Additional symmetries of constrained CKP and BKP hierarchies

ADDITIONA L SYMMETRIES OF CONSTRAINED CKP AND BKP HIERA R CHIES KELEI TIAN † , JINGSONG HE ∗ ‡ , JIPENG CHENG † AND YI CHENG † † Dep artment of Mathematics, Universi ty of Scienc e and T e chno lo gy of China, China ‡ Dep artment of Mathematics, Ningb o University, China Abstract. The additional symmetries of the constra ined CKP (cCKP) and BKP (cBKP) hi- erarchies are given by their actions on the Lax op era tors, and their a ctions on the eigenfunction and adjoint eigenfunction { Φ i , Ψ i } are pres e nted explicitly . F urthermore , we show that acting on the space o f the wa v e op erato r, ∂ ∗ k forms new centerless W cC 1+ ∞ and W cB 1+ ∞ -subalgebra of centerless W 1+ ∞ resp ectively . In order to define above symmetry flows ∂ ∗ k of the cCK P and cBKP hierarchies, tw o vita l o p er ators Y k are intro duced to revise the additional symmetry flows of the CK P and BKP hierarchies. P A CS (2003 ). 02.30.Ik. Mathematics Sub ject Classification (2000). 17B80, 37K05, 37K10. Keyw ords. constrained CKP hierarch y , constrained BKP hierarc h y , additional symme try . 1. Intr oduction The researc h on Kadom tsev-P etviash v ili (KP) hierarch y [1, 2] is one of t he most imp ortan t topics in the dev elopmen t of the theory of in tegrable systems. A sp ecifi c in teresting asp ect on this topic is additional symmetry [3 – 8]. Additional symmetries are sp ecial symmetries whic h are no t con tained in the KP hierarc hy and do not commute with eac h other. Th e additional sym metry flo ws of the KP hierarch y form an infinite dimensional algebra W 1+ ∞ [5, 8]. More recen tly , there are sev eral new results ab out part it io n function in the matrix mo dels and Seib erg-Witten theory asso ciated with additional symmetries, string equation and Virasoro constraints of the KP hierarc hy [9 – 13]. In fact, there is a parallel researc h line on additional symm etries of ev olution equations and their Lie algebraic structures, particularly for 1 + 1 dimensional integrable equations [1 4], and the corr esp o nding symmetries are called τ -symmetries and related L a x op erator s are constructed prett y systematically [15, 16]. It is well known tha t w e can get differen t sub-hierarc hies of the KP by differen t reduction conditions on Lax op erator L . The first tw o imp o rtan t sub-hierarc hies are CKP hierarc h y [1 7] through a restriction L ∗ = − L and BKP hierarch y [1] through a restriction L ∗ = − ∂ L∂ − 1 . So the additio na l symmetries of the CKP (or BKP) hierarc h y ha ve b een constructed [18 – 21] and sho wn to b e a new infinite dime nsional algebra W C 1+ ∞ (or W B 1+ ∞ ), whic h is subalgebra of W 1+ ∞ . The third s ub-hierarc h y is constrained KP (cKP) hierarc hy [22 – 24], whic h is obtained by setting a sp ecial f o rm o f Lax op erat or L = ∂ + P m i =1 Φ i ∂ − i Ψ i , where Φ i is the eigenfunction and Ψ i is the adjoin t e igenfunction of the cKP hierarch y . This is a relativ e new reduction condition and is differen t from the reduction conditions of previous tw o sub-hierarc hies. Hence it is ve ry natura l to explore the additio nal symmetries of t he cKP hierarc h y . How ev er, additional s ymmetry flo ws of the KP hierarc h y break the sp ecial f orm of the L a x op erator for the cKP hierarc h y , a nd so it ∗ Corresp o nding a uthor. email: hejings o ng@nbu.edu.cn. 1 2 KELEI TIA N † , J INGSON G HE ∗ ‡ , JIPENG CHENG † AND YI CHENG † is highly non-trivial to find a suitable form of additional symmetry flo w for this sub-hierarch y . T o this end, it is w ell defined [25] by means of a crucial mo dification o f the