Auxiliary Linear Problem, Difference Fay Identities and Dispersionless Limit of Pfaff-Toda Hierarchy

Recently the study of Fay-type identities revealed some new features of the DKP hierarchy (also known as"the coupled KP hierarchy"and"the Pfaff lattice"). Those results are now extended to a Toda version of the DKP hierarchy (tentatively called"the Pfaff-Toda hierarchy"). Firstly, an auxiliary linear problem of this hierarchy is constructed. Unlike the case of the DKP hierarchy, building blocks of the auxiliary linear problem are difference operators. A set of evolution equations for dressing operators of the wave functions are also obtained. Secondly, a system of Fay-like identities (difference Fay identities) are derived. They give a generating functional expression of auxiliary linear equations. Thirdly, these difference Fay identities have well defined dispersionless limit (dispersionless Hirota equations). As in the case of the DKP hierarchy, an elliptic curve is hidden in these dispersionless Hirota equations. This curve is a kind of spectral curve, whose defining equation is identified with the characteristic equation of a subset of all auxiliary linear equations. The other auxiliary linear equations are related to quasi-classical deformations of this elliptic spectral curve.


Introduction
This paper is a sequel of the study on Fay-type identities of integrable hierarchies, in particular the DKP hierarchy [1]. The DKP hierarchy is a variant of the KP hierarchy and obtained as a subsystem of Jimbo and Miwa's hierarchy of the D ′ ∞ type [2,3]. The same hierarchy was rediscovered later on as "the coupled KP hierarchy" [4] and "the Pfaff lattice" [5,6,7], and has been studied from a variety of points of view [8,9,10,11,12,13,14,15,16]. The term "Pfaff" stems from the fact that Pfaffians play a role in many aspects of this system. The previous study [1] revealed some new features of this relatively less known integrable hierarchy. In this paper, we extend those results to a Toda version of the DKP hierarchy.
The integrable hierarchy in question is a slight modification of the system proposed by Willox [17,18] as an extension of the Jimbo-Miwa D ′ ∞ hierarchy. We call this system, tentatively, "the Pfaff-Toda hierarchy" (as an abbreviation of the "Pfaffian" or "Pfaffianized" Toda hierarchy). Following the construction of Jimbo and Miwa, Willox started from a fermionic definition of the tau function, and derived this hierarchy in a bilinear form. The lowest level of this hierarchy contains a 2 + 2D (2 continuous and 2 discrete) extension 1 2 D x D y τ (s, r, x, y) · τ (s, r, x, y) + τ (s − 1, r, x, y)τ (s + 1, r, x, y) of the usual 2 + 1D Toda equation and an additional 2 + 2D equation D x τ (s, r, x, y) · τ (s + 1, r − 1, x, y) + D y τ (s, r − 1, x, y) · τ (s + 1, r, x, y) = 0 (see the papers of Santini et al. [19], Hu et al. [20] and Gilson and Nimmo [21] for some other sources of these equations). Willox further presented an auxiliary linear problem for these lowest equations, but extending it to the full hierarchy was an open problem. We first address this issue, then turn to issues of Fay-like identities and dispersionless limit.
As we show in this paper, the Pfaff-Toda hierarchy is indeed a mixture of the DKP and Toda hierarchies. Firstly, we can formulate an auxiliary linear problem as a two-component system like that of the DKP hierarchy [1], but building blocks therein are difference (rather than differential) operators as used for the Toda hierarchy. Secondly, the differential Fay identities of the DKP hierarchy are replaced by "difference Fay identities" analogous to those of the Toda hierarchy [22,23]. Lastly, those difference Fay identities have well defined dispersionless limit to the so called "dispersionless Hirota equations". These equations resemble the dispersionless Hirota equations of the Toda hierarchy [22,24,25,26], but exhibits a more complicated structure parallel to the dispersionless Hirota equations of the DKP hierarchy [1].
Among these rich contents, a particularly remarkable outcome is the fact that an elliptic curve is hidden in the dispersionless Hirota equations. A similar elliptic curve was also encountered in the dispersionless Hirota equations of the DKP hierarchy [1], but its true meaning remained to be clarified. This puzzle was partly resolved by Kodama and Pierce [27]. They interpreted the curve as an analogue of the "spectral curve" of the dispersionless 1D Toda lattice. We can now give a more definite answer to this issue. Namely, these curves are defined by the characteristic equations of a subset of the full auxiliary linear equations, hence may be literally interpreted as spectral curves. The other auxiliary linear equations are related to "quasi-classical deformations" [28,29] of these curves. This paper is organized as follows. In Section 2, we formulate the Pfaff-Toda hierarchy as a bilinear equation for the tau function. This bilinear equation is actually a generating functional expression of an infinite number of Hirota equations. In Section 3, we present a full system of auxiliary linear equations that contains Willox's auxiliary linear equations. A system of evolution equations for "dressing operators" of the wave functions are also obtained. The dressing operators are difference operators in a direction (s-direction) of the 2D lattice; another direction (r-direction) plays the role of a discrete time variable. Section 4 deals with the difference Fay identities. These Fay-like identities are derived from the bilinear equation of Section 2 by specializing the values of free variables. We show that they are auxiliary linear equations in disguise, namely, they give a generating functional expression of the auxiliary linear equations of Section 3. Section 5 is devoted to the issues of dispersionless limit. The dispersionless Hirota equations are derived from the differential Fay identities as a kind of "quasi-classical limit". After rewriting these dispsersionless Hirota equations, we find an elliptic curve hidden therein, and identify a set of auxiliary linear equations for which the curve can be interpreted as a spectral curve.

