A strongly coupled extended Toda hierarchy and its Virasoro symmetry
aa r X i v : . [ n li n . S I] J u l A STRONGLY COUPLED EXTENDED TODA HIERARCHY AND ITSVIRASORO SYMMETRY
CHUANZHONG LI
Abstract.
As a generalization of the integrable extended Toda hierarchy and a reductionof the extended multicomponent Toda hierarchy, from the point of a commutative subalgebraof gl (2 , C ), we construct a strongly coupled extended Toda hierarchy(SCETH) which will beproved to possess a Virasoro type additional symmetry by acting on its tau-function. Further wegive the multi-fold Darboux transformations of the strongly coupled extended Toda hierarchy. Mathematics Subject Classifications (2010) : 37K05, 37K10, 37K20.
Key words : strongly coupled extended Toda hierarchy, Additional symmetry, Virasoro Lie algebra. Introduction
In the theory of integrable systems, two important fundamental models are the KP and Todasystems from which one can derive a lot of local and nonlocal integrable equations [1–5]. TheToda lattice hierarchy as a completely integrable system has many important applications inmathematics and physics including the representation theory of Lie algebras, orthogonal poly-nomials and random matrix models [6–10]. The Toda system has many kinds of reductionsor extensions such as the extended Toda hierarchy (ETH) [11, 12], bigraded Toda hierarchy(BTH) [13]- [17] and so on. Considering the application in the Gromov-Witten theory, theToda hierarchy was extended to the extended Toda hierarchy [11] which governs the Gromov-Witten invariants of C P . The extended bigraded Toda hierarchy(EBTH) is the extension ofthe bigraded Toda hierarchy (BTH) which includes additional logarithmic flows [13] with con-sidering its application in the Gromov-Witten theory of orbifolds C N,M . In [18], the extendedflow equations of the multi-component Toda hierarchy were constructed. Meanwhile the Dar-boux transformation and bi-Hamiltonian structure of this new extended multi-component Todahierarchy(EMTH) were given. We considered the Hirota quadratic equation of the commuta-tive subhierarchy of the extended multi-component Toda hierarchy which might be useful inthe theory of Frobenius manifolds in [19]. In [19], we constructed the extended flow equationsof a new Z N -Toda hierarchy which took values in a commutative subalgebra Z N of gl ( N, C ).Meanwhile we gave the Hirota bilinear equations and tau functions of the hierarchy which mightbe useful in the topological field theory and Gromov-Witten theory.The additional symmetry is a universal property for integrable systems [20–22]. Among thealgebraic structures of additional symmetries, the Virasoro symmetry is one kind of importantsymmetries such as [23]. In [24], we provided a kind of Block type algebraic structures for thebigraded Toda hierarchy (BTH) [14, 16]. Later on, this Block type Lie algebra was found againin the dispersionless bigraded Toda hierarchy [17].It was pointed out that the Darboux transformation was an efficient method to generatesoliton solutions of integrable equations. The multi-solitons can be obtained by this Darbouxtransformation from a trivial seed solution.This paper will be arranged as follows. In the next section we recall the extended multi-component Toda hierarchy. In Section 3, we will give the additional symmetry of the extended ulticomponent Toda hierarchy which constitutes a Virasoro type Lie algebra. From the pointof a commutative reduction from Lie algebras, the strongly coupled extended Toda hierarchyis recalled in Section 4. The additional symmetry of the SCETH will be constructed in Section5 and this symmetry has a Virasoro type structure which includes the Virasoro algebra as asubalgebra. The Virasoro action on the tau function of the SCETH will be given in Section6. Further after that, we give its multi-fold Darboux transformations of the strongly coupledextended Toda hierarchy.2. Extended multicomponent Toda hierarchy