correspo nding additional symmetry flows of the KP hierarch y . Note that a new and complicated op erat or X k pla ys a v ery imp ortan t role in this pro cedure. F urthermore, the fourth sub-hierarch y and fifth sub-hierarch y are constrained CKP (cCKP) [26] and constrained BKP (cBKP) hierarc h y [27], whic h are obta ined by putting the com bination of the previous men tioned reduction conditions, i.e., CKP and cKP conditio ns, BKP and cKP conditions. Some results asso ciated cCKP and cBKP hierarc hies ha v e already b een rep orted, for example, a Gramm-ty p e τ -function [26 ] of the cCKP hierarc hy and a Pfaffian-fo rm τ - function [27] of the cBKP are also constructed from the v acuum solution, the dimens ional reductions of the CKP and BKP hierarc hies are presen ted in [28]. Moreo ver, determinan t represen tations o f the τ function, whic h are generated b y the the gauge transformations, of the cCKP a nd cBKP hierarc hies are obtained [29]. Ho w ev er, for our b est knowled ge, there is no an y results on the additional symmetries of the cBKP and cCKP hierarch ies. So, w e shall fill the gap in this pa p er b y constructing a dditional symmetrie s for t he tw o new sub-hierarch ies and t heir actions on eigenfunctions and adjoint eigenfunctions. The basic idea is to revise the additional symmetry flow s of CKP and BKP hierarc hies suc h that the new symmetry flo ws ∂ ∗ k will preserv e the form of Lax op erat or of the cKP hierar ch y . This will b e realized b y in tr o ducing t wo vital op erators Y k in sequel. The pap er is orga nized as follows. W e fir st recall some basic results of t he KP hierarc hy , CKP hierarch y , BKP hierarch y and the constrained K P hierarc h y in Section 2. The main results a re stated and prov ed in Sections 3 and 4, whic h are the a dditio nal symmetries and its action on the eigenfunction and adjoint eigenfunction of the cCKP and cBKP hierarc hies, and w e further sho w tha t acting on the space of the wa v e op erator, ∂ ∗ k forms new centerle ss W cC 1+ ∞ and W cB 1+ ∞ -subalgebra of cente rless W 1+ ∞ resp ectiv ely . Section 5 is dev oted to conclusions and discussions . 2. back ground on KP hierarchy In this section w e first giv e a brief in tro duction of KP hierarch y based on [1, 2]. Let t he pseudo-differen tial op erator L = ∂ + u 1 ∂ − 1 + u 2 ∂ − 2 + u 3 ∂ − 3 + · · · (2.1) b e a Lax op erator of the KP hierarch y , which is described b y the asso ciated Lax equations ∂ L ∂ t n = [ B n , L ] , n = 1 , 2 , 3 , · · · , (2.2) where B n = ( L n ) + = n P k =0 a k ∂ k denotes the non-negativ e p ow ers of ∂ in L n , ∂ = ∂ /∂ x , u i = u i ( x = t 1 , t 2 , t 3 , · · · ). The other notation L n − = L n − L n + will b e used in t he pap er. The Lax op erato r L in eq.(2.1) can b e g enerated by dressing op erator S = 1 + P ∞ k =1 s k ∂ − k in the follo wing w a y L = S ∂ S − 1 . (2.3) The dressing op erato r S satisfies Sato equation ∂ S ∂ t n = − ( L n ) − S, n = 1 , 2 , 3 , · · · . (2.4) ADDITION AL SY MMETRIES OF CON STRAINED CKP AND BKP H IERARCHIES 3 The w av e function w ( t, z ) of the KP hierarc h y is defined b y w ( t, z ) = S e ξ ( t,z ) , (2.5) where ξ ( t, z ) = t 1 z + t 2 z 2 + t 3 z 3 + · · · + t n z n · · · . The w av e function satisfies the equations L k w ( t, z ) = z k w ( t, z ) , ∂ ∂ t n w = B n w , k ∈ Z , n ∈ N . (2.6) Moreo ver w ( t, z ) ha s a v ery simple expression b y τ function of the KP hierar ch y w ( t, z ) = τ ( t 1 − 1 z , t 2 − 1 2 z 2 , t 3 − 1 3 z 3 , · · · ) τ ( t 1 , t 2 , t 3 , · · · ) e ξ ( t,z ) . (2.7) Beside the existence o f the Lax op erator, w a v e function, τ function for the KP hierarc h y , another imp orta n t prop erty is the Zakharov -Shabat equation and asso ciated linear equation. In other words, KP hierarch y a lso has an alternativ e expression, i.e., ∂ B m ∂ t n − ∂ B n ∂ t m + [ B m , B n ] = 0 , m, n = 1 , 2 , 3 , · · · . (2.8) The eigenfunction Φ = Φ( t 1 , t 2 , t 3 , · · · ) and adjoint eigenfunction Ψ = Ψ( t 1 , t 2 , t 3 , · · · ) of KP hierarc hy asso ciated t o (2.8) are defined b y ∂ Φ ∂ t n = ( B n Φ) , ∂ Ψ ∂ t n = − ( B ∗ n Ψ) , (2.9) where the sym b ol ∗ is the formal adjoin t op eratio n. F or an arbitrary pseudo-differential op erat or P = P i p i ∂ i , P ∗ = P i ( − 1) i ∂ i p i , and ( AB ) ∗ = B ∗ A ∗ for pseudo-differen tial op erato r s A and B . F or example, ∂ ∗ = − ∂ , ( ∂ − 1 ) ∗ = − ∂ − 1 . T o construct additional symmetries asso ciated with the KP hierarch y , let us in tro duce Γ and Orlo v-Sh ulman’s M o p erator as Γ = ∞ P i =1 it i ∂ i − 1 and M = S Γ S − 1 . Dressing [ ∂ k − ∂ k , Γ] = 0 gives [ ∂ k − B k , M ] = 0 , i.e. ∂ k M = [ B k , M ] , (2.10) and then ∂ k ( M m L n ) = [ B k , M m L n ] . (2.11) Th us, the additional flows are defined by ∂ S ∂ t ∗ m,n = − ( M m L n ) − S, (2.12) or equiv alently ∂ L ∂ t ∗ m,n = − [( M m L n ) − , L ] . (2.13) Here ( M m L n ) − serv es as the generator of the additiona l flo ws along the new time v ariables t ∗ m,n . The additional flows ∂ ∗ m,n = ∂ ∂ t ∗ m,n comm ute with the hierarch y , i.e. [ ∂ ∗ m,n , ∂ k ] = 0 but do not comm ute with each other, they indeed determine symmetries. Therefore these flo ws ∂ ∗ m,n are also called additional symmetry flo ws, and they forms a centerle ss W 1+ ∞ algebra [5, 8] acting on the spaces of the Lax op erator L a nd w av e op erator S . 4 KELEI TIA N † , JINGSONG HE ∗ ‡ , JIPENG CHENG † AN D YI CHENG † The CKP hierarc h y is a reduction of t he KP hierarc hy through t he constrain t on L g iv en b y eq.(2.1) as L ∗ = − L, (2.14) then L is called the Lax op erator of the CKP hierarc h y , and the asso ciated Lax equation of the CKP hierarc hy is ∂ L ∂ t n = [ B n , L ] , n = 1 , 3 , 5 , · · · . (2.15) whic h compresses all ev en flo ws, i.e. the Lax equation of the CKP hierar ch y has only o dd flows. Additional symmetry flo ws [18] for the CKP hierarch y are defined as ∂ L ∂ t ∗ m,l = − [( A m,l ) − , L ] , (2.16) where A m,l for the CKP hierarc hy should satisfy A ∗ m,l = − A m,l , (2.17) w e can a ssume A m,l b e A m,l = M m L l − ( − 1) l L l M m , (2.18) where M = S ( ∞ X i =0 (2 i + 1) t 2 i +1 ∂ 2 i ) S − 1 . (2.19) Acting on the space of the w av e op erator S , ∂ ∗ m,l of the CKP hierarch y forms a new cen terless W C 1+ ∞ - subalgebra of cen terless W 1+ ∞ . If L a x op erator L giv en by eq.(2.1) satisfies L ∗ = − ∂ L∂ − 1 , ( 2 .20) then L is called the Lax op erator of the BKP hierarc h y , the La x equation of the BKP hierarch y also has only o dd flows ∂ L ∂ t n = [ B n , L ] , n = 1 , 3 , 5 , · · · . (2.21) Additional symmetry flows [19 , 20] for t he BKP hierarc hy are give n b y ∂ L ∂ t ∗ m,l = − [( A m,l ) − , L ] , (2.22) where A m,l for the BKP hierarch y should satisfy the constrain t equation A ∗ m,l = − ∂ A m,l ∂ − 1 , (2.23) then A m,l for the BKP hierarch y could b e c hosen as A m,l = M m L l − ( − 1) l L l − 1 M m L, (2.24) where M = S ( ∞ X i =0 (2 i + 1) t 2 i +1 ∂ 2 i ) S − 1 . (2.25) Acting on the space of the w a v e op era t or S , ∂ ∗ m,l of the BKP hierarc h y forms another new cen terless W B 1+ ∞ - subalgebra of cen terless W 1+ ∞ . ADDITION AL SY MMETRIES OF CON STRAINED CKP AND BKP H IERARCHIES 5 W e no w turn to the cKP hierarc hy [22 – 24]. The Lax op erato r L of the constrained KP hierarc hy is giv en by L = ∂ + m X i =1 