Nonlinear Evolution Equations for Second-order Spectral Problem

Full text

(1)

Figure1. One-soliton solution of the KdV equation

Nonlinear Evolution Equations for Second-order

Spectral Problem

Wei Liu

Department of Mathematics and Physics, ShiJiaZhuang TieDao University, ShiJiaZhuang, China Email: Lwei_1981@126.com

Shujuan Yuan

Qinggong College, Hebei United University, TangShan, China Email: yuanshujuan1980@163.com

Shuhong Wang

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

Abstract—Soliton equations are infinite-dimensional

integrable systems described by nonlinear evolution equations. As one of the soliton equations, long wave equation takes on profound significance of theory and reality. By using the method of nonlinearization, the relation between long wave equation and second-order eigenvalue problem is generated. Based on the nonlinearized Lax pairs, Euler-Lagrange function and Legendre transformations, a reasonable Jacobi-Ostrogradsky coordinate system is obtained. Moreover, by means of the Bargmann constrained condition between the potential function and the eigenfunction, the Lax pairs is equivalent to matrix spectral problem. Furthermore, the involutive representations of the solutions for long wave equation are generated.

Index Terms—spectral problem, Hamilton canonical system,

Bargmann constraint, integrable system, involutive solution

I. INTRODUCTION

In 1895, Korteweg and de Vries [1-2] derived a nonlinear evolution equation as follows:

2 2

2

3 1 2 1

( )

2 2 3 3

g h

η η αη σ η

τ ξ ξ

∂ ∂ ∂

= + +

∂ ∂ ∂

3 1 3

Th h

g

σ

ρ

= −

By making the transformation

1 2

g t

hστ

= ,x= −σ ξ1 2 , 1 1

2 3

u= η+ α

the famous KdV equation

6 0

t x xxx

u + uu +u =

is obtained. It aroused an increasing interest among scientific researchers in the field of mathematics and physics, so more and more scientists have been interested in searching various methods to obtain solutions of some partial differential equation. Many effective methods have been proposed, for example, Hirota method, the inverse scattering, Darboux transformation, Painlevé

expansion, Bäcklund transformation, algebraic method and so on [1-7]. Using the inverse scattering method, we could obtain the N-soliton solution of KdV equation (see Fig. 1 and Fig. 2).

In our paper, by nonlinearization [8-15] of spectral problems, we considered the spectral

2

( ) x

Lϕ= ∂ + ∂ +q pϕ λϕ=

(2)

Figure2. Two-soliton solution of the KdV equation

II. LAX PAIRS AND EVOLUTION EQUATIONS Let us take the 2nd-order problem

Lϕ= ∂ + ∂ +( 2 q p)ϕ λϕ= x (1) where q=q x t( , )∈R p, = p x t( , )∈R , q p, ≠const are potential functions, λ is a complex eigenparameter,

x

∂ = ∂ ∂ ,∂∂ = ∂ ∂ =−1 −1 1.

Suppose Ω is the basic interval of (1), for the sake of simplicity, we assume that if the potentials q p, ,ϕ and all their derivatives with respect to x tend to zero, thenΩ = −∞ +∞( , ); If they are all periodic T functions, then Ω =[0, 2 ]T .

Definition 2.1 Assume that our linear space is equipped with a L2 scalar product

2( ) ( , )⋅ ⋅L Ω :

2

* ( )

( , )ϕ h L Ω ϕh dx Ω

=

< ∞ symbol * denotes the complex conjugate.

