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New Hamiltonian structure of the fractional C-KdV soliton equation hierarchy
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Chaos, Solitons and Fractals 37 (2008) 688–697 www.elsevier.com/locate/chaos
New Hamiltonian structure of the fractional C-KdV soliton equation hierarchy Fajun Yu *, Hongqing Zhang Department of Mathematics, Dalian University of Technology, Dalian 116024, PR China Accepted 19 September 2006
Communicated by Prof. M. Wadati
Abstract A generalized Hamiltonian structure of the fractional soliton equation hierarchy is presented by using of differential forms and exterior derivatives of fractional orders. Example of the fractional Hamiltonian system of the C-KdV soliton equation hierarchy is constructed, which is a new Hamiltonian structure. 2006 Published by Elsevier Ltd.
1. Introduction Derivatives and integrals of fractional orders [1,2] have found many applications in recent studies in physics. The interest in fractional analysis has been growing continually during the past few years. Fractional analysis has numerous applications: kinetic theories [3,4,9], statistical mechanics [10,11], dynamics in complex media [12,13] and many others [5–8]. The theory of derivatives of non-integer order goes back to Leibniz, Liouville, Grun-wald, Letnikov and Riemann. In the past few decades 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 fractional derivatives in comparison with classical integer-order models in which such effects are, in fact, neglected. In this paper, we use a fractional exterior calculus that was suggested in [14]. It allows us to consider the fractional generalized Hamiltonian [15,16]. Then, by using of differential forms and exterior derivatives of fractional orders, we extend the trace identity [17–27] to fractional form and obtain fractional Hamiltonian systems of soliton equation hierarchy. In Section 2, the fractional derivatives and integrals is considered to fix notation. In Section 3, a brief review of differential forms is suggested. In Section 4, the fractional form spaces will be given to fix notations and provide a convenient reference. In Section 5, we consider a fractional Hamilton system. In Section 6, a example of the well-known CKdV equation hierarchy [28] is discussed. In Section 7, we give that the spectral case a = 1.
*
Corresponding author. E-mail address: [email protected] (F. Yu).
0960-0779/$ - see front matter 2006 Published by Elsevier Ltd. doi:10.1016/j.chaos.2006.09.049
F. Yu, H. Zhang / Chaos, Solitons and Fractals 37 (2008) 688–697
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2. Brief review of fractional derivatives and integrals The derivatives of arbitrary real order a can be considered as an interpolation of this sequence of operators. We will use for it the notion suggested and used by Davis [29], namely a a Dt f ðtÞ;
ð1Þ
the short name for derivatives of arbitrary order is fractional derivatives. The subscripts a and t denote the two limits related to the operation of fractional differentiation. Following Ross [30], we will call them the terminals of fractional differentiation. The appearance of the terminals in the symbol of fractional is essential. This helps to avoid ambiguities in applications of fractional derivatives to real problems. Integrals of arbitrary order p > 0 p a Dt f ðtÞ
¼
Z t m X f ðkÞ ðaÞðt aÞpþk 1 þ ðt sÞpþm f ðmþ1Þ ðsÞ ds; Cðp þ k þ 1Þ a Cðp þ k þ 1Þ k¼0
ð2Þ
the formula (2) immediately provides us with the asymptotic of a Dp t f ðtÞ at t = 0. Derivatives of arbitrary order Z t pþk m X f ðkÞ ðaÞðt aÞ 1 p lim fnðpÞ ðtÞ ¼ ðt sÞmp f ðmþ1Þ ðsÞ ds; ð3Þ þ a Dt f ðtÞ ¼ h!1;nh!ta Cðp þ k þ 1Þ a Cðp þ k þ 1Þ k¼0 the formula (3) has been obtained under the assumption that the derivatives f(k)(t) (k = 1, 2, . . . , m + 1) are continuous in closed interval [a, t] and that m is an integer number satisfying the condition m > p + 1. The smallest probeable value for m is determined by the inequality m < p < m + 1. From the pure mathematical point of view of such a class of functions is narrow. However this class of functions is very important for applications. Because the character of the majority of dynamical processes is smooth enough and does not allow discontinuities. Understanding this fact is important for the proper use of the methods of the fractional calculus in applications, especially because of the fact the Riemann–Liouville definition mþ1 Z t d p ðt sÞmp f ðsÞ ds ðm 6 p 6 m þ 1Þ ð4Þ a Dt f ðtÞ ¼ dt a provides an excellent opportunity to weaken the conditions on the function f(t). We will think how the Riemann–Liouville definition (4) appears as the result of the unification of the notions of integer-order integration and differentiation. Let us suppose that the function f(s) is continuous and integrable in every finite interval (a, t); the function f(t) may have on integrable singularity of order r < 1 at the point s = a limðs aÞr f ðtÞ ¼ constð6¼ 0Þ: s!a
Then the integral Z t f ðsÞ ds f 1 ðtÞ ¼ a
exists and has a finite value, namely equal to 0, as t ! a. In the general case we have the Cauchy formula Z t 1 ðt sÞn1 f ðsÞ ds: f n ðtÞ ¼ CðnÞ a
ð5Þ
Let us now suppose that n P 1 is fixed and take integer k P 0. Obviously, we will obtain Z k 1 Dk ðt sÞn1 f ðsÞ ds; f kn ðtÞ ¼ CðnÞ a Z k 1 ðt sÞn1 f ðsÞ ds; Dk f kn ðtÞ ¼ CðnÞ a where the symbol Dk(k P 0) and Dk(k P 0) denote k interacted differentiations. To extend the notion if n-fold integration to non-integer values of n, we can start with the Cauchy formula (5) and replace the integer n in it by a real p > 0,
690
F. Yu, H. Zhang / Chaos, Solitons and Fractals 37 (2008) 688–697 p a Dt f ðtÞ
¼
1 CðpÞ
Z
t
ðt sÞp1 f ðsÞ ds:
ð6Þ
a
In (5) the integer n must satisfy the condition n P 1; the responding condition for p is weak; for the existence of integral (6) we must have p > 0. Moreover, under certain reasonable assumptions lim a Dp t f ðtÞ ¼ f ðtÞ:
ð7Þ
p!0
So we can put a D0t f ðtÞ ¼ f ðtÞ. If f(t) is continuous for t P a, the integration of arbitrary real order defined by (6) has the following important property: p q a Dt ðaDt Þf ðtÞ
¼ a Dpq f ðtÞ: t
ð8Þ
Indeed, we have p q a Dt ðaDt Þf ðtÞ ¼
¼
1 CðqÞ
Z
t
ðt sÞq1 a Dp t f ðsÞ ds ¼
a
1 CðpÞCðqÞ
Z
t
f ðnÞ dn
a
Z
1 CðpÞCðqÞ
Z
t
ðt sÞq1 ds
a
t
ðt sÞq1 ðt nÞp1 ds ¼
a
Z
t
ðt nÞp1 f ðnÞ dn a
1 Cðp þ qÞ
Z
t
ðt nÞpþq1 f ðnÞ dn ¼ a Dpq f ðtÞ: t
a
Obviously, we can interchange p and q, so we have p q alpha Dt ðaDt Þf ðtÞ
p pq ¼ a Dq f ðtÞ: t ðaDt Þf ðtÞ ¼ a Dt
ð9Þ
Let us consider some properties of the Riemann–Liouville fractional derivatives. The first and maybe the most important–property of the Riemann–Liouville fractional derivative is that for p > 0 and t > a p p a Dt ðaDt f ðtÞÞ
¼ f ðtÞ:
ð10Þ
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.
