Integrability of the Reduction Fourth-Order Eigenvalue Problem

(1)

Integrability of the Reduction Fourth-Order

Eigenvalue Problem

Shuhong Wang

College of Mathematics, Inner Mongolia University for Nationalities, Tongliao, China Email: shuhong7682 @163.com

Wei Liu

Department of Mathematics and Physics, ShiJiaZhuang TieDao University, Shijiazhuang, China Email: [email protected]

Shujuan Yuan

Qinggong College, Hebei United University, Tangshan, China Email: [email protected]

Abstract—To study the reduced fourth-order eigenvalue

problem, the Bargmann constraint of this problem has been given, and the associated Lax pairs have been nonlineared. By means of the viewpoint of Hamilton mechanics, the Euler-Lagrange function and the Legendre transformations have been derived, and a reasonable Jacobi-Ostrogradsky coordinate system has been found. Then, the Hamiltonian cannonical coordinate system equivalent to this eigenvalue problem has been obtained on the symplectic manifolds. It is proved to be an infinite-dimensional integrable Hamilton system in the Liouville sense. Moreover the involutive representation of the solutions is generated for the evolution equations hierarchy in correspondence with this reduced fourth-order eigenvalue problem.

Index Terms—constraint flow, Bargmann system, integrable

system, involutive representation

I. INTRODUCTION

The theory of integrable systems has been an interesting and important problem. The finite dimensional integrable system is used to describe the problems in mathematical physics, mechanics, etc., such as Kovalevskia top, geodesic flows on the ellipsoid harmonic oscillator equation on sphere, Calogero-Moser system.etc [1-3]. The soliton equation as an infinite dimensional integrable system is one of the most prominent subjects in the field for the nonlinear science. “Solutions” were first observed by J. Scott Russell in 1834 whilst riding on horseback beside the narrow Union canal near Edinburgh, Scotland. There are a number of discussions in the literature describing Russell’s observations. At the center of these observations is the discovery that these nonlinear waves can interact elastically and continue afterward almost as if there had been no interaction at all (see Fig. 1- Fig. 5). Because of the analogy with particles, Zabusky and Kruskal named these special waves as solitons. Zabusky and Krusk’s remarkable numerical discovery demanded an analytical

explanation and detailed mathematical study of the partial differential equations. In 1968, Lax put the inverse scattering method for solving the KDV equation into a more general framework which subsequently paved the way to generalizations of the technique as a method for solving other partial differential equations [1-4]. The technique of the so-called nonlinearization of Lax pairs [5,8-10] has been developed and applied to various soliton hierarchies, from which a large of interesting finite-dimensional Liouville integrable Hamiltonian systems have been obtained. Recently, this method was generalized to discuss the nonlinearization of Lax pairs and adjoint Lax pairs of soliton hierarchies [8-11]. The nonlinearization approach (or constrained flows) of eigenvalue problems or Lax Pairs has been used to seek the relations between the infinite and finite integrable system. There are series of finite integrable system obtained by this approach [8-19].

(2)

Figure2. Bell-shaped Solitons at different times

Figure5. Bell-shaped Solitons at different times The fourth-order eigenvalue problem

(

4 2 2

)

Lϕ= ∂ + ∂ + ∂ + ∂ + ∂ +q q p p r ϕ λϕ= has been discussed , but its reduced system

(

4

)

Lϕ= ∂ + ∂ ∂ +u v ϕ λϕ= _.

In this paper, we consider the reduced fourth-order eigenvalue problem

(

4

)

Lϕ= ∂ + ∂ ∂ +u v ϕ λϕ= (1) here

x ∂ ∂ =

∂ ，eigenparameter λ∈R , and potential function u v, is a actual function in( , )x t . By means of

the viewpoint of Hamilton mechanics, the Euler-Lagrange function and the Legendre transformations have been derived, and a reasonable Jacobi-Ostrogradsky coordinate system has been found. Then, the Hamiltonian cannonical coordinate system equivalent to the eigen-value problem (1) has been obtained on the symplectic manifold. It is proved to be an infinite-dimensional integrable Hamilton system in the Liouville sense. Moreover the involutive representation of the solutions is generated for the evolution equations hierarchy in correspondence with the reduced fourth-order eigenvalue problem (1).

II. MAIN RESULTS A. Evolution Equations and Lax Pairs

Now, suppose Ω is the basic interval of the reduced eigenvalue problem (1), if the potentials u v, ,ϕ in (1) and their derivates on x _{are all decay at infinity, then}

( , )

Ω = −∞ +∞ ; if they are all periodic T functions, thenΩ =[0, 2 ]T .

