Nonlinear Evolution Equations for Second-order ... - Semantic Scholar
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Nonlinear Evolution Equations for Second-order Spectral Problem 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]
Shuhong Wang College of Mathematics, Inner Mongolia University for Nationalities, TongLiao, China Email: [email protected]
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.
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).
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: ∂η 3 g ∂ 1 2 2 1 ∂ 2η = ( η + αη + σ 2 ) ∂τ 2 h ∂ξ 2 3 3 ∂ξ 1 3 Th σ= h − 3 ρg By making the transformation 1 g 1 1 t= τ , x = −σ 1 2ξ , u = η + α 2 hσ 2 3 the famous KdV equation ut + 6uu x + u xxx = 0 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 © 2012 ACADEMY PUBLISHER doi:10.4304/jcp.7.9.2144-2151
Figure1. One-soliton solution of the KdV equation
In our paper, by nonlinearization [8-15] of spectral problems, we considered the spectral Lϕ = (∂ 2 + ∂q + p)ϕ = λϕ x The paper is structured as follows. In Sect.2, the adjoint Lax pairs of the spectral problem is generalized. In Sect.3, based on the Euler-Lagrange equations and Legendre transformations, a suitable Jacobi-Ostrogradsky coordinate is been found. Section 4 and Sect.5 are devoted to establishing the Liouville integrability of the resulting Hamiltonian systems from the 2nd-order spectral problems.
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⎛0 ∂⎞ J =⎜ ⎟ ⎝∂ 0⎠ ⎛ 2∂ K = ⎜⎜ 2 ⎝ q∂ − ∂
(6)
∂ 2 + ∂q ⎞ ⎟ p∂ + ∂p ⎠⎟
(7)
⎛ aj ⎞ Gj = ⎜ ⎟ ⎜b ⎟ ⎝ j⎠
TABLE I. THE LENARD SEQUENCE
Figure2. Two-soliton solution of the KdV equation
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, then Ω = (−∞, +∞) ; If they are all periodic T functions, then Ω = [0, 2T ] . Definition 2.1 Assume that our linear space is equipped with a L2 scalar product (⋅, ⋅) L2 ( Ω ) : (ϕ , h) L2 ( Ω ) = ∫ ϕ h* dx < ∞ Ω
symbol * denotes the complex conjugate. Definition 2.2 Operator A is an adjoint operator of A , if ( Aϕ , h) L2 ( Ω ) = (ϕ , Ah) L2 ( 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 λω x + [ω , L] = λω x + ω L − Lω (3) Take
ω = ∑ [−a j −1 − b j −1x + b j −1∂ ] λ
−j
j =0
and set ⎛ a−1 ⎞ ⎛ 0 ⎞ G−1 = ⎜ ⎟ = ⎜ ⎟ ⎝ b−1 ⎠ ⎝1 ⎠ therefore, we obtain the recursive relation KG j −1 = JG j , j = 0,1, 2K ,
where
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aj
bj
j = −1 j=0
0 p
1 q
j =1
− px + 2qp
2 p + qx + q 2
L
II. LAX PAIRS AND EVOLUTION EQUATIONS
∞
Gj
Now, we consider the auxiliary spectral problem ϕtm = ωmϕ m = 0,1, 2,K
(8)
with m
ωm = ∑ [− a j −1 − b j −1x + b j −1∂ ] λ m − j
(9)
j =0
Then, the isospectral (i.e. λtm = 0) compatible condition Ltm = λωmx + [ωm , L] = λωmx + ωm L − Lωm
(10)
of the Lax pairs ⎧⎪ Lϕ = λϕ x ⎨ ⎪⎩ϕtm = ωmϕ m = 0,1, 2,K determines a (m + 1) -order long-wave equation
(11)
⎛ qtm ⎞ (12) ⎜⎜ ⎟⎟ = JGm = KGm −1 m = 0,1, 2,K ⎝ ptm ⎠ For example, when m = 1 and 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 = 2 and q ≡ 0 , it is exactly the famous KdV equations. TABLE II. THE FIRST AND THE SECOND NONLINEAR SYSTEM Evolution equation
m =1
m = 2 and q ≡ 0
qtm
2 px + qxx + 2qqx
0
ptm
− pxx + 2(qp) x
pt2 = pxxx + 3( p 2 ) x
(4)
(5)
In order to give the constraints between the potentials and the eigenfunctions, it is necessary to calculate the functional gradient with respect to the potential functions. Proposition 2.3: [11] i) If λ is an eigenvalue of (1), then λ is real.
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ii) If ϕ is an eigenfunction of (1) and ψ is an eigenfunction of (2), then ϕ and ψ can be taken real functions. iii) If ϕ is an eigenfunction corresponding to the eigenvalue λ of (1) and ψ is an eigenfunction corresponding to the eigenvalue λ of (2), then ⎛ δλ ⎞ ⎜δq ⎟ −ϕψ x ⎞ ⎟ = ( ϕ xψ dx) −1 ⎛⎜ ∇λ = ⎜ ⎟ ∫ Ω ⎜ δλ ⎟ ⎝ ϕψ ⎠ ⎜ ⎟ ⎝δ p ⎠ and K ∇λ = λ J ∇ λ (13) Proof: In fact, from (1) and (2), we have λ ∫ ϕ xψ dx = ∫ Lϕ (ψ * )* dx Ω
Ω
= ∫ ϕ ( Lψ ∗ )* dx Ω
= ∫ ϕ ( L∗ψ ∗ )* dx Ω
= −λ * ∫ ϕψ x dx Ω
=λ
*
∫
Ω
ϕ xψ dx
⎛ δλ ⎞ ⎜δq ⎟ −ϕψ x ⎞ ⎟ = ( ϕ xψ dx) −1 ⎛⎜ ∇λ = ⎜ ⎟ ∫ Ω ⎜ δλ ⎟ ⎝ ϕψ ⎠ ⎜ ⎟ ⎝δ p ⎠ by (6) and (7), (13) holds.
