A 2-component $mu$-Hunter-Saxton equation

A 2-COMPONEN T µ -HUNTER-SAXTON EQUA TION DAFENG ZUO Abstract. In this paper, we propose a t wo -component generalization of the generalized Hunte r-Saxton equation obtained in [22]. W e w i ll sho w that this equation is a bihamiltonian Euler equation, and also can be viewed as a bi - v ariational equation. 1. I ntroduction V.I.Arnold in [1] suggested a general framework for the Euler equatio ns on an arbitrar y (possibly infinite-dimensional) Lie algebra G . In many cases, the Euler equations on G describ e geo desic flo ws with re s pec t to a suitable one-side inv a r iant Riemannian metr ic o n the corres po nding group G . Now it is well-kno wn that Arnold’s a pproach to the Euler equation w or ks very well for the Virasoro algebra and its extensions, see [6, 1 0, 13, 14, 15, 19] a nd r e ferences therein. Let D ( S 1 ) b e a gr oup of orientation preserving diff e omorphisms of the circle and G = D ( S 1 ) ⊕ R be the Bott-Virasoro group. In [6], Ovsienko and Khesin showed that the K dV eq uation is a n Eule r eq ua tion, des cribing a geo desic flow o n G with resp ect to a right inv ar iant L 2 metric. Another interesting ex ample is the Camass a- Holm equation, whic h was origina lly de r ived in [4] a s an abstr act equation with a bihamiltonian structure, and indep endently in [9] as a shallow water approximation. In [1 0], Misiolek show ed tha t the Camassa-Holm equation is also an Euler equation for a geo desic flow on G with resp ect to a r ight-in v aria n t Sobolev H 1 -metric. In [13], Khes in and Misiolek extended the Arnold’s approa ch to homogene o us spaces and pr ovided a bea utiful geometric se tting for the Hun ter -Saxton equation, which firstly app ea r ed in [8] as a n a symptotic equation for ro ta tors in liquid crystals, and its rela tives. They showed that the Hunter-Saxton equa tio n is an Euler equa- tion descr ibing the g e o desic flow o n the homogeneous spaces of the Bott-Viraso ro group G mo dulo ro tations with re s pec t to a rig ht inv ar iant homogeneo us ˙ H 1 metric. F urthermor e, by using e xtended Bott-Viraso ro gro ups, Guha etc. [11, 16, 21] generalized the abov e r esults to tw o- c omp o nent in teg rable systems, including s e v- eral co upled KdV type systems and 2-comp onent p eak type systems , especially 2- comp onent Camassa -Holm equa tio n which was introduced by Chen, Liu a nd Zha ng [17] and independently by F alqui [18]. Another int eresting topic is to discuss the sup e r or supe r symmetric a nalogue, see [6, 12, 16, 20, 2 3, 24] and references ther e in. Recently Khesin, Lenells a nd Misiolek in [22] intro duced a generaliz e d Hun ter - Saxton ( µ -HS in brief ) equa tion lying mid-way betw ee n the p erio dic Hunter-Saxton and Camas s a-Holm equations, (1.1) − f txx = − 2 µ ( f ) f x + 2 f x f xx + f f xxx , 2000 Mathematics Subje ct Classific ation. 37K10, 35 Q51. 