Binary Bargmann symmetry constraint associated with 3 × 3 discrete matrix spectral problem

Research Article

Binary Bargmann symmetry constraint associated with 3 × 3 discrete matrix spectral problem

Xin-Yue Li^a,∗, Qiu-Lan Zhao^a, Yu-Xia Li^b, Huan-He Dong^a

aCollege of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, P. R. China.

bShandong Key Laboratory for Robot and Intelligent Technology, Qingdao 266590, P. R. China.

Communicated by B. G. Sidharth

Abstract

Based on the nonlinearization technique, a binary Bargmann symmetry constraint associated with a new discrete 3×3 matrix eigenvalue problem, which implies that there exist infinitely many common commuting symmetries and infinitely many common commuting conserved functionals, is proposed. A new symplectic map of the Bargmann type is obtained through binary nonlinearization of the discrete eigenvalue problem and its adjoint one. The generating function of integrals of motion is obtained, by which the symplectic map is further proved to be completely integrable in the Liouville sense. c⃝2015 All rights reserved.

Keywords: Discrete Hamiltonian structure, binary Bargmann symmetry constraint, finite-dimensional integrable system .

2010 MSC: 35Q51, 37J15.

1. Introduction

Recently in the past decade, an unusual way of using the nonlinearization technique arose in the theory of soliton equations. In general, one considers the complicated nonlinear problems to be solved in such a way to break nonlinear problems into several linear or smaller ones and then to solve these resulting problems. It is following this idea that one has introduced the method of Lax pair to study nonlinear soliton equations. The Lax pairs are always linear with respect to their eigenfunctions. Nevertheless, the nonlinearization technique puts this original object, the Lax pair, into a nonlinear and more complicated object, the nonlinearized Lax system. The main reason why the nonlinearization technique takes effect is that kind of specific symmetry constraints expressed through the variational derivative of the potential.

∗Corresponding author

Email addresses: [email protected](Xin-Yue Li),[email protected](Qiu-Lan Zhao) Received 2015-3-30

The study of symmetry constraints itself is an important part of the kernel of the mathematical theory of nonlinearization, which can manipulate both mono-nonlinearization [3] and binary nonlinearization [12, 25].

However, all examples of application of the nonlinearization technique, discussed so far, are related to lower-order matrix spectral problems of soliton equations, most of which are only concerned with second- order traceless matrix spectral problems. On the other hand, there appears much difficulty in handling the Liouville integrability of the so-called constrained flows generated from spectral problems, in the case of the third and fourth-order matrix spectral problems[5, 10, 15, 16, 29, 34]. It is a challenging task to extend the theory of nonlinearization to the case of higher-order matrix spectral problems. In this article, we would like to establish a concrete example to apply the nonlinearization technique to the case of higher-order matrix spectral problems, by manipulating binary nonlinearization[1, 4, 7, 9, 11, 13, 14, 17, 18, 22, 23, 24, 30, 31, 32]

for arbitrary-order matrix spectral problems associated with 3×3 discrete matrix eigenvalue problem. The resulting theory will show a direct way for generating sufficiently many integrals of motion for the Liouville integrability of the constrained flows resulting from higher-order matrix spectral problems.

This article is organized as follows. In Section 2, a discrete 3×3 matrix spectral problem is introduced, and a hierarchy of lattice soliton equations is derived by the method of discrete zero curvature representation.

A lattice system is proposed, it is a typical lattice system in resulting hierarchy. Infinitely many commuting symmetries and infinitely many commuting conserved functionals for the obtained hierarchy are given. In Section 3,we consider the Bargmann symmetry constraint for the proposed new Lax pairs and adjoint Lax pairs of the discrete soliton hierarchy. Finally in Section 4, conclusions and remarks are given.

