The determinant representation of the gauge transformation for the discrete KP hierarchy

Abstract

A successive gauge transformation operator T n+k for the discrete KP (dKP) hierarchy is defined, which is involved with two types of gauge transformations operators. The determinant representation of the T n+k is established and it is used to get a new τ function τ (n+k) of the dKP hierarchy from an initial τ . In this process, we introduce a generalized discrete Wronskian determinant and some useful properties of discrete difference operators.

Generalized dressing method for the extended two-dimensional Toda lattice hierarchy and its reductions

“…The evolution equations of w i in (6d) and (6e) can be proved by straightforward calculations with the help of (14a), (14b) and (16).…”

“…The constraint (25a) and (25b) are easy to check by passing the dressing form (16) of L and M and the definition (13) of w i to (25a) and (25b). To verify (25c), we remind that one cannot draw the conclusion L * (w * i ) = 0 from Lemma 3.5, even though L * (w * i ) = W −1 …”

“…For discrete and q-deformed integrable hierarchies such as discrete KP hierarchy and q-deformed KP hierarchy etc., dressing methods (or gauge transformations) have been constructed (see [10,16,22,30] and references therein). However, dressing method cannot be applied directly to the extended integrable hierarchies.…”

The compatibility of additional symmetry and gauge transformations for the constrained discrete Kadomtsev-Petviashvili hierarchy

“…The gauge transformation is one kind of powerful method to construct the solutions of the integrable systems for the continuous KP hierarchy [12,13,14,15,16,17], the dKP hierarchy [11,18,19] and the cdKP hierarchy [11], which in fact reflects the intrinsic integrability of the KP hierarchy and dKP hierarchy. Chau et al [12] introduce two kinds of elementary gauge transformation operators: the differential type T D and the integral type T I .…”

Addition Formulae of Discrete KP, q-KP and Two-Component BKP Systems

“…Similar to the pseudo differential operator of KP hierarchy, the difference operator ∆ can be used to define the discrete KP hierarchy [3,5,6]. By using τ (n; t) of the discrete KP hierarchy, we get the form of the Hirota bilinear equations as follows [8,10,11]: If we set y = 1…”