Integrable coupling system of fractional soliton equation hierarchy

Physics Letters A 373 (2009) 3730–3733

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

Integrable coupling system of fractional soliton equation hierarchy Fajun Yu ∗ College of Maths and Systematic Science, Shenyang Normal University, Shenyang 110034, PR China

a r t i c l e

i n f o

Article history: Received 30 April 2009 Received in revised form 21 July 2009 Accepted 6 August 2009 Available online 12 August 2009 Communicated by A.R. Bishop

a b s t r a c t In this Letter, we consider the derivatives and integrals of fractional order and present a class of the integrable coupling system of the fractional order soliton equations. The fractional order coupled Boussinesq and KdV equations are the special cases of this class. Furthermore, the fractional AKNS soliton equation hierarchy is obtained. © 2009 Elsevier B.V. All rights reserved.

PACS: 02.30.Ik 02.30.Jr Keywords: Integrable coupling system Fractional soliton equation AKNS equation hierarchy

1. Introduction The fractional calculus has a long history from 1695, when the derivative of order 1/2 has been described by Leibniz. The theory of integrals and derivatives of non-integer order went back to Leibniz, Liouville, Riemann, Grunwald and Letnikov. In the past few decades, derivatives and integrals of fractional order [1,2] have numerous applications: kinetic theories [3–5], statistical mechanics [6,7], dynamics in complex media [8,9], and many others [10–13]. Many authors have pointed out that fractional-order models are more appropriate than integer-order models for various real materials. Fractional derivatives provide an excellent instrument for the description of memory and hereditary properties of various materials and processes. This is the main advantage of the fractional derivatives in comparison with classical integer-order models in which such effects are neglected. The advantages of the fractional derivatives become apparent in modeling mechanical and electrical properties of real materials, as well as in the description of rheological properties of rocks, and in many other fields. Recently, integrable couplings have been receiving more and more attention. A few ways to establish integrable couplings are presented by using perturbations [14–16], enlarging spectral problems [17,18], semidirect sums of Lie algebras [19,20], and creating

new Lie algebras [21–27]. Moreover, Ma did a series of original work on improving the Tu trace identity to construct a Hamiltonian structure for the more general systems including integrable couplings, discrete integrable systems, and super-integrable systems by the variational identity [28–30]. How to obtain the integrable coupling system of the fractional soliton equation is an important work. In this Letter we present the integrable coupling system of the fractional soliton equation and soliton hierarchy by using differential forms [31–33], then, the fractional coupled Boussinesq and KdV equations are presented. In Section 2, some information of the fractional differential forms is considered to fix notation. In Section 3, the integrable coupling system of the fractional soliton equation is presented. We discuss an example of the fractional AKNS equation hierarchy in Section 4. 2. Derivatives and integrals of fractional order Several definitions of the fractional derivative have been proposed. These definitions include the Riemann–Liouville, the Grunwald–Letnikov, the Weyl, the Caputo, the Riesz, and the Miller and Ross fractional derivatives. In this section, we present the definitions of the Riemann–Liouville integral and derivatives, and their properties. Fractional integral of arbitrary order p > 0,

*

Address for correspondence: Department of Mathematics, Shenyang Normal University, Shenyang 110034, China. Tel.: +86 13840329476; fax: +86 02486574979. E-mail address: [email protected]. 0375-9601/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2009.08.017

p a It

f (t ) =

1

( p )

t a

(t − τ ) p −1 f (τ ) dτ .

(1)

F. Yu / Physics Letters A 373 (2009) 3730–3733

The subscripts a and t denote the two limits related to the operation of fractional differentiation. In formula (2), we define 0 a It

f (t ) = f (t ).

(2)

Fractional derivative of arbitrary order p > 0,

⎧  1 d m +1 t (t − τ )m− p f (τ ) dτ ⎪ a ⎨ (− p +m+1) ( dt ) p (m  p < m + 1), a Dt f (t ) = ⎪ ⎩ dm+1 f (t ) ( p = m + 1). dt m+1

= I,

(3)

I f (t ) = f (t ),

(4)

(γ + 1) γ − p = t , (γ + 1 − p )

p > 0,

γ > −1, t > 0.

