Generalized Tu Formula and Hamilton Structures of Fractional Soliton Equation Hierarchy

Generalized T u F orm ula and H amil ton Structures of F ractio nal Soliton Eq uation Hierarc h y Guo-c heng W u 1 ∗ , She ng Zhang 2 1. Colledge of T extile, Donghua Un iv ersit y , Shanghai 2016 20, P .R. China; 2. Sc ho ol of Mathematical Sciences, D alian Univ ersit y of T ec hnology , Dalian 11 6024, P .R. China. ——————————————————————— — — — — — — — — — — — — — — — – Abstract With the m o dified Riemann-Liouville fractional deriv ati ve , a fr actional T u formula is pre- sen ted to in v estigat e generalized Hamilto n structur e of fractional soliton equations. The obtained results can be reduced to the classical Hamilton hierac h y of ordinary calculus. Key words: F ractionalize d T u formula; F r actional Hamilton system; F ractional ev olutionary equations P ACS: 02. 0 3. Ik; 05.45.Df; 05.30.Pr ——————————————————————— — — — — — — — — — — — — — — — – 1 In tro duction Nob el Laureate Gerardu s’t Ho oft once r emark ed th at discrete space-time is the most radical and logica l viewp oint of realit y . In suc h discon tin uous s p ace-time, fr actional calculus p la ys an imp ortant role which can accurately describ e m any nonlinear phenomena in physics, i.e., Bro w - nian motion, anomalous diffusion, transp ortation in p orous media, c haotic dyn amics, p h ysical kinetics and quantum mec hanics. P ast decades witness the deve lopment of fractional calculus in v arious fi elds , suc h as rheology , quan titativ e biology , electro c hemistry , scattering theory , diffusion, transp ort theory , p r obabilit y p oten tial theory and elasticit y . F or details, see the mon ograph s of Kilbas et al. [1], Kiry ak o v a [2], L akshmik antham and V atsala [3], Miller and Ross [4] , and Podlu bny [5 ]. ∗ Corresponding author, E-mail: wuguo cheng2002@y aho o.com.cn (G.C. W u) 1 Since Riewe [6] prop osed a concept of non-conserv ation mec hanics, f ractional conserv ation la ws [7], fracti onal Lie symmetries [8] and fractio nal Hamilton systems [9 –16] ha v e caugh t m uc h atten tion. In recen t stu dy , F u jiok a et al found that the p ropagation of optical s olitons can b e describ ed by an extend ed n onlinear Shr ¨ o dinger equation wh ic h incorp orates fractional d eriv ativ es [17, 18]. Searc hing for new int egrable hierarchies of soliton equatio ns is an imp ortan t and in teresting topic in s oliton theory . T u sc heme of ordinary calculus [19] is an efficient metho d to generate in tegrable Hamilton systems. I t to ok v arious efficien t approac hes to ha ve obtained man y in te- grable systems s u c h as AKNS hierarc hy , KN h ierarc h y , Schr ¨ o dinger system, and so on [20–26]. In order to consider the Hamilton structur e of fractional soliton equations, some questions ma y naturally arise: (1) Can we ha v e a generalized T u sheme in fr actional case? (2) How to d efine Hamilton equations for fractional soliton hierac hy? In this study , w e start from a Lax p air of fractional ord er in the sense of the mo difi ed Riemann-Liouville’s d eriv ativ e [14] and prop ose a generalized T u sheme to in ve stigate the Hamil- ton str ucture of fractional soliton ev olutionary equations. 