A generalized Tu formula and Hamiltonian structures of fractional AKNS hierarchy

Physics Letters A 375 (2011) 3659–3663

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Physics Letters A www.elsevier.com/locate/pla

A generalized Tu formula and Hamiltonian structures of fractional AKNS hierarchy Guo-cheng Wu a,b,∗ , Sheng Zhang c,∗ a b c

Key Laboratory of Numerical Simulation of Sichuan Province, Neijiang, Sichuan 641112, PR China College of Mathematics and Information Science, Neijiang Normal University, Neijiang, Sichuan 641112, PR China School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, PR China

a r t i c l e

i n f o

Article history: Received 5 April 2011 Received in revised form 17 August 2011 Accepted 17 August 2011 Available online 24 August 2011 Communicated by R. Wu

a b s t r a c t In this Letter, a generalized Tu formula is firstly presented to construct Hamiltonian structures of fractional soliton equations. The obtained results can be reduced to the classical Hamiltonian hierarchy of AKNS in ordinary calculus. © 2011 Elsevier B.V. All rights reserved.

Keywords: Generalized Tu formula Fractional Hamiltonian system Fractional differentiable functions

1. Introduction Nobel Laureate Gerardus ’t Hooft once remarked that discrete space–time is the most radical and logical viewpoint of reality. In such discontinuous space–time, fractional calculus plays an important role which can accurately describe many nonlinear phenomena in physics, i.e., Brownian motion, anomalous diffusion, transportation in porous media, chaotic dynamics, physical kinetics and quantum mechanics. For details, see the monographs of Kilbas et al. [1], Kiryakova [2], Lakshmikantham and Vatsala [3], Miller [4] and Podlubny [5]. Since Riewe [6] proposed a concept of non-conservation mechanics, fractional conservation laws [7], Lie symmetries [8] and fractional Hamiltonian systems [9–16] have caught much attention. In recent study, Fujioka et al. found that the propagation of optical solitons can be described by an extended nonlinear Schrödinger equation (NLS) which incorporates fractional derivatives [17,18]. Searching for new integrable hierarchies of soliton equations is an important and interesting topic in soliton theory. The Tu scheme [19] is an efficient method to generate integrable Hamilton systems such as Ablowitz–Kaup–Newell–Segur (AKNS) hierarchy, Kaup–Newell (KN) hierarchy, Schrödinger system [20–26]. Whether fractional evolution equations admit fractional integrable systems? Some questions may naturally arise: (1) Can we have a generalized Tu scheme for a fractional case? (2) How to define Hamilton’s equations for the fractional soliton hierarchy?

*

Corresponding authors. E-mail addresses: [email protected] (G.-c. Wu), [email protected] (S. Zhang). 0375-9601/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2011.08.040

In this Letter, we start from a Lax pair of fractional order and firstly propose a generalized Tu scheme to investigate the Hamiltonian structure of fractional evolution equations. 2. Properties of fractional differentiable functions As we all know, there are two kinds of fractional derivatives: local fractional derivatives and nonlocal ones. The most used nonlocal operator is the Caputo derivative which requires the defined functions should be differentiable. As a result, the Caputo derivative is not suitable for the functions possessing non-smooth structures since they are on fractal set in real lines [27,28]. Several local versions have been proposed: Kolwankar–Gangal’s local fractional derivative [27–29], Chen’s fractal derivative [30,31], Cresson’s derivative [32], Jumarie’s modified Riemman–Liouville derivative [33] and Parvate’s F α derivative [34], among which Jumarie’s derivative is defined as

Dα x f (x) =

1

d

(1 − α ) dx

0 < α < 1.

x

  (x − ξ )−α f (ξ ) − f (0) dξ,

0

(1)

Both Jumarie’s derivative and Kolwankar’s derivative deal with fractal initial boundary problems and coarse-graining disturbance [8,35]. They have the same results, i.e., fractional differentiable functions, fractional Hamiltonian system [14,36], fractional variational derivative [15,36] although they are defined from different physical backgrounds. Some properties of the fractional differentiable functions are used in this study as follows: (I) The Leibniz product law.

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G.-c. Wu, S. Zhang / Physics Letters A 375 (2011) 3659–3663

If f (x) is an α order differentiable function in the area of point x, from the Jumarie–Kolwankar’s Taylor series [33], one can have

Dα x f (x) = lim

y →x+

(1 + α )( f ( y ) − f (x)) , ( y − x)α

0 < α  1.