(5)

Note the remarkable fact that the fractional derivative f (t ) is not zero for constant function f (t ) ≡ 1 if p is not integer. In fact, (5) with γ = 0 teaches us that

t−p

=

p > 0,

,

(1 − p )

t > 0.

(6)

If f (t ) is continuous derivatives for t  a, the integration of arbitrary real order defined by (1) has the follow important property:

p q a It a It f (t )

p +q a It

=

f (t ).

(7)

Indeed, p q a It a It f (t )

=

=

=

=

t

1

(q)

(t − τ )q−1 a It f (τ ) dτ p

t

( p )(q)

f (ξ ) dξ a

( p + q) p +q

t

(t − ξ ) p −1 f (ξ ) dξ

p +q−1

f (ξ ) dξ

a

f (t )

(8) q

f (t ) = a It

p a It



p +q

f (t ) = a It

f (t ).

(9)

Let us consider some properties of the Riemann–Liouville fractional derivatives. The most important property of the Riemann– Liouville fractional derivative is that for p > 0 and t > a

p p a Dt a It f (t ) = f (t ), p p a It a Dt f (t ) = f (t ),  p n+ p n a Dt a Dt f (t ) = a Dt

(10) (11) f (t ),

n ∈ N,

(12)

which means that the Riemann–Liouville fractional differentiation operator is a left inverse to the Riemann–Liouville fractional integration operation operator of the same order p. The product rule is as follows: p



Dt f (t ) g (t ) =



(14b)

(3) The inverse Fourier transform:



F−1 : u˜ (k, t ) → F−1 u˜ (k, t ) = u (x, t ).

(14c)

The operation Tˆ = F−1 LF is called a transform operation, since it performs a transform of a discrete model of coupled oscillators into the continuous medium model. In Ref. [34], it considers a one-dimensional generalized nonlinear derivative equation system. The transform operation Tˆ maps the discrete equations of motion +∞









J (n, m) un (t ) − um (t ) + F un (t ) ,

(15)

m=−∞,m=n

(16)

α

(t − ξ )



L : uˆ (k, t ) → L uˆ (k, t ) = u˜ (k, t );

 ∂s ∂α u (x, t ) − G α A α u (x, t ) − F u (x, t ) = 0, s α ∂t ∂|x|

(t − τ )q−1 (t − ξ ) p −1 dτ

is obtained. Obviously, interchanging p and q, we have

p q a It a It

(14a)

with noninteger α -interaction into the fractional continuous medium equations:

ξ

t

1

τ a

t

( p )(q)

= a It

(t − τ )q−1 dτ

a

1



(2) The passage to the limit x → 0:

∂ s un (t ) =g ∂ts

a

1

(1) The Fourier series transform:

F : un (t ) → F un (t ) = uˆ (k, t );

p a Dt

p a Dt 1

In this section, we first consider a map of discrete model into the continuous one with transform operation. The transform operation Tˆ is a combination Tˆ = F−1 LF of the operations in Ref. [34], which are as follows:

and p γ a Dt t

The above equations play an important role in fractional calculus. We will apply the fractional calculus to soliton equation and integrable system. 3. Fractional integrable coupling system of some equations

Defining for complementation a Dt0 f (t ) = f (t ), we easily recognize that p p a Dt It

3731

∞ 

p j =0

j

p− j

Dt

j

f (t )Dt g (t ).