Bilinear equations
The Pfaff-Toda hierarchy has two discrete variables s, r ∈ Z and two sets of continuous variables t = (t 1 , t 2 , . . .),t = (t 1 ,t 2 , . . .). In this section, we present this hierarchy in a bilinear form, which comprises various bilinear equations for the tau function τ = τ (s, r, t,t). In the following consideration, we shall frequently use shortened notations such as τ (s, r) for τ (s, r, t,t) to save spaces.

Bilinear equation of contour integral type
The most fundamental bilinear equation is the equation z s ′ +r ′ −s−r e ξ(t ′ −t,z) τ (s ′ , r ′ , t ′ − [z −1 ],t ′ )τ (s, r, t + [z −1 ],t) that is understood to hold for arbitrary values of (s, r, t,t) and (s ′ , r ′ , t ′ ,t ′ ). This equation is a modification of the bilinear equation derived by Willox [17,18] in a fermionic construction of the tau function (see Section 2.3 below). Note that we have used the standard notations and both hand sides of the bilinear equation are contour integrals along simple closed cycles C ∞ (for integrals on the left hand side) and C 0 (for integrals on the right hand side) that encircle the points z = ∞ and z = 0. Actually, since these integrals simply extract the coefficient of z −1 from Laurent expansion at those points, we can redefine these integrals as a genuine linear map from Laurent series to constants: As we show below, this bilinear equation is a generating functional expression of an infinite number of Hirota equations. In some cases, it is more convenient to shift s and r as s → s + 1 and r → r + 1. The outcome is the equation By changing variables as z → z −1 on the right hand side, this equation can be converted to a more symmetric form as though we shall not pursue this line further.

Hirota equations
Following the standard procedure, we now introduce arbitrary constants a = (a 1 , a 2 , . . .),ā = (ā 1 ,ā 2 , . . .) and shift the continuous variables t, t ′ ,t,t ′ in the bilinear equation (2.1) as The bilinear equation thereby takes such a form as With the aid of Hirota's notations the product of two shifted tau functions in each term of this equation can be expressed as = e ξ(Dt,z −1 ) e a,Dt + ā,Dt τ (s, r, t,t) · τ (s ′ , r ′ , t,t), and a, D t and ā, Dt their linear combinations Let us introduce the functions h n (t), n ≥ 0, defined by the generating function h n (t)z n = e ξ(t,z) .
The first few terms read

Pfaff-Toda
The prefactors e ±2ξ(a,z) , etc., can be thereby expanded as Similarly, the exponential operators e ±ξ(Dt,z −1 ) , etc., can be expanded as The bilinear equation thus turns into the Hirota form The last equation is still a generating functional expression, from which one can derive an infinite number of equations by Taylor expansion of both hand sides at a = 0 andā = 0. For example, the linear part of the expansion give the equations for n = 0, 1, . . .. In particular, the special case of (2.4) where s ′ = s, r ′ = r and n = 1 gives the equation Moreover, specializing (2.3) to a =ā = 0, s ′ = s + 1 and r ′ = r − 1 yields the equation D t 1 τ (s, r) · τ (s + 1, r − 1) + Dt 1 τ (s, r − 1) · τ (s + 1, r) = 0. (2.5) These equations give the lowest part of the whole Hirota equations.