2 Mo dified Riemann-Liouville d eriv ativ e Generally , there are t wo kinds o f fractional deriv ativ es: lo cal f ractional deriv ativ es and nonlo cal ones. The most u sed nonlo cal op erator is the Capu to deriv ativ e whic h requires the defined functions should b e differen tiable. The condition is s o str ict that man y engineering problems cannot sati sfy , i.e., fu nctions defined on fractal curv es, fractional diffusion pr oblem. As a result, the C aputo deriv ativ e is not suitable f or such problems theoretically . Sev eral lo cal v ersions ha v e b een prop osed: Kolw ank ar-Gangal’s lo cal fr actional d eriv ativ e [27–29 ], Chen’s fractal deriv ativ e [30, 31 ], Cresson’s deriv ativ e [32], Ju mrie’s mo dified Riemman- Liouville d eriv ativ e [33] and Parv ate’s F α deriv ativ e [34], among which J umari’s mo dified R-L deriv ativ e is defined as D α x f ( x ) = 1 Γ(1 − α ) Z x 0 ( x − ξ ) − α ( f ( ξ ) − f (0)) dξ , 0 < α < 1 . (1) Here the d eriv ativ e on the right- hand side is the Riemann -Liouville fractional deriv ativ e. The nonlinear te c hn iqu es for such fractional differen tiable e quations can b e found in Refs. [8, 35, 3 6] W e can hav e follo wing resu lts for Ju marie’s mo d ified Riemann-Liouville (R-L) deriv ativ e. (a) T he Leibniz pro d uct la w If f ( x ) is an α order d ifferen tiable f unction in the area of p oin t x , from the Rolle-Kolw ank ar- Jumarie’s T a ylor series [31], one can hav e D α x f ( x ) = lim y → x Γ(1 + α )( f ( y ) − f ( x )) ( y − x ) α , 0 < α < 1 . (2) 2 Assuming g ( x ) is an α order different iable function, the Leibniz pro d uct la w can h old D α x ( f ( x ) g ( x )) = g ( x ) D α x f ( x ) + f ( x ) D α x g ( x ) . (3) (b) I n tegration with resp ect to ( dx ) α (Lemma 2.1 of [37]) 0 I α x f ( x ) = 1 Γ( α ) Z x 0 ( x − ξ ) α − 1 f ( ξ ) dξ = 1 Γ( α + 1) Z x 0 f ( ξ )( dξ ) α , 0 < α ≤ 1 . (4) (c) Generalized Newton-Leibniz F ormulat ion Assume D α x f ( x ) is a integrable fun ction in the int erv al [ a, b ]. Obviously , 1 Γ(1 + α ) Z b a D α x f ( x )( dx ) α = f ( b ) − f ( a ) , (5) 1 Γ(1 + α ) Z x a D α ξ f ( ξ )( dξ ) α = f ( x ) − f ( a ) , (6) and D α x Γ(1 + α ) Z x a f ( ξ )( dξ ) α = f ( x ) . (7) (d) I n tegration b y p arts With the prop erties (b) and (c), in tegratio n b y parts for α ord er differen tiable functions f ( x ) and g ( x ) can b e presen ted as 1 Γ(1 + α ) Z b a g ( x ) D α x f ( x )( dx ) α = g ( x ) f ( x ) | b a − 1 Γ(1 + α ) Z b a f ( x ) D α x g ( x )( dx ) α . (8) The ab o ve p rop erties (a)–(d) can b e found in Ref. [33]. (e) F r actional v ariational deriv ativ e F rom Jumarie’s v ariational deriv ativ e [14] a nd Almeida’ fr actional v ariational approac h [15] , the f ractional v ariational d eriv ativ e is d efined as δ L δ y = ∂ L ∂ y + X k =1 ( − 1) k ( D α x ) k ( ∂ L ∂ ( D α x ) k y ) , (9) where k is a p ositiv e inte ger. (f ) F rom Eq. (2), we can h a v e D α x f ( x ) = lim h → 0 Γ(1 + α )( f ( x + h ) − f ( x )) h α = Γ(1 + α ) d f ( x ) ( dx ) α , 0 < α < 1 . (10) As a result, w e can find that f ( b ) − f ( a ) = 1 Γ(1 + α ) Z b a D α x f ( x )( dx ) α = 1 Γ(1 + α ) Z b a d α f ( x ) . (11) 3 3 F ractional Hamilton Structure A num b er of usefu l attempts ha v e b een made to establish fractional v ariational principles and Hamilton system [9–16]. Differen t t yp es of fractional deriv ativ es ma y lead to differen t results, for examples, i.e., Balean u ’s fractional Hamilton system w ith Capu to deriv ativ e [10], Riemann- Liouville type Hamilton m ec hanics [11], Ar