(2)

Eq. (2) is also given by Li and Chen for Kolwankar and Gangal’s derivative [37,38]. We only consider the right limit of Eq. (2) for simplicity in this Letter. Directly from Eq. (2), we recently proofed the following fundamental property from fractal geometry β Dα xx =

(1 + β)xβ−α , (1 + β − α )

(3)

where xβ is a fractional differentiable function. Assuming g (x) is a differentiable function of α order, the Leibniz product law can hold for the non-differentiable functions [26–28]





α α Dα x f (x) g (x) = g (x) D x f (x) + f (x) D x g (x),

(4)

which is invalid for differentiable functions in ordinary calculus. (II) Denoting 0 I xα as the Riemann–Liouville integration in the following form α −α 0 I x f (x) = D x f (x) =

x

1

(α + 1)

f (ξ )(dξ )α ,

0 < α  1, (5)

0

one can have a generalized Newton–Leibniz formulation

1

(1 + α ) 1

(1 + α )

1 α Dα x f (x)(dx) = f (1) − f (0),

(6)

0

3. Fractional Hamiltonian structure A number of useful attempts have been made to establish fractional variational principles and Hamiltonian system [9–16], i.e., Baleanu’s fractional Hamiltonian system with Caputo derivative [10], Riemann–Liouville type Hamiltonian mechanics [11], Agrawal’s Hamiltonian formulation with Riesz derivative [12] and Jumarie’s Lagrange formula [14] and the optimal control of the fractional system [42]. This section revisits Jumarie’s fractional Hamiltonian system [14]. 3.1. A fractional exterior differential approach Since Adda proposed the fractional generalization of differential forms [43,44], several versions of fractional exterior differential approaches and applications related to different forms of fractional derivatives appeared in open literature [45–47]. A brief review is available in Tarasov’s work [48]. Starting from the total derivative in the integer dimensional space, and assuming f = f (u , v ), u = u (x) and v = v (x), where u, v are non-differentiable functions and f is a differentiable function with respect to u and v, we obtain the total derivative as follows

df = f u du + f v dv .

(11)

On multiplying the both sides of Eq. (11) with (1 + α ) and using the equality

x α Dα ξ f (ξ )(dξ ) = f (x) − f (0),

Remark. In the above properties, the functions f (x) and g (x) are generated through Cantor-like set and the Riemann–Liouville integration. They are nowhere differentiable functions or fractional differentiable functions totally different from the smooth functions in ordinary calculus. On the other hand, unlike Cresson and Nottale’s non-differentiable functions depending on scales [32,41], the functions are absolutely non-differentiable functions.

(7) dα u = (1 + α ) du

0

which holds for non-differentiable functions, we can get

and

Dα x

(1 + α )

x f (ξ )(dξ )α = f (x).

(8)

0

(III) With the properties (I) and (II), integration by parts for the differentiable functions f (x) and g (x) of α order can be written as

1

(1 + α )

α α α dα f = f u dα u + f v dα v = f u D α x u (dx) + f v D x v (dx) .

On the other hand, if we assume that u, v are differentiable functions and f is an α order differentiable function with respect to u and v (α )

df =

b α g (x) D α x f (x)(dx)

= g (x) f (x)

−

1

(1 + α )

(α )

fu

(1 + α )

(du )α +

fv

(1 + α )

(dv )α ,

(13)

we can derive dα different from Eq. (13)

a

|ab

(12)

b

(α )

f (x) D α g (x)(dx)α . x

(9)

(14)

3.2. Hamilton’s equations with fractional differentiable functions

a

The above properties (I)–(III) can be found in Refs. [14,33,39,40]. (IV) From Jumarie’s variational derivative [14], Almeida’ fractional variational approach [15] and Yang’s variational principle for fractional differentiable functions [36], the fractional variational derivative is given as

   k ∂L  δL ∂L , = + (−1)k D αx δy ∂ y ∂( D αx )k y

(α )

dα f = f u (u x )α (dx)α + f v ( v x )α (dx)α .

(10)

k =1

where k is a positive integer. In this study, we propose a generalized trace identity for fractional soliton hierarchy from Eq. (10).

We define the fractional functional

J [ p , q] =

1



(1 + α )





p D tα q − H (t , p , q) (dt )α .

(15)

Then, we can readily derive the generalized Poincare–Cartan 1-form, which reads

ω = p dα q − H (dt )α . From Eq. (16), we have

(16)

G.-c. Wu, S. Zhang / Physics Letters A 375 (2011) 3659–3663

(α )

dα ω = p t

(dt )α ∧ dα q + dα p ∧ dα q −

∂H α d p ∧ (dt )α ∂p

Supposing f ( A , B ) = tr( A B ), the following properties can be satisfied:

∂H α − d q ∧ (dt )α ∂q  



∂H ∂H (α ) (dt )α ∧ dα q + = pt + (dt )α − dα q ∧ dα p . ∂q ∂p

(a) The symmetry relationship satisfies

f ( A , B ) = f ( B , A ); (b) The bilinearity can hold

(17) In the above derivation, p and q are fractional differentiable functions with respect to t. The fractional closed condition dα ω = 0 allows the following fractional Hamilton’s equations