Fermionic formula of tau functions
Solutions of these bilinear equations are given by ground state expectation values of operators on the Fock space of 2D complex free fermions. Let us recall basic constituents of the fermion system. ψ j , ψ * j (j ∈ Z) denote the Fourier modes of fermion fields They obey the anti-commutation relations |0 and 0| denote the vacuum states characterized by the annihilation conditions The Fock space and its dual space are generated these vacuum states, and decomposed to eigenspaces of the charge operator The ground states of the charge-s subspace are given by |s = ψ s−1 · · · ψ 0 |0 for s > 0, ψ * s · · · ψ * −1 |0 for s < 0, s| = 0|ψ * 0 · · · ψ * s−1 for s > 0, 0|ψ −1 · · · ψ s for s < 0. where H(t) andH(t) are the linear combinations n H −n of H n 's, and g is an operator of the form Note that this operator, unlike H(t) andH(t), does not preserve charges, hence the foregoing expectation value can take nonzero values for r = 0. Let us mention that Willox's original definition [17,18] of the tau function is slightly different from (2.6). His definition reads This is certainly different from our definition; for example, Hirota equations are thereby modified. The difference is, however, minimal, because the two tau functions are connected by the simple relatioñ so that one can transfer from one definition to the other freely.
The bilinear equation (2.1) is a consequence of the identity satisfied by the operator g. This identity implies the equation where the abbreviated notations are used. This equation implies the bilinear equation (2.1) by the bosonization formulae and their duals

Relation to DKP hierarchy
The Pfaff-Toda hierarchy contains an infinite number of copies of the DKP hierarchy as subsystems. Such a subsystem shows up by restricting the variables in (2.1) as where l is a constant. (2.1) then reduces to the equation which is substantially the bilinear equation characterizing tau functions of the DKP hierarchy. Thus turns out to be a tau function of the DKP hierarchy with respect to t. One can derive another family of subsystems by restricting the variables as where l is a constant. (2.1) thereby reduces to the equation This is again equivalent to the bilinear equation for the DKP hierarchy. Thus is a tau function of the DKP hierarchy with respect tot.
3 Auxiliary linear problem

Wave functions and dressing operators
To formulate an auxiliary linear problem, we now introduce the wave functions Ψ * 1 (s, r, t,t, z) = z −s−r−2 e −ξ(t,z) τ (s + 1, r + 1, t + [z −1 ],t) τ (s, r, t,t) , and their duals Ψ 1 (s, r, t,t, z) = z s−r e ξ(t,z −1 ) τ (s + 1, r, t,t − [z]) τ (s, r, t,t) , These wave functions are divided to two groups with respect to the aforementioned two copies of the DKP hierarchy. When the discrete variables (s, r) are restricted on the line s = l+r, the first four (3.1) may be thought of as wave functions of the DKP hierarchy with tau function (2.7). Similarly, when (s, r) sit on the line s = l − r, the second four (3.2) are to be identified with wave functions of the DKP hierarchy with tau function (2.8). If the tau function is given by the fermionic formula (2.6), these wave functions, too, can be written in a fermionic form as As a consequence of the bilinear equation (2.1), these wave functions satisfy a system of bilinear equations. Those equations can be cast into a matrix form as .
Let us now introduce the dressing operators 1n e −n∂s , where w 1n , etc., are the coefficients of Laurent expansion of the tau-quotient in the wave functions (3.1) and (3.2), namely, The wave function can be thereby expressed as