Φ i ∂ − i Ψ i , (2.26) where Φ i (Ψ i ) is the (adjoint) eigenfunctions of this hierarc hy , and the corresp onding Lax equation is formulated as ∂ L ∂ t n = [ B n , L ] , n = 1 , 2 , 3 , · · · . (2.27) In particular, we stress that eigenfunctions and adjoin t eigenfunctions { Φ i , Ψ i } are imp orta n t dynamical v aria bles in cKP hierarc hy , so it is necessary to find the action of the correct ad- ditional symmetry flows on them. Here the correct flow ∂ τ means its action on L of the cKP hierarc hy should b e the f orm of [25] ( ∂ τ L ) − = m X i =1 ( ˜ A∂ − 1 Ψ i + Φ i ∂ − 1 ˜ B ) , (2.28) whic h will result in its action on ( ∂ τ Φ i = ˜ A ) and ( ∂ τ Ψ i = ˜ B ). How eve r, in general, the additional symmetry flo ws of the K P hierarc hy acting on L of the cKP hierarch y are not the form of eq.(2.28). In other w o r ds, these flows do not preserv e the form of the Lax op erator of the cKP hierarc h y , the fact show s additional symmetry o f the KP hierarc hy is not consisten t with cKP reduction condition automat ically . Therefore, the additiona l symmetry flo ws hav e to b e revise d according to the analysis ab o v e, and then the correct additional symmetry flo ws of the cKP hierarc h y are giv en b y [2 5] ∂ ∗ k L = [ − ( M L k ) − + X k − 1 , L ] , k = 0 , 1 , 2 , 3 , · · · , (2.29) where X k − 1 = 0 , k = 0 , 1 , 2 and X k − 1 = m X i =1 k − 2 X j =0 ( j − 1 2 ( k − 2)) L k − 2 − j (Φ i ) ∂ − 1 ( L ∗ ) j (Ψ i ) , k ≥ 3 . (2.30) F urthermore [25], ∂ ∗ k Φ i = k 2 L k − 1 (Φ i ) + X k − 1 (Φ i ) + ( A 1 ,k ) + (Φ i ) , (2.31) ∂ ∗ k Ψ i = k 2 L ∗ ( k − 1) (Ψ i ) − X ∗ k − 1 (Ψ i ) − ( A 1 ,k ) ∗ + (Ψ i ) . (2.32) 3. Additional symmetries of constraine d CKP hierarchy Let us consider the constrained CKP hierarc hy , whic h is the C-t yp e sub-hierarc hy of cKP hierarc hy . The Lax o p erator L of the cCKP hierarc hy [2 6] is given b y L = ∂ + m X i =1 ( q i ∂ − 1 r i + r i ∂ − 1 q i ) , (3.1) where q i and r i are eigenfunctions. The correspo nding L a x equation of the cCKP hierarc hy is defined b y 6 KELEI TIA N † , JINGSONG HE ∗ ‡ , JIPENG CHENG † AN D YI CHENG † ∂ L ∂ t n = [ B n , L ] , n = 1 , 3 , 5 , · · · . (3.2) F or k = 1 , 3 , 5 , · · · , w e first try t o calculate t he original additional symmetry flo ws of t he CKP hierarc hy as ∂ L ∂ t ∗ 1 ,k = − [( A 1 ,k ) − , L ] , (3.3) where A 1 ,k = M L k − ( − 1) k L k M . Thus ( ∂ L ∂ t ∗ 1 ,k ) − = [( A 1 ,k ) + , L ] − + 2 ( L k ) − . (3.4) In order to g et its action on q i and r i , w e need the form of ( L k ) − . Lemma 3.1. The Lax op erat or L of the constrained CKP hierarch y give n by eq.(3.1 ) satisfies the relation ( L k ) − = m X i =1 k − 1 X j =0 ( L k − j − 1 ( q i ) ∂ − 1 ( L ∗ ) j ( r i ) + L k − j − 1 ( r i ) ∂ − 1 ( L ∗ ) j ( q i )) , k = 1 , 3 , 5 , · · · (3.5) where L ( q i ) = L + ( q i ) + m X j =1 ( q j ∂ − 1 x ( r j q i ) + r j ∂ − 1 x ( q j q i )) . Ho w ev er, ( L k ) − is not in the form of ( ∂ τ L ) − = m X i =1 (( ∂ τ q i ) ∂ − 1 r i + ( ∂ τ r i ) ∂ − 1 q i + q i ∂ − 1 ( ∂ τ r i ) + r i ∂ − 1 ( ∂ τ q i )) , (3.6) as w e exp ected for a cor r ect flo ws. Here correctness means w e can get the actio ns on ∂ τ q i and ∂ τ r i from eq.(3.6). This sho ws that eq.(3.4) can not imply its a ctio n on eigenfunctions ∂ ∗ 1 ,k q i and ∂ ∗ 1 ,k r i . Therefore, in order to get correct additio nal flows of the cCKP hierarc hy , we revise eq.(3.3) and then define new flows by ∂ ∗ k L = [ − ( A 1 ,k ) − + Y k , L ] , k = 1 , 3 , 5 , · · · , (3.7) where A 1 ,k = M L k − ( − 1) k L k M a nd Y k is in tro duced suc h that the left hand side o f eq.(3.7) will b e the form of eq.