Pfaffian form of the Grammian determinant solutions of the BKP hierarchy

The Grammian determinant type solutions of the KP hierarchy, obtained through the vectorial binary Darboux transformation, are reduced, imposing suitable differential constraint on the transformation data, to Pfaffian solutions of the BKP hierarchy. On leave of absence from Beijing Graduate School, CUMT, Beijing 100083, China Supported by Beca para estancias temporales de doctores y tecnólogos extranjeros en España: SB95-A01722297 Partially supported by CICYT: proyecto PB95–0401 1 1. Binary Darboux transformations for the KP hierarchy [7] after iteration give Grammian determinant expressions for new potentials and wave functions. For the BKP hierarchy [2] instead one finds Pfaffian expressions, see [3] for a bilinear approach, dressing the zero background, and [6] for direct Darboux transformation approach (in both papers only the BKP equation was considered, not the hierarchy); in fact the appearance of Pfaffians was described in [2]. As the BKP hierarchy is a reduction of the KP hierarchy [2], it is natural to expect a relation between Grammian and Pfaffian expressions. That is, the Grammian solutions of the KP hierarchy reduces to Pfaffian solutions of the BKP hierarchy, when suitable constraints are impossed in the transformation data. In this short note we show that this correspondece holds. The letter is organized as follows. Next, in §2 we remind the reader about some basic facts regarding the KP hierarchy and the vectorial binary Darboux transformation [4], and also about the BKP hierarchy and how to reduce the mentioned vectorial transformation to it. Section 3 is devoted to show that these Grammian expressions reduce to Pfaffians. 2. The KP hierarchy can be formulated as the compatibility of the following linear system: ∂ψ ∂tn = Bn(ψ), n ≥ 1, (1) where Bn := ∂ n + n−2 ∑ m=0 un,m∂ , with ∂ := ∂ ∂x , x = t1. There is also an adjoint linear system ∂ψ ∂tn = −B n(ψ ), n ≥ 1, (2) where B n := (−1) ∂ + n−2 ∑ m=0 (−1)∂un,m. The functions un,m are differential polynomials in u2,0 =: u. 2 The vectorial binary Darboux transformation is constructed in terms of the potential defined by the following compatible equations: ∂Ω(ψ,ψ) ∂tn = − res(∂ψ ⊗B nψ ∂), n ≥ 1, where we are working in the ring of pseudodifferential operators in ∂, see for example [7], and assuming that ψ,ψ take values in a vectorial spaces V and W ∗ (the dual of W ), respectively. The first equation is ∂(Ω(ψ,ψ)) = ψ ⊗ψ, that for V = V ∗ = C has the form of a Grammian matrix when ψ = ψ. Vectorial Binary Darboux Transformation [7, 4]. For any pair of vectorial wave and adjoint wave functions, ξ ∈ V = C , ξ ∈ V , respectively, we construct a new potential ũ and wave function ψ̃, ũ = u− ∂(ln det Ω(ξ, ξ)), ψ̃ = ψ − Ω(ψ, ξ)Ω(ξ, ξ)ξ, = 1 |Ω(ξ, ξ)| ∣