In this section by following [18], we will denote G m as a group which contains invertibleelements of N × N complex matrices and denote its Lie algebra g m as the associative algebraof N × N complex matrices M N ( C )(Λ) where the shift operator Λ acting on any functions g ( x )as (Λ g )( x ) := g ( x + ǫ ).In this section, firstly let us recall the basic notation of the extended multicomponent Todahierarchy defined in [18].In [18], we define the dressing operators W, ¯ W as follows W := S · W , ¯ W := ¯ S · ¯ W , (2.1)where S, ¯ S have expansions of the form S = I N + ω ( x )Λ − + ω ( x )Λ − + · · · ∈ G m − , ¯ S = ¯ ω ( x ) + ¯ ω ( x )Λ + ¯ ω ( x )Λ + · · · ∈ G m + , (2.2)and G m − and G m + are two subgroups of G m . The free operators W , ¯ W ∈ G m have forms as W := N X k =1 E kk exp ∞ X j =0 t jk Λ j + s j Λ j j ! ( ǫ∂ − c j ) ! , (2.3)¯ W := N X k =1 E kk exp ∞ X j =0 ¯ t jk Λ − j + s j Λ − j j ! ( ǫ∂ − c j ) ! , (2.4)with t jk , ¯ t jk , s j as continuous times. Also we define the symbols of S, ¯ S as S , ¯ SS = I N + ω ( x ) λ − + ω ( x ) λ − + · · · , ¯ S = ¯ ω ( x ) + ¯ ω ( x ) λ + ¯ ω ( x ) λ + · · · . (2.5)Also the inverse operators S − , ¯ S − of operators S, ¯ S have expansions of the form S − = I N + ω ′ ( x )Λ − + ω ′ ( x )Λ − + · · · ∈ G m − , ¯ S − = ¯ ω ′ ( x ) + ¯ ω ′ ( x )Λ + ¯ ω ′ ( x )Λ + · · · ∈ G m + . (2.6)Also we define the symbols of S − , ¯ S − as S − , ¯ S − S − = I N + ω ′ ( x ) λ − + ω ′ ( x ) λ − + · · · , ¯ S − = ¯ ω ′ ( x ) + ¯ ω ′ ( x ) λ + ¯ ω ′ ( x ) λ + · · · . (2.7)The Lax operators L, C kk , ¯ C kk ∈ g m are defined by L := W · Λ · W − = ¯ W · Λ − · ¯ W − , (2.8) C kk := W · E kk · W − , ¯ C kk := ¯ W · E kk · ¯ W − , (2.9) nd have the following expansions L = Λ + u ( x ) + u ( x )Λ − ,C kk = E kk + C kk, ( x )Λ − + C kk, ( x )Λ − + · · · , ¯ C kk = ¯ C kk, ( x ) + ¯ C kk, ( x )Λ + ¯ C kk, ( x )Λ + · · · . (2.10)In fact the Lax operators L, C kk , ¯ C kk ∈ g m can also be equivalently defined by L := S · Λ · S − = ¯ S · Λ − · ¯ S − , (2.11) C kk := S · E kk · S − , ¯ C kk := ¯ S · E kk · ¯ S − . (2.12)The matrix operators B jk , ¯ B jk , D j are defined as follows B jk := W E kk Λ j W − , ¯ B jk := ¯ W E kk Λ − j ¯ W − ,D j := 2 L j j ! (log L − c j ) , c = 0; c j = j X i =1 i , j ≥ . (2.13)To define extended flows of the extended multi-component Toda hierarchy(EMTH), we definethe following logarithmic matrices [18]log + L = ( S · ǫ∂ · S − ) = ǫ∂ + X k< W k ( x )Λ k , (2.14)log − L = − ( ¯ S · ǫ∂ · ¯ S − ) = − ǫ∂ + X k ≥ W k ( x )Λ k , (2.15)where ∂ is the derivative with respect to the spatial variable x . Combining these above loga-rithmic operators together can help us in deriving the following important logarithmic matrixlog L : = 12 log + L + 12 log − L = 12 ( S · ǫ∂ · S − − ¯ S · ǫ∂ · ¯ S − ) , (2.16)which will generate a series of extended flow equations contained in the following Lax equations. Proposition 2.1.
The Lax equations of the EMTH are as follows ǫ∂ t jk L = [( B jk ) + , L ] , ǫ∂ t jk C ss = [ − ( B jk ) − , C ss ] , ǫ∂ t jk ¯ C ss = [( B jk ) + , ¯ C ss ] , (2.17) ǫ∂ ¯ t jk L = [( ¯ B jk ) + , L ] , ǫ∂ ¯ t jk C ss = [ − ( ¯ B jk ) − , C ss ] , ǫ∂ ¯ t jk ¯ C ss = [( ¯ B jk ) + , ¯ C ss ] , (2.18) ǫ∂ s j L = [( D j ) + , L ] , ǫ∂ s j C ss = [ − ( D j ) − , C ss ] , ǫ∂ s j ¯ C ss = [( D j ) + , ¯ C ss ] . (2.19)3. Strongly coupled extended Toda hierarchy