3. The differential forms The calculus of differential forms is an elegant branch of pure mathematics and a powerful tool in applied mathematics. A clear introduction to the field, with emphasis on applications, is given in Flanders [5]. Vector spaces at a point P 2 En (n-dimensional Euclidean space) can be constructed out of expressions of the following type: one forms; a ¼ two forms; b ¼
n X i¼1 n X
ai dxi ;
ð11Þ
bij dxi ^ dxj ;
ð12Þ
i;j¼1
.. . n forms; x ¼ w dx1 ^ dx2 ^ ^ dxn ;
ð13Þ
where the {xi} are the Cartesian coordinates of En. The above sums are taken over all possible values of the indices with the constraint that dxi ^ dxj ¼ dxj ^ dxi :
ð14Þ
The functions ai, bij, etc., depend only on P and may be real or complex depending on the application. If a k form, c, is multiplying an m form, l, the following would be true: c ^ l ¼ ð1Þkm l ^ c:
ð15Þ
The result would be zero if k + m > n. The exterior product, ^, is distributive, and antisymmetric. The dimension of the vector space of k forms over P 2 En is n n! ; ¼ k!ðn kÞ k
F. Yu, H. Zhang / Chaos, Solitons and Fractals 37 (2008) 688–697
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which is zero if k > n. For the purposes of this paper let F(k, k, n) denote the vector space of k forms over P 2 En. The apparently redundant n in the above notation will be needed later for the fractional form case, as there is some additional freedom. The exterior derivative is defined as d¼
n X
dxi
i¼1
o : oxi
ð16Þ
The exterior derivative maps k forms into k + 1 forms and has the following algebraic properties. Let c and k be k forms, and l be an m form, then dðc þ kÞ ¼ dc þ dk;
ð17Þ k
dðc ^ lÞ ¼ ðdcÞ ^ l þ ð1Þ c ^ dl;
ð18Þ
dðdcÞ ¼ 0:
ð19Þ
The last identity is called the Poincare´ lemma. A form, c, is called closed if dc = 0. A form, c, is called exact if there exists a form, l, such that dl = c. The order of l is one less than the order of c. Exact forms are always closed. Closed forms are not always exact.
4. Fractional form spaces If the partial derivatives in the definition of the exterior derivative are allowed to assume fractional orders, a fractional exterior derivative can be defined da ¼
n X
dxai
i¼1
oa : ðoðxi ai ÞÞa
ð20Þ
Note that the subscript i denote number, the superscript a denotes the order of the fractional coordinate differential, and ai is the initial point of the derivative. Sometimes the notation oai will be used to denote oa : ðoðxi ai ÞÞa In two dimension (x, y), the fractional exterior derivative of order a of xp, with the initial point taken to be the origin, is given by da xp ¼ dxa
Cðp þ 1Þ pa xp x þ dy a a : ðp a þ 1Þ y Cð1 aÞ
ð21Þ
For specific values of the derivative parameter the following results are obtained: a ¼ 0;
d0 xp ¼ 2xp ; 1 p
1
ð22Þ p1
a ¼ 1;
d x ¼ dx px
a ¼ 2;
d2 xp ¼ dx2 pðp 1Þxp2 :
;
ð23Þ ð24Þ
By analogy with standard exterior calculus, vector spaces can be constructed using the dxai . Let F(a, m, n) be a vector space at P 2 En. a denotes the sum of the fractional differential orders of the basis elements, m denotes the number of coordinate differentials appearing in the basis element, n the number of coordinates, and {xi} are the Cartesian coordinates for En. For example, a basis set for F(a, 1, n) would be fdxa1 ; dxa2 ; . . . ; dxan g and arbitrary element of F(a, 1, n) would be expressed as n X a¼ ai dxai : i¼1
For a fixed a this is an n-dimensional vector space. Also note that there is a different vector space for each value of a. For a = 1 the one forms from exterior calculus are recovered. Now suppose that the basis elements are made up of two coordinate differentials, F(a, 2, n). In this case the basis set is more complicated, l
fdxl1 11 ^ dxl121 ; dxl1 11 ^ dx231 ; . . . ; dxln n1m ^ dxln nm jlij þ lkj ¼ ag:
ð25Þ
692
F. Yu, H. Zhang / Chaos, Solitons and Fractals 37 (2008) 688–697
Note that dxl1 11 ^ dxl1 21 would be zero if and only of l11 = l21, etc. An arbitrary element of F(a, 2, n) would be expressed as a sum of the form n X n Z a X 1 b¼ ðbij ða1 ; a a1 Þ dxai 1 ^ dxaa Þ da1 ; ð26Þ j i¼1
j¼1
0
where bii(l, l) = 0. Unlike the previous vector space, F(a, 1, n), this is clearly infinite dimensional for any value of a. Not only is it infinite but it is uncountably infinite. An arbitrary element of F(a, 3, n) would be expressed as an integral of the form n X n X n Z a Z aa1 X 1 a2 b¼ ðbijk ða1 ; a a2 ; a a1 a2 Þ dxai 1 ^ dxaj 2 ^ dxaa Þ da2 da: ð27Þ k i¼1
j¼1
k¼1
0
0
With each step up on the middle index of F(a, m, n) another integral and summation is included. A basis set for this vector space would be ( ) m X l11 l21 lm1 F ða; m; nÞ ¼ dxa1 ^ dxa2 ^ ^ dxam ; . . . l ¼a : ð28Þ k¼1 ij The basis elements range over all possible combinations of the fractional coordinate differentials and all possible choices for the l. Note that m need not be less than or equal to n. Let P 2 En, and let A 2 F(a, m, n) and B 2 F(l, k, n) at the point P. Then the exterior product of A and B maintains the antisymmetry property of Eq. (17), A ^ B ¼ ð1Þkm B ^ A 2 F ðl þ a; k þ 1; nÞ:
ð29Þ
If k + m > n, A ^ B need not be zero. Eq. (16) is also maintained due to the linearity of the fractional derivative. Eq. (19) is not maintained due to the product rule for the fractional derivative. Note also that da maps F(l, m, n) into F(l + a, k + 1, n).
