Bilinear equation and additional symmetries for an extension of the Kadomtsev-Petviashvili hierarchy
aa r X i v : . [ n li n . S I] N ov Bilinear Equation and Additional Symmetries for anExtension of the Kadomtsev–Petviashvili Hierarchy
Jiaping Lu Chao-Zhong Wu
School of Mathematics, Sun Yat-Sen University, Guangzhou 510275, P.R. ChinaEmail: [email protected]; [email protected]
Abstract
An extension of the Kadomtsev–Petviashvili (KP) hierarchy was considered in [J.Geom. Phys. 106 (2016), 327–341], which possesses a class of bi-Hamiltonian struc-tures. In this paper, we represent the extended KP hierarchy into the form of bilinearequation of (adjoint) Baker–Akhiezer functions, and construct its additional symme-tries. As a byproduct, we also derive the Virasoro symmetries for the constrained KPhierarchies.
Key words : KP hierarchy; bilinear equation; additional symmetry
As a fundamental model in the theory of integrable systems, the Kadomtsev–Petviashvili(KP) hierarchy is defined as follows. Let L KP = ∂ + v ∂ − + v ∂ − + . . . , ∂ = dd x , (1.1)be a pseudo-differential operator whose coefficients v i are scalar unknown functions of thespacial coordinate x , then the KP hierarchy is composed by the following evolutionaryequations of v i as ∂L KP ∂t k = h ( L KP k ) + , L KP i , k = 1 , , , . . . . (1.2)Here and below the subscript “+” of a pseudo-differential operator means to take itspurely differential part, while the subscript “ − ” means to take its negative part. Supposethat the equations (1.2) are imposed with the constraint ( L KP n ) − = 0 with some positiveinteger n , then they form the Gelfand–Dickey (or the ( n − R -matrix formalism [14], and that it can be represented into theform of bilinear equation of (adjoint) Baker–Akhiezer functions or of a tau function [7].1uch bi-Hamiltonian structures and the bilinear equation can be reduced to that of theGelfand–Dickey hierarchies. What is more, for the KP hierarchy there is a class of non-isospectral symmetries named as the additional symmetries that can be constructed viathe so-called Orlov–Schulman operators [13]. The flows of such additional symmetriescommute with all time flows ∂/∂t k but do not commute between themselves; instead,they compose a so-called W ∞ algebra. As an important property of the additionalsymmetries for the KP hierarchy, part of them can be reduced to the Virasoro symme-tries for its subhierarchies such like the Gelfand–Dickey hierarchies. An analogue of therelationship between the KP hierarchy and the Gelfand–Dickey hierarchies, as well astheir additional/Virasoro symmetries, is that between the two-component BKP (2-BKP)hierarchy and the Drinfeld–Sokolov hierarchies of type D [6, 12, 17]. Note that Virasorosymmetries reveal crucial properties of a large amount of integrable hierarchies includingthe Gelfand–Dickey hierarchies and the Drinfeld–Sokolov hierarchies, see e.g. [8, 9, 18]and references therein.The notion of pseudo-differential operator was extended in [12] to be over a gradeddifferential algebra such that operators may contain infinitely many positive powers in ∂ (see Section 2 below for details). By using these operators, in [19] Zhou and one of theauthors of the present paper considered an integrable hierarchy as follows (cf. [15]). Let P = ∂ + X i ≥ u i ( ∂ − ϕ ) − i , ˆ P = ( ∂ − ϕ ) − ˆ u − + X i ≥ ˆ u i ( ∂ − ϕ ) i (1.3)with u i , ˆ u i and ϕ of certain degrees, then the following evolutionary equations are welldefined: ∂∂t k ( P, ˆ P ) = (cid:16) [( P k ) + , P ] , [( P k ) + , ˆ P ] (cid:17) , ∂∂ ˆ t k ( P, ˆ P ) = (cid:16) [ − ( ˆ P k ) − , P ] , [ − ( ˆ P k ) − , ˆ P ] (cid:17) (1.4)for k = 1 , , , . . . . These equations form an integrable hierarchy, which is named as theextended KP hierarchy for it is reduced to the KP hierarchy (1.2) whenever ˆ P = 0 (notethat the operator P gives an alternative representation of L KP ). On the other hand, ifone lets ϕ → P andˆ P , then the flows in (1.4) with k ∈ Z odd > give the 2-BKP hierarchy. Similar to the cases ofthe KP and the 2-BKP hierarchies [14, 16], the extended KP hierarchy (1.4) was shown topossess infinitely many bi-Hamiltonian structures, whose Hamiltonian density functionswere used to define a tau function [19].In this paper we will show that the above operators P and ˆ P , with ϕ = ∂ ( f ) for somefunction f ∈ A , can be represented in a dressing way as P = Φ ∂ Φ − , ˆ P = ˆΦ ∂ − ˆΦ − (1.5)where Φ and ˆΦ are pseudo-differential operators of the formΦ = 1 + X i ≥ a i ∂ − i , ˆΦ = e f X i ≥ b i ∂ i . ψ ( t , ˆ t ; z ), ˆ ψ ( t , ˆ t ; z ) and their adjoints ψ † ( t , ˆ t ; z ),ˆ ψ † ( t , ˆ t ; z ) that depend on the time variables t = ( t , t , t , . . . ), ˆ t = (ˆ t , ˆ t , ˆ t , . . . ), and anonzero parameter z . Our first main result is the following theorem (see Theorem 3.7 fora more precise version). Theorem 1.1
