Virasoro symmetries of Multi-Component Gelfand-Dickey systems
aa r X i v : . [ n li n . S I] J un Virasoro symmetries of Multi-Component Gelfand-Dickeysystems
Ling An, Chuanzhong Li ∗ School of Mathematics and Statistics, Ningbo University, Ningbo, 315211, China
Abstract
In this paper, we mainly study the additional symmetry and τ functions of a multi-component Gelfand-Dickey hierarchy which includes many classical integrable systems,such as the multi-component KdV hierarchy and the multi-component Boussinesq hier-archy. With other kinds of reductions, we can derive a B type multi-component Gelfand-Dickey hierarchy and a C type multi-component Gelfand-Dickey hierarchy. In our re-search, the additional flows of the additional symmetries can not all survive. By calculat-ing, we find that the generator of the additional symmetry of the C type multi-componentGelfand-Dickey hierarchy is different from that of the B type multi-component Gelfand-Dickey hierarchy, while the forms of their surviving additional flows are the same. Mathematics Subject Classifications (2010) : 37K05, 37K10, 35Q53.
Keywords: multi-component Gelfand-Dickey hierarchy, additional symmetry, string equa-tion, τ function, Virasoro canstraint. Contents τ function and Virasoro constraint of the mcGD hierarchies . . . . . . . . 8 τ function and Virasoro constraint of the mcBGD hierarchies . . . . . . . 14 ∗ Corresponding author:[email protected]. Introduction
Gelfand-Dickey hierarchy was introduced by Gelfand and Dickey [1], which attractedmany people’s attention, and then became one of the hottest topics in classical integrablesystems. The Hamiltonian theory of the Gelfand-Dickey hierarchy was developing grad-ually in terms of the Lax pairs, and has become a powerful supports for the study ofintegrable systems. These theories were introduced in detail in [2]. In fact, many do-mestic and foreign scholars have done extensive research on the soliton solutions of theGelfand-Dickey hierarchy, additional symmetry, τ function, B¨ a cklund transformation andother related properties [3]. In addition, a lot of research has been done on supersymmet-ric Gelfand-Dickey hierarchy, q-deformed Gelfand-Dickey hierarchy, etc [4–10]. Generallyspeaking, little research has been done on the multi-component Gelfand-Dickey (mcGD)hierarchy. In this paper, the additional symmetries and τ functions of a multi-componentGelfand-Dickey hierarchy, a B type multi-component Gelfand-Dickey (mcBGD) hierarchyand a C type multi-component Gelfand-Dickey (mcCGD) hierarchy will be studied basedon the work done by domestic and foreign scholars on the KP hierarchy and GD hierarchy.When the theory of additional symmetries first appeared, it ranked only at the edgeof integrable systems. With the continuous study of string equations [11–13], Virasoroconstraints [14,15] and other theories, it was found that the additional symmetries playedan extremely important role in these theories, and the additional symmetries graduallyattracted people’s attention [16–18]. A direct application of the additional symmetry ofthe mcGD hierarchy is to derive the string equation which appears in the study of thestring theory. The general expression of the string equation is [ P, Q ] = 1, where P and Q are differential operators. For string equation, one of its important characteristics is thatit has a close relationship with hierarchies of some integrable equations. The relationshipbetween them is that the string equation is invariant under the flows generated by theequation in the hierarchy [2]. A portion of the additional symmetries can be reduced toVirasoro symmetries. Virasoro symmetries have a wide range of applications, of whichtheir roles in τ function are particularly important [19, 20], because τ function appearsas a partition function or as a generating function in modern problems of mathematicsand physics. According to the action of Virassoro symmetries on the τ function, we canalso get the explicit solutions of the Virasoro constainted nonlinear equations in the formof matrix integrals. Such as [21, 22], it was the Virasoro costrainted solution in the Todachain case and in the KdV case. The additional symmetries can also be used to find theeigenfunctions of the linearized problem and to solve the stability problems [23].In this paper, the definitions and properties of a mcGD hierarchy, a mcBGD hierarchyand a mcCGD hierarchy are described respectively. The order of the article is as follows:Firstly, we define their Lax equations by the operator L , operator R and appropriate con-straints, and introduce the wave operator φ and discuss some properties of wave functionswhich naturally lead to their Sato equations. Then, by introducing an Orlov-Shulman’soperator M , the definition of additional symmetries is given, and many practical prop-erties are derived from the additional symmetries. In the calculation, we find that onlya small part of the additional flows can survival, so we give the forms of the survivingadditional flows. At the same time, a special additional flow is analyzed and calculated,and an important application of additional symmetries in the string theory is obtained.Finally, through the existence theorem of τ functions, the additional symmetries of τ func-tions are discussed, and the equivalent form of the string equation, Virasoro constraint,is obtained. It should be noted that the mcCGD hierarchy can be divided into odd andeven forms. Multi-Component Gelfand-Dickey hierarchies
The Gelfand-Dickey hierarchy is one of the most important topic in the area of classicalintegrable systems. The definition of the multi-component Gelfand-Dickey hierarchy isbased on a N -order differential operator L and a operator R α like these L = A∂ N + u ∂ N − + u ∂ N − + · · · + u , (2.1)among them, A = diag ( a , a , · · · , a n ), a i are nonzero constants, The diagonal elements of u are all equal to zero, except in the case of n = 1. And the u i is an arbitrary n × n matrix; R α = ∞ X i =1 R iα ∂ − i , α = 1 , , · · · , n, (2.2)among them, R α = E α , E α is a matrix having only one nonzero element on the ( α, α )place which is equal to 1, R α satisfies [ L, R α ] = 0.It can be found that R jα exist. Their elements are differential polynomials composed of u i , and they have the following properties R α R β = δ αβ R α , n X α =1 R α = I. (2.3) L and R α are collectively called the Lax operators of the mcGD hierarchy.The first definition of the mcGD hierarchy are the Lax equations ∂ n,α L = [ B n,α , L ] , ∂ n,α R β = [ B n,α , R β ] , n = 0 mod N, n X α =1 ∂ jN,α L = 0 , n X α =1 ∂ jN,α R β = 0 , j = 1 , , · · · , (2.4)where ∂ n,α = ∂∂t n,α , and B n,α refers to the differential part of the operator L nN R α , B n,α = ( L nN R α ) + . The second definition of the mcGD hierarchy is the zero curvature equation ∂ n,α B m,α − ∂ m,α B n,α + [ B m,α , B n,α ] = 0 , n, m = 0 mod N, (2.5)the zero curvature equation of the mcGD hierarchy can be derived from its Lax equations.Note: The variables x and t n,α are not independent, but they are also not equivalent. Theyhave the following relationship ∂ = n X α =1 a − N α ∂ ,α . When N = 2, we can derive the multi-component KdV hierarchy; when N = 3, we canderive the multi-component Boussinesq hierarchy.The Lax operators of the mcGD hierarchy can also be expressed in dressing form L = φA∂ N φ − , R α = φE α φ − , (2.6)where the quasi-differential operator φ = φ ( A∂ N ) = ∞ P i =1 α i ( A∂ N ) − i , α = I , φ is calleddressing operator or wave operator. roposition 2.1. By dressing transformation, the vector field on A u can be converted to A w ∂ n,α φ = − ( L nN R α ) − φ, n = 0 mod N, n X α =1 ∂ jN,α φ = 0 , j = 1 , , · · · , (2.7) the above equations are called the Sato equations of the multi-component Gelfand-Dickeyhierarchy. The wave function of the mcGD hierarchy is discussed below.Let’s first introduce a series ξ ( t, z ) = ∞ X i =1 n X α =1 t i,α E α z i , (2.8) ξ ( t, z ) has the following properties ∂ n,α e ξ ( t,z ) = z n E α e ξ ( t,z ) , n = 0 mod N, n X α =1 ∂ jN,α e ξ ( t,z ) = z jN e ξ ( t,z ) , j = 1 , , · · · ,∂ m e ξ ( t,z ) = z m A − mN e ξ ( t,z ) . (2.9)Then the wave function of the mcGD hierarchy can be defined as W ( t, z ) = φe ξ ( t,z ) = ω ( t, z ) e ξ ( t,z ) . Corollary 2.2.