Definition 2.2 Operator

A

is an adjoint operator of A, if

2( ) 2( )

(Aϕ, )h L Ω =( ,ϕ Ah)L Ω . Using definition 2.2, we get

Lψ = ∂ − ∂ +( 2 q p)ψ = −λ ψ* x (2) In order to derive the evolution equation related to the spectral problem (1), we consider the stationary zero curvature equation

λωx+[ , ]ω L =λω ωx+ LLω (3) Take

1 1 1

0

[ ] j

j j x j

j

a b b

ω ∞ λ−

− − −

=

=

− − + ∂

and set

1 1

1 0 1

a G

b − −

⎛ ⎞ ⎛ ⎞ = ⎟ ⎜ ⎟= ⎝ ⎠

⎝ ⎠ (4) therefore, we obtain the recursive relation

KGj−1=JGj,j=0,1, 2K, (5) where

0 0 J = ⎜⎛ ∂⎞

⎝ ⎠ (6)

2

2

2 q

K

q p p

⎛ ∂ ∂ + ∂ ⎞

= ⎜

∂ − ∂ ∂ + ∂

⎝ ⎠ (7) j

j j a G

b ⎛ ⎞ = ⎜ ⎟⎜ ⎟ ⎝ ⎠

Now, we consider the auxiliary spectral problem 0,1, 2,

m

t m m

ϕ =ω ϕ = K (8) with

1 1 1

0

[ ]

m

m j

m j j x j

j

a b b

ω λ −

− − −

=

=

− − + ∂ (9) Then, the isospectral ( . . 0)

m

t

i eλ = compatible condition

[ , ]

m

t mx m mx m m

L =λω + ω L =λω +ω LLω (10) of the Lax pairs

0,1, 2,

m

x

t m

L

m

ϕ λϕ

ϕ ω ϕ

= ⎧⎪

= =

⎪⎩ K (11) determines a (m+1)-order long-wave equation

1 0,1, 2,

m

m

t

m m

t q

JG KG m

p

⎛ ⎞

= = =

⎜ ⎟ ⎜ ⎟

⎝ ⎠ K (12)

For example, when m=1and m=2, we can get the first and the second nonlinear systems, the results are shown in TableⅡ. When m=1, it is the long-wave equation. When m=2and q≡0, it is exactly the famous KdV equations.

In order to give the constraints between the potentials and the eigenfunctions, it is necessary to calculate the functional gradientwith respect to the potential functions.

Proposition 2.3: [11]

i) Ifλis an eigenvalue of (1), thenλ is real. TABLE II.

THE FIRST AND THE SECOND NONLINEAR SYSTEM

Evolution

equation m=1 m=2and q≡0

m

t

q 2px+qxx+2qqx 0

m

t

ppxx+2(qp)x

2

2 3( )

t xxx x

p = p + p

TABLE I. THE LENARD SEQUENCE

j

G aj bj

1

j= − 0 1

0

j= p q

1

j= −px+2qp 2

(3)

ii) If ϕ is an eigenfunction of (1) andψ is an eigenfunction of (2), thenϕ and ψ can be taken real functions.

iii) Ifϕ is an eigenfunction corresponding to the eigenvalue λ of (1) and ψ is an eigenfunction corresponding to the eigenvalueλof (2), then

1

( x ) x

q

dx

p

δλ

ϕψ δ

λ ϕ ψ

δλ ϕψ

δ

− Ω

⎛ ⎞

⎜ ⎟

⎜ ⎟

∇ = =

⎜ ⎟

⎜ ⎟ ⎝ ⎠

and

K∇ =λ λJ∇λ (13)

Proof: In fact, from (1) and (2), we have

* * ( )

x dx L dx

λ ϕ ψ ϕ ψ

Ω = Ω

* (L ) dx

ϕ ψ∗

Ω =

* (L ) dx

ϕ ψ∗ ∗ Ω =

*

xdx

λ ϕψ

Ω = −

*

x dx

λ ϕ ψ

Ω =

so

*

λ λ=

If ϕ is a complex eigenfunction of (1) onλ , and a ib

ϕ= + ,a b, are real functions, from Lϕ λϕ= xandλ is real, then

x x

La a

Lb b

λ λ

= ⎧ ⎨ =

soa b, are eigenfunction of (1) onλ, ϕ can be taken real function. Similarly, ψ can be taken real function.