(3.6 ). Next, we shall prov e new flows ∂ ∗ k are correct a dditional symmetry flo ws of the cCKP hierarch y . First o f all, Y k is discussed, whic h is crucial to our purp ose. Lemma 3.2. F or the cCKP hierarch y , the CKP reduction conditio n infers a constraint on Y k , Y ∗ k = − Y k , k = 1 , 3 , 5 , · · · . (3.8) Pro of. The a ctio n of the a dditio nal flows ∂ ∗ k on the adjoint La x op erator L ∗ of the constrained CKP hierarc h y can b e o btained b y t w o differen t wa ys. W e first computet it b y a formal a djoin t op eration on b oth sides of eq. (3.7), i.e. ∂ ∗ k L ∗ = ([ − ( A 1 ,k ) − + Y k , L ]) ∗ = [ L ∗ , − ( A 1 ,k ) ∗ − + Y ∗ k ] , k = 1 , 3 , 5 , · · · . (3.9) ADDITION AL SY MMETRIES OF CON STRAINED CKP AND BKP H IERARCHIES 7 The another w a y is to do a deriv ativ e with resp ect to t ∗ k on L ∗ and use CKP reduction condition L ∗ = − L , then ∂ ∗ k L ∗ = − ∂ ∗ k L = − [ − ( A 1 ,k ) − + Y k , L ] = [ L ∗ , ( A 1 ,k ) − − Y k ] , k = 1 , 3 , 5 , · · · . (3.10) Comparing eq.(3.9) a nd eq.(3.10), w e hav e Y ∗ k = − Y k , k = 1 , 3 , 5 , · · · , with the help of A ∗ 1 ,k = − A 1 ,k .  Th us w e set Y k as Y 1 = 0 , (3.11) Y k = m X i =1 k − 2 X j =0 (2 j − ( k − 2))( L k − 2 − j ( q i ) ∂ − 1 ( L ∗ ) j ( r i ) + L k − 2 − j ( r i ) ∂ − 1 ( L ∗ ) j ( q i )) , k ≥ 3 , (3.12) and shall show Y k satisfy the constraint given b y eq.(3.8) . Lemma 3.3. Y k = − Y ∗ k . (3.13) Pro of. Y ∗ 1 = Y 1 = 0 is obv ious, for k = 3 , 5 , 7 , · · · , w e hav e Y ∗ k = m X i =1 k − 2 X j =0 (2 j − ( k − 2))( − ( L ∗ ) j ( r i ) ∂ − 1 L k − 2 − j ( q i ) − ( L ∗ ) j ( q i ) ∂ − 1 L k − 2 − j ( r i )) = m X i =1 k − 2 X j =0 (2 j − ( k − 2))( − ( − 1) j L j ( r i ) ∂ − 1 ( − 1) k − 2 − j ( L ∗ ) k − 2 − j ( q i ) − ( − 1) j L j ( q i ) ∂ − 1 ( − 1) k − 2 − j ( L ∗ ) k − 2 − j ( r i )) = m X i =1 k − 2 X j =0 (2 j − ( k − 2))( L j ( r i ) ∂ − 1 ( L ∗ ) k − 2 − j ( q i ) + L j ( q i ) ∂ − 1 ( L ∗ ) k − 2 − j ( r i )) = − m X i =1 k − 2 X l =0 (2 l − ( k − 2))( L k − 2 − l ( q i ) ∂ − 1 ( L ∗ ) l ( r i ) + L k − 2 − l ( r i ) ∂ − 1 ( L ∗ ) l ( q i )) = − Y k In the fourth step w e let l = k − 2 − j .  F urthermore, in order to calculate [ Y k , L ] in the eq.(3.7), the following lemma is neces sary . Lemma 3.4. F or the Lax op erator L of the cCKP hierarch y in eq.(3.1) and a pseudo-differen t op erator X = P l k =1 M k ∂ − 1 N k , the equation [ X , L ] − = l X k =1 ( − L ( M k ) ∂ − 1 N k + M k ∂ − 1 L ∗ ( N k )) + m X i =1 ( X ( q i ) ∂ − 1 r i + X ( r i ) ∂ − 1 q i − q i ∂ − 1 X ∗ ( r i ) − r i ∂ − 1 X ∗ ( q i )) holds. 8 KELEI TIA N † , JINGSONG HE ∗ ‡ , JIPENG CHENG † AN D YI CHENG † Based on the ab ov e lemma, we hav e [ Y k , L ] − = − 2( L k ) − + k m X i =1 ( L k − 1 ( q i ) ∂ − 1 r i + L k − 1 ( r i ) ∂ − 1 q i ) + k m X i =1 ( q i ∂ − 1 L ∗ ( k − 1) ( r i ) + r i ∂ − 1 L ∗ ( k − 1) ( q i )) + m X i =1 ( Y k ( q i ) ∂ − 1 r i + Y k ( r i ) ∂ − 1 q i ) − m X i =1 ( q i ∂ − 1 Y ∗ k ( r i ) + r i ∂ − 1 Y ∗ k ( q i )) . (3.14) W e are now in a p osition to calculate the explicit form of the rig ht hand side of the new additional flow give b y eq.(3.7). Theorem 3.1. The additional flo ws acting o n the eigenfunction q i and r i of the constrained CKP hierarc hy are ∂ ∗ k q i = k L k − 1 ( q i ) + Y k ( q i ) + ( A 1 ,k ) + ( q i ) , k = 1 , 3 , 5 , · · · (3.15) and ∂ ∗ k r i = k L k − 1 ( r i ) + Y k ( r i ) + ( A 1 ,k ) + ( r i ) , k = 1 , 3 , 5 , · · · . (3.16) Pro of. F rom the revised definition of the additional symmetry flo ws in eq.(3.7) of the cCKP hierarc hy , b y a short calculation, w e hav e ∂ ∗ k L − = [ − ( A 1 ,k ) − + Y k , L ] − = [( A 1 ,k ) + , L ] − + 2 ( L k ) − + [ Y k , L ] − . Using eq.