gwal’s Hamilton F ormulatio n w ith Riesz deriv ativ e [12] and J umairie’s Lagrange formula [14]. In this section, w e revisit Ju marie’s fractional Hamil- ton sy s tem. 3.1 A F r actional Exterior Differen tial Approac h Since Ben Adda prop osed the fractional generaliza tion of differen tial [38, 39], m an y f ractional ex- terior differen tial approac hes and applications relate d to differen t forms of fr actional deriv ativ es app eared in op en literature [40–42]. A b rief review is a v ailable in T araso v’s work [43]. Starting fr om the total deriv ativ e in the in teger dimensional space, and assuming f = f ( u, v ) , u = u ( x ) and v = v ( x ), where u , v are α order differenti able fu ctions and f is a differen tiable function with resp ect to u and v , we obtain the total deriv ativ e as follo ws d f = f u du + f v dv . (12) Mutiple th e b oth sides of Eq. (12) with Γ(1 + α ), we can get d α f = f u d α u + f v d α v = f u D α x u ( dx ) α + f v D α x v ( dx ) α . (13) On the other hand, if w e assume that u , v are differen tiable fuctions and f is a α order differen tiable function with resp ect to u and v d f = f ( α ) u Γ(1 + α ) ( du ) α + f ( α ) v Γ(1 + α ) ( dv ) α , (14) w e can d er ive a different definition of d α as follo ws d α f = f ( α ) u ( u x ) α ( dx ) α + f ( α ) v ( v x ) α ( dx ) α . (15) Th us, w e ma y ha ve t wo different results f or D α x x 2 , 2Γ(1+ α ) x 1 − α Γ(2 − α ) or 2 x 2 − α Γ(3 − α ) , r esp ectiv ely , if we kno w n othing ab out the differen tiablit y of x 2 . 3.2 F ractional Hamilton E quations W e define the fractional functional J [ p, q ] = 1 Γ(1 + α ) Z [ pD α t q − H ( t, p , q )]( dt ) α (16) Then, we can r eadily d eriv e the fr actional P oincare–Cartan 1-form, whic h reads 4 ω = pd α q − H ( dt ) α . (17) F rom Eq. (17), w e ha v e d α ω = p ( α ) t ( dt ) α ∧ d α q + d α p ∧ d α q − ∂ H ∂ p d α p ∧ ( dt ) α − ∂ H ∂ q d α q ∧ ( dt ) α = [ p ( α ) t + ∂ H ∂ q ]( dt ) α ∧ d α q + [ ∂ H ∂ p ( dt ) α − d α q ] ∧ d α p. (18) The fractional close d condition d α ω = 0 allo ws us to obtain the follo wing fracti onal Hamilton equations D ( α ) t q = ∂ H ∂ p , (19) and D ( α ) t p = − ∂ H ∂ q . (20) W e must p oin t out, the results Eq . (13), Eq. (15), Eq . (19) and Eq. (20) can b e found in Ref. [14]. 4 F ractional T u F orm ula and Its A pplication 4.1 A F r actionalized T u F orm ula Set A n = A = ( a i, j ) , a i, j ∈ C . Assume A and B ∈ C . Deffine [ A, B ] = AB − B A . Hence, A n is a Lie algebra. The corresp ondin g lo op algebra is defin ed as ˜ A n = A ( n ) = A λ n , n ∈ Z . (21) T u F orm ula is a b eautiful identit y to generate in tegral Hamilton equations. In the past decades, man y integral Hamilton hierarc hies are obtained via this tec hnical [20–26] . Consider the f ractional compatibilit y condition, φ ( α ) x ( x, t ) = U φ, φ ( β ) t ( x, t ) = V φ, (22) where the fractional deriv ativ e is in the sense of the mo difi ed R-L deriv ativ e [14, 31], and φ is a n -dimensional fu nction ve ctor. The compatibilit y condition of Eq. (22) leades to the generalized zero curv ature equation U ( β ) t − V ( α ) x + [ U, V ] = 0 , [ U, V ] = U V − V U. (23) When taking α = β = 1, Eq. (22) reduces to the classical zero curv ature equation. Set U = e 0 ( λ ) + n X i =1 e i ( λ ) u i , { e i ( λ ) , 0 ≥ i ≤ n } ⊂ ˜ A n , (24) 5 where u = u ( u 1 , u 2 , ..., u n ) T denotes