2.7). Then

(i) (K − α2j J )δ j = 0, has a solution:

δ j = (−α j p2j ,α j Eq2

j ,−p jq j)T ; (3.1)

(ii) (K − λ2 J )Gλ = 0, has a solution (Q λ(ξ,η) �∑N

j=1ξ jη j

λ2−α2j):

Gλ = (−λ2 Q λ(Ap, p), λ2 E Q λ(Aq,q),

− [1/2 + Q λ

(A2 p,q

)])T. (3.2)

Define the Moser matrix Vλ = σλ(Gλ):

Vλ =( 1

2 0

0 − 12

)+

(Q λ(A2 p,q) −λ2 Q λ(p, p)

Q λ(A2q,q) −Q λ(A2 p,q)

)

�(

V 11λ V 12

λ

V 21λ −V 11

λ

), (3.3)

Vλ and U commute:

(E Vλ)U − U Vλ = 0.

Since det U = 1, we have E det Vλ = det Vλ , thus S-flow has anintegral Fλ ,

Fλ = det Vλ

= −(1/2 + Q λ

(A2 p,q

))2 + λ2 Q λ(p, p)Q λ

(A2q,q

).

Consider Fλ as a Hamiltonian function in the symplectic space(R2N , dp ∧ dq) and note the flow variable by tλ , the canonicalequation is written as

d

dtλ

(p j

q j

)=

(− ∂ Fλ

∂q j

∂ Fλ

∂ p j

)= 2α j

λ2 − α2j

(α j V 11

λ λV 12λ

λV 21λ −α j V 11

λ

). (3.4)

Define Hλ by −4Fλ = (−4Hλ)2, the Hamilton systems are deter-

mined through power series expansions:

Fλ =∞∑

F− jλ2 j, Hλ =

∞∑H− jλ

2 j.

j=0 j=0

1924 X. Yang, J. Wang / Physics Letters A 374 (2010) 1922–1926

The first few members are

F0 = −(

〈p,q〉 − 1

2

)2

,

F−1 = −(2〈p,q〉 − 1

)⟨A−2 p,q

⟩ + ⟨A−1 p, p

⟩⟨A−1q,q

⟩,

H0 = 1

2

(〈p,q〉 − 1

2

),

H−1 = 1

2

⟨A−2 p,q

⟩ − 1

2

〈A−1 p, p〉〈A−1q,q〉2〈p,q〉 − 1

.

Refer to [16], we have

Proposition 3.2. Each of the Hamiltonian systems (F− j), (H− j) is com-pletely integrable in the Liouville sense.

Note the flows of (Hλ), (F− j), (H− j) by variables τλ, t− j, τ− j .Since −4Fλ = (−4Hλ)

2, the Leibniz rule of the Poisson bracketgives rise to:

d

dtλ= −8Hλ

d

dτλ

. (3.5)

By Proposition 3.1, a direct calculation gives:

d

dtλ

(a

b

)= −2λ−2 P J Gλ, (3.6)

Gλ = 4Hλgλ, (3.7)

where gλ = ∑∞j=0 g− jλ

2 j , satisfies (K − λ−2 J )gλ = 0. By Eq. (3.5)and (3.6), we have the evolution equation of (a,b) = f S (p,q) alongthe τλ-flow:

d

dτλ

(a

b

)= P J gλ =

∞∑j=0

X− jλ2 j. (3.8)

Comparing the coefficients of λ2 j , we have Eq. (2.3)

d

dτ j

(a

b

)= X j,

also the lattice hierarchy (2.4).

Proposition 3.3.

(i) Let (p(n, τ− j),q(n, τ− j)) be a compatible solution of the discreteflow (S-flow) and the continuous flow (H− j -flow):

d

dτ− j

(p

q

)= I∇H− j =

(−∂ H− j/∂q

∂ H− j/∂ p

).

Then (a(n, τ− j),b(n, τ− j)) = f S (p(n, τ− j),q(n, τ− j)) solves thelattice hierarchy (2.4).