III. BARGMANN SYSTEMS AND THE HAMILTON CANONICAL FORMS
We suppose λ1 < λ2 < K < λN are the eigenvalues of the eigenvalue problem (1) and (2), ϕ j ,ψ j are the eigenfunction for λ j ( j = 1, 2,L , N ). Let Λ = diag (λ1 , λ2 ,L λN ) , Φ = (ϕ1 , ϕ2 ,K , ϕ N )T , Ψ = (ψ 1 ,ψ 2 ,K ,ψ N )T Now, we consider the Bargmann constraint [16-18] ⎧q =< Φ, Ψ > (14) ⎨ ⎩ p = − < Φ, Ψ x > here N
< Φ, Ψ >= ∑ ϕ jψ j
so
k =1
λ =λ If ϕ is a complex eigenfunction of (1) on λ , and ϕ = a + ib , a, b are real functions, from Lϕ = λϕ x and λ is real, then ⎧ La = λ ax ⎨ ⎩ Lb = λ bx *
so a, b are eigenfunction of (1) on λ , ϕ can be taken real function. Similarly, ψ can be taken real function. Let d h = h(λ + εδλ , q + εδ q, p + εδ p ) , d ε ε =0 by Lϕ = λϕ x ( Lϕ ) = (λϕ x ) L ϕ + Lϕ = λ ϕ x + λϕ x
and
∫
Ω
( Lϕ )ψ dx = ∫ ϕ ( Lψ )dx Ω
= ∫ ϕ (−λψ x )dx Ω
= ∫ λϕ xψ dx Ω
so
∫
Ω
λ ϕ xψ dx = ∫ L ϕψ dx Ω
= ∫ ((δ qϕ ) x + δ pϕ )ψ dx Ω
= ∫ (−δ qϕψ x + δ pϕψ )dx Ω
then
namely ⎛ p ⎞ ⎛ − < Φ, Ψ x > ⎞ G0 = ⎜ ⎟ = ⎜ ⎟ ⎝ q ⎠ ⎝ < Φ, Ψ > ⎠ ⎛ < −Λ j Φ, Ψ x > ⎞ j = 0,1, 2L (15) Gj = ⎜ ⎜ < Λ j Φ, Ψ > ⎟⎟ ⎝ ⎠ so the eigenvalue problem (1) and (2) are equivalent to the systems ⎧Φ xx + (< Φ, Ψ > Φ ) x − < Φ, Ψ x > Φ = ΛΦ x (16) ⎨ ⎩Ψ xx − < Φ, Ψ > Ψ x − < Φ, Ψ x > Ψ = −ΛΨ x
and (16) are called the Bargmann systems for the eigenvalue problem (1) and (2). Let Iˆ = ∫ Idx (17) Ω
where the Lagrange function I is defined as follows: 1 1 I =< Φ x , Ψ x > + < Φ, Ψ >< Φ, Ψ x > + < ΛΦ x , Ψ > 2 2 1 1 − < ΛΦ, Ψ x > − < Φ, Ψ >< Φ x , Ψ > 2 2 Proposition 3.1: The Bargmann systems (16) of the eigenvalue problem (1) and (2) are equivalent to the Euler-Lagrange equation systems: ⎧ δ Iˆ =0 ⎪ ⎪ δΦ (18) ⎨ ⎪ δ Iˆ = 0 ⎪⎩ δΨ Proof: By (17), we have δ Iˆ = ψ jxx − < Φ, Ψ x > ψ j − < Φ, Ψ > ψ jx + λψ jx
δϕ j
=< Φ, Ψ x > ψ j + < Φ, Ψ > ψ jx − λψ jx © 2012 ACADEMY PUBLISHER
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− < Φ, Ψ x > ψ j − < Φ, Ψ > ψ jx + λψ jx =0 similarly, δ Iˆ = −ϕ jxx − (< Φ, Ψ > ϕ j ) x + < Φ, Ψ x > ϕ jx + λϕ jx
Then the Bargmann systems (16) are equivalent to the following Hamilton canonical systems: ∂g ⎧ ⎪u jx = ∂v j ⎪ j = 1, 2 ⎨ ⎪v = − ∂g ⎪ jx ∂u j ⎩ where 1 1 g =< v1 , v2 > + < u1 , u2 >< u2 , v2 > − < λ u2 , v2 > 2 2 1 1 1 − < u1 , u2 >< u1 , v1 > − < u1 , u2 >3 + < λ u1 , v1 > 2 4 2 1 1 + < u1 , u2 >< λ u1 , u2 > − < λ 2 u1 , u2 > 2 4 By (19), the Jacobi-Ostrogradsky coordinates can be written in the following form: ⎧ y1 = Φ, ⎪ ⎪ y = Φ x + 1 qΦ − 1 ΛΦ ⎪ 2 2 2 (20) ⎨ = Ψ z ⎪ 2 ⎪ 1 1 ⎪ z1 = −Ψ x + qΨ − ΛΨ ⎩ 2 2 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: ∂h ⎧ ⎪ y jx = ∂z j ⎪ j = 1, 2 ⎨ ∂ ⎪z = − h ⎪ jx ∂y j ⎩ where 1 1 h =< y2 , z1 > − < y1 , z2 >< y2 , z2 > + < Λy2 , z2 > 2 2 1 1 1 − < y1 , z2 >< y1 , z1 > + < y1 , z2 >3 + < Λy1 , z1 > 2 4 2 1 1 − < Λy1 , z2 >< y1 , z2 > + < Λ 2 y1 , z2 > 2 4 and h = − g .
δψ j
= (< Φ, Ψ > ϕ j ) x − < Φ, Ψ x > ϕ jx − λϕ jx
−(< Φ, Ψ > ϕ j ) x + < Φ, Ψ x > ϕ jx + λϕ jx =0 Now, the Poisson bracket [19] of the real-valued functions F and H in the symplectic [20] space 2
( w = ∑ dz j ∧ dy j , R 4 N ) is defined as follows: j =1
2
N
{F , H } = ∑∑ ( j =1 k =1
∂F ∂H ∂F ∂H − ) ∂y jk ∂z jk ∂z jk ∂y jk
2
= ∑ (< Fyj , H zj > − < Fzj , H yj >) j =1
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
u1 = Φ, u2 = Ψ , g = ∑ < u jx , v j > − I j =1
Our aim is to find that the coordinates {v1 , v2 } and g satisfy the following Hamilton canonical equations: ∂g ⎧ ⎪u jx = {u j , g} = ∂v j ⎪ j = 1, 2 ⎨ ∂ ⎪v = {v , g} = − g j ⎪ jx ∂u j ⎩ By directly computing, we have 1 1 ⎧ ⎪v1 = Ψ x − 2 qΨ + 2 ΛΨ ⎪ ⎪v = Φ + 1 qΦ − 1 ΛΦ x ⎪ 2 2 2 ⎪ 1 ⎪v = < Φ , Ψ > Ψ + 1 < Φ , Ψ > Ψ x x ⎪ 1x 2 2 ⎨ 1 1 ⎪ − < Φ x , Ψ > Ψ − ΛΨ x ⎪ 2 2 ⎪ 1 1 ⎪v2 x = < Φ, Ψ x > Φ − < Φ x , Ψ > Φ 2 2 ⎪ ⎪ 1 1 − < Φ, Ψ > Φ x + ΛΦ x ⎪ ⎩ 2 2 So, if we take the Jacobi-Ostrogradsky coordinates as follows: ⎧u1 = Φ ⎪u = Ψ ⎪ 2 ⎪ 1 1 (19) ⎨v1 = Ψ x − qΨ + ΛΨ 2 2 ⎪ ⎪ 1 1 ⎪v2 = Φ x + qΦ − ΛΦ 2 2 ⎩
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IV.
NONLINEARIZATION OF THE LAX PAIRS
Form (20) and Theorem 3.2, the Bargmann systems (16) have the equivalence form ⎧⎪Yx = MY ⎨ T ⎪⎩ Z x = − M Z where Φ ⎛ ⎞ ⎛ y1 ⎞ ⎜ ⎟ Y =⎜ ⎟= 1 1 ⎝ y2 ⎠ ⎜⎜ Φ x + qΦ − ΛΦ ⎟⎟ ⎝ ⎠ 2 2 1 1 ⎛ ⎞ −Ψ x + qΨ − ΛΨ ⎟ ⎛ z1 ⎞ Z =⎜ ⎟=⎜ 2 2 ⎟⎟ ⎝ z2 ⎠ ⎜⎜ Ψ ⎝ ⎠
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⎛ M 11 M =⎜ ⎝ M 21
E ⎞ ⎟ M 22 ⎠
where
M 21
1 1 M 11 = − qE + Λ 2 2 1 2 1 1 1 M 21 = q E − pE − qx E − Λq + Λ 2 4 2 2 4 1 1 M 22 = − qE + Λ 2 2 E = EN × N = diag (1,1,K ,1)
M 22
Proposition 4.1: The Lax pairs (11) for the (m + 1) order evolution equation (12) is equivalent to ⎧⎪Yx = MY , Z x = − M T Z ; (21) ⎨ T ⎪⎩Ytm = WmY , Z tm = −Wm Z , m = 0,1, 2K , where m ⎛ Am Bm ⎞ m − j Wm = ∑ ⎜ ⎟Λ Dm ⎠ j = 0 ⎝ Cm 1 1 Am = −a j −1 − qb j −1 − b j −1x + Λb j −1 2 2 Bm = b j −1 1 1 1 1 Cm = − b j −1xx − qx b j −1 − pb j −1 + q 2 b j −1 − Λqb j −1 2 2 4 2 1 2 + Λ b j −1 4 1 1 Dm = −a j −1 − qb j −1 + Λb j −1 2 2 By (14) and (20), we have the Bargmann constraint ⎧q =< y1 , z2 > ⎪ ⎨ 1 1 2 ⎪⎩ p =< y1 , z1 > − 2 < y1 , z2 > + 2 < Λy1 , z2 > (22) G11 ⎛aj ⎞ ⎛ ⎞ Gj = ⎜ ⎟ = ⎜ ⎟ , j = 0,1, 2,K , j ⎜b ⎟ ⎝ j ⎠ ⎝ < Λ y1 , z2 > ⎠ (23) where G11 =< Λ j y1 , z1 > −
