Bilinear equations in Darboux transformations by Boson-Fermion correspondence
aa r X i v : . [ n li n . S I] J a n BILINEAR EQUATIONS IN DARBOUX TRANSFORMATIONS BYBOSON-FERMION CORRESPONDENCE
YI YANG, JIPENG CHENG ∗ School of Mathematics, China University of Mining and Technology,Xuzhou, Jiangsu 221116, P. R. China
Abstract.
Bilinear equation is an important property for integrable nonlinear evolution equation.Many famous research objects in mathematical physics, such as Gromov-Witten invariants, can bedescribed in terms of bilinear equations to show their connections with the integrable systems. Herein this paper, we mainly discuss the bilinear equations of the transformed tau functions under thesuccessive applications of the Darboux transformations for the KP hierarchy, the modified KP hi-erarchy (Kupershmidt-Kiso version) and the BKP hierarchy, by the method of the Boson-Fermioncorrespondence. The Darboux transformations are considered in the Fermionic picture, by multiplyingthe different Fermionic fields on the tau functions. Here the Fermionic fields are corresponding to the(adjoint) eigenfunctions, whose changes under the Darboux transformations are showed to be the onesof the squared eigenfunction potentials in the Bosonic picture, used in the spectral representationsof the (adjoint) eigenfunctions. Then the successive applications of the Darboux transformations aregiven in the Fermionic picture. Based upon this, some new bilinear equations in the Darboux chain arederived, besides the ones of ( l − l ′ ) -th modified KP hierarchy. The corresponding examples of thesenew bilinear equations are given. Keywords : bilinear equations; Darboux transformations; Boson-Fermion correspondence; tau func-tions; squared eigenfunction potential.
Contents
1. Introduction 22. Preliminaries on the KP, modified KP and BKP hierarchies 42.1. Boson-Fermion correspondence and ( l − l ′ )-th modified KP hierarchy 42.2. Basic facts on the KP hierarchy 72.3. The mKP hierarchy and the Miura links 92.4. The neutral free Fermions and the BKP hierarchy 132.5. Important relations on free Fermions 153. The Darboux transformations of the KP hierarchy 193.1. Review on the Darboux transformations of the KP hierarchy 193.2. Successive applications of T d in the Fermionic picture 21 *Corresponding author. Email: [email protected]. .3. Successive applications of T i in the Fermionic picture 223.4. Successive applications of the mixed using T d and T i in the Fermionic picture 233.5. The bilinear equations in the Darboux chains of the KP hierarchy 273.6. Examples of the bilinear equations in the KP Darboux transformations 304. The Darboux transformations of the modified KP hierarchy 314.1. Reviews on some facts of the Darboux transformations of the modified KP hierarchy 314.2. Bilinear relations for τ [1]0 and τ [1]1 Introduction
Bilinear equations [6,16,27,28] are an important integrability for the nonlinear evolution equations,which are usually expressed in the Hirota bilinear forms of the tau functions by the Hirota bilinearoperators [27] or bilinear residue identities of tau functions (or wave functions) [16, 28, 29, 41]. Thereexists a geometric interpretation for these kinds of bilinear equations. In fact by viewing tau func-tions as points in the infinite Grassmannian, the bilinear equations can be written as the Pl¨uckerrelations [16, 41, 45, 52]. As for the algebraic aspects, the bilinear equations of the KP hierarchy (orthe BKP hierarchy) are constructed as equivalent conditions of the orbit of GL ∞ (or O ∞ ) acting onthe corresponding highest weight vectors [16, 28, 29, 31, 51, 55]. Bilinear equations play an importantrole in mathematical physics. Besides in seeking various solutions of the nonlinear evolution equations(for example [19, 37]), the bilinear equations can also connect the famous Gromov-Witten invariantswith the integrable systems, for example [7, 14, 39, 40]. Here in this article, we mainly discuss bilinearequations of the transformed tau functions under the Darboux transformations.Darboux transformation is a kind of powerful method to construct solutions of the integrable system[8, 22, 38, 42, 44]. In the KP hierarchy, there exist two types of elementary Darboux transformations[8, 22, 42, 44], that is, the differential type T d (Φ) = Φ ∂ − Φ − and the integral type T i (Ψ) = Ψ − ∂ − Ψ(see Subsection 3.1 for more details). If denote τ [ l ] ( t ) to be the transformed tau functions from τ ( t )under l -step Darboux transformation T d , also called the T d Darboux chain (see (50)), then it can be ound that τ [ l ] ( t ) satisfy [52],Res λ λ l − l ′ τ [ l ] ( t − ε ( λ − )) τ [ l ′ ] ( t ′ + ε ( λ − )) e ξ ( t − t ′ ,λ ) = 0 , l ≥ l ′ , (1)which is just the bilinear equation of the ( l − l ′ )-th modified KP hierarchy [28, 30], or the one of thediscrete KP hierarchy [2,18,21] with the discrete variables l and l ′ only taking the non-negative values.The 0-th modified KP hierarchy is the usual KP hierarchy [16,41], while the 1st modified KP hierarchycan be used to describe the generating functions of open and closed intersection numbers [3]. Notethat the discussion above is only restricted to the case of T d . So it will be very natural to ask whatthe bilinear equations are in the case of T i or in the more generalized case of mixed using T d and T i in the KP hierarchy, and whether there are similar results for other integrable hierarchies. In whatfollows, we will discuss these questions for the KP, modified KP (the Kupershshmidt-Kiso version) and BKP hierarchies, since these three integrable hierarchies are closely connected with each other.Note that there are the Miura links between the KP and modified KP hierarchies [44, 46], and theBKP hierarchy can be viewed as their sub-hierarchies [16,54]. Further these relations lead to the closerelations for their Darboux transformations [23, 24, 46, 54].It is usually very difficult to obtain the bilinear equations of the tau functions in the chain ofDarboux transformations by the Bosonic approach. In fact, the efficient way to construct the bilinearequations is the Boson-Fermion correspondence [4,16,28–31,41], i.e., linking the Fermionic Fock spaceand the Bosonic Fock space (see Section 2 for more details). To establish the bilinear equationsin the Darboux transformations, the primary step is to express the Darboux transformation in theFermionic picture, which is always presented in the changes of the Fermionic tau function [8, 53].Under Darboux transformations, the changes of the Bosonic tau functions are closely related withthe (adjoint) eigenfunctions [8, 12, 22, 24, 42, 44, 54]. So the key is to find suitable Fermionic fieldsto express the (adjoint) eigenfunctions. Fortunately the (adjoint) wave functions are correspondingto the generating functions of Fermions [28, 41] (i.e., ψ ( λ ) and ψ ∗ ( λ ) in (7)), while any (adjoint)eigenfunctions are related to the (adjoint) wave functions through the spectral representations (seeSection 2 for more details) [5, 11, 13, 36, 53] with the spectral density given by squared eigenfunctionpotential (SEP). Here the SEPs [42, 43] are usually defined as the integrals of the products of theeigenfunction and the adjoint eigenfunction with respect to x in the KP case (other cases can befound in Section 2), whose predecessor is so called Cauchy-Baker-Akhiezer kernel [20].By using these facts, any (adjoint) eigenfunction is showed to be corresponding to one Fermionicfield, which further leads to the correspondence of SEP and this Fermionic field. Therefore under theDarboux transformations, the changes of the Fermionic fields corresponding to the (adjoint) eigen-function are in fact the ones of the corresponding SEPs. Based upon these, we establish the Darboux There are many versions of the modified KP hierarchy [12, 13]. In this paper, the modified KP hierarchy alwaysmeans the Kupershmidt-Kiso version [13, 32, 33, 44, 48] except for special illustrations. ransformations for the modified KP and BKP hierarchies in the Fermionic picture. Further we haveobtained the transformed Fermionic tau functions under successive applications of Darboux transfor-mations, especially the mixed using T d and T i in the KP case, which is one of the most importantresults in this paper. With this, the corresponding bilinear equations are obtained by using someimportant relations of fermions in Subsection 2.5. At last, some new examples of the bilinear relationsare given. Particularly, the bilinear equations involving the binary Darboux transformations are veryimportant in the proof of the Adler-Shiota-van Moerbeke (ASvM) formulas in the additional symme-tries [1, 13, 17, 34, 49], and in the derivation of the bilinear equations [10, 15, 35, 47] of the symmetryconstraints of the KP, modified KP and BKP hierarchies.This paper is organized in the way below. Firstly in Section 2, some basics facts on the construc-tions of the KP, modified KP and BKP hierarchies by free Fermions are reviewed, and the spectralrepresentations of these three integrable hierarchies are discussed by using the relations among them.Also some important relations on the free Fermions are established. Next based upon these, thetransformed Fermionic tau functions and the bilinear equations under the successive applications ofDarboux transformations for the KP, modified KP and BKP hierarchies are investigated in Section3-5 respectively. At last, some conclusions and discussions are presented in Section 6.2. Preliminaries on the KP, modified KP and BKP hierarchies
In this section, we firstly review some basic facts on the free Fermions and Boson-Fermion cor-respondence, and construct the ( l − l ′ )-th modified KP hierarchy. Then starting from the bilinearequations, we reviewed the basic facts of the KP, modified KP and BKP hierarchies, including thedressing structures and Lax equations, spectral representations of the eigenfunctions. Different frombefore, the spectral representations for the modified KP and the BKP hierarchy are constructed fromthe ones of the KP hierarchy by considering the relations among them. At last, we derived someimportant relations about the free Fermions, which will be used in the constructions of the bilinearequations in Darboux transformations.2.1. Boson-Fermion correspondence and ( l − l ′ ) -th modified KP hierarchy. In this subsection,we will review the Fermionic approach in the construction of the integrable systems and some factson the ( l − l ′ )-th modified KP hierarchy. One can refer to [28, 29, 41] for more details. Let A be theClifford algebra generated by the free Fermions ψ j and ψ ∗ j ( j ∈ Z ), satisfying the following relations[ ψ i , ψ j ] + = [ ψ ∗ i , ψ ∗ j ] + = 0 , [ ψ i , ψ ∗ j ] + = δ ij , (2)where [ A, B ] + = AB + BA . If define the vacuum vector | i and the dual vacuum vector h | as follows ψ i | i = 0 ( i < , ψ ∗ i | i = 0 ( i ≥ , h | ψ i = 0 ( i ≥ , h | ψ ∗ i = 0 ( i < , (3) hen one can obtain the Fermionic Fock space F = A| i and its dual space F ∗ = h |A . The pairingbetween F and F ∗ are given as follows, F ∗ × F −→ C ( h | a , a | i ) | a a | i , where h | a a | i can be computed by h | | i = 1, the relations (2) (3) and the Wick theorem. TheFermionic Fock space F can be used as the representation space of infinite dimensional Lie algebra gl ∞ and its corresponding group GL ∞ . Here the infinite Lie algebra gl ∞ is defined by gl ∞ = { X i,j ∈ Z a i,j : ψ i ψ ∗ j : | there exists an N such that a ij = 0 , | i − j | > N } ⊕ C and the corresponding group GL ∞ is given by GL ∞ = { e X e X · · · e X k | X i ∈ gl ∞ } . The charges of F and F ∗ can be defined the way below,charge of ψ j = 1, charge of ψ ∗ j = − , charge of | i =0, charge of a | i = that of a, charge of h | =0, charge of h | a = -that of a, then one can decompose F and F ∗ according to different charges F = M m ∈ Z F m , F ∗ = M m ∈ Z F ∗ m . For m >