2. A family of lattice soliton equations and its Liouville integrability Let we define the shift operator E, the inverse ofE by

Ef_n=f_n+1, E⁻¹f_n=f_n−1,∆ =E−E⁻¹, n∈Z,

(1−E)⁻¹=−(1 +E⁻¹)∆⁻¹,(1−E⁻¹)⁻¹ = (1 +E)∆⁻¹. We introduce the new discrete 3×3 matrix spectral problem

Eψn=Un(un, λ)ψn=



 p_n 1 q_n−λ

0 0 1

sn 0 0







 ψ_n¹ ψ_n² ψ_n³



, (2.1)

where the potential vector U_n = (p_n, q_n, s_n)^T, λ_t = 0, and solve the stationary discrete zero curvature equation

(EV_n)U_n−U_nV_n= 0, V_n= (V_n^ij)_3×3, (2.2) where each entry (Vn^ij)_3×3 =Vn^ij(A_n(λ), B_n(λ), D_n(λ)) of the 3×3 matrixV_n is a Laurent expansion ofλ.

When we choose V_n¹²=An(λ), V_n³²=Bn(λ), V_n²²=Dn(λ), we have

V_n¹¹=E⁻¹pnAn(λ)−λE⁻¹Bn(λ) +E⁻¹qnBn(λ) +E⁻¹Dn(λ), V_n¹³=qnAn(λ)−λAn(λ) +_s¹

nEBn(λ), V_n²¹=E⁻¹B_n(λ), V_n²³=E^{−1 1}_s

nE⁻¹s_nA_n(λ)−E^−1p_sⁿ

nB_n(λ), V_n³¹=E⁻¹snAn(λ), V_n³³=−λBn(λ) +qnBn(λ) +EDn(λ).

(2.3)

Substituting the following expressions A_n(λ) =

∑∞ m=−1

A^(m)_n λ^−m, B_n(λ) =

∑∞ m=−1

B_n^(m)λ^−m, D_n(λ) =

∑∞ m=−1

D_n^(m)λ^−m. (2.4) The stationary discrete zero-curvature equation (2.2) is equivalent to the recursion relation:

(s_nE−E⁻¹s_n)A^(j)_n (λ) +p_n(1−E⁻¹)B^(j)_n (λ)

= (p²_n−p_nE⁻¹p_n+s_nEq_n−q_nE⁻¹s_n)A^(jn⁻¹⁾(λ) + (pnqn−pnE⁻¹qn+snE_s¹

nE−E⁻¹)Bn^(j−1)(λ) +pn(1−E⁻¹)D^(jn⁻¹⁾(λ), (1−E)D^jn(λ)

= (E−E^{−1 1}_s

nE⁻¹s_n)A^(j_n⁻¹⁾(λ) + (E^−1p_sⁿ

n −^p_sⁿ_nE)B_n^(j−1)(λ) +q_n(1−E)D_n^(j−1)(λ), s_n(E−E⁻¹)Bn^(j)(λ)

=sn(1−E⁻¹)pnA^(jn⁻¹⁾(λ) +sn(E−E⁻¹)qnBn^(j−1)(λ) +sn(E²−E⁻¹)Dn^(j−1)(λ).

(2.5)

From the above recursion equations, we obtain the initial data A⁽⁻¹⁾_n = 0, B⁽⁻¹⁾_n = 1, D⁽⁻¹⁾_n = 0, A⁽⁰⁾_n = 1

s_n, B⁽⁰⁾_n =q_n, D_n⁽⁰⁾= p_n−1 s_n−1,· · · To obtain Lax integrable equations, we defineFn^j by the following relation:

D^(j)n (λ) =−p_nA^(j)n (λ)−(1 +E⁻¹)s_nFn^(j)(λ). (2.6) It is easy to see that

(s_nE−E⁻¹s_n)A^(j)n (λ) +p_n(1−E⁻¹)Bn^(j)(λ)

= (snEqn−qnE⁻¹sn)A^(jn⁻¹⁾(λ) + (pnqn−pnE⁻¹qn+snE_s¹

nE−E⁻¹)Bn^(j−1)(λ) +p_n(E⁻²−1)s_nF_n^(j−1)(λ),

(E−1)p_nA^(j)n (λ) + ∆s_nFn^j(λ)