In this section, we firstly construct a strongly coupled extended Toda hierarchy. The algebrahas a maximal symmetric commutative subalgebra S = C [Γ] and Γ = (cid:20) (cid:21) ∈ gl (2 , C ) . Denote Λ as a shift operator by acting on any function f ( x ) as Λ f ( x ) = f ( x + ǫ ), and analgebra S (Λ) := g , then the algebra g has the following splitting g = g + ⊕ g − , (3.1)where g + = n X j ≥ X j ( x )Λ j , X j ( x ) ∈ S o , g − = n X j< X j ( x )Λ j , X j ( x ) ∈ S o . he splitting (3.1) leads us to consider the following factorization of g ∈ Gg = g − − ◦ g + , g ± ∈ G ± , (3.2)where G ± have g ± as their Lie algebras. G + is the set of invertible linear operators of the form P j ≥ g j ( x )Λ j ; while G − is the set of invertible linear operators of the form I + P j< g j ( x )Λ j .Now we introduce the following free operators W , ¯ W ∈ G W := exp ∞ X j =0 t j Λ j ǫj ! + y j Λ j ǫj ! ( ǫ∂ − c j ) , ∂ = ∂∂x , (3.3)¯ W := exp ∞ X j =0 t j Λ − j ǫj ! + y j Λ − j ǫj ! ( ǫ∂ − c j ) , c j = j X i =1 i , (3.4)where t j , y j ∈ C will play the role of continuous times.We define the dressing operators W , ¯ W as follows W := P ◦ W , ¯ W := ¯ P ◦ ¯ W , P ∈ G − , ¯ P ∈ G + . (3.5)Given an element g ∈ G and denote t = ( t j ) , y = ( y j ); j ∈ N , one can consider the factorizationproblem in G W ◦ g = ¯ W , (3.6)i.e. the factorization problem P ( t, y ) ◦ W ◦ g = ¯ P ( t, y ) ◦ ¯ W . (3.7)Observe that P, ¯ P have expansions of the form P = I + ω ( x )Λ − + ω ( x )Λ − + · · · ∈ G − , ¯ P = ¯ ω ( x ) + ¯ ω ( x )Λ + ¯ ω ( x )Λ + · · · ∈ G + . (3.8)Also we define the symbols of P, ¯ P as P , ¯ PP = I + ω ( x ) λ − + ω ( x ) λ − + · · · = (cid:20) P P P P (cid:21) , ¯ P = ¯ ω ( x ) + ¯ ω ( x ) λ + ¯ ω ( x ) λ + · · · = (cid:20) ¯ P ¯ P ¯ P ¯ P (cid:21) . (3.9)The inverse operators P − , ¯ P − of operators P, ¯ P have expansions of the form P − = I + ω ′ ( x )Λ − + ω ′ ( x )Λ − + · · · ∈ G − , ¯ P − = ¯ ω ′ ( x ) + ¯ ω ′ ( x )Λ + ¯ ω ′ ( x )Λ + · · · ∈ G + . (3.10)Also we define the symbols of P − , ¯ P − as P − , ¯ P − P − = I + ω ′ ( x ) λ − + ω ′ ( x ) λ − + · · · = (cid:20) P P P P (cid:21) , ¯ P − = ¯ ω ′ ( x ) + ¯ ω ′ ( x ) λ + ¯ ω ′ ( x ) λ + · · · = (cid:20) ¯ P ¯ P ¯ P ¯ P (cid:21) . (3.11)The Lax operators L ∈ G are defined by L := W ◦ Λ ◦ W − = ¯ W ◦ Λ − ◦ ¯ W − = P ◦ Λ ◦ P − = ¯ P ◦ Λ − ◦ ¯ P − , (3.12) nd have the following expansions L = Λ + u ( x ) + v ( x )Λ − = (cid:20) Λ + u ( x ) + v ( x )Λ − u ( x ) + v ( x )Λ − u ( x ) + v ( x )Λ − Λ + u ( x ) + v ( x )Λ − (cid:21) . (3.13)Now we define the following two logarithm matriceslog + L = W ◦ ǫ∂ ◦ W − = P ◦ ǫ∂ ◦ P − , (3.14)log − L = − ¯ W ◦ ǫ∂ ◦ ¯ W − = − ¯ P ◦ ǫ∂ ◦ ¯ P − . (3.15)Combining these above logarithm operators together can derive following important loga-rithm matrixlog L : = 12 (log + L + log − L ) = 12 ( P ◦ ǫ∂ ◦ P − − ¯ P ◦ ǫ∂ ◦ ¯ P − ) := + ∞ X i = −∞ W i Λ i ∈ G, (3.16)which will generate a series of flow equations which contain the spatial flow in later defined Laxequations.Let us first introduce some convenient notations of S -valued matrix operators B j , F j asfollows B j := L j +1 ( j + 1)! , F j := 2 L j j ! (log L − c j ) , c j = j X i =1 i , j ≥ . (3.17)Now we give the definition of the strongly coupled extended Toda hierarchy(SCETH). Definition 3.1.
The strongly coupled extended Toda hierarchy is a hierarchy in which thedressing operators P, ¯ P satisfy the following Sato equations ǫ∂ t j P = − ( B j ) − P, ǫ∂ t j ¯ P = ( B j ) + ¯ P , (3.18) ǫ∂ y j P = − ( F j ) − P, ǫ∂ y j ¯ P = ( F j ) + ¯ P , (3.19) which is equivalent to that the dressing operators W, ¯ W are subject to the following Sato equa-tions ǫ∂ t j W = ( B j ) + W, ǫ∂ t j ¯ W = ( B j ) + ¯ W , (3.20) ǫ∂ y j W = ( L j j ! (log + L − c j ) − ( F j ) − ) W, ǫ∂ y j ¯ W = ( − L j j ! (log − L − c j ) + ( F j ) + ) ¯ W . (3.21)From the previous proposition we derive the following Lax equations for the Lax operators.