1 2 where f = f ( t, x ) is a time-dep endent function on the unit circle S 1 = R / Z a nd µ ( f ) = R S 1 f dx denotes its mean. This equation describ es ev olutio n of rotator s in liquid cr ystals with external mag netic field and self-interaction. Let D s ( S 1 ) b e a group of orientation preserving So bo lev H s diffeomorphisms o f the circle. Th ey prov ed that the µ -HS equation (1 .1) describes a geo desic flow on D s ( S 1 ) with a rig ht-in v ariant metric given at the iden tit y by the inner pro duct (1.2) h f , g i µ = µ ( f ) µ ( g ) + Z S 1 f ′ ( x ) g ′ ( x ) dx. They also show ed that (1.2) is bihamiltonian and admits b oth cusp ed as w ell as smo oth trav eling-wa ve solutions which are natural candidates for so litons. In this pap er, we wan t to generalize these to a t wo-component µ -HS (2- µ HS in brief ) equation. Our main o b ject is the Lie algebra G = V ect s ( S 1 ) ⋉ C ∞ ( S 1 ) and its three-dimensional cen tral extension b G . Firstly , we in tro duce an inner pro duct on b G given b y (1.3) h ˆ f , ˆ g i µ = µ ( f ) µ ( g ) + Z S 1 ( f ′ ( x ) g ′ ( x ) + a ( x ) b ( x )) dx + − → α · − → β , where ˆ f = ( f ( x ) d dx , a ( x ) , − → α ), ˆ g = ( g ( x ) d dx , b ( x ) , − → β ) and − → α , − → β ∈ R 3 . Afterwards, we ha ve Theorem 1.1 . [ =The or em 2.2 ] . The Euler e quation on b G ∗ r eg with r esp e ct to (1.3) is a 2- µ HS e quation (1.4)  − f xxt = 2 µ ( f ) f x − 2 f x f xx − f f xxx + v x v − γ 1 f xxx + γ 2 v xx , v t = ( v f ) x − γ 2 f xx + 2 γ 3 v x , wher e γ j ∈ R , j = 1 , 2 , 3 . Actually fr om the geometric view, if we extend the inner pro duct (1.3) to a left inv aria nt metr ic on b G = D s ( S 1 ) ⋉ C ∞ ( S 1 ) ⊕ R 3 , we co uld view the 2- µ HS equa tio n (1.4) as a geo desic flow on b G with res pec t to this left inv a riant metric. Obviously , if we choo se v = 0 and γ j = 0, j = 1 , 2 , 3 a nd re place t by − t , (1.4) reduces to (1.1). F urthermore, we show that Theorem 1.2. [ =The or em 3.1 and 4.1 ] . The 2- µ HS e quation (1.4) c an b e viewe d as a bihamiltonian and bi-variationa l e quation. This pap er is orga nized as follo ws. In section 2, we calculate the Euler equation on b G ∗ r eg . In section 3, we study the Hamiltonian nature and the Lax pair of the 2- µ HS equation (1.4). Section 4 is devoted to discus s the v ariatio nal nature of (1.4). In the last section we des crib e t he in terr e lation betw een bihamiltonian na tures and bi-v ariationa l natures. Ac kno wledgeme nt. The a uthor would like to thank Prof. Khesin Bo ris a nd P rof. Partha Guha for references [6] a nd [21], respectively , and the anon ymous r eferee for several useful sugg estions. This work is partially s upp or ted by the F undamental Research F unds for the Ce n tral Universities and NSF C(10 9712 09,108 71184). 