Bilinear equation and additional symmetries for an extension of the Kadomtsev-Petviashvili hierarchy

Bilinear Equation and Additional Symmetries for an Extension of the Kadom tsev–P etviash vili Hierarc h y Jiaping Lu Chao-Zhong W u Scho ol of Mathematics, Sun Y at-Sen University, Gu angzhou 510275, P.R. China Abstract An extension of the Kadomtsev–P etviashvili (KP) hierarch y defined via scalar pseudo-differential oper ators was studied in [16, 20]. In this pap er, we represent the extended KP hierar ch y int o the for m of bilinear equation of (a djoint ) Baker–Akhiezer functions, and co nstruct its additional symmetrie s . As a bypro duct, we derive the Virasoro symmetries for the constrained KP hierarchies. Key words : Ka do mt sev–Petviashvili hierarch y; Baker–Akhiezer function; additional symmetry 1 In tro duction As a fundamental mo del in the theory of in tegrable systems, th e K adom tsev–Pe tviash vili (KP) hierarc h y can b e defined as f ollo ws. Let L K P = ∂ + v 1 ∂ − 1 + v 2 ∂ − 2 + . . . , ∂ = d d x , (1.1) b e a p seudo-different ial op erator wh ose coefficients v i are scalar un kno wn functions of the spacial coordinate x , then the KP hierarc h y is comp osed b y the f ollo wing ev olutionary equations of v i as ∂ L K P ∂ t k = h ( L K P k ) + , L K P i , k = 1 , 2 , 3 , . . . . (1.2) Here and b elo w the s u bscript “ +” o f a ps eudo-differen tial o p erator means to tak e its purely differen tial part, wh ile the subscript “ − ” means to tak e its negativ e part. S upp ose that the equ ations (1.2) are imp osed with the constr aint ( L K P n ) − = 0 with some in teger n ≥ 2, then they form the Gelfand–Dic key (or the ( n − 1)-th Korteweg– de V ries) hierarc h y . It is kno wn that the KP hierarch y (1.2) p ossesses a series of bi-Hamiltonia n stru c- tures derived b y the R -matrix formalism [15], and that it can b e represen ted into the form of b ilinear equation of (adjoin t) Bake r–Akhiezer fu nctions or of a tau fu nction [8]. Suc h b i-Hamiltonian stru ctures and the bilinear equation can b e redu ced to that of th e Gelfand–Dic k ey hierarc hies. What is more, f or the KP hierarc h y there is a class of non- isosp ectral symm etries n amed as the additional symmetries th at can b e constructed via 1 the so-called Orlov– Sc hulman op erators [14]. The flo w s of suc h add itional symmetries comm u te with all the time flows ∂ /∂ t k but do n ot commute b et ween themselve s; instead, they generate the W 1+ ∞ algebra. As an application of the add itional symmetries for the KP h ierarc hy , part of these symmetries can be r educed to the Virasoro symmetries for its su bhierarc hies such lik e the Gelfand–Dic key hierarchies. Note that Virasoro symm e- tries rev eal crucial prop er ties of a large amoun t of int egrable hierarchies including the Gelfand–Dic k ey h ierarc hies and the Drinfeld–Sok olo v hierarc h ies, see e.g. [1, 9, 10, 19] and references therein. In th e definition of the KP hierarch y (1.2) , it is used pseudo-differentia l op erators with only finitely man y p ositiv e p o wers in ∂ . The notion of pseudo-differen tial op era- tor w as ge neralized in [13] to b e o v er a certain graded differentia l alg ebra A su c h that these op erators may conta in infinitely many p ositiv e p o w ers in ∂ (see S ection 2 b elo w for details). By using t hese op erators, in [20] Zhou and one of the authors of the present pap er considered an in tegrable h ierarc hy , w h ic h can b e viewe d as a subh ierarc hy of the disp ersionf ul analogue [16] of the universal Whitham hierarc h y . More exactly , let P = ∂ + X i ≥ 1 u i ( ∂ − ϕ ) − i , ˆ P = ( ∂ − ϕ ) − 1 ˆ u − 1 + X i ≥ 0 ˆ u i ( ∂ − ϕ ) i (1.3) with u i , ˆ u i and ϕ b eing certain unkn o wn fun ctions b elong to A , then the follo w ing ev olu- tionary equations are w ell d efined: ∂ ∂ t k ( P , ˆ P ) =  [( P k ) + , P ] , [( P k ) + , ˆ P ]  , ∂ ∂ ˆ t k ( P , ˆ P ) =  [ − ( ˆ P k ) − , P ] , [ − ( ˆ P k ) − , ˆ P ]  , (1.4) where k = 1 , 2 , 3 , . . . . These evol utionary equations comp ose an integ rable hierarc h y , whic h is named as t he extended KP hierarch y . In fact, this hierarc hy is red u ced to the KP hierarch y (1.2) whenev er ˆ P = 0 (note that the op erator P giv es a n alternativ e expression of L K P ). O n th e other hand , if one lets ϕ → 0 and imp ose certain B-t yp e symmetry cond itions to the op erators P and ˆ P , then the fl o w s in (1.4) with k ∈ Z odd > 0 giv e the t wo-c omp onent BKP (2-BKP) h ierarc hy [7, 13]. With the R -matrix metho d applied in the cases of the K P and the 2-BKP hierarc hies [15, 17], the exte nded KP hierarch y (1.4) wa s sho wn to p ossess infinitely man y bi-Hamiltonian stru ctures [20]. In this pap er we assu me ϕ = ∂ ( f ) in (1.3) with some homogeneous fun ction f ∈ A of degree 0. W e will show that the op erators P and ˆ P can b e rep r esen ted in a dressing form as P = Φ ∂ Φ − 1 , ˆ P = ˆ Φ ∂ − 1 ˆ Φ − 1 , (1.5) where Φ an d ˆ Φ are p seudo-different ial op erators of the form Φ = 1 + X i ≥ 1 a i ∂ − i , ˆ Φ = e