0, define the following vectors with charge m , | m i = ψ m − · · · ψ | i , | − m i = ψ ∗− m · · · ψ ∗− | ih m | = h | ψ ∗ · · · ψ ∗ m − , h− m | = h | ψ − · · · ψ − m . By the definitions, ψ n | m i = 0 for n < m, ψ ∗ n | m i = 0 for n ≥ m, h m | ψ n = 0 for n ≥ m, h m | ψ ∗ n = 0 for n < m. (4)If define S = P j ψ j ⊗ ψ ∗ j , one can find S ( | l i ⊗ | l ′ i ) = 0 , S l − l ′ ( | l ′ i ⊗ | l i ) = ( − ( l − l ′ )( l − l ′− ( l − l ′ )! | l i ⊗ | l ′ i , for l ≥ l ′ . Further if set f l = g | l i for g ∈ GL ∞ and assume l ≥ l ′ , then by the fact S can commute with g ⊗ g , S ( f l ⊗ f l ′ ) = 0 , (5) S l − l ′ ( f l ′ ⊗ f l ) = ( − ( l − l ′ )( l − l ′− ( l − l ′ )! f l ⊗ f l ′ . (6) ere (5) is the ( l − l ′ )-th modified KP hierarchy [28–30] in the Fermionic picture. (5) is showedto be equivalent to (6) in [29, 30]. Particularly, the 0-th modified KP hierarchy is the usual KPhierarchy [16, 41] and (6) for l = l ′ + 1 is the modified KP hierarchy in the Kupershmidt-Kiso version[13, 32, 33, 44, 48], which is equivalent to the 1st modified KP hierarchy. Next we will mainly discussthese two particular cases. In order to rewritten (5) and (6) into the usual forms, i.e., the Bosonicforms, we will next review the Boson-Fermion correspondence.Introduce the generating sums of free Fermions ψ ( λ ) = X i ∈ Z ψ i λ i , ψ ∗ ( λ ) = X i ∈ Z ψ ∗ i λ − i (7)and define P n ∈ Z H n λ − n =: ψ ( λ ) ψ ( λ ) ∗ : with the normal order : AB := AB − h | AB | i . Then it canbe proved that H n satisfies Heisenberg algebraic relations[ H m , H n ] = mδ m, − n . (8)For the time variables t = ( t = x, t , t , · · · ), define H ( t ) = P ∞ n =1 t n H n . Then e H ( t ) ψ ( λ ) e − H ( t ) = e ξ ( t,λ ) ψ ( λ ) , e H ( t ) ψ ∗ ( λ ) e − H ( t ) = e − ξ ( t,λ ) ψ ∗ ( λ ) , (9)where ξ ( t, λ ) = P ∞ n =1 t n λ n . If introduce the Bosonic Fock space B = C [ z, z − , t , t , t , · · · ], then thereexists an isomorphism σ t : F −→ B given by σ t ( a | i ) = X j z j h j | e H ( t ) a | i , a ∈ A . Note that if charge of a = l , then the terms j = l on the right hand side will be zero according to theWick Theorem. By using the isomorphism σ t and the following formulas [28, 41], h l | ψ ( λ ) e H ( t ) = λ l − h l − | e H ( t − ε ( λ − )) , h l | ψ ∗ ( λ ) e H ( t ) = λ − l h l + 1 | e H ( t + ε ( λ − )) , (10)with ε ( λ − ) = ( λ − , λ − / , λ − / , · · · ), one can realize the Fermions ψ i and ψ ∗ i in the forms of theBosonic operators [28, 41] σ t · ψ ( λ ) · σ − t = e ξ ( t,λ ) e − ξ ( ˜ ∂,λ − ) e K λ H , σ t · ψ ∗ ( λ ) · σ − t = e − ξ ( t,λ ) e ξ ( ˜ ∂,λ − ) e − K λ − H , where ˜ ∂ = (cid:16) ∂∂x , ∂∂t , ∂∂t , · · · (cid:17) , and the operators λ H and e K act on B by the formulas below (cid:0) λ H f (cid:1) ( z, t ) , f ( λz, t ) , (cid:0) e K f (cid:1) ( z, t ) , zf ( z, t ) . This realization is called the Boson-Fermion correspondence.After the preparation above, now we can write the bilinear equations (5) and (6) into the Bosonicforms. If denote τ l ( t ) = h l | e H ( t ) g | l i , then by Boson-Fermion correspondence, (5) and (6) will becomeinto Res λ λ l − l ′ τ l ( t − ε ( λ − )) τ l ′ ( t ′ + ε ( λ − )) e ξ ( t − t ′ ,λ ) = 0 , (11) es λ · · · Res λ l − l ′ τ l ′ ( t − l − l ′ X i =1 ε ( λ − i )) τ l ( t ′ + l − l ′ X i =1 ε ( λ − i )) × l − l ′ Y i =1 λ l ′ − li Y i In this subsection, we will review the basic facts of theKP hierarchy including the dressing and Lax equations, the spectral representations of the (adjoint)eigenfunctions.Firstly when l = l ′ = 0, (11) is justRes λ τ ( t − ε ( λ − )) τ ( t ′ + ε ( λ − )) e ξ ( t − t ′ ,λ ) = 0 (13)with τ ( t ) = τ ( t ). By introducing the wave function ψ ( t, λ ) and the adjoint wave function ψ ∗ ( t, λ ) inthe way below, ψ ( t, λ ) = τ ( t − ε ( λ − )) τ ( t ) e ξ ( t,λ ) = h | e H ( t ) ψ ( λ ) g | ih | e H ( t ) g | i , (14) ψ ∗ ( t, λ ) = τ ( t + ε ( λ − )) τ ( t ) e − ξ ( t,λ ) = 1 λ h− | e H ( t ) ψ ∗ ( λ ) g | ih | e H ( t ) g | i , (15)the bilinear equations (13) becomes intoRes λ ψ ( t, λ ) ψ ∗ ( t ′ , λ ) = 0 . (16)Introduce the following pseudo-differential operators (denote ∂ = ∂ x ) W = 1 + ∞ X j =1 w j ∂ − j , ˜ W = 1 + ∞ X j =1 ˜ w j ∂ − j , satisfying ψ ( t, λ ) = W ( e ξ ( t,λ ) ) and ψ ∗ ( t, λ ) = ˜ W ( e − ξ ( t,λ ) ), that is w i = p i ( − ˜ ∂ ) τ ( t ) τ ( t ) and ˜ w i = p i ( ˜ ∂ ) τ ( t ) τ ( t ) .Here p i ( t ) is Schur polynomial defined by exp( ξ ( t, λ )) = P i p i ( t ) λ i . Then one can obtain [16]˜ W = ( W − ) ∗ , W t n = − ( W ∂ n W − ) < W, (17)where ∗ is the adjoint operation defined by (cid:16) P i a i ∂ i (cid:17) ∗ = P i ( − i ∂ i a i . If define the Lax operator L as follows L = W ∂W − = ∂ + u ∂ − + u ∂ − + · · · , then we can obtain the Lax equation of the KP hierarchy [16, 41] L t n = [( L n ) ≥ , L ] . (18) emark: By considering the ∂ − -terms in the second equation of (17), one can obtain Res ∂ L n =(log τ ) xt n with Res ∂ P i a i ∂ i = a − . So each u i in the Lax operator L can be expressed in termsof (log τ ) xt n . Thus for a fixed Lax operator L , the corresponding tau function τ can be determinedup to a multiplication of c exp( P i a i t i ). For convenience, we let τ ∼ τ ′ if two tau functions τ and τ ′ determine the same Lax operator of the KP hierarchy. Further, denote τ ≈ τ ′ when τ and τ ′ determinethe same dressing operator W . It is obviously that τ ≈ τ ′ if and only if τ ′ = cτ .In what follows, the spectral representation for the KP hierarchy will be needed in the discussionof the Darboux transformation. That is, for the eigenfunction Φ and the adjoint eigenfunction Ψ ofthe KP hierarchy defined by Φ t n = ( L n ) ≥ (Φ) , Ψ t n = − ( L n ) ∗≥ (Ψ) , it is showed in [5, 53] that they can be expressed by the wave function ψ ( t, λ ) and the adjoint wavefunction ψ ∗ ( t, λ ) of the KP hierarchy respectively, that is,Φ( t ) = Res λ ρ ( λ ) ψ ( t, λ ) , Ψ( t ) = Res λ ρ ∗ ( λ ) ψ ∗ ( t, λ ) , (19)where ρ ( λ ) = − Ω(Φ( t ′ ) , ψ ∗ ( t ′ , λ )) and ρ ∗ ( λ ) = Ω( ψ ( t ′ , λ ) , Ψ( t ′ )) belong to C (( λ − )). Here Ω( f, g ) isthe squared eigenfunction potential (SEP) [42, 43], determined byΩ( f ( t ) , g ( t )) x = f ( t ) g ( t ) , Ω( f ( t ) , g ( t )) t n = Res ∂ ( ∂ − g ( t )( L n ) ≥ f ( t ) ∂ − )for the eigenfunction f ( t ) and the adjoint eigenfunction g ( t ) of the KP hierarchy, up to a constant.The expressions of SEP can be derived in the way below. Note thatΩ(Φ( t ) , ψ ∗ ( t, λ )) = ( − Φ( t ) + O ( λ − )) λ − e − ξ ( t,λ ) , Ω( ψ ( t, λ ) , Ψ( t )) = (Ψ( t ) + O ( λ − )) λ − e ξ ( t,λ ) . So if letting t = t ′ + ε ( λ − ) for in first relation of (19) and t ′ = t + ε ( λ − ) for the second one, thenthe expressions of SEPs can be derived which are given in the proposition [5, 53] below. Proposition 1. Given the eigenfunction Φ( t ) and the adjoint eigenfunction Ψ( t ) of the KP hierarchy, Ω(Φ( t ) , ψ ∗ ( t, λ )) = − λ Φ( t + ε ( λ − )) ψ ∗ ( t, λ ) , Ω( ψ ( t, λ ) , Ψ( t )) = 1 λ Ψ( t − ε ( λ − )) ψ ( t, λ ) . Particularly, Ω( ψ ( t, µ ) , ψ ∗ ( t, λ )) = − λ ψ ( t + ε ( λ − ) , µ ) ψ ∗ ( t, λ ) + δ ( λ, µ )= 1 µ ψ ( t, µ ) ψ ∗ ( t − ε ( µ − ) , λ ) , where δ ( λ, µ ) = µ P n ∈ Z (cid:0) µλ (cid:1) n = λ − µ/λ + µ − λ/µ . herefore the spectral representation (19) can be rewritten intoΦ( t ) = Res λ λ ψ ( t, λ ) ψ ∗ ( t ′ , λ )Φ( t ′ + ε ( λ − )) , Ψ( t ) = Res λ λ ψ ∗ ( t, λ ) ψ ( t ′ , λ )Ψ( t ′ − ε ( λ − )) . (20)If denote τ + ( t ) = Φ( t ) τ ( t ) and τ − ( t ) = Ψ( t ) τ ( t ), then one can find that the spectral representation(19) of Φ( t ) and Ψ( t ) are equivalent to the following bilinear relations involving τ ± ( t ) and τ ( t ), thatis Res λ λ − τ ( t − ε ( λ − )) τ + ( t ′ + ε ( λ − )) e ξ ( t − t ′ ,λ ) = τ + ( t ) τ ( t ′ ) , Res λ λ − τ − ( t − ε ( λ − )) τ ( t ′ + ε ( λ − )) e ξ ( t − t ′ ,λ ) = τ ( t ) τ − ( t ′ ) , (21)which are just the bilinear equations of the modified KP hierarchy of the Kupershmidt-Kiso version.2.3. The mKP hierarchy and the Miura links. Just as showed in the last subsection, the bilinearequation of the modified KP hierarchy is given as follows (that is (12) for l = l ′ + 1 = 1),Res λ λ − τ ( t − ε ( λ − )) τ ( t ′ + ε ( λ − )) e ξ ( t − t ′ ,λ ) = τ ( t ) τ ( t ′ ) . (22)Here ( τ , τ ) is called the tau pair of the modified KP hierarchy. Firstly, we can introduce the wavefunction w ( t, λ ) and the adjoint wave function w ∗ ( t, λ ) of the modified KP hierarchy [13] w ( t, λ ) = τ ( t − ε ( λ − )) τ ( t ) e ξ ( t,λ ) = h | e H ( t ) ψ ( λ ) g | ih | e H ( t ) g | i , (23) w ∗ ( t, λ ) = τ ( t + ε ( λ − )) τ ( t ) λ − e − ξ ( t,λ ) = h | e H ( t ) ψ ∗ ( λ ) g | ih | e H ( t ) g | i λ − , (24)so that (22) can be written into Res λ w ( t, λ ) w ∗ ( t ′ , λ ) = 1 . (25)If introduce the dressing operator Z = P ∞ i =0 z i ∂ − i such that w ( t, λ ) = Z (cid:0) e ξ ( t,λ ) (cid:1) , then by the similarmethod to the KP case, one can get w ∗ ( t, λ ) = ( Z − ∂ − ) ∗ (cid:16) e − ξ ( t,λ ) (cid:17) , Z t n = − ( Z∂ n Z − ) < Z. (26)Then the Lax operator L of the modified KP hierarchy can be introduced as L = Z∂Z − = ∂ + v + v ∂ − + v ∂ − + · · · , satisfying the Lax equation [13, 44] below L t n = [( L n ) ≥ , L ] (27) Remark: Note that [9] Res ∂ L n = (log τ ) xt n and Res ∂ ( ∂ L n ∂ − ) ∗ = − (log τ ) xt n , which can beobtained by comparing the ∂ and ∂ − -terms in the evolution equation (26) of the dressing operator . Thus each v i in L can be expressed by (log τ ) xt n and v = (log( τ /τ )) x , or (log τ ) xt n and v .Therefore the tau pair ( τ ′ , τ ′ ) = e ax (cid:16) c exp( X i> a i t i ) τ , c exp( X i> b i t i ) τ (cid:17) has the same Lax operator of the modified KP hierarchy as the tau pair ( τ , τ ). In this case, wedenote ( τ ′ , τ ′ ) ∼ ( τ , τ ). Further when ( τ ′ , τ ′ ) = c ( τ , τ ), they share the same dressing structure,denoted by ( τ ′ , τ ′ ) ≈ ( τ , τ ).Before further discussion, the lemma [13] below is needed, which can be proved by applying ∂ x ′ onthe both sides of (22) and set t − t ′ = ε ( z − ). Lemma 2. If τ and τ are tau functions of the modified KP hierarchy satisfying (22), then ∂ x (cid:18) τ ( t + ε ( λ − )) λτ ( t ) (cid:19) = τ ( t + ε ( λ − )) τ ( t ) − τ ( t + ε ( λ − )) τ ( t ) τ ( t ) , (28) and therefore the wave function w ( t, λ ) and the adjoint wave function w ∗ ( t, λ ) of the modified KPhierarchy satisfy w ( t, λ ) x = λ τ ( t ) τ ( t − ε ( λ − )) τ ( t ) e ξ ( t,λ ) , w ∗ ( t, λ ) x = − τ ( t + ε ( λ − )) τ ( t ) τ ( t ) e − ξ ( t,λ ) . (29)By this lemma, we can obtain the proposition [13] below. Proposition 3. If τ and τ are tau functions of the modified KP hierarchy, i.e., Res λ λ − τ ( t − ε ( λ − )) τ ( t ′ + ε ( λ − )) e ξ ( t − t ′ ,λ ) = τ ( t ) τ ( t ′ ) , then τ and τ are the tau functions of the KP hierarchy, Res λ τ i ( t − ε ( λ − )) τ i ( t ′ + ε ( λ − )) e ξ ( t − t ′ ,λ ) = 0 , i = 1 , . The eigenfunction ˆΦ( t ) and the adjoint eigenfunction ˆΨ( t ) of the modified KP hierarchy are definedby ˆΦ t n = ( L n ) ≥ ( ˆΦ) , ˆΨ t n = − (cid:16) ∂ − ( L n ) ∗≥ ∂ (cid:17) ( ˆΨ) . The spectral representation of the (adjoint) eigenfunction of the modified KP hierarchy can be derivedfrom the ones in the KP case, by using the Miura links between the modified KP hierarchy (27) andthe KP hierarchy (18), which are given as follows. • Anti-Miura transformations (KP −→ modified KP) L → L = T m LT − m , T m (Φ) = Φ − ,T n LT − n , T n (Ψ) = ∂ − Ψ , where Φ and Ψ are the eigenfunction and the adjoint eigenfunction of the KP hierarchy respectively. Miura transformation (modified KP −→ KP) L → L = T µ L T − µ , T µ = z − ,T ν L T − ν , T ν = z − ∂, where z to be the coefficient of ∂ -term in the dress operator Z of the modified KP hierarchy.The corresponding changes of the dressing operators, the eigenfunctions and the adjoint eigenfunc-tions, the wave and the adjoint wave functions, and tau functions under the Miura links are given inthe two tables below [46].Table I. Anti-Miura transformation: KP → modified KP L → L Z = ˆΦ = ˆΨ = w = w ∗ = τ = τ = T m (Φ) = Φ − Φ − W Φ − Φ − R ΦΨ x dx Φ − ψ − R Φ ψ ∗ x dx τ Φ τT n (Ψ) = ∂ − Ψ ∂ − Ψ W ∂ R ΨΦ dx Ψ − Ψ x λ R Ψ ψdx Ψ − ψ ∗ x /λ Ψ τ τ Table II. Miura transformation: modified KP → KP L → L W = Φ = Ψ = ψ = ψ ∗ = τ = T µ = z − z − Z z − ˆΦ − z ˆΨ x z − w − z w ∗ x τ T ν = z − ∂ z − ∂Z∂ − z − ˆΦ x z ˆΨ z − w x /λ λz w ∗ τ Here Φ = c Φ and Ψ = c Ψ are the eigenfunction and the adjoint eigenfunction of the KP hierar-chy with respect to the Lax operator L , while ˆΦ and ˆΨ mean the eigenfunction and the adjointeigenfunction of the modified KP hierarchy with respect to L . w = w ( t, λ ) and w ∗ = w ( t, λ ) are thewave and the adjoint wave functions of the modified KP hierarchy respectively, while ψ = ψ ( t, λ ) and ψ ∗ = ψ ∗ ( t, λ ) are the ones of the KP hierarchy. The results for tau functions can be obtained bycomparing the coefficients of ∂ and ∂ − in the dressing operators Z and W , which are not consideredin [46]. Remark: In the anti-Miura transformations, ( τ , τ ) = ( τ, Φ τ ) or ( τ , τ ) = (Ψ τ, τ ) still satisfies thebilinear equation (22) of the modified KP hierarchy, according to the results in (21), while τ = τ or τ = τ satisfies the bilinear equation of the KP hierarchy (13) by Proposition 3.After the preparation above, now we can discuss the spectral representations of the modified KPhierarchy, by using the Miura links from the results for the KP hierarchy. For this, define the squaredeigenfunction potential Ω( ˆΦ , ˆΨ x ) and ˆΩ( ˆΦ x , ˆΨ) for the eigenfunction ˆΦ and the adjoint eigenfunctionˆΨ of the modified KP hierarchy in the way below [13, 43].Ω( ˆΦ , ˆΨ x ) x = ˆΦ ˆΨ x , Ω( ˆΦ , ˆΨ x ) t n = Res ∂ ( ∂ − ˆΨ x ( L n ) ≥ ˆΦ ∂ − ) , ˆΩ( ˆΦ x , ˆΨ) x = ˆΦ x ˆΨ , ˆΩ( ˆΦ x , ˆΨ) t n = Res ∂ ( ∂ − ˆΨ ∂ ( L n ) ≥ ∂ − ˆΦ x ∂ − ) , The relation of Ω( ˆΦ , ˆΨ x ) and ˆΩ( ˆΦ x , ˆΨ) is as follows.