Proposition 3.2.
The Lax equations of the SCETH are as follows ǫ∂ t j L = [( B j ) + , L ] , ǫ∂ y j L = [( F j ) + , L ] , ǫ∂ t j log L = [( B j ) + , log L ] , (3.22) ǫ (log L ) y j = [ − ( F j ) − , log + L ] + [( F j ) + , log − L ] . (3.23) .1. Strongly coupled extended Toda equations.
As a consequence of the factorizationproblem (3.6) and Sato equations, after taking into account that S ∈ G − and ¯ S ∈ G + , the t flow of L in the form of L = Λ + U + V Λ − is as ǫ∂ t L = [Λ + U, V Λ − ] , (3.24)which lead to a strongly coupled Toda equation ǫ∂ t U = V ( x + ǫ ) − V ( x ) , (3.25) ǫ∂ t V = U ( x ) V ( x ) − V ( x ) U ( x − ǫ ) . (3.26)Of course, one can switch the order of the matrices because of the commutativity of S . Suppose U = (cid:20) u u u u (cid:21) , V = (cid:20) v v v v (cid:21) , (3.27)then the specific strongly coupled Toda equation is ǫ∂ t u = v ( x + ǫ ) − v ( x ) , (3.28) ǫ∂ t u = v ( x + ǫ ) − v ( x ) , (3.29) ǫ∂ t v = u ( x ) v ( x ) + u ( x ) v ( x ) − v ( x ) v ( x − ǫ ) − v ( x ) v ( x − ǫ ) , (3.30) ǫ∂ t v = ( u ( x ) − u ( x − ǫ )) v ( x ) − v ( x )( u ( x ) − u ( x − ǫ ) . (3.31)To get the standard strongly coupled Toda equation, one need to use the alternative expressions U := ω ( x ) − ω ( x + ǫ ) = ǫ∂ t φ ( x ) ,U := ¯ ω ( x ) − ¯ ω ( x + ǫ ) = ǫ∂ t φ ( x ) ,V = e φ ( x ) − φ ( x − ǫ ) cosh( φ ( x ) − φ ( x − ǫ )) = − ǫ∂ t ω ( x ) ,V = e φ ( x ) − φ ( x − ǫ ) sinh( φ ( x ) − φ ( x − ǫ )) = − ǫ∂ t ¯ ω ( x ) . (3.32)From Sato equations we deduce the following set of nonlinear partial differential-differenceequations ω ( x ) − ω ( x + ǫ ) = ǫ∂ t φ ( x ) , ¯ ω ( x ) − ¯ ω ( x + ǫ ) = ǫ∂ t φ ( x ) ,ǫ∂ t ω ( x ) = − e φ ( x ) − φ ( x − ǫ ) cosh( φ ( x ) − φ ( x − ǫ )) ǫ∂ t ¯ ω ( x ) = − e φ ( x ) − φ ( x − ǫ ) sinh( φ ( x ) − φ ( x − ǫ )) . (3.33)Observe that if we cross the above equations, then we get the following strongly coupled Todasystem ǫ ∂ t φ ( x ) = e φ ( x + ǫ ) − φ ( x ) cosh( φ ( x + ǫ ) − φ ( x )) − e φ ( x ) − φ ( x − ǫ ) cosh( φ ( x ) − φ ( x − ǫ )) ,ǫ ∂ t φ ( x ) = sinh( φ ( x + ǫ ) − φ ( x )) e φ ( x + ǫ ) − φ ( x ) − sinh( φ ( x ) − φ ( x − ǫ )) e φ ( x ) − φ ( x − ǫ ) . Besides above strongly coupled Toda equations, with logarithm flows the SCETH also con-tains some extended flow equations in the next part. Here we consider the extended flowequations in the simplest case, i.e. the y flow for L ,ǫ∂ y L = [( Sǫ∂ x S − ) + , L ] (3.34)= [ ǫ∂ x SS − , L ] (3.35)= ǫ L x , (3.36) hich leads to the following specific equation ∂ y U = U x , ∂ y V = V x . (3.37)To see the extended equations clearly, one need to rewrite the extended flows in the Laxequations of the SCETH as in the following lemma. Lemma 3.3.
