Chains of KP, semi-infinite1-Toda lattice hierarchy and Kontsevich integral

L A Dickey

doi:10.1155/s1110757x01000122

Abstract

There are well-known constructions of integrable systems that are chains of infinitely many copies of the equations of the KP hierarchy “glued” together with some additional variables, for example, the modified KP hierarchy. Another interpretation of the latter, in terms of infinite matrices, is called the1-Toda lattice hierarchy. One way infinite reduction of this hierarchy has all the solutions in the form of sequences of expanding Wronskians. We define another chain of the KP equations, also with solutions of the Wronsksian type, that is characterized by the property to stabilize with respect to a gradation. Under some constraints imposed, the tau functions of the chain are the tau functions associated with the Kontsevich integrals.

Highlights

This paper was motivated by the following arguments
There are well-known chains of infinitely many copies of the equations of the KP hierarchy “glued” together with some variables, like, for example, modified KP (see (2.2a), (2.2b), and (2.2c) below). The latter is a sequence of dressing operators of the KP hierarchy {wN} along with “gluing” variables {uN}
It can be shown that all the solutions are sequences of well-known Wronskian solutions to KP, each wN being represented by a determinant of Nth order

Summary

Introduction

This paper was motivated by the following arguments. There are well-known chains of infinitely many copies of the equations of the KP hierarchy “glued” together with some variables, like, for example, modified KP (see (2.2a), (2.2b), and (2.2c) below). Copyright c 2001 Hindawi Publishing Corporation Journal of Applied Mathematics 1:4 (2001) 175–193 2000 Mathematics Subject Classification: 35Q53, 35Q58, 37K10 URL: http://jam.hindawi.com/volume-1/S1110757X01000122.html It can be shown (see Proposition 2.3) that all the solutions are sequences of well-known Wronskian solutions to KP, each wN being represented by a determinant of Nth order. Every determinant is obtained from the preceding one by an extension of the Wronskian when a new function is added to the existing ones There is another situation where one deals with a sequence of Wronskian solutions of increasing order. The question we try to answer here is whether the sequence of Kontsevich tau functions is interesting by itself, by its limit Is it possible to complete it with “gluing” variables to obtain a chain of related KP equations similar to (2.2a), (2.2b), and (2.2c)? This is the conversion of Itzykson and Zuber’s [7] reasoning, and the reader can find the skipped details there

Semi-infinite 1-Toda lattice hierarchy

Solutions to the chain

Solutions in the form of series in Schur polynomials: stabilization