(3.14) and the tec hnical identit y [ K , f ∂ − 1 g ] − = K ( f ) ∂ − 1 g − f ∂ − 1 K ∗ ( g ) (where f , g are arbitrar y functions and K is a purely differen t ia l op erator), note that k = 1 , 3 , 5 , · · · , it is follo w ed b y ∂ ∗ k L − = m X i =1 (( k L k − 1 ( q i ) + Y k ( q i ) + ( A 1 ,k ) + ( q i )) ∂ − 1 r i + ( k L k − 1 ( r i ) + Y k ( r i ) + ( A 1 ,k ) + ( r i )) ∂ − 1 q i + q i ∂ − 1 ( k ( L ∗ ) k − 1 ( r i ) − Y ∗ k ( r i ) − ( A 1 ,k ) ∗ + ( r i )) + r i ∂ − 1 ( k ( L ∗ ) k − 1 ( q i ) − Y ∗ k ( q i ) − ( A 1 ,k ) ∗ + ( q i )) th us ∂ ∗ k q i = k L k − 1 ( q i ) + Y k ( q i ) + ( A 1 ,k ) + ( q i ) = k ( L ∗ ) k − 1 ( q i ) − Y ∗ k ( q i ) − ( A 1 ,k ) ∗ + ( q i ) , ∂ ∗ k r i is also obtained at the same time.  Corollary 3.1. The additional flows act on w av e op erator S of the constrained CKP hier- arc hy as ∂ ∗ k S = ( − ( A 1 ,k ) − + Y k ) S, k = 1 , 3 , 5 , · · · . (3.17) Theorem 3.2. The additional flo ws ∂ ∗ k comm ute with the constrained CKP hierarch y flo ws ∂ t 2 n +1 = ∂ ∂ t 2 n +1 , i.e. [ ∂ ∗ k , ∂ t 2 n +1 ] = 0 . (3.18) ADDITION AL SY MMETRIES OF CON STRAINED CKP AND BKP H IERARCHIES 9 Th us ∂ ∗ k ( k = 1 , 3 , 5 , · · · ) are indeed the additional symmetry flo ws of t he cCKP hierarc h y . Pro of. The pro o f starts with the definition [ ∂ ∗ k , ∂ t 2 n +1 ] S = ∂ ∗ k ( ∂ t 2 n +1 S ) − ∂ t 2 n +1 ( ∂ ∗ k S ) , and using the action of the additional flow s, we g et [ ∂ ∗ k , ∂ t 2 n +1 ] S = − ∂ ∗ k  L 2 n +1 − S  + ∂ t 2 n +1 ((( A 1 ,k ) − − Y k ) S ) = − ( ∂ ∗ k L 2 n +1 ) − S − ( L 2 n +1 ) − ( ∂ ∗ k S ) + ∂ t 2 n +1 (( A 1 ,k ) − − Y k ) S + (( A 1 ,k ) − − Y k )( ∂ t 2 n +1 S ) . T a king eq.(3.17) of Corollary 1 in to the ab o v e form ula, it is not difficult to compute [ ∂ ∗ k , ∂ t 2 n +1 ] S = [( A 1 ,k ) − − Y k , L 2 n +1 ] − S + ( L 2 n +1 ) − (( A 1 ,k ) − − Y k ) S + [( L 2 n +1 ) + , ( A 1 ,k ) − − Y k ] − S − (( A 1 ,k ) − − Y k )( L 2 n +1 ) − S = [(( A 1 ,k ) − − Y k ) , L 2 n +1 ] − S − [( A 1 ,k ) − − Y k , L 2 n +1 + ] − S + [ L 2 n +1 − , ( A 1 ,k ) − − Y k ] S = [( A 1 ,k ) − − Y k , L 2 n +1 − ] − S + [ L 2 n +1 − , ( A 1 ,k ) − − Y k ] S = 0 W e hav e used the fact that [ L 2 n +1 + , ( A m,l ) − Y k ] − = [ L 2 n +1 + , (( A 1 ,k ) − − Y k )] − in the second step of the ab ov e deriv ation, since ( Y k ) − = Y k and [ L 2 n +1 + , ( A m,l ) + ] − = 0. The last equalit y holds b y virtue of ( P − ) − = P − for arbitrary pseduo-differential op erator P .  T a king in to accoun t of Y ∗ k = − Y k , k = 1 , 3 , 5 , · · · , w e can giv e the next theorem by a straigh tforw ard and tedious calculation. Theorem 3.3. Acting on the space of the w a v e op erator S of the constrained CKP hierarc hy , ∂ ∗ k forms a new cen terless W cC 1+ ∞ -subalgebra of cente rless W 1+ ∞ . 4. Additional symmetries of constraine d B KP hierarchy W e now turn to the case of the constrained BKP hierarc h y , this follo ws in a similar w ay of the case of the constrained CKP hierarc h y . T he Lax operato r L of the constrained BK P hierarc hy [27] is g iv en by L = ∂ + m X i =1 ( q i ∂ − 1 r i,x − r i ∂ − 1 q i,x ) , (4.1) where t he q i and r i are eigenfunc tions. The corresp onding Lax equ ation of the constrained BKP hierarc h y is defined by ∂ L ∂ t n = [ B n , L ] , n = 1 , 3 , 5 , · · · . (4.2) F or k = 1 , 3 , 5 , · · · , at first w e calculate the o riginal additio na l symmetry flo ws of the BKP hierarc hy as ∂ L ∂ t ∗ 1 ,k = − [( A 1 ,k ) − , L ] , (4.3) where A 1 ,k = M m L l − ( − 1) l L l − 1 M m L. Th us ( ∂ L ∂ t ∗ 1 ,k ) − = [( A 1 ,k ) + , L ] − + 2 ( L k ) − . (4.4) 10 KELEI TIA N † , JINGSONG HE ∗ ‡ , JIPENG CHENG † AN D YI CHENG † F or the cBKP hierarc h y , t he