The extended KP hierarchy (1.4) can be represented equivalently to a bi-linear equation as res z (cid:16) ψ ( t , ˆ t ; z ) ψ † ( t ′ , ˆ t ′ ; z ) (cid:17) = res z (cid:16) ˆ ψ ( t , ˆ t ; z ) ˆ ψ † ( t ′ , ˆ t ′ ; z ) (cid:17) . (1.6)Our second main result is the construction of additional symmetries for the extendedKP hierarchy, with the help of certain Orlov–Schulman operators given by the abovedressing operators Φ and ˆΦ. Moreover, such additional symmetries will be shown to forma W ∞ × W ∞ algebra. These results will be applied to study the ( n, n, ; see, e.g., [1, 3, 5, 10]), which is a reduction of the extendedKP hierarchy (1.4) under the constraint P n = ˆ P with a positive integer n . In fact, the Virasoro symmetries for the cKP n, hierarchy(even for more general cases) were proposed by Aratyn, Nissimov and Pacheva [1], withthe method of adding certain “ghost” symmetry flows that contain nonlocal actions onfunctions. As to be seen, the Virasoro symmetries appearing in [1] for the cKP n, hierarchycan be recovered in a relatively straightforward way, with the help of pseudo-differentialoperators of the second type (see Collorary 4.9 below).This paper is arranged as follows. In the next section, we will recall the pseudo-differential operators of the first and the second types over a graded differential algebra.In Section 3, we will recall the definition of the extended KP hierarchy, and representit into the form of a bilinear equation. In Section 4, we will construct the additionalsymmetries for the extended KP hierarchy, and then study the Virasoro symmetries forthe cKP n, hierarchy. The final section is devoted to some remarks. Let A be a commutative associative algebra, and ∂ : A → A be a derivation. We considerthe linear space (cid:8)P i ∈ Z f i ∂ i | f i ∈ A (cid:9) and its subsets. For instance, the set of pseudo-differential operators is A (( ∂ − )) = X i ≤ k f i ∂ i | f i ∈ A , k ∈ Z , and it becomes an associated algebra if a product is defined by f ∂ i · g∂ j = X r ≥ (cid:18) ir (cid:19) f ∂ r ( g ) ∂ i + j − r , f, g ∈ A . (2.1)3or any two pseudo-differential operators A and B , their commutator means [ A, B ] = AB − BA . Clearly, one has [ ∂, f ] = ∂ ( f ) for any f ∈ A .In the present paper we assume the algebra A to be a graded one. Namely, A = Q i ≥ A i , such that A i · A j ⊂ A i + j , ∂ ( A i ) ⊂ A i +1 . Denote D − = A (( ∂ − )), which is called the algebra of pseudo-differential operators of thefirst type over A . In contrast, by the algebra of pseudo-differential operators of the secondtype over A it means [12] D + = X i ∈ Z X j ≥ max { ,k − i } a i,j ∂ i | a i,j ∈ A j , k ∈ Z , (2.2)whose product is also defined by (2.1). One observes that an operator in D + may containinfinitely many positive powers in ∂ .Given an operator A = P i ∈ Z f i ∂ i ∈ D ∓ , its differential part, negative part and residueare defined respectively as: A + = X i ≥ f i ∂ i , A − = X i< f i ∂ i , res A = f − . (2.3)It is known that on each D ∓ there is an anti-automorphism defined by ∂ ∗ = − ∂, f ∗ = f with f ∈ A . Clearly, for any A ∈ D ∓ , one has res A ∗ = − res A. In what follows we will use the notation A ≥ r = Q i ≥ r A i with r ∈ Z ≥ . Given anelement ϕ ∈ A ≥ , the following two maps are well defined with ∂ replaced by ∂ − ϕ , thatis, S ϕ : D ∓ → D ∓ , X f i ∂ i X f i ( ∂ − ϕ ) i . (2.4)For instance, we have S ϕ ( ∂ − ) =( ∂ − ϕ ) − = ∂ − (1 − ϕ∂ − ) − = ∂ − + ∂ − ϕ∂ − + ∂ − ϕ∂ − ϕ∂ − + . . . . (2.5)In [19], it was verified that the maps S ϕ are automorphisms on each D ∓ . Accordingly,the algebras D ∓ can be represented as follows: D − = X i ≤ k g i ( ∂ − ϕ ) i | g i ∈ A , k ∈ Z , (2.6)4 + = X i ∈ Z X j ≥ max { ,k − i } b i,j ( ∂ − ϕ ) i | b i,j ∈ A j , k ∈ Z , (2.7)and their product can be defined equivalently by f ( ∂ − ϕ ) i · g ( ∂ − ϕ ) j = X r ≥ (cid:18) ir (cid:19) f ∂ r ( g ) ( ∂ − ϕ ) i + j − r , f, g ∈ A . Moreover, it is easy to verify the following properties: for any A ∈ D ∓ ,( S ϕ A ) ± = S ϕ ( A ± ) , res ( S ϕ A ) = res A, ( S ϕ A ) ∗ = S − ϕ A ∗ . (2.8) We proceed to recall the definition of the extended KP hierarchy, and then capsule it intoa bilinear equation of certain (adjoint) Baker–Akhiezer functions.