3 2. E ulerian na ture of the 2- µ HS equa tion Let D s ( S 1 ) b e a gr oup of orientation pres erving Sob ole v H s diffeomorphisms of the cir cle and let T id D s ( S 1 ) b e the cor r esp onding Lie alge bra of vector fields, denoted by V ect s ( S 1 ) = { f ( x ) d dx | f ( x ) ∈ H s ( S 1 ) } . The ma in o b jects in our pap er will b e the gro up D s ( S 1 ) ⋉ C ∞ ( S 1 ), its Lie algebra G = V ect s ( S 1 ) ⋉ C ∞ ( S 1 ) with the Lie brack e t given b y [( f ( x ) d dx , a ( x )) , ( g ( x ) d dx , b ( x ))] =  ( f ( x ) g ′ ( x ) − f ′ ( x ) g ( x )) d dx , f ( x ) b ′ ( x ) − a ′ ( x ) g ( x )  , and their central extensions . It is well known in [3, 7] that the alg e bra G has a three dimensiona l central extension g iven b y the following nontrivial co cycles ω 1  ( f ( x ) d dx , a ( x )) , ( g ( x ) d dx , b ( x ))  = Z S 1 f ′ ( x ) g ′′ ( x ) dx, (2.1) ω 2  ( f ( x ) d dx , a ( x )) , ( g ( x ) d dx , b ( x ))  = Z S 1 [ f ′′ ( x ) b ( x ) − g ′′ ( x ) a ( x )] dx, ω 3  ( f ( x ) d dx , a ( x )) , ( g ( x ) d dx , b ( x ))  = 2 Z S 1 a ( x ) b ′′ ( x ) dx, where f ( x ), g ( x ) ∈ H s ( S 1 ) a nd a ( x ), b ( x ) ∈ C ∞ ( S 1 ). Notice that the first co- cycle ω 1 is the well-kno wn Gelfand-F uchs co cycle [2 , 5]. The Vira soro algebra V ir = V ec t s ( S 1 ) ⊕ R is the unique non-tr ivial central extension o f V ect s ( S 1 ) via the Gelfand-F uchs co cycle ω 1 . Sometimes we would like to use the mo dified Gelfand- F uchs co cycle (2.2) ˜ ω 1  ( f ( x ) d dx , a ( x )) , ( g ( x ) d dx , b ( x ))  = Z S 1 ( c 1 f ′ ( x ) g ′′ ( x ) + c 2 f ′ ( x ) g ( x )) dx, which is cohomolo geous to the Gelfand-F uchs co cycle ω 1 , wher e c 1 , c 2 ∈ R . Definition 2.1. The alg ebr a b G is an ex tension of G define d by (2.3) b G = V e ct s ( S 1 ) ⋉ C ∞ ( S 1 ) ⊕ R 3 with the c ommu tation r elation (2.4) [ ˆ f , ˆ g ] =  ( f g ′ − f ′ g ) d dx , f b ′ − a ′ g , − → ω  , wher e ˆ f = ( f ( x ) d dx , a ( x ) , − → α ) , ˆ g = ( g ( x ) d dx , b ( x ) , − → β ) and − → α , − → β ∈ R 3 and − → ω = ( ω 1 , ω 2 , ω 3 ) ∈ R 3 . Let (2.5) b G ∗ r eg = C ∞ ( S 1 ) ⊕ C ∞ ( S 1 ) ⊕ R 3 denote the regular par t of the dual space b G ∗ to the Lie alg ebra b G , under the pa iring (2.6) h ˆ u, ˆ f i ∗ = Z S 1 ( u ( x ) f ( x ) + a ( x ) v ( x )) dx + − → α · − → γ , where ˆ u = ( u ( x )( dx ) 2 , v ( x ) , − → γ ) ∈ b G ∗ . Of particular interest are the coadjoint orbits in b G ∗ r eg . On b G , let us in tro duce an inner pro duct (2.7) h ˆ f , ˆ g i µ = µ ( f ) µ ( g ) + Z S 1 ( f ′ ( x ) g ′ ( x ) + a ( x ) b ( x )) dx + − → α · − → β . 4 A direc t computation gives h ˆ f , ˆ g i µ = h ˆ f , (Λ ( g )( dx ) 2 , b ( x ) , − → β ) i ∗ , Λ( g ) = µ ( g ) − g ′′ ( x ) , which induces an iner tia op era tor A : b G − → b G ∗ r eg given b y (2.8) A ( ˆ g ) = (Λ( g )( dx ) 2 , b ( x ) , − → β ) . Theorem 2.2. The 2- µ HS e quation (1.4) is an Euler e quation o n b G ∗ r eg with r esp e ct to the inner pr o duct (2.7) . Pr o of. By definition, h ad ∗ ˆ f ( ˆ u ) , ˆ g i ∗ = −h ˆ u, [ ˆ f , ˆ g ] i ∗ by using int egration by parts = h  (2 uf x + u x f + a x v − α 1 f xxx + α 2 a xx )( dx ) 2 , ( v f ) x − α 2 f xx + 2 α 3 a x , 0  , ˆ g i ∗ . This gives ad ∗ ˆ f ( ˆ u ) = ( (2 uf x + u x f + a x v − α 1 f xxx + α 2 a xx )( dx ) 2 , ( v f ) x − α 2 f xx + 2 α 3 a x , 0) . By definition in [1 3], the Euler equation on b G ∗ r eg is g iven b y (2.9) d ˆ u dt = ad ∗ A − 1 ˆ u