5. Fractional generalized Hamiltonian systems Fractional phase space can be considered as a phase space of the systems that are described by the fractional powers of coordinates and momenta. Let us consider the class of Hamiltonian systems that are constructed in the fractional phase space. Nowadays it is well known that a great number of nonlinear evolution equations, arising form various branches of physics, they admit the zero curvature equation U t V x þ ½U ; V ¼ 0:
ð30Þ
Our target is to write the fractional zero curvature equation, which has more excellent instrument than the integer form Dat U Dax V ðnÞ þ ½U ; V ðnÞ ¼ 0;
ð31Þ
which is generalized fractional zero curvature equation. Fractional generalized differential form [12]: b = Gi dpi Fi dqi (the differential 1-form), which is used in the definition of the Hamiltonian system, can be defined in the following form: ba ¼ Gi ðdpi Þa F i ðdqi Þa : Let us consider the canonical coordinates (x1, . . . , xn, xn+1, . . . ,x2n) = (q1, . . . , qn,p1, . . . , pn) in the phase space E2n and a dynamical systems can be defined by the equations ! Dapi 0 Hn Hn a qi ¼ ¼ J ; ð32Þ Dt 0 Daqi pi Hn Hn with Dapi H n ¼ Gi ðp; qÞ;
Daqi H n ¼ F i ðp; qÞ:
ð33Þ
The fractional generalization of Hamiltonian system can be defined by the equations generalization of differential forms [14].
F. Yu, H. Zhang / Chaos, Solitons and Fractals 37 (2008) 688–697
693
A dynamical system (31) on the phase space E2n, it called a fractional Hamiltonian system if the fractional differential 1-form ba = Gi(dpi)a Fi(dqi)a is a closed fractional form daba = 0, where da is the fractional exterior derivative. A dynamical system is called a fractional non-Hamiltonian system if the fractional differential 1-form ba is a non-closed fractional form daba 5 0. A central and very important subject in the theory of integrable system is to search for a sequence of scalar functions {Hn} such that Eq. (29) can be cast in the Hamiltonian form ! Dapi 0 Hn a qi Dt ¼ : 0 Daqi pi Hn The Hamiltonian {Hn} constitute, in fact, an infinite number of conserved densities of the hierarchy (29).
6. The generalized fractional C-KdV equation hierarchy To illustrate our method, we want to apply the fractional zero curvature equation and fractional Hamiltonian structure to construct the generalized fractional C-KdV equation hierarchy. We consider the following matrix spectral problem: ! k2 þ q2 r /x ¼ U ðu; kÞ/; U ðu; kÞ ¼ ð34Þ ¼ U 0k þ U 1; k 1 q2 2 where k is a spectral parameter. Because U0 has multiple eigenvalues, the spectral problem (31) is degenerated. The spectral problem (31) is called C-KdV spectral problem. To derive an associated soliton hierarchy, we first solve the adjoint equation Dax W ¼ ½U ; W
ð35Þ
of the spectral problem (44) through the generalized Tu scheme [21]. We assume that a solution W is given by a b W ¼ : c a
ð36Þ
Then we have ½U ; W ¼
rc b kb þ qb þ 2ra : kc qc þ 2a rc þ b
Therefore, the adjoint equation (32) is equivalent to Dax a ¼ rc b; Dax b ¼ kb þ qb þ 2ra; Dax c ¼ kc qc þ 2a: Let us seek a formal solution of the type X 1 1 X a b aðkÞ W ¼ ¼ W k kk ¼ c d cðkÞ k¼0 k¼0
ð37aÞ ð37bÞ
bðkÞ d ðkÞ
! kk :
Therefore, the condition (34) becomes the following recursion relation: 8 að0Þ ¼ 12 ; bð0Þ ¼ cð0Þ ¼ 0; > > > > < Da aðnÞ ¼ rcðnÞ bðnÞ ; x ðnþ1Þ > > b ¼ ðDax bðnÞ þ qbðnÞ þ 2raðnÞ Þ; > > : ðnþ1Þ c ¼ ðDax cðnÞ þ qcðnÞ 2aðnÞ Þ: We can obtain the following recursion relation for b(n+1) and c(n+1): ! ! aðnþ1Þ aðnÞ ¼W ; n P 1; cðnþ1Þ cðnÞ where the 2 · 2 matrix operator W is given by
ð38Þ
ð39Þ
ð40Þ
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F. Yu, H. Zhang / Chaos, Solitons and Fractals 37 (2008) 688–697
W¼
a Dax þ Da x qDx 2
a 2Da x rDx þ r : Dax þ q
ð41Þ
As usual, for any integer n P 0, we take V ðnÞ ¼ ðkn W Þþ ¼
n X
W j kðnjÞ ;
ð42Þ
j¼0