Let M be an infinite-dimensional manifold with coordinate a = ( a , a , a , . . . ; f, b , b , b , . . . ) . We consider the following graded algebra of differential polynomials: A = C ∞ ( S → M )[[ ∂ r ( a ) | r ≥ , (3.1)in which the derivation is ∂ = d / d x with x being the coordinate of the loop S , anddeg a = 0 , deg ∂ r ( a ) = r. Over the graded differential A , it is defined the algebras D − and D + of pseudo-differentialoperators of the first type and of the second type, respectively.We introduce two pseudo-differential operators:Φ = 1 + X i ≥ a i ∂ − i ∈ D − , (3.2)ˆΦ = e f X i ≥ b i ∂ i ∈ D + . (3.3)Observe that these operators have inverses of the formΦ − = 1 + X i ≥ ˜ a i ∂ − i ∈ D − , (3.4)5Φ − = X i ≥ ˜ b i ∂ i e − f ∈ D + . (3.5)In fact, by expanding Φ − Φ = 1 and ˆΦ − ˆΦ = 1 one sees that the coefficients ˜ a i and ˜ b i take the form ˜ a i = − a i + g i ( a , a , . . . , a i − ) , ˜ b i = − b i + h i ( b , b , b , . . . )with g i , h i ∈ A ≥ . For instance, one has ˜ a = − a , and that ˜ b is given recursively by˜ b (cid:12)(cid:12)(cid:12) A = − b , ˜ b (cid:12)(cid:12)(cid:12) A j = − j X r =1 b r (cid:18) ˜ b (cid:12)(cid:12)(cid:12) A j − r (cid:19) ( r ) for j ≥ . Now let us introduce two pseudo-differential operators as follows: P = Φ ∂ Φ − ∈ D − , ˆ P = ˆΦ ∂ − ˆΦ − ∈ D + . (3.6) Proposition 3.1
The operators P and ˆ P given above can be represented in the form: P = ∂ + X i ≥ v i ∂ − i , ˆ P = ( ∂ − f ′ ) − ρ + X i ≥ ˆ v i ∂ i , (3.7) where v i +1 , ˆ v i ∈ A ≥ for i ∈ Z ≥ , and ρ = e f (cid:16) ˆΦ − (cid:17) ∗ (1) . (3.8) Proof:
The representation of P is well known, so let us verify the case of ˆ P . To thisend, firstly let us check that ˆ P given in (3.6) takes the formˆ P = ( ∂ − f ′ ) − X i ≥ ˜ v i ( ∂ − f ′ ) i , ˜ v i − δ i ∈ A ≥ . (3.9)We substitute this expansion and (3.3) into an equivalent version of ˆΦ ∂ − ˆΦ = ˆ P , say, e − f ( ∂ − f ′ ) ˆΦ ∂ − = e − f ( ∂ − f ′ ) ˆ P ˆΦ . Note e − f ( ∂ − f ′ ) e f = ∂ , then we have1 + b ′ + X i ≥ ( b i + b ′ i +1 ) ∂ i = X i ≥ ˜ v i ∂ i X i ≥ b i ∂ i , (3.10)in which the coefficients of ∂ i lead to i = 0 : 1 + b ′ = ˜ v , (3.11)6 ≥ b i + b ′ i +1 = ˜ v i + i X r =1 X s ≥ (cid:18) i − r + ss (cid:19) ˜ v i − r + s b ( s ) r . (3.12)The equations (3.12) yield b i = ˜ v i | A + i X r =1 ˜ v i − r | A b r , i ≥ , which together with (3.11) gives ˜ v i | A = δ i , i ≥ . The equations (3.12) also yield b ′ i +1 = ˜ v i | A + i X r =1 ˜ v i − r | A b r , i ≥ , which together with (3.11) determine ˜ v i | A recursively. In fact, we can obtain a generatingfunction for ˜ v i | A of a parameter z as X i ≥ ˜ v i | A z i +1 = ∂ log X i ≥ b i z i . For j ≥
2, it follows from (3.12) that˜ v i | A j + i X r =1 j − X s =0 (cid:18) i − r + ss (cid:19) ˜ v i − r + s | A j − s · b ( s ) r = 0 , i ≥ , hence ˜ v i | A j are determined recursively. So the operator ˆ P takes the form (3.7), in which ρ = res (cid:16) ˆΦ ∂ − ˆΦ − (cid:17) = res (cid:16) e f ∂ − ˆΦ − (cid:17) = e f (cid:16) ˆΦ − (cid:17) ∗ (1) . (3.13)Therefore the proposition is proved. (cid:3) Remark 3.2
From (3.8) it implies that ρ − ∈ A ≥ . In fact, by using (3.5) one has ρ = e f (cid:16) ˆΦ − (cid:17) ∗ (1) = 1 + X i ≥ ( − i ˜ b ( i ) i . (3.14)One also sees that, the coefficients of P and ˆ P have different degrees from that in [19],since now the graded differential algebra A is chosen differently. (cid:3) P and ˆ P , let us define a class of evolutionary equationson M : for k ∈ Z > , ∂ Φ ∂t k = − ( P k ) − Φ , ∂ ˆΦ ∂t k = (cid:0) ( P k ) + − δ k ˆ P − (cid:1) ˆΦ , (3.15) ∂ Φ ∂ ˆ t k = − ( ˆ P k ) − Φ , ∂ ˆΦ ∂ ˆ t k = ( ˆ P k ) + ˆΦ , (3.16)Here we note that the right hand sides make sense since the operators ( P k ) + , ( ˆ P k ) − ∈D − ∩ D + , and we assume that the flows ∂/∂t k and ∂/∂ ˆ t k commutate with ∂/∂x . Inparticular, it can be seen ∂/∂t = ∂/∂x , so in what follows we will just take t = x . Proposition 3.3
The flows (3.15) , (3.16) satisfy, for k ∈ Z > , ∂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) Moreover, these flows commute with each other.Proof:
The proposition can be verified case by case. For instance, we have ∂ ˆ P∂t k = " ∂ ˆΦ ∂t k ˆΦ − , ˆ P = h ( P k ) + − δ k ˆ P − , ˆ P i = h ( P k ) + , ˆ P i , (cid:20) ∂∂t k , ∂∂ ˆ t l (cid:21) ˆΦ = ∂∂t k (cid:16) ( ˆ P l ) + ˆΦ (cid:17) − ∂∂ ˆ t l (cid:16)(cid:0) ( P k ) + − δ k ˆ P − (cid:1) ˆΦ (cid:17) = h ( ˆ P l ) + , ( P k ) + − δ k ˆ P − i ˆΦ + h ( P k ) + , ˆ P l i + ˆΦ − (cid:18)h − ( ˆ P l ) − , P k i + − δ k h − ( ˆ P l ) − , ˆ P − i(cid:19) ˆΦ= h ( ˆ P l ) + , ( P k ) + i ˆΦ − h ˆ P l , ( P k ) + i + ˆΦ+ h ( ˆ P l ) − , ( P k ) + i + ˆΦ − δ k h ˆ P l , ˆ P − i ˆΦ=0 . The other cases are almost the same. So we complete the proof. (cid:3)