f   1 + X i ≥ 1 b i ∂ i   . With th e h elp of these t w o dressing op erators in the extend ed KP h ierarc hy , w e w ill in tro duce t wo Bak er–Akhiezer fun ctions ψ ( t , ˆ t ; z ), ˆ ψ ( t , ˆ t ; z ) and their adjoin ts ψ † ( t , ˆ t ; z ), 2 ˆ ψ † ( t , ˆ t ; z ) that dep end on the time v ariables t = ( t 1 , t 2 , t 3 , . . . ), ˆ t = ( ˆ t 1 , ˆ t 2 , ˆ t 3 , . . . ), and a nonzero parameter z . Ou r first main result is the follo win g theorem (see Theorem 3.8 for a more precise version). Theorem 1.1 The e xtende d KP hier ar chy (1.4) c an b e r epr esente d e qu ivalently to a bi- line ar e quation as res z  ψ ( t , ˆ t ; z ) ψ † ( t ′ , ˆ t ′ ; z )  = res z  ˆ ψ ( t , ˆ t ; z ) ˆ ψ † ( t ′ , ˆ t ′ ; z )  (1.6) with arbitr ary time variables ( t , ˆ t ) and ( t ′ , ˆ t ′ ) . Our s econd main result is the construction of additional symmetries for the extended KP hierarc hy , with the help of certain Or lo v–Sch ulman op erators given by the ab o ve dressing op er ators Φ and ˆ Φ. S uc h additional sym metries will b e shown to generate a W 1+ ∞ × W 1+ ∞ algebra. Th ese results will b e app lied to stud y the ( n, 1)-constrained KP hierarc hy (see, e.g., [2, 4, 6, 11, 12]), which is d en oted as cKP n, 1 and can b e red u ced from the extended KP hierarc hy (1.4) un der the constrain t P n = ˆ P with a giv en int eger n ≥ 1. In fact , the Virasoro symmetries for the cKP n, 1 hierarc hy (ev en for more general cases) w ere prop osed b y Arat yn , Nissimov and Pa c hev a [2], with the met ho d o f adding certain “ghost” symmetry fl o w s su c h that some nonlocal ac tions on fu nctions are in v olv ed . They also sh o wed that, the cKP n, 1 hierarc hy sub ject to the subsidiary condition of inv ariance under the lo west Virasoro symmetry flo w can b e applied to compute explicit W ronskian solution for the t w o-matrix mo del partition fu nction [3]. As to b e seen, the Virasoro symm etries app earing in [2, 3] for the cKP n, 1 hierarc hy can b e obtained in an alternativ e w a y , starting fr om the additional s y m metries for the extended KP hierarc hy (see Prop osition 4.9 b elo w). This pap er is arranged as follo ws. In th e next section, w e will recall the pseu d o- differen tial op erators of the first and the second t yp es ov er a graded differen tial algebra. In Section 3, we will recall the definition of th e extend ed KP hierarch y , and represent it into the form of a b ilinear equation. In Section 4, w e will construct the add itional symmetries for th e extended KP h ierarc hy , and then study th e Virasoro symm etries for the cKP n, 1 hierarc hy . T he fi nal section is devote d to some remarks. 2 Pseudo-differen tial op erators Let A b e a commutativ e asso ciativ e algebra, and ∂ : A → A b e a deriv ation. W e consider the li near space D ( A ) =  P i ∈ Z f i ∂ i | f i ∈ A  and its subsets. F o r instance, the set of pseudo-differenti al op erators is A (( ∂ − 1 )) =    X i ≤ k f i ∂ i | f i ∈ A , k ∈ Z    , 3 and it b ecomes an asso ciativ e algebra if a pro d uct is defin ed by f ∂ i · g ∂ j = X r ≥ 0  i r  f ∂ r ( g ) ∂ i + j − r , f , g ∈ A . (2.1) F or any tw o p seudo-differen tial op erators A and B , th eir comm u tator means [ A, B ] = AB − B A . Clearly , one has [ ∂ , f ] = ∂ ( f ) for any f ∈ A . In the pr esen t pap er w e assume the algebra A to b e a graded one. Namely , A = Q i ≥ 0 A i , suc h that A i · A j ⊂ A i + j , ∂ ( A i ) ⊂ A i +1 . Denote D − = A (( ∂ − 1 )), whic h is called the algebra of pseudo-differenti al op erators of the first type o v er A . In con trast, by th e algebra of pseud o-differen tial op erators of the second t yp e o ver A it means [13] D + =    X i ∈ Z X j ≥ max { 0 ,k − i } a i,j ∂ i | a i,j ∈ A j , k ∈ Z    , (2.2) whose p r o duct is also defin ed by (2.1). One observ es that an op erator in D + ma y conta in infinitely many p ositiv e p ow ers in ∂ . Giv en an element A = P i ∈ Z f i ∂ i ∈ D ( A ), its differen tial part, negativ e p art and residue are defined resp ectiv ely as: A + = X i ≥ 0 f i ∂ i , A − = X i< 0 f i ∂ i , res A = f − 1 . (2.3) It is easy to see that  D ∓  ± ⊂ D − ∩ D + . What is more, on eac h D ∓ there is an an ti-automorphism defined b y ∂ ∗ = − ∂ , f ∗ = f with f ∈ A . Clearly , for an y A ∈ D ∓ , one h as res A ∗ = − res A. In what follo ws w e will use the notation A ≥ r = Q i ≥ r A i with r ∈ Z ≥ 0 . Giv en an elemen t ϕ ∈ A ≥ 1 , the follo wing tw o maps are we ll d efined with ∂ replaced by ∂ − ϕ , that is, S ϕ : D ∓ → D ∓ , X f i ∂ i 7→ X f i ( ∂ − ϕ ) i . (2.4) F or instance, we ha v e S ϕ ( ∂ − 1 ) =( ∂ − ϕ ) − 1 = ∂ − 1 (1 − ϕ∂ − 1 ) − 1 4 = ∂ − 1 + ∂ − 1 ϕ∂ − 1 + ∂ − 1 ϕ∂ − 1 ϕ∂ − 1 + . . . . (2.5) In [20 ], it was v erified that the maps S ϕ are automorph ism s on eac h D ∓ . Accordingly , the algebras D ∓ can b e represente d as follo ws: D − =    X i ≤ k g i ( ∂ − ϕ ) i | g i ∈ A , k ∈ Z    , (2.6) D + =    X i ∈ Z X j ≥ max { 0 ,k − i } b i,j ( ∂ − ϕ ) i | b i,j ∈ A j , k ∈ Z    , (2.7) and their pro d uct can b e d efined equiv alently by f ( ∂ − ϕ ) i · g ( ∂ − ϕ ) j = X r ≥ 0  i r  f ∂ r ( g ) ( ∂ − ϕ ) i + j − r , f , g ∈ A . Moreo ve r, it is easy to v erify the follo wing p rop erties: for an y A ∈ D ∓ , ( S ϕ A ) ± = S ϕ ( A ± ) , res ( S ϕ A ) = r es A, ( S ϕ A ) ∗ = S − ϕ A ∗ . (2.8) 3 The extended KP hierarc h y and its bilinear equation W e p ro ceed to recall the definition of the extend ed KP h ierarc hy , and then capsule it into a bilinear equation of certain (adjoin t) Bak er–Akhiezer functions. 