Definition 1 [8, 9] Assume that our linear space is equipped with aL2scalar product

(

)

2

* ( )

, _L

f g _Ω fg dx

Ω

=

_∫

< ∞

the symbol * denotes the complex conjugate.

Definition 2 [8, 9] Operator A is called dual operator ofA, if

(

)

_{( )}

(

)

_{( )}

2 ₂

,g , g

L _L

Af _Ω f A

Ω

=

and A_{is called a self-adjioint operator, if} A=A.

We consider the following fourth-order operator Lon the interval Ω

4

L= ∂ + ∂ ∂ +u v

here u v, is potential function of the eigenvalue problem (1). λis called eigenvalue of the eigenvalue problem (1), andϕis called eigenfunction to eigenvalueλ, if

Lϕ λϕ= , 2( )

L

ϕ∈ Ω is non-trivial solution of the eigenvalue problem (1) .

Theorem 1 _L_{= ∂ + ∂ ∂ +}4 _u _v

is a self-adjioint operator onΩ.

Proof. By Definition 2, we easily derive Theorem1.

If

0

( )

L L u

ε

εδμ ε =

∂

′ = +

∂ is called the derived function

of differential operator L=L u( ) , here u is potential function of L, then we have

Theorem 2 If ϕ is an eigenfunction corresponding to the eigenvalue λof (1), then functional Gradient [20]

2

2 1

2

( )

u x

v

dx

δλ δ δλ

δ

ϕ

λ ϕ

ϕ

− Ω

⎛− ⎞

⎛ ⎞

∇ =_⎜ _⎟= _⎜ _⎟

⎝ ⎠

∫

⎝ ⎠ (2)

(3)

(

Lϕ ϕ

)

dx ϕ

( )

Lϕ dx λϕ ϕdx

Ω ′ = Ω ′ = Ω ′

∫

then

Lϕ′=λϕ′

hence

L′ϕ λ ϕ= ′ Namely:

( )

(

)

(

)

(

)

2

2 2

x

xx x xx

dx L dx

u u v dx

v u dx

λ ϕϕ ϕ ϕ

δ δ δ ϕ ϕ

δ ϕ ϕ δ ϕ ϕ ϕ δ ϕ

δ ϕ δ ϕ

Ω Ω

Ω Ω Ω

′ = ′

⎡ ⎤

= _⎣ ∂ + ∂ + _⎦

⎡ ⎤

= _⎣ − + + _⎦

= −

∫

so

2

2 1

2

( )

u x

v

dx

δλ_δ δλ

δ

ϕ

λ ϕ

ϕ

− Ω

⎛− ⎞

⎛ ⎞

∇ =⎜ ⎟= ⎜ ⎟

⎝ ⎠

∫

⎝ ⎠

Definition 3 [21]. The commutator of operators W

and Lis defined as follows [W L, ]=WL−LW

. Set

(

)

(

)

3 2

0

1 1 1

2 2

4 8 8

1

3 , 0,1, 2

8

, 0,1, 2, m

m j jx j jxx j

j

m j

jx jxxx jx

j j

j

W b b a b ub

a b ub L m

a

G j m

b

=

−

⎧ ₌ ⎡ _{∂ −} _{∂ +} ₊ ₊ _∂

⎪ _⎢⎣

⎪

⎪ _⎤

⎪ ₋ ₊ ₊ ₌

⎨ _⎥⎦

⎪

⎪ _{⎛ ⎞}

= =

⎪ ⎜ ⎟

⎪ ⎝ ⎠

⎩

∑

L

(3)

3 0

3 1 1

4 2 4 x

J

u u

∂

⎛ ⎞

⎜ ⎟

= ⎜_⎜_∂ _{∂ +} _{∂ +} ⎟_⎟

⎝ ⎠

(4)

11 12

21 22

K K

K

K K

⎛ ⎞

= ⎜ ⎟

⎝ ⎠ (5)

here

3 11

5 3 2

12

4 1 1

,

5 2 4

3 3 3 3 3

,

8 8 4 8 4

x

x xx x

K u u

K u u u v v

= ∂ + ∂ +

⎛ ⎞

= ∂ + ∂ + ∂ +_⎜ + _⎟∂ +

⎝ ⎠

5 3 2

21

3 3 3 1

,

8 8 8 x 4 x

K = ∂ + u∂ + u∂ + ∂ +v v

7 5 4 2 3

22

2

1 1 5 3 1 3

8 4 8 4 8 4

1 3 9 7 1

2 8 8 8 8

1 1 1 1 1 1

8 2 8 4 4 4

x xx

xxx x x xx xxxx

xx x x x xxx

K u u u u v

u uu v v u

uu uv u uv vu v

⎛ ⎞

= ∂ + ∂ + ∂ +_⎜ + + _⎟∂

⎝ ⎠

⎛ ⎞ ⎛

+_⎜ + + _⎟∂ +_⎜ +

⎝ ⎠ ⎝

⎞ ⎛ ⎞

+ + + _⎟∂ +_⎜ + + _⎟

⎠ ⎝ ⎠

Theorem 3 Operators J andK are the bi-Hamilton operators [5, 8, 9], moreover if λ and ϕ are the eigenvalue and eigenfunction of (1), then