The system of equations (3.17), (3.18) was studied in [19] by Zhou and one of theauthors (cf. [15]), with the set of unknown functions as { v i +1 , ˆ v i | i ∈ Z ≥ } ∪ { ρ, ϕ = f ′ } . This system is called the extended KP hierarchy for the reason that the flows ∂P/∂t k in (3.17) with k ∈ Z > compose the well-known KP hierarchy. By virtue of the aboveproposition, we will also call the system of equations (3.15), (3.16) the extended KPhierarchy. 8 emark 3.4 By using (3.3) and (3.15) we have, for k ∈ Z > , ∂f∂t k = e − f ∂ ˆΦ ∂t k (1) = e − f (cid:0) ( P k ) + − δ k ˆ P − (cid:1) ˆΦ(1) = e − f ( P k ) + e f (1)=res (cid:16) e − f P k e f ∂ − (cid:17) = res S − f ′ (cid:16) P k e f ∂ − e − f (cid:17) =res (cid:16) P k e f ∂ − e − f (cid:17) = res (cid:16) P k ( ∂ − f ′ ) − (cid:17) . (3.19)Here in the fifth equality we have used (2.8). With the same method, we deduce ∂f∂ ˆ t k = res (cid:16) ˆ P k ( ∂ − f ′ ) − (cid:17) , k ∈ Z > . (3.20)So we recover the equations (2.23) in [19] by letting ϕ = f ′ . Moreover, as was derived in[19] one has ∂ρ∂ ˙ t k = X i ≥ ( − i − ∂ i (cid:16) ρ res ˙ P k ( ∂ − f ′ ) − i − (cid:17) . (3.21) (cid:3) With z being a nonzero parameter, we assign ∂ i ( e xz ) = z i e xz for any i ∈ Z , and moregenerally, ∂ i ( ge xz ) = ( ∂ i g ) ( e xz ) , g ∈ A , i ∈ Z . Namely, it is the usual action of a differential operator on a function whenever i ≥ i <
0. In order to simplifynotations, for any A ∈ D ± and exponential functions of the the form e ± xz , we will justwrite Ae ± xz instead of A ( e ± xz ).Denote t = ( t = x, t , t , . . . ) and ˆ t = (ˆ t , ˆ t , ˆ t , . . . ), and let ξ be given by ξ ( t ; z ) = X k ∈ Z > t k z k . Given a solution of the extended KP hierarchy (3.15), (3.16), let us introduce two Baker–Akhiezer functions: ψ ( t , ˆ t ; z ) = Φ e ξ ( t ; z ) , ˆ ψ ( t , ˆ t ; z ) = ˆΦ e xz − ξ (ˆ t ; z − ) , (3.22)and two adjoint Baker–Akhiezer functions: ψ † ( t , ˆ t ; z ) = (cid:0) Φ − (cid:1) ∗ e − ξ ( t ; z ) , ˆ ψ † ( t , ˆ t ; z ) = (cid:16) ˆΦ − (cid:17) ∗ e − xz + ξ (ˆ t ; z − ) . (3.23)When there is no confusion, we 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 straight forward to verify the the following Lemma.9 emma 3.5 The (adjoint) Baker–Akhiezer functions given above satisfy: (i)
P ψ ( z ) = zψ ( z ) , P ∗ ψ † ( z ) = zψ † ( z ) , (3.24)ˆ P ˆ ψ ( z ) = z − ˆ ψ ( z ) , ˆ P ∗ ˆ ψ † ( z ) = z − ˆ ψ † ( z ); (3.25)(ii) ∂ ˙ ψ∂t k = (cid:16) P k (cid:17) + ˙ ψ, ∂ ˙ ψ∂ ˆ t k = − (cid:16) ˆ P k (cid:17) − ˙ ψ, (3.26) ∂ ˙ ψ † ∂t k = − (cid:16) P k (cid:17) ∗ + ˙ ψ † , ∂ ˙ ψ † ∂ ˆ t k = (cid:16) ˆ P k (cid:17) ∗− ˙ ψ † (3.27) where ˙ ψ ∈ n ψ ( z ) , ˆ ψ ( z ) o and ˙ ψ † ∈ n ψ † ( z ) , ˆ ψ † ( z ) o . For any formal series P i ∈ Z g i z i , its residue is defined byres z X i ∈ Z g i z i = g − . The following lemma is useful.
Lemma 3.6 (see, for example, [7])
For any pseudo-differential operators
Q, R ∈ D ± ,the following equality holds true whenever both sides make sense: res z (cid:0) Qe zx · R ∗ e − zx (cid:1) = res ( QR ) . (3.28) Theorem 3.7
The (adjoint) Baker–Akhiezer functions of the extended KP hierarchy sat-isfy the following bilinear equation res z (cid:16) ψ ( t , ˆ t ; z ) ψ † ( t ′ , ˆ t ′ ; z ) (cid:17) = res z (cid:16) ˆ ψ ( t , ˆ t ; z ) ˆ ψ † ( t ′ , ˆ t ′ ; z ) (cid:17) , (3.29) for arbitrary time variables ( t , ˆ t ) and ( t ′ , ˆ t ′ ) . Conversely, suppose that four functions ofthe form ψ ( t , ˆ t ; z ) = X i ≥ a i ( t , ˆ t ) z − i e ξ ( t ; z ) , (3.30)ˆ ψ ( t , ˆ t ; z ) = e f ( t , ˆ t ) X i ≥ b i ( t , ˆ t ) z i e xz − ξ (ˆ t ; z − ) , (3.31) ψ † ( t , ˆ t ; z ) = X i ≥ a † i ( t , ˆ t ) z − i e − ξ ( t ; z ) , (3.32)ˆ ψ † ( t , ˆ t ; z ) = e − f ( t , ˆ t ) X i ≥ b † i ( t , ˆ t ) z i e − xz + ξ (ˆ t ; z − ) (3.33) satisfy the bilinear equation (3.29) , then they are the Baker–Akhiezer functions and theadjoint Baker–Akhiezer functions of the extended KP hierarchy. roof: As a preparation, we introduce the set of indices as I = { ( m , m , m , . . . ) | m i ∈ Z ≥ such that m i = 0 for i ≫ } . For m = ( m , m , m , . . . ) ∈ I , denote ∂ tm = Y k ≥ (cid:18) ∂∂t k (cid:19) m k , ∂ ˆ tm = Y k ≥ (cid:18) ∂∂ ˆ t k (cid:19) m k . In order to show the equality (3.29), we only need to check that the Baker–Akhiezerfunctions and the adjoint Baker–Akhiezer functions satisfyres z (cid:16) ∂ tm ∂ ˆ tn ψ ( t , ˆ t ; z ) · ψ † ( t , ˆ t ; z ) (cid:17) = res z (cid:16) ∂ tm ∂ ˆ tn ˆ ψ ( t , ˆ t ; z ) · ˆ ψ † ( t , ˆ t ; z ) (cid:17) (3.34)for any m , n ∈ I . In fact, according to Lemma 3.5 and Proposition 3.3 one sees thatthe following two equalities hold simultaneously for the same pseudo-differential operator A m , n ∈ D − ∩ D + : ∂ tm ∂ ˆ tn ψ ( z ) = A m , n ψ ( z ) , ∂ tm ∂ ˆ tn ˆ ψ ( z ) = A m , n ˆ ψ ( z ) . Then, by using (3.23) and Lemma 3.6, the equality (3.34) is recast tores (cid:0) A m , n ΦΦ − (cid:1) = res (cid:16) A m , n ˆΦ ˆΦ − (cid:17) , which is clearly valid. Hence the equality (3.34) holds true, and the first assertion isverified.For the second assertion, one sees that there are uniquely pseudo-differential operatorsΦ, ˆΦ, Ψ and ˆΨ such that the functions (3.30)–(3.33) are represented as ψ ( t , ˆ t ; z ) = Φ e ξ ( t ; z ) , ˆ ψ ( t , ˆ t ; z ) = ˆΦ e xz − ξ (ˆ t ; z − ) ,ψ † ( t , ˆ t ; z ) = Ψ ∗ e − ξ ( t ; z ) , ˆ ψ † ( t , ˆ t ; z ) = ˆΨ ∗ e − xz + ξ (ˆ t ; z − ) . Moreover, the operators Φ and ˆΦ take the form (3.2) and (3.3) respectively, while Ψ andˆΨ take the form (3.4) and (3.5) respectively. The bilinear equation (3.29) leads to thefollowing facts.