1 1 < y1 , z2 >< Λ j y1 , z2 > + < Λ j +1 y1 , z2 > 2 2
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: ⎧⎪Yx = MY , Z x = − M T Z (24) ⎨ T ⎪⎩ Ytm = WmY Z tm = −Wm Z m = 0,1, 2,K where E ⎞ ⎛ M 11 M =⎜ ⎟ ⎝ M 21 M 22 ⎠ ⎛ Am Wm = ⎜ ⎜C ⎝ m
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Bm ⎞ ⎟ Dm ⎟⎠
1 1 < y1 , z2 > + Λ 2 2 1 2 3 1 1 = Λ + < y1 , z2 > 2 − < y1 , z1 > − < y2 , z2 > 4 4 2 2 1 1 − < Λy1 , z2 > − Λ < y1 , z2 > 2 2 1 1 = − < y1 , z2 > + Λ 2 2
M 11 = −
m 1 1 Am = ∑ [− < Λ j −1 y2 , z2 >]Λ m − j + Λ m +1 − < Λ m y1 , z2 > 2 2 j =1 m
Bm = ∑ [< Λ j −1 y1 , z2 >]Λ m − j + Λ m j =1
m 1 Cm = ∑ [< Λ j −1 y2 , z1 >]Λ m − j − (< y1 , z1 > + < y2 , z2 > ）Λ m 2 j =1
1 1 1 < y1 , z2 > Λ m +1 − < Λy1 , z2 > Λ m + Λ m + 2 4 4 4 1 1 − < Λ m +1 y1 , z2 > − < Λ m y1 , z2 > Λ 4 4 1 1 m + < y1 , z2 >< Λ y1 , z2 > + < y1 , z2 > 2 Λ m 2 4 m 1 1 Dm = ∑ [− < Λ j −1 y1 , z1 >]Λ m − j + Λ m +1 − < Λ m y1 , z2 > 2 2 j =1 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]: −
∂h ∂h ⎧ ⎪⎪Yx = ∂Z , Z x = − ∂Y ; ⎨ ⎪Y = ∂hm , Z = − ∂hm , m = 0,1, 2,K , t tm ∂Y ⎩⎪ m ∂Z
(25) where 1 1 1 hm = < Λ m +1 y1 , z1 > + < Λ m +1 y2 , z2 > + < Λ m + 2 y1 , z2 > 2 2 4 1 1 + < y1 , z2 > 2 < Λ m y1 , z2 > − < Λ m y1 , z2 >< Λy1 , z2 > 4 4 1 − < Λ m +1 y1 , z2 >< y1 , z2 > + < Λ m y2 , z1 > 4 1 − < Λ m y1 , z2 > (< y1 , z1 > + < y2 , z2 >) 2 m < Λ j −1 y , z > < Λ m − j y2 , z 2 > 1 2 +∑ j −1 y1 , z1 > < Λ m − j y2 , z1 > j =1 < Λ and h0 = h . Proof:
∂h 1 1 = y2 − < y1 , z2 > y1 + Λy1 2 2 ∂z1 = y1x
∂h 1 1 1 = Λy2 − < y1 , z2 > y2 − < y2 , z2 > y1 2 2 ∂z2 2
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1 3 1 < y1 , z1 > y1 + < y1 , z2 > 2 y1 + Λ 2 y1 2 4 4 1 1 − < Λy1 , z2 > y1 − < y1 , z2 > Λy1 2 2 = y2 x −
∂hm 1 m +1 1 = Λ y1 − < Λ m y1 , z2 > y1 + Λ m y2 2 ∂z1 2 m
+ ∑ [< Λ j −1 y1 , z2 > y2 − < Λ m − j y2 , z2 > y1 ]Λ m − j
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= − z1tm
∂hm 1 m +1 1 = Λ z2 + Λ m z1 − < Λ m y1 , z2 > z2 ∂y2 2 2 m
+ ∑ [< Λ j −1 y1 , z2 > z1 − < Λ j −1 y1 , z1 > z2 ]Λ m − j j =1
= − z 2 tm
so Zx = −
j =1
= y1tm
∂hm 1 m +1 1 1 = Λ y2 + Λ m + 2 y1 − < y1 , z1 > Λ m y1 4 2 ∂z2 2 1 1 < Λ m y1 , z2 > y2 − < y2 , z2 > Λ m y1 2 2 1 1 + < y1 , z2 > 2 Λ m y1 + < y1 , z2 >< Λ m y1 , z2 > y1 4 2 1 1 m − < Λ y1 , z2 > Λy1 − < Λy1 , z2 > Λ m y1 4 4 1 1 − < Λ m +1 y1 , z2 > y1 − < y1 , z2 > Λ m +1 y1 4 4 −
m
+ ∑ [< Λ j −1 y2 , z1 > y1 − < Λ j −1 y1 , z1 > y2 ]Λ m − j j =1
= y2 t m
so Yx =
∂h ∂h , Ytm = m ∂Z ∂Z
Similarly, we have ∂h 1 1 3 = Λz1 − < y2 , z2 > z2 + < y1 , z2 > 2 z2 ∂y1 2 2 4 1 1 1 + Λ 2 z2 − < y1 , z2 > z1 − < y1 , z1 > z2 4 2 2 1 1 − < Λy1 , z2 > z2 − < y1 , z2 > Λz2 2 2 = − z1x ∂h 1 1 = z1 − < y1 , z2 > z2 + Λz2 ∂y2 2 2 = − z2 x ∂hm 1 m +1 1 1 = Λ z1 − < Λ m y1 , z2 > z1 − < y1 , z1 > Λ m z2 ∂y1 2 2 2
1 1 < y1 , z2 > 2 Λ m z2 + < y1 , z2 >< Λ m y1 , z2 > z2 4 2 1 1 − < Λ m +1 y1 , z2 > z2 − < y1 , z2 > Λ m +1 z2 4 4 1 1 m − < Λ y1 , z2 > Λz2 − < Λy1 , z2 > Λ m z2 4 4 1 1 − < y2 , z 2 > Λ m z 2 + Λ m + 2 z 2 2 4 +
m
+ ∑ [< Λ j −1 y2 , z1 > z2 − < Λ j −1 y2 , z2 > z1 ]Λ m − j j =1
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∂h ∂h , Z tm = − m ∂Y ∂Y
V. LIOUVILLE COMPLETELY INTEGRABLE SYSTEMS Now we discuss the completely integrability for the Bargmann systems (25). Let 1 ⎧ (1) 1 ⎪ Ek = 2 λk y1k z1k + 2 λk y2 k z2 k ⎪ ⎪ E (2) = − 1 y z (< y , z > + < y , z >) + y z 1k 2 k 1 1 2 2 2 k 1k ⎪ k 2 ⎨ 1 1 ⎪ − λk y1k z2 k < y1 , z2 > + y1k z2 k < y1 , z2 > 2 ⎪ 4 4 ⎪ 1 2 1 ⎪ + λk y1k z2 k − y1k z2 k < Λy1 , z2 > −Γ k(1,2) ⎩ 4 4 where N y1k y1l z1k z1l 1 Γ (1,2) = ∑ k l =1, l ≠ k λk − λl y2 k y2 l z 2 k z 2 l Proposition 5.1: (1) (1) (2) (2) (2) i) {E (1) j , Ek } = 0,{E j , Ek } = 0,{E j , Ek } = 0 ∀j , k = 1, 2,K N
(26)
ii) {dE , j = 1, 2,K N ; l = 1, 2} are the linear independ(l ) j
ence. ∞ 1 − m −1 (2) hm ( E (1) j + Ej ) = ∑ μ j =1 μ − λ j m=0
N
ⅲ) H μ = ∑ N
(2) hm = ∑ λ jm ( E (1) j + Ej ) j =1
m = 0,1, 2,K
(27)
(28)
Theorem 5.2: The Bargmann [8] systems (25) are the completely integrable systems in the Liouville sense. i.e. {h, E (j l ) } = 0, l = 1, 2; j = 1, 2,K , N (29) {hm , E (j l ) } = 0, l = 1, 2; j = 1, 2,K , N {hm , hn } = 0, m, n = 0,1, 2,K {h, hm } = 0, m = 0,1, 2,K Proof: By (26) and (28), we have {hm , hn } = 0, m, n = 0,1, 2,K from (27), then {H λ , H μ } = 0
using h = h0 , we have {h, E (j l ) } = 0, l = 1, 2; j = 1, 2,K , N
(30) (31) (32)
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According to the above Theorem, the Hamiltonian phase flows g htnn and g htmm are commutable. Now, we arbitrarily choose an initial value ( yi (0, 0), zi (0, 0))T i = 1, 2 , let y (0, 0) ⎞ y (0, 0) ⎞ ⎛ yi (tm , tn ) ⎞ tm tn ⎛ i t n tm ⎛ i ⎜ ⎟ = g hm g hn ⎜ ⎟ = g hn g hm ⎜ ⎟ ⎝ zi (tm , tn ) ⎠ ⎝ zi (0, 0) ⎠ ⎝ zi (0, 0) ⎠ From (8), (9) and Theorem 4.2, the following theorem holds. Theorem 5.3: Suppose ( y1 , y2 , z1 , z2 ) is an involutive solution of the Hamiltonian [11] canonical equation systems (25), then ⎧q =< y1 , z2 > ⎪ ⎨ 1 1 2 ⎪⎩ p =< y1 , z1 > − 2 < y1 , z2 > + 2 < Λy1 , z2 > 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].