1 0,1, 2

m m

JG ₊ =KG m= _L (6) K∇ =λ Jλ λ∇ ₍₇₎ Proof. By means of the Definition of the bi-Hamilton operators, it is easily derived that J andK are the

bi-Hamilton operators by the definition of the bi-bi-Hamilton operators.

By complex computing, we can obtain (6), (7). Set JG0=0, then

0 4 0 G = ⎜ ⎟⎛ ⎞

⎝ ⎠ (8)

or

0 4

u G = ⎜ ⎟⎛− ⎞

⎝ ⎠ (9)

So, we have

* *

0

1

* * 0

* * 1

[ , ] [ ( ) ( ) ]

( ) ( )

m

m j

t m j j

j

m m

L W L L KG L JG L L

L KG L JG L L KG L JG

− =

+ +

= = −

= −

= =

∑

here, * 2

0

( ) d q :

L R R

p d ε

ς ες

ε =

⎛⎛ ⎞ ⎞

= ⎜⎜ ⎟+ ⎟ →

⎝ ⎠

⎝ ⎠ is one to one

[5,8-10].

By the Hamilton operatorsJ and K (4), (5), we have: Theorem 4 SetJG0 =0, then the m-th-order evolution equation of (1) are:

1 m

t m m m

q =X =JG + =KG

(10) here q=( , ) ,u v T m=0,1, 2L , and (10) become the isospectral compatible condition of the following Lax pairs

m

t m

L W

ϕ λϕ

ϕ ϕ

= ⎧⎪

⎨ ₌

⎪⎩ (11)

here m=0,1, 2L.

So, we have the first evolution equation

0

t x t x

u u

v v

= ⎧⎪

⎨ ₌

⎪⎩ (12)

and its Lax pairs

0

t x

Lϕ λϕ

ϕ ϕ

= ⎧⎪

⎨ ₌

⎪⎩ (13) or the first evolution equation

0

5 3

3

4 4

3 3 3 3

8 8 8 4

t xxx x x

t xxxxx xxx x xx xxx x

u u uu v

v u uu u u v uv

⎧ _{= −} ₋ ₊

⎪⎪ ⎨

⎪ _{= −} ₋ ₋ ₊ ₊

⎪⎩

(14)

and its Lax pairs

0

3 3

8 4

t x x

L

u u

ϕ λϕ

ϕ ϕ ϕ

= ⎧ ⎪

⎨ ₌ ₊

(4)

B. Jacobi-Ostrogradsky Coordinates and Hamilton Canonical Forms of Bargmann System

Now, suppose {λ ϕ_j, _j}(j=1, 2,_L,N)are eigenvalues and eigenfunctions for the fourth-order eigenvalue problem (1), andλ1<λ2 <L<λN,then

4

( )

LΦ = ∂ + ∂ ∂ + Φ = ΛΦu v (16) here, Λ =diag( ,λ λ1 2,L,λN), ( ,1 2, , )

T N

diag ϕ ϕ ϕ

Φ = _L .

From (7), we have

1

1 1

,

N N

k k

j j j j

j j

K λ λ J λ + λ

= =

⎛ ⎞ ⎛ ⎞

∇ = ∇

⎜ ⎟ ⎜ ⎟

⎝

∑

⎠ ⎝

∑

⎠

here k=0,1, 2,_L. So

1

, ,

,

, ,

k k

x x x x

k k

K J

+ +

⎛− Λ Φ Φ ⎞ ⎛− Λ Φ Φ ⎞

⎜ ⎟₌ ⎜ ⎟

⎜ _{Λ Φ Φ} ⎟ ⎜ _{Λ Φ Φ} ⎟

⎝ ⎠ ⎝ ⎠

(17)

here k=0,1, 2,_L.