= (E−E^{−1 1}_s

nE⁻¹sn+qnEpn−pnqn)A^(jn⁻¹⁾(λ) + (E^−1p_sⁿ

n −^p_sⁿ_nE)Bn^(j−1)(λ) +qn∆snFn^(j−1)(λ),

s_n∆Bn^(j)(λ) =s_n(1−E²)p_nA^(jn⁻¹⁾(λ) +s_n∆q_nBn^(j−1)(λ) +sn(E⁻²−E²+E⁻¹−E)snFn^(j−1)(λ).

(2.7)

Using the matrix notation, the above expressions (2.3) can be written as

KG^j_n⁻¹ =J G^j_n, G^j_n= (A^(j)_n , B^(j)_n , F_n^(j))^T, j ≥0, (2.8) where so-called Lenards operator pair J and K are two skew-symmetric operators

J =



 snE−E⁻¹sn pn(1−E⁻¹) 0

(E−1)pn 0 ∆sn

0 s_n∆ 0





and K =



 ^snEqn−qnE

−1sn pnqn−pnE⁻¹qn+snE_s¹

nE−E⁻¹ pn(E⁻²−1)sn

E−E^{−1 1}_s

nE⁻¹s_n+q_nEp_n−p_nq_n E⁻¹pn

sn −pn

snE q_n∆s_n

s_n(1−E²)p_n s_n∆q_n s_n(E⁻²−E²+E⁻¹−E)s_n



.

From (2.8), we have

G⁻_n¹= (A⁽n⁻¹⁾, Bn⁽⁻¹⁾, Fn⁽⁻¹⁾)^T = (0,1,0)^T, G⁰_n= (A⁽⁰⁾n , B⁽⁰⁾n , Fn⁽⁰⁾)^T = ( 1sn, q_n,0)^T, G¹_n= (A⁽¹⁾n , B⁽¹⁾n , Fn⁽¹⁾)^T = (q_n+q_n+1

s_n , q²+pn

s_n +p_n−1

s_n−1,pn(qn+qn+1)−1 s_n ),· · · Letψ_n(λ) satisfy (2.1) and its auxiliary problem

∂ψ_n(λ)

∂t_n =V_n^(m)ψ_n(λ), (2.9)

where

V_n^(m) = (V_n^(ijm))_3×3, V_n^(ijm) =V_n^(ij)(A^(m)_n (λ), B_n^(m)(λ), D^(m)_n (λ)) and

A^(m)_n (λ) =

∑∞ i=−1

A^(m)_n λ^m−i, B_n^(m)(λ) =

∑∞ i=−1

B^(m)_n λ^m−i, D_n^(m)(λ) =

∑∞ i=−1

D_n^(m)λ^m−i.

Then the compatibility conditions of (2.1) and (2.9) are

∂Un

∂tnm

= (EV_n^(m))Un−UnV_n^(m), m≥ −1, (2.10) which implies the lattice solition equations

∂Un

∂t_nm =X_n^m+1, U_n= (p_n, q_n, s_n)^T, m≥ −1, and

X_n^j =J G^j_n=KG^j_n⁻¹, j ≥0 which give rise to the following hierarchy of lattice soliton equations















∂p_n

∂t_nm = (s_nE−E⁻¹s_n)A^(j)_n (λ) +p_n(1−E⁻¹)B_n^(j)(λ),

∂qn

∂tnm

= (E−1)pnA^(j)_n (λ) + ∆snF_n^j(λ),

∂s_n

∂tnm

=s_n∆B_n^(j)(λ).

j ≥ −1. (2.11)

So the (2.10) are discrete zero curvature representation of (2.11), the discrete spectral problem (2.3) and (2.9) constitute the Lax pair of (2.11), and (2.11) is a hierarchy of Lax integrable nonlinear lattice equations. It is easy to verify that the new first Liouville integrable differential-difference equation in (2.11), whenm= 0, is











∂t∂_np_n=p_n(q_n−q_n−1) +s_n−s_n+1 sn+1 ,

∂

∂tnqn= pn+1

s_n+1 −p_n s_n,

∂t∂_ns_n=s_n(q_n+1−q_n−1).