action of one flow ∂ τ on the eigenfunctions ( ∂ τ q i ) and ( ∂ τ r i ) may b e deriv ed fro m its action on the L if ( ∂ τ L ) − = m X i =1 (( ∂ τ q i ) ∂ − 1 r i,x − ( ∂ τ r i ) ∂ − 1 q i,x + q i ∂ − 1 ( ∂ τ r i,x ) − r i ∂ − 1 ( ∂ τ q i,x )) . (4.5) A t the same time, ( ∂ τ q i,x ) should b e consisten t with ( ∂ τ q i ), ( ∂ τ r i,x ) should b e consisten t with ( ∂ τ r i ). Ho w ev er, the fo llo wing lemma show s tha t ∂ t ∗ 1 ,k is not the case due t o the form of ( L k ) − . Lemma 4.1. The Lax op erat or L of the constrained BKP hierarc hy giv en by eq.(4.1 ) satisfies the relation ( L k ) − = m X i =1 k − 1 X j =0 ( L k − j − 1 ( q i ) ∂ − 1 ( L ∗ ) j ( r i,x ) − L k − j − 1 ( r i ) ∂ − 1 ( L ∗ ) j ( q i,x )) , k = 1 , 3 , 5 , · · · (4.6) where L ( q i ) = L + ( q i ) + m X j =1 ( q j ∂ − 1 x ( r j,x q i ) − r j ∂ − 1 x ( q j,x q i )) . According to the analysis ab ov e, w e ma y revise the flow s in eq.(4.3) to define a correct additional symmetry flo ws of the cBKP hierarc hy . Similar to the case of cCKP hierarch y , w e define a new additional flow ∂ ∗ k L = [ − ( A 1 ,k ) − + Y k , L ] , k = 1 , 3 , 5 , · · · , (4.7) where A 1 ,k = M m L l − ( − 1) l L l − 1 M m L is the generato r o f the a dditional symmetry of the BKP hierarc hy . W e shall sho w in this section that ∂ ∗ k is a correct additional symmetry flow for the cBKP hierarch y . First of all, by a similar discussion as the case of the cCKP hierarc h y , w e hav e the followin g lemma. Lemma 4.2. F or the cBKP hierarc hy , the BKP reduction conditio n infers a constraint on Y k as Y ∗ k = − ∂ Y k ∂ − 1 , k = 1 , 3 , 5 , · · · (4.8) Th us w e can set Y 1 = 0 , (4.9) Y k = m X i =1 k − 2 X j =0 (2 j − ( k − 2))( L k − 2 − j ( q i ) ∂ − 1 ( L ∗ ) j ( r i,x ) − L k − 2 − j ( r i ) ∂ − 1 ( L ∗ ) j ( q i,x )) , k ≥ 3 . (4.10) The follo wing lemma show s t ha t { Y k , k = 1 , 3 , 5 , · · · } satisfy t he prop erty w e need. Lemma 4.3. Y ∗ k = − ∂ Y k ∂ − 1 . (4.11) Pro of. Y ∗ 1 = Y 1 = 0 is trivial. F or k = 3 , 5 , 7 , · · · , w e hav e ∂ Y k ∂ − 1 = m X i =1 k − 2 X j =0 (2 j − ( k − 2))( ∂ L k − 2 − j ( q i ) ∂ − 1 ( L ∗ ) j ( r i,x ) ∂ − 1 − ∂ L k − 2 − j ( r i ) ∂ − 1 ( L ∗ ) j ( q i,x ) ∂ − 1 ) = m X i =1 k − 2 X j =0 (2 j − ( k − 2))(( − 1) k − 2 − j ( L ∗ ) k − 2 − j ( q i,x )( − 1) j ( L j ( r i ) ∂ − 1 − ∂ − 1 L j ( r i )) − ( − 1) k − 2 − j ( − L ∗ ) k − 2 − j ( r i,x )( − 1) j ( L j ( q i ) ∂ − 1 − ∂ − 1 L j ( q i ))) ADDITION AL SY MMETRIES OF CON STRAINED CKP AND BKP H IERARCHIES 11 = − m X i =1 k − 2 X j =0 (2 j − ( k − 2))(( L ∗ ) k − 2 − j ( q i,x ) L j ( r i ) ∂ − 1 − ( L ∗ ) k − 2 − j ( q i,x ) ∂ − 1 L j ( r i ) − ( L ∗ ) k − 2 − j ( r i,x ) L j ( q i ) ∂ − 1 + ( L ∗ ) k − 2 − j ( r i,x ) ∂ − 1 L j ( q i )) = − m X i =1 k − 2 X l =0 (2 l − ( k − 2))( − ( L ∗ ) l ( r i,x ) ∂ − 1 L k − 2 − l ( q i ) + ( L ∗ ) l ( q i,x ) ∂ − 1 L k − 2 − l ( r i )) − m X i =1 k − 2 X j =0 (2 j − ( k − 2))( − 1) k − 2 − j ( ∂ L k − 2 − j ( q i ) L j ( r i ) ∂ − 1 − ∂ L k − 2 − j ( r i ) L j ( q i ) ∂ − 1 ) = − Y ∗ k w e ha v e used the iden tit y ∂ − 1 f x ∂ − 1 = f ∂ − 1 − ∂ − 1 f as f = ( L ∗ ) j ( r i,x ) and f = ( L ∗ ) j ( q i,x ) in the deriv ation.  In order to get the explicit f o rm of the right hand side of eq.(4.7), the follo wing lemma is necessary . Lemma 4.4. F or the Lax o p erator L o f the cBKP hierarc h y a nd X = P l k =1 M k ∂ − 1 N k , [ X , L ] − = l X k =1 ( − L ( M k ) ∂ − 1 N k + M k ∂ − 1 L ∗ ( N k )) + m X i =1 ( X ( q i ) ∂ − 1 r i,x − X ( r i ) ∂ − 1 q i,x − q i ∂ − 1 X ∗ ( r i,x ) + r i ∂ − 1 X ∗ ( q i,x )) holds. Applying the ab ov e lemma we conclude that [ Y k , L ] − = − 2( L k ) − + k m X i =1 ( L k − 1 ( q i ) ∂ − 1 r i,x − L k − 1 ( r i ) ∂ − 1 q i,x ) + k m X i =1 ( q i ∂ − 1 L ∗ ( k − 1) ( r i,x ) − r i ∂ − 1 L ∗ ( k − 1) ( q i,x )) + m X i =1 ( Y k ( q i ) ∂ − 1 r i,x − Y k ( r i ) ∂ − 1 q i,x ) − m X i =1 ( − q i ∂ − 1 Y ∗ k ( r i,x ) + r i ∂ − 1 Y ∗ k ( q i,x )) . Theorem 4.1. F or the cBKP hierarch y , the additional flows defined b y eq.