Set G₀ =(4, 0)T, using (6) , we have: 2

1

3 3

4uxx 8u v

G

u

⎛₋ ₋ ₊ ⎞

⎜ ⎟

= ⎜_⎜ ⎟_⎟

⎝ ⎠

Now we define the constrained system to the Bargmann system

2

1

3 3

,

4 8

,

x x

xx

u u v

G

u

⎛₋ ₋ ₊ _{⎞ ⎛}_{− Φ Φ} _⎞

⎜ _{⎟ ⎜} _⎟

=_⎜ _{⎟ ⎜}=

⎟ Φ Φ

⎜ _{⎟ ⎝} _⎠

⎝ ⎠

(18)

then, we have the relation between the potential( , )u v and the eigenvectorΦas follows

2 ,

3 1 3

, , ,

2 xx 2 x x 8

u v

⎛ Φ Φ ⎞

⎛ ⎞ ⎜₌ ⎟

⎜ ⎟ ⎜ _{Φ Φ} ₊ _{Φ Φ +} _{Φ Φ} ⎟

⎝ ⎠ ⎜ ⎟

⎝ ⎠

(19)

From (6), (17), (18) and (19), we have

1

1 ,

, , k

x x

k _k

G

− −

⎛− Λ Φ Φ ⎞

⎜ ⎟

=

⎜ _{Λ Φ Φ} ⎟

⎝ ⎠

here k=1, 2_L.

Based on the constrained system (19), the eigenvalue problem

LΦ = ΛΦ (20) is equivalent to the following system

2

, 2 ,

3 1 3

, , ,

2 2 8

xxxx xx x x

xx x x

Φ + Φ Φ Φ + Φ Φ Φ

⎛ ⎞

+_⎜ Φ Φ + Φ Φ + Φ Φ _⎟Φ = ΛΦ

⎝ ⎠

(21)

Remark 2 we call the equation system (21) to be the Bargmann system for the fourth-order eigenvalue problem (1).

In order to obtain the Hamilton canonical forms which is equivalent to the Bargmann system (21), the Lagrange

functionI

∧

[8-9] is defined as follows:

I Idx

∧ Ω

=

∫

(22)

here,

2

3

1 1 1

, , , ,

2 2 4

3 1

, ,

48 2

xx xx xx x

I= Φ Φ + Φ Φ Φ Φ + Φ Φ

+ Φ Φ − ΛΦ Φ

(23)

Theorem 5. The Bargmann system (21) for the fourth-order eigenvalue problem (1) is equivalent to the Euler-Lagrange equations

0 I

δ δ

∧

=

Φ (24)

Proof. By (22), we have

(

2

)

1

3 , 4 , 12 ,

8

2 , ,

x x xx xx

x xx

x

x x xx xxxx

I I I I

δ δ

∧

⎛ ⎞ ⎛ ⎞

∂ ∂ ∂

= −_⎜ _⎟ +_⎜ _⎟ Φ ∂Φ _⎝∂Φ _⎠ _⎝∂Φ _⎠

= −ΛΦ + Φ Φ + Φ Φ + Φ Φ Φ

+ Φ Φ Φ + Φ Φ Φ + Φ

so δ I L 0 δ

∧

= Φ − ΛΦ = Φ

Set y1= −Φ,y2 = Φx , and

2

1 ,

jx j j

h y z I

=

∑

− , our

aims are to find the coordinates z z1, 2and the Hamilton functionh, that satisfy the following Hamilton canonical equations

{ }

,

jx j

j

jx j

j

h

y y h

z h

z z h

y

∂

⎧ ₌ ₌

⎪ _∂

⎪

⎨ _∂

⎪ ₌ _{= −}

⎪ ∂

⎩

(25)

here j=1, 2, the symbol

{ }

,

is the Poisson bracket in the symplectic space

2

4

1

, N

j j

j

dy dz R

ω

=

⎛ ⎞

= ∧

⎜ ⎟

⎝

∑

⎠, and the

Poisson bracket of Hamilton functions F H, in the symplectic space is defined as follows [1, 3, 4, 5]

{

}

(

)

2

1 1

2

1 ,

, ,

j j j j

N

j k jk jk jk jk

y z z y

j

F H H F

F H

y z y z

F H F H

= = =

⎛ _∂ _∂ _∂ _∂ ⎞

= ⎜_⎜ − ⎟_⎟

∂ ∂ ∂ ∂

⎝ ⎠

= −

∑∑

∑

By 2

1 ,

jx j j

h y z I

=

∑

− , then

2

1

( jx, j j, jx )

j

dh y dz z dy dI

=

∑

+ − .