3.1 The extended KP hierarc h y Let M b e an in finite-dimensional manifold with co ordinate a = ( a 1 , a 2 , a 3 , . . . ; f , b 1 , b 2 , b 3 , . . . ) . W e consider the follo win g graded algebra of formal d ifferential p olynomials: A = C ∞ ( S 1 → M )[[ ∂ r ( a ) | r ≥ 1]] , (3.1) in w hic h the deriv ation is ∂ = d / d x with x b eing the co ordinate of the lo op S 1 , and deg a = 0 , deg ∂ r ( a ) = r. Ov er the graded differential algebra A , it is defined the algebras D − and D + of pseudo- differen tial op erators of the first t yp e and of the second type, resp ectiv ely . Let us consider tw o p seudo-different ial op erators as follo ws : Φ = 1 + X i ≥ 1 a i ∂ − i ∈ D − , (3.2) 5 ˆ Φ = e f   1 + X i ≥ 1 b i ∂ i   ∈ D + . (3.3) These tw o op erators ha ve inv erses of the form Φ − 1 = 1 + X i ≥ 1 ˜ a i ∂ − i ∈ D − , (3.4) ˆ Φ − 1 =   1 + X i ≥ 1 ˜ b i ∂ i   e − f ∈ D + . (3.5) In fact, by expanding Φ − 1 Φ = 1 and ˆ Φ − 1 ˆ Φ = 1, one sees that the co efficien ts ˜ a i and ˜ b i are represent ed as ˜ a i = − a i + g i ( a 1 , a 2 , . . . , a i − 1 ) , ˜ b i = − b i + h i ( b 1 , b 2 , b 3 , . . . ) with g i , h i ∈ A ≥ 1 . F or instance, one has ˜ a 1 = − a 1 , and that ˜ b 1 is give n recursiv ely b y ˜ b 1    A 0 = − b 1 , ˜ b 1    A j = − j X r =1 b r  ˜ b 1    A j − r  ( r ) for j ≥ 1 . Here ˜ b 1    A j means the degree- j comp onen t of ˜ b 1 . No w let us in tro duce t wo pseudo-differentia l op erators as follo ws: P = Φ ∂ Φ − 1 ∈ D − , ˆ P = ˆ Φ ∂ − 1 ˆ Φ − 1 ∈ D + . (3.6) Prop osition 3.1 The op er ators P and ˆ P give n ab ove c an b e r epr esente d in the form: P = ∂ + X i ≥ 1 v i ∂ − i , ˆ P = ( ∂ − f ′ ) − 1 ρ + X i ≥ 0 ˆ v i ∂ i , (3.7) wher e v i +1 , ˆ v i ∈ A ≥ 1 for i ∈ Z ≥ 0 , and ρ = e f  ˆ Φ − 1  ∗ (1) . (3.8) Pr o of: The represent ation of P is we ll kno wn, so let us v erify the case of ˆ P . T o this end, firstly let u s chec k th at ˆ P giv en in (3.6) tak es the form ˆ P = ( ∂ − f ′ ) − 1   X i ≥ 0 ˜ v i ( ∂ − f ′ ) i   , ˜ v i − δ i 0 ∈ A ≥ 1 . (3.9) W e subs titute this expansion and (3.3) in to an equ iv alen t v ers ion of ˆ Φ ∂ − 1 ˆ Φ − 1 = ˆ P , s a y , e − f ( ∂ − f ′ ) ˆ Φ ∂ − 1 = e − f ( ∂ − f ′ ) ˆ P ˆ Φ . 6 Note e − f ( ∂ − f ′ ) e f = ∂ , th en we h av e 1 + b ′ 1 + X i ≥ 1 ( b i + b ′ i +1 ) ∂ i =   X i ≥ 0 ˜ v i ∂ i     1 + X i ≥ 1 b i ∂ i   , (3.10) in w hic h the co efficients of ∂ i lead to i = 0 : 1 + b ′ 1 = ˜ v 0 , (3.11) i ≥ 1 : b i + b ′ i +1 = ˜ v i + i X r =1 X s ≥ 0  i − r + s s  ˜ v i − r + s b ( s ) r . (3.12) The equations (3.12) yield b i = ˜ v i | A 0 + i X r =1 ˜ v i − r | A 0 b r , i ≥ 1 , whic h together with (3.11) giv es ˜ v i | A 0 = δ i 0 , i ≥ 0 . The equations (3.12) also yield b ′ i +1 = ˜ v i | A 1 + i X r =1 ˜ v i − r | A 1 b r , i ≥ 1 , whic h together with (3.11) determine ˜ v i | A 1 recursiv ely . In fact, we can obtain a generating function for ˜ v i | A 1 of a parameter z as X i ≥ 0 ˜ v i | A 1 z i +1 = ∂ log   1 + X i ≥ 1 b i z i   . F or j ≥ 2, it follo ws from (3.12 ) that ˜ v i | A j + i X r =1 j − 1 X s =0  i − r + s s  ˜ v i − r + s | A j − s · b ( s ) r = 0 , i ≥ 1 , hence ˜ v i | A j are determined rec ursively . By using (2. 4 ), we deriv e that the op erator ˆ P tak es the form (3.7), in wh ic h ρ = res  ˆ Φ ∂ − 1 ˆ Φ − 1  = res  e f ∂ − 1 ˆ Φ − 1  = e f  ˆ Φ − 1  ∗ (1) . (3.13) Therefore the prop osition is p ro v ed.  7 Remark 3.2 F rom (3.8) and (3.5) it follo ws that ρ = e f  ˆ Φ − 1  ∗ (1) = 1 + X i ≥ 1 ( − 1) i ˜ b ( i ) i , (3.14) whic h implies ρ − 1 ∈ A ≥ 1 . One also sees that, the co efficien ts of P and ˆ P hav e different degrees f rom that in [20], since no w the graded differen tial algebra A is chosen differently .  With the help of the op erators P and ˆ P , let us defin e a class of evolutio nary equations on M : for k ∈ Z > 0 , ∂ Φ ∂ t k = − ( P k ) − Φ , ∂ ˆ Φ ∂ t k =  ( P k ) + − δ k 1 ˆ P − 1  ˆ Φ , (3.15) ∂ Φ ∂ ˆ t k = − ( ˆ P k ) − Φ , ∂ ˆ Φ ∂ ˆ t k = ( ˆ P k ) + ˆ Φ , (3.16) Here we note that t he righ t h an d sides m ak e sense since the operators ( P k ) + , ( ˆ P k ) − ∈ D − ∩ D + , an d we assume that the flo ws ∂ /∂ t k and ∂ /∂ ˆ t k comm u tate w ith ∂ /∂ x . In particular, it can b e seen ∂ /∂ t 1 = ∂ /∂ x , so in wh at follo w s we will just tak e t 1 = x . Prop osition 3.3 The flows (3.15) , (3.16) satisfy, for k ∈ Z > 0 , ∂ P ∂ t k = h ( P k ) + , P i , ∂ ˆ P ∂ t k = h ( P k ) + , ˆ P i , (3.17) ∂ P ∂ ˆ t k = h − ( ˆ P k ) − , P i , ∂ ˆ P ∂ ˆ t k = h − ( ˆ P k ) − , ˆ P i . (3.18) Mor e over, these flows c ommute with e ach other. Pr o of: The p rop osition can b e ve rified case by case. F or in stance, we hav e ∂ ˆ P ∂ t k = " ∂ ˆ Φ ∂ t k ˆ Φ − 1 , ˆ P # = h ( P k ) + − δ k 1 ˆ P − 1 , ˆ P i = h ( P k ) + , ˆ P i ,  ∂ ∂ t k , ∂ ∂ ˆ t l  ˆ Φ = ∂ ∂ t k  ( ˆ P l ) + ˆ Φ  − ∂ ∂ ˆ t l   ( P k ) + − δ k 1 ˆ P − 1  ˆ Φ  = h ( ˆ P l ) + , ( P k ) + − δ k 1 ˆ P − 1 i ˆ Φ + h ( P k ) + , ˆ P l i + ˆ Φ −  h − ( ˆ P l ) − , P k i + − δ k 1 h − ( ˆ P l ) − , ˆ P − 1 i  ˆ Φ = h ( ˆ P l ) + , ( P k ) + i ˆ Φ − h ˆ P l , ( P k ) + i + ˆ Φ + h ( ˆ P l ) − , ( P k ) + i + ˆ Φ − δ k 1 h ˆ P l , ˆ P − 1 i ˆ Φ 8 =0 . The other cases are almost the same. So we complete the p ro of.  