(4.7) acting on the eigenfunction q i and r i are ∂ ∗ k q i = k L k − 1 ( q i ) + Y k ( q i ) − ( A 1 ,k ) + ( q i ) , (4.12) ∂ ∗ k r i = k L k − 1 ( r i ) + Y k ( r i ) + ( A 1 ,k ) + ( r i ) , (4.13) ∂ ∗ k q i,x = − k L ∗ ( k − 1) ( q i,x ) − Y ∗ k ( q i,x ) + ( A 1 ,k ) ∗ + ( q i,x ) , (4.14) ∂ ∗ k r i,x = − k L ∗ ( k − 1) ( r i,x ) − Y ∗ k ( r i,x ) − ( A 1 ,k ) ∗ + ( r i,x ) , k = 1 , 3 , 5 , · · · . (4.15) 12 KELEI TIA N † , JINGSONG HE ∗ ‡ , JIPENG CHENG † AN D YI CHENG † Pro of. F rom the revised definition o f the additional flows in eq.(4.7) of the cBKP hierarc h y , by a short calculation, we ha v e ∂ ∗ k L − = [ − ( A 1 ,k ) − + Y k , L ] − = [( A 1 ,k ) + , L ] − + 2 ( L k ) − + [ Y k , L ] − = m X i =1 ( k L k − 1 ( q i ) + Y k ( q i ) − ( A 1 ,k ) + ( q i )) ∂ − 1 r i,x − ( k L k − 1 ( r i ) + Y k ( r i ) + ( A 1 ,k ) + ( r i )) ∂ − 1 q i,x + q i ∂ − 1 ( − k L ∗ ( k − 1) ( r i,x ) − Y ∗ k ( r i,x ) − ( A 1 ,k ) ∗ + ( r i,x )) − r i ∂ − 1 ( − k L ∗ ( k − 1) ( q i,x ) − Y ∗ k ( q i,x ) + ( A 1 ,k ) ∗ + ( q i,x )) th us ∂ ∗ k q i = k L k − 1 ( q i ) + Y k ( q i ) − ( A 1 ,k ) + ( q i ) , the other three identities could b e also obt a ined in the same wa y .  Remark 4.1. By a simple calculation, ∂ ∗ k q i,x and ∂ ∗ k r i,x are not essen tial and necessary , b ecause it can b e o btained from ∂ ∗ k q i and ∂ ∗ k r i , r esp ective ly . Corollary 4.1. The additional flows a ct on w a v e o p erator S of the constrained BKP hier- arc hy as ∂ ∗ k S = ( − ( A 1 ,k ) − + Y k ) S, k = 1 , 3 , 5 , · · · . (4.16) Theorem 4.2. The additional flow s ∂ ∗ k comm ute with constrained BKP hierarc hy ∂ ∂ t 2 n +1 , i.e. [ ∂ ∗ k , ∂ t 2 n +1 ] = 0 . (4.17) This theorem shows ∂ ∗ k ( k = 1 , 3 , 5 , · · · ) are indeed the additional symmetry flows of the cBKP hierarc hy . Using the identit y Y ∗ k = − ∂ Y k ∂ − 1 , k = 1 , 3 , 5 , · · · , we can presen t the next theorem omitting the pro of . Theorem 4.3. Acting on the space of the w a v e op erator S of the constrained BKP hierarch y , ∂ ∗ k forms new cen t erless W cB 1+ ∞ -subalgebra of cente rless W 1+ ∞ . 5. Conclusions and Discuss ions T o summarize, we hav e constructed the additional symmetries of the constrained CKP hi- erarc hy in eq.(3.7) and theorem 3.2. The additional flo ws a ction on the eigenfunction of the cCKP hierarc h y are giv en in theorem 3.1. Acting on the space o f the wa v e op erator S of the cCKP hierarc hy , ∂ ∗ k forms a new cen terless W cC 1+ ∞ -subalgebra of cen terless W 1+ ∞ algebra in theorem 3.3. Similarly , the conclusions for the constrained BKP hierarch y are obta ined using the analogous tec hnique in the case of constrained CKP hierarc h y . The main results of t he cBKP hierarc hy are presen ted in eq.(4.7), theorem 4 .1, 4.2 and 4.3. Our results sho w that t he cCKP a nd cBKP hierarc hies ha v e, indeed, some different prop erties for additional symmetry comparing with the KP , BKP , CKP and constrained KP hierarchie s. F or example, the defini- tions of additional symmetry flows f or the cCKP and cBKP hierar chies are different f rom the ab ov e hierarc hies, t he revised op erator s Y k of the cCKP and cBKP hierarc hies are differen t from the corresponding op erat o r X k of the cKP hierarch y give n in [25]. 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