ˆ u as an evolution of a p oint ˆ u ∈ b G ∗ r eg . That is to say , the Euler equation o n b G ∗ r eg is u t = 2 uf x + u x f + v x v − γ 1 f xxx + γ 2 v xx , v t = ( v f ) x − γ 2 f xx + 2 γ 3 v x , where u ( x, t ) = Λ( f ( x, t )) = µ ( f ) − f xx . By int egrating b oth sides of this equation ov er the c ir cle and using p er io dicity , we obtain µ ( f t ) = µ ( f ) t = 0 . This yields that − f xxt = 2 µ ( f ) f x − 2 f x f xx − f f xxx + v x v − γ 1 f xxx + γ 2 v xx , v t = ( v f ) x − γ 2 f xx + 2 γ 3 v x , which is the 2- µ HS equation (1.4).  Remark 2. 3 . If we r eplac e the Gelfand-F uchs c o cycle ω 1 by the mo difie d c o cycle ˜ ω 1 , the Euler e quation b G ∗ r eg is of the form − f xxt = 2 µ ( f ) f x − 2 f x f xx − f f xxx + v x v − γ 1 c 1 f xxx + γ 2 v xx + γ 1 c 2 f x , v t = ( v f ) x − γ 2 f xx + 2 γ 3 v x . 3. H amil tonian na ture of the 2- µ HS equa tion In this section, we w ant to s tudy the Hamiltonian nature of the 2- µ HS equation (1.4) and its ge o metric meaning . W e will show that Theorem 3.1. The 2- µ HS e quation (1.4) is bihamiltonia n . 5 Pr o of. Let us define u ( x, t ) = Λ( f ) = µ ( f ) − f xx and (3.1) H 1 = 1 2 Z S 1 ( uf + v 2 ) dx and (3.2) H 2 = Z S 1 ( µ ( f ) f 2 + 1 2 f f 2 x + 1 2 f v 2 − γ 2 v f x + γ 3 v 2 − γ 1 2 f f xx ) dx. It is easy to chec k that the 2- µ HS equatio n can b e written as (3.3)  u v  t = J 1  δH 2 δu δH 2 δv  = J 2  δH 1 δu δH 1 δv  , where the Hamiltonian o per ators ar e (3.4) J 1 =  ∂ x Λ 0 0 ∂ x  , J 2 =  u∂ x + ∂ x u − γ 1 ∂ 3 x v ∂ x + γ 2 ∂ 2 x ∂ x v − γ 2 ∂ 2 x 2 γ 3 ∂ x  . By a dir ect a nd lengthy calcula tion we co uld show tha t Hamiltonia n op er ators J 1 and J 2 are co mpatible.  Next we w ant to explain the geometric meaning of the bihamiltonian structures of the 2- µ HS equatio n (1.4). Let F i : b G ∗ r eg → R , i = 1 , 2, b e tw o ar bitrary s mo oth functionals. It is well-kno wn that the dual spa ce b G ∗ r eg carries the canonical Lie- Poisson brac ket (3.5) { F 1 , F 2 } 2 ( ˆ u ) = h ˆ u, [ δ F 1 δ ˆ u , δ F 2 δ ˆ u ] i ∗ , where ˆ u = ( u ( x, t )( dx ) 2 , v ( x, t ) , ~ γ ) ∈ b G ∗ r eg and δF i δ ˆ u = ( δF i δu , δF i δv , δF i δ~ γ ) ∈ b G , i = 1 , 2 . By definition of the Euler equation (2.9), w e kno w that the Lie-Poisson str ucture (3.5) is exac tly the second Poisson bracket, induced b y J 2 , of the 2- µ HS equation (1.4). T o explain the fir s t Hamiltonia n s tr ucture, in the fo llowing we will use the “frozen Lie-Poisson” metho d intro duced in [13]. Let us define a fro zen (or constant) Poisson brack et (3.6) { F 1 , F 2 } 1 ( ˆ u ) = h ˆ u 0 , [ δ F 1 δ ˆ u , δ F 2 δ ˆ u ] i ∗ , where ˆ u 0 = ( u 0 ( dx ) 2 , v 0 , ~ γ 0 ) ∈ b G ∗ r eg . The corresp onding Hamiltonia n equation for any fu nctional F : b G ∗ r eg → R reads (3.7) d ˆ u dt = ad ∗ δF δ ˆ u ˆ u 0 which giv es u t = 2 u 0 ( δ F δ u ) x + ( δ F δ v ) x v 0 − γ 0 1 ( δ F δ u ) xxx + γ 0 2 ( δ F δ v ) xx , v t = ( v 0 δ F δ u ) x − γ 0 2 ( δ F δ u ) xx + 2 γ 0 3 ( δ F δ v ) x , (3.8) ~ γ 0 ,t = 0 . Let us take the Hamiltonian