Vertex Operator Solutions to the Discrete KP-Hierarchy

Abstract

Contents 1 The KP τ -functions, Grassmannians and a residue formula 7 2 The existence of a τ -vector and the discrete KP bilinear identity 13 3 Sequences of τ -functions, flags and the discrete KP equation 18 4 Discrete KP-solutions generated by vertex operators 23 5 Example of vertex generated solutions: the q-KP equation 24Vertex operators, which are disguised Darboux maps, transform solutions of the KP equation into new ones. In this paper, we show that the bi-infinite * The final version appeared in: Comm.

The Spectrum of Coupled Random Matrices

“…In [2], we discussed the discrete KP hierarchy and found a general method for generating its solutions, in both, the bi-and semi-infinite situations; this paper mainly deals with the semi-infinite case. In [2] and [4], we gave an application of the bi-infinite discrete KP to the q-KP equation.…”

“…of the weights; they imply for the matrix L the so-called "discrete KPhierarchy" in t; this hierarchy is fully described in [2], and a large class of solutions is explained in section 1.…”

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

“…which satisfies L(n)Φ(n; t, z) = zΦ(n; t, z), L * (n)Φ * (n; t, z) = zΦ * (n; t, z), (19) and ∂ t j Φ(n; t, z) = B j (n)Φ(n; t, z), ∂ t j Φ * (n; t, z) = −B * j (n − 1)Φ * (n; t, z). (20) Also one has a tau function τ (n; t) for the discrete KP hierarchy [20], which satisfies…”

Semiclassical expansions in the Toda hierarchy and the Hermitian matrix model

“…is a given set of complex constants. It turns out [3] that the functions ψ n (z, t) := P n (z, t) exp…”

“…As a consequence of the activity in this field a rich theory of the different facets of the Toda hierarchy has been developed [1][2][3][4][5][6][7][8][9][10].…”