(13)

∂ where ∂| is the Riesz fractional derivative and G α = g |x|α is a x|α finite parameter. In the continuous limit (|ai | → 0), the α -interaction in the multi-dimensional lattice gives the continuous medium equations α3 α1 α2 with derivatives ∂∂xα1 , ∂∂y α2 , ∂∂zα3 , . . . : n

 ∂ s U (r , t ) ∂ αi u (r , t ) = − g A + F u (r , t ) . α i s α i ∂t ∂|x|

(17)

i =1

In this section, we extend the model of system (17) to the enlarged integrable equation systems and present a multidimensional generalized integrable coupling system of the nonlinear derivative equation as follows:

⎧ s n ∂ U (r ,t ) ∂ αi u (r ,t ) ⎪ ⎪ ⎪ ∂ t s = − g i =1 A αi ∂|x|αi + F (u (r , t )), ⎨   αi αi ∂ s V (r ,t ) = − g ni=1 A αi ∂ ∂|ux|(αr i,t ) − f ni=1 B αi ∂ ∂|vx|(αri,t ) (18) s ∂ t ⎪ ⎪   α α ⎪ i i ⎩ − k ni=1 C αi ∂ ∂|ux|(αr i,t ) ni=1 ∂ ∂|vx|(αr i,t ) + E (u (r , t )). Let us consider an example of the integrable coupling system for quadratic-nonlinear long-rang interactions:

3732

F. Yu / Physics Letters A 373 (2009) 3730–3733

(I) The continuous limit of the integrable coupling system of the lattice equation

⎧ 2  =+∞ ∂ un (t ) ⎪ = g2 m ⎪ m=−∞,m=n J 2 (n, m)[ f (un ) − f (um )] ⎪ ∂t2 ⎪ ⎪  ⎪ =+∞ ⎪ + g4 m ⎪ m=−∞,m=n J 4 (n, m)[un − u m ], ⎪ ⎪ ⎪  ⎨ ∂ v n (t ) m=+∞ m=−∞,m=n J 2 (n, m)[ f (un ) − f (um )] ∂ t = g2 ⎪ ⎪ × J 1 (n, m)[ f ( v n ) − f ( v m )] ⎪ ⎪ ⎪  =+∞ ⎪ ⎪ ⎪ + g4 m ⎪ m=−∞,m=n J 4 (n, m)[un − u m ] ⎪ ⎪ ⎩ 2 × J 3 (n, m)[ v n2 − v m ],

(19)

2 ∂2 ⎪ ⎪ v (x, t ) − G 2 ∂∂x2 [u (x, t ) v (x, t )] ⎪ ∂t2 ⎪ ⎪ ⎩ 2 4 + gG 2 ∂∂x2 [u 2 (x, t ) v 2 (x, t )] + G 4 ∂∂x4 u (x, t ) v (x, t ) = 0.

α

(20)

v t + v (uv )x + u xxx = 0; ut + u x + 2uu x − 3α u xxx = 0, v t + v v x + u x − 3α v xxt = 0; ut + u x + uu x + α u xx + β u xxt = 0, v t + (uv )x + 2α v x + β u xxx = 0.

(21)

(22)

(23)

(24)

(29)

Super MKdV equations (iii):



ut = (6v 2 − 6v x )u x + (6v v x − 3v xx )u − uu xxx ,

(30)

4. The generalized fractional AKNS equation hierarchy In this section, we want to apply the fractional zero curvature equation to construct the generalized fractional AKNS equation hierarchy. Consider the following matrix spectral problem

U (u , λ) =

−λ q , r λ

∂tαn λ = 0,

(31)

where λ is a spectral parameter. The spectral problem (31) is called AKNS spectral problem [35]. To derive an associated soliton hierarchy, we first solve the adjoint equation

W x = [U , W ]

(32)

of the spectral problem (31) through the generalized Tu scheme [36]. Assume that a solution W is given by



W =

a c

b −a



.

⎧ ⎨ ax = qc − rb, b = −2λb − 2qa, ⎩ x c x = 2λc + 2ra.