The extended flows in Lax formulation of the SCETH can be equivalently givenby ǫ ∂ L ∂y j = [ D j , L ] , (3.38) D j = ( L j j ! (log + L − c j )) + − ( L j j ! (log − L − c j )) − , (3.39) which can also be rewritten in the form ǫ ∂ L ∂y n = [ ¯ D n , L ] , (3.40)¯ D j = L j j ! ǫ∂ + [ L j j ! ( X k< W k ( x )Λ k − c j )] + − [ L j j ! ( X k ≥ W k ( x )Λ k − c j )] − . Then one can derive the y flow equation of the SCETH as ǫu y = ǫ (1 − Λ)( v (Λ − − − (log( v + v ) + log( v − v )) x ) − − v + ǫ (1 − Λ)( v (Λ − − − (log( v + v ) − log( v − v )) x ) + ǫ u x + ǫ u x + ǫv x , (3.41) ǫu y = ǫ (1 − Λ)( v (Λ − − − (log( v + v ) − log( v − v )) x ) − − v + ǫ (1 − Λ)( v (Λ − − − (log( v + v ) + log( v − v )) x ) + ǫu x u x + ǫv x , (3.42) ǫv y = ((Λ − − − ǫ (log( v + v ) + log( v − v )) x ) + 2)[( u ( x − ǫ ) − u ( x )) v + ( u ( x − ǫ ) − u ( x )) v ](Λ − − − ǫ (log( v + v ) − log( v − v )) x )[( u ( x − ǫ ) − u ( x )) v + ( u ( x − ǫ ) − u ( x )) v ]+ ǫv x u ( x − ǫ ) + ǫv x u ( x − ǫ ) + ǫ ( u x ( x − ǫ ) + u x ( x )) v + ǫ ( u x ( x − ǫ ) + u x ( x )) v ,ǫv y = ((Λ − − − ǫ (log( v + v ) − log( v − v )) x ) + 2)[( u ( x − ǫ ) − u ( x )) v + ( u ( x − ǫ ) − u ( x )) v ](Λ − − − ǫ (log( v + v ) + log( v − v )) x )[( u ( x − ǫ ) − u ( x )) v + ( u ( x − ǫ ) − u ( x )) v ]+ ǫv x u ( x − ǫ ) + ǫv x u ( x − ǫ ) + ǫ ( u x ( x − ǫ ) + u x ( x )) v + ǫ ( u x ( x − ǫ ) + u x ( x )) v , where u i , v i without bracket behind them means u i ( x ) , v i ( x ) respectively.4. Virasoro symmetries of the strongly coupled extended Toda hierarchy
In this section, we will put constrained condition eq.(3.12) into a construction of the flows ofadditional symmetries which form the well-known Virasoro algebra.With the dressing operators given in eq.(3.12), we introduce Orlov-Schulman operators asfollowing M = P Γ P − , ¯ M = ¯ P ¯Γ ¯ P − , (4.1)Γ = xǫ Λ − + X n ≥ n Λ n − t n + X n ≥ n − n − ( ǫ∂ − c n − ) y n , (4.2) Γ = − xǫ Λ − X n ≥ n − − n +1 ( − ǫ∂ − c n ) y n . (4.3)Therefore we can get M − ¯ M = P xǫ Λ − P − + X n ≥ nt n B n − + ¯ P xǫ
Λ ¯ P − + X n ≥ n − n − (log L − c n ) y n , is a pure difference operator.Then one can prove the Lax operator L and Orlov-Schulman operators M , ¯ M satisfy thefollowing theorem. Proposition 4.1.
The S -valued Lax operator L and Orlov-Schulman operators M , ¯ M of theSCETH satisfy the following [ L , M ] = L , [ L , ¯ M ] = 1 , [log + L , M ] = S Λ − S − , [log − L , ¯ M ] = ¯ P Λ ¯ P − , (4.4) ǫ∂ t k M m L k = [( B k ) + , M m L k ] , ǫ∂ ¯ t k M m L k = [ − ( ¯ B k ) − , M m L k ] , (4.5) ǫ∂ t k ¯ M m L k = [( B k ) + , ¯ M m L k ] , ǫ∂ ¯ t k ¯ M m L k = [ − ( ¯ B k ) − , ¯ M m L k ] , (4.6) ∂ M m L k ∂y j = [ L j j ! (log + L − c j ) − ( F j ) − , M m L k ] , ∂ ¯ M m L k ∂y j = [ − L j j ! (log − L − c j ) + ( F j ) + , ¯ M m L k ] . (4.7) Proof.