The wave function of the multi-component Gelfand-Dickey hierarchy sat-isfies L mN W = z m W, ∂ n,α W = ( L nN R α ) + W, n = 0 mod N, n X α =1 ∂ jN,α W = L j W, j = 1 , , · · · , (2.10) Proof.
According to the formula (2.9), we can obtain A mN ∂ m e ξ ( t,z ) = z m e ξ ( t,z ) , so we can get A mN ∂ m = z m , therefore L mN W = φA mN ∂ m e ξ ( t,z ) = A mN ∂ m φe ξ ( t,z ) = z m W,∂ n,α W = ∂ n,α ( φe ξ ( t,z ) ) = ( ∂ n,α φ ) e ξ ( t,z ) + φ ( ∂ n,α e ξ ( t,z ) )= − ( L nN R α ) − φe ξ ( t,z ) + φz n E α e ξ ( t,z ) = − ( L nN R α ) − W + φA nN ∂ n E α φ − φe ξ ( t,z ) = − ( L nN R α ) − W + L nN R α W =( L nN R α ) + W. n X α =1 ∂ jN,α W = n X α =1 ∂ jN,α ( φe ξ ( t,z ) ) = φ ( n X α =1 ∂ jN,α e ξ ( t,z ) )= φ ( z jN e ξ ( t,z ) ) = L j W. ext, let’s consider the adjoint wave function W ∗ ( t, z ), where the adjoint symbol “ ∗ ”represents a formal adjoint operator, such as ∂ ∗ = − ∂, ( ∂ − ) ∗ = − ∂ − , ( AB ) ∗ = B ∗ A ∗ .The adjoint wave function W ∗ ( t, z ) of the mcGD hierarchy is defined as W ∗ ( t, z ) = ( φ ∗ ) − e − ξ ( t,z ) . (2.11)We have given the Lax operators of the mcGD hierarchy before, and it is easy to provethat the operators ∂ n,α − ( L nN R α ) + and n P α =1 ∂ jN,α − L j can be expressed in a dressing form ∂ n,α − ( L nN R α ) + = φ ( ∂ n,α − A nN ∂ n E α ) φ − , n = 0 mod N, n X α =1 ∂ jN,α − L j = φ ( n X α =1 ∂ jN,α − A j ∂ jN ) φ − , j = 1 , , · · · . (2.12)Dressing transformation on both sides of [ ∂ n,α − A nN ∂ n E α , A∂ N ] = 0 and [ n P α =1 ∂ jN,α − A j ∂ jN , A∂ N ] = 0, then we can get the Lax equations of the mcGD hierarchy[ ∂ n,α − ( L nN R α ) + , L ] = 0 , n = 0 mod N, [ n X α =1 ∂ jN,α − L j , L ] = 0 , j = 1 , , · · · . (2.13) We introduce an operatorΓ = ∞ X k =1 n X α =1 kt k,α A k − N ∂ k − E α , k = 0 mod N. (2.14)The operator Γ has the following properties ∂ n,α Γ = nA n − N ∂ n − E α , ∂ k Γ = Γ ∂ k + kA − N ∂ k − , n = 0 mod N. It is not difficult to verify that the operators Γ and ∂ n,α − A nN ∂ n E α are commutative,that is to say, [ ∂ n,α − A nN ∂ n E α , Γ] = 0 , n = 0 mod N. (2.15)Dressing transformation on both sides of [ ∂ n,α − A nN ∂ n E α , Γ] = 0, then we can obtain ∂ n,α M = [( L nN R α ) + , M ] , n = 0 mod N, (2.16)where M = φ Γ φ − , we call it an Orlov-Shulman’s operator. Definition 2.3.
The solution of the differential equation ∂φ∂ ∗ l,m,α = − ( M m L lN R α ) − φ, ( ∂ ∗ l,m,α = ∂∂t ∗ l,m,α ) (2.17) is called the additional symmetry of the multi-component Gelfand-Dickey hierarchy. ext we consider a case related to the restriction of Virasoro. For the special dif-ferential operator M m L lN R α , assuming its negative part disappears and let the operator( M m L lN R α ) + act on W , we can get an equation related to z ( M m L lN R α ) + W = z l E α ∂ mz W. (2.18)Note:This system can be rewritten into a linear equation for the isomonodromy problem. Corollary 2.4.
Combining with the definition of the additional symmetry of the mcGDhierarchy, we can obtain ∂ ∗ l,m,α L = − [( M m L lN R α ) − , L ] ,∂ ∗ l,m,α R β = − [( M m L lN R α ) − , R β ] , (2.19) these imply ∂ ∗ l,m,α ( L nN R β ) − = − [( M m L lN R α ) − , ( L nN R β ) − + ∂ n,β ] − , n = 0 mod N. (2.20) Proof.
By calculating, we can get ∂ ∗ l,m,α L = ∂ ∗ l,m,α ( φA∂ N φ − )=( ∂ ∗ l,m,α φ ) A∂ N φ − + φA∂ N ( ∂ ∗ l,m,α φ − )= − ( M m L lN R α ) − L + L ( M m L lN R α ) − = − [( M m L lN R α ) − , L ] , The second equation can be proved by the same principle. Then we can find ∂ ∗ l,m,α ( L nN R β ) − = − [( M m L lN R α ) − , ( L nN R β ) − ] − − [( M m L lN R α ) − , ∂ n,β ] − = − [( M m L lN R α ) − , ( L nN R β ) − + ∂ n,β ] − . For the additional flows of the mcGD hierarchy, only a few special additional flowscan survive.
Theorem 2.5.
In the additional flows of the multi-component Gelfand-Dickey hierar-chy, only the flows which satisfy the condition ( M m − L N + l − N ) − = 0 and are shaped like n P α =1 ∂ ∗ l,m,α can survive.Proof. For the Lax operator L of the mcGD hierarchy, it has no negative part, that is tosay, its negative part is equal to zero.From above, we can know ∂ ∗ l,m,α L = − [( M m L lN R α ) − , L ], so let’s consider ∂ ∗ l,m,α L − = − [( M m L lN R α ) − , L ] − = − ( φ [Γ m A lN ∂ l E α , A∂ N ] φ − ) − , after some deductions, the upper formula can be reduced to ∂ ∗ l,m,α L − = NN + l (cid:2) L N + lN R α , M m (cid:3) − = mN ( M m − L N + l − N R α ) − , f and only if ( M m − L N + l − N ) − = 0, we can get n X α =1 ∂ ∗ l,m,α L − = mN ( M m − L N + l − N ) − = 0 . So we just need to thing about the surviving additional flows n P α =1 ∂ ∗ l,m,α which satisfythe condition ( M m − L N + l − N ) − = 0.According to the theorem 2.5, we can find that assuming that a solution of the mcGDhierarchy is defined by the Virasoro condition n P α =1 ( M m − L N + l − N R α ) − = 0, if the con-straint ( L N + lN ) − = 0 is imposed on this solution, it must also satisfy the W ∞ symmetry( M m ) − = 0. That is, the W ∞ symmetry is compatible both with the Virasoro condition( M m − L N + l − N ) − = 0 and the constraint ( L N + lN ) − = 0. Proposition 2.6.