Figure3. Interaction of two solitary waves at different times
Figure3. Interaction of two solitary waves at different times
Especially, if ( y1 , y2 , z1 , z2 ) satisfies ⎧⎛ y1 ⎞ ⎛ y1 ⎞ ⎪⎜ ⎟ = M ⎜ ⎟ ⎝ y2 ⎠ ⎪⎝ y2 ⎠ x ⎪⎛ z ⎞ z ⎪⎜ 1 ⎟ = − M T ⎛⎜ 1 ⎞⎟ ⎪⎪⎝ z2 ⎠ x ⎝ z2 ⎠ ⎨ ⎪⎛ y1 ⎞ = W ⎛ y1 ⎞ ⎟ 1⎜ ⎪⎜⎝ y2 ⎟⎠ ⎝ y2 ⎠ t1 ⎪ ⎪⎛ z1 ⎞ z T ⎛ 1 ⎞ ⎪⎜ ⎟ = −W1 ⎜ ⎟ z ⎪⎝ ⎝ z2 ⎠ ⎩ 2 ⎠t1
where 1 ⎛1 ⎞ E ⎜ 2 Λ − 2 < y1 , z2 > ⎟ ⎟ M =⎜ 1 1 ⎜ M 21 Λ − < y1 , z2 > ⎟⎟ ⎜ ⎝ ⎠ 2 2 1 2 3 1 1 M 21 = Λ + < y1 , z2 > 2 − < y1 , z1 > − < y2 , z2 > 4 4 2 2 ⎛ A1 B1 ⎞ W1 = ⎜ ⎜ C D ⎟⎟ 1⎠ ⎝ 1 1 1 A1 = Λ 2 − < Λy1 , z2 > − < y2 , z2 > 2 2 B1 = Λ+ < y1 , z2 > 1 C1 =< y2 , z1 > − (< y1 , z1 > + < y2 , z2 > ）Λ 2
1 1 1 < y1 , z2 > Λ 2 − < Λy1 , z2 > Λ + Λ 3 4 4 4 1 1 − < Λ 2 y1 , z2 > − < Λy1 , z2 > Λ 4 4 1 1 + < y1 , z2 >< Λy1 , z2 > + < y1 , z2 > 2 Λ 2 4 1 2 1 D1 = Λ − < Λy1 , z2 > − < y1 , z1 > 2 2 then −
Figure3. Interaction of two solitary waves at different times
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⎧q =< y1 , z2 > ⎪ ⎨ 1 1 2 ⎪⎩ p =< y1 , z1 > − 2 < y1 , z2 > + 2 < Λy1 , z2 > satisfies long-wave equation ⎛ qt1 ⎞ ⎛ 2 px + qxx + 2qqx ⎞ ⎜⎜ ⎟⎟ = ⎜ ⎟ ⎝ pt1 ⎠ ⎝ − pxx + 2(qp) x ⎠
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[14]
[15]
[16]
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). REFERENCES [1] M. J. Ablowitz and P. A. Clarkson, Solitons, Evolution Equations and Inverse Scattering, Cambridge University Press, 1991. [2] D. Y. Chen, Soliton Theory, Science Press, 2006. [3] V. I. Arnold, Mathematical Methods of Classical Mechanics, 2nd ed., Spring-Verlag, New York Berlin Heidelberg, 1999. [4] C. W. Cao, “A classical integrable system and involutive representation of solutions of the KdV equation,” Acta Math. Sinia, 1991, pp. 436–440. [5] D. S. Wang, “Integrability of a coupled KdV system: Painleve property, Lax pair and Bäcklund transformation,” Appl. Math. and Comp., vol. 216, 2010, pp. 1349–1354. [6] Y. H. Cao and D. S. Wang, “Prolongation structures of a generalized coupled Korteweg-de Vries equation and Miura transformation,” Commun. in Nonl. Sci. and Num. Simul., vol.15, 2010, pp.2344–2349. [7] D. S. Wang, “Complete integrability and the Miura transformation of a coupled KdV equation,” Appl. Math. Lett., vol. 23, 2010, pp. 65–669. [8] X. Zeng and D. S. Wang, “A generalized extended rational expansion method and its application to (1+1)-dimensional dispersive long wave equation,” Appl. Math. Comp., vol. 212, 2009, pp. 296–304. [9] Y. T. Wu and X. G. Geng, “A finite-dimensional integrable system associated with the three-wave interaction equations,” J. Math. Phys., vol. 40, 1999, pp. 3409–3430. [10] C. W. Cao, Y. T. Wu, and X. G. Geng, “Relation between the Kadometsev-Petviashvili equation and the confocal involution system,” J. Math. Phys., vol. 40, 1999, pp. 3948–3970. [11] Z. Q. Gu, “The Neumann system for 3rd-order eigenvalue problems related to the Boussinesq equation. IL Nuovo Cimento, vol. 117B(6),” 2002, pp. 615–632. [12] W. X. Ma and R. G. Zhou, “Binary nonlinearization of spectral problems of the perturbation AKNS systems,” Chaos Solitons &Fractals 13, 2002, pp. 1451–1463. [13] X. G. Geng and D. L. Du, “Two hierarchies of new nonlinear evolution equations associated with 3 × 3 matrix
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spectral problems,” Chaos Solitons &Fractals 29, 2006, pp. 1165–1172. X. G. Geng and H. H. Dai, “A hierarchy of new nonlinear differential-difference equations”. Journal of the Physical Socoety of Japan 75, vol. 1, 2006. J. S. He, J. Yu, Y. Cheng, and R. G. Zhou, “Binary Nonlinearization of the Super AKNS System,” Mod. Phys. Lett. B, vol.22, 2008, pp. 275-288. Z. Q. Gu and J. X. Zhang, “A new constrained flow for Boussinesq-Burgers’ hierarchy,” IL Nuovo Cimento, vol. 122B(8), 2007, pp. 871–884. W. Liu, S. J. Yuan, and Q. Wang, “The second-order spectral problem with the speed energy and its completely integrable system”, Journal of Shijiazhuang Railway Institute, vol. 22(1), 2009. Z. Q. Gu, J. X. Zhang, and W. Liu, “Two new completely integrable systems related to the KdV equation hierarchy,” IL Nuovo Cimento, vol. 123B(5), 2008, pp. 605–622. O. J. Peter, Applications of Lie Groups to Differential Equations, 2nd ed., Spring-Verlag, New York Berlin Heidelberg, 1999, pp. 13–53, 452–462. B. Q. Xia and R. G. Zhou, “Integrable deformations of integrable symplectic maps,” Phys. Lett. A, vol.373, 2009, pp.1121–1129. Y. Q. Yao, Y. H. Huang, Y. Wei, and Y. B. Zeng, “Some new integrable systems constructed from the biHamiltonian systems with pure differential Hamiltonian operators,” Applied Mathematics and Computation, vol. 218, 2011, pp. 583-591. Y. Keskin and G. Oturanc, “Numerical solution of regularized long wave equation by reduced differential transform method,” Applied Mathematical Sciences, vol. 4, 2010, pp. 1221-1231. J. X. Cai, “A multisymplectic explicit scheme for the modified regularized long-wave equation,” Journal of Computational and Applied Mathematics 234, 2010, pp. 899-905. J. X. Cai, “Multisymplectic numerical method for the regularized long-wave equation,” Computer Physics Communications 180, 2009, pp. 1821-1831.