On the other hand, h=h y z( j, j), (j=1, 2), so

(

)

2

1

2

1

, ,

j j

j j j

jx j jx j

j

h h

dh dy dz

y z

z dy y dz

=

⎛ _∂ _∂ ⎞

⎜ ⎟

= _⎜ + _⎟

∂ ∂

⎝ ⎠

= − +

∑

(5)

1 1 2 2 1 1 2 2

1 2 1 2

, , , ,

, , ,

x x x x

x xx x x x

x x x xx

dI z dy z dy z dy z dy

z d z d z d z d

z d z z d z d

= + + +

= − Φ + Φ − Φ + Φ

= Φ + − Φ + Φ

we have the relations:

1

2

1 1 1 1

, ,

2 2 2 4

1 1

,

2 2

xxx x x xxx x x

xx xx

z u u

z u ⎧ = Φ + Φ Φ Φ + Φ Φ Φ = Φ + Φ + Φ ⎪⎪ ⎨ ⎪ = Φ + Φ Φ Φ = Φ + Φ ⎪⎩

Theorem 6. The Jacobi-Ostrogradsky coordinates are as follows 1 2 1 2 1 1 2 4 1 2 x

xxx x x

xx

y y

z u u

z u = −Φ ⎧ ⎪ _{= Φ} ⎪ ⎪ ⎨ = Φ + Φ + Φ ⎪ ⎪ = Φ + Φ ⎪ ⎩ (26)

and the Bargmann system (21) for the fourth-order eigenvalue problem (1) is equivalent to the Hamilton canonical system

{ }

, , jx j j jx j j h

y y h

z h

z z h

y ∂ ⎧ ₌ ₌ ⎪ _∂ ⎪ ⎨ _∂ ⎪ ₌ _{= −} ⎪ ∂ ⎩ (27)

herej=1, 2 , and the Hamilton functionhis 2

1 1 2 1 2 2 1 2

3

1 1 1 2 1 1

1 1 1

, , , ,

2 2 4

1 1

, , ,

2 16

h y y y z z z y y

y y y y y y

= Λ − + −

+ +

(28)

Remark 3. Based on the Jacobi-Ostrogradsky coordinate system (26), the Bargmann constrained equations associated with the fourth-order eigenvalue problem (1) is

1 1

2

1 2 2 2 1 1

,

3 1 3

, , ,

2 2 8

u y y

v y z y y y y

⎧ = ⎪ ⎨ = − + − ⎪⎩ (29)

C. Hamilton Equations of Bargmann System

Theorem 7 The Bargmann system (21) for the fourth-order eigenvalue problem (1) is equivalent to:

x

Y =MY (30) where,

1 2 3 4

( , , , )

1 1 1

, , ,

2 2 4

T

x xx xxx x x

Y y y y y

u u u

= ⎛ ⎞ = −Φ Φ Φ +_⎜ Φ Φ + Φ + Φ_⎟ ⎝ ⎠ (31) 2

0 0 0

1 0 0 2 1 0 0 4

1 1 1 1

0

4 4 4 2

x

xx x

E

uE E

M

u E E

v u u E u E uE

− ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ = ⎜ − ⎟ ⎜ ⎟ ⎜ _⎛ _⎞ ⎟ −Λ + − − − − ⎜ _⎜ _⎟ ⎟ ⎝ ⎠ ⎝ ⎠ (32) and the system of the Lax pairs for the evolution equation hierarchy (10) is equivalent to

m

t m

Y =W Y (33) where m=0,1, 2,L , and

4 4

( m) 0,1, 2

m ij

W = ω × m= L (34)

11 1 1 1 1

0

1 1 1 3

8 16 16 8

m

m m j

j xxx j x x j j x

j

b ub u b a

ω − − − − − = ⎛ ⎞ = _⎜ + + + _⎟Λ ⎝ ⎠

∑

12 1 1 1

0

1 1 1

8 8 4

m

m m j

j xx j j

j

b ub a

ω − − − − = ⎛ ⎞ = _⎜ + + _⎟Λ ⎝ ⎠

∑

13 1 0 1 8 m

m m j

j x j b ω − − = ⎛ ⎞ = _⎜− _⎟Λ ⎝ ⎠

∑

14 1 0 1 4 m

m m j

j j b ω − − = ⎛ ⎞ = _⎜ _⎟Λ ⎝ ⎠

∑

21 1 1 1

0

1 1 1

1 1 5

8 8 32

1 3 1

4 8 8

m m

j xxxx j xx x j x

j

m j

j j xx j

b ub u b

vb a ua

ω − − − = − − − − ⎛ = _⎜− − − ⎝ ⎞ − − − _⎟Λ ⎠

∑

22 1 1

0

1 1

16 8

m

m m j

j x j x

j ub a ω − − − = ⎛ ⎞ = _⎜− + _⎟Λ ⎝ ⎠

∑

23 1 0 1 4 m

m m j

j j a ω − − = ⎛ ⎞ = _⎜− _⎟Λ ⎝ ⎠

∑

24 1 0 1 8 m

m m j

j x j b ω − − = ⎛ ⎞ = _⎜− _⎟Λ ⎝ ⎠

∑

2

31 1 1 1

0

1 1 1

1 1 0

1 3 1 1 1

16 32 32 8 16

1 1 1 1

8 32 4 16

1 8

m m

j xxxxx x j xx xx j x

j

m j

x x j j xxx x j

m

m j j x j

b u b u v u b

v uu b a u a

b ω ₋ ₋ ₋ = − − − − − + − = ⎡ ⎛ ⎞ = _⎢ + + −_⎜ + + _⎟ ⎝ ⎠ ⎣ ⎤ ⎛ ⎞ +_⎜ − _⎟ + + _⎥Λ ⎝ ⎠ ⎦ ⎛ ⎞ − _⎜ _⎟Λ ⎝ ⎠