The system of equations (3 .17) , (3.18) was stu died in [2 0] by Zhou and one of the authors (cf. [16]), with the set of unknown functions as { v i +1 , ˆ v i | i ∈ Z ≥ 0 } ∪ { ρ, ϕ = f ′ } . This system is called the extended KP hierarch y for the reason that the flo w s ∂ P /∂ t k in (3.17) with k ∈ Z > 0 comp ose the well-kno wn KP h ierarc hy . By virtue o f the ab o v e prop osition, we w ill also call the system of equations (3.15) , (3.16 ) th e extended KP hierarc hy . Example 3.4 F rom the extended KP hierarc hy one can wr ite do wn some equations ex- plicitly as follo ws : ∂ 2 v 1 ∂ t 2 2 =  4 3 ∂ v 1 ∂ t 3 − 4 v 1 v ′ 1 − 1 3 v ′′′ 1  ′ , ∂ f ∂ t 2 = 2 v 1 + ( f ′ ) 2 + f ′′ , ∂ ρ ∂ t 2 =  2 ρf ′ − ρ ′  ′ . (3.19) Example 3.5 By using (3.3) and (3.15) we h a ve, for k ∈ Z > 0 , ∂ f ∂ t k = e − f ∂ ˆ Φ ∂ t k (1) = e − f  ( P k ) + − δ k 1 ˆ P − 1  ˆ Φ(1) = e − f ( P k ) + e f (1) =res  e − f P k e f ∂ − 1  = res S − f ′  P k e f ∂ − 1 e − f  =res  P k e f ∂ − 1 e − f  = res  P k ( ∂ − f ′ ) − 1  . (3.20) Here in th e fifth equalit y we h a ve u sed (2.8). With the same metho d, w e obtain ∂ f ∂ ˆ t k = res  ˆ P k ( ∂ − f ′ ) − 1  , k ∈ Z > 0 . (3.21) Moreo ve r, for ˙ t k = t k or ˆ t k (corresp ondingly , ˙ P = P or ˆ P ) , w e ha v e ∂ ρ ∂ ˙ t k = X i ≥ 1 ( − 1) i − 1 ∂ i  ρ res ˙ P k ( ∂ − f ′ ) − i − 1  . (3.22) By letting ϕ = f ′ , w e reco ver the equations (2.23) and (2.24) in [20]. 3.2 Bak er–A khiezer functions and t he bilinea r equation With z b eing a n onzero parameter, we assign ∂ i ( e xz ) = z i e xz for an y i ∈ Z , and more generally , ∂ i ( g e xz ) = ( ∂ i g ) ( e xz ) , g ∈ A , i ∈ Z . 9 Namely , it is the usu al action of a differentia l op erator on a function wh enev er i ≥ 0, and the int egral constan ts are fixed in a sp ecial w a y w henev er i < 0. In ord er to s im p lify notations, for an y A ∈ D ± and exp onen tial fun ctions of the the f orm e ± xz , w e w ill just write Ae ± xz instead of A ( e ± xz ). Denote t = ( t 1 = x, t 2 , t 3 , . . . ) and ˆ t = ( ˆ t 1 , ˆ t 2 , ˆ t 3 , . . . ), and let ξ b e giv en b y ξ ( t ; z ) = X k ∈ Z > 0 t k z k . Giv en a solution of the extended KP hierarc h y (3.15), (3.16), let us in tr o duce t wo Bak er– Akhiezer functions: ψ ( t , ˆ t ; z ) = Φ e ξ ( t ; z ) , ˆ ψ ( t , ˆ t ; z ) = ˆ Φ e xz − ξ ( ˆ t ; z − 1 ) , (3.23) and t wo adjoin t Bak er–Akhiezer functions: ψ † ( t , ˆ t ; z ) =  Φ − 1  ∗ e − ξ ( t ; z ) , ˆ ψ † ( t , ˆ t ; z ) =  ˆ Φ − 1  ∗ e − xz + ξ ( ˆ t ; z − 1 ) . (3.24) When there is n o confusion, w e will just write η ( z ) = η ( t , ˆ t ; z ) with η ∈ { ψ , ˆ ψ , ψ † , ˆ ψ † } . Based on (3.15) and (3.16), it is straigh tforwa rd to ve rify the th e follo w ing Lemma. Lemma 3.6 The (adjoint) Baker–Akhiezer functions given ab ove satisfy: (i) P ψ ( z ) = z ψ ( z ) , P ∗ ψ † ( z ) = z ψ † ( z ) , (3.25) ˆ P ˆ ψ ( z ) = z − 1 ˆ ψ ( z ) , ˆ P ∗ ˆ ψ † ( z ) = z − 1 ˆ ψ † ( z ); (3.26) (ii) ∂ ˙ ψ ( z ) ∂ t k =  P k  + ˙ ψ ( z ) , ∂ ˙ ψ ( z ) ∂ ˆ t k = −  ˆ P k  − ˙ ψ ( z ) , (3.27) ∂ ˙ ψ † ( z ) ∂ t k = −  P k  ∗ + ˙ ψ † ( z ) , ∂ ˙ ψ † ( z ) ∂ ˆ t k =  ˆ P k  ∗ − ˙ ψ † ( z ) (3.28) wher e ˙ ψ ∈ n ψ , ˆ ψ o and ˙ ψ † ∈ n ψ † , ˆ ψ † o . F or any formal series P i ∈ Z g i z i , its residue is d efined by res z X i ∈ Z g i z i = g − 1 . The follo wing lemma is us efu l. Lemma 3.7 (see, for example, [8]) F or any pseudo-differ ential op er ators Q, R ∈ D ± , the fol lowing e quality holds true whenever b oth sides make sense: res z  Qe z x · R ∗ e − z x  = res ( QR ) . (3.29) 10 Theorem 3.8 The (adjoint) Baker–Akhiezer functions of the extende d KP hier ar chy sat- isfy the fol lowing biline ar e quation res z  ψ ( t , ˆ t ; z ) ψ † ( t ′ , ˆ t ′ ; z )  = res z  ˆ ψ ( t , ˆ t ; z ) ˆ ψ † ( t ′ , ˆ t ′ ; z )  , (3.3 0) for arbitr ary time variables ( t , ˆ t ) and ( t ′ , ˆ t ′ ) . Conversely, supp ose that four functions of the form ψ ( t , ˆ t ; z ) =   1 + X i ≥ 1 a i ( t , ˆ t ) z − i   e ξ ( t ; z ) , (3.31) ˆ ψ ( t , ˆ t ; z ) = e f ( t , ˆ t )   1 + X i ≥ 1 b i ( t , ˆ t ) z i   e xz − ξ ( ˆ t ; z − 1 ) , (3.32) ψ † ( t , ˆ t ; z ) =   1 + X i ≥ 1 a † i ( t , ˆ t ) z − i   e − ξ ( t ; z ) , (3.33) ˆ ψ † ( t , ˆ t ; z ) = e − f ( t , ˆ t )   1 + X i ≥ 1 b † i ( t , ˆ t ) z i   e − xz + ξ ( ˆ t ; z − 1 ) (3.34) satisfy the b iline ar e quation (3.30) , then they ar e th e Baker–A khiezer functions and the adjoint B aker–Akhiezer fu nctions of the extende d KP hier ar chy. Pr o of: As