On tau-functions for the Toda lattice hierarchy

Abstract

We extend a recent result of [13] for the KdV hierarchy to the Toda lattice hierarchy. Namely, for an arbitrary solution to the Toda lattice hierarchy, we define a pair of wave functions, and use them to give explicit formulae for the generating series of k-point correlation functions of the solution. Applications to computing GUE correlators and Gromov-Witten invariants of the Riemann sphere are under consideration.

The matrix-resolvent method to tau-functions for the nonlinear Schrödinger hierarchy

“…Following the uniqueness argument of [43], in order to prove Lemma 1.2, we will first prove the following lemma.…”

On tau-functions for the KdV hierarchy

“…Remark 3. Both the formula (19) and the formula (21) are generalized to the Toda lattice hierarchy in [68]. For example, the following theorem is proved in [68].…”

“…See [68] for the precise definitions of the functions S i (n, t), Ω Toda i 1 ,...,i k (n, t), and of a pair of wave functions for the Toda lattice hierarchy.…”

Tau-functions for the Ablowitz--Ladik hierarchy: the matrix-resolvent method

“…By using a uniqueness argument given in [28] (see the Lemma 3 and the Proposition 2 of [28]) we will prove in Section 2 the following lemma.…”

“…Recently, the matrix-resolvent method for studying tau-structures (in the sense of [15]) of differential integrable systems was introduced in [6,7,29]. This method was extended to certain differential-difference integrable systems in [13,14,28]. Our aim is to further extend the matrix-resolvent method to the study of the tau-structure of the AL hierarchy.…”