The additional flows n P α =1 ∂ ∗ l,m,α which satisfy the condition ( M m − L N + l − N ) − =0 commute with the flows ∂ n,β ( n = 0 mod N ) of the multi-component Gelfand-Dickeyhierarchy.Proof. (cid:2) n X α =1 ∂ ∗ l,m,α , ∂ n,β (cid:3) φ = (cid:16) n X α =1 (cid:2) ∂ ∗ l,m,α , ∂ n,β (cid:3)(cid:17) φ = n X α =1 (cid:16)(cid:2) ∂ ∗ l,m,α , ∂ n,β (cid:3) φ (cid:17) , [ ∂ ∗ l,m,α , ∂ n,β ] φ = ∂ ∗ l,m,α ( ∂ n,β φ ) − ∂ n,β ( ∂ ∗ l,m,α φ )= − ∂ ∗ l,m,α (( L nN R β ) − φ ) + ∂ n,β (( M m L lN R α ) − φ )= − ( ∂ ∗ l,m,α ( L nN R β ) − ) φ − ( L nN R β ) − ( ∂ ∗ l,m,α φ )+ ( ∂ n,β ( M m L lN R α ) − ) φ + ( M m L lN R α ) − ( ∂ n,β φ )=[( M m L lN R α ) − , ( L nN R β ) − + ∂ n,β ] − φ + ( L nN R β ) − ( M m L lN R α ) − φ + [ ∂ n,β , ( M m L lN R α ) − ] − φ − ( M m L lN R α ) − ( L nN R β ) − φ =[( L nN R β ) − , ( M m L lN R α ) − ] φ − [( L nN R β ) − , ( M m L lN R α ) − ] − φ =0 , then we can obtain (cid:2) n P α =1 ∂ ∗ l,m,α , ∂ n,β (cid:3) φ = 0.From the above calculation we can find they commute on φ , so they will commute on thewhole differential algebra generated by coefficients of φ . The proposition is proved.A direct application of the additional symmetry of the mcGD hierarchy is to derivethe string equation which appears in the study of the string theory. First of all, we foundthe following relationships between the Lax operators and the Orlov-Shulman’s operatorof the mcGD hierarchy[ L N , M ] = φ [ A N ∂, Γ] φ − = φ ( A N ∂ Γ) φ − = φ ( A N A − N ) φ − = I ,[ R α , M ] = φ [ E α , Γ] φ − = O. hen, we can get[ L nN , M ] = φ [ A nN ∂ n , Γ] φ − = φ ( A nN nA − N ∂ n − ) φ − = nL n − N . (2.21)Furthermore, we can obtain [ L nN , M L − n − N R α ] = nR α . Next, we consider a special additional flow (cid:0) l = − ( n − , l − mod N (cid:1) n X α =1 ∂ ∗− ( n − , ,α L nN = − n X α =1 (cid:2) ( M L − n − N R α ) − , L nN (cid:3) , n = 0 mod N. Combined with the formula (2.21), we can rewrite it to n X α =1 ∂ ∗− ( n − , ,α L nN = n X α =1 (cid:2) ( M L − n − N R α ) + , L nN (cid:3) + nI, and when n = 0 mod N , there is L nN = ( L nN ) + , therefore n X α =1 (cid:2) L nN , n ( M L − n − N R α ) + (cid:3) = I. so h L nN , n (cid:0) n X α =1 ( M L − n − N R α ) + (cid:1)i = I. (2.22)The equation (2.22) is called the string equation of the mcGD hierarchy. From the abovededuction process, we can find that the string equation refers to the condition that theoperator is independent of the additional variables. τ function and Virasoro constraint of the mcGD hier-archies The existence theorem of the τ function is given below. Theorem 2.7. [2] Suppose the τ function is a matrix τ = ( τ α,β ) and τ α,α is abbreviatedas τ , then there exists a function τ ( · · · , t s,γ , · · · ) , which makes ω α,α ( t, z ) = τ ( ··· ,t s,γ − δ γ,β szs , ··· ) τ ( ··· ,t s,γ , ··· ) ,ω α,β ( t, z ) = τ α,β ( ··· ,t s,γ − δ γ,β szs , ··· ) z · τ ( ··· ,t s,γ , ··· ) , α = β, s = 0 mod N, (2.23) only when γ = β , the time variable t s,γ will be moved.Except for the above case s = 0 mod N , the other cases all satisfy n X α =1 ∂ kN,α τ = 0 , k = 1 , , · · · . (2.24)Next, starting from the additional symmetry of the wave operator φ and combiningwith the existence theorem of the τ function, we study the additional symmetry of the τ unction. To facilitate subsequent calculations, we first split n P α =1 ∂ ∗ l, ,α φ ( l − mod N )appropriately, where n X α =1 ∂ ∗ l, ,α φ = − n X α =1 ( M L lN R α ) − φ = − n X α =1 ( φ ∞ X k =1 kt k,α A k + l − N ∂ k + l − E α φ − ) − φ,k = 0 mod N, based on the value of k , we can divide ( φ ∞ P k =1 kt k,α A k + l − N ∂ k + l − E α φ − ) − φ ( l <
0) of theabove formula into three parts O = [ φ, t ,α ] A lN ∂ l E α + − l X k =1 kt k,α φA k + l − N ∂ k + l − E α ,P = ( φ ( − l + 1) t − l +1 ,α E α φ − ) − φ = ( − l + 1)[ φ, t − l +1 ,α E α ] ,Q = − ∞ X k = − l +2 kt k,α ∂ k + l − ,α φ, from the above splitting results, we can get n X α =1 ∂ ∗ l, ,α ω = − z l ∂ z ωI − n X α =1 (cid:0) − l X k =1 kt k,α z k + l − ωE α + ( − l + 1)[ ω, t − l +1 ,α E α ] − ∞ X k = − l +2 kt k,α ∂ k + l − ,α ω (cid:1) , k = 0 mod N, (2.25)accordingly, we can obtain n X α =1 ∂ ∗ l, ,α ω γ,β = n X α =1 (cid:0) − z l ∂ z ω γ,β δ α,β − − l X k =1 kt k,α z k + l − ω γ,β δ α,β − ( − l + 1)[ ω, t − l +1 ,α δ α,β ]+ ∞ X k = − l +2 kt k,α ∂ k + l − ,α ω γ,β (cid:1) , k = 0 mod N. (2.26)After the above study of relevant knowledge, it is easy to obtain the proposition on theadditional symmetry of the τ function. Proposition 2.8. [2] The action of additional flows n P α =1 ∂ ∗ l, ,α ( l − mod N ) with l < on ( τ γ,β ) is given by n X α =1 ∂ ∗ l, ,α τ γ,β = n X α =1 ( ∞ X k = − l +1 kt k,α ∂ k + l − ,α + 12 X k + s = − l +1 kst k,α t s,α ) τ γ,β + ( n X α =1 c α,β ) τ γ,β . (2.27) where t k,α , t s,α are arguments of the element τ γ,β . he method of proving the above proposition is similar to that in the literature [2],and it will not be repeated here. Because the string equation is the condition that theoperator is independent of the additional variables, we can also obtain the condition thatthe operators are independent of the additional variables