and then introduce the time evolution law for the eigenfunction /: ð43Þ /tn ¼ V ðnÞ / ¼ V ðnÞ ðu; ux ; . . . ; kÞ/: ðnþ1Þ c 0 Taking Dn ¼ 12 ; V ðnÞ ¼ ðkn CÞþ þ Dn , the compatibility condition, i.e., the zero curvature equation 0 cðnþ1Þ Datn U Dax V ðnÞ þ ½U ; V ðnÞ ¼ 0, leads to a system of evolution equation ! q Dax cðnþ1Þ a Dt n ; ð44Þ ¼ r Dax aðnþ1Þ which is the generalized fractional version of the C-KdV equation. To construct the Hamiltonian structure of the fractional C-KdV hierarchy (44), we extent the trace identity [17] to the fractional form Z d oU o tr W dx ¼ kc ½kc trðW Dau U Þ; ð45Þ du ok ok with c being a constant to be determined. We defined the fractional variational derivatives X d ðoÞa DauðkÞ ; ðuðkÞ ¼ d k uÞ: ¼ du kP0 An application of (45) yields ~n dH ¼ Gnþ1 ; du
~n ¼ 1 H n
Z
ðnþ1Þ
a
In fact, it is easy to compute that X oU ¼ a ¼ ðak Þkk ; tr W ok kP0
dx;
Gnþ1 ¼
aðnþ1Þ cðnþ1Þ
ð46Þ
! ;
n P 0:
ð47Þ
X a oU ¼ ¼ tr W Gk1 kk : ou c kP0
Upon inserting these two expressions into the fractional trace identity (45) and considering the case of k = 2, we know c = 0, and thus we have (47). Now it follows form (47) that the fractional C-KdV equation (44) have the following biHamiltonian structure: Dat u ¼ JGnþ1 ¼
dH n dH n1 ¼ JW ; du du
n ¼ 1; 2; . . . ;
where the Hamiltonian pair of J is defined by 0 Dax J¼ : Dax 0
ð48Þ
ð49Þ
7. The spectral case: when a = 1 When a = 1, we consider the following matrix spectral problem: ! k2 þ q2 r : /x ¼ U ðu; kÞ/; U ðu; kÞ ¼ k 1 q2 2 When a = 1, the adjoint equation is as followed:
ð50Þ
F. Yu, H. Zhang / Chaos, Solitons and Fractals 37 (2008) 688–697
695
W x ¼ ½U ; W
ð51Þ
of the spectral problem (50) through the Tu scheme [17]. We assume that a solution W is given by a b W ¼ : c a
ð52Þ
Then we have ½U ; W ¼
rc b kb þ qb þ 2ra : kc qc þ 2a rc þ b
Therefore, the adjoint equation (51) is equivalent to 8 > < ax ¼ rc b; > :
bx ¼ kb þ qb þ 2ra; cx ¼ kc qc þ 2a:
Let us seek a formal solution of the type X 1 1 X a b aðkÞ W k kk ¼ W ¼ ¼ c d cðkÞ k¼0 k¼0
bðkÞ aðkÞ
ð53Þ
! kk :
ð54Þ
Therefore, the condition (51) becomes the following recursion relation: 8 ð0Þ a ¼ 1; bð0Þ ¼ cð0Þ ¼ 0; > > > ð1Þ > ð1Þ ð1Þ > > < a ¼ 0; b ¼ r; c ¼ 1; ðnÞ ðnÞ ðnÞ ax ¼ rc b ; > > ðnÞ > > bðnþ1Þ ¼ bðnÞ þ 2raðnÞ ; > x þ qb > : ðnþ1Þ ðnÞ ðnÞ c ¼ cx þ qc 2aðnÞ ;
ð55Þ
when n = 2, 3, we obtain 8 ð2Þ a ¼ r; bð2Þ ¼ rx þ qr; cð2Þ ¼ q; > > > ð3Þ < a ¼ 2qr rx ; > bð3Þ ¼ rxx ðqrÞx qrx þ q2 r þ ar2 ; > > : ð3Þ c ¼ qx q2 2r: We can obtain the following recursion relation for a(n+1) and c(n+1): ! ! aðnþ1Þ aðnÞ ¼L ; n P 1; cðnþ1Þ cðnÞ
ð56Þ
where the matrix operator L is given by ! o þ o1 qo 2o1 o þ r L¼ : 2 oþq
ð57Þ
As usual, for any integer n P 0, we take n X V ðnÞ ¼ ðkn W Þþ ¼ W j kðnjÞ
ð58Þ
j¼0
and then introduce the time evolution law for the eigenfunction /: /tn ¼ V ðnÞ / ¼ V ðnÞ ðu; ux ; . . . ; kÞ/: The compatibility condition of (51) and (59),i.e., the zero curvature equation U tn V system of evolution equations ! q cðnþ1Þ x utn ¼ ¼ : r tn aðnþ1Þ x
ð59Þ ðnÞ x
þ ½U ; V
ðnÞ
¼ 0, leads to a
ð60Þ
696
F. Yu, H. Zhang / Chaos, Solitons and Fractals 37 (2008) 688–697
Among this soliton hierarchy (60), the first nonlinear system reads as ( qt2 ¼ qxx þ ðq2 þ 2rÞx ; rt2 ¼ rxx 2ðqrÞx ; which are the C-KdV equations. To construct the Hamiltonian structure of the C-KdV hierarchy (60), we apply the trace identity [17]: Z d oU o c oU tr W dx ¼ kc k tr W ; du ok ok ou with c being a constant to be determined. An application of (62) yields ! Z ðnþ1Þ ~n a dH 1 ðnþ1Þ ~n ¼ a ¼ Gnþ1 ; H dx; Gnþ1 ¼ ; n P 0: n du cðnþ1Þ In fact, it is easy to compute that oU ¼ a ¼ ðak Þkk ; tr W ok
ð61Þ
ð62Þ
ð63Þ
X a oU ¼ ¼ tr W Gk1 kk : ou c kP0
Upon inserting these two expressions into the trace identity (62) and considering the case of k = 0, we know c = 0, and thus we have (63). Now it follows form (63) that the C-KdV equation (60) have the following bi-Hamiltonian structure utn ¼ JGnþ1 ¼ J
dH n dH n1 ¼M ; du du
n ¼ 1; 2; . . . ;
where the Hamiltonian pair of J is defined by 0 o J¼ : o 0
ð64Þ
ð65Þ
It is easy to verify that JL = L*J. Therefore, the system (64) is Liouville integrable.
Acknowledgements One of authors Fajun Yu would like to express his sincere thanks to Professor Wadati M. for his enthusiastic guidance and help. This work was supported by the National Key Basic Research Development of China(G.N1998030600) and the National Nature Science Foundation of China (G.N10072013).