(i) For i ∈ Z , we have (recall that when i < z (cid:16) ∂ i ψ ( t , ˆ t ; z ) ψ † ( t ′ , ˆ t ′ ; z ) (cid:17) = res z (cid:16) ∂ i ˆ ψ ( t , ˆ t ; z ) ˆ ψ † ( t ′ , ˆ t ′ ; z ) (cid:17) . Let ( t ′ , ˆ t ′ ) = ( t , ˆ t ), and then with the help of (3.28) we obtainres ∂ i ΦΨ = res ∂ i ˆΦ ˆΨ = 0 , i ≥ ∂ i ˆΦ ˆΨ = res ∂ i ΦΨ = δ i, − , i < . which implies ΦΨ = ˆΦ ˆΨ = 1. So we deriveΨ = Φ − , ˆΨ = ˆΦ − . (3.35)11ii) Denote X k = ∂ Φ ∂t k Φ − , ˆ X k = ∂ ˆΦ ∂t k ˆΦ − . Clearly, one has ( X k ) + = ( ˆ X k ) − = 0, and that ∂ψ ( z ) ∂t k = (cid:16) X k Φ + Φ ∂ k (cid:17) e ξ ( t ; z ) = (cid:16) X k + Φ ∂ k Φ − (cid:17) ψ ( z ) ,∂ ˆ ψ ( z ) ∂t k = (cid:16) ˆ X k ˆΦ + δ k ˆΦ ∂ (cid:17) e xz − ξ (ˆ t ; z − ) = (cid:16) ˆ X k + δ k ˆΦ ∂ ˆΦ − (cid:17) ˆ ψ ( z ) . For any i ∈ Z , we let ∂ i act on the derivative of (3.29) with respect to t k , and let( t ′ , ˆ t ′ ) = ( t , ˆ t ), then by using (3.28) again we obtainres ∂ i (cid:16) X k + Φ ∂ k Φ − (cid:17) ΦΦ − = res ∂ i (cid:16) ˆ X k + δ k ˆΦ ∂ ˆΦ − (cid:17) ˆΦ ˆΦ − . Hence X k + Φ ∂ k Φ − = ˆ X k + δ k ˆΦ ∂ ˆΦ − , and we arrive at X k = − (cid:16) Φ ∂ k Φ − (cid:17) − , ˆ X k = (cid:16) Φ ∂ k Φ − (cid:17) + − δ k ˆΦ ∂ ˆΦ − . (3.36)Similarly, we can derive ∂ Φ ∂ ˆ t k Φ − = − (cid:16) Φ ∂ − k Φ − (cid:17) − , ∂ ˆΦ ∂ ˆ t k ˆΦ − = (cid:16) Φ ∂ − k Φ − (cid:17) + . (3.37)Taking (i) and (ii) together we achieve the second assertion. The theorem is proved. (cid:3) In this section, we want to construct a class of additional symmetries for the extended KPhierarchy (3.15), (3.16), following the approach of [13, 17], and then study the Virasorosymmetries for the constrained KP hierarchies.
Suppose that the operators Φ and ˆΦ solve the hierarchy (3.15), (3.16). Let us introducetwo Orlov–Schulman operators as follows: M = ΦΞΦ − , ˆ M = ˆΦˆΞ ˆΦ − , (4.1)where Ξ = X k ∈ Z > kt k ∂ k − , ˆΞ = x + X k ∈ Z > k ˆ t k ∂ − k − . Here we assume that all t k and ˆ t k vanish except finitely many of them, such that theoperators M and ˆ M are well defined. 12 emark 4.1 There is another way to ensure the Orlov–Schulman operators M and ˆ M (even with all time variables nontrivial) to be well defined. Indeed, as what was donein [17], one can extend the graded algebra A to include also { t k , ˆ t k | k ∈ Z > } withdeg t k = deg ˆ t k = k , so that a new graded algebra ˜ A is obtained. Then, the set D − ofpseudo-differential operators of the first type can be extended to˜ D − = X i ∈ Z X j ≥ max { ,i − k } a i,j ∂ i | a i,j ∈ ˜ A j , k ∈ Z . So, the operators M and ˆ M are elements of ˜ D − and D + (with A replaced by ˜ A ) respec-tively. (cid:3) Lemma 4.2
The Orlov–Schulman operators M and ˆ M satisfy: [ P, M ] = 1 , [ ˆ P − , 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) where ˙ M = M or ˆ M .Proof: Based on the definition of M and ˆ M in (4.1), the first line (4.2) follows from (3.6),the second line (4.3) follows from (3.22), while the third line (4.4) follows from (3.15) and(3.16). The lemma is proved. (cid:3) For any pair of integers ( m, p ) with m ≥
0, let B mp = M m P p , ˆ B mp = ˆ M m ˆ P − p , (4.5)and we introduce the following evolutionary equations: ∂ Φ ∂β mp = − ( B mp ) − Φ , ∂ ˆΦ ∂β mp = ( B mp ) + ˆΦ , (4.6) ∂ Φ ∂ ˆ β mp = − ( ˆ B mp ) − Φ , ∂ ˆΦ ∂ ˆ β mp = ( ˆ B mp ) + ˆΦ . (4.7)As before, such flows are assumed to commute with ∂/∂x . Lemma 4.3
The flows (4.6) , (4.7) satisfy, for any m, m ′ ∈ Z ≥ and p, p ′ ∈ Z , ∂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)13 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) where ˙ β mp = β mp , ˆ β mp correspond to ˙ B mp = B mp , ˆ B mp respectively.Proof: The equalities (4.8)–(4.10) follow from the definition (4.6), (4.7). Subsequently,the equalities (4.11) follow from (4.9) and (4.10). The lemma is proved. (cid:3)
Theorem 4.4
The flows (4.6) , (4.7) acting on Φ and ˆΦ commute with those in (3.15) , (3.16) that compose the extended KP hierarchy. More 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 = 0 , m ∈ Z ≥ , p ∈ Z , k ∈ Z > . (4.12) Proof:
Firstly, from (3.17), (3.18) and (4.4) it follows that ∂ ˙ B mp ∂t k = [( P k ) + , ˙ B mp ] , ∂ ˙ B mp ∂ ˆ t k = [ − ( ˆ P k ) − , ˙ B mp ] , with ˙ B mp = B mp , ˆ B mp . Then the proposition is checked case by case with the help ofLemmas 4.2 and 4.3. For instance, " ∂∂ ˆ β mp , ∂∂t k ˆΦ = ∂∂ ˆ β mp (cid:16) (( P k ) + − δ k ˆ P − ) ˆΦ (cid:17) − ∂∂t k (cid:16) ( ˆ B mp ) + ˆΦ (cid:17) =[( P k ) + − δ k ˆ P − , ( ˆ B mp ) + ] ˆΦ+ (cid:16) [ − ( ˆ B mp ) − , P k ] + − δ k [( ˆ B mp ) + , ˆ P − ] (cid:17) ˆΦ − [( P k ) + , ˆ B mp ] + ˆΦ= (cid:16) [( P k ) + , ( ˆ B mp ) + ] + [( P k ) + , ( ˆ B mp ) − ] + − [( P k ) + , ˆ B mp ] + (cid:17) ˆΦ = 0 . The other cases are similar. Thus the proposition is proved. (cid:3)
Proposition 4.4 means that the flows (4.6), (4.7) give a class of symmetries for theextended KP hierarchy. Following the notions in the literature, such symmetries arecalled the additional symmetries .Let us study the commutation relation between the additional symmetries themselves.Observe that each commutator [ B mp , B m ′ p ′ ] is a polynomial in M and P ± , hence, byvirtue of (4.2), there exist certain constants c nqmp,m ′ p ′ such that[ B mp , B m ′ p ′ ] = X n,q c nqmp,m ′ p ′ B nq . (4.13)14n fact, one has c nqmp,m ′ p ′ = 0 whenever n ≥ m + m ′ or | q − ( p + p ′ ) | > max( m, m ′ ), whichimplies that all but finitely many structure constants on the right hand side of (4.13)vanish. For instance, when m + m ′ ≤ c nq p, p ′ = 0 , c nq p, p ′ = pδ n δ q,p + p ′ − , c nq p, p ′ = ( p − p ′ ) δ n δ q,p + p ′ − ,c nq p, p ′ = p ( p − δ n δ q,p + p ′ − + 2 pδ n δ q,p + p ′ − . By virtue of (4.2), it holds for the same structure constants that[ ˆ B mp , ˆ B m ′ p ′ ] = X n,q c nqmp,m ′ p ′ ˆ B nq . (4.14) Proposition 4.5
When acting on the operators Φ and ˆΦ (or on the Baker–Akhiezerfunctions ψ ( z ) and ˆ ψ ( z ) ) of the extended KP hierarchy, the additional symmetries (4.6) , (4.7) satisfy: (cid:20) ∂∂β mp , ∂∂β m ′ p ′ (cid:21) = − X n,q c nqmp,m ′ p ′ ∂∂β nq , (4.15) " ∂∂ ˆ β mp , ∂∂ ˆ β m ′ p ′ = X n,q c nqmp,m ′ p ′ ∂∂ ˆ β nq , (4.16) " ∂∂β mp , ∂∂ ˆ β m ′ p ′ = 0 . (4.17) Proof:
The conclusion can be checked case by case with the help of Lemma 4.3. Forinstance, (cid:20) ∂∂β mp , ∂∂β m ′ p ′ (cid:21) ˆΦ=[( B m ′ p ′ ) + , ( B mp ) + ] ˆΦ + [ − ( B mp ) − , B m ′ p ′ ] + ˆΦ − [ − ( B m ′ p ′ ) − , B mp ] + ˆΦ= − [( B mp ) + , ( B m ′ p ′ ) + ] ˆΦ − [( B mp ) − , ( B m ′ p ′ ) + ] + ˆΦ − [ B mp , ( B m ′ p ′ ) − ] + ˆΦ= − [ B mp , B m ′ p ′ ] + ˆΦ = − X n,q c nqmp,m ′ p ′ ( B nq ) + ˆΦ= − X n,q c nqmp,m ′ p ′ ∂ ˆΦ ∂β nq , (4.18) " ∂∂ ˆ β mp , ∂∂ ˆ β m ′ p ′ ˆΦ=[( ˆ B m ′ p ′ ) + , ( ˆ B mp ) + ] ˆΦ + [( ˆ B mp ) + , ˆ B m ′ p ′ ] + ˆΦ − [( ˆ B m ′ p ′ ) + , ˆ B mp ] + ˆΦ=[( ˆ B mp ) + , ( ˆ B m ′ p ′ ) − ] + ˆΦ + [ ˆ B mp , ( ˆ B m ′ p ′ ) + ] + ˆΦ=[ ˆ B mp , ˆ B m ′ p ′ ] + ˆΦ = X n,q c nqmp,m ′ p ′ ( ˆ B nq ) + ˆΦ15 X n,q c nqmp,m ′ p ′ ∂ ˆΦ ∂ ˆ β nq , (4.19) " ∂∂β mp , ∂∂ ˆ β m ′ p ′ ˆΦ=[( ˆ B m ′ p ′ ) + , ( B mp ) + ] ˆΦ + [( B mp ) + , ˆ B m ′ p ′ ] + ˆΦ − [ − ( ˆ B m ′ p ′ ) − , B mp ] + ˆΦ= (cid:16) [( B mp ) + , ( ˆ B m ′ p ′ ) − ] + + [( ˆ B m ′ p ′ ) − , ( B mp ) + ] + (cid:17) ˆΦ = 0 . (4.20)The other cases are checked in the same way. Thus the proposition is proved. (cid:3) This proposition means that the additional symmetries (4.6), (4.7) for the extendedKP hierarchy form a W ∞ × W ∞ algebra. Given a positive integer n , as in [19] we considered a constraint (cf. [4]) P n = ˆ P (4.21)imposed to the extended KP hierarchy (3.17), (3.18). Under this reduction, note ∂/∂t nk = ∂/∂ ˆ t k for k ≥
1, so without loss of generality we will not consider the flows ∂/∂ ˆ t k in thissection. As a result, the reduced hierarchy reads ∂L∂t k = [( P k ) + , L ] , k = 1 , , , . . . , (4.22)where L := P n = ˆ P takes the form L = ∂ n + u ∂ n − + u ∂ n − + · · · + u n − ∂ + u n − + ( ∂ − f ′ ) − ρ. (4.23)The hierarchy (4.22) is called the constrained KP hierarchy, denoted by cKP n, (see[1, 3, 10]). In fact, if we write v = e f and w = ρe − f , then L − = v∂ − w (4.24)and it is yielded by the equations (4.22) that ∂v∂t k = (cid:16) P k (cid:17) + ( v ) , ∂w∂t k = − (cid:16) P k (cid:17) ∗ + ( w ) . (4.25) Example 4.6
