Symmetries Of The Discrete KP Hierarchy And Their Applications

A successive gauge transformation operator Tn+k for the discrete KP(dKP) hi-erarchy is defined, which includes two types of gauge transformation operators. The determinant representation of the operator Tn+k is established and it is used to obtain a newτfunctionτ(n+k)Δof the discrete-KP hierarchy from an initialτΔfunction. In this process, a generalized discrete Wronskian determinant is introduced and some useful properties for the discrete difference operators are obtained. Another two kinds of sym-metries, Sato's Backlund transformations and additional symmetries, for the discrete KP(dKP) hierarchy are also introduced. And the ASvM formula which demonstrates the equivalence of these two kinds of symmetries is obtained. In additional, the Fay identity and the difference Fay identity of the discrete-KP hierarchy are introduced and the ASvM formula in the form of theτfunction is calculated.