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The gauge transformation is a powerful method to solve integrable hierarchies. In the discrete KP hierarchy, there are two basic gauge transformations, T D (q) = Λ(q) ∘ Δ ∘ q −1 and T I (r) = Λ−1 (r −1) ∘ Δ−1 ∘ r. This paper is concerned with several integrable properties of the discrete KP hierarchy found by means of gauge transformations. The successive applications of T I to the constrained discrete KP hierarchy have been presented. Meanwhile, we derive two novel discrete equations as the orbits of gauge transformations T D and T I for the constrained KP hierarchy. Moreover, with the help of the spectral representations and the gauge transformations, Fay-similar equations of the tau functions for the constrained discrete KP have been investigated. In addition, we also discuss the relation between the gauge transformation and an additional symmetry for the discrete KP hierarchy. [ABSTRACT FROM AUTHOR]

In this paper, we mainly study the gauge transformations of the constrained q-deformed Kadomtsev–Petviashvili (q-KP) hierarchy. Different from the usual case, we have to consider the additional constraints on the Lax operator of the constrained q-deformed KP hierarchy, since the form of the Lax operator must be kept when constructing the gauge transformations. For this reason, the selections of generating functions in elementary gauge transformation operators TD and TI must be very special, which are from the constraints in the Lax operator. At last, we consider the successive applications of n-step of TD and k-step of TI gauge transformations. [ABSTRACT FROM AUTHOR]

In this article, the successive application of three gauge transformation operators for the modified semidiscrete Kadomtsev–Petviashvili(mdKP) hierarchy has been provided. The commutativity of the Bianchi diagram of these gauge transformation operators is investigated. [ABSTRACT FROM AUTHOR]

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 -dimensional Toda lattice equation structure. Then the tau function of the BKP hierarchy generated by the gauge transformation is shown 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 proven. [Copyright &y& Elsevier]

In this paper, the gauge transformation of the constrained semi-discrete KP(cdKP) hierarchy is constructed explicitly by the suitable choice of the generating functions. Under the m-step successive gauge transformation Tm, we give the transformed (adjoint) eigenfunctions and the τ-function of the transformed Lax operator of the cdKP hierarchy. [ABSTRACT FROM AUTHOR]