Let

0

( , , )

d

h h q q p p

d ε

λ εδλ εδ εδ

ε =

= + + + ,

by

x Lϕ λϕ= (Lϕ) =(λϕx)

x x

Lϕ+Lϕ =λ ϕ +λϕ and

(Lϕ ψ) dx ϕ ψ(L )dx

Ω = Ω

( x)dx

ϕ λψ

Ω =

x dx

λϕ ψ

Ω =

so

x dx L dx

λ ϕ ψ ϕψ

Ω = Ω

((δ ϕq )x δ ϕ ψp ) dx Ω

=

+

( δ ϕψq x δ ϕψp )dx Ω

=

− +

then

q

p

δλ δ λ

δλ δ

⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ∇ =

⎜ ⎟ ⎜ ⎟ ⎝ ⎠

1

( ϕ ψx dx) ϕψx

ϕψ

− Ω

⎛ ⎞

=

⎝ ⎠

by (6) and (7), (13) holds. III. BARGMANN SYSTEMS AND THE HAMILTON

CANONICAL FORMS

We suppose λ12 <KN are the eigenvalues of the eigenvalue problem (1) and (2), ϕ ψj, j are the eigenfunction forλj( j=1, 2,L,N). Let

1 2

( , , N)

diag λ λ λ

Λ = L ,

1 2

( , , , )T N

ϕ ϕ ϕ

Φ = K ,

1 2

( , , , )T

N

ψ ψ ψ

Ψ = K

Now, we consider the Bargmann constraint [16-18] ,

, x q

p

=< Φ Ψ > ⎧

⎨ = − < Φ Ψ >

⎩ (14)

here

1 ,

N j j k

ϕ ψ

= < Φ Ψ >=

namely

0

, ,

x p

G q

− < Φ Ψ >

⎛ ⎞

⎛ ⎞

=⎜ ⎟= ⎜

< Φ Ψ >

⎝ ⎠ ⎝ ⎠

, ,

j x

j j

G = ⎜⎛< −Λ Φ Ψ >⎞⎟ < Λ Φ Ψ >

⎝ ⎠ j=0,1, 2L (15) so the eigenvalue problem (1) and (2) are equivalent to the systems

( , ) ,

, ,

xx x x x

xx x x x

Φ + < Φ Ψ > Φ − < Φ Ψ > Φ = ΛΦ ⎧

⎨Ψ − < Φ Ψ > Ψ − < Φ Ψ > Ψ = −ΛΨ

⎩ (16)

and (16) are called the Bargmann systems for the eigenvalue problem (1) and (2).

Let

Iˆ=

ΩIdx (17) where the Lagrange function Iis defined as follows:

1 1

, , , ,

2 2

x x x x

I=< Φ Ψ > + < Φ Ψ >< Φ Ψ > + < ΛΦ Ψ >

1 1

, , ,

2 x 2 x

− < ΛΦ Ψ > − < Φ Ψ >< Φ Ψ >

Proposition 3.1: The Bargmann systems (16) of the eigenvalue problem (1) and (2) are equivalent to the Euler-Lagrange equation systems:

ˆ 0 ˆ

0 I

I

δ δ δ δ

⎧ = ⎪⎪ Φ ⎨

=

⎪ Ψ ⎩

(18)

Proof: By (17), we have

ˆ

, ,

jxx x j jx jx

j I

δ ψ ψ ψ λψ

δϕ = − < Φ Ψ > − < Φ Ψ > +

, x ψj , ψjx λψjx

(4)

, x ψj , ψjx λψjx

− < Φ Ψ > − < Φ Ψ > +

0

= similarly,

ˆ

( , ) ,

jxx j x x jx jx

j I

δ ϕ ϕ ϕ λϕ

δψ = − − < Φ Ψ > + < Φ Ψ > +

( , ϕj)x , x ϕjx λϕjx

= < Φ Ψ > − < Φ Ψ > − ( , ϕj)x , x ϕjx λϕjx

− < Φ Ψ > + < Φ Ψ > +

0

= Now, the Poisson bracket [19] of the real-valued functions F and H in the symplectic [20] space (

2

4

1

, N

j j

j

w dz dy R

=

=

∧ ) is defined as follows:

2

1 1

{ , } ( )

N

j k jk jk jk jk

F H F H

F H

y z z y

= =

∂ ∂ ∂ ∂

= −

∂ ∂ ∂ ∂

∑∑

2

1

( yj, zj zj, yj )

j

F H F H

=

=

< > − < >

Based on the the Euler-Lagrange equation (18), the Jacobi-Ostrogradsky coordinates can be found, and the Bargmann systems (16) can be written in the Hamilton canonical equation systems [21].