(2.12)

The variational derivative, the Gateaux derivative and the inner product are defined, respectively, by δHn

δu_n = ∑

m∈Z

E^−m( ∂Hn

∂u_n+m), J^′(un)[vn] = ∂

∂εJ(un+εvn)|ε=0,⟨fn, gn⟩=∑

n∈Z

(fn, gn)_R², (2.13) wherefn, gnare required to be rapidly vanished at the infinity, and (fn, gn)_R² denotes the standard inner product off_nand g_n in the Euclidean spaceR₂. OperatorJ^∗ is defined by⟨f, J^∗g⟩=⟨J f_n, g_n⟩; it is called adjoint operator of J with respect to (2.8). If an operator J has the property J^∗ =−J, then J is called to be a skew-symmetric. A linear operator J is called a Hamiltonian operator, if J is a skew-symmetric operator and satisfies the Jacobi identity, i.e., it satisfies that

⟨f, J g⟩=−⟨J f, g⟩, ⟨J^′(un)[J f]g, h⟩+Cycle(f, g, h) = 0 (2.14) based on a given Hamiltonian operator J, we can define a corresponding Poisson bracket

{f, g}J =⟨ δf δu_n, J δg

δu_n⟩=∑

n∈Z

( δf δu_n, J δg

δu_n). (2.15)

To establish the Hamiltonian structures for (2.11), we define

Rn=VnU_n⁻¹ =







V_n¹² (λ−qn)V_n¹²+V_n¹³ V_n¹¹−p_nV_n¹² s_n V_n²² (λ−q_n)V_n²²+V_n²³ V_n²¹−p_nV_n²²

sn

V_n³² (λ−qn)V_n³²+V_n³³ V_n³¹−pnV_n³² sn







and ⟨A, B⟩=T r(AB), where A and B are the some order square matrices. We have

∂Un

∂λ =



 0 0 −1

0 0 0

0 0 0



,∂Un

∂p_n =



 1 0 0 0 0 0 0 0 0



,∂Un

∂q_n =



 0 0 1 0 0 0 0 0 0



,∂Un

∂s_n =



 0 0 0 0 0 0 1 0 0



.

Hence

⟨R_n,∂Un

∂λ ⟩=−V_n³²=−B_n(λ), ⟨R_n,∂Un

∂pn⟩=V_n¹²=A_n(λ), ⟨R_n,∂Un

∂qn⟩=V_n³²=B_n(λ),

⟨R_n,∂Un

∂sn⟩= V_n¹¹−p_nV_n¹² s_n = _s¹

n[(E⁻¹−1)p_nA_n(λ)−E⁻¹(λ−q_n)B_n(λ) +E⁻¹D_n(λ)].

(2.16)

By virtue of the discrete trace identity δ

δu

∑

n∈Z

⟨R_n,∂Un

∂λ ⟩= (

λ^−ε ( ∂

∂λ )

λ^ε )

⟨R_n,∂Un

∂uⁱ_n⟩, i= 1,2,3. (2.17) The substitution of (2.4) into (2.17), and comparing the coefficients ofλ^−m−1 in (2.17), we get

( δ δpn

, δ δqn

, δ δsn

) (

B_n^(m+1) )

= (ε−m)







A^(m)_n B_n^(m) 1

sn

(E⁻¹−1)A^(m)_n −E⁻¹B^(m+1)_n +E⁻¹(qnB_n^(m)+D_n^(m))





. (2.18)

When m= 0 in the (2.18), through a direct calculation, we find thatε= 0. So we have

( δ δsn

, δ δwn

, δ δpn

) (

−B_n^(m+1) m+ 1

)

=





A^(m)_n B_n^(m) 1

sn

(E⁻¹−1)A^(m)_n −E⁻¹B_n^(m+1)+E⁻¹(q_nB^(m)_n +D^(m)_n )



, m≥ −1.