One can prove the proposition by dressing the following several commutative Lie brackets[ ∂ t n − Λ n , Γ]= [ ∂ t n − Λ n , xǫ Λ − + X n ≥ n Λ n − t n + X n ≥ n − n − ( ǫ∂ − c n − ) y n ]= 0 , [ ∂ y n − n ! Λ n ( ǫ∂ − c n ) , Γ]= [ ∂ y n − n ! Λ n ( ǫ∂ − c n ) , xǫ Λ − + X n ≥ n Λ n − t n + X n ≥ n − n − ( ǫ∂ − c n − ) y n ]= 0 , [ ∂ y n + 1 n ! Λ − n ( − ǫ∂ − c n ) , ¯Γ]= [ ∂ y n + 1 n ! Λ − n ( − ǫ∂ − c n ) , − xǫ Λ − X n ≥ n Λ − n +1 ¯ t n − X n ≥ n − − n +1 ( − ǫ∂ − c n ) y n ]= 0 . (cid:3) We are now to define the additional flows, and then to prove that they are symmetries, whichare called additional symmetries of the SCETH. We introduce additional independent variables t ∗ l and define the actions of the additional flows on the wave operators as ∂P∂t ∗ l = − (cid:0) ( M − ¯ M ) L l (cid:1) − P, ∂ ¯ P∂t ∗ l = (cid:0) ( M − ¯ M ) L l (cid:1) + ¯ P , (4.8) here m ≥ , l ≥ Proposition 4.2.
The additional derivatives act on M , ¯ M as ∂ M ∂t ∗ l = [ − (cid:0) ( M − ¯ M ) L l (cid:1) − , M ] , (4.9) ∂ ¯ M ∂t ∗ l = [ (cid:0) ( M − ¯ M ) L l (cid:1) + , ¯ M ] . (4.10)By the propositions above, we can find for n, k, l ≥
0, the following identities hold ∂ M n L k ∂t ∗ l = − [(( M − ¯ M ) L l ) − , M n L k ] , ∂ ¯ M n L k ∂t ∗ l = [(( M − ¯ M ) L l ) + , ¯ M n L k ] . (4.11)Basing on above results, the following theorem can be proved. Theorem 4.3.
The additional flows ∂ t ∗ l commute with the SCETH flows, i.e., [ ∂ t ∗ l , ∂ t n ]Φ = 0 , [ ∂ t ∗ l , ∂ y n ]Φ = 0 , (4.12) where Φ can be P , ¯ P or L , ≤ γ ≤ N ; n ≥ and ∂ t ∗ l = ∂∂t ∗ l , ∂ t n = ∂∂t n .Proof. Here we also give the proof for commutativity of additional symmetries with the extendedflow ∂ y n . To be an example, we only let the Lie bracket act on ¯ P ,[ ∂ t ∗ l , ∂ y n ] ¯ P = ∂ t ∗ l ( F j ) + ¯ P − ∂ y n (cid:0) (( M − ¯ M ) L l ) + ¯ P (cid:1) = ∂ t ∗ l ( F j ) + ¯ P + ( F j ) + ( ∂ t ∗ l ) ¯ P − ( ∂ y n (( M − ¯ M ) L l )) + ¯ P − (( M − ¯ M ) L l ) + ( ∂ y n ¯ P ) , which further leads to[ ∂ t ∗ l , ∂ y n ] ¯ P = [(( M − ¯ M ) L l ) + , L j j ! (log − L − c j )] + ¯ P − [(( M − ¯ M ) L l ) − , L j j ! (log + L − c j )] + ¯ P +( F j ) + (( M − ¯ M ) L l ) + ¯ P − (( M − ¯ M ) L l ) + ( F j ) + ¯ P − [ (cid:18) L j j ! (log + L − c j ) (cid:19) + − (cid:18) L j j ! (log − L − c j ) (cid:19) − , ( M − ¯ M ) L l ] + ¯ P = [(( M − ¯ M ) L l ) + , ( L j j ! (log − L − c j )) + ] + ¯ P − [(( M − ¯ M ) L l ) − , L j j ! (log + L − c j )] + ¯ P +( F j ) + (( M − ¯ M ) L l ) + ¯ P − (( M − ¯ M ) L l ) + ( F j ) + ¯ P +[( M − ¯ M ) L l , (cid:18) L j j ! (log + L − c j ) (cid:19) + ] + ¯ P = 0 . The other cases in the theorem can be proved in similar ways. (cid:3) he commutative property in Theorem 4.3 means that additional flows are symmetries ofthe SCETH. As a special reduction from the EMTH to the SCETH, it is easy to derive thealgebraic structures among these additional symmetries in the following important theorem. Theorem 4.4.