Let

2

1 2

1

, , jx, j

j

u u g u v I

=

= Φ = Ψ =

< > −

Our aim is to find that the coordinates { ,v v1 2}and g

satisfy the following Hamilton canonical equations:

{ , }

1, 2 { , }

jx j

j

jx j

j g

u u g

v

j g

v v g

u

= =

=

= = −

⎪ ∂

By directly computing, we have

1

2

1

2

1 1

2 2

1 1

2 2

1 1

, ,

2 2

1 1

,

2 2

1 1

, ,

2 2

1 1

,

2 2

x

x

x x x

x x

x x x

x x

v q

v q

v

v

⎧ = Ψ − Ψ + ΛΨ ⎪

⎪ = Φ + Φ − ΛΦ ⎪

= < Φ Ψ > Ψ + < Φ Ψ > Ψ

⎪ ⎨

− < Φ Ψ > Ψ − ΛΨ

⎪ ⎪

⎪ = < Φ Ψ > Φ − < Φ Ψ > Φ

⎪ ⎪

− < Φ Ψ > Φ + ΛΦ ⎪

So, if we take the Jacobi-Ostrogradsky coordinates as follows:

1

2

1

2

1 1

2 2

1 1

2 2

x

x u u

v q

v q

= Φ ⎧ ⎪ = Ψ ⎪ ⎪

⎨ = Ψ − Ψ + ΛΨ

⎪ ⎪

= Φ + Φ − ΛΦ

⎪ ⎩

(19)

Then the Bargmann systems (16) are equivalent to the following Hamilton canonical systems:

1, 2 jx

j

jx j g u

v

j g v

u

=

=

= −

⎪ ∂

where

1 2 1 2 2 2 2 2

1 1

, , , ,

2 2

g=<v v > + <u u ><u v > − <λu v >

3

1 2 1 1 1 2 1 1

1 1 1

, , , ,

2 u u u v 4 u u 2 λu v

− < >< > − < > + < >

2

1 2 1 2 1 2

1 1

, , ,

2 u u λu u 4 λ u u

+ < >< > − < >

By (19), the Jacobi-Ostrogradsky coordinates can be written in the following form:

1

2

2

1

,

1 1

2 2

1 1

2 2

x

x y

y q

z

z q

= Φ ⎧ ⎪

= Φ + Φ − ΛΦ

⎪ ⎨ = Ψ ⎪ ⎪

= −Ψ + Ψ − ΛΨ

⎪ ⎩

(20)

then, we have:

Theorem 3.2: The Bargmann systems (16) of the eigenvalue problem (1) and (2) are equivalent to the following Hamilton canonical systems:

1, 2 jx

j

jx j h y

z

j h z

y

=

=

= −

⎪ ∂

where

2 1 1 2 2 2 2 2

1 1

, , , ,

2 2

h=< y z > − <y z ><y z > + < Λy z >

3

1 2 1 1 1 2 1 1

1 1 1

, , , ,

2 y z y z 4 y z 2 y z

− < >< > + < > + < Λ >

2

1 2 1 2 1 2

1 1

, , ,

2 y z y z 4 y z

− < Λ >< > + < Λ >

and h= −g.