by the way of proving the aboveproposition, as shown below, L l τ = 0 , l − mod N, l < , (2.28)where L l = n X α =1 (cid:0) ∞ X k = − l +1 kt k,α ∂ k + l − ,α + 12 X k + s = − l +1 kst k,α t s,α + c α,β (cid:1) , it is obviously equivalent to the string equation, We call it the Virasoro constraint of themcGD hierarchy. And L l satisfies the Virasoro exchange relation[ L − m , L − n ] = ( − m + n ) L − ( m + n ) , m, n = 1 , , · · · . There are non-autonomous ODEs, related to the solutions of the Virasoro constraints,similar to the Painlev´ e equations appearing in the theory of equations NLS and KdV.Next, we’ll give a brief description based on the classical KdV equation. Example 1. [24] The classical KdV equation u t + 6 uu x + u xxx = 0 (2.29) can be obtained by Lax pairs L = A∂ + u and A = − ∂ − u∂ − u x , where A = diag (1) , u = diag ( u ( x, t )) . Now we introduce a Virasoro operator ˆ L = L − = t , ∞ X k =2 kt k,α ∂ k − ,α . Let τ ( x ) be a solution of the mcGD hierarchy, which satisfies ˆ Lτ ( x ) = 0 , add the limit ∂τ∂t i, = 0 , ( i = 1 , , · · · ) to τ ( x ) to make it also a solution of the mcKdV hierarchy. Forexample, τ ( x ) satisfies the bilinear differential equation ( D t , + D t , D t , ) τ · τ = 0 (2.30) of equation (2.29) Then let δ ( x ) = τ ( x ) under the constraints t , = x, t , = − , and t j, =0 , ( j = 1 , , it can be found that δ ( x ) satisfies the bilinear differential equation ( D x − x ) δ · δ = 0 . (2.31) We take p = − d log ( δ ) dx , then equation (2.31) can be rewritten into the first Painlev ´ e equation P I d pdx = 6 p + x. The second part of this paper has given the Lax operators of the mcGD hierarcy L = A∂ N + u ∂ N − + u ∂ N − + · · · + u , α = ∞ X i =1 R iα ∂ − i , α = 1 , , · · · , n, if L and R α also satisfy L ∗ = − ∂L∂ − , R ∗ α = ∂R α ∂ − , then L and R α at this time arecalled the Lax operators of the mcBGD hierarchy.From the above Lax operators, we can define the Lax equations of the mcBGD hierarchy ∂ n,α L = [ B n,α , L ] , ∂ n,α R β = [ B n,α , R β ] , n = 0 mod N, n X α =1 ∂ jN,α L = 0 , n X α =1 ∂ jN,α R β = 0 , j = 1 , , · · · , (3.1)where L ∗ = − ∂L∂ − , R ∗ α = ∂R α ∂ − , ∂ n,α = ∂∂t n,α and B n,α = ( L nN R α ) + . Proposition 3.1.
The B type multi-component Gelfand-Dickey hierarchy has only oddflows.Proof.
From the foregoing, we can see that the Lax equations of the mcBGD hierarchyare ∂ n,α L = [ B n,α , L ] , ∂ n,α R β = [ B n,α , R β ] , n = 0 mod N, where L ∗ = − ∂L∂ − , R ∗ α = ∂R α ∂ − , ∂ n,α = ∂∂t n,α and B n,α = ( L nN R α ) + .Combined with the constraints of its Lax equations, we can get( L nN ) ∗ = ( L ∗ ) nN = ( − n ∂L nN ∂ − , ( L nN R α ) ∗ = R ∗ α ( L ∗ ) nN = ( − n ∂R α L nN ∂ − . Then we simplify ∂ n,α L ∗ and ∂ n,α R ∗ α in two ways ∂ n,α L ∗ = ( ∂ n,α L ) ∗ = [ B n,α , L ] ∗ = − ∂ [ B n,α , L ] ∂ − = [ ∂B n,α ∂ − , L ∗ ] ,∂ n,α L ∗ = [ B n,α , L ] ∗ = ( B n,α L − LB n,α ) ∗ = L ∗ B ∗ n,α − B ∗ n,α L ∗ = [ − B ∗ n,α , L ∗ ] , (3.2) ∂ n,α R ∗ β = ( ∂ n,α R β ) ∗ = [ B n,α , R β ] ∗ = ∂ [ B n,α , R β ] ∂ − = [ ∂B n,α ∂ − , R ∗ β ] ,∂ n,α R ∗ β = [ B n,α , R β ] ∗ = ( B n,α R β − R β B n,α ) ∗ = R ∗ β B ∗ n,α − B ∗ n,α R ∗ β = [ − B ∗ n,α , R ∗ β ] , (3.3)comparing equation (3.2) and (3.3), we can find B ∗ n,α = − ∂B n,α ∂ − , n = 0 mod N. (3.4)Then we start with the definition of B n,α ( n = 0 mod N ) and solve its adjoint B ∗ n,α = (( L nN R α ) + ) ∗ = ( − n ∂ ( R α L nN ) + ∂ − = ( − n ∂B n,α ∂ − , n = 0 mod N. (3.5)By analyzing equation (3.4) and (3.5), we can find that n can only take odd numbers.Therefore, a more concise form of the Lax equations of the mcBGD hierarchy can beobtained ∂ n +1 ,α L = [ B n +1 ,α , L ] , ∂ n +1 ,α R β = [ B n +1 ,α , R β ] , n + 1 = 0 mod N, n X α =1 ∂ (2 j ′ +1) N,α L = 0 , n X α =1 ∂ (2 j ′ +1) N,α R β = 0 , j ′ = 0 , , , · · · , (3.6) here ∂ n +1 ,α = ∂∂t n +1 ,α and B n +1 ,α = ( L n +1 N R α ) + .The Lax operators of the mcBGD hierarchy can also be expressed in dressing form L = φA∂ N φ − , R α = φE α φ − , where dressing operator φ = φ ( A∂ N ) = ∞ P i =1 α i ( A∂ N ) − i , α = I and φ ∗ = ∂φ − ∂ − whichis different from the mcGD hierarchy.Subsequently, the Sato equations ∂ n +1 ,α φ = − ( B n +1 ,α ) − φ (2 n + 1 = 0 mod N ) and n P α =1 ∂ (2 j ′ +1) N,α φ = 0 ( j ′ = 0 , , , · · · ) of the mcBGD hierarchy can also be obtained.Then, the wave function W ( t, z ) and the adjoint wave function W ∗ ( t, z ) of the mcBGDhierarchy are given W ( t, z ) = φe ξ ( t,z ) = ω ( t, z ) e ξ ( t,z ) ,W ∗ ( t, z ) = ( φ ∗ ) − e − ξ ( t,z ) = ( ∂φ∂ − ) e − ξ ( t,z ) , (3.7)where ξ ( t, z ) = ∞ X i =1 n X α =1 t i − ,α E α z i − . It should be noted that the Lax equations of the mcBGD hierarchy can also be deducedfrom L n +1 N W = z n +1 W, ∂ n +1 ,α W = B n +1 ,α W, n + 1 = 0 mod N, n X α =1 ∂ (2 j ′ +1) N,α W = L j ′ +1 W, j ′ = 0 , , , · · · . Firstly, we give the Orlov-Shulman’s operator M = φ Γ φ − of the mcBGD hierarchy,among them, Γ = ∞ P i =1 n P α =1 (2 i − t i − ,α A i − N ∂ i − E α , i − = 0 mod N. Combining the Orlov-Shulman’s operator M , we can easily get [ M, L n +1 N ] = − (2 n +1) L nN . Definition 3.2.
