Chains of KP, semi-infinite $1$-Toda lattice hierarchy and Kontsevich integral

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 the $1$-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.