IV. NONLINEARIZATION OF THE LAX PAIRS

Form (20) and Theorem 3.2, the Bargmann systems (16) have the equivalence form

x T x

Y MY

Z M Z

= ⎧⎪ ⎨

= − ⎪⎩

where

1

2

1 1

2 2

x y

Y

y q

Φ

⎛ ⎞

⎛ ⎞ ⎜ ⎟

=⎜ ⎟ ⎜=

Φ + Φ − ΛΦ

⎜ ⎟

⎝ ⎠ ⎝ ⎠

1

2

1 1

2 2

x

z q

Z z

−Ψ + Ψ − ΛΨ

⎛ ⎞ ⎜ ⎟

=⎜ ⎟ ⎜=

⎜ ⎟

(5)

11 21 22 M E M M M ⎛ ⎞ = ⎜ ⎟ ⎝ ⎠ where 11 1 1 2 2

M = − qE+ Λ

2 2

21

1 1 1 1

4 2 x 2 4

M = q EpEq E− Λ + Λq

22

1 1

2 2

M = − qE+ Λ

(1,1, ,1)

N N

E=E × =diag K

Proposition 4.1: The Lax pairs (11) for the(m+1) -order evolution equation (12) is equivalent to

, ;

, , 0,1, 2 ,

m m

T

x x

T

t m t m

Y MY Z M Z

Y W Y Z W Z m

⎧ = = −

⎪ ⎨

= = − =

⎪⎩ K (21)

where

0

m

m m m j

m

j m m

A B W C D − = ⎛ ⎞ = Λ ⎝ ⎠

1 1 1 1

1 1

2 2

m j j j x j

A = −aqbb + Λb

1

m j

B =b

2

1 1 1 1 1

1 1 1 1

2 2 4 2

m j xx x j j j j

C = − bq bpb + q b − Λqb

2 1 1 4 bj

+ Λ

1 1 1

1 1

2 2

m j j j

D = −a qb + Λb

By (14) and (20), we have the Bargmann constraint

1 2

2

1 1 1 2 1 2

,

1 1

, , ,

2 2

q y z

p y z y z y z

=< > ⎧

⎪ ⎨

=< > − < > + < Λ >

⎪⎩

(22)

11

1 2

, 0,1, 2, , , j j j j a G G j

b y z

⎛ ⎞ ⎛ ⎞

=⎜ ⎟ ⎜⎜ ⎟= = < Λ >

⎝ ⎠

⎝ ⎠ K

(23) where

1 11 1 1 1 2 1 2 1 2

1 1

, , , ,

2 2

j j j

G =< Λ y z > − <y z >< Λ y z > + < Λ+ y z >

Substituting the Bargmann constraint (22) and (23) into (21), the Lax pairs (24) for the (m+1)-order evolution equation (12) is equivalent to the following forms:

, 0,1, 2, m m T x x T

t m t m

Y MY Z M Z

Y W Y Z W Z m

⎧ = = −

= = − =

⎪⎩ K (24) where 11 21 22 M E M M M ⎛ ⎞ = ⎜ ⎟ ⎝ ⎠ m m m m m A B W C D ⎛ ⎞ = ⎜ ⎝ ⎠

11 1 2

1 1

,

2 2

M = − <y z > + Λ

2 2

21 1 2 1 1 2 2

1 3 1 1

, , ,

4 4 2 2

M = Λ + <y z > − <y z > − <y z >

1 2 1 2

1 1

, ,

2 y z 2 y z

− < Λ > − Λ < >

22 1 2

1 1

,

2 2

M = − <y z > + Λ

1 1

2 2 1 2

1

1 1

[ , ] ,

2 2

m

j m j m m

m j

Ay z − + y z

=

=

− < Λ > Λ + Λ − < Λ >

1 1 2 1

[ , ]

m

j m j m

m j

By z

=

=

< Λ > Λ + Λ

1

2 1 1 1 2 2

1

1

[ , ] ( , ,

2

m

j m j m

m j

Cy zy z y z

=

=

< Λ > Λ − < > + < > Λ)