∑

2

32 1 1 1

0

1

1 1

0

1 1 1 1

( )

8 8 4 16

1 1

2 4

m m

j xxxx x j x j

j

m

m j m j

j xx j

j

b u b v u b

(6)

33 1 1 0

1 1

16 8

m

m m j

j x j x

j ub a ω − − − = ⎛ ⎞ = _⎜ − _⎟Λ ⎝ ⎠

∑

34 1 1 1

0

1 1 1

8 8 4

m

m m j

j xx j j

j

b ub a

ω − − − − = ⎛ ⎞ = _⎜− − − _⎟Λ ⎝ ⎠

∑

41 1 1 1

0

1 1

2

1 1 1

2

1

1 1 5

32 32 32

3 1 1 3 1

)

32 4 32 16 32

1 1 1 1

16 64 16 8

1 1 1

]

16 4 16

1 4

m m

j xxxxxx j xxxx x j xxx

j

xx j xx x x xxx j x

x

xx j j xxxx j xx

m j

xx j

j xx

ub ub u b

u v b uu v u

v u b a ua

u v u a

b ω ₋ ₋ ₋ = − − − − − − − − ⎡ = − − − ⎢⎣ ⎛ ⎞ ⎛ ⎞ −_⎜ + _⎟ −_⎜ + + _⎟ ⎝ ⎠ ⎝ ⎠ ⎛ ⎞ −_⎜ + _⎟ − + ⎝ ⎠ ⎤ ⎛ ⎞ +_⎜ − + _⎟ _⎥Λ ⎝ ⎠ ⎦ + +

∑

1 1 0 1 8 m m j j j

a− − + = ⎛ _⎞Λ ⎜ ⎟ ⎝ ⎠

∑

2

42 1 1 1

0

1 1 1

1 1 0

1 3 1 1 1

16 32 32 8 16

1 1 1 1

8 32 4 16

1 8 m m

j xxxxx x j xx xx j x

j

m j

x x j j xxx x j

m

m j j x j

b u b u v u b

v uu b a u a

b ω ₋ ₋ ₋ = − − − − − + − = ⎡ ⎛ ⎞ = + + −_⎜ + + _⎟ ⎢⎣ ⎝ ⎠ ⎛ ⎞ ⎤ +_⎜ − _⎟ + + _⎥Λ ⎝ ⎠ ⎦ ⎛ ⎞ − _⎜ _⎟Λ ⎝ ⎠

∑

43 1 1 1 1

0

1

1 1 1

0

1 1 5 1

8 8 32 4

3 1 1

8 8 4

m m

j xxxx j xx x j x j

j

m

m j m j

j xx j j

j

b ub u b vb

a ua b

ω ₋ ₋ ₋ ₋ = − − + − − − = ⎡ = + + ⎢⎣ ⎤ ⎛ ⎞ + + Λ − _⎜ _⎟Λ ⎥⎦ ⎝ ⎠

∑

44 1 1 1 1

0

1 1 1 3

8 16 16 8

m

m m j

j xxx j x x j j x

j

b ub u b a

ω − − − − − = ⎡ ⎤ = _⎢− − − − _⎥Λ ⎣ ⎦

∑

Proof. By direct computing, Theorem 7 is derived. Now, substituting (18)-(20) into (30) and (31), then

m

x

t m

Y MY Y W Y

= ⎧⎪ ⎨ ₌

⎪⎩ (35) there m=0,1, 2,L,

1 1

1 2

2

1 2 1 1 1 2 1 1

0 0 0

1

, 0 0

2 1

, 0 0

2

3 1 1

, , , , 0

8 2 2

E

y y E E

M

y y E E

y z y y E y y E y y E

− ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ = ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ −Λ− − − ⎜ ⎟ ⎝ ⎠ (36) and

( )

4 4 0,1, 2

m

m ij

W = ω _× m= L (37)