∂H (α ) qt = , ∂p

(18)

(α )

=−

f (c 1 A 1 + c 2 A 2 , B ) = c 1 f ( A 1 , B ) + c 2 f ( A 2 , B ); (c) In the sense of the local fractional derivative, the gradient ∇ B f ( A , B ) of the functional f ( A , B ) is defined by

  ∂ f ( A , B + C ) = f δ B f ( A , B ), C , ∂

∀ A , B , C ∈ A˜ n ,

(27)

where δ B is variational derivative with respect to B. With the fractional variational derivative (10), we can derive

and

pt

3661

∂H . ∂q

(19)

The results Eq. (12), Eq. (14), Eq. (18) and Eq. (19) can be found in Ref. [14].

 (kα )  (kα ) = (−1)k A x , δB f A, B x

(28)

α where k is a positive integer and D kxα = D α x · · · Dx ;







k

(d) The communication relationship can be used as



4. A generalized Tu formula and its application







f [ A , B ], C = f A , [ B , C ] ,

∀ A , B , C ∈ A˜ n .

(29)

4.1. A generalized Tu formula Constructing a functional Set A n = A = (ai , j ), ai , j ∈ C . Assume A and B ∈ C . Define the bracket [ A , B ] = A B − B A. Hence, A n is a Lie algebra. The corresponding loop algebra is defined as

˜ n = A (n) = A λn , A

n ∈ Z.

(20)

Consider the fractional compatibility condition, (α )

(β)

φx (x, t ) = U φ,

φt (x, t ) = V φ,

(21)

where the fractional derivative is in the sense of the modified derivative [14,31] or the local fractional derivative (2) and φ is a n-dimensional function vector. The compatibility condition of Eq. (21) leads to the generalized zero curvature equation (β)

Ut

− V x(α ) + [U , V ] = 0,

[U , V ] = U V − V U .

(22)

When taking α = β = 1, Eq. (22) reduces to the classical zero curvature equation. Set

U = e 0 (λ) +

n 

e i (λ)u i ,



˜ n, e i (λ), 0  i  n ⊂ A

(23)

i =1

where u = u (u 1 , u 2 , . . . , un ) T denotes a non-differentiable vector function. Define rank(λ) = deg(λ), then rank(e 0 (λ)) = ξ , rank(e i (λ)) = αi , 0  i  n, can be obtained. If the ranks of u i are taken as ξ − αi , 1  i  n, then each term in U has the homogeneous rank α which is denoted by

rank(U ) = rank



 α  ∂ = ξ. ∂ xα

(24)

Set V = m0 V m λ−m , ( V m )λ = 0, m  0, as a solution of the stationary zero curvature equation (α )

−V x

+ [U , V ] = 0

(25)

and rank( V m )λ is assumed to be given so that rank( V m )λ = m  0, each team in V has the same rank as follows



rank( V ) = rank

∂β ∂tβ

η,



= η.

(26)



(α )

W = f (V , U λ) + f K , V x

 − [U , V ] ,

(30)

˜ n , rank K = − rank λ, and using where U , V meet Eq. (22), K ∈ A the fractional variational derivative (10) we can derive

δW = V x(α ) − [U , V ], δK

δW = U λ − K x(α ) + [U , K ], δV

(31) (α )

where δδW is served as a constraint variation. Since [ K , V ]x V be calculated as

can

  [ K , V ](xα ) = K x(α ) , V + K , V x(α )   = U λ + [U , K ], V + K , [U , V ]   = [U λ , V ] + [U , K ], V + [ V , U ], K  = [U λ , V ] + U , [ K , V ] , V = [ K , V ] − V λ and Vλ can satisfy V x = [U , V ], respectively. γ They also have the same rank. As a result, we can get V = λ V where γ is a constant. A fractional trace identity can be presented from Eqs. (25) and (30) (α )

    ∂Uλ ∂Uλ δ f (V , U λ) δ W + f [ K , V ], = = f V, δui δui ∂ ui ∂ ui       ∂Uλ ∂Uλ ∂U γ = f V, + f V λ, + f V, ∂ ui ∂ ui λ ∂ ui     ∂ ∂U ∂Uλ = + f V λ, f V, ∂λ ∂ ui ∂ ui     ∂ ∂ U + λ−γ λγ f V , ∂λ ∂ ui

  ∂ ∂ U λγ f V , = λ− γ , 0  i  n. (32) ∂λ ∂ ui The variational derivative here is defined through Eq. (10). To the best of our knowledge, the above trace identity for fractional differentiable functions is completely new and it doesn’t appear in open literature.