It is known that, when n = 1 the hierarchy (4.22) is just the nonlinearShr¨odinger hierarchy (see, e.g., [4, 19]), and when n = 2 the hierarchy (4.22) is theYajima–Oikawa hierarchy [20]. More exactly, when n = 2 the first nontrivial equations in(4.22) with L = ∂ + u + ( ∂ − f ′ ) − ρ. read ∂u∂t = 2 ρ ′ , ∂ρ∂t = (2 ρf ′ − ρ ′ ) ′ , ∂f∂t = u + ( f ′ ) + f ′′ , (4.26)16hich is, in terms of the variables { u, v = e f , w = ρe − f } , the Yajima–Oikawa system: ∂u∂t = 2( vw ) ′ , ∂v∂t = v ′′ + uv, ∂w∂t = − w ′′ − uw. (4.27)We proceed to construct a series of Virasoro symmetries for the cKP n, hierarchy(4.22). Given an operator L in (4.23), there exist two dressing operators Φ ∈ D − andˆΦ ∈ D + of the form (3.2) and (3.3) such that L = Φ ∂ n Φ − = ˆΦ ∂ − Φ − . (4.28)Note that Φ and ˆΦ are determined up to multiplication to the right by some operator withconstant coefficients, and we fix a choice of them. The flows ∂/∂t k can be represented viathe dressing operators as ∂ Φ ∂t k = − ( P k ) − Φ , ∂ ˆΦ ∂t k = (cid:0) ( P k ) + − δ k ˆ P − (cid:1) ˆΦ . (4.29)Now let us introduce M = Φ X k ∈ Z > kt k ∂ k − Φ − , ˆ M = ˆΦ x ˆΦ − . As before, here all but finitely many time variables are assumed to be zero, so that M andˆ M can be regarded as pseudo-differential operators of the first and of the second typesrespectively. Accordingly, we introduce S p = 1 n M P np +1 + ˆ M ˆ P p − ∈ D − ∪ D + , p ∈ Z . (4.30)When p ≥ −
1, by using (4.2) it is straight forward to verify[ S p , L ] = 1 n (cid:2) M P np +1 , P n (cid:3) + h ˆ M ˆ P p − , ˆ P i = − P n ( p +1) + ˆ P p +1 = − L p +1 + L p +1 = 0 . (4.31)It is worthwhile to emphasize that, here the condition p ≥ − L hasdifferent inverses in the rings D − and D + .Let us define the following evolutionary equations ∂ Φ ∂s p = − ( S p ) − Φ , ∂ ˆΦ ∂s p = ( S p ) + ˆΦ . (4.32)Indeed, these flows are consistent with the constraint (4.21). More precisely, the equations(4.32) lead to (recall (4.28) and (4.31)) ∂L∂s p = [ − ( S p ) − , L ] = [( S p ) + , L ] , p ≥ − , (4.33)which are well defined by comparing the coefficients of pseudo-differential operators. Weremark that, if one takes another choice of Φ and ˆΦ, then the flows ∂/∂s p in (4.33) areup to addition of a linear combination of the flows ∂/∂t k .17 roposition 4.7 For the constrained KP hierarchy (4.22) , the flows ∂/∂s p defined by (4.32) satisfy: (i) (cid:20) ∂∂t k , ∂∂s p (cid:21) L = 0 , (4.34)(ii) (cid:20) ∂∂s p , ∂∂s q (cid:21) L = ( q − p ) ∂L∂s p + q , (4.35) where k ∈ Z > and p, q ∈ Z ≥− .Proof: Firstly, by using (4.29) we have ∂S p ∂t k = 1 n h − ( P k ) − , M P np +1 i + kn P k + np + h ( P k ) + − δ k ˆ P − , ˆ M ˆ P p − i + δ k ˆ P p − = 1 n h − ( P k ) − + P k , M P np +1 i + h ( P k ) + , ˆ M ˆ P p − i = h ( P k ) + , S p i . (4.36)What is more, by using (4.32) it is easy to verify: ∂ ( 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)The first item is checked by (cid:20) ∂∂t k , ∂∂s p (cid:21) L = ∂∂t k [( S p ) + , L ] − ∂∂s p h ( P k ) + , L i = (cid:20)h ( P k ) + , S p i + , L (cid:21) + h ( S p ) + , h ( P k ) + , L ii − (cid:20)h − ( S p ) − , P k i + , L (cid:21) − 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 . For the second item, we have (cid:20) ∂∂s p , ∂∂s q (cid:21) L = ∂∂s p [( S q ) + , L ] − ∂∂s q [( S p ) + , L ]= (cid:20)(cid:20) − ( S p ) − , n M P nq +1 (cid:21) + + h ( S p ) + , ˆ M ˆ P q − i + , L (cid:21) + [( S q ) + , [( S p ) + , L ]] − ( p ↔ q )= h (cid:20) − ( S p ) − , n M P nq +1 (cid:21) + + h ( S p ) + , ˆ M ˆ P q − i + − (cid:20) − ( S q ) − , n M P np +1 (cid:21) + h ( S q ) + , ˆ M ˆ P p − i + + [( S q ) + , ( S p ) + ] , L i = (cid:20) n X + 1 n Y + Z, L (cid:21) where, with (4.30) substituted, X = (cid:2) − ( M P np +1 ) − , M P nq +1 (cid:3) + − (cid:2) − ( M P nq +1 ) − , M P np +1 (cid:3) + + (cid:2) ( M P nq +1 ) + , ( M P np +1 ) + (cid:3) = (cid:2) M P nq +1 , M P np +1 (cid:3) + = n ( q − p )( M P n ( p + q )+1 ) + ,Y = h − ( ˆ M ˆ P p − ) − , M P nq +1 i + + h ( M P np +1 ) + , ˆ M ˆ P q − i + − h − ( ˆ M ˆ P q − ) − , M P np +1 i + − h ( M P nq +1 ) + , ˆ M ˆ P p − i + + h ( M P nq +1 ) + , ( ˆ M ˆ P p − ) + i + h ( ˆ M ˆ P q − ) + , ( M P np +1 ) + i = h ( M P nq +1 ) + , ˆ M ˆ P p − , i + − h ( M P nq +1 ) + , ˆ M ˆ P p − i + + h ( M P np +1 ) + , ( ˆ M ˆ P q − ) + i + + h ( ˆ M ˆ P q − ) + , ( M P np +1 ) + i = 0 ,Z = h ( ˆ M ˆ P p − ) + , ˆ M ˆ P q − i + − h ( ˆ M ˆ P q − ) + , ˆ M ˆ P p − i + + h ( ˆ M ˆ P q − ) + , ( ˆ M ˆ P p − ) + i = h ( ˆ M ˆ P p − ) + , ˆ M ˆ P q − i + − h ( ˆ M ˆ P q − ) + , ( ˆ M ˆ P p − ) − i + = h ˆ M ˆ P p − , ˆ M ˆ P q − i + = ( q − p )( ˆ M ˆ P p + q − ) + . So we derive (cid:20) ∂∂s p , ∂∂s q (cid:21) L = ( q − p ) [( S p + q ) + , L ] = ( q − p ) ∂L∂s p + q . The proposition is proved. (cid:3)