a vecto r fu nction. By th e gradation of ˜ B n , define r ank λ =deg( λ ), then rank ( e 0 ( λ )) = α , r ank ( e i ( λ )) = α i , 0 ≤ i ≤ n are all kn o wn. If w e tak e the ranks of u i as α − α i , 1 ≤ i ≤ n , then eac h term in U is of the same r ank α , denoted by rank rank (U) = r ank ( ∂ α ∂ x α ) = α. (25) If a solution of the stationary zero curv ature equation − V ( α ) x + [ U, V ] = 0 , (26) is give n b y V = P m ≥ 0 V m λ − m , ( V m ) λ = 0, m ≥ 0. rank ( V m ) λ is assum ed to b e giv en so that rank ( V m ) λ = β , m ≥ 0 , then eac h team in V h as th e same rank, d en oted by rank(V) = r ank( ∂ β ∂ t β ) = η . (27) Supp ose f ( A, B ) = tr(AB). The f ollwoing prop erties can b e satisfied (a) S ymmetry relationship f ( A, B ) = f ( B , A ); (b) T he b ilinearity can hold 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 lo cal f r actional d eriv ativ e, the gradient ∇ B f ( A, B ) of the functional f ( A, B ) is defined by ∂ ∂ ǫ f ( A, B + ǫC ) = f ( δ B f ( A, B ) , C ) , ∀ A, B , C ∈ ˜ A n , (28) where δ B is v ariational d eriv ativ e with resp ect to B . With the fractional v ariational deriv ativ e, we can h av e th e follo wing results, δ B f ( A, B k α x ) = ( − 1) k A ( kα ) x , (29) where k is a p ositiv e inte ger and D k α x = D α x ...D α x | {z } k . (d) C omm unication relationship f ([ A, B ] , C ) = f ( A, [ B , C ]) , ∀ A, B , C ∈ ˜ A n . (30) Construct a fun ctional W = f ( V , U λ ) + f ( K , V ( α ) x − [ U, V ]) , (31) where U , V meet Eq. (22), K ∈ ˜ A n , rank K =-r an k λ . 6 With the defin ed fractional v ariational deriv ative , δ W δ K = V ( α ) x − [ U, V ] , δ W δ V = U λ − K ( α ) x + [ U, V ] , (32) from the ab o ve equations, w e can derive [ K, V ] ( α ) x = [ K ( α ) x , V ] + [ K, V ( α ) x ] = [ U λ + [ U, K ] , V ] + [ K , [ U, V ]] = [ U λ , V ] + [[ U, K ] , V ] + [[ V , U ] , K ] = [ U λ , V ] + [ U, [ K, V ]] . W e can c hec k V ′ = [ K, V ] − V λ satisfies Eq. (26) and V λ also satisfies Eq. (26) s ince rank( Z )= r ank( V λ )=rank( V )-rank( λ )=rank( V λ ). Therefore, if t wo solutions of Eq. (23), V and V ′ are linearly dep enden t, w e can get V ′ = γ λ V . Using E q . (31) again, we can ha ve a fractional trace ident y as follo ws δ f ( V , U λ ) δ u i = f ( V , ∂ U λ ∂ u i ) + f ([ K, V ] , ∂ U λ ∂ u i ) = f ( V , ∂ U λ ∂ u i ) + f ( V λ , ∂ U λ ∂ u i ) + γ λ f ( V , ∂ U ∂ u i ) = ∂ ∂ λ f ( V , ∂ U ∂ u i ) + f ( V λ , ∂ U λ ∂ u i ) + ( λ − γ ∂ ∂ λ λ γ ) f ( V , ∂ U ∂ u i ) = λ − γ ∂ ∂ λ [ λ γ f ( V , ∂ U ∂ u i )] , 0 ≤ i ≤ n. W e must p oint ou t, the v ariational d eriv ativ e here is d efined in the sense of the mo dified R-L fractional deriv ativ e. 4.2 F ractional Soliton Hierarchie s and Their Hamilton Structures Recen tly , F ujiok a et al [17] f ound the pr opagation of optical solitons can b e describ ed b y an extended NLS equation whic h incorp orates fractional deriv ativ es. The d etailed r eview can b e found in Ref. [18]. In view of this p oin t, we consider f r actional AKNS hierarc h y strating from the generalized sp ectral problem Φ ( α ) x = U ( λ, u ) = − λ q r λ ! Φ , u =    q r    , Φ = φ 1 φ 2 ! . (33) Cho ose a simple subalgebra of A 1 e 1 (0) = 1 0 0 − 1 ! , e 2 (0) = 0 1 0 0 ! , e 3 (0) = 0 0 1 0 ! , (34) 7 equipp ed with the comm utativ e relations [ e 1 ( m ) , e 2 ( n )] = 2 e 2 ( m + n ) , [ e 1 ( m ) , e 3 ( n )] = − 2 e 3 ( m + n ) , [ e 2 ( m ) , e 3 ( n )] = e 1 ( m + n ) . (35) Then, we find that the adjoin t representa tion equation V ( α ) x = [ U, V ] = U V − V U yields a ( α ) 0 x = q c 0 − r b 0 , b 0 = 0 , c 0 = 0 , a ( α ) ix = q c i − r b i , b ( α ) ix = − 2 b i +1 − 2 q a i , c ( α ) ix = 2 r a i + 2 c i +1 , i ≥ 1 , the fi rst few of wh ic h r eads a 0 = − 1 , b 0 = 0 , c 0 = 0 , a 1 = 0 , b 1 = q , c 1 = r , b 2 = − 1 2 q ( α ) x , c 2 = 1 2 r ( α ) x , a 2 = 1 2 q r. W e can derive the recurenece relationship c n +1 b n +1 ! = 1 2 D α x − 2 r D − α x q 2 r D − α x r − 2 q D − α x q − D α x + 2 q D − α x ! c n b n ! = L c n b n ! , (36) a n = D − α x ( q c n − r b n ) , n = 0 , 1 , 2 , 3 · · · . Denoting ( V ( n ) + ) ( α ) x = n X i =0 a ( i ) e 1 ( n − i ) + b ( i ) e 2 ( n − i ) + c ( i ) e 3 ( n − i ) , V ( n ) − = λ n V − V ( n ) + , Eq. (26) can b e written as − ( V ( n ) + ) ( α ) x + [ U, V ( n ) + ] = ( V ( n ) − ) ( α ) x − [ U, V ( n ) − ] . (37) It is easy to verify that the terms on the left-hand side in (37) are of degree ≥ 0, while th e terms on the right -hand side in Eq. (37) are of degree ≤ 0. T h us, w e ha v e − ( V ( n ) + ) ( α ) x + [ U, V ( n ) + ] = 2 b ( n + 1) e 2 (0) − 2 c ( n + 1) e 3 (0) . T aking an arbitrary mo difi ed term f or V ( n ) + as △ n = 0. Notice V ( n ) = V ( n ) + , it is easy to compute, the zero curv atur e equation U ( β ) t − ( V ( n ) ) ( α ) x + [ U, V ( n ) ] = 0 , (38) 8 whic h giv es rise to u β t =  q ( β ) r ( β )  t =  0 − 2 2 0   c ( n + 1) b ( n + 1)  = J L  c ( n ) b ( n )  = J L n r q ! . (39) Here J is Hamiltonian op erator. F or n = 2, we obtain the fractional AK NS equations ( D β t q t 1 = − 1 2 D α x D α x q + q 2 r , D β t r t 1 = 1 2 D α x D α x r − q r 2 . (40) When α = β = 1, Eq. (40) can b e reduced to the classical AKNS system, ( q t 1 = − 1 2 q xx + q 2 r , r t 1 = 1 2 r xx − q r 2 . (41) In ord er to use the prop osed trace identit y , a deirect compute leads to f ( V , U λ ) = − 2 a n , f ( V , ∂ U ∂ q ) = c n , f ( V , ∂ U ∂ r ) = b n . (42) As a result, δ ( − 2 a ) δ u = λ − γ ∂ ∂ λ λ γ  c b  . (43) Compared w ith the coefficients of λ − n − 1 , δ ( − 2 a n +1 ) δ u = ( γ − n )  c n b n  . (44) Setting n = 1 , w e can d etermine γ = 0 from the initial v alues. Then w e c an deriv e the f ractional Hamilton function H n = 2 a n +1 n , δ H n δ u =  c n b n  . (45) The generalized ev olutionary equations can b e giv en as u ( β ) t n = q ( β ) t n r ( β ) t n ! = J δ H n δ u . (46) Ac kno wledgmen ts The fi rst author feels grateful to Dr. Xiao-Jun Y ang for h is h elpf ul discussion ab out the physic al meaning of the lo cal fractional deriv ativ e and its p ossible u se in other fields. 9 5 Conclusion Inspired b y the previous w ork [44], i n this study we use a different fr actional der iv ativ e, modified Riemann-Liouville deriv ativ e, e stablish a fractionalized T u sheme for fractional d ifferen tial equa- tions and defin e a lo cal fractional Hamilton system and deriv e fractional evolutio nary soliton hierac hies. Ho w ev er, there are still other int eresting questions needed to b e add ressed i.e., ph ysical mearning of fractional soliton whic h may b e related to fractal media, fractio nal i nteg ral co up ling metho d, nonlinear tec h niques for fractional soliton equations. S uc h work is un der consideration. 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