functional F to b e (3.9) H 2 = Z S 1 ( µ ( f ) f 2 + 1 2 f f 2 x + 1 2 f v 2 − γ 2 v f x + γ 3 v 2 − γ 1 2 f f xx ) dx 6 and set u ( x, t ) = Λ( f ( x, t )) = µ ( f ) − f xx . Then we hav e δ F δ u = Λ − 1 ( µ ( f 2 ) + 2 f µ ( f ) − 1 2 f 2 x − f f xx − γ 1 f xx + γ 2 v x ) , δ F δ v = v f − γ 2 f x + 2 γ 3 v . (3.10) Let us choose a fixed p oint ˆ u 0 = ( u 0 , v 0 , ~ γ 0 ) = (0 , 0 , (1 , 0 , 1 2 )) . Observe that ∂ 3 x Λ − 1 = − ∂ x . By substituting ( 3.10) in to (3.8 ), we o bta in the 2- µ HS equation (1.4). According to the Prop ositi on 5.3 in [13], { , } 1 and { , } 2 are compatible for every freezing p oint ˆ u 0 . Consequently w e hav e Theorem 3.2. The 2- µ HS e quation (1.4) is Hamiltonian with r esp e ct to two c om- p atible Poisson stru ctur es (3.5) and (3.6 ) on b G ∗ r eg , wher e the first br acket is fr ozen at the p oint ˆ u 0 = ( u 0 , v 0 , ~ γ 0 ) = (0 , 0 , (1 , 0 , 1 2 )) . Let us point out that the co nstant brack et dep ends o n the choice of the fre e z - ing p oint ˆ u 0 , while the Lie-Poisso n brac ket is only determined by the Lie algebra structure. T o this end we w ant to derive a Lax pair of 2- µ HS eq uation (1.4) with − → γ = 0, i.e., (3.11) − f xxt = 2 µ ( f ) f x − 2 f x f xx − f f xxx + v x v , v t = ( v f ) x . Motiv ated by the Lax pair of the t wo-comp onent Camassa- Ho lm equa tion in [17], we could assume that the Lax pair of (3.11) has the following form (3.12) Ψ x = U Ψ , Ψ t = V Ψ with U =  0 1 λ Λ( f ) − λ 2 v 2 0  and V =  p r q − p  , where λ is a spe ctral par ameter. The co mpatibility condition U t − V x + U V − V U = 0 in comp onent wise form rea ds p = − r x 2 , q = p x + r ( λ Λ( f ) − λ 2 v 2 ) , 2 λ 2 v v t + λf xxt + q x − 2 p ( λ Λ( f ) − λ 2 v 2 ) = 0 . By choo s ing r = f − 1 2 λ , we hav e p = − f x 2 , q = − f xx 2 + ( f − 1 2 λ )( λ Λ( f ) − λ 2 v 2 ) and f xxt + 2 µ ( f ) f x − 2 f x f xx − f f xxx + v x v + 2 λv ( v t − ( v f ) x ) = 0 which yields the system (3.11). Let us write Ψ =  ψ ψ x  , we hav e Prop ositio n 3.3 . The s ystem (3.11) has a L ax p air give n by ψ xx = ( λ Λ( f ) − λ 2 v 2 ) ψ , ψ t = ( f − 1 2 λ ) ψ x − 1 2 f x ψ , wher e λ ∈ C − { 0 } is a sp e ctr al p ar ameter. 7 4. V aria tional na ture of the 2- µ HS equa tion In [22], they ha ve s hown that the µ -HS equation (1.1 ) can b e obtaine d from t wo distinct v ar iational principles. In this section we will show that the 2- µ HS equation (1.4) also arises a s the equation δ S = 0 for the action functiona l S = Z ( Z L dx ) dt with tw o different densities L . That is to say , Theorem 4.1. T he 2- µ HS e quation (1 .4) satisfi es two differ ent variational prin- ciples. Pr o of. Motiv ated by the Lagrangia n densities for the µ -HS equation (1.1) in [22], by some conjectural computations w e find tw o gener alized La grangia n densities for the 2- µ HS equation (1 .4). Mor e precise ly , Case I . Let us consider the fir s t Lagra ngian densit y (4.1) L 1 = 1 2 f 2 x + 1 2 µ ( f ) f + 1 2 v 2 − v z x + w ( f z x − z t + ˜ γ 3 v ) + γ 2 w x f − 2 γ 1 f , where ˜ γ 3 = γ 3 − 1 2 γ 1 . V a rying the corr e spo nding actio n with resp ect to f , v , w and z respec tively , w e get (4.2) f xx = µ ( f ) + w z x + γ 2 w x − 2 γ 1 , z x = v + ˜ γ 3 w, z t = f z x + ˜ γ 3 v − γ 2 f x , w t = ( wf ) x − v x . By using (4.2), we hav e v t = z xt − γ 3 w t = [ f ( v + ˜ γ 3 w ) + ˜ γ 3 v − γ 2 f x ] x − ˜ γ 3 (( wf ) x − v x ) , = ( v f ) x − γ 2 f xx + (2 γ 3 − γ 1 ) v x , (4.3) and − f xxt + f x f xx + f f xxx = − ( µ ( f ) + w z x + γ 2 w x ) t + f x ( µ ( f ) + wz x + γ 2 w x − 2 γ 1 ) + f ( µ ( f ) + wz x + γ 2 w x ) x = − w t z x − w z xt + γ w xt + f x wz x + f w x z x + f w z xx + γ 2 f w xx + 2 γ 1 f x = v v x + 2 µ ( f ) f x + γ 2 v xx − 2 γ 1 f x . (4.4) Notice that if we replace f by f + γ 1 in the system (4.3) and (4.4), this g ives the 2- µ HS equatio n (1.4). Case I I . The second v aria tional repr esentation can b e obtained from t he La- grangia n densit y (4.5) L 2 = − f x f t + 2 µ ( f ) f 2 + f f 2 x + f φ 2 x − γ 1 f f xx − 2 γ 2 φ x f x + 2 γ 3 φ 2 x − φ x φ t . The v ariatio nal principle δ S = 0 gives the E uler-Lag r ange equation (4.6) − f xt = 2 µ ( f ) f + µ ( f 2 ) − 1 2 f 2 x − f f xx + 1 2 φ 2 x − γ 1 f x + γ 2 φ xx , φ xt = ( f φ x ) x − γ 2 f xx + 2 γ 3 φ xx . 8 If we set φ x = v and take the x -deriv ative of the first term in (4.6), this yields the 2- µ HS equatio n (1.4).  5. Rela tion between Hamil tonian na ture and V aria tional na ture Recall that w e hav e shown that the 2 - µ HS equa tion (1.4) is bihamiltonian and has tw o different v ariationa l principles. In the las t section we wan t to study the relation b etw een Hamiltonian na tur es and bi-v ariational principles and prov e that Theorem 5.1 . The two variational formulations for the 2- µ HS e quation (1.4) for- mal ly c orr esp ond to the t wo Hamiltonian formulations of t his e quation with Hamil- tonian functionals H 1 and H 2 . Pr o of. The a c tio n is rela ted to the Lagr angian by S = R ( R L dx ) dt . The first v ariational principle has the Lagr angian density , L 1 = 1 2 f 2 x + 1 2 µ ( f ) f + 1 2 v 2 − v z x + w ( f z x − z t + ˜ γ 3 v ) + γ 2 w x f − 2 γ 1 f . The momenta conjugate to the velocities f t , v t , z t and w t , r esp ectively , are ∂ L 1 ∂ f t = 0 , ∂ L 1 ∂ w t = 0 , ∂ L 1 ∂ z t = − w , ∂ L 1 ∂ w t = 0 . Consequently , the Hamiltonian density is H = − z t w − L 1 = − 1 2 f 2 x − 1 2 µ ( f ) f − 1 2 v 2 + v z x − w ( f z x + ˜ γ 3 v ) − γ 2 w x f + 2 γ 1 = 1 2 µ ( f ) f − 1 2 f 2 x + 1 2 v 2 − f f xx , by using (4.2) . Therefore, the Hamiltonian is H = Z H dx = Z ( 1 2 µ ( f ) f − 1 2 f 2 x + 1 2 v 2 − f f xx ) dx = 1 2 Z ( µ ( f ) f − f f xx + v 2 ) dx, which is exactly H 1 defined in (3.1). In the second principle the Lagra ng ian densit y is L 2 = − f x f t + 2 µ ( f ) f 2 + f f 2 x + f φ 2 x − γ 1 f f xx − 2 γ 2 φ x f x + 2 γ 3 φ 2 x − φ x φ t . 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Dep ar tment of M a thema tics,University of Science and Technology of China , Hefei 230026, P.R. China E-mail addr ess : dfzuo@ustc.edu .cn