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. Shuhong Wang, native of Liaoning, was born in January 1980. In 2006, master of science in Applied Mathematics at Hebei University of Technology, Tianjin, China. She mainly engaged in domain of differential equations, Inequality and so on.
JOURNAL OF COMPUTERS, VOL. 7, NO. 9, SEPTEMBER 2012
Nonlinear Evolution Equations for Second-order Spectral Problem 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]
Shuhong Wang College of Mathematics, Inner Mongolia University for Nationalities, TongLiao, China Email: [email protected]
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.
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).
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: ∂η 3 g ∂ 1 2 2 1 ∂ 2η = ( η + αη + σ 2 ) ∂τ 2 h ∂ξ 2 3 3 ∂ξ 1 3 Th σ= h − 3 ρg By making the transformation 1 g 1 1 t= τ , x = −σ 1 2ξ , u = η + α 2 hσ 2 3 the famous KdV equation ut + 6uu x + u xxx = 0 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 © 2012 ACADEMY PUBLISHER doi:10.4304/jcp.7.9.2144-2151
Figure1. One-soliton solution of the KdV equation
In our paper, by nonlinearization [8-15] of spectral problems, we considered the spectral Lϕ = (∂ 2 + ∂q + p)ϕ = λϕ x The paper is structured as follows. In Sect.2, the adjoint Lax pairs of the spectral problem is generalized. In Sect.3, based on the Euler-Lagrange equations and Legendre transformations, a suitable Jacobi-Ostrogradsky coordinate is been found. Section 4 and Sect.5 are devoted to establishing the Liouville integrability of the resulting Hamiltonian systems from the 2nd-order spectral problems.
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⎛0 ∂⎞ J =⎜ ⎟ ⎝∂ 0⎠ ⎛ 2∂ K = ⎜⎜ 2 ⎝ q∂ − ∂
(6)
∂ 2 + ∂q ⎞ ⎟ p∂ + ∂p ⎠⎟
(7)
⎛ aj ⎞ Gj = ⎜ ⎟ ⎜b ⎟ ⎝ j⎠
TABLE I. THE LENARD SEQUENCE
Figure2. Two-soliton solution of the KdV equation
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, then Ω = (−∞, +∞) ; If they are all periodic T functions, then Ω = [0, 2T ] . Definition 2.1 Assume that our linear space is equipped with a L2 scalar product (⋅, ⋅) L2 ( Ω ) : (ϕ , h) L2 ( Ω ) = ∫ ϕ h* dx < ∞ Ω
symbol * denotes the complex conjugate. Definition 2.2 Operator A is an adjoint operator of A , if ( Aϕ , h) L2 ( Ω ) = (ϕ , Ah) L2 ( 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 λω x + [ω , L] = λω x + ω L − Lω (3) Take
ω = ∑ [−a j −1 − b j −1x + b j −1∂ ] λ
−j
j =0
and set ⎛ a−1 ⎞ ⎛ 0 ⎞ G−1 = ⎜ ⎟ = ⎜ ⎟ ⎝ b−1 ⎠ ⎝1 ⎠ therefore, we obtain the recursive relation KG j −1 = JG j , j = 0,1, 2K ,
where
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aj
bj
j = −1 j=0
0 p
1 q
j =1
− px + 2qp
2 p + qx + q 2
L
II. LAX PAIRS AND EVOLUTION EQUATIONS
∞
Gj
Now, we consider the auxiliary spectral problem ϕtm = ωmϕ m = 0,1, 2,K
(8)
with m
ωm = ∑ [− a j −1 − b j −1x + b j −1∂ ] λ m − j
(9)
j =0
Then, the isospectral (i.e. λtm = 0) compatible condition Ltm = λωmx + [ωm , L] = λωmx + ωm L − Lωm
(10)
of the Lax pairs ⎧⎪ Lϕ = λϕ x ⎨ ⎪⎩ϕtm = ωmϕ m = 0,1, 2,K determines a (m + 1) -order long-wave equation
(11)
⎛ qtm ⎞ (12) ⎜⎜ ⎟⎟ = JGm = KGm −1 m = 0,1, 2,K ⎝ ptm ⎠ For example, when m = 1 and 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 = 2 and q ≡ 0 , it is exactly the famous KdV equations. TABLE II. THE FIRST AND THE SECOND NONLINEAR SYSTEM Evolution equation
m =1
m = 2 and q ≡ 0
qtm
2 px + qxx + 2qqx
0
ptm
− pxx + 2(qp) x
pt2 = pxxx + 3( p 2 ) x
(4)
(5)
In order to give the constraints between the potentials and the eigenfunctions, it is necessary to calculate the functional gradient with respect to the potential functions. Proposition 2.3: [11] i) If λ is an eigenvalue of (1), then λ is real.
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ii) If ϕ is an eigenfunction of (1) and ψ is an eigenfunction of (2), then ϕ and ψ can be taken real functions. iii) If ϕ is an eigenfunction corresponding to the eigenvalue λ of (1) and ψ is an eigenfunction corresponding to the eigenvalue λ of (2), then ⎛ δλ ⎞ ⎜δq ⎟ −ϕψ x ⎞ ⎟ = ( ϕ xψ dx) −1 ⎛⎜ ∇λ = ⎜ ⎟ ∫ Ω ⎜ δλ ⎟ ⎝ ϕψ ⎠ ⎜ ⎟ ⎝δ p ⎠ and K ∇λ = λ J ∇ λ (13) Proof: In fact, from (1) and (2), we have λ ∫ ϕ xψ dx = ∫ Lϕ (ψ * )* dx Ω
Ω
= ∫ ϕ ( Lψ ∗ )* dx Ω
= ∫ ϕ ( L∗ψ ∗ )* dx Ω
= −λ * ∫ ϕψ x dx Ω
=λ
*
∫
Ω
ϕ xψ dx
⎛ δλ ⎞ ⎜δq ⎟ −ϕψ x ⎞ ⎟ = ( ϕ xψ dx) −1 ⎛⎜ ∇λ = ⎜ ⎟ ∫ Ω ⎜ δλ ⎟ ⎝ ϕψ ⎠ ⎜ ⎟ ⎝δ p ⎠ by (6) and (7), (13) holds.