(ii) Let (p(n, τ−1),q(n, τ−1)) be a compatible solution of the S-flowand the H−1-flow with τ−1 = t. Then (a(n, τ−1),b(n, τ−1)) =f S (p(n, τ−1),q(n, τ−1)) is a solution of the lattice equation (1.1).

4. Straightening out of the continuous (H− j) flow

Note ζ = λ2. Since Fλ, V 12λ , V 21

λ are rational functions of ζ withsimple poles at ζ j = α2

j , they can be factorized as:

Fλ = −V 12λ V 21

λ − (V 11

λ

)2 = − β(ζ )

4α(ζ )= − R(ζ )

4α2(ζ ); (4.1)

V 12λ = −λQ λ(Ap, p) = −λ〈Ap, p〉m(ζ )

α(ζ ); (4.2)

V 21λ = λQ λ(Aq,q) = λ〈Aq,q〉 n(ζ ) ; (4.3)

α(ζ )

where

α(ζ ) =N∏

j=1

(ζ − α2

j

) =N∏

j=1

(ζ − ζ j),

β(ζ ) =N∏

j=1

(ζ − ζN+ j),

R(ζ ) = α(ζ )β(ζ ) =2N∏j=1

(ζ − ζ j),

m(ζ ) =N−1∏j=1

(ζ − μ j), n(ζ ) =N−1∏j=1

(ζ − γ j).

The elliptic coordinates {μ j}, {γ j} are defined as zeros of m(ζ ),

n(ζ ).Resorting to Eqs. (3.4) and (4.1)–(4.3), the evolution equations

of the elliptic coordinates along the Fλ-flow are obtained:

1

2√

R(μ2j )

dμ2j

dtλ= ζm(ζ )

α(ζ )(ζ − μ2j )m

′(μ2j )

,

1

2√

R(γ 2j )

dγ 2j

dtλ= −ζn(ζ )

α(ζ )(ζ − γ 2j )n′(γ 2

j ).

By the interpolation formula, we have ( j = 1,2, . . . , N):

N−1∑k=1

(μ2k )N−1− j

2√

R(μ2k )

dμ2k

dtλ= ζ N− j

α(ζ ),

N−1∑k=1

(γ 2k )N−1− j

2√

R(γ 2k )

dγ 2k

dtλ= −ζ N− j

α(ζ ). (4.4)

Consider the algebraic curve Γ given by the affine equationΓ : ξ2 − R(ζ ) = 0. The genus of the curve is g = N − 1, sincedeg R = 2N . Corresponding to each ζ ∈ C, there are two pointson Γ : P (ζ ) = (ζ, ξ = ±√

R(ζ )). Take the usual holomorphic dif-ferentials on Γ : ω j and define the quasi-Abel–Jacobi variable φ j :

ω j = ζ g− jdζ

2√

R(ζ ); φ j =

g∑k=1

P (μ2k )∫

P0

ω j,

ψ j =g∑

k=1

P (γ 2k )∫

P0

ω j ( j = 1, . . . , g);

with fixed point P0 ∈ Γ , which make Eq. (4.4) very simple:

dφ j

dtλ= ζ g− j+1

α(ζ ),

dψ j

dtλ= −ζ g− j+1

α(ζ ).

Hence, by Eqs. (3.5) and (4.1),

dφ j

dτλ

= ζ g− j+1

2√

R(ζ ),

dψ j

dτλ

= − ζ g− j+1

2√

R(ζ ). (4.5)

Let a1, . . . ,ag,b1, . . . ,bg , be the normalized basis of homolog-ical cycles of Γ , and C = (A jk)

−1g×g , where A jk is equal to the

integral of ω j along the path ak . Then ω = Cω is the normalizedholomorphic differential. The periodic vector δk, Bk are the inte-grals of ω along ak,bk , respectively. They span the lattice T in C

g .Let B be the matrix with the matrix with the vector Bk . It is used

X. Yang, J. Wang / Physics Letters A 374 (2010) 1922–1926 1925

to construct the theta function θ(ζ, B) of the curve Γ . Define theAbel map as:

A : Div(Γ ) → J (Γ ) = Cg/T , A(P ) =

P∫P0

ω,

A(∑

k

Pk

)=

∑k

A(Pk).