The additional flows ∂ t ∗ l of the SCETH form a Virasoro type Lie algebra withthe following relation [ ∂ t ∗ l , ∂ t ∗ k ] = ( k − l ) ∂ ∗ k + l − , (4.13) which holds in the sense of acting on P , ¯ P or L and l, k ≥ . Virasoro action on tau-functions of SCETH
Introduce the following sequence: t − [ λ ] := ( t j − ǫ ( j − λ j , ≤ j ≤ ∞ ) . (5.1)A S -valued function (cid:20) τ σσ τ (cid:21) ∈ S depending only on the dynamical variables t and ǫ is calledthe S -valued tau-function of the SCETH if it provides symbols related to matrix-valued waveoperators as following, P : = τ ( t − [ λ − ]) τ ( t ) − σ ( t − [ λ − ]) σ ( t ) τ ( t ) − σ ( t ) , (5.2) P : = σ ( t − [ λ − ]) τ ( t ) − τ ( t − [ λ − ]) σ ( t ) τ ( t ) − σ ( t ) , (5.3) P : = τ ( x + ǫ, t + [ λ − ]) τ ( x + ǫ, t ) − σ ( x + ǫ, t + [ λ − ]) σ ( x + ǫ, t ) τ ( x + ǫ, t ) − σ ( x + ǫ, t ) , (5.4) P : = σ ( x + ǫ, t + [ λ − ]) σ ( x + ǫ, t ) − τ ( x + ǫ, t + [ λ − ]) τ ( x + ǫ, t ) τ ( x + ǫ, t ) − σ ( x + ǫ, t ) , (5.5)¯ P : = τ ( x + ǫ, t + [ λ ]) τ ( t ) − σ ( x + ǫ, t + [ λ ]) σ ( t ) τ ( t ) − σ ( t ) , (5.6)¯ P : = σ ( x + ǫ, t + [ λ ]) τ ( t ) − τ ( x + ǫ, t + [ λ ]) σ ( t ) τ ( t ) − σ ( t ) , (5.7)¯ P : = τ ( x, t − [ λ − ]) τ ( x + ǫ, t ) − σ ( x, t − [ λ − ]) σ ( x + ǫ, t ) τ ( x + ǫ, t ) − σ ( x + ǫ, t ) , (5.8)¯ P : = σ ( x, t − [ λ − ]) σ ( x + ǫ, t ) − τ ( x, t − [ λ − ]) τ ( x + ǫ, t ) τ ( x + ǫ, t ) − σ ( x + ǫ, t ) . (5.9)Then according to the ASvM formula in [25] and a commutative algebraic reduction, we canget the following formula( ∂ t ∗ k P ) P − ( ∂ t ∗ k P ) P P − P = ( e − P ∞ i =1 ǫ ( i − λ − i ∂ ti −
1) ( L k − τ ) τ − ( L k − σ ) στ − σ , (5.10)( ∂ t ∗ k P ) P − ( ∂ t ∗ k P ) P P − P = ( e − P ∞ i =1 ǫ ( i − λ − i ∂ ti −
1) ( L k − σ ) τ − ( L k − τ ) στ − σ , (5.11)( ∂ t ∗ k ¯ P )¯ P − ( ∂ t ∗ k ¯ P )¯ P ¯ P − ¯ P = ( e ǫ∂ x + P ∞ i =1 ǫ ( i − λ i ∂ ti −
1) ( L k − τ ) τ − ( L k − σ ) στ − σ , (5.12) ∂ t ∗ k ¯ P )¯ P − ( ∂ t ∗ k ¯ P )¯ P ¯ P − ¯ P = ( e ǫ∂ x + P ∞ i =1 ǫ ( i − λ i ∂ ti −
1) ( L k − σ ) τ − ( L k − τ ) στ − σ , (5.13)where L − = ∞ X n =1 ( t n − ∂ t n − + 2 c n − ( n − y n ∂ t n − − c n − y n ∂ t n − + y n ∂ y n − ) + t y , (5.14) L = ∞ X n =1 ( nt n − ∂ t n − + 2 c n ( n − y n ∂ t n − c n − y n ∂ t n − + ny n ∂ y n ) + y , (5.15) L p = ∞ X n =1 ( ( n + p )!( n − t n − ∂ t n + p − + 2 c n + p ( n − y n ∂ t n + p − n + p )!( n − c n − y n ∂ t n + p − + ( n + p )!( n − y n ∂ y n + p ) + 2 p ! y ∂ t p − + p − X n =1 n !( p − n )! ∂ t n − ∂ t p − n − , p ≥ . (5.16)These operators { L k , k ≥ − } constitute a Virasoro algebra [23, 26] (one half without thecental extension) as [ L m , L n ] = ( m − n ) L m + n . (5.17)The central extension appears only if we consider the action on the tau function as it was donein [27, 28]. 6. Multi-fold Darboux transformations of the SCETH