Algebraic relations among dressing operators
A technical clue of the following consideration is a formula that connects wave functions and dressing operators. This formula is an analogue of the formula for the case where the dressing operators are pseudo-differential operators [30,31,32]. Let us introduce a few notations. For a pair of difference operators of the form let Ψ(s, z) and Φ(s, z) denote the wave functions Moreover, let P * denote the formal adjoint and (P ) s ′ s the "matrix elements" With these notations, the formula reads One can derive this formula by straightforward calculations, which are rather simpler than the case of pseudo-differential operators [30,31,32].
To illustrate the usage of this formula, we now derive a set of algebraic relations satisfied by the dressing operators from the bilinear equation (3.3) specialized to t ′ = t andt ′ =t. Since these relations contain dressing operators for two different values of r, let us indicate the (s, r) dependence explicitly as W (s, r), etc.
Proof . The (1, 1) component of the specialized bilinear equation reads By the key formula (3.4), each term of this equation can be expressed as Thus we find that the (1, 1) component of the specialized bilinear equation is equivalent to (3.5) for α = β = 1. The other components can be treated in the same way.
In particular, letting r ′ = r in (3.5), we obtain a set of algebraic relations satisfied by W , V , W ,V . We can rewrite these relations in the following matrix form, which turns out to be useful later on. Note that the formal adjoint of a matrix of operators is defined to be the transposed matrix of the formal adjoints of matrix elements as Corollary 1. The dressing operators satisfy the algebraic relation or, equivalently, Proof . Let us examine (3.5) in the case where r ′ = r. The (1, 1) component reads Among the four terms in this relation, W 1 e ∂s V * 1 andV 1 e −∂sW * 1 are linear combinations of e −∂s , e −2∂s , . . ., and V 1 e −∂s W * 1 andW 1 e ∂sV * 1 are linear combinations of e ∂s , e 2∂s , . . .. Therefore this relation splits into the two relations which are actually equivalent. In the same way, we can derive the relations from the (2, 2) component of (3.5). Let us now consider the (1, 2) component, which we rewrite as The left hand side is a sum of e ∂s and a linear combination of 1, e −∂s , . . ., and the right hand side is a sum of e ∂s and a linear combination of e 2∂s , e 3∂s , . . .. Therefore both hand sides should be equal to e ∂s , namely, By the same reasoning, we can derive the relations from the (2, 1) component of (3.5). These relations can be cast into a matrix form as (3.6) and (3.7).
(3.6) and (3.7) may be thought of as constraints preserved under time evolutions with respect to t andt. Actually, the discrete variable r, too, has to be interpreted as a time variable. Letting r ′ = r + 1 in (3.5), we can see how the dressing operators evolve in r.
Corollary 2. The dressing operators with r shifted by one are related to the unshifted dressing operators as Proof . When r ′ = r + 1, (3.5) reads These equations can be cast into a matrix form as Noting that we can use (3.6) and (3.7) to rewrite this equation as By the definition of the dressing operators, the left hand side of this equation is a matrix of operators of the form , and the right hand side take such a form as    w 10 (s, r + 1) w 10 (s − 1, r) e −∂s + · · · •e ∂s + •e 2∂s + · · · w 20 (s, r + 1) w 10 (s − 1, r) e −∂s + · · · •e ∂s + •e 2∂s + · · ·     . Consequently, Rewriting this result in terms of the tau functions, we obtain the formulae (3.9) of A, B, C.

Evolution equations of dressing operators
The dressing operators turn out to satisfy a set of evolution equations with respect to the continuous time variables t andt as well. To present the result, let us introduce the notations p n e n∂s , p n e n∂s of truncated operators for difference operators of the form Theorem 2. The dressing operators satisfy the equations where the subscript t n in W 1,tn , etc., stands for differentiating the coefficients of the difference operators by t n , and A n , B n , C n and D n are defined as Proof . Differentiate the bilinear equation (3.3) by t ′ n and specialize the variables to t ′ = t, t ′ =t, s ′ = s and r ′ = r. This leads to the equation 2×2 (s, r, z) · tΨ * 2×2 (s, r, z).
The t n -derivatives of the wave functions in this equation can be expressed as ∂ tn Ψ α (s, r, z) = (W α,tn + W α e n∂s )z s+r e ξ(t,z) , ∂ tn Ψ * α (s, r, z) = (V α,tn − V α e −n∂s )z −s−r e −ξ(t,z) , ∂ tnΨα (s, r, z) =W α,tn z s−r e ξ(t,z −1 ) , ∂ tnΨ * α (s, r, z) =V α,tn e −s+r e −ξ(t,z −1 ) . We now use the key formula (3.4) to convert the last equation to equations for the dressing operators. Those equations can be cast into a matrix form as By (3.6) and (3.7), this equation can be rewritten as e ∂sV * 2 e −∂s −e ∂sV * 1 e ∂s −e −∂s W * 2 e −∂s e −∂s W * 1 e ∂s of the matrix of these operators that can be derived from the foregoing construction and the algebraic relations (3.6) and (3.7). As regards A n , this implies that In much the same way, the following evolution equations int can be derived.