ˆΩ( ˆΦ x , ˆΨ) = − Ω( ˆΦ , ˆΨ x ) + ˆΦ ˆΨ . (30) hen one can have the following spectral representations of the eigenfunction and the adjoint eigen-function for the modified KP hierarchy in the proposition below. Proposition 4. Let w ( t, λ ) and w ∗ ( t, λ ) be the wave function and the adjoint wave function for themodified KP hierarchy respectively, then for the eigenfunction ˆΦ( t ) and the adjoint eigenfunction ˆΨ( t ) of the modified KP hierarchy, Res λ w ( t ′ , λ ) ˆΩ( ˆΦ( t ) x , w ∗ ( t, λ )) = ˆΦ( t ) − ˆΦ( t ′ ) , (31)Res λ w ∗ ( t ′ , λ )Ω( w ( t, λ ) , ˆΨ( t ) x ) = ˆΨ( t ) − ˆΨ( t ′ ) . (32) Proof. Firstly, if denote Φ( t ) = z − ˆΦ( t ) and ψ ∗ ( t, λ ) = − z w ∗ ( t, λ ) x , then Φ and ψ ∗ ( t, λ ) are theeigenfunction and the adjoint wave function of the KP hierarchy, according to the results of the Miuratransformations T µ = z − , showed in Table II. Then one can compute ˆΩ( ˆΦ( t ) x , w ∗ ( t, λ )) in the waybelow, ˆΩ( ˆΦ( t ) x , w ∗ ( t, λ )) = Z ˆΦ( t ) x w ∗ ( t, λ ) dx = ˆΦ( t ) w ∗ ( t, λ ) + Ω(Φ( t ) , ψ ∗ ( t, λ )) . (33)Further by considering z − w ( t, λ ) is the wave function of the KP hierarchy (see Table I), one canprove (31) according to (25) and (19). (32) can be proved by the similar method. (cid:3) By considering the relation of Ω(Φ , Ψ x ) and ˆΩ(Φ x , Ψ) in (30), one can obtain the corollary [13]below. Corollary 5. Under the same condition in Proposition 4, ˆΦ( t ) = Res λ w ( t, λ )ˆ ρ ( λ ) , ˆΨ( t ) = Res λ w ∗ ( t, λ )ˆ ρ ∗ ( λ ) , (34) with ˆ ρ ( λ ) = Ω( ˆΦ( t ′ ) , w ∗ ( t ′ , λ ) x ′ ) and ˆ ρ ∗ ( λ ) = ˆΩ( w ( t ′ , λ ) x ′ , ˆΨ( t ′ )) . According to (33), Proposition 1 and Lemma 2, one can obtain the expressions of SEPs in themodified KP hierarchy. Proposition 6. Given the eigenfunction ˆΦ( t ) and ˆΨ( t ) , the expressions of SEPs for the modified KPhierarchy are listed below. Ω( ˆΦ( t ) , w ∗ ( t, λ ) x ) = w ∗ ( t, λ ) ˆΦ( t + ε ( λ − )) , ˆΩ( ˆΦ( t ) x , w ∗ ( t, λ )) = w ∗ ( t, λ ) (cid:16) ˆΦ( t ) − ˆΦ( t + ε ( λ − )) (cid:17) , ˆΩ( w ( t, µ ) x , ˆΨ( t )) = w ( t, µ ) ˆΨ( t − ε ( µ − )) , Ω( w ( t, µ ) , ˆΨ( t ) x ) = w ( t, µ ) (cid:16) ˆΨ( t ) − ˆΨ( t − ε ( µ − )) (cid:17) . Particularly, Ω( w ( t, µ ) , w ∗ ( t, λ ) x ) = w ∗ ( t, λ ) w ( t + ε ( λ − ) , µ ) , ˆΩ( w ( t, µ ) x , w ∗ ( t, λ )) = w ( t, µ ) w ∗ ( t − ε ( µ − ) , λ ) . .4. The neutral free Fermions and the BKP hierarchy. The BKP hierarchy can be seen asthe sub-hierarchy of the KP hierarchy or the modified KP hierarchy [16, 54], which can be expressedby the neutral free Fermions [16, 28, 29, 31, 50, 51, 55] φ n = ψ n + ( − n ψ ∗− n √ , n ∈ Z , satisfying [ φ m , φ n ] + = ( − m δ m, − n . (35)Denote the A B as the Clifford algebra generated by { φ n } n ∈ Z . Then define the vacuum | i and h | inthe way below φ − j | i = 0 ( j > , h | φ j = 0 ( j > ,φ | i = 1 √ | i , h | φ = 1 √ h | . (36)One can thus obtain the Fock space for neutral Fermions F B = A B | i and its dual space F ∗ B = h |A B .Also there is a bilinear pairing F ∗ B × F B → C , denoted by h | ab | i and computed according to (35),(36) and h | i = 1.If define S B = P j ∈ Z ( − j φ j ⊗ φ − j and g ∈ O ∞ = n exp (cid:16) X m,n ∈ Z b mn : φ m φ n : (cid:17) |∃ N , b mn = 0, if | m + n | > N o , then one can obtain the bilinear equations of the BKP hierarchy in the Fermionic version [29, 31, 50] S B ( g | i ⊗ g | i ) = 12 g | i ⊗ g | i . (37)By introducing the following two generating series: φ ( z ) = X j ∈ Z φ j z j , X k ∈ Z +1 H B,k z − k − =: 12 z φ ( z ) φ ( − z ) :and defining H B ( t ) = P ∞ k =0 t k +1 H B, k +1 , then we can define the isomorphism between F B and C [ t , t , t · · · ] as follows [31, 51, 55], σ B,t : F B → C [ t , t , t · · · ] a | i 7→ h | e H B ( t ) a | i , a ∈ A B . Here the normal order of neutral Fermions : · : is the same as before. Then the neutral Boson-Fermion correspondence can be introduced in the way below by using the relation [51] √ h | φ ( z ) = h | e − H B (2 e ε ( λ − )) , σ B,t · φ ( λ ) · σ − B,t = 1 √ e e ξ ( t,λ ) e − e ξ ( ˆ ∂,λ − )13 here e ξ ( t, λ ) = P k =0 t k +1 λ k +1 and ˆ ∂ = (cid:16) ∂∂x , ∂∂t , ∂∂t , · · · (cid:17) . Now if denote τ B ( t ) = h | e H B ( t ) g | i ,we can rewritten (37) intoRes λ λ − τ B ( t − e ε ( λ − )) τ B ( t ′ + 2 e ε ( λ − )) e e ξ ( t − t ′ ,λ ) = τ B ( t ) τ B ( t ′ ) , (38)where e ε ( λ − ) = ( λ − , λ − / , λ − / , · · · ). By introducing the wave functions of the BKP hierarchy inthe way below [16] ψ B ( t, λ ) = τ B ( t − e ε ( λ − )) τ B ( t ) e e ξ ( t,λ ) = √ h | e H B ( t ) φ ( λ ) g | ih | e H B ( t ) g | i , (39)then (38) will become into Res λ λ − ψ B ( t, λ ) ψ B ( t ′ , − λ ) = 1 . (40)So if let W B = 1 + P j =1 v j ∂ − j satisfying ψ B ( t, λ ) = W B ( e e ξ ( t,λ ) ), then it can be proved that W B ∂ − W ∗ B = ∂ − , ( W B ) t k +1 = − ( W B ∂ k +1 W − B ) < W B . (41)Further define L B = W B ∂W − B , one can obtain the Lax equation of the BKP hierarchy L ∗ B = − ∂L B ∂ − , ( L B ) t k +1 = [( L k +1 B ) ≥ , L B ] . (42) Remark: Given a fixed Lax operator L B of the BKP hierarchy, the tau function τ B is determined upto a multiplication of c exp( P i odd a i t i ). τ ′ B ∼ τ B and τ ′ B ≈ τ B have similar meanings to the KP andmodified KP cases.In [54], the BKP hierarchy is viewed as the Kupershmidt reduction of the modified KP hierarchy,i.e., L ∗ = ∂ L ∂ − . As a sub-hierarchy of the modified KP hierarchy, the wave function ψ B ( t, λ ) of theBKP hierarchy is related with the adjoint wave function w ∗ ( t, λ ) of the mKP hierarchy in the waybelow [54], w ∗ ( t, λ ) = 1 λ ψ B ( t, − λ ) . (43)Therefore the bilinear equation (25) of the modified KP hierarchy can be naturally reduced into theone of the BKP hierarchy given in (40). What’s more, any eigenfunction Φ B ( t ) of the BKP hierarchy(satisfying Φ B,t k +1 = ( L k +1 B ) ≥ (Φ B )) can also be viewed as the adjoint eigenfunction of the BKPhierarchy seen as the Kupershmidt reduction of the modified KP hierarchy [54], that is,Φ B ( t ) t k +1 = − ( ∂ ( L k +1 B ) ≥ ∂ − ) ∗ (Φ B ( t )) . By using these facts, the corresponding spectral representation for the BKP hierarchy can be de-rived naturally from the one of the modified KP hierarchy (see Proposition 4), which is given in theproposition below [11, 36]. roposition 7. For the eigenfunction Φ B ( t ) of the BKP hierarchy, Res λ λ Ω(Φ B ( t ) x , ψ B ( t, − λ )) ψ B ( t ′ , λ ) = Φ B ( t ) − Φ B ( t ′ ) , (44) Further, Φ B ( t ) = Res λ λ ρ B ( λ ) ψ B ( t, λ ) , ρ B ( λ ) = Ω(Φ B ( t ′ ) , ψ B ( t ′ , − λ ) x ) . (45)If assume Ω( ψ B ( t, λ ) , Φ B ( t ) x ) = K ( t, λ ) e ξ ( t,λ ) with K ( t, λ ) ∈ C (( λ − )), then by choosing t − t ′ =2 e ε ( λ − ) in (44) one can obtain the following expressions of SEPs for the BKP hierarchy [11, 36]. Proposition 8. Assume Φ B ( t ) to be the eigenfunction of the BKP hierarchy, then the expression ofSEPs for the BKP hierarchy are given below. Ω( ψ B ( t, λ ) , Φ B ( t ) x ) = 12 ψ B ( t, λ ) (cid:16) Φ B ( t ) − Φ B ( t − e ε ( λ − )) (cid:17) Ω(Φ B ( t ) , ψ B ( t, λ ) x ) = 12 ψ B ( t, λ ) (cid:16) Φ B ( t ) + Φ B ( t − e ε ( λ − )) (cid:17) . In particular, Ω( ψ B ( t, µ ) , ψ B ( t, − λ ) x ) = 12 ψ B ( t, µ ) (cid:16) ψ B ( t, − λ ) − ψ B ( t − e ε ( µ − ) , − λ ) (cid:17) = 12 ψ B ( t, − λ ) (cid:16) ψ B ( t, µ ) + ψ B ( t + 2 e ε ( λ − ) , µ ) (cid:17) + 12 ( λ + µ ) δ ( λ, µ ) . Further if define the SEP of B type (BSEP) [11,36] Ω B (Φ B , Φ B ) , Ω(Φ B , Φ B ,x ) − Ω(Φ B , Φ B ,x ),then one can find that Ω B (Φ B ( t ) , ψ B ( t, λ )) = − ψ B ( t, λ )Φ B ( t − e ε ( λ − )) . (46)2.5. Important relations on free Fermions. In this section, we will give important relations aboutfree Fermions used to construct the bilinear equations. Firstly denote H n,a = { ( j a , j a − , · · · , j ) | n ≥ j a > · · · > j ≥ } ,m + H n,a = { ( m + j a , m + j a − , · · · , m + j ) | n ≥ j a > · · · > j ≥ } . Note that H n,n = ~ n , H n, = { } and H n,a = ∅ with a > n or a < 0. For ~ n = ( n, n − , · · · , , β ~ n = β n β n − · · · β β and ~β ~ n = ( β n , β n − , · · · , β ). Define ~α ~ m ∪ ~β ~ n , ( α m , · · · , α , β n , · · · , β ) , ~α ~ m \ { α i } , ( α m , · · · , α i +1 , α i − , · · · , α ) . and | ~α | , α m + · · · + α . .Next, let us discuss the relations on the free Fermions ψ i and ψ ∗ j , which will be used to the bilinearequations of the Darboux transformations of the KP and modified KP hierarchies. emma 9. For β i ∈ V = ⊕ l C ψ l and β ∗ j ∈ V ∗ = ⊕ l C ψ ∗ l , one has the following relations S ( β ~ n ⊗ 1) = ( − n ( β ~ n ⊗ S, S (1 ⊗ β ∗ ~ k ) = ( − k (1 ⊗ β ∗ ~ k ) S,S (1 ⊗ β ~ n ) = n X l =1 ( − n − l β l ⊗ β ~ n \{ l } + ( − n (1 ⊗ β ~ n ) S,S ( β ∗ ~ k ⊗ 1) = k X l =1 ( − k − l β ∗ ~ k \{ l } ⊗ β ∗ l + ( − k ( β ∗ ~ k ⊗ S. Proof. Note that S ( β ~ n ⊗ 1) and S (1 ⊗ β ∗ ~ k ) are obvious. Next we only discuss S (1 ⊗ β ~ n ), since S ( β ∗ ~ k ⊗ P j [ ψ ∗ j , β l ] + ψ j = β l , S (1 ⊗ β ~ n ) = X j ( ψ j ⊗ ψ ∗ j ) · (1 ⊗ β ~ n )= X j n X l =1 ( − n − l [ ψ ∗ j , β l ] + ψ j ⊗ β ~ n \{ l } + ( − n X j ( ψ j ⊗ β ~ n ψ ∗ j )= n X l =1 ( − n − l β l ⊗ β ~ n \{ l } + ( − n (1 ⊗ β ~ n ) S. (cid:3) Lemma 10. For β i ∈ V = ⊕ l C ψ l and β ∗ j ∈ V ∗ = ⊕ l C ψ ∗ l , S l ( β ~ n ⊗ 1) = ( − nl ( β ~ n ⊗ S l , S l (1 ⊗ β ∗ ~ k ) = ( − kl (1 ⊗ β ∗ ~ k ) S l ,S l (1 ⊗ β ~ n ) = ( − nl l X j =0 C jl j ! A + n,j S l − j , S l ( β ∗ ~ k ⊗ 1) = ( − kl l X j =0 C jl j ! A − k,j S l − j , where for n ≥ j ≥ , A + n,j = X ~γ ∈ H n,j ( − −| ~γ | β ~γ ⊗ β ~ n \ ~γ , A − n,j = X ~γ ∈ H n,j ( − −| ~γ | β ∗ ~ n \ ~γ ⊗ β ∗ ~γ . In particular, A + n, = 1 ⊗ β ~ n , A − n, = β ∗ ~ n ⊗ and A ± n,j = 0 for n < j or j < .Proof. The results for S l ( β ~ n ⊗ 1) and S l (1 ⊗ β ∗ ~ k ) are obviously by using Lemma 9. Next we try to prove S l (1 ⊗ β ~ n ), where the key is to compute SA + n,j . Firstly note that A + n,j = P nl = j (cid:16) β l ⊗ β ~ n \ ~l (cid:17) A + l − ,j − ,then by Lemma 9, SA + n,j = ( − n A + n,j +1 + n X l = j ( − n − (cid:16) β l ⊗ β ~ n \ ~l (cid:17) SA + l − ,j − . Repeat the procedure above, one at last obtains SA + n,j =( − n jA + n,j +1 + X ~γ ∈ H n,j ( − n − −| ~γ | + γ (cid:16) β γ ⊗ β ( ~ n \ ~γ ) \−−−→ γ − (cid:17) SA + γ − , ( − n (cid:16) ( j + 1) A + n,j +1 + A + n,j S (cid:17) , where ~γ = ( γ j , · · · , γ ) ∈ H n,j . After the preparation above, one can easily prove the results for S l (1 ⊗ β ~ n ) by induction on l . Similar discussion can be done for S l ( β ∗ ~ k ⊗ (cid:3) By Lemma 10, one can obtain the lemma below Lemma 11. For ˜ β i , ¯ β i , β i ∈ V and ˜ β ∗ j , ¯ β ∗ j , β ∗ j ∈ V ∗ , the actions of S l on ˜ β ∗−→ k β ∗ ~ k ˜ β −→ n β ~ n ⊗ ¯ β ∗−→ k β ∗ ~ k ¯ β −→ n β ~ n and ˜ β −→ n β ~ n ˜ β ∗−→ k β ∗ ~ k ⊗ ¯ β −→ n β ~ n ¯ β ∗−→ k β ∗ ~ k are given by S l (cid:16) ˜ β ∗−→ k β ∗ ~ k ˜ β −→ n β ~ n ⊗ ¯ β ∗−→ k β ∗ ~ k ¯ β −→ n β ~ n (cid:17) = l X j =0 C jl ( − ( l − j )( n + n + k + k ) j ! B + k ,n ,j S l − j , l ≥ , (47) S l (cid:16) ˜ β −→ n β ~ n ˜ β ∗−→ k β ∗ ~ k ⊗ ¯ β −→ n β ~ n ¯ β ∗−→ k β ∗ ~ k (cid:17) = l X j =0 C jl ( − ( l − j )( n + n + k + k ) j ! B − k ,n ,j S l − j , l ≥ . (48) Here B ± k ,n ,j is defined in the way below, B + k ,n ,j = j X a =0 X ~γ ∈ H k ,j − a X ~δ ∈ H n ,a ( − a ( k + n )+ jk −| ~γ |−| ~δ | ˜ β ∗−→ k \ ~γ β ∗ ~ k ¯ β ~δ ˜ β −→ n β ~ n ⊗ ˜ β ∗ ~γ ¯ β ∗−→ k β ∗ ~ k ¯ β −→ n \ ~δ β ~ n ,B − k ,n ,j = j X a =0 X ~γ ∈ H n ,j − a X ~δ ∈ H k ,a ( − a ( k + n )+ jn −| ~γ |−| ~δ | ¯ β ~γ ˜ β −→ n β ~ n ˜ β ∗−→ k \ ~δ β ∗ ~ k ⊗ ¯ β −→ n \ ~γ β ~ n ˜ β ∗ ~δ ¯ β ∗−→ k β ∗ ~ k . Particularly, B ± k ,n ,j = 0 for k + n < j and B + k ,n ,k + n = ( − n ( k + k )+ k k − + n n − β ∗ ~ k ¯ β −→ n ˜ β −→ n β ~ n ⊗ ˜ β ∗−→ k ¯ β ∗−→ k β ∗ ~ k β ~ n ,B − k ,n ,k + n = ( − k ( n + n )+ k k − + n n − ¯ β −→ n ˜ β −→ n β ~ n β ∗ ~ k ⊗ β ~ n ˜ β ∗−→ k ¯ β ∗−→ k β ∗ ~ k . Now let us switch to discussing the relations of the neutral free Fermions, which will be needed inthe constructions of the bilinear equations in the BKP Darboux transformations. Lemma 12. For β i ∈ V B = ⊕ l C φ l , one has the following relations S B ( β ~ n ⊗ 1) = n X l =1 ( − n − l β ~ n \{ l } ⊗ β l + ( − n ( β ~ n ⊗ S B ,S B (1 ⊗ β ~ n ) = n X l =1 ( − n − l β l ⊗ β ~ n \{ l } + ( − n (1 ⊗ β ~ n ) S B . Proof. This lemma can be proved by the similar method in Lemma 9 with β l = P j ( − j [ β l , φ − j ] + φ j . (cid:3) For β i ∈ V B with i = 1 , , · · · , n and n ≥ j ≥ 0, denote A + n,j = X ~γ ∈ H n,j ( − −| ~γ | β ~ n \ ~γ ⊗ β ~γ , A − k,j = X ~γ ∈ H k,j ( − −| ~γ | β ~γ ⊗ β ~ k \ ~γ . t is obviously that A ± n,j = 0 for j > n or j < A + n, = β ~ n ⊗ A − n, = 1 ⊗ β ~ n . Lemma 13. For β i ∈ V B = ⊕ l C φ l , S lB ( β ~ n ⊗ and S lB (1 ⊗ β ~ n ) can be computed in the way below. S lB ( A ± n, ) = ( − nl l X j =0 C jl [ j/ X k =0 a n,j,k A ± n,j − k S l − j , where [ j/ means the integer that is no greater than j/ , and a n,j,k is some constant satisfying thefollowing recursion relation, that is, a n,j +1 ,k = na n,j,k − + ( a n,j,k − a n,j,k − )( j − k + 1) with a n,j, = j ! and a n,j,k = 0 if k < or k > [ j/ .Proof. Let us firstly compute S B A + n,j . By using A + n,j = P nl = j (cid:16) β ~ n \ ~l ⊗ β l (cid:17) A + l − ,j − and the formula[ S B ( β ~ n ⊗ , ⊗ β l ] + = β l β ~ n ⊗ S B A + n,j =( − n A + n,j +1 + n X l = j ( − n − ( β ~ n \ ~l ⊗ β ~l ) S B A + l − ,j − + X ~γ ∈ H n,j − ( − n −| ~γ | ( n − γ j − )( β ~ n \ ~γ ⊗ β ~γ ) . Further one can obtain S B A + n,j =( − n (cid:16) ( j + 1) A + n,j +1 + A + n,j S B (cid:17) + X ~γ ∈ H n,j − j X l =1 ( − n −| ~γ | ( γ l − γ l − − (cid:16) β ~ n \ ~γ ⊗ β ~γ (cid:17) =( − n (cid:16) ( j + 1) A + n,j +1 + A + n,j S B + ( n − j + 1) A + n,j − (cid:17) , where γ j = n and γ = 0. Similarly, one can prove S B A − n,j = ( − n (cid:16) ( j + 1) A − n,j +1 + A − n,j S B + ( n − j + 1) A − n,j − (cid:17) . After the preparation above, one can easily obtain by induction on l , S lB ( A ± n, ) = ( − nl l X j =0 A ± n,j ; l S l − jB , where A ± n,j ; l = P [( j +1) / k =0 a n,j,k ; l A ± n,j − k and a n,j,k ; l is some constant satisfying the following recursionrelations a n, , l +1 = a n, , l , a n,l +1 ,k ; l +1 = na n,l,k − l + ( l − k + 1)( a n,l,k ; l − a n,l,k − l ) ,a n,j,k ; l +1 = a n,j,k ; l + na n,j − ,k − l + ( j − k )( a n,j − ,k ; l − a j − ,k − l ) , k = 0 , , · · · , [( j + 1) / . Here a n, , = 1 and if j, k and l do not satisfy 0 ≤ j ≤ l and 0 ≤ k ≤ [ j/ a n,j,k ; l = 0.If set a n,j,k = a n,j,k ; j , then a n,j, = j ! can be proved by induction on j . A ± n,j ; l satisfies the followingrecursion relations A ± n,j ; l +1 =( − n ( l +1) (cid:16) S l +1 B ( A ± n, ) S − l − jB (cid:17) [0] = ( − n (cid:16) S B l X j =0 A ± n,j ; l S j − j − B (cid:17) [0]18 ( − n (cid:16) l X j =0 (cid:0) A ± n,j ; l S B + ( S B A ± n,j ; l ) [0] (cid:1) S j − j − B (cid:17) [0] =( − n ( S B A ± n,j − l ) [0] + A ± n,j ; l , where ( P i b i S iB ) [0] = b . In particular, ( − n ( S B A ± n,j − j − ) [0] = A ± n,j ; j since A ± n,j ; j − = 0 by noting a n,j,k ; j − = 0.At last this lemma can be proved if a n,j,k ; l = C jl a n,j,k , which is equivalent to A ± n,j ; l = C jl A ± n,j ; j . Infact, if it holds for l , then by the recursion relation of A ± n,j ; l , one can find A ± n,j ; l +1 =( − n ( S B A ± n,j − l ) [0] + A ± n,j ; l = ( − n C j − l ( S B A ± n,j − j − ) [0] + C jl A ± n,j ; j =( C j − l + C jl ) A ± n,j ; j = C jl +1 A ± n,j ; j still holds for l + 1. (cid:3) The Darboux transformations of the KP hierarchy In this section, we firstly review some basic facts on the Darboux transformations of the KP hier-archy, especially the fermionic forms. Then we discuss the transformed tau functions in the fermionicforms under the successive applications of the Darboux transformations. At last, the bilinear equa-tions in the Darboux chains are investigated and some examples are given. Here we would like topoint out that the result in Subsection 3.4 for the Fermionic tau functions under the mixed using T D and T I is the key to find the bilinear equations.3.1. Review on the Darboux transformations of the KP hierarchy. Just as we know, thereare two basic Darboux transformations [8, 22, 42, 44] of the KP hierarchy, that is, T d (Φ) = Φ ∂ Φ − , T i (Ψ) = Ψ − ∂ − Ψ , where Φ and Ψ are the eigenfunction and the adjoint eigenfunction of the KP hierarchy respectively. T d and T i can be derived [46] by the Miura links between the KP and modified KP hierarchies through L anti-Miura −−−−−−→ L Miura −−−→ L [1] . Under the Darboux transformation, the corresponding changes of thedressing operator W , eigenfunction Φ , the adjoint eigenfunction Ψ and the tau function τ are listedin Table III [8, 22, 42, 44]. Table III. Darboux transformation: KP → KP L → L [1] W [1] = Φ [1]1 = Ψ [1]1 = τ [1] T d (Φ) = Φ ∂ Φ − Φ ∂ Φ − W ∂ − Φ(Φ / Φ) x − R ΦΨ dx/ Φ Φ τT i (Ψ) = Ψ − ∂ − Ψ Ψ − ∂ − Ψ W ∂ R ΨΦ dx/ Ψ − Ψ(Ψ / Ψ) x Ψ τ Here we use A [ n ] to denote the transformed object A under n -step Darboux transformations. Some-times in order to distinguish the actions of T d and T i , A +[ n ] is used to indicate the transformed A under n -step T d , while A − [ n ] is for T i . rom (21) and Proposition 3, one can obtain the corollary [30, 53] below. Corollary 14. If τ ( t ) is a tau function of the KP hierarchy, Φ( t ) is any eigenfunction and Ψ( t ) isany adjoint eigenfunction, then τ +[1] ( t ) = Φ( t ) τ ( t ) and τ − [1] ( t ) = Ψ( t ) τ ( t ) are still the tau functionsof the KP hierarchy, that is Res λ τ ± [1] ( t − ε ( λ − )) τ ± [1] ( t ′ + ε ( λ − )) e ξ ( t − t ′ ,λ ) = 0 , By direct computation, one can obtain the lemma [44, 52] below Lemma 15. The Bianchi diagram for Darboux transformations T d and T i can commute, L [1] T [1] β ' ' ◆◆◆◆◆◆◆◆◆◆◆◆◆ L [0] T α ♣♣♣♣♣♣♣♣♣♣♣♣♣ T β ' ' ◆◆◆◆◆◆◆◆◆◆◆◆◆ L [2] L [1] T [1] α ♣♣♣♣♣♣♣♣♣♣♣♣♣ that is, T d (Φ +[1]1 ) T d (Φ ) = T d (Φ +[1]2 ) T d (Φ ) , T i (Ψ − [1]1 ) T i (Ψ ) = T i (Ψ − [1]2 ) T i (Ψ ) ,T d (Φ − [1] ) T i (Ψ) = T i (Ψ +[1] ) T d (Φ) . Here T [1] β means the Darboux transformation generated by the (adjoint) eigenfunction of L [1] . Though almost everything in the KP hierarchy can be kept under the Darboux transformation,there is still some facts that are not clear after the Darboux transformation, for example, the relationsbetween the tau functions and the wave functions, which is given in the lemma below. Lemma 16. Under the Darboux transformation T d (Φ) and T i (Ψ) , T d (Φ)( ψ ( t, λ )) = λτ +[1] ( t − ε ( λ − )) τ +[1] ( t ) e ξ ( t,λ ) , ( T ∗ d (Φ)) − ( ψ ∗ ( t, λ )) = τ +[1] ( t + ε ( λ − )) λτ +[1] ( t ) e − ξ ( t,λ ) ; T i (Ψ)( ψ ( t, λ )) = τ − [1] ( t − ε ( λ − )) λτ − [1] ( t ) e ξ ( t,λ ) , ( T ∗ i (Ψ)) − ( ψ ∗ ( t, λ )) = λτ − [1] ( t + ε ( λ − )) τ − [1] ( t ) e − ξ ( t,λ ) Proof. Here we will only discuss the case T d , since it is almost the same in the case T i . Firstly byusing Lemma 2, one can obtain the result for T d (Φ)( ψ ( t, λ )). As for ( T ∗ d (Φ)) − ( ψ ∗ ( t, λ )), one can usethe expression of Ω(Φ( t ) , ψ ∗ ( t, λ )) in Proposition 1. (cid:3) Next we will express the Darboux transformation in the Fermionic language. For this, if introduce α = Res λ ρ ( λ ) ψ ( λ ) , α ∗ = Res λ λ ρ ∗ ( λ ) ψ ∗ ( λ ) , ith ρ ( λ ) and ρ ∗ ( λ ) given in Subsection 2.2, then by (14), (15) and (19),Φ( t ) = h | e H ( t ) αg | ih | e H ( t ) g | i , Ψ( t ) = h− | e H ( t ) α ∗ g | ih | e H ( t ) g | i . (49)Note that α ∈ V = ⊕ i ∈ Z C ψ i and α ∗ ∈ V ∗ = ⊕ i ∈ Z C ψ ∗ i . Therefore one can obtain the followingproposition [8, 53] by Table III. Proposition 17. Assume τ ( t ) = σ ( g | i ) = h | e H ( t ) g | i with g ∈ GL ∞ , then • Under T d (Φ) , g | i → αg | i , i.e., τ ( t ) → τ +[1] ( t ) = h | e H ( t ) αg | i . • Under T i (Ψ) , g | i → αg | i , i.e., τ ( t ) → τ − [1] ( t ) = h− | e H ( t ) α ∗ g | i . Remark : Slightly abusing the notation, we will use τ for g | i if τ ( t ) = σ ( g | i ) and let z = 1 in theisomorphism σ t between F and B . In these notations, τ T d (Φ) −−−→ τ +[1] = ατ and τ T i (Ψ) −−−→ τ − [1] = α ∗ τ .In what follows, we always believe the initial tau function is τ [0] = g | i , since other cases are almostthe same. Under the Darboux transformations, one can easily find F l T d −→ F l +1 and F l T i −→ F l − .3.2. Successive applications of T d in the Fermionic picture. Before further discussion, let us seethe changes of SEPs for the KP hierarchy, which can be computed by direct computations accordingto Table III. Lemma 18. Under the Darboux transformation T d (Φ) , Ω(Φ ( t ) +[1] , ψ ∗ ( t, λ ) +[1] ) = λ (cid:16) Ω(Φ ( t ) , ψ ∗ ( t, λ )) − Φ ( t )Φ( t ) Ω(Φ( t ) , ψ ∗ ( t, λ )) (cid:17) , Ω( ψ ( t, λ ) +[1] , Ψ ( t ) +[1] ) = 1 λ (cid:16) Ω( ψ ( t, λ ) , Ψ ( t )) − Ω(Φ( t ) , Ψ ( t ))Φ( t ) ψ ( t, λ ) (cid:17) . Under the Darboux transformation T i (Ψ) , Ω(Φ ( t ) − [1] , ψ ∗ ( t, λ ) − [1] ) = 1 λ (cid:16) Ω(Φ ( t ) , ψ ( t, λ )) − Ω(Φ ( t ) , Ψ( t ))Ψ( t ) ψ ∗ ( t, λ ) (cid:17) , Ω( ψ ( t, λ ) − [1] , Ψ ( t ) − [1] ) = λ (cid:16) Ω( ψ ( t, λ ) , Ψ ( t )) − Ψ ( t )Ψ( t ) Ω( ψ ( t, λ ) , Ψ( t )) (cid:17) . Firstly, consider the change of τ ( t ) under successive applications of T d . For this, let us discuss thefollowing Darboux transformation T d chain. L T d (Φ ( t )) −−−−−−→ L [1] T d (Φ [1]2 ( t )) −−−−−−→ L [2] → · · · → L [ n − T d (Φ [ n − n ( t )) −−−−−−−−→ L [ n ] , (50)where Φ , · · · , Φ n are independent eigenfunctions of the KP hierarchy. Further we assume α i ∈ V such that Φ i ( t ) = h | e H ( t ) α i g | ih | e H ( t ) g | i and denote T [ ~ n ] d = T d (Φ [ n − n ( t )) · · · T d (Φ ( t )). In what follows, A +[ ~ n ] (for example τ +[ ~ n ] ) is also used as the transformed research objects A under T [ ~ n ] d .For the Darboux chain (50), denote ρ [ j ] i ( λ ) and α [ j ] i ∈ V in the way below, ρ [ j ] i ( λ ) = − Ω(Φ [ j ] i ( t ′ ) , ψ ∗ ( t ′ , λ ) [ j ] ) , α [ j ] i = Res λ λ j ρ [ j ] i ( λ ) ψ ( λ ) . y Lemma 18, one can find for i > jρ [ j ] i ( λ ) = λ (cid:0) ρ [ j − i ( λ ) + c [ j − i ρ [ j − j ( λ ) (cid:1) , α [ j ] i = α [ j − i + c [ j − i α [ j − j , ρ [ i ] i ( λ ) = 0 , α [ i ] i = 0 . (51)where c [ j − i = − Φ [ j − i ( t ′ )Φ [ j − j ( t ′ ) can be viewed as a constant independent of t . Then one has following lemma. Proposition 19. Under the n -step Darboux transformation T d (see chain (50)), τ +[ ~ n ] = α [ n − n · · · α [0]1 τ [0] = α n · · · α τ [0] , that is, τ +[ ~ n ] ( t ) = h n | e H ( t ) α [ n − n · · · α [0]1 g | i = h n | e H ( t ) α n · · · α g | i , where we have set τ [0] ( t ) = h | e H ( t ) g | i .Proof. Obviously, it is correct for n = 0 and n = 1. Next assume τ [ n ] ( t ) = h n | e H ( t ) α [ n − n · · · α [0]1 g | i holds n . Then τ [ n +1] ( t ) =Φ n +1 ( t ) [ n ] τ [ n ] ( t ) = Res λ ρ [ n ] n +1 ( λ ) ψ ( t, λ ) [ n ] τ [ n ] ( t ) = Res λ ρ [ n ] n +1 ( λ ) τ [ n ] ( t − ε ( λ − )) e ξ ( t,λ ) =Res λ ρ [ n ] n +1 ( λ ) h n | e H ( t − ε ( λ − )) α [ n − n · · · α [0]1 g | i e ξ ( t,λ ) =Res λ λ − n ρ [ n ] n +1 ( λ ) h n + 1 | e H ( t ) ψ ( λ ) α [ n − n · · · α [0]1 g | i = h n + 1 | e H ( t ) α [ n ] n +1 α [ n − n · · · α [0]1 g | i . Based upon this, τ [ n ] ( t ) = h n | e H ( t ) α n · · · α g | i can be obtained by (51) and the fact α [ j ] i anti-commuteswith each other. (cid:3) Remark: For ~ n = ( n, n − , · · · , , ~α ~ n = ( α n , · · · , α ) and τ ~α ~ n = α n · · · α τ . According toProposition 19, one can know τ [ ~ n ] = τ ~α ~ n .3.3. Successive applications of T i in the Fermionic picture. Next consider the following chainof Darboux transformation T i , L T i (Ψ ( t )) −−−−−−→ L [1] T i (Ψ [1]2 ( t )) −−−−−−→ L [2] → · · · → L [ n − T i (Ψ [ n − n ( t )) −−−−−−−−→ L [ n ] . (52)Here Ψ , · · · , Ψ n are the independent adjoint eigenfunctions of the KP hierarchy. Denote α ∗ j ∈ V ∗ suchthat Ψ j ( t ) = h− | e H ( t ) α ∗ j g | ih | e H ( t ) g | i and T [ ~ n ] i = T i (Ψ [ n − n ( t )) · · · T i (Ψ ( t )). The transformed research object A of the KP hierarchy under T [ ~ n ] i will be denoted as A − [ ~ n ] . Let ρ ∗ [ j ] i ( λ ) = Ω( ψ ( t ′ , λ ) [ j ] , Ψ [ j ] i ( t ′ )) , α ∗ [ j ] i = Res λ λ j +1 ρ ∗ [ j ] i ( λ ) ψ ∗ ( λ ) , i > j satisfying ρ ∗ [ j ] i ( λ ) = λ (cid:0) b [ j − i ρ ∗ [ j − j ( λ ) + ρ ∗ [ j − i ( λ ) (cid:1) , α ∗ [ j ] i = b [ j − i α ∗ [ j − j + α ∗ [ j − i , here b [ j − i = − Ψ [ j − i ( t ′ )Ψ [ j − j ( t ′ ) is also believed as a constant independent of t . And also ρ ∗ [ i ] i = 0 and α ∗ [ i ] i = 0. Similarly, one can obtain the following proposition. Proposition 20. For the chain (52) of the Darboux transformation T i (Ψ) , τ − [ ~ n ] = α [ n − ∗ n · · · α ∗ τ [0] = α ∗ n · · · α ∗ τ [0] . (53) Remark: By Lemma 15, α ∗ α ∗ τ ≈ α ∗ α ∗ τ . If denote ~α ∗ ~ n = ( α ∗ n , · · · , α ∗ ) and τ ~α ∗ ~ n = α ∗ n · · · α ∗ τ with τ = g | i , then τ − [ ~ n ] = τ ~α ∗ ~ n .3.4. Successive applications of the mixed using T d and T i in the Fermionic picture. Nowlet us discuss the mixed using T d (Φ) and T i (Ψ). It is more complicated in this case. The possiblereason is that ψ i can not always anticommute or commute with ψ ∗ j , by noting that [ ψ i , ψ ∗ j ] + = δ ij .Consider the following Darboux chain of T d and T i , L T d (Φ ( t )) −−−−−−→ L [1] T d (Φ [1]2 ( t )) −−−−−−→ L [1] → · · · → L [ n − T d (Φ [ n − n ( t )) −−−−−−−−→ L [ n ] T i (Ψ [ n ]1 ( t )) −−−−−−→ L [ n +1] T i (Ψ [ n +1]2 ( t )) −−−−−−−−→ · · · → L [ n + k − T i (Ψ [ n + k − n + k ( t )) −−−−−−−−−−→ L [ n + k ] . (54)Denote T +[ ~ n ,~ k ] = T i (Ψ [ n + k − k ) · · · T i (Ψ [ n ]1 ) T [ ~ n ] d and T − [ ~ n ,~ k ] = T d (Φ [ n + k − n ) · · · T i (Φ [ k ]1 ) T [ ~ k ] i , and let τ ± [ ~ n ,~ k ] be the transformed tau functions under T ± [ ~ n ,~ k ] . Then it is obviously that T +[ ~ n ,~ k ] = T − [ ~ n ,~ k ] and τ +[ ~ n ,~ k ] ∼ τ − [ ~ n ,~ k ] .Let ρ ∗ +[ ~ n ] j and ρ − [ ~ k ] j be the transformed results of ρ ∗ j and ρ j under T [ ~ n ] d and T [ ~ k ] i respectively. Set α ∗ +[ ~ n ] j = Res λ λ n − ρ ∗ +[ ~ n ] j ( λ ) ψ ∗ ( λ ) and α − [ ~ k ] j = Res λ λ k ρ − [ ~ k ] j ( λ ) ψ ( λ ), which can also be written as α ∗ [ n ] j and α [ k ] j for short. Proposition 21. τ ± [ ~ n ,~ k ] are expressed in the following forms τ +[ ~ n ,~ k ] = α ∗ +[ ~ n ] k · · · α ∗ +[ ~ n ]1 α n · · · α τ [0] , τ − [ ~ n ,~ k ] = α − [ ~ k ] n · · · α − [ ~ k ]1 α ∗ k · · · α ∗ τ [0] . (55) Further τ +[ ~ n ,~ k ] = k X a =0 X ~γ ∈ H k,a X ~δ ∈ H n,n − a C +[ n,k ] a,~γ,~δ ~α ∗ ~ k \ ~γ ~α ~δ τ [0] , τ − [ ~ n ,~ k ] = n X a =0 X ~γ ∈ H n,a X ~δ ∈ H k,k − a C − [ n,k ] a,~γ,~δ ~α ~ n \ ~γ ~α ∗ ~δ τ [0] . (56) Here C ± [ n,k ] a,~γ,~δ is some constant independent of t , which can be determined by C +[ n,k ]0 , ,~ n = C − [ n,k ]0 , ,~ k = 1 ,and the following recursion relations, C +[ n,k +1] a,~γ,~δ = C +[ n,k ] a,~γ,~δ , C +[ n,k +1] a +1 , { k +1 }∪ ~γ,~δ \{ δ l } = C +[ n,k ] a,~γ,~δ n X i =1 ( − n + k − l Ψ [ i ] k +1 ( t ′ ) τ ~α δl ∪ ~α −−→ i − ( t ′ ) τ ~α −−→ i − ( t ′ ) , (57) C − [ n +1 ,k ] a,~ξ,~η = C − [ n,k ] a,~ξ,~η , C − [ n +1 ,k ] a +1 , { n +1 }∪ ~ξ,~η \{ η l } = C − [ n,k ] a,~ξ,~η k X i =1 ( − n + k − l Φ [ i ] n +1 ( t ′ ) τ ~α ∗ ηl ∪ ~α ∗−−→ i − ( t ′ ) τ ~α ∗−−→ i − ( t ′ ) , (58) with ~γ ∈ H k,a , ~δ ∈ H n,n − a , ξ ∈ H n,a , ~η ∈ H k,k − a . roof. By the similar way in Proposition 20, one can obtain τ +[ ~ n ,~ k ] = α ∗ +[ ~ n ] k · · · α ∗ +[ ~ n ]1 α n · · · α τ [0] . Next we will try to prove the result for τ +[ ~ n ,~ k ] by induction on k . Firstly by Lemma 18 one can obtain ρ ∗ +[ ~ n ] j ( λ ) = 1 λ n ρ ∗ [0] j ( λ ) + n X i =1 λ n +1 − i Ψ [ i ] j ( t ′ ) ψ [ i − ( t ′ , λ ) , which implies that α ∗ +[ ~ n ] j = α ∗ j + X l ∈ Z n X i =1 Ψ [ i ] j ( t ′ ) τ ~α −−→ i − ( t ′ ) σ t ′ ( ψ l α −−→ i − τ [0] ) · ψ ∗ l . (59)When k = 1, one can find τ +[ ~ n , = α ∗ +[ ~ n ]1 α ~ n τ [0] = α ∗ α ~n τ [0] + n X i =1 Ψ [ i ]1 ( t ′ ) τ ~α −−→ i − ( t ′ ) ( σ t ′ ⊗ · S ( α −−→ i − τ [0] ⊗ α ~ n τ [0] ) . Next compute S ( α −−→ i − τ [0] ⊗ α ~ n τ [0] ) by Lemma 9 and the fact S ( τ [0] ⊗ τ [0] ) = 0, that is, S ( α −−→ i − τ [0] ⊗ α ~ n τ [0] ) = S (1 ⊗ α ~ n ) · ( α −−→ i − τ [0] ⊗ τ [0] ) = n X l =1 ( − n − l α { l }∪−−→ i − τ [0] ⊗ α ~ n \{ l } τ [0] . Therefore by setting C +[ n, , ,~ n \{ l } = ( − n − l P li =1 Ψ [ i ]1 ( t ′ ) τ ~α l ∪ ~α −−→ i − ( t ′ ) /τ ~α −−→ i − ( t ′ ), τ +[ ~ n , = α ∗ α ~ n τ [0] + n X l =1 C +[ n, , ,~ n \{ l } α ~ n \{ l } τ [0] , which has the form of (56) with C +[ n, , ,~ n = 1.If τ +[ ~ n ,~ k ] in (56) is correct for k , then apply α ∗ +[ ~ n ] k +1 on τ +[ ~ n ,~ k ] , τ +[ ~ n , −−→ k + ] = k X a =0 X ~γ ∈ H k,a X ~δ ∈ H n,n − a C +[ n,k ] a,~γ,~δ (cid:16) α ∗ [ n ] k +1 α ∗ ~ k \ ~γ α ~δ τ [0] (cid:17) . Further by Lemma 9, α ∗ [ n ] k +1 α ∗ ~ k \ ~γ α ~δ τ [0] = α ∗ k +1 α ∗ ~ k \ ~γ α ~δ τ [0] + n X i =1 Ψ [ i ] k +1 ( t ′ ) τ ~α −−→ i − ( t ′ ) ( σ t ′ ⊗ · S ( α −−→ i − τ [0] ⊗ α ∗ ~ k \ ~γ α ~δ τ [0] )= α ∗ k +1 α ∗ ~ k \ ~γ α ~δ τ [0] + n X i =1 n − a X l =1 ( − n + k − l Ψ [ i ] k +1 ( t ′ ) τ ~α −−→ i − ( t ′ ) τ ~α { δl }∪−−→ i − ( t ′ ) α ∗ ~ k \ ~γ α ~δ \{ δ l } τ [0] After inserting the expression of α ∗ [ n ] k +1 α ∗ ~ k \ ~γ α ~δ τ [0] into τ +[ ~ n , −−→ k + ] and noting that H k +1 ,a can be dividedinto two groups: H k,a and { k + 1 } ∪ H k,a − , one can obtain that (56) for τ +[ ~ n ,~ k ] is also correct for k + 1 with the recursion relation (57).The results of τ − [ ~ n ,~ k ] can be proved in the similar method. (cid:3) emark : Here we would like to point out that the structures of τ ± [ ~ n ,~ k ] should be connected with theproduct of β n · · · β β ∗ k · · · β ∗ for β i ∈ V and β ∗ j ∈ V ∗ , that is, β n · · · β β ∗ k · · · β ∗ = n X a =0 X ~γ ∈ H k,a X η ∈ S n ( − sgnη [ β η ( a ) , β ∗ γ a ] + · · · [ β η (1) , β ∗ γ ] + β ∗ ~n \ ~γ β η ( n ) · · · β η ( a +1) ( − ak −| γ | +( n − a ) k / ( n − a )! . (60)Here S n is the n -th permutation group. Corollary 22. For any ~γ ∈ H k,a and ~δ ∈ H n,b , one has the following relations α ∗ +[ ~ n ] ~γ τ ~α ~δ = α ∗ +[ ~δ ] ~γ τ ~α ~δ , α − [ ~ n ] ~γ τ ~α ∗ ~δ = α − [ ~δ ] ~γ τ ~α ∗ ~δ . Proof. We firstly prove the case ~γ = { j } ∈ H k, . Assume ~δ = ~ b = ( b, b − , · · · , α ∗ +[ ~ n ] j τ α ~ b = α ∗ j α ~ b τ [0] + n X l =1 Ψ [ l ] j ( t ′ ) τ [ l − ( t ′ ) (cid:16) h l − | e H ( t ′ ) ⊗ (cid:17) · S ( α −−→ l − τ [0] ⊗ α ~ b τ [0] ) . Note that S ( α −−→ l − τ [0] ⊗ α ~ b τ [0] ) = 0 for l ≥ b + 1. So the sum P nl =1 will become into P bl =1 , which canlead to α ∗ +[ ~ n ] j τ ~α ~ b = α ∗ +[ ~ b ] j τ ~α ~ b . Therefore for general ~δ ∈ H n,b , one has α ∗ +[ ~ n ] j τ ~α ~ n \ ~δ = α ∗ +[( ~ n \ ~δ ) ∪ ~δ ] j τ ~α ~δ = α ∗ +[ ~δ ] j τ ~α ~δ . Here we have used α ∗ +[ ~ n ] j = α ∗ +[( ~ n \ ~δ ) ∪ ~δ ] j derived from ρ ∗ +[ ~ n ] j = ρ ∗ +[( ~ n \ ~δ ) ∪ ~δ ] j by Lemma 15 and τ +[ ~ n ] ≈ τ +[( ~ n \ ~δ ) ∪ ~δ ] . Next assume we have proved α ∗ +[ ~ n ] ~γ τ ~α ~δ = α ∗ +[ ~δ ] ~γ τ ~α ~δ for any ~γ ∈ H k,a . Then if ~γ =( γ a +1 , γ a , · · · , γ ) ∈ H k,a +1 , α ∗ +[ ~ n ] ~γ τ ~α ~δ = α ∗ +[ ~ n ] γ a +1 α ∗ +[ ~ n ] ~γ \{ γ a +1 } τ ~α ~δ = α ∗ +[ ~ n ] γ a +1 α ∗ +[ ~δ ] ~γ \{ γ a +1 } τ ~α ~δ = ( − a α ∗ +[ ~δ ] ~γ \{ γ a +1 } α ∗ +[ ~ n ] γ a +1 τ ~α ~δ =( − a α ∗ +[ ~δ ] ~γ \{ γ a +1 } α ∗ +[ ~δ ] γ a +1 τ ~α ~δ = α ∗ +[ ~δ ] ~γ τ ~α ~δ Similarly, one can prove α − [ ~ n ] ~γ τ ~α ∗ ~δ = α − [ ~δ ] ~γ τ ~α ∗ ~δ . (cid:3) Corollary 23. α ∗ +[ ~ n ] ~ k α −−−→ n + n and α − [ ~ k ] ~ n α ∗−−−→ k + k act on τ [0] in the way below α ∗ +[ ~ n ] ~ k α −−−→ n + n τ [0] = k X a =0 X ~γ ∈ H k,a X ~δ ∈ n + H n ,n − a C +[ k,n ,n ] a,~γ,~δ α ∗ +[ ~δ ∪ ~ n ] ~ k \ ~γ α ~δ ∪ ~ n τ [0] ,α − [ ~ k ] ~ n α ∗−−−→ k + k τ [0] = n X a =0 X ~γ ∈ H n,a X ~δ ∈ k + H k ,k − a C − [ n,k ,k ] a,~γ,~δ α − [ ~δ ∪ ~ k ] ~ n \ ~γ α ∗ ~δ ∪ ~ k τ [0] . Here C +[ k,n ,n ] a,~γ,~δ and C − [ n,k ,k ] a,~γ,~δ are some constant independent of t , which can be fixed by with C +[ k,n ,n ]0 , , −→ n = C − [ n,k ,k ]0 , , −→ k = 1 and the recursion relations below C +[ k +1 ,n ,n ] a,~γ,~δ = C +[ k,n ,n ] a,~γ,~δ , C +[ k +1 ,n ,n ] a +1 , { k +1 }∪ ~γ,~δ \{ δ l } = C +[ k,n ,n ] a,~γ,~δ l X i =1 ( − n + k − l − Ψ +[ ~δ ~ i ∪ ~ n ] k +1 ( t ′ ) τ +[ δ l ∪ ~δ −−→ i − ∪ ~ n ] ( t ′ ) τ +[ ~δ −−→ i − ∪ ~ n ] ( t ′ ) , − [ n +1 ,k ,k ] a,~ξ,~η = C − [ n,k ,k ] a,~ξ,~η , C − [ n +1 ,k ,k ] a +1 , { n +1 }∪ ~ξ,~η \{ η l } = C − [ n,k ,k ] a,~ξ,~η l X i =1 ( − k + n − l − Φ − [ ~η ~ i ∪ ~ n ] n +1 ( t ′ ) τ − [ η l ∪ ~η −−→ i − ∪ ~ n ] ( t ′ ) τ − [ ~η −−→ i − ∪ ~ n ] ( t ′ ) , with ~γ ∈ H k,a , ~δ ∈ n + H n ,n − a , ~ξ ∈ H n,a , ~η ∈ k + H k ,k − a .Proof. Here we only prove the first one, since the second is almost the same. Firstly, the result of α ∗ +[ ~ n ] ~ k α −−−→ n + n τ [0] for k = 0 is correct. So we can assume it holds for k . Then by the similar way inProposition 21 for ~δ ∈ n + H n ,n − a , ρ ∗ +[ ~δ ∪ ~ n ] k +1 = 1 λ n − a ρ ∗ +[ ~ n ] k +1 + n − a X i =1 λ n − a +1 − i Ψ +[ ~δ ~ i ∪ ~ n ] k +1 ( t ′ ) ψ +[ ~δ −−→ i − ∪ ~ n ] ( t ′ , λ ) ,α ∗ +[ ~δ ∪ ~ n ] k +1 = α ∗ +[ ~ n ] k +1 + X l ∈ Z n − a X i =1 Ψ +[ ~δ ~ i ∪ ~ n ] k +1 ( t ′ ) τ ~α δ −−→ i − ∪ α ~ n ( t ′ ) σ t ′ ( ψ l α δ −−→ i − ∪ α ~ n τ [0] ) ψ ∗ l , and α ∗ +[ ~ n ] k +1 α ~δ ∪ ~ n τ [0] = α ∗ +[ ~δ ∪ ~ n ] k +1 α ~δ ∪ ~ n τ [0] + n − a X l =1 ˜ C l α ~δ \{ δ l } α ~ n τ [0] , with ˜ C l = P li =1 ( − n − a − l − Ψ +[ δ ~ i ∪ ~ n ] k +1 ( t ′ ) τ ~αδl ∪ ~αδ −−→ i − ∪ ~α~ n ( t ′ ) τ ~αδ −−→ i − ∪ ~α~ n ( t ′ ) and ~δ = ( δ n − a , δ n − a − , · · · , δ ) ∈ n + H n ,n − a . Therefore α ∗ +[ ~ n ] −−→ k + α −−−→ n + n τ [0] = k X a =0 X ~γ ∈ H k,a X ~δ ∈ n + H n ,n − a C +[ k,n ,n ] a,~γ,~δ α ∗ +[ ~ n ] k +1 α ∗ +[ ~δ ∪ ~ n ] ~ k \ ~γ α ~δ ∪ ~ n τ [0] = X a,~γ,~δ C +[ k,n ,n ] a,~γ,~δ (cid:16) α ∗ +[ ~δ ∪ ~ n ] k +1 α ∗ +[ ~δ ∪ ~ n ] ~ k \ ~γ α ~δ ∪ ~ n + ( − k − a n − a X l =1 ˜ C l α ∗ +[( ~δ \{ δ l } ) ∪ ~ n ] ~ k \ ~γ α ~δ \{ δ l } α ~ n (cid:17) τ [0] , which is just the result for k + 1. Here we have used α ∗ +[ ~δ ∪ ~ n ] ~ k \ ~γ α ~δ \{ δ l } α ~ n τ [0] = α ∗ +[( ~δ \{ δ l } ) ∪ ~ n ] ~ k \ ~γ α ~δ \{ δ l } α ~ n τ [0] derived by Corollary 22. (cid:3) Remark: Denote O ± n,k as the linear combination of the transformed tau functions τ ± [ −→ n , −→ k ] for n ≤ n and k ≤ k , under no more than n -step T d and no more than k -step T i with the the generatingeigenfunctions and adjoint eigenfunctions in T ± [ ~ n ,~ k ] , then α ∗ +[ ~ n ] ~ k α −−−→ n + n τ [0] = α ∗ +[ −−−→ n + n ] ~ k α −−−→ n + n τ [0] + O + n + n − ,k − ,α − [ ~ k ] ~ n α ∗−−−→ k + k τ [0] = α − [ −−−→ k + k ] ~ n α ∗−−−→ k + k τ [0] + O − n − ,k + k − . Further it is important to note that by Lemma 9, S ( O ± n ′ ,k ′ ⊗ O ± n,k ) = O ± n ′ ,k ′ − ⊗ O ± n,k +1 + O ± n ′ +1 ,k ′ ⊗ O ± n − ,k . (61) .5. The bilinear equations in the Darboux chains of the KP hierarchy. After the preparationabove, now we can consider the generalization of the results about n -step T d showed in the bilinearequations (1) of the ( l − l ′ )-th modified KP hierarchy. Firstly by Lemma 11, Corollary 22 and Corollary23, one can easily obtain the following theorem. Theorem 24. Given n ′ ≥ n ≥ , k ′ ≥ k ≥ , S l (cid:16) τ +[ −→ n ′ , −→ k ′ ] ⊗ τ +[ ~ n ,~ k ] (cid:17) = l ! X ~γ ∈ k + H k ′− k,l ( − lk ′ −| ~γ | τ +[ −→ n ′ , −→ k ′ \ ~γ ] ⊗ τ +[ ~ n ,~γ ∪ ~ k ] , (62) S l (cid:16) τ − [ ~ n ,~ k ] ⊗ τ − [ −→ n ′ , −→ k ′ ] (cid:17) = l ! X ~γ ∈ n + H n ′− n,l ( − ln ′ −| ~γ | τ − [ ~γ ∪ ~ n ,~ k ] ⊗ τ − [ −→ n ′ \ ~γ, −→ k ′ ] . (63) In particular, S k ′ − k (cid:16) τ +[ −→ n ′ , −→ k ′ ] ⊗ τ +[ ~ n ,~ k ] (cid:17) = ( − ( k ′− k )( k ′− k − ( k ′ − k )! τ +[ −→ n ′ ,~ k ] ⊗ τ +[ ~ n , −→ k ′ ] , (64) S n ′ − n (cid:16) τ − [ ~ n ,~ k ] ⊗ τ − [ −→ n ′ , −→ k ′ ] (cid:17) = ( − ( n ′− n )( n ′− n − ( n ′ − n )! τ − [ −→ n ′ ,~ k ] ⊗ τ − [ ~ n , −→ k ′ ] , (65) S k ′ − k +1 (cid:16) τ +[ −→ n ′ , −→ k ′ ] ⊗ τ +[ ~ n ,~ k ] (cid:17) = S n − n ′ +1 (cid:16) τ − [ ~ n ,~ k ] ⊗ τ − [ −→ n ′ , −→ k ′ ] (cid:17) = 0 . (66) Remark: Here we do not discuss S l (cid:16) τ − [ −→ n ′ , −→ k ′ ] ⊗ τ − [ ~ n ,~ k ] (cid:17) and S l (cid:16) τ +[ ~ n ,~ k ] ⊗ τ +[ −→ n ′ , −→ k ′ ] (cid:17) . Actually accordingto Corollary 23, there will be many additional terms in them, just as Theorem 26 below. Remark: (64) and (65) are the generalization of (6). What’s more, we have S (cid:16) τ ⊗ τ α ~ n (cid:17) = n X l =1 ( − n − l τ α l ⊗ τ α ~ n \ α l , S ( τ α ∗ ~ n ⊗ τ ) = n X l =1 ( − n − l τ α ∗ ~ n \ α ∗ l ⊗ τ α ∗ l , which implies thatRes λ λ − n τ ( t ′ − ε ( λ − )) τ α ~ n ( t + ε ( λ − )) = n X l =1 ( − n − l τ α l ( t ′ ) τ α ~ n \ α l ( t ) , Res λ λ − n τ α ∗ ~ n ( t ′ − ε ( λ − )) τ ( t + ε ( λ − )) = n X l =1 ( − n − l τ α ∗ ~ n \ α ∗ l ( t ′ ) τ α ∗ l ( t ) . In particularly when n = 1, the relations above are just the bilinear equations (22) of the modifiedKP hierarchy of Kupershmidt-Kiso version. Corollary 25. τ ± [ ~ n ] satisfies the following bilinear equations S ( τ +[ −→ n ′ ] ⊗ τ +[ ~ n ] ) = 0 , S ( τ − [ ~ n ] ⊗ τ − [ −→ n ′ ] ) = 0 , n ′ ≥ n. (67) The corresponding Bosonic forms are Res λ λ n ′ − n τ +[ −→ n ′ ] ( t − ε ( λ − )) τ +[ ~ n ] ( t ′ + ε ( λ − )) e ξ ( t − t ′ ,λ ) = 0 , (68)Res λ λ n ′ − n τ − [ ~ n ] ( t − ε ( λ − )) τ − [ −→ n ′ ] ( t ′ + ε ( λ − )) e ξ ( t − t ′ ,λ ) = 0 , n ′ ≥ n. (69) emark: Note that relations (69) describe the transformed tau functions in the chain of the Darbouxtransformation T i . In fact, (69) is also the bilinear equations of ( n ′ − n )-th modified KP hierarchy,just by setting τ − n = τ − [ ~ n ] . What’s more, (68) and (69) constitute the whole discrete KP hierarchy[2, 18, 21], since the discrete variables in (68) only takes the non-negative values, while the ones (69)are all non-positive.Now we have discussed the bilinear equations between the transformed tau functions under n ′ -step T d and k ′ -step T i , and the ones under n -step T d and k -step T i for n ′ ≥ n , k ′ ≥ k . In fact, theserelations are not the whole, there should be other bilinear equations involving (cid:16) τ ± [ −→ n ′ ,~ k ] , τ ± [ ~ n , −→ k ′ ] (cid:17) and (cid:16) τ ± [ ~ n , −→ k ′ ] , τ ± [ −→ n ′ ,~ k ] (cid:17) with n ′ ≥ n , k ′ ≥ k , which are given in the theorem below. Theorem 26. For n ′ ≥ n , k ′ ≥ k , S (cid:16) τ ± [ −→ n ′ ,~ k ] ⊗ τ ± [ ~ n , −→ k ′ ] (cid:17) =0 , (70) S l ( τ ± [ ~ n , −→ k ′ ] ⊗ τ ± [ −→ n ′ ,~ k ] ) = l ! l X a =0 X ~γ ∈ k + H k ′− k,l − a X ~δ ∈ n + H n ′− n,a ( − a ( n ′ − k )+ lk ′ + ∓ (cid:16) ( k − k ′ ) a +( n ′ − n )( l − a ) (cid:17) × ( − | ~γ | + | ~δ | (cid:16) τ ± [ ~δ ∪ ~ n , −→ k ′ \ ~γ ] ⊗ τ ± [ −→ n ′ \ ~δ,~γ ∪ ~ k ] + X µ = ± M ± a + n,k ′ − l + a,n ′ − a,l − a + k ; µ (cid:17) . (71) Particularly, S k ′ − k + n ′ − n ( τ ± [ ~ n , −→ k ′ ] ⊗ τ ± [ −→ n ′ ,~ k ] ) = ( − ( n ′ − n )( k ′ − k )+ ( n ′− n )( n ′− n − + ( k ′− k )( k ′− k − × ( n ′ − n + k ′ − k )! (cid:16) τ ± [ −→ n ′ ,~ k ] ⊗ τ ± [ ~ n , −→ k ′ ] + M ± n ′ knk ′ ; ± (cid:17) , (72) S k ′ − k + n ′ − n +1 ( τ ± [ ~ n , −→ k ′ ] ⊗ τ ± [ −→ n ′ ,~ k ] ) = 0 , (73) where M ± n ′ knk ′ ;+ = O ± n ′ − ,k − ⊗ O ± n,k ′ and M ± n ′ knk ′ ; − = O ± n ′ ,k ⊗ O ± n − ,k ′ − .Proof. In fact, this theorem can be proved by using Lemma 11 and Corollary 23. Here we should pointout that there is no O + n ′ ,k ⊗ O + n − ,k ′ − -term in S k ′ − k + n ′ − n ( τ +[ ~ n , −→ k ′ ] ⊗ τ +[ −→ n ′ ,~ k ] ), since it is proportionalto α ∗ [ ~ n ] ~ k α −→ n ′ τ [0] ⊗ α ∗ [ ~ n ] −→ k ′ α ~ n τ [0] . Similar discussion can be done for S k ′ − k + n ′ − n ( τ − [ ~ n , −→ k ′ ] ⊗ τ − [ −→ n ′ ,~ k ] ) and S k ′ − k + n ′ − n +1 ( τ ± [ ~ n , −→ k ′ ] ⊗ τ ± [ −→ n ′ ,~ k ] ). (cid:3) Remark: Note that the terms M + n ′ knk ′ ;+ in (72) are usually the linear combination of the tau functionswith some constants as the coefficients. If we set these constants to be zero, then these terms vanish.In fact, this can be realized. For example, we can consider the Darboux transformations generated bythe (adjoint) wave functions, instead of the (adjoint) eigenfunctions.Next let us consider the Darboux chain generated by the wave functions. L T d ( ψ ( t,λ )) −−−−−−−→ L [1] T d ( ψ ( t,λ ) [1] ) −−−−−−−−→ L [2] → · · · → L [ n − T d ( ψ ( t,λ n ) [ n − ) −−−−−−−−−−→ L [ n ]28 i ( ψ ∗ ( t,µ ) [ n +1] ) −−−−−−−−−−→ L ( n +1) T i ( ψ ∗ ( t,µ ) [ n +2] ) −−−−−−−−−−→ · · · → L [ n + k − T i ( ψ ∗ ( t,µ k ) [ n + k − ) −−−−−−−−−−−−→ L [ n + k ] . Denote T [ ~ n ,~ k ] λµ = T i ( ψ ∗ ( t, µ k ) [ n + k − ) · · · T i ( ψ ∗ ( t, µ ) [ n ] ) T d ( ψ ( t, λ n ) [ n − ) · · · T d ( ψ ( t, λ )) as the succes-sive applications of n -step T d and k -step T i , and τ [ ~ n ,~ k ] λµ as the transformed tau functions under T [ ~ n ,~ k ] λµ .By the similar method as before, one can obtain τ [ ~ n ,~ k ] λµ = n Y j =2 λ − j +1 j k Y l =1 µ n − ll ψ λ ~ n ; µ ~ k τ [0] , (74)where ψ λ ~ n ; µ ~ k , ψ ∗ ( µ k ) · · · ψ ∗ ( µ ) ψ ( λ n ) · · · ψ ( λ ). Note that τ [ ~ n ,~ k ] λµ ≈ ψ λ ~ n ; µ ~ k τ [0] , i.e.,they determine thesame (adjoint) wave functions, which means that the corresponding T d and T i are the same in theDarboux chain above. So in what follows, we can always use τ [ ~ n ,~ k ] λµ = ψ λ ~ n ; µ ~ k τ [0] instead. Proposition 27. For n ′ ≥ n , k ′ ≥ k , S l (cid:16) τ [ −→ n ′ , −→ k ′ ] λµ ⊗ τ [ ~ n ,~ k ] λµ (cid:17) = l ! X ~γ ∈ k + H k ′− k,l ( − lk ′ −| ~γ | τ [ −→ n ′ , −→ k ′ \ ~γ ] λµ ⊗ τ [ ~ n ,~γ ∪ ~ k ] λµ ,S l (cid:16) τ [ ~ n ,~ k ] λµ ⊗ τ [ −→ n ′ , −→ k ′ ] λµ (cid:17) = l ! X ~γ ∈ n + H n ′− n,l ( − l ( k ′ − k + n ′ ) −| ~γ | τ [ ~γ ∪ ~ n ,~ k ] λµ ⊗ τ [ −→ n ′ \ ~γ, −→ k ′ ] λµ ,S (cid:16) τ [ −→ n ′ ,~ k ] λµ ⊗ τ [ ~ n , −→ k ′ ] λµ (cid:17) =0 ,S l (cid:16) τ [ ~ n , −→ k ′ ] λµ ⊗ τ [ −→ n ′ ,~ k ] λµ (cid:17) = l ! l X a =0 X ~γ ∈ k + H k ′− k,l − a X ~δ ∈ n + H n ′− n,a ( − a ( n ′ − k )+ lk ′ −| ~γ |−| ~δ | (cid:16) τ [ ~δ ∪ ~ n , −→ k ′ \ ~γ ] λµ ⊗ τ [ −→ n ′ \ ~δ,~γ ∪ ~ k ] λµ (cid:17) . Particularly, S k ′ − k (cid:16) τ [ −→ n ′ , −→ k ′ ] λµ ⊗ τ [ ~ n ,~ k ] λµ (cid:17) =( − ( k ′− k )( k ′− k − ( k ′ − k )! τ [ −→ n ′ ,~ k ] λµ ⊗ τ [ ~ n , −→ k ′ ] λµ ,S n ′ − n (cid:16) τ [ ~ n ,~ k ] λµ ⊗ τ [ −→ n ′ , −→ k ′ ] λµ (cid:17) =( − ( n ′ − n )( k ′ − k )+ ( n ′− n )( n ′− n − ( n ′ − n )! τ [ −→ n ′ ,~ k ] λµ ⊗ τ [ ~ n , −→ k ′ ] λµ ,S n ′ − n + k ′ − k (cid:16) τ [ ~ n , −→ k ′ ] λµ ⊗ τ [ −→ n ′ ,~ k ] λµ (cid:17) =( − ( n ′ − n )( k ′ − k )+ ( n ′− n )( n ′− n − + ( k ′− k )( k ′− k − × ( n ′ − n + k ′ − k )! τ [ −→ n ′ ,~ k ] λµ ⊗ τ [ ~ n , −→ k ′ ] λµ . Remark: Note that after several applications of the operator S , τ [ −→ n ′ , −→ k ′ ] λµ ⊗ τ [ ~ n ,~ k ] λµ , τ [ ~ n ,~ k ] λµ ⊗ τ [ −→ n ′ , −→ k ′ ] λµ and τ [ ~ n , −→ k ′ ] λµ ⊗ τ [ −→ n ′ ,~ k ] λµ will become into τ [ −→ n ′ ,~ k ] λµ ⊗ τ [ ~ n , −→ k ′ ] λµ up to a multiplication