In this section, we will consider the Darboux transformation of the SCETH on the Laxoperator L [1] = W LW − , (6.1)where W is the Darboux transformation operator.That means after the Darboux transformation, the spectral problem Lφ = Λ φ + uφ + v Λ − φ = λφ, (6.2)will become L [1] φ [1] = Λ φ [1] + u [1] φ [1] + v [1] Λ − φ [1] = λφ [1] . (6.3)To keep the Lax equation of the SCETH invariant, i.e. The Lax equations of the SCETHare as follows ǫ∂ t j L [1] = [( B [1] j ) + , L [1] ] , ǫ∂ y j L [1] = [( F [1] j ) + , L [1] ] , (6.4) B [1] j := B j ( L [1] ) , F [1] j := F j ( L [1] ) , (6.5)the dressing operator W should satisfy the following equation W t j = − W ( B j ) + + ( W B j W − ) + W, W y j = − W ( F j ) + + ( W F j W − ) + W, j ≥ . (6.6)Now, we will give the following important theorem which will be used to generate newsolutions. heorem 6.1. If φ = (cid:20) φ φ φ φ (cid:21) is the first wave function of the SCETH, the Darboux trans-formation operator of the SCETH W ( λ ) = ( I − φ Λ − φ Λ − ) = φ ◦ ( I − Λ − ) ◦ φ − , (6.7) will generater new solutions u [1]0 = u + (Λ − I ) φ ( x ) φ ( x − ǫ ) − φ ( x ) φ ( x − ǫ ) φ ( x − ǫ ) − φ ( x − ǫ ) , (6.8) u [1]1 = u + (Λ − I ) φ ( x ) φ ( x − ǫ ) − φ ( x ) φ ( x − ǫ ) φ ( x − ǫ ) − φ ( x − ǫ ) , (6.9) v [1]0 = ( φ ( x ) φ ( x − ǫ ) + φ ( x ) φ ( x − ǫ ))( φ ( x − ǫ ) + φ ( x − ǫ )) − ( φ ( x ) φ ( x − ǫ ) + φ ( x ) φ ( x − ǫ ))(2 φ ( x − ǫ ) φ ( x − ǫ ))( φ ( x − ǫ ) + φ ( x − ǫ )) − (2 φ ( x − ǫ ) φ ( x − ǫ )) v ( x − ǫ )+ ( φ ( x ) φ ( x − ǫ ) + φ ( x ) φ ( x − ǫ ))( φ ( x − ǫ ) + φ ( x − ǫ )) − ( φ ( x ) φ ( x − ǫ ) + φ ( x ) φ ( x − ǫ ))(2 φ ( x − ǫ ) φ ( x − ǫ ))( φ ( x − ǫ ) + φ ( x − ǫ )) − (2 φ ( x − ǫ ) φ ( x − ǫ )) v ( x − ǫ ) ,v [1]1 = ( φ ( x ) φ ( x − ǫ ) + φ ( x ) φ ( x − ǫ ))( φ ( x − ǫ ) + φ ( x − ǫ )) − ( φ ( x ) φ ( x − ǫ ) + φ ( x ) φ ( x − ǫ ))(2 φ ( x − ǫ ) φ ( x − ǫ ))( φ ( x − ǫ ) + φ ( x − ǫ )) − (2 φ ( x − ǫ ) φ ( x − ǫ )) v ( x − ǫ )+ ( φ ( x ) φ ( x − ǫ ) + φ ( x ) φ ( x − ǫ ))( φ ( x − ǫ ) + φ ( x − ǫ )) − ( φ ( x ) φ ( x − ǫ ) + φ ( x ) φ ( x − ǫ ))(2 φ ( x − ǫ ) φ ( x − ǫ ))( φ ( x − ǫ ) + φ ( x − ǫ )) − (2 φ ( x − ǫ ) φ ( x − ǫ )) v ( x − ǫ ) . Define φ i = φ [0] i := φ | λ = λ i = (cid:20) φ i φ i φ i φ i (cid:21) , then after iteration on Darboux transformations, wecan generalize the Darboux transformation to the n -fold case.Taking seed solution u = u = 0 , v = 1 , v = 0, then after iteration on Darboux transfor-mations, one can get the n -th new solution of the SCETH as u [ n ] = 12 (1 − Λ − ) ∂ t , (log( τ n + σ n ) + log( τ n − σ n )) , (6.10) v [ n ] = e (1 − Λ − ) (log( τ n + σ n ) − log( τ n − σ n )) , (6.11) τ n = [ n ] X m =0 X P ni =1 α i =2 m W r ( φ ( α )1 , φ ( α )2 , . . . φ ( α n ) n ) , (6.12) σ n = [ n ] X m =0 X P ni =1 α i =2 m +1 W r ( φ ( α )1 , φ ( α )2 , . . . φ ( α n ) n ) , α i = 0 , φ (0) i = φ i , φ (1) i = φ i , (6.13)where W r ( φ ( α )1 , φ ( α )2 , . . . φ ( α n ) n ) is the Wronskian, i.e. a Casorati determinant W r ( φ ( α )1 , φ ( α )2 , . . . φ ( α n ) n ) = det (Λ − j +1 φ ( α n +1 − i ) n +1 − i ) ≤ i,j ≤ n , (6.14)Particularly for the SCETH, choosing appropriate wave function φ , the n -th new solutions canbe solitary wave solutions, i.e. n -soliton solutions. Acknowledgements:
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