From another point of view, the flows ∂/∂s p in (4.32) can be considered as reductionsof the linear combinations 1 n ∂∂β ,np +1 + ∂∂ ˆ β , − p +1 of the additional symmetries (4.6), (4.7) for the extended KP hierarchy constrained by(4.28). The commutation relation between the flows ∂/∂s p agrees with that in Propo-sition 4.5. This observation motivates us to consider whether the symmetries flows like ∂/∂β p or ∂/∂ ˆ β p for the extended KP hierarchy could be reduced to that for the cKP n, hierarchy. In the same way as above, the following evolutionary equations are well defined ∂L∂s ′ p = h − ( S p + ( κp + λ ) ˆ P p ) − , L i , p ≥ − , (4.38)19or arbitrary constants κ and λ . In other words, the flows ∂/∂s p are modified with linearterms in ∂/∂ ˆ β , − p , that is, (cid:18) ∂∂s ′ p − ∂∂s p (cid:19) L = ( , p = − , − ( κp + λ ) [( L p ) − , L ] , p ≥ . (4.39) Proposition 4.8
For the constrained KP hierarchy (4.22) , the flows ∂/∂s ′ p defined by (4.38) with arbitrary constants κ and λ satisfy: (i) (cid:20) ∂∂t k , ∂∂s ′ p (cid:21) L = 0 , (4.40)(ii) (cid:20) ∂∂s ′ p , ∂∂s ′ q (cid:21) L = ( q − p ) ∂L∂s ′ p + q , (4.41) where k ∈ Z > and p, q ∈ Z ≥− .Proof: The results is checked in the same way as Proposition 4.8, by doing the replace-ments S p S p + ( κp + λ ) ˆ P p . The proposition is proved. (cid:3) On the other hand, for the cKP n, hierarchy (4.22), a series of the Virasoro symmetrieswas constructed by Aratyn, Nissimov and Pacheva in [1] by adding certain “ghost” sym-metry flows related to the eigenfunctions characteristic for the operators L and L ∗ . (infact the cases cKP n,m with general m were studied there). More exactly, in Proposition 2of [1] the following symmetry flows were given: ∂L∂ ˜ s p = (cid:20) − n ( M P np +1 ) − + Y p , L (cid:21) , p ≥ − , (4.42)where Y p = 0 for p = − , ,
1, and Y p = p − X j =0 j − p + 12 L p − j − ( v ) ∂ − ( L ∗ ) j ( w ) , p ≥ . (4.43)Here we recall v = e f = ˆΦ(1) , w = ρe − f = (cid:16) ˆΦ − (cid:17) ∗ (1) , and note that there are nonlocal-action terms in Y p with p ≥ L − = 0. Further-more, Aratyn, Nissimov and Pacheva showed that, the cKP n, hierarchy subject to thesubsidiary condition of invariance under the lowest Virasoro symmetry flow can be appliedto compute explicit Wronskian solution for the two-matrix model partition function [2]. Corollary 4.9
The symmetry flows in (4.42) satisfy ∂L∂ ˜ s p = ∂L∂s ′ p , p ≥ − , where ∂/∂s ′ p are given in (4.38) with κ = − and λ = . roof: Let us fix κ = − and λ = in (4.38), then (cid:18) ∂∂ ˜ s p − ∂∂s ′ p (cid:19) L = (cid:20) − n (cid:0) M P np +1 (cid:1) − + Y p + (cid:18) S p − p −
12 ˆ P p (cid:19) − , L (cid:21) = [ Y p + Z p , L ] , where Z p := (cid:18) ˆ M ˆ P p − − p −
12 ˆ P p (cid:19) − = (cid:16) ˆΦ x∂ − p +1 ˆΦ − (cid:17) − − p − (cid:16) ˆΦ ∂ − p ˆΦ − (cid:17) − . (4.44)When p = − , ,
1, clearly Z p = 0 = Y p , then the flows ∂/∂ ˜ s p and ∂/∂s ′ p acting on L coincide. When p = 2, on one hand from (4.43) we have Y = 12 (cid:0) − L ( v ) ∂ − w + v∂ − L ∗ ( w ) (cid:1) = − L ( v ) ∂ − w + 12 (cid:0) L ( v ) ∂ − w + v∂ − L ∗ ( w ) (cid:1) = − L ( v ) ∂ − w + 12 (cid:0) L (cid:1) − , where the last equality is due to the formula (61) in the appendix of [1], say, in the presentcase ( L k ) − = k − X j =0 L k − j − ( v ) ∂ − ( L ∗ ) j ( w ) , k ≥ . On the other hand, since (cid:16) ˆΦ x∂ − ˆΦ − (cid:17) − = ˆΦ( x ) · ∂ − · (cid:16) ˆΦ − (cid:17) ∗ (1) = ˆΦ ∂ − ˆΦ − ˆΦ ∂ ( x ) · ∂ − · (cid:16) ˆΦ − (cid:17) ∗ (1)= L ˆΦ(1) · ∂ − · w = L ( v ) ∂ − w, then from (4.44) we have Z = L ( v ) ∂ − w − (cid:0) L (cid:1) − . Clearly Y + Z = 0, then we obtain ∂L/∂ ˜ s = ∂L/∂s ′ . When p ≥
3, the flows ∂/∂ ˜ s p and ∂/∂ ˜ s ′ p can be determined by those flows with p = − , , , (cid:3) In this paper we have introduced the (adjoint) Baker–Akhiezer functions of the extendedKP hierarchy, and capsuled the hierarchy into a bilinear equation satisfied by them. Fur-thermore, we have constructed a class of additional symmetries for this hierarchy, andconsidered the reduction properties. Such results, which are analogue to those for the KPand the 2-BKP hierarchies, are expected to extend our understanding to the theory ofintegrable hierarchies. 21t the end of [19], a tau function of the extended KP hierarchy was introduced byusing the densities of Hamiltonian functionals. Similar to the KP hierarchy, one can derivea Sato formula that links the tau function to the (adjoint) Baker–Akhiezer functions ψ ( z )and ψ † ( z ). However, the relationship between the tau function and ˆ ψ ( z ) (or ˆ ψ † ( z )) stillneeds to be clarified. That is why a bilinear equation of tau function is still missing. Wealso hope that the tau function of the extended KP hierarchy could be applied in thetwo-matrix model such as in the context of [2]. This will be studied elsewhere. Acknowledgments.
The authors thank Professor Baofeng Feng for helpful discussions.This work is partially supported by the National Natural Science Foundation of ChinaNos. 11771461 and 11831017.
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