1

11 1 1

1

, 4

m

m j m j

j y z

ω

− − = ⎛ ⎞ = _⎜− Λ _⎟Λ ⎝ ⎠

∑

1

12 1 2

1

, 4

m

m j m j m

j y z ω − − = ⎛ ⎞ = − _⎜ Λ _⎟Λ + Λ ⎝ ⎠

∑

1

13 1 2

1

, 4

m

m j m j

j y y

ω

− − = ⎛ ⎞ = _⎜ Λ _⎟Λ ⎝ ⎠

∑

1

14 1 1

1

, 4

m

m j m j

j y y

ω

− − = ⎛ ⎞ = _⎜ Λ _⎟Λ ⎝ ⎠

∑

1

21 2 1 1 1

1

1 1

, ,

4 2

m

m j m j m

j

y z y y

ω − − = ⎛ ⎞ = − _⎜ Λ _⎟Λ − Λ ⎝ ⎠

∑

1

22 2 2

1

, 4

m

m j m j

j y z ω − − = ⎛ ⎞ = − _⎜ Λ _⎟Λ ⎝ ⎠

∑

1

23 2 2

1

1 ,

4

m

m j m j m

j

y y

ω

− − =

⎛

⎞

=

_⎜

Λ

_⎟

Λ

− Λ

⎝

⎠

∑

1

24 2 1

1

1 ,

4

m

m j m j

j

y y

ω

− − =

⎛

⎞

=

_⎜

Λ

_⎟

Λ

⎝

⎠

∑

1

31 2 1 1 2

1

1 ,

,

4

2

m

m j m j m

j

z z

y y

ω

− − =

⎛

⎞

= −

_⎜

Λ

_⎟

Λ

−

Λ

⎝

⎠

∑

1

32 2 2

1

, 4

m

m j m j

j z z

ω

− − = ⎛ ⎞ = − _⎜ Λ _⎟Λ ⎝ ⎠

∑

1

33 2 2

1

, 4

m

m j m j

j y z

ω

− − = ⎛ ⎞ = _⎜ Λ _⎟Λ ⎝ ⎠

∑

1

34 1 2

1

1 ,

4

m

m j m j m

j

y z

ω

− − =

⎛

⎞

=

_⎜

Λ

_⎟

Λ

− Λ

⎝

⎠

∑

1 1

41 1 1 1 2

1 2 1 1 1 , , 4 3 , 8 m

m j m j m m

j

m

z z y z

y y

ω

− − + = ⎛ ⎞ = − _⎜ Λ _⎟Λ + Λ + Λ ⎝ ⎠ + Λ

∑

1

42 2 1 1 2

1

1 1

, ,

4 2

m

m j m j m

j

z z y y

ω

− − = ⎛ ⎞ = − _⎜ Λ _⎟Λ − Λ ⎝ ⎠

∑

1

43 2 1 1 1

1

1 ,

,

4

2

m

m j m j m

j

y z

y y

ω

− − =

⎛

⎞

=

_⎜

Λ

_⎟

Λ

+

Λ

⎝

⎠

∑

1

44 1 1

1

1 ,

4

m

m j m j

j

y z

ω

− − =

⎛

⎞

=

_⎜

Λ

_⎟

Λ

⎝

⎠

∑

So that, the following theorem holds:

Theorem 8 On the Bargmann constrained equation (29), the evolution equation hierarchy (10) of the fourth-order eigenvalue problem (1) are nonlinearized as the following Hamilton canonical equation system

, m m x x m m t t h h Y Z Z Y h h Y Z Z Y ∂ ∂ ⎧ ₌ ⎧ ₌ ⎪ ⎪ ⎪ ∂ ⎪ ∂ ⎨ _∂ ⎨ _∂ ⎪ ₌ ⎪ _{= −} ⎪ ∂ ⎪ ∂

(7)

(

)

(

1

1 1 2 1 1 2 1 2 2 1 1 1 2 1 1 1 1

1 2 2 1 1 1 1

1

1 1

2 2 2 2 1 1 1 1 1

2 2 2 2 1

2 1 1 2 2 1 1

1 1

, , , ,

2 4

1 1

, , , ,

2 16

1 1

, , ,

2 8

, , , ,

, ,

1

, , ,

4

m m m

m

m m

m

m m j j

j

m j j m j j

m j j

m

m j j m j j

h y y y z y y y y

y y y z y y y y

z z z z y y

z z y y y z y z

y z y z

z z y y y z

+

− −

=

− − − −

− −

− − −

=

= − Λ + Λ + Λ

− Λ − Λ

− Λ + Λ Λ

+ Λ Λ − Λ Λ

− Λ Λ

+ Λ Λ − Λ Λ

∑

1

)

1, 2

j

y z −

D. Integrable System of Hamilton Equations Set

2

(1) 2 2

1 1 2 1 1 1

2

2 1 2 1 2 1 2

(2) ( , )