(33)

Let us seek a formal solution of the type

 W =

a c

b −a

=

∞

W k λ−k =

k =0

∞  (k)

a k =0

c (k)

b(k) −a(k)



λ−k ,

(34)

the condition (33) gives rise to the following recursions relations:

(25)

where J i (n) (i = 1, 3) define the αi -interactions with α1 = 1 and α3 = 3, gives the integrable coupling system of KdV equations

⎧ 3 ∂ ⎪ u (x, t ) − G 1 u (x, t ) ∂∂x u (x, t ) + G 3 ∂∂x3 u (x, t ) = 0, ⎪ ⎨ ∂t ∂ ∂ ∂ t v (x, t ) − G 1 u (x, t ) v (x, t ) ∂ x [u (x, t ) v (x, t )] ⎪ ⎪ 3 ⎩ + G 3 ∂∂x3 [u (x, t ) v (x, t )] = 0.

ut − 14 (u xxx + 6uu x + 12uu xx ) = 0,

Therefore, the adjoint equation (32) is equivalent to

(II) The continuous limit of the integrable coupling system of equations

m=+∞ ⎧ ∂ un (t ) 2 2 ⎪ m=−∞,m=n J 1 (n, m)[un − u m ] ∂ t = g1 ⎪ ⎪  ⎪ m =+∞ ⎨ + g3 m=−∞,m=n J 3 (n, m)[un − u m ], m=+∞ ∂ v n (t ) 2 2 2 2 ⎪ ⎪ m=−∞,m=n J 1 (n, m)[un − u m ][ v n − v m ] ∂ t = g1 ⎪ ⎪ m=+∞ ⎩ + g 3 m=−∞,m=n J 3 (n, m)[un − um ][ v n − v m ],



φx = U (u , λ)φ,

Coupled Boussinesq equation (iii):



SKdV equations (ii):



Coupled Boussinesq equation (ii):



(28)

v t + 3α uv x + α v xxx = 0;

v t = 6v 2 v x − v xxx + 34 (uu xx )x + 32 u ( vu x )x .

Coupled Boussinesq equation (i):

ut + u x + uu x = 0,

ut + 6α uu x + β u xxx = 0,

v t − 14 ( v xxx + 6uv x + 3u x v ) = 0;

Through the reductions, from the system (21) we obtain the integrable coupling equations as follows:



α 3 ∂ xα3



⎧ 2 ∂ ∂ α2 ⎪ ⎪ 2 u (x, t ) − G α2 ∂ xα2 u (x, t ) ⎪ ∂ t ⎪ ⎪ α4 ⎪ ∂ α2 2 ⎪ α ∂ ⎪ ⎨ + gG α2 ∂ xα2 u (x, t ) + G α 4 ∂ xα4 u (x, t ) = 0, 2

(27)

Coupled KdV equation (i):

It is used in fluid dynamics as simplified model for turbulence, boundary layer behaviour, shock wave formation and mass transport. In general, the integrable coupling system of the fractional Boussinesq equations are given as follows:

2 ∂ v (x, t ) − G α2 ∂∂xα2 [u (x, t ) v (x, t )] ∂t2 ⎪ ⎪ α ⎪ ⎪ ⎪ + gG α2 ∂∂xα22 [u 2 (x, t ) v 2 (x, t )] ⎪ ⎪ ⎪ α ⎩ + G α α4 ∂∂xα44 [u (x, t ) v (x, t )] = 0.

⎧ ∂ ∂ α1 ∂ α3 ⎪ ⎪ ⎨ ∂ t u (x, t ) − G α1 u (x, t ) ∂ xα1 u (x, t ) + G α 3 ∂ xα3 u (x, t ) = 0, ∂ ∂ α1 ∂ t v (x, t ) − G α1 u (x, t ) v (x, t ) ∂ xα1 [u (x, t ) v (x, t )] ⎪ ⎪ ⎩ + G ∂ α3 [u (x, t ) v (x, t )] = 0.