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Chaos, Solitons and Fractals 37 (2008) 688–697 www.elsevier.com/locate/chaos
New Hamiltonian structure of the fractional C-KdV soliton equation hierarchy Fajun Yu *, Hongqing Zhang Department of Mathematics, Dalian University of Technology, Dalian 116024, PR China Accepted 19 September 2006
Communicated by Prof. M. Wadati
Abstract A generalized Hamiltonian structure of the fractional soliton equation hierarchy is presented by using of differential forms and exterior derivatives of fractional orders. Example of the fractional Hamiltonian system of the C-KdV soliton equation hierarchy is constructed, which is a new Hamiltonian structure. 2006 Published by Elsevier Ltd.
1. Introduction Derivatives and integrals of fractional orders [1,2] have found many applications in recent studies in physics. The interest in fractional analysis has been growing continually during the past few years. Fractional analysis has numerous applications: kinetic theories [3,4,9], statistical mechanics [10,11], dynamics in complex media [12,13] and many others [5–8]. The theory of derivatives of non-integer order goes back to Leibniz, Liouville, Grun-wald, Letnikov and Riemann. In the past few decades 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 fractional derivatives in comparison with classical integer-order models in which such effects are, in fact, neglected. In this paper, we use a fractional exterior calculus that was suggested in [14]. It allows us to consider the fractional generalized Hamiltonian [15,16]. Then, by using of differential forms and exterior derivatives of fractional orders, we extend the trace identity [17–27] to fractional form and obtain fractional Hamiltonian systems of soliton equation hierarchy. In Section 2, the fractional derivatives and integrals is considered to fix notation. In Section 3, a brief review of differential forms is suggested. In Section 4, the fractional form spaces will be given to fix notations and provide a convenient reference. In Section 5, we consider a fractional Hamilton system. In Section 6, a example of the well-known CKdV equation hierarchy [28] is discussed. In Section 7, we give that the spectral case a = 1.
*
Corresponding author. E-mail address: [email protected] (F. Yu).
0960-0779/$ - see front matter 2006 Published by Elsevier Ltd. doi:10.1016/j.chaos.2006.09.049
F. Yu, H. Zhang / Chaos, Solitons and Fractals 37 (2008) 688–697
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2. Brief review of fractional derivatives and integrals The derivatives of arbitrary real order a can be considered as an interpolation of this sequence of operators. We will use for it the notion suggested and used by Davis [29], namely a a Dt f ðtÞ;
ð1Þ
the short name for derivatives of arbitrary order is fractional derivatives. The subscripts a and t denote the two limits related to the operation of fractional differentiation. Following Ross [30], we will call them the terminals of fractional differentiation. The appearance of the terminals in the symbol of fractional is essential. This helps to avoid ambiguities in applications of fractional derivatives to real problems. Integrals of arbitrary order p > 0 p a Dt f ðtÞ
¼
Z t m X f ðkÞ ðaÞðt aÞpþk 1 þ ðt sÞpþm f ðmþ1Þ ðsÞ ds; Cðp þ k þ 1Þ a Cðp þ k þ 1Þ k¼0
ð2Þ
the formula (2) immediately provides us with the asymptotic of a Dp t f ðtÞ at t = 0. Derivatives of arbitrary order Z t pþk m X f ðkÞ ðaÞðt aÞ 1 p lim fnðpÞ ðtÞ ¼ ðt sÞmp f ðmþ1Þ ðsÞ ds; ð3Þ þ a Dt f ðtÞ ¼ h!1;nh!ta Cðp þ k þ 1Þ a Cðp þ k þ 1Þ k¼0 the formula (3) has been obtained under the assumption that the derivatives f(k)(t) (k = 1, 2, . . . , m + 1) are continuous in closed interval [a, t] and that m is an integer number satisfying the condition m > p + 1. The smallest probeable value for m is determined by the inequality m < p < m + 1. From the pure mathematical point of view of such a class of functions is narrow. However this class of functions is very important for applications. Because the character of the majority of dynamical processes is smooth enough and does not allow discontinuities. Understanding this fact is important for the proper use of the methods of the fractional calculus in applications, especially because of the fact the Riemann–Liouville definition mþ1 Z t d p ðt sÞmp f ðsÞ ds ðm 6 p 6 m þ 1Þ ð4Þ a Dt f ðtÞ ¼ dt a provides an excellent opportunity to weaken the conditions on the function f(t). We will think how the Riemann–Liouville definition (4) appears as the result of the unification of the notions of integer-order integration and differentiation. Let us suppose that the function f(s) is continuous and integrable in every finite interval (a, t); the function f(t) may have on integrable singularity of order r < 1 at the point s = a limðs aÞr f ðtÞ ¼ constð6¼ 0Þ: s!a
Then the integral Z t f ðsÞ ds f 1 ðtÞ ¼ a
exists and has a finite value, namely equal to 0, as t ! a. In the general case we have the Cauchy formula Z t 1 ðt sÞn1 f ðsÞ ds: f n ðtÞ ¼ CðnÞ a
ð5Þ
Let us now suppose that n P 1 is fixed and take integer k P 0. Obviously, we will obtain Z k 1 Dk ðt sÞn1 f ðsÞ ds; f kn ðtÞ ¼ CðnÞ a Z k 1 ðt sÞn1 f ðsÞ ds; Dk f kn ðtÞ ¼ CðnÞ a where the symbol Dk(k P 0) and Dk(k P 0) denote k interacted differentiations. To extend the notion if n-fold integration to non-integer values of n, we can start with the Cauchy formula (5) and replace the integer n in it by a real p > 0,
690
F. Yu, H. Zhang / Chaos, Solitons and Fractals 37 (2008) 688–697 p a Dt f ðtÞ
¼
1 CðpÞ
Z
t
ðt sÞp1 f ðsÞ ds:
ð6Þ
a
In (5) the integer n must satisfy the condition n P 1; the responding condition for p is weak; for the existence of integral (6) we must have p > 0. Moreover, under certain reasonable assumptions lim a Dp t f ðtÞ ¼ f ðtÞ:
ð7Þ
p!0
So we can put a D0t f ðtÞ ¼ f ðtÞ. If f(t) is continuous for t P a, the integration of arbitrary real order defined by (6) has the following important property: p q a Dt ðaDt Þf ðtÞ
¼ a Dpq f ðtÞ: t
ð8Þ
Indeed, we have p q a Dt ðaDt Þf ðtÞ ¼
¼
1 CðqÞ
Z
t
ðt sÞq1 a Dp t f ðsÞ ds ¼
a
1 CðpÞCðqÞ
Z
t
f ðnÞ dn
a
Z
1 CðpÞCðqÞ
Z
t
ðt sÞq1 ds
a
t
ðt sÞq1 ðt nÞp1 ds ¼
a
Z
t
ðt nÞp1 f ðnÞ dn a
1 Cðp þ qÞ
Z
t
ðt nÞpþq1 f ðnÞ dn ¼ a Dpq f ðtÞ: t
a
Obviously, we can interchange p and q, so we have p q alpha Dt ðaDt Þf ðtÞ
p pq ¼ a Dq f ðtÞ: t ðaDt Þf ðtÞ ¼ a Dt
ð9Þ
Let us consider some properties of the Riemann–Liouville fractional derivatives. The first and maybe the most important–property of the Riemann–Liouville fractional derivative is that for p > 0 and t > a p p a Dt ðaDt f ðtÞÞ
¼ f ðtÞ:
ð10Þ
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.