The solution of the differential equation ∂φ∂ ∗ l,m,α = − ( D l,m,α ) − φ, (3.8) where ∂ ∗ l,m,α = ∂∂t ∗ l,m,α , D l,m,α = M m L lN R α − ( − l R α L l − N M m L N , is called the additional symmetry of the B type multi-component Gelfand-Dickey hierarchy. Similarly, for the differential operator D l,m,α , assuming its negative part disappearsand let the operator ( D l,m,α ) + act on W , we can get an equation related to z ( D l,m,α ) + W = ( z l − ( − l z l − ) E α ∂ mz W − ( − l m ( l − z l − E α ∂ m − z W. (3.9)Note:This system can also be rewritten into a linear equation for the isomonodromyproblem. ccording to the definition of the additional symmetry of the mcBGD hierarchy, someformulas can be obtained by simple calculation ∂ ∗ l,m,α L = − [( D l,m,α ) − , L ] , ∂ ∗ l,m,α R β = − [( D l,m,α ) − , R β ] . (3.10)For the additional flows of the mcBGD hierarchy, they have a fraction of the flows thatcan survive. Theorem 3.3.
In the additional flows of the B type multi-component Gelfand-Dickeyhierarchy, the flows which satisfy the condition ( M m − L N + l − N ) − = 0 and are shaped like n P α =1 ∂ ∗ l,m,α or the condition l = 2 i ( i ∈ Z ) and are shaped like n P α =1 ∂ ∗ l,m,α can survive.Proof. For Lax operator L of the mcBGD hierarchy, its negative part is equal to zero.And we know ∂ ∗ l,m,α L = − [( D l,m,α ) − , L ], so let’s consider ∂ ∗ l,m,α L − = − [( D l,m,α ) − , L ] − = − ( φ [Γ m A lN ∂ l E α , A∂ N ] φ − ) − + ( − l ( φ [ E α A l − N ∂ l − Γ m A N ∂, A∂ N ] φ − ) − , after some deductions, the upper formula can be reduced to ∂ ∗ l,m,α L − = (1 − ( − l ) mN ( M m − L N + l − N R α ) − , only when ( M m − L N + l − N ) − = 0 or l = 2 i ( i ∈ Z ), we can obtain n X α =1 ∂ ∗ l,m,α L − = (1 − ( − l ) mN ( M m − L N + l − N ) − = 0 . Proposition 3.4.
The additional symmetric flows n P α =1 ∂ ∗ l,m,α which satisfy the condition ( M m − L N + l − N ) − = 0 or the condition l = 2 i ( i ∈ Z ) commute with the flows ∂ k +1 ,β (2 k +1 = 0 mod N ) of the B type multi-component Gelfand-Dickey hierarchy.Proof. (cid:2) n X α =1 ∂ ∗ l,m,α , ∂ k +1 ,β (cid:3) φ = (cid:16) n X α =1 (cid:2) ∂ ∗ l,m,α , ∂ k +1 ,β (cid:3)(cid:17) φ = n X α =1 (cid:16)(cid:2) ∂ ∗ l,m,α , ∂ k +1 ,β (cid:3) φ (cid:17) , [ ∂ ∗ l,m,α , ∂ k +1 ,β ] φ = − ∂ ∗ l,m,α (( B k +1 ,β ) − φ ) + ∂ k +1 ,β (( D l,m,α ) − φ )=[( D l,m,α ) − , ( B k +1 ,β ) − + ∂ k +1 ,β ] − φ + ( B k +1 ,β ) − ( D l,m,α ) − φ + [ ∂ k +1 ,β , ( D l,m,α ) − ] − φ − ( D l,m,α ) − ( B k +1 ,β ) − φ =[( D l,m,α ) − , ( B k +1 ,β ) − ] φ − [( D l,m,α ) − , ( B k +1 ,β ) − ] − φ =0 , then (cid:2) n P α =1 ∂ ∗ l,m,α , ∂ k +1 ,β (cid:3) φ = 0 , so the proposition is proved. hen we consider a special additional flow (cid:0) l = − (2 k − , l − mod N (cid:1) n X α =1 ∂ ∗− (2 k − , ,α L kN = − n X α =1 (cid:2) ( D − (2 k − , ,α ) − , L kN (cid:3) , k = 0 mod N, after a series of calculations, we can get n X α =1 ∂ ∗− (2 k − , ,α L kN = n X α =1 (cid:2) ( D − (2 k − , ,α ) + , L kN (cid:3) + 4 kI, When 2 k = 0 mod N , L kN is a differential operator, then n X α =1 (cid:2) L kN , k ( D − (2 k − , ,α ) + (cid:3) = I, thus h L kN , k (cid:0) n P α =1 ( D − (2 k − , ,α ) + (cid:1)i = I is the string equation of the mcBGD hierarchy. τ function and Virasoro constraint of the mcBGD hier-archies The existence theorem of a τ function of the mcBGD hierarchy is given below. Theorem 3.5. [2] Suppose the τ function is a matrix T=( τ α,β ) and τ α,α is abbreviatedas τ , then there exists a function τ ( · · · , t s − ,γ , · · · ) , which makes ω α,α ( t, z ) = τ ( ··· ,t s − ,γ − δ γ,β s − z s − , ··· ) τ ( ··· ,t s − ,γ , ··· ) ,ω α,β ( t, z ) = τ α,β ( ··· ,t s − ,γ − δ γ,β s − z s − , ··· ) z · τ ( ··· ,t s − ,γ , ··· ) , α = β, s − = 0 mod N, (3.11) only when γ = β , the time variable t s − ,γ will be moved.Except for the above case s − = 0 mod N , the other cases satisfy n P α =1 ∂ k,α τ = 0 , where k =0 mod N or k = 0 mod . Next, starting from the additional symmetry of the wave operator ω and combiningwith the existence theorem of the τ function, we study the additional symmetry of the τ function. Proposition 3.6. D − (2 k − , ,α (2 k = 0 mod N, k > can be reduced to the followingform ( D − (2 k − , ,α ) − =2 φ ( k X i =1 (2 i − t i − ,α A i − k − N ∂ i − k − E α ) φ − + 2 ∞ X i = k +1 (2 i − t i − ,α ( L i − k − N R α ) − − kL − kN R α , i − = 0 mod N. (3.12) roof. According to the additional symmetry of the mcBGD hierarchy, we will have( D − (2 k − , ,α ) − =( M L − k − N R α − ( − − (2 k − R α L l − N M m L N ) − =( M L − k − N R α ) − + ( R α L l − N M m L N ) − =2( φ Γ A − k − N ∂ − (2 k − E α φ − ) − − k ( φE α A − kN ∂ − k φ − ) − =2( φ ∞ X i =1 (2 i − t i − ,α A i − k − N ∂ i − k − E α φ − ) − − kL − kN R α =2 φ ( k X i =1 (2 i − t i − ,α A i − k − N ∂ i − k − E α ) φ − + 2 ∞ X i = k +1 (2 i − t i − ,α ( L − i − k − N R α ) − − kL − kN R α . Based on the additional symmetry of the mcBGD hierarchy and the proposition 3.6,we can calculate n X α =1 ∂ ∗− (2 k − , ,α φ = − n X α =1 ( D − (2 k − , ,α ) − φ = n X α =1 (cid:0) − φ k X i =1 (2 i − t i − ,α A i − k − N ∂ i − k − E α + 2 ∞ X i = k +1 (2 i − t i − ,α ( ∂ i − k ) − ,α φ ) (cid:1) + 2 kφA − kN ∂ − k I, (2 k = 0 mod N ) . (3.13)Obviously, when both sides of equation (3.13) are simultaneously applied to a function exp ( A − N xz ), the equation is still valid. By further calculating with[ φ, t ,α ] exp ( A − N xz ) = ( ∂ z ω ) exp ( A − N xz ) ,φ∂ − l exp ( A − N xz ) = φz − l exp ( A − N xz ) = z − l ω ( exp ( A − N xz )) , e can get ( n X α =1 ∂ ∗− (2 k − , ,α ω ) exp ( A − N xz )=( − n X α =1 ( D − (2 k − , ,α ) − φ ) exp ( A − N xz )= − z − k +1 ( ∂ z ω ) exp ( A − N xz ) + 2 kz − k ω ( exp ( A − N xz )) − k X i =1 n X α =1 (2 i − t i − ,α A i − k − N ∂ i − k − E α ω ) exp ( A − N xz )+ 2 ∞ X i = k +1 n X α =1 (2 i − t i − ,α ( ∂ i − k ) − ,α ω ) exp ( A − N xz ) , therefore n X α =1 ∂ ∗− (2 k − , ,α ω = − z − k +1 ( ∂ z ω ) I + 2 kz − k ωI − k X i =1 n X α =1 (2 i − t i − ,α A i − k − N ∂ i − k − E α ω )+ 2 ∞ X i = k +1 n X α =1 (2 i − t i − ,α ( ∂ i − k ) − ,α ω ) , (3.14)correspondingly, we can obtain n X α =1 ∂ ∗− (2 k − , ,α ω γ,β = n X α =1 (cid:0) − z − k +1 δ α,β ( ∂ z ω γ,β ) + 2 kz − k δ α,β ω γ,β − k X i =1 (2 i − t i − ,α A i − k − N ∂ i − k − δ α,β ω γ,β )+ 2 ∞ X i = k +1 (2 i − t i − ,α ( ∂ i − k ) − ,α ω γ,β ) (cid:1) . (3.15)We can find that n P α =1 ∂ ∗− (2 k − , ,α φ = 0 and n P α =1 ∂ ∗− (2 k − , ,α ω = 0 are equivalent. Thus,we can get the constraints of the tau function by equation (3.11).For n P α =1 ∂ ∗− (2 k − , ,α ω (2 k = 0 mod N, k > k is a positive integer. Proposition 3.7.
The action of additional symmetries n P α =1 ∂ ∗− (2 k − , ,α (2 k = 0 mod N, k > ) on ( τ γ,β ) is given by n X α =1 ∂ ∗− (2 k − , ,α τ γ,β = n X α =1 (cid:0) ( 12 ∞ X m =2 k (2 m − t m − ,α ∂ m − k ) − ,α + 18 X m + s =2 k (2 m − s − t m − ,α t s − ,α ) (cid:1) τ γ,β + ( n X α =1 c α,β ) τ γ,β . (3.16) where t m − ,α , t s − ,α are arguments of the element τ γ,β . Taking L − (2 k − = n X α =1 (cid:0) ∞ X m =2 k (2 m − t m − ,α ∂ m − k ) − ,α + 18 X m + s =2 k (2 m − s − t m − ,α t s − ,α + c α,β (cid:1) , (3.17)where 2 k = 0 mod N, k > L − (2 k − satisfies the Virasoro exchange relation[ L − m , L − n ] = ( − m + n ) L − ( m + n ) , m, n = 1 , , · · · . (3.18)so the Virasoro constraint of the τ function of the mcBGD hierarchy is L − (2 k − τ = 0 , k = 0 mod N, k > . (3.19) The Lax operators of the mcCGD hierarchy are composed of the operators L and R α of the mcGD hierarchy and the constraint conditions L ∗ = ( − N L, R ∗ α = R α . It is clearthat the mcCGD hierarchy can be divided into two types according to the parity of N .The odd mcCGD hierarchy can be constructed according to the differential operator L = A∂ N + u ∂ N − + u ∂ N − + · · · + u , N = 0 mod ,R α = ∞ X i =1 R iα ∂ − i , α = 1 , , · · · , n, which satisfy L ∗ = − L and R ∗ α = R α .The even mcCGD hierarchy can also be constructed according to the differential operator L = A∂ N + u ∂ N − + u ∂ N − + · · · + u , N = 0 mod ,R α = ∞ X i =1 R iα ∂ − i , α = 1 , , · · · , n, which satisfy L ∗ = L and R ∗ α = R α .Though their constraints are different, their additional symmetric generators, survivingflows, string equation and so on are the same.Similar to the Lax equations of the mcBGD hierarchy, the Lax equations of the mcCGDhierarchy does not have even flows, that is, u i = u i ( x ; t , t , · · · ). roposition 4.1. The C type multi-component Gelfand-Dickey hierarchy has only oddflows.Proof.