1 2

1 2 1 2

1 1 1

, ,

4 4 4

m m m

y z + y z +

− < > Λ − < Λ > Λ + Λ

1

1 2 1 2

1 1

, ,

4 4

m m

y z y z

+

− < Λ > − < Λ > Λ

1 2 1 2

1

, ,

2

m

y z y z

+ < >< Λ > 2

1 2 1 , 4 m y z

+ < > Λ

1 1

1 1 1 2

1

1 1

[ , ] ,

2 2

m

j m j m m

m j

Dy z − + y z

=

=

− < Λ > Λ + Λ − < Λ >

Theorem 4.2: On the Bargmann constraint (22), the nonlinearized Lax pairs (24) for the (m+1)-order long wave equation (12) can be written as the following Hamilton systems [11-12]:

, ;

, , 0,1, 2, ,

m m x x m m t t h h Y Z Z Y h h

Y Z m

Z Y ∂ ∂ ⎧ = = − ⎪⎪ ∂ ∂ ⎨ = = − = ⎪ ∂ ∂ ⎩ K (25) where

1 1 2

1 1 2 2 1 2

1 1 1

, , ,

2 2 4

m m m

m

h = < Λ + y z > + < Λ +y z > + < Λ + y z >

2

1 2 1 2 1 2 1 2

1 1

, , , ,

4 4

m m

y z y z y z y z

+ < > < Λ > − < Λ >< Λ >

1 1 2 1 2

1

, ,

4 m

y z y z

+

− < Λ >< >+ < Λmy z2, 1>

1 2 1 1 2 2

1

, ( , , )

2 m

y z y z y z

− < Λ > < > + < >

1

1 2 2 2

1

1 1 1 2 1

, ,

, ,

j m j

m

j m j

j

y z y z

y z y z

− −

− −

=

< Λ > < Λ >

+

< Λ > < Λ >

andh0 =h.

Proof: 2 1 2 1 1 1

1 1

,

2 2

h

y y z y y

z

= − < > + Λ

1x y

=

2 1 2 2 2 2 1

2

1 1 1

, ,

2 2 2

h

y y z y y z y

z

= Λ − < > − < >

(6)

2 2

1 1 1 1 2 1 1

1 3 1

, ,

2 y z y 4 y z y 4 y

− < > + < > + Λ

1 2 1 1 2 1

1 1

, ,

2 y z y 2 y z y

− < Λ > − < > Λ

2x y

=

1

1 1 2 1 2

1

1 1

,

2 2

m m m

m h

y y z y y

z

+

= Λ − < Λ > + Λ

1

1 2 2 2 2 1

1

[ , , ]

m

j m j m j

j

y z y y z y

− − −

=

+

< Λ > − < Λ > Λ

1tm

y =

1 2

2 1 1 1 1

2

1 1 1

,

2 4 2

m m m

m h

y y y z y

z

+ +

= Λ + Λ − < > Λ

1 2 2 2 2 1

1 1

, ,

2 2

m m

y z y y z y

− < Λ > − < > Λ

2

1 2 1 1 2 1 2 1

1 1

, , ,

4 2

m m

y z y y z y z y

+ < > Λ + < >< Λ >

1 2 1 1 2 1

1 1

, ,

4 4

m m

y z y y z y

− < Λ > Λ − < Λ > Λ

1 1

1 2 1 1 2 1

1 1

, ,

4 4

m m

y z y y z y

+ +

− < Λ > − < > Λ

1 1

2 1 1 1 1 2

1

[ , , ]