3. The differential forms The calculus of differential forms is an elegant branch of pure mathematics and a powerful tool in applied mathematics. A clear introduction to the field, with emphasis on applications, is given in Flanders [5]. Vector spaces at a point P 2 En (n-dimensional Euclidean space) can be constructed out of expressions of the following type: one forms; a ¼ two forms; b ¼
n X i¼1 n X
ai dxi ;
ð11Þ
bij dxi ^ dxj ;
ð12Þ
i;j¼1
.. . n forms; x ¼ w dx1 ^ dx2 ^ ^ dxn ;
ð13Þ
where the {xi} are the Cartesian coordinates of En. The above sums are taken over all possible values of the indices with the constraint that dxi ^ dxj ¼ dxj ^ dxi :
ð14Þ
The functions ai, bij, etc., depend only on P and may be real or complex depending on the application. If a k form, c, is multiplying an m form, l, the following would be true: c ^ l ¼ ð1Þkm l ^ c:
ð15Þ
The result would be zero if k + m > n. The exterior product, ^, is distributive, and antisymmetric. The dimension of the vector space of k forms over P 2 En is n n! ; ¼ k!ðn kÞ k
F. Yu, H. Zhang / Chaos, Solitons and Fractals 37 (2008) 688–697
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which is zero if k > n. For the purposes of this paper let F(k, k, n) denote the vector space of k forms over P 2 En. The apparently redundant n in the above notation will be needed later for the fractional form case, as there is some additional freedom. The exterior derivative is defined as d¼
n X
dxi
i¼1
o : oxi
ð16Þ
The exterior derivative maps k forms into k + 1 forms and has the following algebraic properties. Let c and k be k forms, and l be an m form, then dðc þ kÞ ¼ dc þ dk;
ð17Þ k
dðc ^ lÞ ¼ ðdcÞ ^ l þ ð1Þ c ^ dl;
ð18Þ
dðdcÞ ¼ 0:
ð19Þ
The last identity is called the Poincare´ lemma. A form, c, is called closed if dc = 0. A form, c, is called exact if there exists a form, l, such that dl = c. The order of l is one less than the order of c. Exact forms are always closed. Closed forms are not always exact.
4. Fractional form spaces If the partial derivatives in the definition of the exterior derivative are allowed to assume fractional orders, a fractional exterior derivative can be defined da ¼
n X
dxai
i¼1
oa : ðoðxi ai ÞÞa
ð20Þ
Note that the subscript i denote number, the superscript a denotes the order of the fractional coordinate differential, and ai is the initial point of the derivative. Sometimes the notation oai will be used to denote oa : ðoðxi ai ÞÞa In two dimension (x, y), the fractional exterior derivative of order a of xp, with the initial point taken to be the origin, is given by da xp ¼ dxa
Cðp þ 1Þ pa xp x þ dy a a : ðp a þ 1Þ y Cð1 aÞ
ð21Þ
For specific values of the derivative parameter the following results are obtained: a ¼ 0;
d0 xp ¼ 2xp ; 1 p
1
ð22Þ p1
a ¼ 1;
d x ¼ dx px
a ¼ 2;
d2 xp ¼ dx2 pðp 1Þxp2 :
;
ð23Þ ð24Þ
By analogy with standard exterior calculus, vector spaces can be constructed using the dxai . Let F(a, m, n) be a vector space at P 2 En. a denotes the sum of the fractional differential orders of the basis elements, m denotes the number of coordinate differentials appearing in the basis element, n the number of coordinates, and {xi} are the Cartesian coordinates for En. For example, a basis set for F(a, 1, n) would be fdxa1 ; dxa2 ; . . . ; dxan g and arbitrary element of F(a, 1, n) would be expressed as n X a¼ ai dxai : i¼1
For a fixed a this is an n-dimensional vector space. Also note that there is a different vector space for each value of a. For a = 1 the one forms from exterior calculus are recovered. Now suppose that the basis elements are made up of two coordinate differentials, F(a, 2, n). In this case the basis set is more complicated, l
fdxl1 11 ^ dxl121 ; dxl1 11 ^ dx231 ; . . . ; dxln n1m ^ dxln nm jlij þ lkj ¼ ag:
ð25Þ
692
F. Yu, H. Zhang / Chaos, Solitons and Fractals 37 (2008) 688–697
Note that dxl1 11 ^ dxl1 21 would be zero if and only of l11 = l21, etc. An arbitrary element of F(a, 2, n) would be expressed as a sum of the form n X n Z a X 1 b¼ ðbij ða1 ; a a1 Þ dxai 1 ^ dxaa Þ da1 ; ð26Þ j i¼1
j¼1
0
where bii(l, l) = 0. Unlike the previous vector space, F(a, 1, n), this is clearly infinite dimensional for any value of a. Not only is it infinite but it is uncountably infinite. An arbitrary element of F(a, 3, n) would be expressed as an integral of the form n X n X n Z a Z aa1 X 1 a2 b¼ ðbijk ða1 ; a a2 ; a a1 a2 Þ dxai 1 ^ dxaj 2 ^ dxaa Þ da2 da: ð27Þ k i¼1
j¼1
k¼1
0
0
With each step up on the middle index of F(a, m, n) another integral and summation is included. A basis set for this vector space would be ( ) m X l11 l21 lm1 F ða; m; nÞ ¼ dxa1 ^ dxa2 ^ ^ dxam ; . . . l ¼a : ð28Þ k¼1 ij The basis elements range over all possible combinations of the fractional coordinate differentials and all possible choices for the l. Note that m need not be less than or equal to n. Let P 2 En, and let A 2 F(a, m, n) and B 2 F(l, k, n) at the point P. Then the exterior product of A and B maintains the antisymmetry property of Eq. (17), A ^ B ¼ ð1Þkm B ^ A 2 F ðl þ a; k þ 1; nÞ:
ð29Þ
If k + m > n, A ^ B need not be zero. Eq. (16) is also maintained due to the linearity of the fractional derivative. Eq. (19) is not maintained due to the product rule for the fractional derivative. Note also that da maps F(l, m, n) into F(l + a, k + 1, n).