Now we can rewrite the (2.11) in the following Hamiltonian forms

∂Un

∂t_nm =X_n^m+1 =J( δ δp_n, δ

δq_n, δ

δs_n)H_n^m+1=J L( δ δp_n, δ

δq_n, δ

δs_n)H_n^m, m≥ −1. (2.19) Let

L=





L₁₁ 1

sn∆⁻¹ L₁₃

∆⁻¹ 1

s_n 0 0

L31 0 (Esn−snE⁻¹)⁻¹



, (2.20)

where

L11=−1

s_n∆⁻¹pn(Esn−snE⁻¹)⁻¹pn(E−1)∆⁻¹ 1 s_n, L13=−1

s_n∆⁻¹(1−E⁻¹)pn(Esn−snE⁻¹)⁻¹, L31=−(Esn−snE⁻¹)⁻¹pn(E−1)∆⁻¹ 1

sn.

It is easy to verify that K is a skew-symmetric operator in this way and the positive hierarchy (2.10) is derived. It is easy to verify that the positive hierarchy has the discrete zero-curvature representation (2.9). And, every soliton equation in (2.11) or the discrete Hamiltonian system (2.19) is a discrete Liouville integrable system.

3. A binary Symmetry constraint by binary nonlinearization

In order to impose the Bargmann symmetry constraint by binary nonlinearization, we consider the adjoint spectral problem of spectral problem (2.1)

E⁻¹ψ_n= (E⁻¹U˜_n^T(u_n, λ)ψ_n), ψ_n= (ψ_n^1j, ψ^2j_n, ψ_n^3j)^T (3.1) and temporal spectral problem

ψ_ntm =−( ˜V_n^m(u_n, λ))^Tψ_n. (3.2) From the compatibility condition(E⁻¹ψ_n)_tm =E⁻¹ψ_ntm, we know that

E⁻¹U˜n T

tm= (E⁻¹U˜n

T)( ˜V_n^m)^T −(E⁻¹( ˜V_n^m)^T)(E⁻¹U˜n

T) (3.3)

Let λ1, λ2, ..., λN be N distinct eigenvalues of spectral problem (1) and λj ̸= 0, j= 1,2, ..., N, we have {

(Eφ^1jn, Eφ^2jn, Eφ^3jn) = (φ^1jn, φ^2jn, φ^3jn)U_n^T(u_n, λ), (φ^1jn, φ^2jn, φ^3jn)_tm= (φ^1jn, φ^2jn, φ^3jn)V_n^T(u_n, λ), {

(Eψn^1j, Eψ^2jn, Eψn^3j) = (ψn^1j, ψn^2j, ψ^3jn)(U_n(u_n, λ))⁻¹, (ψn^1j, ψ^2jn, ψn^3j)_tm= (ψn^1j, ψ^2jn, ψn^3j)(−V_n(u_n, λ)).

(3.4)

We can compute the variational derivative of the spectral parameter λwith respect to the potential u δλj

δun

=αj(Eψ_n^1j, Eψ_n^2j, Eψ_n^3j)∂Un(un, λj)

∂un

(φ^1j_n, φ^2j_n, φ^3j_n)^T. (3.5) Namely

∇λj =







 δλ_j δpn

δλ_j δq_n δλj

δsn







=αj





φ^2jnψ^3jn

φ^2jnψ^1jn

s⁻_n¹φ^4jnψn^4j



, (3.6)

whereδλ_j

δu_n is a variational derivative for eigenvalue λ_j, α_j is a constant and φⁱ_n, ψ_nⁱ, i = 1,2,3,4 are required to be rapidly vanishing at the infinity, and we denote the inner product inR^N by < ., . > and use the following notations

Φⁱ_n= (φⁱ¹_n, φⁱ²_n, ..., φ^iN_n ), Ψⁱ_n= (ψⁱ¹_n, ψ_nⁱ², ..., ψ_n^iN), i= 1,2,3, ∧=diag(λ1, λ2, ..., λN).