Define the Abel–Jacobi coordinates as:

φ = A( g∑

k=1

P(μ2

k

)) = C φ, ψ = A( g∑

k=1

P(γ 2

k

)) = Cψ.

Then Eq. (4.5) can be rewritten as:

dφ

dτλ

= ζω

dζ,

dψ

dτλ

= −ζω

dζ.

Considering the expansion of the normalized basis ω of holomor-phic differentials in the neighborhood of 0l = (0, ξ =(−1)l√R(0)) with coefficient Ω− j , j = 1,2, . . .:

ω = (−1)l∞∑j=1

Ω− jζj−1dζ, Ω−1 = 1

2√

R(0)C g,

we have

Proposition 4.1. The H− j -flow is straightened out by the Abel–Jacobicoordinates:

dφ

dτ− j= Ω j,

dψ

dτ− j= −Ω j . (4.6)

5. Straightening out of the discrete S flow and the explicitsolutions in Abel–Jacobi coordinates

Let an,bn be finite-gap potentials, Un = U (an,bn, λ), and M(n)

be the fundamental solution matrix of Eq. (2.1):

E M(n) = Un M(n); M(0) =(

1 00 1

),

M(n) =(

p(1)(n, ζ ) p(2)(n, ζ )

q(1)(n, ζ ) q(2)(n, ζ )

). (5.1)

By induction, we have

M(n) = Un−1Un−2 · · · U0; Vλ(n)M(n) = M(n)Vλ(0),

M(n) =(

m11n λ−n+2[1 + O (λ2)] m12

n λ−n+1[1 + O (λ2)]m21

n λ−n+1[1 + O (λ2)] m22n λ−n[1 + O (λ2)]

),

ζ → 0;

M(n) =(

m11n λn[1 + O (λ−2)] m12

n λn−1[1 + O (λ−2)]m21

n λn−1[1 + O (λ−2)] m22n λn−2[1 + O (λ−2)]

),

ζ → ∞.

By Eq. (5.1), the solution space Eλ of Eq. (2.1) is invariant under theaction of Vλ . It has a eigenvalue ρ± = ±√−Fλ with the associatedeigenvector in Eλ as

χ±(n) =(

p±(n, λ)

q±(n, λ)

)= M(n)

(1

d±)

=(

p(1)(n, λ) + d±p(2)(n, λ)

q(1)(n, λ) + d±q(2)(n, λ)

). (5.2)

Let n = 0, we have

d± = − V 11λ (0) − ρ±

V 12λ (0)

= V 21λ (0)

V 11λ (0) + ρ± . (5.3)

By Eqs. (5.2) and (5.3), a direct calculation gives the discreteDubrovin–Novikov formula:

p+(n, λ)p−(n, λ) =g∏

j=1

ζ − μ2j (n)

ζ − μ2j (0)

,

1

d+d− q+(n, λ)q−(n, λ) =g∏

j=1

ζ − γ 2j (n)

ζ − γ 2j (0)

.

Thus we have the asymptotic behaviors for λ → 0, λ → ∞, respec-tively:

λn p+ = 〈A−1 p, p〉n

(2〈p,q〉 − 1)m12n

λ2n[1 + O(λ2)],

λn p− = (2〈p,q〉 − 1)m12n

〈A−1 p, p〉0

[1 + O

(λ2)],

λn−1q+ = 1

m21n

λ2(n−1)[1 + O

(λ2)],

λn−1q− = m21n

[1 + O

(λ2)],

λn p+ = m11n λ2n[1 + O

(λ−2)],

λn p− = 1

m11n

[1 + O

(λ−2)],

λn−1q+ = 1

m21n

[1 + O

(λ−2)],

λn−1q− = m21n

[1 + O

(λ2)].

Resorting to these results, in a similar way as in [8], we have

Proposition 5.1. The Abel–Jacobi coordinates straighten out the sym-plectic flow S.