From the foregoing, we already know the Lax equations of the mcCGD hierarchy.Combined with the constraints of its Lax equations, we can get( L nN ) ∗ = ( − n L nN , ( L nN R α ) ∗ = ( − n R α L nN . Then we simplify ∂ n,α L ∗ and ∂ n,α R ∗ α in two ways ∂ n,α L ∗ = ( ∂ n,α L ) ∗ = [ B n,α , L ] ∗ = − [ B n,α , L ] = [ B n,α , L ∗ ] ,∂ n,α L ∗ = [ B n,α , L ] ∗ = ( B n,α L − LB n,α ) ∗ = L ∗ B ∗ n,α − B ∗ n,α L ∗ = [ − B ∗ n,α , L ∗ ] , (4.1) ∂ n,α R ∗ β = ( ∂ n,α R β ) ∗ = [ B n,α , R β ] ∗ = [ B n,α , R β ] = [ B n,α , R ∗ β ] ,∂ n,α R ∗ β = [ B n,α , R β ] ∗ = ( B n,α R β − R β B n,α ) ∗ = R ∗ β B ∗ n,α − B ∗ n,α R ∗ β = [ − B ∗ n,α , R ∗ β ] , (4.2)comparing (4.1) and (4.2), we can find B ∗ n,α = − B n,α , n = 0 mod N. (4.3)Then we combine the definition of B n,α ( n = 0 mod N ) to solve its adjoint B ∗ n,α = (( L nN R α ) + ) ∗ = ( − n ( R α L nN ) + = ( − n B n,α , n = 0 mod N. (4.4)By analyzing equation (4.3) and (4.4), we find that n can only take odd numbers.So its Lax equations can be defined as ∂ n +1 ,α L = [ B n +1 ,α , L ] , ∂ n +1 ,α R β = [ B n +1 ,α , R β ] , n + 1 = 0 mod N, n X α =1 ∂ (2 j ′ +1) N,α L = 0 , n X α =1 ∂ (2 j ′ +1) N,α R β = 0 , j ′ = 0 , , , · · · , (4.5)where ∂ n +1 ,α = ∂∂t n +1 ,α and B n +1 ,α = ( L n +1 N R α ) + .Taking a dressing operator φ = φ ( A∂ N ) = ∞ P i =1 α i ( A∂ N ) − i ( α = I, φ ∗ = φ − ), then thewave function of the mcCGD hierarchy is W ( t, z ) = φe ξ ( t,z ) = ω ( t, z ) e ξ ( t,z ) , where ξ ( t, z ) = ∞ X i =1 n X α =1 t i − ,α E α z i − . In addition, the adjoint wave function of the mcCGD hierarchy is W ∗ ( t, z ) = ( φ ∗ ) − e − ξ ( t,z ) = φe − ξ ( t,z ) . Then we can get the Sato equations ∂ n +1 ,α φ = − ( B n +1 ,α ) − φ (2 n + 1 = 0 mod N ) and n P α =1 ∂ (2 j ′ +1) N,α φ = 0 ( j ′ = 0 , , , · · · ) of the mcCGD hierarchy.And the Lax equations of the mcCGD hierarchy can also be obtained by the compatibilityconditions of the following linear partial differential equations L n +1 N W = z n +1 W, ∂ n +1 ,α W = B n +1 ,α W, n + 1 = 0 mod N, n X α =1 ∂ (2 j ′ +1) N,α W = L (2 j ′ +1) W, j ′ = 0 , , , · · · , (4.6)where W ( t, z ) is the wave function of the mcCGD hierarchy. .1 Additional symmetry of the mcCGD hierarchies From the above knowledge points related to the mcCGD hierarchy, we can find thatmany of its definitions and properties are similar to those of the mcBGD hierarchy, andthe biggest difference lies in the related definitions and operations of their adjoint. Next,we mainly discuss the knowledge of adjoint in detail. Similarly, we first give the Orlov-Shulman’s operator M = φ Γ φ − of the mcCGD hierarchy, whereΓ = ∞ X i =1 n X α =1 (2 i − t i − ,α A i − N ∂ i − E α , i − = 0 mod N, then we can get M ∗ = ( φ Γ φ − ) ∗ = φ Γ φ − = M. After some necessary knowledge reserve, we will give the definition of the additional sym-metry of the mcCGD hierarchy. The C l,m,α in the additional symmetry of the mcCGDhierarchy is obviously different from the D l,m,α in the definition of the additional symme-try of the mcBGD hierarchy. The fundamental reason is that their Lax operators havedifferent constraints. Proposition 4.2.
In the C type multi-component Gelfand-Dickey hierarchy, the addi-tional symmetric generators C l,m,α should satisfy ( C l,m,α ) ∗ = − C l,m,α , then C l,m,α can betaken as C l,m,α = M m L lN R α − ( − l R α L lN M m . (4.7) Proof.
According to the the additional symmetry of the mcCGD hierarchy, we can knowthat ∂ ∗ l,m,α φ ∗ = ( ∂ ∗ l,m,α φ ) ∗ = ( − ( C l,m,α ) − φ ) ∗ = − φ ∗ ( C l,m,α ) ∗− , and according to φ ∗ = φ − , we can deduce that ∂ ∗ l,m,α φ ∗ = ∂ ∗ l,m,α φ − = − φ − ( ∂ ∗ l,m,α φ ) φ − = φ − ( C l,m,α ) − = φ ∗ ( C l,m,α ) − , by comparing the results of the above two different methods, we can find that( C l,m,α ) − = − ( C l,m,α ) ∗− , so C l,m,α = − ( C l,m,α ) ∗ . Combined with the additional symmetric generators M m L lN R α of the mcGD hierarchy,we will have ( M m L lN R α ) ∗ = R ∗ α ( L lN ) ∗ ( M m ) ∗ = ( − l R α L lN M m , then it’s easy to construct C l,m,α = M m L lN R α − ( − l R α L lN M m . Definition 4.3.
The solution of the differential equation ∂φ∂ ∗ l,m,α = − ( C l,m,α ) − φ, (4.8) where ∂ ∗ l,m,α = ∂∂t ∗ l,m,α , C l,m,α = M m L lN R α − ( − l R α L lN M m , (4.9) is called the additional symmetry of the C type multi-component Gelfand-Dickey hierarchy. imilarly, for the differential operator C l,m,α , assuming its negative part disappearsand let the operator ( C l,m,α ) + act on W , we can get an equation related to z ( C l,m,α ) + W = (1 − ( − l ) z l E α ∂ mz W − ( − l mlz l − E α ∂ m − z W. (4.10)Note:This system can also be rewritten into a linear equation for the isomonodromyproblem.The following is a brief description of the construction method of C l,m,α . According tothe additional symmetry of the mcCGD hierarchy, we can deduce ∂ ∗ l,m,α L = − [( C l,m,α ) − , L ] , ∂ ∗ l,m,α R β = − [( C l,m,α ) − , R β ] . (4.11)For the additional flows of the mcCGD hierarchy, they have a fraction of the flows thatcan survive. Theorem 4.4.
In the additional flows of the C type multi-component Gelfand-Dickeyhierarchy, the flows which satisfy the condition ( M m − L N + l − N ) − = 0 and are shaped like n P α =1 ∂ ∗ l,m,α or the condition l = 2 i ( i ∈ Z ) and are shaped like n P α =1 ∂ ∗ l,m,α can survive.Proof. For the Lax operator L of the mcCGD hierarchy, its negative part is also equal tozero.And we know ∂ ∗ l,m,α L = − [( C l,m,α ) − , L ], so let’s consider ∂ ∗ l,m,α L − = − [( C l,m,α ) − , L ] − = − ( φ [Γ m A lN ∂ l E α , A∂ N ] φ − ) − + ( − l ( φ [ E α A lN ∂ l Γ m , A∂ N ] φ − ) − , after some deductions, the upper formula can be reduced to ∂ ∗ l,m,α L − = ∂ ∗ l,m,α L − = (1 − ( − l ) mN ( M m − L N + l − N R α ) − , only when ( M m − L N + l − N ) − = 0 or l = 2 i ( i ∈ Z ), we can obtain n X α =1 ∂ ∗ l, ,α L − = (1 − ( − l ) mN ( M m − L N + l − N ) − = 0 . The surviving additional flows of the mcCGD hierarchy satisfy the following proposi-tions.
Proposition 4.5.
