Multi-component Toda lattice hierarchy

We give a detailed account of the $N$-component Toda lattice hierarchy, which can be regarded as a generalization of the well-known Toda chain model and its non-abelian version. This hierarchy is an extension of the one introduced earlier by Ueno and Takasaki. Our version contains $N$ discrete variables rather than one. We start from the Lax formalism, deduce the bilinear relation for wave functions from it, and then, based on the latter, prove the existence of the tau-function. We also show how the multi-component Toda lattice hierarchy is embedded into the universal hierarchy, which is basically the multi-component Kadomtsev–Petviashvili hierarchy. Finally, we show how the bilinear integral equation for the tau-function can be obtained using the free fermion technique. An example of exact solutions (a multi-component analogue of one-soliton solutions) is given.