m

j j m j

j

y z y y z y

− − −

=

+

< Λ > − < Λ > Λ

2tm

y = so x h Y Z ∂ =

∂ , m

m t h Y Z ∂ = ∂

Similarly, we have

2

1 2 2 2 1 2 2

1

1 1 3

, ,

2 2 4

h

z y z z y z z

y

= Λ − < > + < >

2

2 1 2 1 1 1 2

1 1 1

, ,

4 z 2 y z z 2 y z z

+ Λ − < > − < >

1 2 2 1 2 2

1 1

, ,

2 y z z 2 y z z

− < Λ > − < > Λ

1x z

= −

1 1 2 2 2

2

1 1

,

2 2

h

z y z z z

y

= − < > + Λ

2x z

= −

1

1 1 2 1 1 1 2

1

1 1 1

, ,

2 2 2

m m m

m h

z y z z y z z

y

+

= Λ − < Λ > − < > Λ

2

1 2 2 1 2 1 2 2

1 1

, , ,

4 2

m m

y z z y z y z z

+ < > Λ + < >< Λ >

1 1

1 2 2 1 2 2

1 1

, ,

4 4

m m

y z z y z z

+ +

− < Λ > − < > Λ

1 2 2 1 2 2

1 1

, ,

4 4

m m

y z z y z z

− < Λ > Λ − < Λ > Λ

2

2 2 2 2

1 1

,

2 4

m m

y z z + z

− < > Λ + Λ

1 1

2 1 2 2 2 1

1

[ , , ]

m

j j m j

j

y z z y z z

− − −

=

+

< Λ > − < Λ > Λ

1tm

z = −

1

2 1 1 2 2

2

1 1

,

2 2

m m m

m h

z z y z z

y

+

= Λ + Λ − < Λ >

1 1

1 2 1 1 1 2

1

[ , , ]

m

j j m j

j

y z z y z z

− − −

=

+

< Λ > − < Λ > Λ

2tm

z = − so x h Z Y ∂ = −

∂ , m

m t h Z Y ∂ = − ∂ V.LIOUVILLE COMPLETELY INTEGRABLE SYSTEMS Now we discuss the completely integrability for the Bargmann systems (25).

Let

(1)

1 1 2 2

(2)

1 2 1 1 2 2 2 1

2

1 2 1 2 1 2 1 2

2 (1,2)

1 2 1 2 1 2

1 1 2 2 1 ( , , ) 2 1 1 , , 4 4 1 1 , 4 4

k k k k k k k

k k k k k

k k k k k

k k k k k k

E y z y z

E y z y z y z y z

y z y z y z y z

y z y z y z

λ λ λ λ ⎧ = + ⎪ ⎪

= − < > + < > +

⎪ ⎨

< > + < >

⎪ ⎪

⎪ + − < Λ > −Γ

where

1 1 1 1

(1,2)

1, 2 2 2 2

1 N

k l k l

k

l l k k l k l k l

y y z z

y y z z

λ λ = ≠ Γ = −

Proposition 5.1:

i) {Ej(1),Ek(1)}=0,{E(1)j ,Ek(2)}=0,{E(2)j ,Ek(2)}=0 , 1, 2,

j k N

∀ = K (26) ii) {dE( )jl ,j=1, 2,KN l; =1, 2} are the linear independ- ence.

ⅲ) (1) (2) 1

1 0

1

( )

N

m

j j m

j j m

Hμ μ λ E E μ h

∞ − −

= =

= + =

(27)

(1) (2)

1

( )

N m

m j j j

j

h λ E E

=

=

+ m=0,1, 2,K (28)

Theorem 5.2: The Bargmann [8] systems (25) are the completely integrable systems in the Liouville sense. i.e.

( )

{ , l } 0, 1, 2; 1, 2, , j

h E = l= j= K N (29)

( )

{h Em, jl }=0, l=1, 2;j=1, 2,K,N (30) {h hm, n}=0, m n, =0,1, 2,K (31) { ,h hm}=0, m=0,1, 2,K (32)

Proof: By (26) and (28), we have {h hm, n}=0, m n, =0,1, 2,K from (27), then

{Hλ,Hμ}=0 using h=h0, we have

( )

{ ,h Ejl}=0, l=1, 2;j=1, 2,K,N

(7)

Figure3. Interaction of two solitary waves at different times

Figure3. Interaction of two solitary waves at different times

Figure3. Interaction of two solitary waves at different times According to the above Theorem, the Hamiltonian

phase flows n n

t h

g and m m

t h

g are commutable. Now, we arbitrarily choose an initial value

( (0, 0), (0, 0))yi zi T i=1, 2, let

( , ) (0, 0) (0, 0)

( , ) (0, 0) (0, 0)

m n n m

m n n m

i m n t t i t t i

h h h h

i m n i i

y t t y y

g g g g

z t t z z

⎛ ⎞ ⎛ ⎞ ⎛ ⎞

= =

⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ ⎝ ⎠

From (8), (9) and Theorem 4.2, the following theorem holds.

