Several Integrable Properties of the Discrete KP Hierarchy Obtained by Means of Gauge Transformations

Abstract

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.