Auxiliary linear equations
The evolution equations (3.10), (3.11) and (3.8) for the dressing operators can be readily cast into auxiliary linear equations for the wave functions.
Corollary 3. The wave functions satisfy the following linear equations: (3.14) Note that each of (3.13), (3.14) and (3.12) is a collective expression of four sets of linear equations, namely, for the four pairs Φ α = Ψ α , Ψ * α ,Ψ α ,Ψ * α (α = 1, 2) of wave functions. The lowest (n = 1) equations of (3.13) and (3.14) agree with Willox's result [17,18]: Thus we have obtained auxiliary linear equations for the Pfaff-Toda hierarchy. Apart from the fact that difference operators play a central role, this auxiliary linear problem resembles that of the DKP hierarchy (see Appendix). This is a manifestation of the common Lie algebraic structure [2,3] that underlies these hierarchies.
Let us specify an algebraic structure in the building blocks of the auxiliary linear problem. Let U ,Ū , P n ,P n and J denote the matrix operators With these notations, (3.6) can be rewritten as This exhibits a Lie group structure (now realized in terms of difference operators). Moreover, (3.10) and (3.11) imply that P n andP n can be expressed as We can confirm by straightforward calculations that P n andP n satisfy the algebraic relations P * n = −JP n J −1 ,P * n = −JP n J −1 , which are obviously a Lie algebraic version of the foregoing constraints for U andŪ . In components, these relations read

Difference Fay identities 4.1 How to derive difference Fay identities
We now derive six Fay-like identities with parameters λ and µ from the bilinear equation (2.2) by specializing the free variables therein as follows: To clarify the meaning of calculations, we now assume that the integrals in (2.2) are contour integrals along simple closed curves C ∞ and C 0 encircling the points z = ∞ and z = 0 respectively. λ and µ are understood to be in a particular position specified below. 1a) and 1b). λ and µ are assumed to sit on the far side (closer to z = ∞) of the contour C ∞ . The exponential factors in the integrand thereby become rational functions as

Thus the bilinear equation (2.2) reduce to the equations
in the case of 1a) and C∞ dz 2πi in the case of 1b). The contour integrals in these equation can be calculated by residue calculus. For example, the first integral on the left hand side is given by the sum of residues of the integrand at z = λ, µ; the other contour integrals can be treated in the same way. The outcome are the equations ,t)τ (s, r + 1, t,t).

t + [λ] + [µ] + [z])τ (s + 1, r, t,t − [z])
in the case of 2a) and C∞ dz 2πi in the case of 2b). By residue calculus, we obtain the equations 3a) and 3b). λ and µ are assumed to be on the far side of C ∞ and inside C 0 respectively.

The bilinear equation (2.2) turn into the equations
in the case of 3a) and C∞ dz 2πi in the case of 3b), and boil down to the equations Actually, these calculations are meaningful even if the contour integrals are understood to be genuine algebraic operators that extract the coefficient of z −1 from Laurent series. Thus we are led to the following conclusion: In the rest of this paper, (4.1)-(4.6) are referred to as "difference Fay identities". The structure of these Fay-like equations is similar to the difference Fay identities of the Toda hierarchy [22,23,1], though the latter are three-term relations and given on a 1D lattice.