5. Fractional generalized Hamiltonian systems Fractional phase space can be considered as a phase space of the systems that are described by the fractional powers of coordinates and momenta. Let us consider the class of Hamiltonian systems that are constructed in the fractional phase space. Nowadays it is well known that a great number of nonlinear evolution equations, arising form various branches of physics, they admit the zero curvature equation U t V x þ ½U ; V ¼ 0:
ð30Þ
Our target is to write the fractional zero curvature equation, which has more excellent instrument than the integer form Dat U Dax V ðnÞ þ ½U ; V ðnÞ ¼ 0;
ð31Þ
which is generalized fractional zero curvature equation. Fractional generalized differential form [12]: b = Gi dpi Fi dqi (the differential 1-form), which is used in the definition of the Hamiltonian system, can be defined in the following form: ba ¼ Gi ðdpi Þa F i ðdqi Þa : Let us consider the canonical coordinates (x1, . . . , xn, xn+1, . . . ,x2n) = (q1, . . . , qn,p1, . . . , pn) in the phase space E2n and a dynamical systems can be defined by the equations ! Dapi 0 Hn Hn a qi ¼ ¼ J ; ð32Þ Dt 0 Daqi pi Hn Hn with Dapi H n ¼ Gi ðp; qÞ;
Daqi H n ¼ F i ðp; qÞ:
ð33Þ
The fractional generalization of Hamiltonian system can be defined by the equations generalization of differential forms [14].
F. Yu, H. Zhang / Chaos, Solitons and Fractals 37 (2008) 688–697
693
A dynamical system (31) on the phase space E2n, it called a fractional Hamiltonian system if the fractional differential 1-form ba = Gi(dpi)a Fi(dqi)a is a closed fractional form daba = 0, where da is the fractional exterior derivative. A dynamical system is called a fractional non-Hamiltonian system if the fractional differential 1-form ba is a non-closed fractional form daba 5 0. A central and very important subject in the theory of integrable system is to search for a sequence of scalar functions {Hn} such that Eq. (29) can be cast in the Hamiltonian form ! Dapi 0 Hn a qi Dt ¼ : 0 Daqi pi Hn The Hamiltonian {Hn} constitute, in fact, an infinite number of conserved densities of the hierarchy (29).
6. The generalized fractional C-KdV equation hierarchy To illustrate our method, we want to apply the fractional zero curvature equation and fractional Hamiltonian structure to construct the generalized fractional C-KdV equation hierarchy. We consider the following matrix spectral problem: ! k2 þ q2 r /x ¼ U ðu; kÞ/; U ðu; kÞ ¼ ð34Þ ¼ U 0k þ U 1; k 1 q2 2 where k is a spectral parameter. Because U0 has multiple eigenvalues, the spectral problem (31) is degenerated. The spectral problem (31) is called C-KdV spectral problem. To derive an associated soliton hierarchy, we first solve the adjoint equation Dax W ¼ ½U ; W
ð35Þ
of the spectral problem (44) through the generalized Tu scheme [21]. We assume that a solution W is given by a b W ¼ : c a
ð36Þ
Then we have ½U ; W ¼
rc b kb þ qb þ 2ra : kc qc þ 2a rc þ b
Therefore, the adjoint equation (32) is equivalent to Dax a ¼ rc b; Dax b ¼ kb þ qb þ 2ra; Dax c ¼ kc qc þ 2a: Let us seek a formal solution of the type X 1 1 X a b aðkÞ W ¼ ¼ W k kk ¼ c d cðkÞ k¼0 k¼0
ð37aÞ ð37bÞ
bðkÞ d ðkÞ
! kk :
Therefore, the condition (34) becomes the following recursion relation: 8 að0Þ ¼ 12 ; bð0Þ ¼ cð0Þ ¼ 0; > > > > < Da aðnÞ ¼ rcðnÞ bðnÞ ; x ðnþ1Þ > > b ¼ ðDax bðnÞ þ qbðnÞ þ 2raðnÞ Þ; > > : ðnþ1Þ c ¼ ðDax cðnÞ þ qcðnÞ 2aðnÞ Þ: We can obtain the following recursion relation for b(n+1) and c(n+1): ! ! aðnþ1Þ aðnÞ ¼W ; n P 1; cðnþ1Þ cðnÞ where the 2 · 2 matrix operator W is given by
ð38Þ
ð39Þ
ð40Þ
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F. Yu, H. Zhang / Chaos, Solitons and Fractals 37 (2008) 688–697
W¼
a Dax þ Da x qDx 2
a 2Da x rDx þ r : Dax þ q
ð41Þ
As usual, for any integer n P 0, we take V ðnÞ ¼ ðkn W Þþ ¼
n X
W j kðnjÞ ;
ð42Þ
j¼0