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Recently, Fujioka et al. [17,18] found the propagation of optical solitons can be described by an extended NLS equation which incorporates fractional derivatives. In view of this point, we consider fractional AKNS hierarchy starting from the generalized spectral problem β

λt = 0,





1 0 0 −1



Dt r = 0 1 0 0



 ,

e 3 (0) =

0 0 1 0

 ,

equipped with the commutative relations





e 1 (m), e 3 (n) = −2e 3 (m + n),  e 2 (m), e 3 (n) = e 1 (m + n).

(34) (α )

We find that the adjoint representation equation V x U V − V U yields

b 0 = 0,

= [U , V ] =

(α )

b ix = −2b i +1 − 2qai , i  1,

c n +1 b n +1



bn

an = D x (qcn − rbn ),

f ( V , U λ ) = −2a,



∂U f V, ∂q

Hn =

   −α α α 1 Dα 2r D − cn x − 2r D x q x r Dx = αq α + 2qD −α r D α −2qD − − D b 2 n x x x x   c =L n , (35)

−α

(40)





∂U f V, ∂r

= c,

 =b

(41)

  c b

(42)

,

(43)

Setting n = 1, we can determine γ = 0 from the initial values and derive the fractional Hamiltonian function

Therefore, we can derive the recurrence relationship



q = − 12 q xx + q2 r ,

  δ(−2an+1 ) c = (γ − n) n . bn δu

b 0 = 0, c 0 = 0, a 1 = 0, b 1 = q, 1 (α ) 1 (α ) 1 b2 = − qx , c2 = rx , a2 = qr . 2 2 2

c1 = r ,



where u = (q, r ). Compared with the coefficients of λ−n−1 , we obtain

the first few of which reads

a0 = −1,

When α = β = 1, Eq. (39) can be reduced to the classical AKNS system,

δ(−2a) ∂ γ = λ−γ λ δu ∂λ

(α )

c ix = 2rai + 2c i +1 ,

(39)

− qr 2 .

and

c 0 = 0,

aix = qc i − rb i , (α )

(38)

Using the proposed trace identity (32), a direct compute leads to



(α )

1 α α D D r 2 x x

r = 12 r xx − qr 2 .

e 1 (m), e 2 (n) = 2e 2 (m + n),

a0x = qc 0 − rb0 ,

(β) 

qt

β β



e 2 (0) =

,

=

α 2 D t q = − 12 D α x D x q + q r,

r

e 1 (0) =

   0 −2 c (n + 1) = (β) 2 0 b(n + 1) rt     c (n) r . = JL = J Ln q b(n) 

(β)

ut

Here J and L are a Hamiltonian operator, respectively. For n = 2, we obtain the generalized AKNS equations

(33)

Choose a simple subalgebra of A 1

(37)

which gives rise to



 −λ q (α ) Φx = U (λ, u ) = Φ, r λ     q φ1 . u= , Φ= φ2

( α )   − V (n) x + U , V (n) = 0,

(β)

Ut

4.2. Fractional soliton hierarchy and their Hamiltonian structures

2an+1 n

,

δ Hn = δu



cn bn



.

(44)

The generalized evolutionary equations can be given as



(β)

ut

=

(β) 

qt

(β)

rt

= J

δ Hn . δu

(45)

5. Conclusion

n = 0, 1 , 2 , 3 , . . . .

Denoting

The terms on the left-hand side in (36) are of degree  0, while the terms on the right-hand side in Eq. (36) are of degree  0. Thus, we have

Much effort has been dedicated to the relationship between the fractional calculus, fractals and non-differentiable system [27– 29,39,49–55]. Since the fractional differentiable functions possess non-smooth or non-analytical structures, ordinary calculus are invalid while Kolwankar’s local fractional derivative and Juamrie’s derivative can make sense. In this Letter, from a generalized Lax pair, we establish a generalized Tu scheme for fractional differential equations and derive fractional evolutionary soliton hierarchy. However, there are still other interesting questions needed to be addressed i.e., physical meaning of fractional soliton which may be related to fractal media, fractional integral coupling method, nonlinear techniques for fractional soliton equations. Such work is under consideration.

 (n) (α )  (n) − V + x + U , V + = 2bn+1 e 2 (0) − 2cn+1 e 3 (0).

Acknowledgements



(n) (α )

V+

(n)

x

=

n 

ai e 1 (n − i ) + b i e 2 (n − i ) + c i e 3 (n − i ),

i =0

(n)

V − = λn V − V + , we can write Eq. (26) as

 (n) (α )   (n) (α )  (n) (n) − V+ x + U, V+ = V− x − U, V− .

(n)

(36)

If we take an arbitrary modified term for V + as n = 0 and notice (n) V (n) = V + , it is easy to compute the zero curvature equation

Many thanks for the referees’ helpful suggestions. This work is financially supported by NSFC (No. 10872085).

G.-c. Wu, S. Zhang / Physics Letters A 375 (2011) 3659–3663

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