a pr ep aration, we in tro duce the set of indices as I = { ( m 1 , m 2 , m 3 , . . . ) | m i ∈ Z ≥ 0 suc h that m i = 0 f or i ≫ 0 } . F or m = ( m 1 , m 2 , m 3 , . . . ) ∈ I , denote ∂ t m = Y k ≥ 1  ∂ ∂ t k  m k , ∂ ˆ t m = Y k ≥ 1  ∂ ∂ ˆ t k  m k . In order to show the equalit y (3.30), we only need to c hec k that th e Bak er–Akhiezer functions and the adjoin t Bak er–Akhiezer functions satisfy res z  ∂ t m ∂ ˆ t n ψ ( t , ˆ t ; z ) · ψ † ( t , ˆ t ; z )  = res z  ∂ t m ∂ ˆ t n ˆ ψ ( t , ˆ t ; z ) · ˆ ψ † ( t , ˆ t ; z )  (3.35) for any indices m , n ∈ I . In fact, giv en an y m , n ∈ I , according to Lemm a 3.6 and Prop osition 3.3 there is a p seudo-differen tial op erator A m , n ∈ D − ∩ D + suc h that the follo wing t wo equalities hold sim u ltaneously: ∂ t m ∂ ˆ t n ψ ( z ) = A m , n ψ ( z ) , ∂ t m ∂ ˆ t n ˆ ψ ( z ) = A m , n ˆ ψ ( z ) . Then, b y using (3.24) and Lemma 3.7, the equalit y (3.35) is recast to res  A m , n ΦΦ − 1  = res  A m , n ˆ Φ ˆ Φ − 1  , 11 whic h is clearly v alid. Hence the equalit y (3.35) holds true, and the first assertion is v er ifi ed. F or the second assertion, one sees that there are uniquely p seudo-differen tial op erators Φ, ˆ Φ, Ψ and ˆ Ψ s uc h that th e fu n ctions (3.31)–(3.34) are repr esented as ψ ( t , ˆ t ; z ) = Φ e ξ ( t ; z ) , ˆ ψ ( t , ˆ t ; z ) = ˆ Φ e xz − ξ ( ˆ t ; z − 1 ) , ψ † ( t , ˆ t ; z ) = Ψ ∗ e − ξ ( t ; z ) , ˆ ψ † ( t , ˆ t ; z ) = ˆ Ψ ∗ e − xz + ξ ( ˆ t ; z − 1 ) . Moreo ve r, the op erators Φ and ˆ Φ tak e the form (3.2) and (3.3) r esp ectiv ely , while Ψ and ˆ Ψ take the form (3.4) a nd (3.5 ) resp ectiv ely . The bilinear equation (3 .30) le ads to the follo wing facts. (i) F or i ∈ Z , w e hav e (note that wh en i < 0 the in tegral constan ts on b oth sides are fixed in the same wa y) res z  ∂ i ψ ( t , ˆ t ; z ) ψ † ( t ′ , ˆ t ′ ; z )  = res z  ∂ i ˆ ψ ( t , ˆ t ; z ) ˆ ψ † ( t ′ , ˆ t ′ ; z )  . Let ( t ′ , ˆ t ′ ) = ( t , ˆ t ), and then with th e help of (3.2 9 ) we obtain res ∂ i ΦΨ = res ∂ i ˆ Φ ˆ Ψ = 0 , i ≥ 0; res ∂ i ˆ Φ ˆ Ψ = res ∂ i ΦΨ = δ i, − 1 , i < 0 . whic h implies Φ Ψ = ˆ Φ ˆ Ψ = 1. So w e deriv e Ψ = Φ − 1 , ˆ Ψ = ˆ Φ − 1 . (3.36) (ii) Denote X k = ∂ Φ ∂ t k Φ − 1 , ˆ X k = ∂ ˆ Φ ∂ t k ˆ Φ − 1 . Clearly , one has ( X k ) + = ( ˆ X k ) − = 0, and that ∂ ψ ( z ) ∂ t k =  X k Φ + Φ ∂ k  e ξ ( t ; z ) =  X k + Φ ∂ k Φ − 1  ψ ( z ) , ∂ ˆ ψ ( z ) ∂ t k =  ˆ X k ˆ Φ + δ k 1 ˆ Φ ∂  e xz − ξ ( ˆ t ; z − 1 ) =  ˆ X k + δ k 1 ˆ Φ ∂ ˆ Φ − 1  ˆ ψ ( z ) . F or an y i ∈ Z , w e let ∂ i act on the deriv ativ e of (3.30) with r esp ect to t k , and let ( t ′ , ˆ t ′ ) = ( t , ˆ t ), then b y using (3.29) again w e obtain res ∂ i  X k + Φ ∂ k Φ − 1  ΦΦ − 1 = res ∂ i  ˆ X k + δ k 1 ˆ Φ ∂ ˆ Φ − 1  ˆ Φ ˆ Φ − 1 . Hence X k + Φ ∂ k Φ − 1 = ˆ X k + δ k 1 ˆ Φ ∂ ˆ Φ − 1 , and we arrive at X k = −  Φ ∂ k Φ − 1  − , ˆ X k =  Φ ∂ k Φ − 1  + − δ k 1 ˆ Φ ∂ ˆ Φ − 1 . (3.37) Similarly , w e can d eriv e ∂ Φ ∂ ˆ t k Φ − 1 = −  ˆ Φ ∂ − k ˆ Φ − 1  − , ∂ ˆ Φ ∂ ˆ t k ˆ Φ − 1 =  ˆ Φ ∂ − k ˆ Φ − 1  + . (3.38) T aking (i) and (ii) together w e ac h iev e the second assertion. Th e theorem is prov ed.  12 4 Additional symmetries v ersus Virasoro sy mmetries In this secti on, w e wa nt to construct a class o f a dditional sy m metries for the extended KP h ierarc hy , follo w ing the app roac h of [14, 18], and th en study the Virasoro s y m metries for th e constrained KP hierarc hies. 4.1 Additional symmetries for t he extended KP hierarch y Supp ose that the op erators Φ and ˆ Φ solve the hierarc hy (3.15), (3.16). Let us introd uce t wo Orlo v–Sc hulman op erators as follo ws: M = ΦΞΦ − 1 , ˆ M = ˆ Φ ˆ Ξ ˆ Φ − 1 , (4.1) where Ξ = X k ∈ Z > 0 k t k ∂ k − 1 , ˆ Ξ = x + X k ∈ Z > 0 k ˆ t k ∂ − k − 1 . Here w e assume that all t k and ˆ t k v anish except finitely man y of them, suc h that the op erators M and ˆ M are w ell defin ed . Remark 4.1 There is another wa y to ensu re the Orlo v–Sc h ulman op erators M and ˆ M (ev en with infinitely man y time v ariables) to b e well defined . Indeed, as what w as done in [18], one can extend the graded algebra A to include also { t k , ˆ t k | k ∈ Z > 0 } w ith deg t k = deg ˆ t k = k , so that a new graded algebra ˜ A is obtained. T hen, th e set D − of pseudo-differenti al op erators of the first t yp e can b e extend ed to ˜ D − =    X i ∈ Z X j ≥ max { 0 ,i − k } a i,j ∂ i | a i,j ∈ ˜ A j , k ∈ Z    . So, the op erators M and ˆ M are elemen ts of ˜ D − and D + (with A replaced by ˜ A ) resp ec- tiv ely .  