and then introduce the time evolution law for the eigenfunction /: ð43Þ /tn ¼ V ðnÞ / ¼ V ðnÞ ðu; ux ; . . . ; kÞ/: ðnþ1Þ c 0 Taking Dn ¼ 12 ; V ðnÞ ¼ ðkn CÞþ þ Dn , the compatibility condition, i.e., the zero curvature equation 0 cðnþ1Þ Datn U Dax V ðnÞ þ ½U ; V ðnÞ ¼ 0, leads to a system of evolution equation ! q Dax cðnþ1Þ a Dt n ; ð44Þ ¼ r Dax aðnþ1Þ which is the generalized fractional version of the C-KdV equation. To construct the Hamiltonian structure of the fractional C-KdV hierarchy (44), we extent the trace identity [17] to the fractional form Z d oU o tr W dx ¼ kc ½kc trðW Dau U Þ; ð45Þ du ok ok with c being a constant to be determined. We defined the fractional variational derivatives X d ðoÞa DauðkÞ ; ðuðkÞ ¼ d k uÞ: ¼ du kP0 An application of (45) yields ~n dH ¼ Gnþ1 ; du
~n ¼ 1 H n
Z
ðnþ1Þ
a
In fact, it is easy to compute that X oU ¼ a ¼ ðak Þkk ; tr W ok kP0
dx;
Gnþ1 ¼
aðnþ1Þ cðnþ1Þ
ð46Þ
! ;
n P 0:
ð47Þ
X a oU ¼ ¼ tr W Gk1 kk : ou c kP0
Upon inserting these two expressions into the fractional trace identity (45) and considering the case of k = 2, we know c = 0, and thus we have (47). Now it follows form (47) that the fractional C-KdV equation (44) have the following biHamiltonian structure: Dat u ¼ JGnþ1 ¼
dH n dH n1 ¼ JW ; du du
n ¼ 1; 2; . . . ;
where the Hamiltonian pair of J is defined by 0 Dax J¼ : Dax 0
ð48Þ
ð49Þ
7. The spectral case: when a = 1 When a = 1, we consider the following matrix spectral problem: ! k2 þ q2 r : /x ¼ U ðu; kÞ/; U ðu; kÞ ¼ k 1 q2 2 When a = 1, the adjoint equation is as followed:
ð50Þ
F. Yu, H. Zhang / Chaos, Solitons and Fractals 37 (2008) 688–697
695
W x ¼ ½U ; W
ð51Þ
of the spectral problem (50) through the Tu scheme [17]. We assume that a solution W is given by a b W ¼ : c a
ð52Þ
Then we have ½U ; W ¼
rc b kb þ qb þ 2ra : kc qc þ 2a rc þ b
Therefore, the adjoint equation (51) is equivalent to 8 > < ax ¼ rc b; > :
bx ¼ kb þ qb þ 2ra; cx ¼ kc qc þ 2a:
Let us seek a formal solution of the type X 1 1 X a b aðkÞ W k kk ¼ W ¼ ¼ c d cðkÞ k¼0 k¼0
bðkÞ aðkÞ
ð53Þ
! kk :
ð54Þ
Therefore, the condition (51) becomes the following recursion relation: 8 ð0Þ a ¼ 1; bð0Þ ¼ cð0Þ ¼ 0; > > > ð1Þ > ð1Þ ð1Þ > > < a ¼ 0; b ¼ r; c ¼ 1; ðnÞ ðnÞ ðnÞ ax ¼ rc b ; > > ðnÞ > > bðnþ1Þ ¼ bðnÞ þ 2raðnÞ ; > x þ qb > : ðnþ1Þ ðnÞ ðnÞ c ¼ cx þ qc 2aðnÞ ;
ð55Þ
when n = 2, 3, we obtain 8 ð2Þ a ¼ r; bð2Þ ¼ rx þ qr; cð2Þ ¼ q; > > > ð3Þ < a ¼ 2qr rx ; > bð3Þ ¼ rxx ðqrÞx qrx þ q2 r þ ar2 ; > > : ð3Þ c ¼ qx q2 2r: We can obtain the following recursion relation for a(n+1) and c(n+1): ! ! aðnþ1Þ aðnÞ ¼L ; n P 1; cðnþ1Þ cðnÞ
ð56Þ
where the matrix operator L is given by ! o þ o1 qo 2o1 o þ r L¼ : 2 oþq
ð57Þ
As usual, for any integer n P 0, we take n X V ðnÞ ¼ ðkn W Þþ ¼ W j kðnjÞ
ð58Þ
j¼0
and then introduce the time evolution law for the eigenfunction /: /tn ¼ V ðnÞ / ¼ V ðnÞ ðu; ux ; . . . ; kÞ/: The compatibility condition of (51) and (59),i.e., the zero curvature equation U tn V system of evolution equations ! q cðnþ1Þ x utn ¼ ¼ : r tn aðnþ1Þ x
ð59Þ ðnÞ x
þ ½U ; V
ðnÞ
¼ 0, leads to a
ð60Þ
696
F. Yu, H. Zhang / Chaos, Solitons and Fractals 37 (2008) 688–697
Among this soliton hierarchy (60), the first nonlinear system reads as ( qt2 ¼ qxx þ ðq2 þ 2rÞx ; rt2 ¼ rxx 2ðqrÞx ; which are the C-KdV equations. To construct the Hamiltonian structure of the C-KdV hierarchy (60), we apply the trace identity [17]: Z d oU o c oU tr W dx ¼ kc k tr W ; du ok ok ou with c being a constant to be determined. An application of (62) yields ! Z ðnþ1Þ ~n a dH 1 ðnþ1Þ ~n ¼ a ¼ Gnþ1 ; H dx; Gnþ1 ¼ ; n P 0: n du cðnþ1Þ In fact, it is easy to compute that oU ¼ a ¼ ðak Þkk ; tr W ok
ð61Þ
ð62Þ
ð63Þ
X a oU ¼ ¼ tr W Gk1 kk : ou c kP0
Upon inserting these two expressions into the trace identity (62) and considering the case of k = 0, we know c = 0, and thus we have (63). Now it follows form (63) that the C-KdV equation (60) have the following bi-Hamiltonian structure utn ¼ JGnþ1 ¼ J
dH n dH n1 ¼M ; du du
n ¼ 1; 2; . . . ;
where the Hamiltonian pair of J is defined by 0 o J¼ : o 0
ð64Þ
ð65Þ
It is easy to verify that JL = L*J. Therefore, the system (64) is Liouville integrable.
Acknowledgements One of authors Fajun Yu would like to express his sincere thanks to Professor Wadati M. for his enthusiastic guidance and help. This work was supported by the National Key Basic Research Development of China(G.N1998030600) and the National Nature Science Foundation of China (G.N10072013).
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