Such a gradient satisfies the following equation

K∇λ_j =λ_jJ∇λ_j. (3.7)

Consider the discrete symmetry constraint G₋₁ =

∑N j=1

∇λ_j. (3.8)

That is

δH_n⁽⁰⁾ δun =





1 sn

q_n 0



=





φ^1j_nψ_n^2j φ^3j_nψ_n^2j

s⁻¹_n (φ^1j_nψ_n^1j−p_nφ^1j_nψ^2j_n)



. (3.9)

Note that the explicit constraints of potential functions and eigenvalue functions can not be obtained with the express above. Under the constraint (3.8), we obtain a discrete binary constrained flows

Eφ^1j_n =p_nφ^1j_n +φ^2j_n + (q_n−λ)φ^3j_n, 1≤j≤N, Eφ^2j_n =φ^3j_n, 1≤j ≤N,

Eφ^3j_n =snφ^1j_n, 1≤j ≤N, Eψ_n^1j =ψ_n^2j, 1≤j ≤N,

Eψ_n^2j = (λ−q_n)ψ_n^2j+ψ_n^3j, 1≤j≤N, Eψ_n^3j =s⁻_n¹(ψ_n^1j−pnψ_n^2j), 1≤j≤N,

(3.10)

Here,< ., . >is the standard inner product ofR^N. The symmetry constraint (3.8) yields explicit expressions from (3.9):





pn=<Φ¹_n,Ψ¹_n><Φ¹_n,Ψ²_n>⁻¹, q_n=<Φ³_n,Ψ²_n>,

sn=<Φ¹_n,Ψ²_n>⁻¹.

(3.11) So the discrete symmetry constraint (3.8) is a Bargmann constraint. Setting

Pn= (φ¹¹_n, φ¹²_n ,· · ·, φ^1N_n ,· · · , φ³¹_n, φ³²_n,· · ·, φ^3N_n )^T, Qn= (ψ_n¹¹, ψ_n¹²,· · ·, ψ_n^1N,· · ·, ψ_n³¹, ψ_n³²,· · · , ψ^3N_n )^T

and ∂

∂Pn = ( ∂

∂φ¹¹_n , ∂

∂φ¹²_n ,· · · , ∂

∂φ^1N_n ,· · ·, ∂

∂φ³¹_n , ∂

∂φ³²_n ,· · · ∂

∂φ^3N_n ,)^T,

∂Q∂_n = ( ∂

∂ψ¹¹_n , ∂

∂ψ¹²_n ,· · ·, ∂

∂ψ^1N_n ,· · ·, ∂

∂ψ³¹_n , ∂

∂φ³²_n ,· · · , ∂

∂ψ_n^3N,)^T,

the Poisson bracket of between two arbitrary function ofα, β in symplectic apaceR^6N is defined by {α, β}=⟨∂α

∂P, ∂β

∂Q⟩ − ⟨∂β

∂P,∂α

∂Q⟩= (∂α

∂P)^T(∂β

∂Q)−(∂β

∂P)^T(∂α

∂Q).

This is skew-symmetric, bilinear, and satisfies the Jacobi identity. In particular, any two of α, β is called involutive if{α, β}= 0.

The mapH defined as

H(φ¹_n, φ²_n, φ³_n, ψ_n¹, ψ²_n, ψ_n³) = (Eφ¹_n, Eφ²_n, Eφ³_n, Eψ_n¹, Eψ²_n, Eψ_n³) (3.12) is a symplectic map. Through laborious but direct computation, we get

{α_i, α_j}={β_i, β_j}= 0,{α_i, β_j}=δ_ij,1≤i, j ≤N and theγi, δj are of the same forms. Furthmore, we can deduce

d(EP_n)∧d(EQ_n) =dP_n∧dQ_n.