Through the reduction, the following coupled equations are obtained in the system (27):

where f (u ) = u − gu 2 , f ( v ) = v − g v 2 , and J i (n) define the αi interactions with α2 = 2 and α4 = 4, gives the integrable coupling system of Boussinesq equation which is a nonlinear partial differential equation of fourth order:

⎧ ∂2 2 u (x, t ) − G 2 ∂∂x2 u (x, t ) ⎪ ⎪ ⎪ ∂t2 ⎪ ⎪ ⎨ + gG 2 ∂ 2 u 2 (x, t ) + G 4 ∂ 4 3 u (x, t ) = 0, ∂ x2 ∂ x4

If non-integer αi -interactions is used for J i (n), we get the fractional integrable coupling system of the KdV equations

(26)

⎧ ( 0) a = −1, b ( 0) = 0, c ( 0) = 0, ⎪ ⎪ ⎪ (1 ) ⎪ ⎪ b (1 ) = q , c (1 ) = r , ⎪ ⎪ a = 01, ⎪ 1 ( 2) ( 2) ⎪ a = 2 qr , b = − 2 qx , c (2) = 12 r x , ⎪ ⎪ ⎪ ⎪ 1 1 ⎪ a(3) = (qr ) , ⎨ b(3) = 4 (q xx − 2q2 r ), x 4 c (3) = 14 (r xx − 2r 2 q), ⎪ ⎪ ⎪ ⎪ (n) ⎪ (n) (n) ⎪ ⎪ ax = qc − rb , ⎪ ⎪ ⎪ b(n+1) = − 1 (b(n) − 2qa(n) ), ⎪ ⎪ 2 x ⎪ ⎪ ⎩ 1 (n) ( n +1 ) c = 2 (c x − 2ra(n) ).

(35)

F. Yu / Physics Letters A 373 (2009) 3730–3733

Substituting

1

W =

− 12 q x − 12 qr

qr

2 1 r 2 x

and

[U , W ] =

1 2

qr x +

(36)

1 q r 2 x

q r + λq x − 12 qr x − 12 q x r 2

qr 2 + λr x

(37)

  ∂tαn U − ∂xα V (n) + U , V (n) = 0,

∂tαn



=

−∂xα q , ∂xα r

(39)

which is the generalized fractional version of the AKNS nonlinear equation hierarchy. The equations



∂tαn q + 12 ∂xα q x + q2 r = 0,

(40)

∂tαn r − 12 ∂xα r x + qr 2 = 0,

are the fractional Shrödinger equations. Moreover, we obtain the Hamiltonian structure of the fractional AKNS hierarchy (39)

H = −(qr ),

(41)

where the Hamiltonian operator J is defined by



J=

1 α ∂ 2 x

0

− 12 ∂xα

0





W1 =

1 (qr )x 4

1 (r 4 xx

(42)

.

If we choose

− 2r 2 q)

1 (q − 2q2 r ) 4 xx , − 14 (qr )x

the equation [U , W 1 ] gives



[U , W 1 ] =

1 (qr xx −rq xx ) 4 1 ( r − 2r 2 q)λ+ 12 r (qr )x xx 2

The author Fajun Yu would like to express his sincere thanks to referees for their enthusiastic guidance and help. This work was supported by the Research Work of Liaoning Provincial Development of Education, China (Grant No. 2008670).

(38)

comparing the coefficient of the λ in Eq. (38), we obtain a system of evolution equations

q r

In this Letter, by using the fractional derivatives, we construct the fractional integrable coupling system of soliton equations. Furthermore, the fractional AKNS equation hierarchy is obtained. Acknowledgements



into the generalized zero curvature equation



3733

− 12 (q xx −2q2 r )λ− 12 q(qr )x 1 4 (rq xx −qr xx )

(43)

.

(44)

Comparing the coefficient of the λ in Eq. (38), we obtain the evolution equations

 α ∂tn q − 14 ∂xα (q xx − 2q2 r ) − 12 q(qr )x = 0, ∂tαn r − 14 ∂xα (r xx − 2r 2 q) + 12 r (qr )x = 0,

which are the new fractional nonlinear equations.

(45)

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