of a constant. Different fromthe usual (adjoint) eigenfunctions, we can believe the corresponding fields ψ ( λ ) and ψ ∗ ( µ ) for thewave function and the adjoint wave function respectively can anticommute with each other, since[ ψ ( λ ) , ψ ∗ ( µ )] + = µδ ( λ, µ ) can be viewed as zero when λ = µ . .6. Examples of the bilinear equations in the KP Darboux transformations. In this section,we will give some examples of the bilinear equations. Firstly for ( n ′ , k ′ ) = (1 , 1) and ( n, k ) = (0 , S ( τ ± [1 , ⊗ τ ± [0 , ) = ± τ ± [1 , ⊗ τ ± [0 , , S ( τ ± [0 , ⊗ τ ± [1 , ) = ∓ τ ± [1 , ⊗ τ ± [0 , , (75) S ( τ ± [0 , ⊗ τ ± [1 , ) = ± τ ± [0 , ⊗ τ ± [1 , ∓ τ ± [1 , ⊗ τ ± [0 , , (76) S ( τ ± [1 , ⊗ τ ± [0 , ) = 0 , (77)where in (76) we have used Corollary 23. The corresponding Bosonic forms are listed belowRes λ τ ± [1 , (cid:16) t − ε ( λ − ) (cid:17) τ ± [0 , (cid:16) t ′ + ε ( λ − ) (cid:17) e ξ ( t − t ′ ,λ ) = ± τ ± [1 , ( t ) τ ± [0 , ( t ′ ) , (78)Res λ τ ± [0 , (cid:16) t − ε ( λ − ) (cid:17) τ ± [1 , (cid:16) t ′ + ε ( λ − ) (cid:17) e ξ ( t − t ′ ,λ ) = ∓ τ ± [1 , ( t ) τ ± [0 , ( t ′ ) , (79)Res λ τ ± [0 , (cid:16) t − ε ( λ − ) (cid:17) τ ± [1 , (cid:16) t ′ + ε ( λ − ) (cid:17) λ − e ξ ( t − t ′ ,λ ) = ± τ ± [0 , ( t ) τ ± [1 , ( t ′ ) ∓ τ ± [1 , ( t ) τ ± [0 , ( t ′ ) , (80)Res λ τ ± [1 , (cid:16) t − ε ( λ − ) (cid:17) τ ± [0 , (cid:16) t ′ + ε ( λ − ) (cid:17) λ e ξ ( t − t ′ ,λ ) = 0 . (81)It should be noted that (77) or (81) is the bilinear equation of 2-nd modified KP hierarchy hierarchies,which is equivalent to the following equations [29] S ( τ ± [0 , ⊗ τ ± [1 , ) = − τ ± [1 , ⊗ τ ± [0 , , with the corresponding Bosonic formRes λ Res λ (cid:16) λ − − λ − (cid:17) τ ± [0 , (cid:16) t − [ λ − ] − [ λ − ] (cid:17) τ ± [1 , (cid:16) t ′ + [ λ − ] + [ λ − ] (cid:17) × e ξ ( t − t ′ ,λ )+ ξ ( t − t ′ ,λ ) = − τ ± [1 , ( t ) τ ± [0 , ( t ′ ) . Next we will try to understand (78), (79) and (80). Actually according to Table III and Proposition1, τ ± [1 , ( t ) = ∓ Ω(Φ( t ) , Ψ( t )) τ ( t ) , τ ± [1 , ( t ) = Φ( t ) τ ( t ) , τ ± [0 , ( t ) = Ψ( t ) τ ( t ) , τ [0 , = τ ( t ) . Thus (78), (79) and (80) will become intoRes λ Ω (cid:16) Φ( t − ε ( λ − )) , Ψ( t − ε ( λ − )) (cid:17) ψ ( t, λ ) ψ ∗ ( t ′ , λ ) = − Φ( t )Ψ( t ′ ) , (82)Res λ ψ ( t, λ ) ψ ∗ ( t ′ , λ )Ω (cid:16) Φ( t ′ + ε ( λ − )) , Ψ( t ′ + ε ( λ − )) (cid:17) = Φ( t )Ψ( t ′ ) . (83)Res λ Ω (cid:0) ψ ( t, λ ) , Ψ( t ) (cid:1) Ω (cid:0) Φ( t ′ ) , ψ ∗ ( t ′ , λ ) (cid:1) = Ω (cid:0) Φ( t ′ ) , Ψ( t ′ ) (cid:1) − Ω (cid:0) Φ( t ) , Ψ( t ) (cid:1) . (84)Note that (84) is also obtained in [35], but the method is very complicated. If further set t − t ′ = [ µ − ]in the three relations above (82)-(84), one can obtainΩ (cid:16) Φ( t − ε ( λ − )) , Ψ( t − ε ( λ − )) (cid:17) − Ω (cid:16) Φ( t ) , Ψ( t ) (cid:17) = − λ Φ( t )Ψ( t − ε ( λ − )) , (85) hich is an important property also given in [5, 53]. Conversely by considering (16), the relations(82)-(84) can also be derived from (85). In what follows, we will point out that (85) is very crucial inthe proof of ASvM formula [1,17], which connects the actions of the additional symmetries on the wavefunctions and the tau functions. Also (85) is very useful in the derivations of the bilinear equationsof the constrained KP hierarchy [15, 35].We will end this section with the example of Theorem 26. let ( n ′ , k ′ ) = (2 , 2) and ( n, k ) = (0 , τ [ n,k ] = τ +[ ~ n ,~ k ] , then we will have S ( τ [0 , ⊗ τ [2 , ) = − τ [2 , ⊗ τ [0 , + ( c +1 τ [1 , + c +0 τ [0 , ) ⊗ ( c − τ [0 , + c − τ [0 , + c − τ [0 , ) , (86)where c ± i are some constants. If apply another S on the above relation, one can find S ( τ [0 , ⊗ τ [2 , ) = 0 . (87)4. The Darboux transformations of the modified KP hierarchy In this section, the Darboux transformations of the modified KP hierarchy are discussed in theFermionic pictures. The transformed tau functions in the Fermionic forms under successive applica-tions of the Darboux transformations are given. Then based upon these results, the bilinear equationsin the Darboux chain are obtained. Also some examples are listed in the last subsection.4.1. Reviews on some facts of the Darboux transformations of the modified KP hierarchy. In the modified KP hierarchy, there are three elementary Darboux transformation operators [12,44,46],which are T ( ˆΦ) , ˆΦ − , T ( ˆΦ) , ˆΦ − x ∂, T ( ˆΨ) , ∂ − ˆΨ x , which can also be obtained by the Miura links [46] between the KP and modified KP hierarchiesthrough L Miura −−−→ L anti-Miura −−−−−−→ L [1] . Here the meaning of A [ n ] is the same as the KP case, i.e., thetransformed object A under n -step Darboux transformation. It is noted that T , T and T can notcommute with each other, i.e., T j T l = T l T j with j, l = 1 , , 3. So another two types of Darbouxtransformation operators are introduced [12, 44, 46], that is, T D ( ˆΦ) , T (1 [1] ) T ( ˆΦ) = ( ˆΦ − ) − x ∂ ˆΦ − , T I ( ˆΨ) , T (1 [1] ) T ( ˆΨ) = ˆΨ − ∂ − ˆΨ x , which are showed that they can commute with each other, i.e., T µ T ν = T ν T µ with µ, ν = D, I , andtherefore they are more applicable. One can see [12] for more details.Under the Darboux transformations of the modified KP hierarchy, the objects in the modified KPhierarchy are transformed in the way shown in Table IV [12]. able IV. Elementary Darboux transformations: modified KP → modified KP L mKP → L [1] mKP Z [1] = ˆΦ [1]1 = ˆΨ [1]1 = τ [1]0 = τ [1]1 = T = ˆΦ − ˆΦ − Z ˆΦ − ˆΦ R ˆΦ ˆΨ x dx τ ˆΦ τ T = ˆΦ − x ∂ ˆΦ − x ∂Z∂ − ˆΦ − x ˆΦ x − R ˆΦ x Ψ dx τ ˆΦ x τ /τ T = ∂ − ˆΨ x ∂ − ˆΨ x Z∂ R ˆΨ x ˆΦ dx − ˆΨ − x ˆΨ x ˆΨ x τ /τ τ T D = ( ˆΦ − ) − x ∂ ˆΦ − T D ( ˆΦ) Z∂ − ( ˆΦ / ˆΦ) x / ( ˆΦ − ) x R ˆΦ x ˆΨ dx/ ˆΦ ˆΦ τ − ˆΦ x τ /τ T I = ˆΨ − ∂ − ˆΨ x T I ( ˆΨ) Z∂ R ˆΨ x ˆΦ dx/ ˆΨ ( ˆΨ / ˆΨ) x / ( ˆΨ − ) x ˆΨ x τ /τ ˆΨ τ Remark: The changes of the tau functions under the Darboux transformation presented in the tableabove, are obtained by comparing the changes of the dressing operator. These proofs are not strict.In fact, we need to further show that the new tau functions satisfy the bilinear equations (22) of themodified KP hierarchy. We will discuss this questions in the next subsection.Before further discussion, the lemma below are needed. Lemma 28. Assume ˆΦ and ˆΨ are the eigenfunction and the adjoint eigenfunction of the modifiedKP hierarchy respectively, w ( t, λ ) and w ∗ ( t, λ ) are the wave and adjoint wave functions, and τ ( t ) and τ ( t ) are the tau functions. Then T ( ˆΦ)( w ( t, λ )) = τ [1]0 ( t − ε ( λ − )) τ [1]1 ( t ) e ξ ( t,λ ) , ∂ − ( T ∗ ( ˆΦ)) − ∂ ( w ∗ ( t, λ )) = τ [1]1 ( t + ε ( λ − )) λτ [1]0 ( t ) e − ξ ( t,λ ) ; T ( ˆΦ)( w ( t, λ )) = λτ [1]0 ( t − ε ( λ − )) τ [1]1 ( t ) e ξ ( t,λ ) , ∂ − ( T ∗ ( ˆΦ)) − ∂ ( w ∗ ( t, λ )) = τ [1]1 ( t + ε ( λ − )) λ τ [1]0 ( t ) e − ξ ( t,λ ) ; T ( ˆΨ)( w ( t, λ )) = τ [1]0 ( t − ε ( λ − )) λτ [1]1 ( t ) e ξ ( t,λ ) , ∂ − ( T ∗ ( ˆΨ)) − ∂ ( w ∗ ( t, λ )) = τ [1]1 ( t + ε ( λ − )) τ [1]0 ( t ) e − ξ ( t,λ ) and T D ( ˆΦ)( w ( t, λ )) = λτ [1]0 ( t − ε ( λ − )) τ [1]1 ( t ) e ξ ( t,λ ) , (cid:16) ∂ − ( T ∗ D ( ˆΦ)) − ∂ (cid:17) ( w ∗ ( t, λ )) = τ [1]1 ( t + ε ( λ − )) λ τ [1]0 ( t ) e − ξ ( t,λ ) ; T I ( ˆΨ)( w ( t, λ )) = τ [1]0 ( t − ε ( λ − )) λτ [1]1 ( t ) e ξ ( t,λ ) , (cid:16) ∂ − ( T ∗ I ( ˆΨ)) − ∂ (cid:17) ( w ∗ ( t, λ )) = τ [1]1 ( t + ε ( λ − )) τ [1]0 ( t ) e − ξ ( t,λ ) . Proof. Note that the results for T D and T I can be derived from the ones for T , T and T . So we hereonly need to discuss the case of T , T and T .Firstly T ( ˆΦ)( w ) can be obtained by direct computation. The results for (cid:16) ∂ − ( T ∗ ( ˆΦ)) − ∂ (cid:17) ( w ∗ ) canbe proved by using Proposition 6. As for T ( ˆΦ)( w ) and (cid:16) ∂ − ( T ∗ ( ˆΨ)) − ∂ (cid:17) ( w ∗ ), they can be derived byLemma 2. Next, we will concentrate on the proofs of (cid:16) ∂ − ( T ∗ ( ˆΦ)) − ∂ (cid:17) ( w ∗ ) and T ( ˆΨ)( w ). Actuallyaccording to Proposition 6 and Lemma 2, Z w ( t, z ) x w ∗ ( t, λ ) dx = − τ ( t + ε ( λ − )) τ ( t + ε ( λ − ) τ ( t ) w ( t + ε ( λ − ) , z ) x λ − e − ξ ( t,λ ) , here we have used λ − e ξ ( ε ( λ − ) ,z ) = − z − e ξ ( ε ( z − ) ,λ ) . Therefore by the spectral representation of theeigenfunction ˆΦ( t ) = Res λ ˆ ρ ( λ ) w ( t, λ ) (see Corollary 5). (cid:16) ∂ − ( T ∗ (Φ)) − ∂ (cid:17) ( w ∗ ( t, λ )) = − Res z ˆ ρ ( z ) Z w ( t, z ) x w ∗ ( t, λ ) dx = τ [1]1 ( t + ε ( λ − )) λ τ [1]0 ( t ) e − ξ ( t,λ ) . The result of T ( ˆΨ)( w ) can be proved in the similar way. (cid:3) Bilinear relations for τ [1]0 and τ [1]1 . Just as showed in the last section, one can assume the taupair of the modified KP hierarchy to be ( τ , τ ) = (cid:16) g | i , ˆ α g | i (cid:17) with g ∈ GL ∞ and ˆ α ∈ V . Thenif denote ˆ α = Res λ ˆ ρ ( λ ) ψ ( λ ) , ˆ α ∗ = Res λ λ − ˆ ρ ∗ ( λ ) ψ ∗ ( λ ) , with ˆ ρ ( λ ) and ˆ ρ ∗ ( λ ) given in Corollary 5, then the eigenfunction ˆΦ and the adjoint eigenfunction ˆΨcan be written in the forms below.ˆΦ( t ) = σ t ( ˆ ατ ) σ t ( τ ) , ˆΨ( t ) = σ t ( ˆ α ∗ τ ) σ t ( τ ) . (88)Then one can rewrite the Darboux transformations of the modified KP hierarchy in the Fermionicforms by (88), which are given in the table below.Table V. Fermionic Darboux transformations: modified KP → modified KP L mKP → L [1] mKP τ [1]0 = τ [1]1 = T = ˆΦ − τ ˆ ατ T = ˆΦ − x ∂ τ ˆ ατ T = ∂ − ˆΨ x − ˆ α ∗ τ τ T D = ( ˆΦ − ) − x ∂ ˆΦ − ˆ ατ − ˆ ατ T I = ˆΨ − ∂ − ˆΨ x − ˆ α ∗ τ ˆ α ∗ τ Here we only illustrate how to convert ˆΦ x τ /τ into the Fermionic form, and the case ˆΨ x τ /τ isalmost the same. In fact by Lemma 2 and the spectral representation,ˆΦ( t ) x τ ( t ) τ ( t ) =Res λ λ ˆ ρ ( λ ) τ ( t − ε ( λ − )) e ξ ( t,λ ) = Res λ h | e H ( t ) ψ ( λ ) ˆ α g | i = h | e H ( t ) ˆ α ˆ α g | i = σ t ( ˆ ατ ) . Then according to Theorem 24, one can obtain S (cid:16) τ [1]0 ⊗ τ [1]1 (cid:17) = τ [1]1 ⊗ τ [1]0 , (89)where for the case of T I , it is proved by Lemma 9 and the fact S ( τ ⊗ τ ) = τ ⊗ τ . The correspondingBosonic forms are presented in the theorem below. In fact, other forms such as ( τ , τ ) = (cid:16) ˆ α ∗ g | i , g | i (cid:17) with ˆ α ∗ ∈ V ∗ can also be allowed and the corresponding resultsare the same. heorem 29. Let ( τ , τ ) be the tau functions of the modified KP hierarchy, and denote ˆΦ and ˆΨ tobe the eigenfunction and the adjoint eigenfunction respectively. Then the following relations hold. Res λ λ − ˆΦ( t ′ + ε ( λ − )) τ ( t − ε ( λ − )) τ ( t ′ + ε ( λ − )) e ξ ( t − t ′ ,λ ) = ˆΦ( t ) τ ( t ) τ ( t ′ ) , Res λ λ − ˆΦ x ( t ′ + ε ( λ − )) τ ( t − ε ( λ − )) τ ( t ′ + ε ( λ − )) τ ( t ′ + ε ( λ − )) e ξ ( t − t ′ ,λ ) = ˆΦ x ( t ) τ ( t ) τ ( t ) τ ( t ′ ) , Res λ λ − ˆΨ x ( t − ε ( λ − )) τ ( t − ε ( λ − )) τ ( t − ε ( λ − )) τ ( t ′ + ε ( λ − )) e ξ ( t − t ′ ,λ ) = τ ( t ) ˆΨ x ( t ′ ) τ ( t ′ ) τ ( t ′ ) , and Res λ λ − ˆΦ( t − ε ( λ − )) τ ( t − ε ( λ − )) ˆΦ x ( t ′ + ε ( λ − )) τ ( t ′ + ε ( λ − )) τ ( t ′ + ε ( λ − )) e ξ ( t − t ′ ,λ ) = ˆΦ x ( t ) τ ( t ) τ ( t ) ˆΦ( t ′ ) τ ( t ′ ) , Res λ λ − ˆΨ x ( t − ε ( λ − )) τ ( t − ε ( λ − )) τ ( t − ε ( λ − )) ˆΨ( t ′ + ε ( λ − )) τ ( t ′ + ε ( λ − )) e ξ ( t − t ′ ,λ ) = ˆΨ( t ) τ ( t ) ˆΨ x ( t ′ ) τ ( t ′ ) τ ( t ′ ) . Remark: The relations in this theorem are usually very hard to prove in the Bosonic form, whilethey can be easily obtained in the Fermionic approach. By this theorem, it is again confirmed thatthe transformed tau functions under the Darboux transformations are correct.4.3. Successive applications of the Darboux transformations. By using the results in TableIV, one can obtain the lemma below. Lemma 30. Under the Darboux transformation of the modified KP hierarchy, the SEPs: Ω( ˆΦ , w ∗ ( x, λ ) x ) and ˆΩ( w ( t, λ ) x , ˆΨ ) are transformed in the way below.Table VI. The transformed SEPs of the modified KP hierarchy L mKP → L [1] mKP Ω( ˆΦ [1]1 , w ∗ [1] x ) = ˆΩ( w [1] x , ˆΨ [1]1 ) = T = ˆΦ − Ω( ˆΦ , w ∗ x ) ˆΩ( w x , ˆΨ ) − ˆΩ(ˆΦ x , ˆΨ )ˆΦ wT = ˆΦ − x ∂ λ (cid:16) Ω( ˆΦ , w ∗ x ) − ˆΦ w ∗ (cid:17) λ − (cid:16) ˆΩ( w x , ˆΨ ) − ˆΩ(ˆΦ x , ˆΨ )ˆΦ x w x (cid:17) T = ∂ − ˆΨ x λ − (cid:16) Ω( ˆΦ , w ∗ x ) − Ω(ˆΦ , ˆΨ x )ˆΨ x w ∗ x (cid:17) λ (cid:16) ˆΩ( w x , ˆΨ ) − ˆΨ w (cid:17) T D = ( ˆΦ − ) − x ∂ ˆΦ − λ (cid:16) Ω( ˆΦ , w ∗ x ) − ˆΦ ˆΦ Ω( ˆΦ , w ∗ x ) (cid:17) λ − (cid:16) ˆΩ( w x , ˆΨ ) − ˆΩ(ˆΦ x , ˆΨ )ˆΦ x w x (cid:17) T I = ˆΨ − ∂ − ˆΨ x λ − (cid:16) Ω( ˆΦ , w ∗ ) x ) − w ∗ x ˆΨ x Ω( ˆΦ , ˆΨ x ) (cid:17) λ (cid:16) ˆΩ( w x , ˆΨ ) − ˆΨ ˆΨ ˆΩ( w x , ˆΨ) (cid:17) Next the successive applications of the Darboux transformations are given in the proposition below. Proposition 31. Given τ and τ for the modified KP hierarchy, the corresponding changes underthe n -step Darboux transformation are given in the following table. able VII. Fermionic Darboux transformations: mKP → mKP L mKP → L [ ~ n ] mKP τ [ ~ n ]0 = τ [ ~ n ]1 = T = ˆΦ − τ ˆ α n τ T = ˆΦ − x ∂ ˆ α −−→ n − τ ˆ α ~ n τ T = ∂ − ˆΨ x ( − n ˆ α ∗ ~ n τ ( − n − ˆ α ∗−−→ n − τ T D = ( ˆΦ − ) − x ∂ ˆΦ − ˆ α ~ n τ ( − n ˆ α ~ n τ T I = ˆΨ − ∂ − ˆΨ x ( − n ˆ α ∗ ~ n τ ˆ α ∗ ~ n τ Here A [ ~ n ] is the same as the one in Section 4, ˆ α i ∈ V and ˆ α ∗ j ∈ V ∗ are corresponding to the eigenfunc-tion Φ i and the adjoint eigenfunction Ψ j respectively. Sometimes we use A +[ ~ n ] and A − [ ~ n ] to denotethe transformed object A under T D and T I respectively.Proof. Firstly, the results in case T are obviously by using Lemma 30. Next we will try to prove thecases of T and T D , and other two cases are similar. According to Table V, one can find τ [ n ]0 = τ [ n − for T . So here we only need to investigate τ [ n ]1 . By using Lemma 30,ˆ ρ [ n − n ( λ ) = Ω( ˆΦ ( n − n ( t ′ ) , w ( n − ∗ ( t ′ , λ ) x ) = λ n − ˆ ρ n ( λ ) − n − X i =1 λ i Φ [ n − i − n ( t ′ ) w [ n − i − ∗ ( t ′ , λ ) . Further by the similar way in Proposition 19,ˆ α [ n − n = Res λ λ n − ˆ ρ [ n − n ψ ( λ ) , τ [ n ]1 = α [ n − n τ [ n − . After inserting the expression of ˆ ρ [ n − n ( λ ),ˆ α [ n − n = ˆ α n − X l ∈ Z n − X i =1 Φ [ n − i − n ( t ′ ) τ [ n − i − ( t ′ ) σ t ′ ( ψ ∗ l ˆ α −−−−→ n − i − τ ) · ψ l Therefore if assume τ [ n − = ˆ α −−→ n − τ , then τ [ n ]1 = ˆ α [ n − n ˆ α −−→ n − τ = ˆ α n ˆ α −−→ n − τ − n − X i =1 Φ [ n − i − n ( t ′ ) τ [ n − i − ( t ′ ) S ( ˆ α −−→ n − τ ⊗ ˆ α −−−−→ n − i − τ )= ˆ α ~ n τ − n − X i =1 Φ [ n − i − n ( t ′ ) τ [ n − i − ( t ′ ) ( − i (cid:16) ˆ α −−→ n − \−−−−→ n − i − ⊗ (cid:17) · S ( ˆ α −−−−→ n − i − τ ⊗ ˆ α −−−−→ n − i − τ ) = ˆ α ~ n τ . As for T D , we only need to show ˆ α [ n − n · · · ˆ α [1]2 ˆ α = ˆ α n · · · ˆ α ˆ α , with ˆ α [ n − n = Res λ λ − n ˆ ρ [ n − n ( λ ) ψ ( λ ) and ˆ ρ [ n − n ( λ ) = Ω( ˆΦ [ n − n ( t ′ ) , w [ n − ∗ ( t ′ , λ ) x ). In fact, this canbe proved by using the relations below derived by Lemma 30ˆ ρ [ n − n ( λ ) = λ ˆ ρ [ n − n ( λ ) − ˆΦ [ n − n ( t ′ )ˆΦ [ n − n − ( t ′ ) ˆ ρ [ n − n − ( λ ) ! , ˆ α [ n − n = ˆ α [ n − n − ˆΦ [ n − n ( t ′ )ˆΦ [ n − n − ( t ′ ) ˆ α [ n − n − . (cid:3) ext consider the successive mixed applications of T D and T I . And let ˆ T ± [ ~ n ,~ k ] has the same meaningsas T ± [ ~ n ,~ k ] in the KP case, just replacing T d and T i with T D and T I respectively. A ± [ ~ n ,~ k ] denote thetransformed object A under ˆ T ± [ ~ n ,~ k ] such as the transformed tau pair (cid:16) τ ± [ ~ n ,~ k ]0 , τ ± [ ~ n ,~ k ]1 (cid:17) . Proposition 32. The expressions of (cid:16) τ ± [ ~ n ,~ k ]0 , τ ± [ ~ n ,~ k ]1 (cid:17) are given as follows (cid:16) τ +[ ~ n ,~ k ]0 , τ +[ ~ n ,~ k ]1 (cid:17) = ˆ α ∗ [ n ] k · · · ˆ α ∗ [ n ]1 ˆ α n · · · ˆ α (cid:16) ( − k τ [0]0 , ( − n τ [0]1 (cid:17) , (cid:16) τ − [ ~ n ,~ k ]0 , τ − [ ~ n ,~ k ]1 (cid:17) = ˆ α [ k ] n · · · ˆ α [ k ]1 ˆ α ∗ k · · · ˆ α ∗ (cid:16) ( − k τ [0]0 , ( − n τ [0]1 (cid:17) , where ˆ α ∗ [ n ] j = Res λ λ n − ˆ ρ ∗ [ n ] j ( λ ) ψ ∗ ( λ ) and ˆ α [ k ] l = Res λ λ k ˆ ρ [ k ] l ( λ ) ψ ( λ ) . Further (cid:16) τ +[ ~ n ,~ k ]0 , τ +[ ~ n ,~ k ]1 (cid:17) = k X a =0 X ~γ ∈ H k,a X ~δ ∈ H n,n − a ˆ C +[ n,k ] a,~γ,~δ ˆ α ∗ ~ k \ ~γ ˆ α ~δ (cid:16) ( − k τ [0]0 , ( − n τ [0]1 (cid:17) , (cid:16) τ − [ ~ n ,~ k ]0 , τ − [ ~ n ,~ k ]1 (cid:17) = n X a =0 X ~γ ∈ H n,a X ~δ ∈ H k,k − a ˆ C − [ n,k ] a,~γ,~δ ˆ α ~ n \ ~γ ˆ α ∗ ~δ (cid:16) ( − k τ [0]0 , ( − n τ [0]1 (cid:17) . Here ˆ C ± [ n,k ] a,~γ,~δ with ˆ C +[ n,k ]0 , ,~ n = ˆ C − [ n,k ]0 , ,~ k = 1 can be viewed as the constants independent of t , which satisfiesthe following recursion relations, ˆ C +[ n,k +1] a,~γ,~δ = ˆ C +[ n,k ] a,~γ,~δ , ˆ C +[ n,k +1] a +1 , { k +1 }∪ ~γ,~δ \{ δ l } = ˆ C +[ n,k ] a,~γ,~δ n X i =1 ( − n + k − l ˆΩ (cid:16) ˆΦ [ i − i,x ( t ′ ) , ˆΨ [ i − k +1 ( t ′ ) (cid:17) τ +[ { δ l }∪−−→ i − ]1 ( t ′ )ˆΦ [ i − i,x ( t ′ ) τ +[ i − ( t ′ ) /τ +[ i − ( t ′ ) , ˆ C − [ n +1 ,k ] a,~ξ,~η = ˆ C − [ n,k ] a,~ξ,~η , ˆ C − [ n +1 ,k ] a +1 , { n +1 }∪ ~ξ,~η \{ η l } = ˆ C − [ n,k ] a,~ξ,~η k X i =1 ( − n + k − l Ω (cid:16) ˆΦ [ i − n +1 ( t ′ ) , ˆΨ [ i − i,x ( t ′ ) (cid:17) τ − [ { η l }∪−−→ i − ]0 ( t ′ )ˆΦ [ i − i,x ( t ′ ) τ − [ i − ( t ′ ) /τ − [ i − ( t ′ ) , where ~γ ∈ H k,a , ~δ ∈ n + H n ,n − a , ~ξ ∈ H n,a , ~η ∈ k + H k ,k − a .Proof. Firstly according to Lemma 2 and Lemma 30, one can obtainˆ ρ ∗ [ n ] j ( λ ) = λ − n ˆ ρ ∗ [0] j ( λ ) − n X i =1 ˆΩ (cid:16) ˆΦ [ i − i,x ( t ′ ) , ˆΨ [ i − j ( t ′ ) (cid:17) λ n +1 − i ˆΦ [ i − i,x ( t ′ ) w [ i − x ( t ′ , λ ) , ˆ α ∗ [ n ] j = ˆ α ∗ [0] j − X l ∈ Z n X i =1 ( − i − ˆΩ (cid:16) ˆΦ [ i − i,x ( t ′ ) , ˆΨ [ i − j ( t ′ ) (cid:17) τ +[ i − ( t ′ )ˆΦ [ i − i,x ( t ′ ) τ +[ i − ( t ′ ) σ t ′ ( ψ l ˆ α −−→ i − τ [0]1 ) · ψ ∗ l . Then by the similar methods in Proposition 21, one can prove this proposition. (cid:3) Remark: Note that in the modified KP case, the tau pairs ( τ , τ ) and ( τ ′ , τ ′ ) can determine thesame dressing structures if and only if ( τ , τ ) = c ( τ ′ , τ ′ ). Therefore the sign in Proposition 31 andProposition 32 can not be moved to ensure that the transformed modified KP systems share the sameLax and dressing structures.By similar methods as Corollary 22 and 23, one can obtain the two corollaries below. orollary 33. For any ~γ ∈ H k,a and ~δ ∈ H n,b , one has the following relation ˆ α ∗ +[ ~ n ] ~γ ˆ α ~δ (cid:16) τ [0]0 , τ [0]1 (cid:17) = ˆ α ∗ +[ ~δ ] ~γ ˆ α ~δ (cid:16) τ [0]0 , τ [0]1 (cid:17) , ˆ α − [ ~ n ] ~γ ˆ α ∗ ~δ (cid:16) τ [0]0 , τ [0]1 (cid:17) = ˆ α − [ ~δ ] ~γ ˆ α ∗ ~δ (cid:16) τ [0]0 , τ [0]1 (cid:17) . Corollary 34. ˆ α ∗ +[ ~ n ] ~ k ˆ α −−−→ n + n and ˆ α − [ ~ k ] ~ n ˆ α ∗−−−→ k + k act on (cid:16) τ [0]0 , τ [0]1 (cid:17) in the way below ˆ α ∗ +[ ~ n ] ~ k ˆ α −−−→ n + n (cid:16) τ [0]0 , τ [0]1 (cid:17) = k X a =0 X ~γ ∈ H k,a X ~δ ∈ n + H n ,n − a ˆ C +[ k,n ,n ] a,~γ,~δ ˆ α ∗ +[ ~δ ∪ ~ n ] ~ k \ ~γ ˆ α ~δ ∪ ~ n (cid:16) τ [0]0 , τ [0]1 (cid:17) , ˆ α − [ ~ k ] ~ n ˆ α ∗−−−→ k + k (cid:16) τ [0]0 , τ [0]1 (cid:17) = n X a =0 X ~γ ∈ H n,a X ~δ ∈ k + H k ,k − a ˆ C − a,~γ,~δ ˆ α − [ ~δ ∪ ~ k ] ~ n \ ~γ ˆ α ∗ ~δ ∪ ~ k (cid:16) τ [0]0 , τ [0]1 (cid:17) . Here ˆ C +[ k,n ,n ] a,~γ,~δ and ˆ C − [ n,k ,k ] a,~γ,~δ with ˆ C +[ k,n ,n ]0 , , −→ n = ˆ C − [ n,k ,k ]0 , , −→ k = 1 are some constants independent of t ,satisfying ˆ C +[ k +1 ,n ,n ] a,~γ,~δ = ˆ C +[ k,n ,n ] a,~γ,~δ , ˆ C − [ n +1 ,k ,k ] a,~γ,~δ = ˆ C − [ n,k ,k ] a,~γ,~δ and ˆ C +[ k +1 ,n ,n ] a +1 , { k +1 }∪ ~γ,~δ \{ δ l } = ˆ C +[ k,n ,n ] a,~γ,~δ l X i =1 ( − n + k − l − ˆΩ (cid:16) ˆΦ +[ δ −−→ i − ∪ ~ n ] n + δ i ,x ( t ′ ) , ˆΨ +[ δ −−→ i − ∪ ~ n ] k +1 ( t ′ ) (cid:17) τ +[ { δ l }∪ δ −−→ i − ∪ ~ n ]1 ( t ′ )ˆΦ +[ δ −−→ i − ∪ ~ n ] n + δ i ,x ( t ′ ) τ +[ δ −−→ i − ∪ ~ n ]1 ( t ′ ) /τ +[ δ −−→ i − ∪ ~ n ]0 ( t ′ ) . ˆ C − [ n +1 ,k ,k ] a +1 , { n +1 }∪ ~ξ,~η \{ η l } = ˆ C − [ n,k ,k ] a,~ξ,~η l X i =1 ( − k + n − l − Ω (cid:16) ˆΦ − [ η −−→ i − ∪ ~ n ] n +1 ( t ′ ) , ˆΨ − [ η −−→ i − ∪ ~ n ] k + η i ,x ( t ′ ) (cid:17) τ − [ { η l }∪ η −−→ i − ∪ ~ n ]0 ( t ′ )ˆΨ − [ η −−→ i − ∪ ~ n ] k + η i ,x ( t ′ ) τ − [ η −−→ i − ∪ ~ n ]0 ( t ′ ) /τ − [ η −−→ i − ∪ ~ n ]1 ( t ′ ) , with ~γ ∈ H k,a , ~δ ∈ n + H n ,n − a , ~ξ ∈ H n,a , ~η ∈ k + H k ,k − a . Bilinear equations in the Darboux chains of the modified KP hierarchy. The bilinearequations in the Darboux chains of the modified KP hierarchy are mainly the ones for τ ± [ i,j ]0 ⊗ τ ± [ l,m ]0 , τ ± [ i,j ]1 ⊗ τ ± [ l,m ]1 , τ ± [ i,j ]0 ⊗ τ ± [ l,m ]1 and τ ± [ i,j ]1 ⊗ τ ± [ l,m ]0 with i, l ∈ { n ′ , n } and j, m ∈ { k ′ , k } , whichare in fact all contained in the KP case. But there are still some particular bilinear relations formodified KP itself, that is the ones for τ ± [ i,j ]0 ⊗ τ ± [ l,m ]1 and τ ± [ i,j ]1 ⊗ τ ± [ l,m ]0 . In fact by noting the facts S ( τ ⊗ τ ) = τ ⊗ τ and S ( τ ⊗ τ ) = 0, one can get the following results by using similar methods inTheorem 24 and Theorem 26. Theorem 35. Given n ′ ≥ n , k ′ ≥ k , j = 0 , , l ≥ , S l (cid:16) τ +[ −→ n ′ , −→ k ′ ] j ⊗ τ +[ ~ n ,~ k ]1 − j (cid:17) = l ! − j X i =0 X ~γ ∈ k + H k ′− k,l − i ( − ( l − i )( k ′ − −| ~γ | τ +[ −→ n ′ , −→ k ′ \ ~γ ] i + j ⊗ τ +[ ~ n ,~γ ∪ ~ k ]1 − i − j ,S l (cid:16) τ − [ ~ n ,~ k ] j ⊗ τ − [ −→ n ′ , −→ k ′ ]1 − j (cid:17) = l ! − j X i =0 X ~γ ∈ n + H n ′− n,l − i ( − ( l − i )( n ′ − −| ~γ | τ [ ~γ ∪ ~ n ,~ k ] i + j ⊗ τ [ −→ n ′ \ ~γ, −→ k ′ ]1 − i − j ,S (cid:16) τ ± [ −→ n ′ ,~ k ] j ⊗ τ ± [ ~ n , −→ k ′ ]1 − j (cid:17) = (1 − j ) τ ± [ −→ n ′ ,~ k ]1 − j ⊗ τ ± [ ~ n , −→ k ′ ] j ,S l ( τ ± [ ~ n , −→ k ′ ] j ⊗ τ ± [ −→ n ′ ,~ k ]1 − j ) = l ! − j X i =0 l − i X a =0 X ~γ ∈ k + H k ′− k,l − i − a X ~δ ∈ n + H n ′− n,a ( − a ( n ′ − k )+( l − i )( k ′ − −| ~γ |−| ~δ | × − ∓ (cid:16) ( k − k ′ ) a +( n ′ − n )( l − i − a ) (cid:17)(cid:16) τ ± [ ~δ ∪ ~ n , −→ k ′ \ ~γ ] i + j ⊗ τ ± [ −→ n ′ \ ~δ,~γ ∪ ~ k ]1 − i − j + X µ = ± M ± ( i + j ) a + n,k ′ − l + i + a,n ′ − a,l − i − a + k ; µ (cid:17) , In particular, S k ′ − k +1 − j (cid:16) τ +[ −→ n ′ , −→ k ′ ] j ⊗ τ +[ ~ n ,~ k ]1 − j (cid:17) = ( − ( k ′− k )( k ′− k − ( k ′ − k + 1 − j )! τ +[ −→ n ′ ,~ k ]1 ⊗ τ +[ ~ n , −→ k ′ ]0 ,S n ′ − n +1 − j (cid:16) τ − [ ~ n ,~ k ] j ⊗ τ − [ −→ n ′ , −→ k ′ ]1 − j (cid:17) = ( − ( n ′− n )( n ′− n − ( n ′ − n + 1 − j )! τ − [ −→ n ′ ,~ k ]1 ⊗ τ − [ ~ n , −→ k ′ ]0 ,S k ′ − k + n ′ − n +1 − j ( τ ± [ ~ n , −→ k ′ ] j ⊗ τ ± [ −→ n ′ ,~ k ]1 − j ) = ( − ( n ′ − n )( k ′ − k )+ ( n ′− n )( n ′− n − + ( k ′− k )( k ′− k − × ( n ′ − n + k ′ − k + 1 − j )! (cid:16) τ ± [ −→ n ′ ,~ k ]1 ⊗ τ ± [ ~ n , −→ k ′ ]0 + M ± n ′ knk ′ ; ± (cid:17) ,S k ′ − k +2 − j (cid:16) τ +[ −→ n ′ , −→ k ′ ] j ⊗ τ +[ ~ n ,~ k ]1 − j (cid:17) = 0 , S n ′ − n +2 − j (cid:16) τ − [ −→ n ′ , −→ k ′ ] j ⊗ τ − [ ~ n ,~ k ]1 − j (cid:17) = 0 ,S k ′ − k + n ′ − n +2 − j ( τ ± [ ~ n , −→ k ′ ] j ⊗ τ ± [ −→ n ′ ,~ k ]1 − j ) = 0 , where M ± ( i ) n ′ knk ′ ;+ = O ± ( i ) n ′ − ,k − ⊗ O ± (1 − i ) n,k ′ , M ± ( i ) n ′ knk ′ ; − = O ± ( i ) n ′ ,k ⊗ O ± (1 − i ) n − ,k ′ − and O ± ( i ) n,k means the linearcombinations of the transformed tau functions in the form of τ ± [ −→ n , −→ k ] i for n ≤ n , k ≤ k and i = 0 , ,under no more than n -step T D and no more than k -step T I with the the generating eigenfunctions andadjoint eigenfunctions in ˆ T ± [ ~ n ,~ k ] . We end this subsection with the bilinear equations in the Darboux chain generated by the wavefunctions. Denote ˆ T [ ~ n ,~ k ] λµ as the successive applications of n -step T D and k -step T I in the followingway, ˆ T [ ~ n ,~ k ] λµ = T I ( w ∗ ( t, µ k ) [ n + k − ) · · · T I ( w ∗ ( t, µ ) [ n ] ) T D ( w ( t, λ n ) [ n − ) · · · T D ( w ( t, λ ) [0] ) . And let (cid:16) τ ± [ ~ n ,~ k ]0 ,λµ , τ ± [ ~ n ,~ k ]1 ,λµ (cid:17) be the transformed tau functions under T ± [ ~ n ,~ k ] λµ starting from (cid:16) τ , τ (cid:17) . Bythe similar method as before, one can obtain (cid:16) τ [ ~ n ,~ k ]0 ,λµ , τ [ ~ n ,~ k ]1 ,λµ (cid:17) = n Y j =2 λ − j +1 j k Y l =1 µ n − ll ψ λ ~ n ; µ ~ k (cid:16) ( − k τ [0]0 , ( − n τ [0]1 (cid:17) , where ψ λ ~ n ; µ ~ k is the same as the KP case. Note that (cid:16) τ [ ~ n ,~ k ]0 ,λµ , τ [ ~ n ,~ k ]1 ,λµ (cid:17) ≈ ψ λ ~ n ; µ ~ k (cid:16) ( − k τ [0]0 , ( − n τ [0]1 (cid:17) , Therefore they determine the same (adjoint) wave functions. So in what follows, we can alwaysbelieve (cid:16) τ +[ ~ n ,~ k ]0 ,λµ , τ +[ ~ n ,~ k ]1 ,λµ (cid:17) = ψ λ ~ n ; µ ~ k (cid:16) ( − k τ [0]0 , ( − n τ [0]1 (cid:17) instead. Similarly as before, one can obtainthe proposition below. Proposition 36. For n ′ ≥ n , k ′ ≥ k , j = 0 , , S l (cid:16) τ [ −→ n ′ , −→ k ′ ] j,λµ ⊗ τ [ ~ n ,~ k ]1 − j,λµ (cid:17) = l ! − j X i =0 X ~γ ∈ k + H k ′− k,l − i ( − ( l − i )( k ′ − −| ~γ | τ [ −→ n ′ , −→ k ′ \ ~γ ] i + j,λµ ⊗ τ [ ~ n ,~γ ∪ ~ k ]1 − i − j,λµ , l (cid:16) τ [ ~ n ,~ k ] j,λµ ⊗ τ [ −→ n ′ , −→ k ′ ]1 − j,λµ (cid:17) = l ! − j X i =0 X ~γ ∈ n + H n ′− n,l − i ( − ( l − i )( k ′ − k + n ′ − −| ~γ | τ [ ~ n ∪ ~γ,~ k ] i + j,λµ ⊗ τ [ −→ n ′ \ ~γ, −→ k ′ ]1 − i − j,λµ ,S (cid:16) τ [ −→ n ′ ,~ k ] j,λµ ⊗ τ [ ~ n , −→ k ′ ]1 − j,λµ (cid:17) =(1 − j ) τ [ −→ n ′ ,~ k ]1 − j,λµ ⊗ τ [ ~ n , −→ k ′ ] j,λµ ,S l (cid:16) τ [ ~ n , −→ k ′ ] j,λµ ⊗ τ [ −→ n ′ ,~ k ]1 − j,λµ (cid:17) = l ! − j X i =0 l X a =0 X ~γ ∈ k + H k ′− k,l − i − a X ~δ ∈ n + H n ′− n,a ( − a ( n ′ − k )+( l − i )( k ′ − × ( − −| ~γ |−| ~δ | (cid:16) τ [ ~δ ∪ ~ n , −→ k ′ \ ~γ ] i + j,λµ ⊗ τ [ −→ n ′ \ ~δ,~γ ∪ ~ k ]1 − i − j,λµ (cid:17) . In particular, S k ′ − k +1 − j (cid:16) τ [ −→ n ′ , −→ k ′ ] j,λµ ⊗ τ [ ~ n ,~ k ]1 − j,λµ (cid:17) =( − ( k ′− k )( k ′− k − ( k ′ − k + 1 − j )! τ [ −→ n ′ ,~ k ]1 ,λµ ⊗ τ [ ~ n , −→ k ′ ]0 ,λµ ,S n ′ − n +1 − j (cid:16) τ [ ~ n ,~ k ] j,λµ ⊗ τ [ −→ n ′ , −→ k ′ ]1 − j,λµ (cid:17) =( − ( n ′ − n )( k ′ − k )+ ( n ′− n )( n ′− n − ( n ′ − n + 1 − j )! τ [ −→ n ′ ,~ k ]1 ,λµ ⊗ τ [ ~ n , −→ k ′ ]0 ,λµ ,S n ′ − n + k ′ − k +1 − j (cid:16) τ [ ~ n , −→ k ′ ] j,λµ ⊗ τ [ −→ n ′ ,~ k ]1 − j,λµ (cid:17) =( − ( n ′ − n )( k ′ − k )+ ( n ′− n )( n ′− n − + ( k ′− k )( k ′− k − × ( n ′ − n + k ′ − k + 1 − j )! τ [ −→ n ′ ,~ k ]1 ,λµ ⊗ τ [ ~ n , −→ k ′ ]0 ,λµ . Examples of the bilinear equations in the modified KP Darboux transformation. Herewe will present some examples of the bilinear equations in the subsection above. For ( n ′ , k ′ , n, k ) =(1 , , , n ′ , k ′ , n, k ) = (0 , , , 0) and ( n ′ , k ′ , n, k ) = (1 , , , S ( τ [1 , ⊗ τ [0 , ) = τ [1 , ⊗ τ [0 , , S ( τ [0 , ⊗ τ [1 , ) = τ [0 , ⊗ τ [1 , − τ [1 , ⊗ τ [0 , ,S ( τ [0 , ⊗ τ [0 , ) = τ [0 , ⊗ τ [0 , , S ( τ [0 , ⊗ τ [0 , ) = τ [0 , ⊗ τ [0 , − τ [0 , ⊗ τ [0 , ,S ( τ [1 , ⊗ τ [0 , ) = τ [1 , ⊗ τ [0 , − τ [1 , ⊗ τ [0 , ,S ( τ [0 , ⊗ τ [1 , ) = τ − [0 , ⊗ τ [1 , + τ [1 , ⊗ τ [0 , ,S ( τ [1 , ⊗ τ [0 , ) = τ [1 , ⊗ τ [0 , , S ( τ [0 , ⊗ τ [1 , ) = τ [1 , ⊗ τ [0 , − τ [0 , ⊗ τ [1 , + τ [0 , ⊗ τ [1 , , and S ( τ [1 , ⊗ τ [0 , ) = 0 , S ( τ [0 , ⊗ τ [1 , ) = − τ [1 , ⊗ τ [0 , ,S ( τ [0 , ⊗ τ [0 , ) = 0 , S ( τ [0 , ⊗ τ [0 , ) = − τ [0 , ⊗ τ [0 , ,S ( τ [1 , ⊗ τ [0 , ) = − τ [1 , ⊗ τ [0 , = − S ( τ [0 , ⊗ τ [1 , ) ,S ( τ [1 , ⊗ τ [0 , ) = 0 , S ( τ [0 , ⊗ τ [1 , ) = − τ [0 , ⊗ τ [1 , + τ [1 , ⊗ τ [0 , . Here we have set τ [0 , j = τ ± [0 , j , τ [1 , j = τ ± [1 , j and τ [1 , j = τ +[1 , j . These bilinear relations can bewritten into S ( τ [ l, j ⊗ τ [1 − l, − j ) = − δ l τ [1 , j ⊗ τ [0 , − j + (1 − j ) τ [ l, − j ⊗ τ [1 − l, j , ( τ [0 ,m ] j ⊗ τ [0 , − m ]1 − j ) = − δ m τ [0 , j ⊗ τ [0 , − j + (1 − j ) τ [0 ,m ]1 − j ⊗ τ [0 , − m ] j ,S ( τ [ l,m ] j ⊗ τ [1 − l, − m ]1 − j ) = − δ m τ [ l, j ⊗ τ [1 − l, − j + δ l τ [1 ,m ] j ⊗ τ [0 , − m ]1 − j + (1 − j ) τ [ l,m ]1 − j ⊗ τ [1 − l, − m ] j . The corresponding Bosonic forms are • Res λ λ j + l − τ [ l, j ( t − ε ( λ − )) τ [1 − l, − j ( t ′ + ε ( λ − )) e ξ ( t,λ ) − ξ ( t ′ ,λ ) = − δ l τ [1 , j ( t ) τ [0 , − j ( t ′ ) + (1 − j ) τ [ l, − j ( t ) τ [1 − l, j ( t ′ ) , • Res λ λ j − m ) τ [0 ,m ] j ( t − ε ( λ − )) τ [0 , − m ]1 − j ( t ′ + ε ( λ − )) e ξ ( t,λ ) − ξ ( t ′ ,λ ) = − δ m τ [0 , j ( t ) τ [0 , − j ( t ′ ) + (1 − j ) τ [0 ,m ]1 − j ( t ) τ [0 , − m ] j ( t ′ ) , • Res λ λ j + l − m ) − τ [ l,m ] j ( t − ε ( λ − )) τ [1 − l, − m ]1 − j ( t ′ + ε ( λ − )) e ξ ( t,λ ) − ξ ( t ′ ,λ ) = − δ m τ [ l, j ( t ) τ [1 − l, − j ( t ′ ) + δ l τ [1 ,m ] j ( t ) τ [0 , − m ]1 − j ( t ′ ) + (1 − j ) τ [ l,m ]1 − j ( t ) τ [1 − l, − m ] j ( t ′ ) . If we insert the relations below derived by Table II into the relation above, τ [1 , ( t ) = Φ( t ) τ ( t ) , τ [1 , ( t ) = − Φ x ( t ) τ ( t ) τ ( t ) ,τ [0 , ( t ) = Ψ x ( t ) τ ( t ) τ ( t ) , τ [0 , ( t ) = Ψ( t ) τ ( t ) ,τ [1 , ( t ) = − Ω(Φ( t ) , Ψ x ( t )) τ ( t ) , τ [1 , ( t ) = ˆΩ(Φ x ( t ) , Ψ( t )) τ ( t ) , then one can obtain many important relations about the modified KP hierarchy. Here we only take S ( τ [1 , ⊗ τ [0 , ), S ( τ [1 , ⊗ τ [0 , ) and S ( τ [0 , ⊗ τ [1 , ) as examples, which areRes λ Φ( t − ε ( λ − )) τ ( t − ε ( λ − )) τ ( t ′ + ε ( λ − )) e ξ ( t,λ ) − ξ ( t ′ ,λ ) = − Φ x ( t ) τ ( t ) τ ( t ) τ ( t ′ ) , (90)Res λ Ω(Φ( t − ε ( λ − )) , Ψ x ( t − ε ( λ − ))) w ( t, λ ) w ∗ ( t ′ , λ ) = Φ( t )Ψ( t ′ ) − ˆΩ(Φ x ( t ) , Ψ( t )) , (91)Res λ Ψ( t − ε ( λ − ))Φ( t ′ + ε ( λ − )) w ( t, λ ) w ∗ ( t ′ , λ ) = Ω(Φ( t ′ ) , Ψ x ( t ′ )) + ˆΩ(Φ x ( t ) , Ψ( t )) . (92)Here we would like to point out (91) or (92) can lead toΩ(Φ( t − ε ( λ − )) , Ψ x ( t − ε ( λ − ))) + ˆΩ(Φ x ( t ) , Ψ( t )) = Φ( t )Ψ( t − ε ( λ − )) , (93)which can be viewed as the analogue of (85) in the modified KP hierarchy. (93) is also very importantin the proof of the ASvM formula in the mKP case and derivation of the bilinear equations of the l -constrained modified KP hierarchy: L l = ( L l ) ≥ +Φ ∂ − Ψ ∂ (one can refer to [10,13] for these results).5. The Darboux transformations of the BKP hierarchy Reviews on the Darboux transformations of the BKP hierarchy. The Darboux trans-formation of the BKP hierarchy [23, 24, 54] can be constructed by the union of T d and T i (or T D and T I ), that is T (Φ B ) = T i (Φ B ) T d (Φ B ) = T I (Φ B ) T D (Φ B ) = 1 − B ∂ − Φ B,x , (94) here Φ B is the eigenfunction of the BKP hierarchy, satisfying Φ B,t k +1 = ( L k +1 B ) ≥ (Φ B ). Under T (Φ B ), the eigenfunction Φ B ( t ) and the tau function τ B ( t ) will become intoΦ B ( t ) T (Φ B ( t )) −−−−−−→ Φ B ( t ) [1] = Ω B (Φ B ( t ) , Φ B ( t ))Φ B ( t ) , τ B ( t ) T (Φ B ( t )) −−−−−−→ τ B ( t ) [1] = Φ B ( t ) τ B ( t ) , (95)where A [ ~ k ] or A [ k ] means the transformed objects under k -step T (Φ B ), i.e., T [ ~ k ] B = T (Φ [ n − B,n ) · · · T (Φ [0] B ),while A [ { i } ] denotes the transformed one under T (Φ Bi ). In particular by using (46), ψ B ( t, λ ) [1] = T (Φ B ( t ))( ψ B ( t, λ )) = τ [1] B ( t − e ε ( λ − )) τ [1] B ( t ) e ˜ ξ ( t,λ ) . Next we try to express the Darboux transformation of the BKP hierarchy in the Fermionic pictures.Firstly, assume τ B ( t ) = h | e H B ( t ) g | i for g ∈ O ∞ (we also use τ B = g | i ). Then if denote α B = √ λ λ − ρ B ( λ ) φ ( λ ) ∈ V B = ⊕ l ∈ Z C φ l with ρ B ( λ ) given in Proposition 7, then according to (39) and (45) one can findΦ B ( t ) = h | e H B ( t ) α B g | ih | e H B ( t ) g | i . Therefore in the Fermionic picture, τ B T (Φ B ( t )) −−−−−−→ τ [1] B = α B τ B . (96)Note that according to Lemma 12, S B ( α B τ B ⊗ α B τ B ) = τ B ⊗ α B τ B − α B τ B ⊗ τ B + 12 α B τ B ⊗ α B τ B = 12 α B τ B ⊗ α B τ B , where we have used α B is a constant. The corresponding Bosonic form is as follows,Res λ λ − Φ B ( t − e ε ( λ − )) τ B ( t − e ε ( λ − ))Φ B ( t ′ + 2 e ε ( λ − )) τ B ( t ′ − e ε ( λ − )) e ξ ( t,λ ) − ξ ( t ′ ,λ ) = Φ B ( t ) τ B ( t )Φ B ( t ′ ) τ B ( t ′ ) . This relation is also obtained in [36].5.2. Bilinear equations in the Darboux transformation of the BKP hierarchy. First of all,let’s discuss the transformed BSEP under the BKP Darboux transformation, in order to see thechanges of the BKP tau functions under the successive applications of the Darboux transformation.By using (95), Lemma 37. Under the BKP Darboux transformation T (Φ B ) , Ω(Φ [1] B ( t ) , ψ [1] B ( t, λ ) x ) = Ω(Φ B ( t ) , ψ B ( t, λ ) x ) − B ( t ) , Φ B ( t ) x )Φ B ( t ) Ω(Φ B ( t ) , ψ B ( t, λ ) x ) . roof. By using T (Φ B ) ∗ · ∂ · T (Φ B ) = ∂ , one can find ψ [1] B ( t, λ ) x = ( T (Φ B ) − ) ∗ ( ψ B ( t, λ ) x ). ThenΩ(Φ [1] B ( t ) , ψ [1] B ( t, λ ) x ) = − Φ − B · ( T d (Φ B ) − ) ∗ ( ψ B ( t, λ ) x ) · Ω (cid:16) T d (Φ B ) (cid:0) Φ B (cid:1) , Φ B (cid:17) + Ω (cid:16) T d (Φ B ) (cid:0) Φ B (cid:1) , ( T d (Φ B ) − ) ∗ ( ψ B ( t, λ ) x ) (cid:17) , leads to the corresponding result. (cid:3) Proposition 38. Under n -step BKP Darboux transformations, τ [ ~ n ] B = α [ n − B,n · · · α [1] B, α [0] B, τ [0] B = [ n/ X j =0 X ~γ ∈ H n,n − j C [ ~ n ] j,~γ α B,~γ τ [0] B , where α B,~γ = α B,γ n − j · · · α B,γ and α B,i is corresponding to the BKP eigenfunction Φ Bi ( t ) and C [ ~ n ] j,~γ is the constant independent of t with C [ ~ n ]0 ,~ n = 1 .Proof. According to Lemma 37 and α [ j ] B,i = √ λ λ − ρ [ j ] B,i ( λ ) φ ( λ ), one can find for i > j , ρ [ j ] B,i ( λ ) = ρ [ j − B,i ( λ ) + c [ j − B,i ρ [ j − B,j ( λ ) , α [ j ] B,i = α [ j − B,i + c [ j − B,i α [ j − B,j , where c [ j − B,i = − [ j − Bi ( t ′ ) , Φ [ j − Bj ( t ′ ) x ) / Φ [ j − Bj ( t ′ ) is some constant independent of t . Therefore α [ j ] B,i = α [0] B,i + j X l =0 a [ j ] i,l α [0] B,l , i > j. Here a [ j ] i,l is the constant satisfying a [ j ] i,j = c [ j − B,i and a [ j +1] i,l = a [ j ] i,l + C [ j ] B,i a [ j ] j +1 ,l . Then this propositioncan be proved by induction on n . The recursion relations for constants C [ ~ n ]0 ,~ n = 1 are very complicated.Since in what follows we only need to know C [ ~ n ] j,~γ is constant, so we omit its recursion relation here. (cid:3) Remark: Different from the KP and mKP case, α Bi can not commute or anticommute with eachother, and α Bi = 0. So the expression of τ [ ~ n ] B is more complicated, compared with τ ± [ ~ n ] in KP caseand τ ± [ ~ n ] i ( i = 0 , 1) in mKP case. Remark: Since τ [ ~ n ] B is the tau function of the BKP hierarchy, it satisfies S B (cid:16) τ [ n ] B ⊗ τ [ n ] B (cid:17) = τ [ n ] B ⊗ τ [ n ] B .Particularly for n = 2, the corresponding Bosonic form isRes λ λ − Ω B ( φ B ( t − e ε ( λ − )) , φ B ( t − e ε ( λ − ))) τ B ( t − e ε ( λ − )) × Ω B ( φ B ( t ′ − e ε ( λ − )) , φ B ( t ′ − e ε ( λ − ))) τ B ( t ′ + ε ( λ − )) e ξ ( t,λ ) − ξ ( t ′ ,λ ) =Ω B ( φ B ( t ) , φ B ( t ))Ω B ( φ B ( t ′ ) , φ B ( t ′ )) τ B ( t ) τ B ( t ′ ) . This relation is also obtained in [36], where the corresponding proof is very complicated. But herein the Fermionic approach, we can easily get it, which tells us that the Fermionic method is moreefficient. orollary 39. α B,~ n τ [0] B can be written into α B,~ n τ [0] B = [ n/ X l =0 X ~δ ∈ H n,n − l ˜ C [ ~ n ] l,~δ τ [ ~δ ] B , where ˜ C [ ~ n ] l,~δ satisfies ˜ C [ ~ n ]0 ,~ n = 1 and ˜ C [ ~ n ] l,~δ = − P ji =0 P ~γ ∈ H n,n − i C [ ~ n ] i,~γ ˜ C [ ~γ ] l − i,~δ for l ≥ , ~δ ∈ H j = { ~γ =( γ n − l , · · · , γ ) | ~γ ∈ H n − j,n − l , ~γ / ∈ H n − j − ,n − l } with H n,n − l = ∪ lj =0 H j .Proof. It is obviously correct for n = 0 and n = 1. If we assume it is corrected for < n , then byProposition 38, α B,~ n τ [0] B = τ [ ~ n ] B − [ n/ X i =1 X ~γ ∈ H n,n − i C [ ~ n ] i,~γ α B,~γ τ [0] B = τ [ ~ n ] B − [ n/ X i =1 [ n/ − i X l =0 X ~γ ∈ H n,n − i X ~δ ∈ H n − i,n − i − l C [ ~ n ] i,~γ ˜ C [ ~γ ] l,~δ τ [ γ ~δ ] B = τ [ ~ n ] B − [ n/ X l =1 l X j =0 X ~δ ∈ H j j X i =0 X ~γ ∈ H n,n − i C [ ~ n ] i,~γ ˜ C [ ~γ ] l − i,~δ τ [ γ ~δ ] B Then by noting that H n,n − l = ∪ lj =0 H j , this corollary can be proved. (cid:3) Denote O n to be the linear combination of the transformed tau functions, under no more than n -step T B with the corresponding generating eigenfunctions in T [ ~ n ] B . Then one can obtain the followingtheorem on the bilinear equations of τ [ n ] B . Theorem 40. Given n ′ ≥ n , one has the following relations S lB (cid:16) τ [ n ′ ] B ⊗ τ [ n ] B (cid:17) =( − ( n ′ − n ) l l X j =0 − ( l − j ) C jl [ j/ X k =0 a n ′ − n,j,k X ~γ ∈ n + H n ′− n,j − k ( − −| ~γ | + n ( j − k ) × (cid:16) τ [ −→ n ′ \ ~γ ] B ⊗ τ [ ~γ ∪ ~ n ] B + O n ′ − j +2 k − ⊗ O n + j − k + O n ′ − j +2 k ⊗ O n + j − k − (cid:17) ,S lB (cid:16) τ [ n ] B ⊗ τ [ n ′ ] B (cid:17) =( − ( n ′ − n ) l l X j =0 − ( l − j ) C jl [ j/ X k =0 a n ′ − n,j,k X ~γ ∈ n + H n ′− n,j − k ( − −| ~γ | + n ( j − k ) × (cid:16) τ [ ~γ ∪ ~ n ] B ⊗ τ [ −→ n ′ \ ~γ ] B + O n + j − k − ⊗ O n ′ − j +2 k + O n + j − k ⊗ O n ′ − j +2 k − (cid:17) . Proof. Since S lB (cid:16) τ [ n ] B ⊗ τ [ n ] B (cid:17) = 2 − l (cid:16) τ [ n ] B ⊗ τ [ n ] B (cid:17) , so we can only consider S lB (cid:16) τ [ n ] B ⊗ τ [0] B (cid:17) . Accordingto Lemma 13, Proposition 38 and Corollary 39, S lB (cid:16) τ [ n ] B ⊗ τ [0] B (cid:17) = [ n/ X j =0 X ~γ ∈ H n,n − j S lB ( α B,~γ ⊗ τ [0] B ⊗ τ [0] B ) ( − nl [ n/ X j =0 X ~γ ∈ H n,n − j C [ ~ n ] j,~γ l X i =0 − ( l − j ) C jl [ i/ X k =0 a n − j,i,k X ~δ ∈ H n − j,i − k ( − −| ~δ | × (cid:16) τ [ ~γ \ γ ~δ ] B ⊗ τ [ ~δ ] B + O n − j − i +2 k − ⊗ O i − k + O n − j − i +2 k ⊗ O i − k − ) . Further by using C [ ~ n ]0 ,~ n = 1, one can obtain the corresponding result. (cid:3) Remark: It can be proved that the results of S lB (cid:16) τ [ n ′ ] B ⊗ τ [ n ] B (cid:17) and S lB (cid:16) τ [ n ] B ⊗ τ [ n ′ ] B (cid:17) are equivalent inthe Bosonic form by using the fact Res λ f ( λ ) = − Res λ f ( − λ ).5.3. Examples of the bilinear equations in the BKP Darboux transformation. In this sec-tion, one give some example for the bilinear equations for the BKP hierarchy. S B ( τ [0] B ⊗ τ [1] B ) = τ [0] B ⊗ τ [1] B − τ [1] B ⊗ τ [0] B ,S B ( τ [2] B ⊗ τ [0] B ) = τ [ { } ] B ⊗ τ [ { } ] B − τ [ { } ] B ⊗ τ [ { } ] B + 12 τ [2] B ⊗ τ [0] B S B ( τ [ { } ] B ⊗ τ [ { } ] B ) = τ [2] B ⊗ τ [0] B − τ [0] B ⊗ τ [2] B + 12 τ [ { } ] B ⊗ τ [ { } ] B . The corresponding Bosonic forms areRes λ λ − τ [1] B ( t − e ε ( λ − )) τ [0] B ( t ′ + ε ( λ − )) e e ξ ( t,λ ) − e ξ ( t ′ ,λ ) = 2 τ [0] B ( t ) τ [1] B ( t ′ ) − τ [1] B ( t ) τ [0] B ( t ′ ) , (97)Res λ λ − Ω B (Φ B ( t − e ε ( λ − )) , Φ B ( t − e ε ( λ − ))) w ( t, λ ) w ( t ′ , λ )= 2Φ B ( t )Φ B ( t ′ ) − B ( t )Φ B ( t ′ ) + Ω B (Φ B ( t ) , Φ B ( t )) , (98)Res λ λ − Φ B ( t − e ε ( λ − ))Φ B ( t ′ + 2 e ε ( λ − )) w ( t, λ ) w ( t ′ , λ )= 2Ω B (Φ B ( t ) , Φ B ( t )) − B (Φ B ( t ′ ) , Φ B ( t ′ )) + Φ B ( t )Φ B ( t ′ ) , (99)where we have used τ [ { } ] B ( t ) = Φ B ( t ) τ B ( t ), τ [ { } ] B ( t ) = Φ B ( t ) τ B ( t ) and τ [2] B = Ω B (Φ B ( t ) , Φ B ( t )) τ B ( t ).Here the first relation (97) is just the bilinear equation of the modified BKP hierarchy. By notingthat Ω B (Φ B , 1) = Φ B , so if we set Φ B = 1 (one should recall that 1 is also the eigenfunction of theBKP hierarchy), then the relation (98) is transformed into the bilinear equations of the modified BKPhierarchy [28,51]. Further it can be found that (98) and (99) can give rise to the following relation [11].Ω B ( φ B ( t − e ε ( λ − )) , φ B ( t − e ε ( λ − ))) − Ω B ( φ B ( t ) , φ B ( t ))=Φ B ( t )Φ B ( t − e ε ( λ − )) − Φ B ( t )Φ B ( t − e ε ( λ − )) , (100)which is also very important in deriving the ASvM formula in the additional symmetries of theBKP hierarchy [34, 49] and getting the bilinear equations [47] of the l constrained BKP hierarchy: L lB = ( L lB ) ≥ + Φ B ∂ − Φ B ,x − Φ B ∂ − Φ B ,x by using the method in Subsection 3.6. . Conclusions and Discussions In this paper, we have established various bilinear equations of the transformed tau functionsunder the Darboux transformations for the KP, modified KP and BKP hierarchies, which are givenin Subsection 3.5, Subsection 4.4 and Subsection 5.2 respectively. All these results are based upontwo key things. One is the important relations about the free Fermions in Subsection 2.5, another isthe transformed tau functions in Fermionic forms under the successive applications of the Darbouxtransformations (see Proposition 21, Proposition 32 and Proposition 38). Here we would like to pointout the major difficulty in this paper. Note that free Fermionic field α ∈ V do not commute oranticommute with α ∗ ∈ V ∗ , and similarly the neutral free Fermionic field α B ∈ V B also do notcommute or anticommute with each other. Also α B = 0, different from α = α ∗ = 0. All thesefacts bring much difficulty for mixed using T d and T i (or T D and T I ), and for the BKP Darbouxtransformation. The bilinear equations in KP case involving mixed using T d and T i , and the onesfor the modified KP and BKP hierarchy should be new ones. These bilinear equations are usuallyvery hard to prove in the Bosonic forms, while they can be easily obtained by using the Fermionicapproach. The corresponding examples are given in Subsection 3.6, Subsection 4.5 and Subsection 5.3respectively.There should be some questions needing further discussions. Though we have obtained all thepossible bilinear equations in this paper, it will be very interesting to discuss the relations among themand check which bilinear equations are essential in determining the transformed tau functions in theDarboux chain. 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