Lemma 4.2 The Orlov–Schulman op er ators M and ˆ M satisfy: [ P , M ] = 1 , [ ˆ P − 1 , ˆ M ] = 1 , (4.2) M ψ ( z ) = ∂ ψ ( z ) ∂ z , ˆ M ˆ ψ ( z ) = ∂ ˆ ψ ( z ) ∂ z , ( 4.3) ∂ ˙ M ∂ t k = [( P k ) + , ˙ M ] , ∂ ˙ M ∂ ˆ t k = [ − ( ˆ P k ) − , ˙ M ] , (4.4) wher e ˙ M = M or ˆ M . Pr o of: Based on the defin ition of M and ˆ M in (4.1), the first line (4.2) follo ws from (3.6), the second line (4.3) follo ws from (3.23), while th e third line (4.4) follo w s from (3.15) and (3.16). The lemma is prov ed.  13 F or any pair of in tegers ( m, p ) with m ≥ 0, let B mp = M m P p , ˆ B mp = ˆ M m ˆ P − p , (4 .5) and w e in tro duce the follo win g ev olutionary equations: ∂ Φ ∂ β mp = − ( B mp ) − Φ , ∂ ˆ Φ ∂ β mp = ( B mp ) + ˆ Φ , (4.6) ∂ Φ ∂ ˆ β mp = − ( ˆ B mp ) − Φ , ∂ ˆ Φ ∂ ˆ β mp = ( ˆ B mp ) + ˆ Φ . (4.7) As b efore, suc h flo ws are assumed to commute with ∂ /∂ x . Lemma 4.3 F or any m, m ′ ∈ Z ≥ 0 and p, p ′ ∈ Z , the fol lowing e qualities hold: ∂ ψ ( z ) ∂ ˙ β mp = − ( ˙ B mp ) − ψ ( z ) , ∂ ˆ ψ ( z ) ∂ ˙ β mp = ( ˙ B mp ) + ˆ ψ ( z ) , (4.8) ∂ P ∂ ˙ β mp = [ − ( ˙ B mp ) − , P ] , ∂ ˆ P ∂ ˙ β mp = [( ˙ B mp ) + , ˆ P ] , (4.9) ∂ M ∂ ˙ β mp = [ − ( ˙ B mp ) − , M ] , ∂ ˆ M ∂ ˙ β mp = [( ˙ B mp ) + , ˆ M ] , (4.10) ∂ B m ′ p ′ ∂ ˙ β mp = [ − ( ˙ B mp ) − , B m ′ p ′ ] , ∂ ˆ B m ′ p ′ ∂ ˙ β mp = [( ˙ B mp ) + , ˆ B m ′ p ′ ] , (4.11) wher e ˙ β mp = β mp , ˆ β mp c orr esp ond to ˙ B mp = B mp , ˆ B mp r esp e ctively. Pr o of: The equalities (4.8) –(4.10) follo w f r om the definition (4.6 ), (4.7). Sub sequen tly , the equalities (4.11) follo w from (4.9 ) and (4.10). The lemma is p ro v ed .  Theorem 4.4 The flows define d by (4.6) , (4.7) c ommute with those in (3.15 ) , (3.16) that c omp ose the extende d KP hier ar chy. Mor e exactly, for any ˙ β mp = β mp , ˆ β mp and ¯ t k = t k , ˆ t k it holds that " ∂ ∂ ˙ β mp , ∂ ∂ ¯ t k # ˜ Φ = 0 , m ∈ Z ≥ 0 , p ∈ Z , k ∈ Z > 0 , (4.12) wher e ˜ Φ = Φ or ˆ Φ . Pr o of: Firstly , f rom (3.17 ), (3.18) and (4.4) it follo ws that ∂ ˙ B mp ∂ t k = [( P k ) + , ˙ B mp ] , ∂ ˙ B mp ∂ ˆ t k = [ − ( ˆ P k ) − , ˙ B mp ] , 14 with ˙ B mp = B mp , ˆ B mp . Then the p rop osition is c hec ked c ase by case with the help of Lemmas 4.2 an d 4.3. F or instance, " ∂ ∂ ˆ β mp , ∂ ∂ t k # ˆ Φ = ∂ ∂ ˆ β mp  (( P k ) + − δ k 1 ˆ P − 1 ) ˆ Φ  − ∂ ∂ t k  ( ˆ B mp ) + ˆ Φ  =[( P k ) + − δ k 1 ˆ P − 1 , ( ˆ B mp ) + ] ˆ Φ +  [ − ( ˆ B mp ) − , P k ] + − δ k 1 [( ˆ B mp ) + , ˆ P − 1 ]  ˆ Φ − [( P k ) + , ˆ B mp ] + ˆ Φ =  [( P k ) + , ( ˆ B mp ) + ] + [( P k ) + , ( ˆ B mp ) − ] + − [( P k ) + , ˆ B mp ] +  ˆ Φ = 0 . The other cases are s im ilar. Thus the prop osition is pro v ed.  Prop osition 4.4 mea ns that the flo ws (4.6), (4.7) giv e a class of symmetries for the extended KP hierarc hy , which are called the additiona l symmetries . Let us stud y the comm utation relation b etw een the add itional symmetries thems elves. Observe that eac h commutato r [ B mp , B m ′ p ′ ] is a p olynomial in M and P ± 1 , hence, b y virtue of (4.2 ), there exist certain constants c nq mp,m ′ p ′ suc h that [ B mp , B m ′ p ′ ] = X n,q c nq mp,m ′ p ′ B nq . (4.13) In fact, one h as c nq mp,m ′ p ′ = 0 whenever n ≥ m + m ′ or | q − ( p + p ′ ) | > m ax( m, m ′ ), wh ich implies that all but finitely man y structur e constan ts on the right hand side o f (4.13) v anish. F or in stance, w hen m + m ′ ≤ 2 one has c nq 0 p, 0 p ′ = 0 , c nq 0 p, 1 p ′ = pδ n 0 δ q ,p + p ′ − 1 , c nq 1 p, 1 p ′ = ( p − p ′ ) δ n 1 δ q ,p + p ′ − 1 , c nq 0 p, 2 p ′ = p ( p − 1) δ n 0 δ q ,p + p ′ − 2 + 2 pδ n 1 δ q ,p + p ′ − 1 . By virtue of (4.2), it h olds f or the same str u cture constants that [ ˆ B mp , ˆ B m ′ p ′ ] = X n,q c nq mp,m ′ p ′ ˆ B nq . (4.14) Prop osition 4.5 F or the extende d KP hier ar chy (3.15) , (3.16 ) , its additional symmetries define d by (4.6) , (4.7) satisfy:  ∂ ∂ β mp , ∂ ∂ β m ′ p ′  ˜ Φ = − X n,q c nq mp,m ′ p ′ ∂ ˜ Φ ∂ β nq , (4.15) " ∂ ∂ ˆ β mp , ∂ ∂ ˆ β m ′ p ′ # ˜ Φ = X n,q c nq mp,m ′ p ′ ∂ ˜ Φ ∂ ˆ β nq , (4.16) " ∂ ∂ β mp , ∂ ∂ ˆ β m ′ p ′ # ˜ Φ = 0 , (4.17) wher e ˜ Φ = Φ or ˆ Φ . 15 Pr o of: The conclusion c an b e c hec k ed case b y case with the help of Lemma 4.3. F or instance,  ∂ ∂ β mp , ∂ ∂ β m ′ p ′  ˆ Φ =[( B m ′ p ′ ) + , ( B mp ) + ] ˆ Φ + [ − ( B mp ) − , B m ′ p ′ ] + ˆ Φ − [ − ( B m ′ p ′ ) − , B mp ] + ˆ Φ = − [( B mp ) + , ( B m ′ p ′ ) + ] ˆ Φ − [( B mp ) − , ( B m ′ p ′ ) + ] + ˆ Φ − [ B mp , ( B m ′ p ′ ) − ] + ˆ Φ = − [ B mp , B m ′ p ′ ] + ˆ Φ = − X n,q c nq mp,m ′ p ′ ( B nq ) + ˆ Φ = − X n,q c nq mp,m ′ p ′ ∂ ˆ Φ ∂ β nq , (4.18) " ∂ ∂ ˆ β mp , ∂ ∂ ˆ β m ′ p ′ # ˆ Φ =[( ˆ B m ′ p ′ ) + , ( ˆ B mp ) + ] ˆ Φ + [( ˆ B mp ) + , ˆ B m ′ p ′ ] + ˆ Φ − [( ˆ B m ′ p ′ ) + , ˆ B mp ] + ˆ Φ =[( ˆ B mp ) + , ( ˆ B m ′ p ′ ) − ] + ˆ Φ + [ ˆ B mp , ( ˆ B m ′ p ′ ) + ] + ˆ Φ =[ ˆ B mp , ˆ B m ′ p ′ ] + ˆ Φ = X n,q c nq mp,m ′ p ′ ( ˆ B nq ) + ˆ Φ = X n,q c nq mp,m ′ p ′ ∂ ˆ Φ ∂ ˆ β nq , (4.19) " ∂ ∂ β mp , ∂ ∂ ˆ β m ′ p ′ # ˆ Φ =[( ˆ B m ′ p ′ ) + , ( B mp ) + ] ˆ Φ + [( B mp ) + , ˆ B m ′ p ′ ] + ˆ Φ − [ − ( ˆ B m ′ p ′ ) − , B mp ] + ˆ Φ =  [( B mp ) + , ( ˆ B m ′ p ′ ) − ] + + [( ˆ B m ′ p ′ ) − , ( B mp ) + ] +  ˆ Φ = 0 . (4.20) The other cases are chec k ed in the same wa y . Thus the prop osition is pro v ed.  This p rop osition m eans that the additional symmetries (4.6), (4.7) for the extended KP hierarc hy generate a W 1+ ∞ × W 1+ ∞ algebra. 