Relation to auxiliary linear problem
We now show that the difference Fay identities are closely related to the auxiliary linear equations. To this end, let us rewrite the identities in the language of the wave functions as follows.
Theorem 5. The difference Fay identities (4.1)-(4.6) are equivalent to the system of the following four equations: for the four pairs Φ α = Ψ α , Ψ * α ,Ψ α ,Ψ * α (α = 1, 2) of wave functions, where D(z) andD(z) denote the differential operators Proof . One can derive the four difference Fay identities from (4.7)-(4.10) by straightforward calculations. Actually, this turns out to be largely redundant, namely, each difference Fay identity appears more than once while processing the twelve equations of (4.7)-(4.10). Nevertheless the calculations on the whole, are reversible, proving the converse simultaneously. Since the whole calculations are considerably lengthy, let us demonstrate it by deriving (4.1) and (4.2) from (4.7) and (4.8) for Φ α = Ψ α (α = 1, 2); the other cases are fully parallel. Recall that Ψ α 's can be expressed as we thus obtain the equation for the tau function. Rewriting this equation as and shifting the variables as t → t + [λ −1 ] + [µ −1 ] and r → r + 1, we arrive at (4.2). Let us now consider (4.8), namely, This equation turns into the equation for the tau function, and upon shifting the variables as t → t + [µ −1 ], s → s + 2 and r → r + 1, reduces to (4.1). It will be obvious that these calculations are reversible.
Expanded in powers of λ, (4.7)-(4.10) generate an infinite set of linear equations for the wave functions Φ α (s, r, µ). As we show below, these linear equations are equivalent to the auxiliary linear equations (3.13) and (3.14). Thus (4.7)-(4.10) turn out to give a generating functional expression of these auxiliary linear equations.
Now rescale the variables s, r, t,t as The last equation thereby becomes an equation for the rescaled tau function τ , in which the derivatives are rescaled as Under the quasi-classical ansatz (5.1), we can take the limit of this equation as → 0. The outcome is the equation In much the same way, we can derive the following equations from the other differential Fay identities (4.2)-(4.6): Proof . We can rewrite (5.4) and (5.5) as Eliminating e D(λ)D(µ)F from these equations yields the equation which can be expanded as We can separate λ-dependent and µ-dependent terms to each side of the equation as Therefore both hand sides of the last equation are independent of λ and µ. Letting λ, µ → ∞, we can readily see that this quantity is equal to (∂ r − ∂ s )∂ t 1 F . Thus we obtain (5.10). In much the same way, we can derive (5.11) from (5.6) and (5.7). If we start from (5.8) and (5.9), we end up with the equation µ −1 e ∂sD(µ)F − e −∂rD(µ)F + µ e −∂s(∂r−∂s+D(µ))F − e ∂r (∂r−∂s+D(µ))F = e −∂r∂sF λ e −∂sD(λ)F − e −∂rD(λ)F + λ −1 e ∂s(∂r +∂s+D(λ))F − e ∂r(∂r+∂s+D(λ))F .
It is easy to see from this proof that the converse is also true. Namely, if (5.5), (5.7) and (5.9) holds, the other three dispersionless Hirota equations (5.4), (5.6) and (5.8) can be recovered from (5.10), (5.11) and (5.12). Thus we are led to the following conclusion. Thus the dispersionless Hirota equations of the Pfaff-Toda hierarchy can be reduced to two sets of equations that have quite different appearance and nature. Among the second set of equations, the last equation (5.12) is nothing but the dispersionless limit of (2.5). The other two equations (5.10) and (5.11) appear to be more mysterious. As it turns out below, they are related to special auxiliary linear equations for the wave functions.
A clue of the subsequent consideration is to introduce the auxiliary functions We can thereby rewrite these equations into the "Hamilton-Jacobi" form for the four pairs Φ α (z) = Ψ α (z), Ψ * α (z),Ψ α (z),Ψ * α (z) (α = 1, 2), where A = e ∂s + log τ (s + 1, r) τ (s, r + 1) t 1 + τ (s + 1, r + 1)τ (s − 1, r) τ (s, r + 1)τ (s, r) e −∂s , The second equation implies that Φ 1 (z) and Φ 2 (z) are connected by the relation which one can see from the definition (3.1) and (3.2) of the wave functions as well. Using this relation, we can eliminate Φ 2 (z) from the first equation and obtain the equation After some algebra, this equation boils down to (5.16). Moreover, though we omit details, the coefficients turn out to take such a form as It is easy to see that these coefficients do have the anticipated quasi-classical limit. The other auxiliary linear equations (3.13) and (3.14), too, can be converted to scalar linear equations of the form Note that the 2D difference operators K n (e ∂s , e ∂r ) = A n + B n e −∂r C,K n (e ∂s , e ∂r ) =Ā n +B n e −∂r C show up on the right hand sides of these equations. Consequently, the S-functions S(z) andS(z) satisfy the Hamilton-Jacobi equations ∂ tn S(z) = K n e ∂sS(z) , e ∂rS(z) , ∂t n S(z) =K n e ∂sS(z) , e ∂r S(z) , (5.17) ∂ tnS (z) = K n e ∂sS(z) , e ∂rS(z) , ∂t nS (z) =K n e ∂sS(z) , e ∂rS(z) .