III. BARGMANN SYSTEMS AND THE HAMILTON CANONICAL FORMS
We suppose λ1 < λ2 < K < λN are the eigenvalues of the eigenvalue problem (1) and (2), ϕ j ,ψ j are the eigenfunction for λ j ( j = 1, 2,L , N ). Let Λ = diag (λ1 , λ2 ,L λN ) , Φ = (ϕ1 , ϕ2 ,K , ϕ N )T , Ψ = (ψ 1 ,ψ 2 ,K ,ψ N )T Now, we consider the Bargmann constraint [16-18] ⎧q =< Φ, Ψ > (14) ⎨ ⎩ p = − < Φ, Ψ x > here N
< Φ, Ψ >= ∑ ϕ jψ j
so
k =1
λ =λ If ϕ is a complex eigenfunction of (1) on λ , and ϕ = a + ib , a, b are real functions, from Lϕ = λϕ x and λ is real, then ⎧ La = λ ax ⎨ ⎩ Lb = λ bx *
so a, b are eigenfunction of (1) on λ , ϕ can be taken real function. Similarly, ψ can be taken real function. Let d h = h(λ + εδλ , q + εδ q, p + εδ p ) , d ε ε =0 by Lϕ = λϕ x ( Lϕ ) = (λϕ x ) L ϕ + Lϕ = λ ϕ x + λϕ x
and
∫
Ω
( Lϕ )ψ dx = ∫ ϕ ( Lψ )dx Ω
= ∫ ϕ (−λψ x )dx Ω
= ∫ λϕ xψ dx Ω
so
∫
Ω
λ ϕ xψ dx = ∫ L ϕψ dx Ω
= ∫ ((δ qϕ ) x + δ pϕ )ψ dx Ω
= ∫ (−δ qϕψ x + δ pϕψ )dx Ω
then
namely ⎛ p ⎞ ⎛ − < Φ, Ψ x > ⎞ G0 = ⎜ ⎟ = ⎜ ⎟ ⎝ q ⎠ ⎝ < Φ, Ψ > ⎠ ⎛ < −Λ j Φ, Ψ x > ⎞ j = 0,1, 2L (15) Gj = ⎜ ⎜ < Λ j Φ, Ψ > ⎟⎟ ⎝ ⎠ so the eigenvalue problem (1) and (2) are equivalent to the systems ⎧Φ xx + (< Φ, Ψ > Φ ) x − < Φ, Ψ x > Φ = ΛΦ x (16) ⎨ ⎩Ψ xx − < Φ, Ψ > Ψ x − < Φ, Ψ x > Ψ = −ΛΨ x
and (16) are called the Bargmann systems for the eigenvalue problem (1) and (2). Let Iˆ = ∫ Idx (17) Ω
where the Lagrange function I is defined as follows: 1 1 I =< Φ x , Ψ x > + < Φ, Ψ >< Φ, Ψ x > + < ΛΦ x , Ψ > 2 2 1 1 − < ΛΦ, Ψ x > − < Φ, Ψ >< Φ x , Ψ > 2 2 Proposition 3.1: The Bargmann systems (16) of the eigenvalue problem (1) and (2) are equivalent to the Euler-Lagrange equation systems: ⎧ δ Iˆ =0 ⎪ ⎪ δΦ (18) ⎨ ⎪ δ Iˆ = 0 ⎪⎩ δΨ Proof: By (17), we have δ Iˆ = ψ jxx − < Φ, Ψ x > ψ j − < Φ, Ψ > ψ jx + λψ jx
δϕ j
=< Φ, Ψ x > ψ j + < Φ, Ψ > ψ jx − λψ jx © 2012 ACADEMY PUBLISHER
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− < Φ, Ψ x > ψ j − < Φ, Ψ > ψ jx + λψ jx =0 similarly, δ Iˆ = −ϕ jxx − (< Φ, Ψ > ϕ j ) x + < Φ, Ψ x > ϕ jx + λϕ jx
Then the Bargmann systems (16) are equivalent to the following Hamilton canonical systems: ∂g ⎧ ⎪u jx = ∂v j ⎪ j = 1, 2 ⎨ ⎪v = − ∂g ⎪ jx ∂u j ⎩ where 1 1 g =< v1 , v2 > + < u1 , u2 >< u2 , v2 > − < λ u2 , v2 > 2 2 1 1 1 − < u1 , u2 >< u1 , v1 > − < u1 , u2 >3 + < λ u1 , v1 > 2 4 2 1 1 + < u1 , u2 >< λ u1 , u2 > − < λ 2 u1 , u2 > 2 4 By (19), the Jacobi-Ostrogradsky coordinates can be written in the following form: ⎧ y1 = Φ, ⎪ ⎪ y = Φ x + 1 qΦ − 1 ΛΦ ⎪ 2 2 2 (20) ⎨ = Ψ z ⎪ 2 ⎪ 1 1 ⎪ z1 = −Ψ x + qΨ − ΛΨ ⎩ 2 2 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: ∂h ⎧ ⎪ y jx = ∂z j ⎪ j = 1, 2 ⎨ ∂ ⎪z = − h ⎪ jx ∂y j ⎩ where 1 1 h =< y2 , z1 > − < y1 , z2 >< y2 , z2 > + < Λy2 , z2 > 2 2 1 1 1 − < y1 , z2 >< y1 , z1 > + < y1 , z2 >3 + < Λy1 , z1 > 2 4 2 1 1 − < Λy1 , z2 >< y1 , z2 > + < Λ 2 y1 , z2 > 2 4 and h = − g .
δψ j
= (< Φ, Ψ > ϕ j ) x − < Φ, Ψ x > ϕ jx − λϕ jx
−(< Φ, Ψ > ϕ j ) x + < Φ, Ψ x > ϕ jx + λϕ jx =0 Now, the Poisson bracket [19] of the real-valued functions F and H in the symplectic [20] space 2
( w = ∑ dz j ∧ dy j , R 4 N ) is defined as follows: j =1
2
N
{F , H } = ∑∑ ( j =1 k =1
∂F ∂H ∂F ∂H − ) ∂y jk ∂z jk ∂z jk ∂y jk
2
= ∑ (< Fyj , H zj > − < Fzj , H yj >) j =1
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
u1 = Φ, u2 = Ψ , g = ∑ < u jx , v j > − I j =1
Our aim is to find that the coordinates {v1 , v2 } and g satisfy the following Hamilton canonical equations: ∂g ⎧ ⎪u jx = {u j , g} = ∂v j ⎪ j = 1, 2 ⎨ ∂ ⎪v = {v , g} = − g j ⎪ jx ∂u j ⎩ By directly computing, we have 1 1 ⎧ ⎪v1 = Ψ x − 2 qΨ + 2 ΛΨ ⎪ ⎪v = Φ + 1 qΦ − 1 ΛΦ x ⎪ 2 2 2 ⎪ 1 ⎪v = < Φ , Ψ > Ψ + 1 < Φ , Ψ > Ψ x x ⎪ 1x 2 2 ⎨ 1 1 ⎪ − < Φ x , Ψ > Ψ − ΛΨ x ⎪ 2 2 ⎪ 1 1 ⎪v2 x = < Φ, Ψ x > Φ − < Φ x , Ψ > Φ 2 2 ⎪ ⎪ 1 1 − < Φ, Ψ > Φ x + ΛΦ x ⎪ ⎩ 2 2 So, if we take the Jacobi-Ostrogradsky coordinates as follows: ⎧u1 = Φ ⎪u = Ψ ⎪ 2 ⎪ 1 1 (19) ⎨v1 = Ψ x − qΨ + ΛΨ 2 2 ⎪ ⎪ 1 1 ⎪v2 = Φ x + qΦ − ΛΦ 2 2 ⎩
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IV.