φ(n) − φ(0) = kΩS ,

ψ(n) − ψ(0) = (k − 1)ΩS , (mod T ), (5.4)

ΩS =∞2∫

02

w, (mod T ). (5.5)

In the light of the Abel–Jacobi coordinates, the continuous H− j -flow and discrete S-flow are straightened out and have uniformvelocity. Therefore the compatible solution of various flow is ob-tained through a linear superposition. Specifically, for the latticehierarchy (2.3), we have

φ(n, τ− j) = nΩS + τ− jΩ− j + φ0,

ψ(n, τ− j) = (n − 1)ΩS − τ− jΩ− j + ψ0. (5.6)

For the lattice (1.2), we have

φ(n, t) = nΩS + tΩ−1 + φ0,

ψ(n, t) = (n − 1)ΩS − tΩ−1 + ψ0. (5.7)

The explicit solution of the lattice (1.2) in the Abel–Jacobi coor-dinates (φ,ψ) has been given, in order to get the solution in theoriginal coordinate an,bn , the following section is necessary.

1926 X. Yang, J. Wang / Physics Letters A 374 (2010) 1922–1926

6. Inversion

According to the Riemann theorem [17], there exists constantvectors K1 and K2, such that θ(A(P (ζ ))−φ− K1) has exactly zerosμ1, . . . ,μg , and θ(A(P (ζ )) − ψ − K2) has exactly zeros γ1, . . . , γg

as well. Consider the expansion near 0l:

A(

P (ζ )) =

P (ζ )∫P0

ω = (−1)l∞∑j=1

1

jΩ− jζ

j − η0l , η0l =0l∫

P0

ω,

through a standard treatment [8], we obtain:

g∑j=1

μ−2j = I−1(Γ ) −

2∑l=1

Res0l

ζ−1d ln θ(

A(

P (ζ )) − φ − K1

)= I−1(Γ ) − Ω

j−1∂ j ln

θ(φ + K1 + η01)

θ(φ + K1 + η02), (6.1)

g∑j=1

γ −2j = I−1(Γ ) −

2∑l=1

Res0l

ζ−1d ln θ(

A(

P (ζ )) − ψ − K2

)= I−1(Γ ) + Ω

j−1∂ j ln

θ(−ψ − K2 − η01)

θ(−ψ + K2 − η02), (6.2)

where ∂ j denotes the differentiation with respect to the jth argu-ment of the theta function and

I1(Γ ) =g∑

j=1

∫a j

ζ−1ω j .

In [16], the authors have given

1

2

2N∑j=1

α−2j −

g∑j=1

μ−2j = −an−1,t

an−1. (6.3)

Similarly, the following formula holds

1

2

2N∑j=1

α−2j −

g∑j=1

γ −2j = bn,t

bn. (6.4)

Based on Eqs. (6.1)–(6.4), we have

−an−1,t

an−1= 1

2

2N∑j=1

α−2j − I−1(Γ ) + ∂t ln

θ(φ + K1 + η01)

θ(φ + K1 + η02), (6.5)

bn,t

bn= 1

2

2N∑j=1

α−2j − I−1(Γ ) − ∂t ln

θ(−ψ − K2 − η01)

θ(−ψ + K2 − η02). (6.6)

Submitting Eq. (5.7) into Eqs. (6.5) and (6.6), through a direct cal-culation, we have

Proposition 6.1. The lattice equation (1.2) has an algebro-geometric so-lution:

a(n, t) = a(n,0)e−D−1t

× θ((n + 1)ΩS + φ0 + K1 + η01)θ(tΩ−1 + (n + 1)ΩS + φ0 + K1 + η02

)

θ((n + 1)ΩS + φ0 + K1 + η02)θ(tΩ−1 + (n + 1)ΩS + ψ0 + K1 + η01

),

(6.7)

b(n, t) = b(n,0)e−D−1 y

× θ(−nΩS − ψ0 − K2 − η1)θ(tΩ−1 − nΩS − ψ0 − K2 − η2)

θ(nΩS − ψ0 − K2 − η2)θ(tΩ−1 − nΩS − ψ0 − K2 − η1),

(6.8)

with D−1 = 12

∑2Nj=1 α−2

j − I−1(Γ ).

Acknowledgements

The work of Xiao Yang is supported by the National NaturalScience Foundation of Henan, the Project number is 2008B110003.

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