Therefore, (3.12) determine a symplectic map.

Now, we will solve recursion equations (2.7). Whenm >1, we have

U_ntm =



 pn

q_n sn





tm

=





(s_nE−E⁻¹s_n)A^(j)n (λ) +p_n(1−E⁻¹)Bn^(j)(λ) (E−1)pnA^(j)n (λ) + ∆snFn^j(λ)

sn∆Bn^(j)(λ)





=JδH_n^m

δu_n =JΦ^m_n⁻¹δH_n¹ δu_n =J

∑N j=1

λ^m_j ⁻¹δλj

δu_n.

(3.13)

Using (3.8) and (3.10) and the constraint (3.11), we take the following restriction:

Gj−1=

∑N k=1

λ^j_k∇λ_k. (3.14)

That is to say,





(snE−E⁻¹sn)A^(j)n (λ) +pn(1−E⁻¹)Bn^(j)(λ) (E−1)pnA^(j)n (λ) + ∆snFn^j(λ)

s_n∆Bn^(j)(λ)



=J

∑N j=1

λ^m_j





φ^1jnψ^2jn

φ^3jnψ^2jn 1

sn(φ^1jnψn^1j−p_nφ^1jnψn^2j)



. (3.15)

From (3.15), we can conclude

A^jn=<∧^jΦ¹_n,Ψ²_n>, Bn^j =<∧^jΦ³_n,Ψ²_n>,

Fn^j =s⁻_n¹(<∧^jΦ¹_n,Ψ¹_n>−p_n<∧^jΦ¹_n,Ψ²_n>). (3.16) Substituting (3.16) into the relation (2.6), we obtain a solution ofD^j_n, that is

D_n^j =<∧^jΦ²_n,Ψ²_n> . (3.17) By using (3.16), (3.17) and (2.7), we have

E⁻¹s_nA_n(λ) =<∧^jΦ³_n,Ψ¹_n>, E⁻¹B_n(λ) =<∧^jΦ²_n,Ψ¹_n>, E^{−1 1}_s

nE⁻¹snAn(λ)−E^−1p_sⁿ

nBn(λ) =<∧^jΦ²_n,Ψ³_n>,

E⁻¹p_nA_n(λ)−λE⁻¹B_n(λ) +E⁻¹q_nB_n(λ) +E⁻¹D_n(λ) =<∧^jΦ¹_n,Ψ¹_n>, q_nA_n(λ)−λA_n(λ) + _s¹

nEB_n(λ) =<∧^jΦ¹_n,Ψ³_n>,

−λBn(λ) +qnBn(λ) +EDn(λ) =<∧^jΦ³_n,Ψ³_n> .

(3.18)

In the following, we would like to discuss the Louville integrability on the nonlinearized temporal parts of the Lax pairs and adjoint Lax pairs.

Under the control of (3.11), the temporal parts of the Lax pairs and the adjoint Lax pairs by substituting (3.18) into (3.4) become

∂

∂t(φ^1jn, φ^2jn, φ^3jn)^T =V|B(φ^1jn, φ^2jn, φ^3jn)^T, j= 1,2,· · ·, N,

∂

∂t(ψn^1j, ψ^2jn, ψn^3j)^T =−V^T|B(ψn^1j, ψ^2jn, ψ3j)^T, j = 1,2,· · ·, N. (3.19) We arrive at the finite-dimensional Hamiltonian systems. Here, the subscript B means substitution of (3.18) into the expression.

The temporal parts of the nonlinearized Lax pairs and the adjoint Lax pairs (3.19) may be rewritten as

∂

∂tΦⁱ_n= ∂F_n^m

∂Ψⁱ_n, ∂

∂tΨⁱ_n=−∂F_n^m

∂Φⁱ_n, (3.20)