4.2 Virasoro symmetries for t he constrained KP hierarch ies Giv en an arbitrary p ositiv e integ er n , let us consider the extended KP h ierarc hy (3.15), (3.16) imp osed with the follo wing constrain t Φ ∂ n Φ − 1 = ˆ Φ ∂ − 1 Φ − 1 . (4.21) Under this constrain t, one has P n = ˆ P and hence ∂ /∂ t nk = ∂ /∂ ˆ t k for k ≥ 1. Consequen tly , the extended KP hierarc hy is reduced to ∂ L ∂ t k = [( P k ) + , L ] , k = 1 , 2 , 3 , . . . , (4.22) 16 where L := P n = ˆ P tak es the form L = ∂ n + u 1 ∂ n − 2 + u 2 ∂ n − 3 + · · · + u n − 2 ∂ + u n − 1 + ( ∂ − f ′ ) − 1 ρ. (4.23) The system of equations (4.22) is called the constrained K P hierarc hy , denoted b y cKP n, 1 (see [2, 4, 5, 11]). In fact, if w e wr ite v = e f and w = ρe − f , then L − = v ∂ − 1 w (4.24) and it is yielded by the equations (4.22) that ∂ v ∂ t k =  P k  + ( v ) , ∂ w ∂ t k = −  P k  ∗ + ( w ) . (4.25) Example 4.6 The hierarch y (4.22) is kno wn as th e nonlinear Sh r¨ odin ger h ierarc hy (see, e.g., [5]) when n = 1, and it is the Y a jima–Oik a wa hierarch y [21] when n = 2. F or instance, when n = 2 th e op er ator L tak es the form L = ∂ 2 + u + ( ∂ − f ′ ) − 1 ρ = ∂ 2 + u + v ∂ − 1 w, then the first non trivial equations defined b y (4.22) read (cf. (3.19)) ∂ u ∂ t 2 = 2 ρ ′ , ∂ ρ ∂ t 2 = (2 ρf ′ − ρ ′ ) ′ , ∂ f ∂ t 2 = u + ( f ′ ) 2 + f ′′ , (4.26) or equiv alen tly , ∂ u ∂ t 2 = 2( v w ) ′ , ∂ v ∂ t 2 = v ′′ + uv , ∂ w ∂ t 2 = − w ′′ − uw, (4.27) whic h is called th e Y a jima–Oik a wa system. W e p r o ceed to constru ct a series of Virasoro symmetries f or the cKP n, 1 hierarc hy (4.22) with the help of the extended KP hierarc h y (3.15 ), (3.16). T o this end, let us in tro duce a class of op erators as S p = 1 n M P np +1 + ˆ M ˆ P p − 1 , p ∈ Z , (4.28) where M = Φ   X k ∈ Z > 0 k t k ∂ k − 1   Φ − 1 , ˆ M = ˆ Φ x ˆ Φ − 1 . Here th e op erator M tak es the same form as in (4.1), wh ile for con venience ˆ M do es n ot con tain the time v ariables ˆ t k as b efore. On e obs erv es that the op erators S p b elong to th e space D − + D + , h en ce ( S p ) − ∈ D − and ( S p ) + ∈ D + . The follo wing ev olutionary equ ations are w ell d efined: ∂ Φ ∂ s p = − ( S p ) − Φ , ∂ ˆ Φ ∂ s p = ( S p ) + ˆ Φ , p ≥ − 1 . (4.29) 17 Prop osition 4.7 The fl ows define d by (4.29) ar e c onsistent with the c onstr aint (4.21) , and they ar e r e duc e d to ∂ L ∂ s p = [ − ( S p ) − , L ] = [( S p ) + , L ] , p ≥ − 1 , (4.3 0) Mor e over, the r e duc e d flows define d by (4.29 ) and by (3.15) under the c onstr aint (4.21 ) satisfy: (i)  ∂ ∂ t k , ∂ ∂ s p  L = 0 , (4.31) (ii)  ∂ ∂ s p , ∂ ∂ s q  L = ( q − p ) ∂ L ∂ s p + q , (4.32) wher e k ∈ Z > 0 and p, q ∈ Z ≥− 1 . Pr o of: Since L = Φ ∂ n Φ − 1 = ˆ Φ ∂ − 1 Φ − 1 , (4.33) then the equations in (4.29 ) lead resp ectiv ely to ∂ L ∂ s p = [ − ( S p ) − , L ] , ∂ L ∂ s p = [( S p ) + , L ] . (4.34) When p ≥ − 1, it is straigh tforw ard to verify [ S p , L ] = 1 n  M P np +1 , P n  + h ˆ M ˆ P p − 1 , ˆ P i = − P n ( p +1) + ˆ P p +1 = − L p +1 + L p +1 = 0 , (4.35) whic h implies th at th e equations in (4.34) coincide with eac h other. Thus th e first assertion is obtained. Let u s show the second assertion. On the one hand, by using (3.15) we ha ve , for p ≥ − 1 and k ≥ 1, ∂ S p ∂ t k = 1 n h − ( P k ) − , M P np +1 i + k n P k + np + h ( P k ) + − δ k 1 ˆ P − 1 , ˆ M ˆ P p − 1 i + δ k 1 ˆ P p − 1 = 1 n h − ( P k ) − + P k , M P np +1 i + h ( P k ) + , ˆ M ˆ P p − 1 i = h ( P k ) + , S p i , (4.36) On the other h and, b y using (4.29) we ha ve, f or p, q ≥ − 1, ∂ ( M P p ) ∂ s q = [ − ( S q ) − , M P p ] , ∂ ( ˆ M ˆ P p ) ∂ s q = h ( S q ) + , ˆ M ˆ P p i . (4.37) Then the fi rst item is chec k ed as  ∂ ∂ t k , ∂ ∂ s p  L = ∂ ∂ t k [( S p ) + , L ] − ∂ ∂ s p h ( P k ) + , L i 18 =  h ( P k ) + , S p i + , L  + h ( S p ) + , h ( P k ) + , L ii −  h − ( S p ) − , P k i + , L  − h ( P k ) + , [( S p ) + , L ] i = hh ( P k ) + , ( S p ) + i , L i − hh ( P k ) + , L i , ( S p ) + i − h ( P k ) + , [( S p ) + , L ] i = 0 . F or the second item, it is s tr aigh tforward to calculate  ∂ ∂ s p , ∂ ∂ s q  L = ∂ ∂ s p [( S q ) + , L ] − ∂ ∂ s q [( S p ) + , L ] =  − ( S p ) − , 1 n M P nq +1  + + h ( S p ) + , ˆ M ˆ P q − 1 i + , L  + [( S q ) + , [( S p ) + , L ]] − ( p ↔ q ) = h  − ( S p ) − , 1 n M P nq +1  + + h ( S p ) + , ˆ M ˆ P q − 1 i + −  − ( S q ) − , 1 n M P np +1  + − h ( S q ) + , ˆ M ˆ P p − 1 i + + [( S q ) + , ( S p ) + ] , L i =  1 n 2 X + 1 n Y + Z, L  where, with (4.28) substituted, X =  − ( M P np +1 ) − , M P nq +1  + −  − ( M P nq +1 ) − , M P np +1  + +  ( M P nq +1 ) + , ( M P np +1 ) +  =  M P nq +1 , M P np +1  + = n ( q − p )( M P n ( p + q )+1 ) + , Y = h − ( ˆ M ˆ P p − 1 ) − , M P nq +1 i + + h ( M P np +1 ) + , ˆ M ˆ P q − 1 i + − h − ( ˆ M ˆ P q − 1 ) − , M P np +1 i + − h ( M P nq +1 ) + , ˆ M ˆ P p − 1 i + + h ( M P nq +1 ) + , ( ˆ M ˆ P p − 1 ) + i + h ( ˆ M ˆ P q − 1 ) + , ( M P np +1 ) + i = h ( M P nq +1 ) + , ˆ M ˆ P p − 1 , i + − h ( M P nq +1 ) + , ˆ M ˆ P p − 1 i + + h ( M P np +1 ) + , ( ˆ M ˆ P q − 1 ) + i + + h ( ˆ M ˆ P q − 1 ) + , ( M P np +1 ) + i = 0 , Z = h ( ˆ M ˆ P p − 1 ) + , ˆ M ˆ P q − 1 i + − h ( ˆ M ˆ P q − 1 ) + , ˆ M ˆ P p − 1 i + + h ( ˆ M ˆ P q − 1 ) + , ( ˆ M ˆ P p − 1 ) + i = h ( ˆ M ˆ P p − 1 ) + , ˆ M ˆ P q − 1 i + − h ( ˆ M ˆ P q − 1 ) + , ( ˆ M ˆ P p − 1 ) − i + = h ˆ M ˆ P p − 1 , ˆ M ˆ P q − 1 i + = ( q − p )( ˆ M ˆ P p + q − 1 ) + . So w e obtain  ∂ ∂ s p , ∂ ∂ s q  L = ( q − p ) [ ( S p + q ) + , L ] = ( q − p ) ∂ L ∂ s p + q . 