Comparison with DKP hierarchy
Let us compare the foregoing results with the case of the DKP hierarchy [1,27]. The situation of the DKP hierarchy is similar to the KP hierarchy rather than the Toda hierarchy. The role of the difference Fay identities (4.1)-(4.6) are played by the differential Fay identities (5.20) In the dispersionless limit, they turn into the dispersionless Hirota equations for the F -function F (r, t). The first equation ( and eliminate e D(λ)D(µ)F from these two equations. This leads to the equality hence both hand sides are independent of λ and µ. Letting λ, µ → ∞, we can determine this quantity explicitly. Thus we obtain the equation In other words, the partially exponentiated gradient vector (∂ t 1 S(z), e ∂r S(z) ) of the S-function satisfies the algebraic equation of an elliptic curve on the (p, Q) plane. This curve was studied by Kodama and Pierce [27] as an analogue of the spectral curve of the 1D Toda hierarchy.
We can derive this equation from one of auxiliary linear equations as follows. Let us again consider the case where = 1. As in the case of the Pfaff-Toda hierarchy, we can eliminate Ψ 2 (z) and Ψ * 2 (z) from the auxiliary linear equation (A.7) by the relation that holds for Φ α (z) = Ψ α (z), Ψ * α (z) (α = 1, 2). The matrix equation (A.7) thereby reduces to a scalar equation of the form (5.25) for Φ 1 (z) = Ψ 1 (z), Ψ * 1 (z). The coefficients a, b, c can be determined explicitly as Rescaling the variables as (5.2), one can correctly recover the coefficients of (5.23) and (5.24) in the quasi-classical limit:
All these equations have counterparts in the DKP hierarchy. We have thus demonstrated that the Pfaff-Toda hierarchy is indeed a Toda version of the DKP hierarchy (or a Pfaffian version of the Toda hierarchy). Actually, this is not the end of the story. Let us note a few open problems. Firstly, although the other auxiliary linear equations (3.13) and (3.14) have been encoded to the difference Fay identities, the status of the remaining equation (3.12) is still obscure. Since its counterpart in the dispersionless limit are (5.13) and (5.14), and these equations are obtained from the dispersionless Hirota equations, it seems likely that (3.12), too, can be derived from the difference Fay identities. Unfortunately, we have been unable to find a direct proof. If this conjecture is true, it leads to an important conclusion that the difference Fay identities are, on the whole, equivalent to the Pfaff-Toda hierarchy itself, as it is indeed the case for the KP hierarchy [1,33] and the Toda hierarchy [23].
Secondly, very little is known about special solutions of the Pfaff-Toda hierarchy. Of course one can freely generate solutions by the fermionic formula. Finding an interesting class of solutions is, however, a nontrivial problem. A possible strategy will be to seek, again, for analogy with the DKP hierarchy.
The wave functions are associated with dressing operators of the form The coefficients are determined by Laurent expansion of the tau-quotient in the wave functions as The wave functions are thereby expressed as Ψ α (r, t, z) = W α z 2r e ξ(t,z) , Ψ * α (r, t, z) = V α z −2r e −ξ(t,z) for α = 1, 2.
Various equations for the dressing operators can be derived from this bilinear equation. A technical clue is an analogue of (3.4) for pseudo-differential operators [30,31,32]. For a pair of pseudo-differential operators of the form P = Then one has the identity where ( ) −k−1 stands for the coefficient of ∂ −k−1 x of a pseudo-differential operator. With the aid of this formula (A.3), one can derive the algebraic constraint the discrete evolution equation W 1 (r + 1) V 1 (r + 1) W 2 (r + 1) V 2 (r + 1) .5) and the continuous evolution equations from the bilinear equation (A.2) of the wave functions. Here A, B, C are differential operators of the form A = ∂ 2 t 1 + a∂ t 1 + b, B = − τ (r + 1) τ (r) , C = τ (r) τ (r + 1) , where a and b are the same quantities as shown in (5.26). A n , B n , C n are given by where ( ) ≥0 stands for the projection onto nonnegative powers of ∂ t 1 . These operators satisfy the algebraic relations A * n = −D n , B * n = B n , C * n = C n , D * n = −A n , which may be thought of as Lie algebraic counterparts of the constraint (A.4) for the dressing operators. This algebraic structure is generalized by Kac and van de Leur [3] to multicomponent hierarchies. Let us mention that these algebraic relations among A n , B n , C n , D n are also derived by Kakei [9] in a inverse scattering formalism. The evolution equations (A.5) and (A.6) can be readily converted to the evolution equations and for the wave functions.