NONLINEARIZATION OF THE LAX PAIRS
Form (20) and Theorem 3.2, the Bargmann systems (16) have the equivalence form ⎧⎪Yx = MY ⎨ T ⎪⎩ Z x = − M Z where Φ ⎛ ⎞ ⎛ y1 ⎞ ⎜ ⎟ Y =⎜ ⎟= 1 1 ⎝ y2 ⎠ ⎜⎜ Φ x + qΦ − ΛΦ ⎟⎟ ⎝ ⎠ 2 2 1 1 ⎛ ⎞ −Ψ x + qΨ − ΛΨ ⎟ ⎛ z1 ⎞ Z =⎜ ⎟=⎜ 2 2 ⎟⎟ ⎝ z2 ⎠ ⎜⎜ Ψ ⎝ ⎠
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⎛ M 11 M =⎜ ⎝ M 21
E ⎞ ⎟ M 22 ⎠
where
M 21
1 1 M 11 = − qE + Λ 2 2 1 2 1 1 1 M 21 = q E − pE − qx E − Λq + Λ 2 4 2 2 4 1 1 M 22 = − qE + Λ 2 2 E = EN × N = diag (1,1,K ,1)
M 22
Proposition 4.1: The Lax pairs (11) for the (m + 1) order evolution equation (12) is equivalent to ⎧⎪Yx = MY , Z x = − M T Z ; (21) ⎨ T ⎪⎩Ytm = WmY , Z tm = −Wm Z , m = 0,1, 2K , where m ⎛ Am Bm ⎞ m − j Wm = ∑ ⎜ ⎟Λ Dm ⎠ j = 0 ⎝ Cm 1 1 Am = −a j −1 − qb j −1 − b j −1x + Λb j −1 2 2 Bm = b j −1 1 1 1 1 Cm = − b j −1xx − qx b j −1 − pb j −1 + q 2 b j −1 − Λqb j −1 2 2 4 2 1 2 + Λ b j −1 4 1 1 Dm = −a j −1 − qb j −1 + Λb j −1 2 2 By (14) and (20), we have the Bargmann constraint ⎧q =< y1 , z2 > ⎪ ⎨ 1 1 2 ⎪⎩ p =< y1 , z1 > − 2 < y1 , z2 > + 2 < Λy1 , z2 > (22) G11 ⎛aj ⎞ ⎛ ⎞ Gj = ⎜ ⎟ = ⎜ ⎟ , j = 0,1, 2,K , j ⎜b ⎟ ⎝ j ⎠ ⎝ < Λ y1 , z2 > ⎠ (23) where G11 =< Λ j y1 , z1 > −
1 1 < y1 , z2 >< Λ j y1 , z2 > + < Λ j +1 y1 , z2 > 2 2
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: ⎧⎪Yx = MY , Z x = − M T Z (24) ⎨ T ⎪⎩ Ytm = WmY Z tm = −Wm Z m = 0,1, 2,K where E ⎞ ⎛ M 11 M =⎜ ⎟ ⎝ M 21 M 22 ⎠ ⎛ Am Wm = ⎜ ⎜C ⎝ m
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Bm ⎞ ⎟ Dm ⎟⎠
1 1 < y1 , z2 > + Λ 2 2 1 2 3 1 1 = Λ + < y1 , z2 > 2 − < y1 , z1 > − < y2 , z2 > 4 4 2 2 1 1 − < Λy1 , z2 > − Λ < y1 , z2 > 2 2 1 1 = − < y1 , z2 > + Λ 2 2
M 11 = −
m 1 1 Am = ∑ [− < Λ j −1 y2 , z2 >]Λ m − j + Λ m +1 − < Λ m y1 , z2 > 2 2 j =1 m
Bm = ∑ [< Λ j −1 y1 , z2 >]Λ m − j + Λ m j =1
m 1 Cm = ∑ [< Λ j −1 y2 , z1 >]Λ m − j − (< y1 , z1 > + < y2 , z2 > ）Λ m 2 j =1
1 1 1 < y1 , z2 > Λ m +1 − < Λy1 , z2 > Λ m + Λ m + 2 4 4 4 1 1 − < Λ m +1 y1 , z2 > − < Λ m y1 , z2 > Λ 4 4 1 1 m + < y1 , z2 >< Λ y1 , z2 > + < y1 , z2 > 2 Λ m 2 4 m 1 1 Dm = ∑ [− < Λ j −1 y1 , z1 >]Λ m − j + Λ m +1 − < Λ m y1 , z2 > 2 2 j =1 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]: −
∂h ∂h ⎧ ⎪⎪Yx = ∂Z , Z x = − ∂Y ; ⎨ ⎪Y = ∂hm , Z = − ∂hm , m = 0,1, 2,K , t tm ∂Y ⎩⎪ m ∂Z
(25) where 1 1 1 hm = < Λ m +1 y1 , z1 > + < Λ m +1 y2 , z2 > + < Λ m + 2 y1 , z2 > 2 2 4 1 1 + < y1 , z2 > 2 < Λ m y1 , z2 > − < Λ m y1 , z2 >< Λy1 , z2 > 4 4 1 − < Λ m +1 y1 , z2 >< y1 , z2 > + < Λ m y2 , z1 > 4 1 − < Λ m y1 , z2 > (< y1 , z1 > + < y2 , z2 >) 2 m < Λ j −1 y , z > < Λ m − j y2 , z 2 > 1 2 +∑ j −1 y1 , z1 > < Λ m − j y2 , z1 > j =1 < Λ and h0 = h . Proof:
∂h 1 1 = y2 − < y1 , z2 > y1 + Λy1 2 2 ∂z1 = y1x
∂h 1 1 1 = Λy2 − < y1 , z2 > y2 − < y2 , z2 > y1 2 2 ∂z2 2
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1 3 1 < y1 , z1 > y1 + < y1 , z2 > 2 y1 + Λ 2 y1 2 4 4 1 1 − < Λy1 , z2 > y1 − < y1 , z2 > Λy1 2 2 = y2 x −
∂hm 1 m +1 1 = Λ y1 − < Λ m y1 , z2 > y1 + Λ m y2 2 ∂z1 2 m
+ ∑ [< Λ j −1 y1 , z2 > y2 − < Λ m − j y2 , z2 > y1 ]Λ m − j
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= − z1tm
∂hm 1 m +1 1 = Λ z2 + Λ m z1 − < Λ m y1 , z2 > z2 ∂y2 2 2 m
+ ∑ [< Λ j −1 y1 , z2 > z1 − < Λ j −1 y1 , z1 > z2 ]Λ m − j j =1
= − z 2 tm
so Zx = −
j =1
= y1tm
∂hm 1 m +1 1 1 = Λ y2 + Λ m + 2 y1 − < y1 , z1 > Λ m y1 4 2 ∂z2 2 1 1 < Λ m y1 , z2 > y2 − < y2 , z2 > Λ m y1 2 2 1 1 + < y1 , z2 > 2 Λ m y1 + < y1 , z2 >< Λ m y1 , z2 > y1 4 2 1 1 m − < Λ y1 , z2 > Λy1 − < Λy1 , z2 > Λ m y1 4 4 1 1 − < Λ m +1 y1 , z2 > y1 − < y1 , z2 > Λ m +1 y1 4 4 −
m
+ ∑ [< Λ j −1 y2 , z1 > y1 − < Λ j −1 y1 , z1 > y2 ]Λ m − j j =1
= y2 t m
so Yx =
∂h ∂h , Ytm = m ∂Z ∂Z
Similarly, we have ∂h 1 1 3 = Λz1 − < y2 , z2 > z2 + < y1 , z2 > 2 z2 ∂y1 2 2 4 1 1 1 + Λ 2 z2 − < y1 , z2 > z1 − < y1 , z1 > z2 4 2 2 1 1 − < Λy1 , z2 > z2 − < y1 , z2 > Λz2 2 2 = − z1x ∂h 1 1 = z1 − < y1 , z2 > z2 + Λz2 ∂y2 2 2 = − z2 x ∂hm 1 m +1 1 1 = Λ z1 − < Λ m y1 , z2 > z1 − < y1 , z1 > Λ m z2 ∂y1 2 2 2
1 1 < y1 , z2 > 2 Λ m z2 + < y1 , z2 >< Λ m y1 , z2 > z2 4 2 1 1 − < Λ m +1 y1 , z2 > z2 − < y1 , z2 > Λ m +1 z2 4 4 1 1 m − < Λ y1 , z2 > Λz2 − < Λy1 , z2 > Λ m z2 4 4 1 1 − < y2 , z 2 > Λ m z 2 + Λ m + 2 z 2 2 4 +
m
+ ∑ [< Λ j −1 y2 , z1 > z2 − < Λ j −1 y2 , z2 > z1 ]Λ m − j j =1
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∂h ∂h , Z tm = − m ∂Y ∂Y
V. LIOUVILLE COMPLETELY INTEGRABLE SYSTEMS Now we discuss the completely integrability for the Bargmann systems (25). Let 1 ⎧ (1) 1 ⎪ Ek = 2 λk y1k z1k + 2 λk y2 k z2 k ⎪ ⎪ E (2) = − 1 y z (< y , z > + < y , z >) + y z 1k 2 k 1 1 2 2 2 k 1k ⎪ k 2 ⎨ 1 1 ⎪ − λk y1k z2 k < y1 , z2 > + y1k z2 k < y1 , z2 > 2 ⎪ 4 4 ⎪ 1 2 1 ⎪ + λk y1k z2 k − y1k z2 k < Λy1 , z2 > −Γ k(1,2) ⎩ 4 4 where N y1k y1l z1k z1l 1 Γ (1,2) = ∑ k l =1, l ≠ k λk − λl y2 k y2 l z 2 k z 2 l Proposition 5.1: (1) (1) (2) (2) (2) i) {E (1) j , Ek } = 0,{E j , Ek } = 0,{E j , Ek } = 0 ∀j , k = 1, 2,K N
(26)
ii) {dE , j = 1, 2,K N ; l = 1, 2} are the linear independ(l ) j