Theorem 5.3: Suppose ( ,y y z z1 2, ,1 2) is an involutive solution of the Hamiltonian [11] canonical equation systems (25), then

1 2

2

1 1 1 2 1 2

,

1 1

, , ,

2 2

q y z

p y z y z y z

=< >

⎧ ⎪ ⎨

=< > − < > + < Λ >

⎪⎩

satisfies the (m+1)-order long-wave equation (12).

Remark: By Theorem 5.2, soliton waves have the following properties: when two of them interact, the larger soliton has been shifted to the right of where it would have been no interaction, and the smaller shifted to the left by the same time (see Fig. 3) [22-24].

Especially, if ( ,y y z z1 2, ,1 2) satisfies

1

1

1 1

2 2

1 1

2 2

1 1

1

2 2

1 1

1

2 2

x T x

t T t

y y

M

y y

z z

M

z z

y y

W

y y

z z

W

z z

⎧⎛ ⎞ ⎛ ⎞

=

⎪⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

⎪⎛ ⎞ ⎛ ⎞

⎜ ⎟ = − ⎜ ⎟

⎪⎝ ⎠ ⎝ ⎠

⎨⎛ ⎞ ⎛ ⎞

⎜ ⎟ = ⎜ ⎟

⎪⎝ ⎠ ⎝ ⎠

⎪⎛ ⎞ ⎛ ⎞

= −

⎪⎜ ⎟ ⎜ ⎟

⎪⎝ ⎠ ⎝ ⎠

where

1 2

21 1 2

1 1

,

2 2

1 1

,

2 2

y z E

M

M y z

Λ − < >

⎜ ⎟

= ⎜ ⎟

Λ − < >

⎜ ⎟

⎝ ⎠

2 2

21 1 2 1 1 2 2

1 3 1 1

, , ,

4 4 2 2

M = Λ + <y z > − <y z > − <y z >

1 1 1

1 1

A B

W

C D

⎛ ⎞

= ⎜

⎝ ⎠

2

1 1 2 2 2

1 1

, ,

2 2

A = Λ − < Λy z > − <y z >

1 1, 2

B = Λ+ < y z >

1 2 1 1 1 2 2

1

, ( , ,

2

C =<y z > − <y z > + <y z > Λ)

2 3

1 2 1 2

1 1 1

, ,

4 y z 4 y z 4

− < > Λ − < Λ > Λ + Λ

2

1 2 1 2

1 1

, ,

4 y z 4 y z

− < Λ > − < Λ > Λ

1 2 1 2

1

, ,

2 y z y z

+ < >< Λ > 2

1 2

1 ,

4 y z

+ < > Λ

2

1 1 2 1 1

1 1

, ,

2 2

D = Λ − < Λy z > − <y z >

(8)

1 2

2

1 1 1 2 1 2

,

1 1

, , ,

2 2

q y z

p y z y z y z

=< >

⎧ ⎪ ⎨

=< > − < > + < Λ >

⎪⎩

satisfies long-wave equation

1

1

2 2

2( )

t x xx x

t xx x

q p q qq

p p qp

⎛ ⎞ ⎛ + + ⎞

=

⎟ ⎜

⎜ ⎟ +

⎝ ⎠

ACKNOWLEDGMENT

The author is grateful to anonymous referees for their valuable suggestions. This work is supported in part by a grant from the Youth Science Foundation of Hebei Province (Project No. A2011210017).

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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. She mainly engaged in control and application of differential equations. Current research interests include integrable systems and computational geometry.

Shujuan Yuan was born in Tangshan, Hebei, September, 1980. Received M.S. degree in 2007 from School of Sciences, Hebei University of Technology, Tianjin, China. Current research interests include integrable systems and computional geometry. She is a lecturer in department of Qinggong College, Hebei United University.

Figure

TABLE I.   ENARD SEQUENCE

TABLE I.

ENARD SEQUENCE p.2

References

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