The additional symmetric flows n P α =1 ∂ ∗ l,m,α which satisfy the condition ( M m − L N + l − N ) − = 0 or the condition l = 2 i ( i ∈ Z ) commute with the flows ∂ k +1 ,β (2 k +1 = 0 mod N ) of the C type multi-component Gelfand-Dickey hierarchy.Proof. (cid:2) n X α =1 ∂ ∗ l,m,α , ∂ k +1 ,β (cid:3) φ = (cid:16) n X α =1 (cid:2) ∂ ∗ l,m,α , ∂ k +1 ,β (cid:3)(cid:17) φ = n X α =1 (cid:16)(cid:2) ∂ ∗ l,m,α , ∂ k +1 ,β (cid:3) φ (cid:17) , ∂ ∗ l,m,α , ∂ k +1 ,β ] φ = − ∂ ∗ l,m,α (( B k +1 ,β ) − φ ) + ∂ k +1 ,β (( C l,m,α ) − φ )=[( C l,m,α ) − , ( B k +1 ,β ) − + ∂ k +1 ,β ] − φ + ( B k +1 ,β ) − ( C l,m,α ) − φ + [ ∂ k +1 ,β , ( C l,m,α ) − ] − φ − ( C l,m,α ) − ( B k +1 ,β ) − φ =[( C l,m,α ) − , ( B k +1 ,β ) − ] φ − [( C l,m,α ) − , ( B k +1 ,β ) − ] − φ =0 , then (cid:2) n P α =1 ∂ ∗ l,m,α , ∂ k +1 ,β (cid:3) φ = 0 , so the proposition is proved.After some calculations, we find that whether the odd mcCGD hierarchy or the evenmcCGD hierarchy, their string equations are the same. The derivation process is givenbelow. First we consider a additional flow n X α =1 ∂ ∗− (2 k − , ,α L kN = − n X α =1 (cid:2) ( C − (2 k − , ,α ) − , L kN (cid:3) , k = 0 mod N, after a series of calculations, we will obtain n X α =1 ∂ ∗− (2 k − , ,α L kN = n X α =1 (cid:2) ( C − (2 k − , ,α ) + , L kN (cid:3) + 4 kI, When 2 k = 0 mod N , L kN is a differential operator, then n X α =1 (cid:2) L kN , k ( C − (2 k − , ,α ) + (cid:3) = I, thus h L kN , k (cid:0) n P α =1 ( C − (2 k − , ,α ) + (cid:1)i = I is the string equation of the mcCGD hierarchy.In some literatures, the string equation is described as the form of [ P, Q ] = 1, where P and Q are differential operators, which is equivalent to the string equation derived by us,and refers to the condition that the operator is independent of variables. Acknowledgements:
Chuanzhong Li is supported by the National Natural ScienceFoundation of China under Grant No. 11571192 and K. C. Wong Magna Fund in NingboUniversity.
References [1] I. M. Gelfand and L. A. Dickey, Fractional powers of operators and Hamiltoniansystems, Functional Analysis and its Applications, 10(1976), 259-273.[2] L. A. Dickey, Additional symmetries of KP, grassmannian, and the string equation,Modern Physics Letters A, 8(1993), 1259-1272.[3] Z. Zheng, J. S. He and Y. Cheng, B¨ a cklund transformation of the noncommutativeGelfand-Dickey hierarchy, Journal of High Energy Physics, 2(2003), 69.[4] J. Figueroa-O’Farrill and E. Ramos, W -superalgebras from supersymmetric Laxoperators, Physics Letters B, 262(1991), 265-270.
5] C. Z. Li, Symmetries and Reductions on the noncommutative Kadomtsev-Petviashvili and Gelfand-Dickey hierarchies, Journal of Mathematical Physics,59(2018), 123503.[6] L. Feh´ e r and I. Marshall, Extensions of the matrix Gelfand-Dickey hierarchy fromgeneralized Drinfeld-Sokolov reduction, Communications in Mathematical Physics,183(1997), 423-461.[7] L. Haine and P. Iliev, The bispectral property of a q -deformation of the Schurpolynomials and the q -KdV hierarchy, Journal of Physics A: Mathematical andGeneral, 30(1997), 7217.[8] P. Etingof, I. M. Gelfand and V. Retakh, Factorization of differential operators,quasideterminants, and nonabelian Toda field equations, Mathematical ResearchLetters, 5(1997), 413-425.[9] J. S. He, Y. H. Li and Y. Cheng, q -deformed Gelfand-Dickey hierarchy and the deter-minant representation of its gauge transformation, Chinese Annals of Mathematicsseries A, 3(2004), 373-382.[10] J. L. Miramontes, τ -Functions generating the Conservation Laws for GeneralizedIntegrable Hierarchies of KdV and Affine-Toda type, Nuclear Physics B, 547(2012),623-663.[11] S. P. Novikov, Theory of the string equation in the double-scaling limit of 1-matrixmodels, International Journal of Modern Physics B, 10(1996), 2249-2271.[12] M. A. Awada and S. J. Sin, The string difference equation of the d = 1 matrix modeland W ∞ symmetry of the KP hierarchy, International Journal of Modern PhysicsA, 7(1992), 12.[13] J. van de Leur, KdV type hierarchies, the string equation and W ∞ constraints,Journal of Geometry and Physics, 17(1995), 95-124.[14] S. Panda and S. Roy, The Lax operator approach for the Virasoro and the W -constraints in the generalized KdV hierarchy, International Journal of ModernPhysics A, 8(1993), 3457-3478.[15] H. Aratyn, E. Nissimov and S. Pacheva, Virasoro Symmetry of Constrained KPHierarchies, Physics Letters A, 228(1996), 164-175.[16] A. Y. Orlov and E. I. Shul’man, Additional symmetries for integrable and conformalalgebra representation, Letters in Mathematical Physics, 12(1986), 171-179.[17] L. A. Dickey, On additional symmetries of the KP hierarchy and Sato’s B¨ a cklundtransformation, Communications in Mathematical Physics, 167(1995), 227-233.[18] H. Aratyn, E. Nissimov and S. Pacheva, Supersymmetric Kadomtsev-Petviashvilihierarchy: “ Ghost ” symmetry structure, reductions, and Darboux-B¨ a cklund solu-tions, Journal of Mathematical Physics, 40(1999), 2922-2932.[19] P. G. Grinevich and A. Yu Orlov, Virasoro Action on Riemann Surfaces, Grassman-nians, det ∂ J and Segal-Wilson τ Function, Problems of Modern Quantum FieldTheory, (1989), 86-106.[20] A. Yu Orlov, Vertex operator, ¯ ∂ -problem, symmetries, variational identities andHamiltonian formalism for 2 + 1 integrable systems, Nonlinear and Turbulent Pro-cesses in Physics/ed, (1988).[21] A. Gerasimov, A. Marshakov and A. Mironov, Matrix models of two-dimensionalgravity and Toda theory, Nuclear Physics B, 357(1991), 565-618.[22] M. Kontsevich, Intersection theory on the moduli space of curves and matrix Airyfunction, Communications in Mathematical Physics, 147(1992), 1-23.
23] A. Yu. Orlov and E. I. Shul’man, Additional symmetries of the nonlinear Schr¨ o dingerequation, Theoretical and Mathematical Physics, 64(1986), 862-865.[24] T. Tsuda, From KP/UC hierarchies to Painlev´ e equations, International Journal ofMathematics, 23(2012), 1250010.equations, International Journal ofMathematics, 23(2012), 1250010.