ence. ∞ 1 − m −1 (2) hm ( E (1) j + Ej ) = ∑ μ j =1 μ − λ j m=0
N
ⅲ) H μ = ∑ N
(2) hm = ∑ λ jm ( E (1) j + Ej ) j =1
m = 0,1, 2,K
(27)
(28)
Theorem 5.2: The Bargmann [8] systems (25) are the completely integrable systems in the Liouville sense. i.e. {h, E (j l ) } = 0, l = 1, 2; j = 1, 2,K , N (29) {hm , E (j l ) } = 0, l = 1, 2; j = 1, 2,K , N {hm , hn } = 0, m, n = 0,1, 2,K {h, hm } = 0, m = 0,1, 2,K Proof: By (26) and (28), we have {hm , hn } = 0, m, n = 0,1, 2,K from (27), then {H λ , H μ } = 0
using h = h0 , we have {h, E (j l ) } = 0, l = 1, 2; j = 1, 2,K , N
(30) (31) (32)
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According to the above Theorem, the Hamiltonian phase flows g htnn and g htmm are commutable. Now, we arbitrarily choose an initial value ( yi (0, 0), zi (0, 0))T i = 1, 2 , let y (0, 0) ⎞ y (0, 0) ⎞ ⎛ yi (tm , tn ) ⎞ tm tn ⎛ i t n tm ⎛ i ⎜ ⎟ = g hm g hn ⎜ ⎟ = g hn g hm ⎜ ⎟ ⎝ zi (tm , tn ) ⎠ ⎝ zi (0, 0) ⎠ ⎝ zi (0, 0) ⎠ From (8), (9) and Theorem 4.2, the following theorem holds. Theorem 5.3: Suppose ( y1 , y2 , z1 , z2 ) is an involutive solution of the Hamiltonian [11] canonical equation systems (25), then ⎧q =< y1 , z2 > ⎪ ⎨ 1 1 2 ⎪⎩ p =< y1 , z1 > − 2 < y1 , z2 > + 2 < Λy1 , z2 > 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].
Figure3. Interaction of two solitary waves at different times
Figure3. Interaction of two solitary waves at different times
Especially, if ( y1 , y2 , z1 , z2 ) satisfies ⎧⎛ y1 ⎞ ⎛ y1 ⎞ ⎪⎜ ⎟ = M ⎜ ⎟ ⎝ y2 ⎠ ⎪⎝ y2 ⎠ x ⎪⎛ z ⎞ z ⎪⎜ 1 ⎟ = − M T ⎛⎜ 1 ⎞⎟ ⎪⎪⎝ z2 ⎠ x ⎝ z2 ⎠ ⎨ ⎪⎛ y1 ⎞ = W ⎛ y1 ⎞ ⎟ 1⎜ ⎪⎜⎝ y2 ⎟⎠ ⎝ y2 ⎠ t1 ⎪ ⎪⎛ z1 ⎞ z T ⎛ 1 ⎞ ⎪⎜ ⎟ = −W1 ⎜ ⎟ z ⎪⎝ ⎝ z2 ⎠ ⎩ 2 ⎠t1
where 1 ⎛1 ⎞ E ⎜ 2 Λ − 2 < y1 , z2 > ⎟ ⎟ M =⎜ 1 1 ⎜ M 21 Λ − < y1 , z2 > ⎟⎟ ⎜ ⎝ ⎠ 2 2 1 2 3 1 1 M 21 = Λ + < y1 , z2 > 2 − < y1 , z1 > − < y2 , z2 > 4 4 2 2 ⎛ A1 B1 ⎞ W1 = ⎜ ⎜ C D ⎟⎟ 1⎠ ⎝ 1 1 1 A1 = Λ 2 − < Λy1 , z2 > − < y2 , z2 > 2 2 B1 = Λ+ < y1 , z2 > 1 C1 =< y2 , z1 > − (< y1 , z1 > + < y2 , z2 > ）Λ 2
1 1 1 < y1 , z2 > Λ 2 − < Λy1 , z2 > Λ + Λ 3 4 4 4 1 1 − < Λ 2 y1 , z2 > − < Λy1 , z2 > Λ 4 4 1 1 + < y1 , z2 >< Λy1 , z2 > + < y1 , z2 > 2 Λ 2 4 1 2 1 D1 = Λ − < Λy1 , z2 > − < y1 , z1 > 2 2 then −
Figure3. Interaction of two solitary waves at different times
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JOURNAL OF COMPUTERS, VOL. 7, NO. 9, SEPTEMBER 2012
⎧q =< y1 , z2 > ⎪ ⎨ 1 1 2 ⎪⎩ p =< y1 , z1 > − 2 < y1 , z2 > + 2 < Λy1 , z2 > satisfies long-wave equation ⎛ qt1 ⎞ ⎛ 2 px + qxx + 2qqx ⎞ ⎜⎜ ⎟⎟ = ⎜ ⎟ ⎝ pt1 ⎠ ⎝ − pxx + 2(qp) x ⎠
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[14]
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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). REFERENCES [1] M. J. Ablowitz and P. A. Clarkson, Solitons, Evolution Equations and Inverse Scattering, Cambridge University Press, 1991. [2] D. Y. Chen, Soliton Theory, Science Press, 2006. [3] V. I. Arnold, Mathematical Methods of Classical Mechanics, 2nd ed., Spring-Verlag, New York Berlin Heidelberg, 1999. [4] C. W. Cao, “A classical integrable system and involutive representation of solutions of the KdV equation,” Acta Math. Sinia, 1991, pp. 436–440. [5] D. S. Wang, “Integrability of a coupled KdV system: Painleve property, Lax pair and Bäcklund transformation,” Appl. Math. and Comp., vol. 216, 2010, pp. 1349–1354. [6] Y. H. Cao and D. S. Wang, “Prolongation structures of a generalized coupled Korteweg-de Vries equation and Miura transformation,” Commun. in Nonl. Sci. and Num. Simul., vol.15, 2010, pp.2344–2349. [7] D. S. Wang, “Complete integrability and the Miura transformation of a coupled KdV equation,” Appl. Math. Lett., vol. 23, 2010, pp. 65–669. [8] X. Zeng and D. S. Wang, “A generalized extended rational expansion method and its application to (1+1)-dimensional dispersive long wave equation,” Appl. Math. Comp., vol. 212, 2009, pp. 296–304. [9] Y. T. Wu and X. G. Geng, “A finite-dimensional integrable system associated with the three-wave interaction equations,” J. Math. Phys., vol. 40, 1999, pp. 3409–3430. [10] C. W. Cao, Y. T. Wu, and X. G. Geng, “Relation between the Kadometsev-Petviashvili equation and the confocal involution system,” J. Math. Phys., vol. 40, 1999, pp. 3948–3970. [11] Z. Q. Gu, “The Neumann system for 3rd-order eigenvalue problems related to the Boussinesq equation. IL Nuovo Cimento, vol. 117B(6),” 2002, pp. 615–632. [12] W. X. Ma and R. G. Zhou, “Binary nonlinearization of spectral problems of the perturbation AKNS systems,” Chaos Solitons &Fractals 13, 2002, pp. 1451–1463. [13] X. G. Geng and D. L. Du, “Two hierarchies of new nonlinear evolution equations associated with 3 × 3 matrix
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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. Shuhong Wang, native of Liaoning, was born in January 1980. In 2006, master of science in Applied Mathematics at Hebei University of Technology, Tianjin, China. She mainly engaged in domain of differential equations, Inequality and so on.