19 The prop osition is prov ed.  According to this pr op osition, the flo w s (4.30) giv e a series of Virasoro s ymmetries for the cKP n, 1 hierarc hy (4.22). It is w orth while to indicate that, here the condition p ≥ − 1 is necessary , otherwise (4.35) ma y not v anish since L has d ifferen t inv erse as an op erator in D − or in D + . One observ es that the flo ws ∂ /∂ s p in (4.29) can be co nsidered as reductions of the linear com b in ations 1 n ∂ ∂ β 1 ,np +1 + ∂ ∂ ˆ β 1 , − p +1 of the additional sym metries (4.6), (4.7) for th e extended K P hierarc hy constrained b y (4.33). T his observ ation motiv ates us to consider whether the symmetries lik e ∂ /∂ β 0 p or ∂ /∂ ˆ β 0 p for the extended KP h ierarc hy could b e reduced to that f or the cKP n, 1 hierarc hy . Similar as in (4.29), for arbitrary constan ts κ and λ we can defin e the follo wing equa- tions: ∂ Φ ∂ s ′ p = −  S p + ( κp + λ ) ˆ P p  − Φ , ∂ ˆ Φ ∂ s ′ p =  S p + ( κp + λ ) ˆ P p  + ˆ Φ , p ≥ − 1 . (4.38) In the same wa y as ab o v e, by doing the replacemen t S p 7→ S p + ( κp + λ ) ˆ P p , we ac h iev e that the flo ws ∂ /∂ s ′ p are consisten t w ith the constraint (4.21), and they are reduced to ∂ L ∂ s ′ p = h − ( S p + ( κp + λ ) ˆ P p ) − , L i , p ≥ − 1 . (4.39) Moreo ve r, suc h reduced flo ws satisfy: (i)  ∂ ∂ t k , ∂ ∂ s ′ p  L = 0 , (4.40) (ii)  ∂ ∂ s ′ p , ∂ ∂ s ′ q  L = ( q − p ) ∂ L ∂ s ′ p + q , (4.41) where k ∈ Z > 0 and p, q ∈ Z ≥− 1 . Remark 4.8 The redu ced flows defined by (4.29) and by (4.38) u nder the constrain t (4.21) satisfy  ∂ ∂ s ′ p − ∂ ∂ s p  L = ( 0 , p = − 1 , 0; − ( κp + λ ) [( L p ) − , L ] , p ≥ 1 . (4.42)  It is kno wn that, for the cKP n, 1 hierarc hy (4.22 ), a s eries of the Virasoro symmetries w as constructed by Arat yn , Nissimo v and Pac hev a in [2] b y adding certain “ghost” sym - metry fl o w s related to the eigenfunctions charact eristic for the op erators L and L ∗ (in f act 20 more general cases we re stu died there). More exactly , in Prop osition 2 of [2] the follo wing Virasoro symmetries w ere giv en: ∂ L ∂ ˜ s p =  − 1 n ( M P np +1 ) − + Y p , L  , p ≥ − 1 , (4.43) where Y p = 0 f or p = − 1 , 0 , 1, and Y p = p − 1 X j =0 2 j − p + 1 2 L p − j − 1 ( v ) ∂ − 1 ( L ∗ ) j ( w ) , p ≥ 2 . (4.44) Here w e rep eat v = e f = ˆ Φ(1) , w = ρe − f =  ˆ Φ − 1  ∗ (1) , and note that th ere are nonlo cal-action terms in Y p whenev er p ≥ 2. Prop osition 4.9 The flows define d by (4.43) satisfy ∂ L ∂ ˜ s p = ∂ L ∂ s ′ p , p ≥ − 1 , wher e ∂ /∂ s ′ p ar e gi v en in (4.39) with κ = − 1 2 and λ = 1 2 . Pr o of: Let u s fix κ = − 1 2 and λ = 1 2 in (4.39 ), then  ∂ ∂ ˜ s p − ∂ ∂ s ′ p  L =  − 1 n  M P np +1  − + Y p +  S p − p − 1 2 ˆ P p  − , L  = [ Y p + Z p , L ] , where Z p :=  ˆ M ˆ P p − 1 − p − 1 2 ˆ P p  − =  ˆ Φ x∂ − p +1 ˆ Φ − 1  − − p − 1 2  ˆ Φ ∂ − p ˆ Φ − 1  − . (4.45) When p = − 1 , 0 , 1, clearly Z p = 0 = Y p , then th e fl o w s ∂ /∂ ˜ s p and ∂ /∂ s ′ p acting on L coincide. When p = 2, on one h and from (4.44) we hav e Y 2 = 1 2  − L ( v ) ∂ − 1 w + v ∂ − 1 L ∗ ( w )  = − L ( v ) ∂ − 1 w + 1 2  L ( v ) ∂ − 1 w + v ∂ − 1 L ∗ ( w )  = − L ( v ) ∂ − 1 w + 1 2  L 2  − , where the last equalit y is due to the formula (61) in the app end ix of [2], say , in the present case ( L k ) − = k − 1 X j =0 L k − j − 1 ( v ) ∂ − 1 ( L ∗ ) j ( w ) , k ≥ 1 . 21 On the other h and, since  ˆ Φ x∂ − 1 ˆ Φ − 1  − = ˆ Φ( x ) · ∂ − 1 ·  ˆ Φ − 1  ∗ (1) = ˆ Φ ∂ − 1 ˆ Φ − 1 ˆ Φ ∂ ( x ) · ∂ − 1 ·  ˆ Φ − 1  ∗ (1) = L ˆ Φ(1) · ∂ − 1 · w = L ( v ) ∂ − 1 w, then from (4.45 ) w e ha ve Z 2 = L ( v ) ∂ − 1 w − 1 2  L 2  − . Clearly Y 2 + Z 2 = 0, then w e obtain ∂ L/∂ ˜ s 2 = ∂ L/∂ s ′ 2 . When p ≥ 3, the flows ∂ /∂ ˜ s p and ∂ /∂ ˜ s ′ p can b e d etermined by those fl o ws with p = − 1 , 0 , 1 , 2 via the Virasoro comm utation relations. Th erefore we complete the pr o of.  5 Concluding remarks In this pap er we ha ve represen ted the extended KP hierarc hy into a b ilinear equation satisfied b y the (adjoint) Bak er–Akhiezer fun ctions. What is more, we hav e constructed a class of additional sym metries for this hierarc hy , and stud ied their r eduction prop erties. Suc h results, whic h are analogue to those for the KP an d the 2-BKP hierarc hies, are exp ected to extend our under s tanding to th e th eory of integ rable sy s tems. A t the e nd of [20], a tau function of the extended KP hierarc hy w as intro d uced by using the densities of Hamiltonian functionals. Similar to the KP hierarc hy , one can d er ive a Sato form ula that links the tau fu nction to th e (adjoin t) Bak er–Akhiezer fun ctions ψ ( z ) and ψ † ( z ). Ho w ev er, the relationship b et w een the tau function and ˆ ψ ( z ) (or ˆ ψ † ( z )) still needs to b e clarified. It is why a bilinear equation of tau function is still miss in g. 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