1 1 1

, ,

2 2 8

1 1

,

2 4

1

, , 1, 2 8

j j j j

j j j j j

l k

j j

E y y z y y y

y z z y y y y

E l k

λ

⎧ _{= −} ₋ ⎛ ₊ ⎞

⎜ ⎟

⎪ _⎝ _⎠

⎪ ⎪

⎨ + − +

⎪ ⎪

= Γ =

⎪ ⎩

(39)

there

( , ) 1,

1

, , 1, 2

N

lj lj kj kj l k

j

i i j j i li li ki ki

y z y z

l k

y z y z

λ λ

= ≠

Γ = =

−

∑

Theorem 9. (ⅰ)

{

(m)_, _{1, 2,} _, _; _{1, 2}

}

j

E j= _L N m= are the involutive system, i.e.

{

( ) ( )

}

, 0

m n

j j

E E =

here j i, =1, 2,_L, ;N m=1, 2

(ⅱ)

(

(1) (2)

)

1

1 0

1 N

m j j m

j j m

H_λ E E λ h

λ λ

∞ − −

= =

= + =

−

∑

here

(

(1) (2)

)

1

, 0,1, 2 N

m

m j j j

j

h λ E E m

=

∑

+ = _L

Proof. (ⅰ) By the definition of the Poisson bracket, it easily is derived.

(ⅱ)

(

)

(

)

(

)

(

)

(1) (2) (1) (2)

1 1

(1) (2)

1 0

1 (1) (2)

0 1

1

0

1 1 1

1

N N

j j j j

j

j j j

m N

j

j j

j m

N

m m

j j j

m j

m m m

H E E E E

E E

h

λ _{λ λ} _λ _λ

λ λ

λ

= =

∞ = = ∞

− − = = ∞

− − =

= + = +

− ₋

⎛ ⎞

= _⎜ _⎟ +

⎝ ⎠

= +

=

∑

∑ ∑

∑

here

(

(1) (2)

)

1

, 0,1, 2

N m

m j j j

j

h

λ

E E m

=

∑

+ = _L

Theorem 10. (ⅰ)

{

h E, ( )jk

}

=0 (40) here j=1,2,L, ;N k=1, 2 .

(ⅱ)

{

h h, _m

}

=0. (41)

here m=0,1, 2,_L.

(ⅲ)

{

hm,hn

}

=0. (42) here m n, =0,1, 2,_L

(ⅳ)

2 4

1

, ,

N

j j

j

R dy dz h

=

⎛ ⎞

Λ

⎜ ⎟

⎝

∑

⎠and

2 4

1

, ,

N

j j m

j

R dy dz h

=

⎛ ⎞

Λ

⎜ ⎟

⎝

∑

⎠

are integrable system in the Liouville sense.

Proof. In fact, we obtain (40) by direct computing. So, (41) holds by Theorem 9 and (40). Using (40) and (41), we have (42). According to the Liouville theorem[1, 3, 4], (ⅳ) is holds.

Theorem 11. If

{

y zj, j j=1, 2

}

is an involutive

solution of the integrable systems (38), then (30)

1 1

2

1 2 2 2 1 1

,

3 1 3

, , ,

2 2 8

u y y

v y z y y y y

⎧ = ⎪ ⎨

= − + −

⎪⎩

is the involutive representation of solutions for the evolution equation hierarchy (10) .

Proof. According to (38) is the Lax pairs of the evolution equation hierarchy (10), by means of theorem 10, then theorem 11 holds.

III. CONCLUSIONS

The Bargmann constraint problem is studied in this paper, and the associated Lax pairs have been nonlineared. By means of the viewpoint of Hamilton mechanics, the Euler-Lagrange function and the Legendre transformations have been derived, and a reasonable Jacobi-Ostrogradsky coordinate system has been found. Then, the Hamiltonian cannonical coordinate system equivalent to this eigenvalue problem has been obtained on the symplectic manifolds. It is proved to be an infinite-dimensional integrable Hamilton system in the Liouville sense. Moreover the involutive representation of the solutions is generated for the evolution equations hierarchy in correspondence with this reduced fourth-order eigenvalue problem.

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ShuHong Wang was born in Huludao, Liaoning, China,

February 1980. She received M. S. degree in 2006 from School of Sciences, Hebei University of Technology, Tianjin, China. Current research interests include integrable systems and Inequality.

Wei Liu was born in Shijiazhuang, Hebei, January 1981,

Received M.S. degree in 2007 from School of Sciences, Hebei University of Technology, Tianjin, China. Current research interests include integrable systems and computational geometry.

Shujuan Yuan was born in Tangshan, Hebei, September 1980.