Multi-component Toda lattice hierarchy

Russian Mathematical Surveys. 2025. Vol. 80. No. 4(484). P. 47–120.

Takebe T., A. V. Zabrodin

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 intro-
duced earlier by Ueno and Takasaki. Our version contains N discrete vari-
ables 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.

Keywords: wave functions, Lax formalism, multi-component version, Zakharov–Shabat equation, tau-function, fermion technique, universal hierarchy

Publication based on the results of: Integrability in mathematical physics, representation theory, algebraic and differential geometry (2025)