The applications of the gauge transformation for the BKP hierarchy

Abstract

In this paper, we investigated four applications of the gauge transformation for the BKP hierarchy. Firstly, it is found that the orbit of the gauge transformation for the constrained BKP hierarchy defines a special (2 + 1)-dimensional Toda lattice equation structure. Then the tau function of the BKP hierarchy generated by the gauge transformation is showed to be the Pfaffian. And the higher Fay-like identities for the BKP hierarchy is also obtained through the gauge transformation.At last, the compatibility between the additional symmetry and the gauge transformation of the BKP hierarchy is proved. . 1 Here ∂ = ∂x and the asterisk stands for the conjugation operation: (AB) * = B * A * , ∂ * = −∂, f * = f with f be a function.1 gauge transformation operator can be the one of the usual BKP hierarchy just by letting the generating function be one of the original eigenfunctions in the definition. Furthermore, it is an interesting problem to explore the non-trivial relevance of the gauge transformation to the other integrable properties of the BKP besides the explicit solutions, which shall be illustrated from the following four concerns. We firstly show that the orbit of the gauge transformation for the constrained BKP hierarchy defines a special (2 + 1)-dimensional Toda lattice equation structure (see (46)), proposed by Cao et al in [16]. This equation has some importance in mathematics and physics, whose continuous analogue is equivalent to the Ito equation [17]. And some interesting integrable properties of this special (2 + 1)-dimensional Toda lattice equation can be seen in [16][17][18][19].Then starting from the Grammian determinant solutions of the BKP hierarchy, we derived two types of the Pfaffian structures [20,21] for the BKP hierarchy (see Proposition 4). Since the BKP hierarchy is the subhierarchy of the KP hierarchy, there are two types of the tau functions for the BKP hierarchy [15]: one is inherited from the KP hierarchy (denoted by τ KP ) by cancelling the even flow variables, the other is of its own (denoted by τ ) defined directly by the odd variables according to the flow equations. In [9], the transformed tau function under the gauge transformation of the BKP hierarchy is only provided for τ KP in the form of the Grammian determinant [22], while the transformed tau function of its own is not considered. The crucial point to derive the transformed tau function of BKP's own is to root the Grammian determinant according to the relation of these two types of the tau functions [15]. In this paper, we derived the two types of the Pfaffian structures for the BKP hierarchy, according to the even or odd steps of the gauge transformation. In particular, for the case of the even times, it can be obtained directly by using the definition of the Pfaffian, while the case of the odd times is some complicated.Further through the gauge transformation of the BKP hierarchy, the higher order Fay-like identities [23][24][25] are derived. The transformed tau function under the k + 1-step gauge transformation can be e...

The gauge transformation of the q-deformed modified KP hierarchy

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BKP tau-functions as square roots of KP tau-functions

“…Remark 9. In the paper [9], Cheng and He, obtain BKP tau-functions via gauge transformations starting with some initial BKP tau-function. The Pfaf an BKP tau-functions they obtain are the same ones as appear in theorem 5, if one starts with the initial tau-function τ 0 = 1 and puts additional constraints on the BKP eigenfunctions, viz that each Φ j is of the form…”

The gauge transformation of the modified KP hierarchy

“…The gauge transformation [2,26,27], also called Darboux transformation, provides a simple way to construct solutions for integrable hierarchies. By now, the gauge transformations of many integrable hierarchies have been studied, for example, the KP and mKP hierarchies [1][2][3]9,26,27,29,30,34], the BKP and CKP hierarchies [4,10,12,25], the discrete KP and modified discrete KP hierarchies [13,20,21,24,28], the q-KP and modified q-KP hierarchies [5,11,22,23,33] and so on. Here in this paper, we will investigate the gauge transformation of the mKP hierarchy in Kupershmidt-Kiso version.…”