On the (2+1)-dimensional extension of 1-dimensional Toda lattice hierarchy

Abstract

We introduce a (2+1)-dimensional extension of the 1-dimensional Toda lattice hierarchy. The hierarchy is given by two different formulations. For the first formulation, we obtain the bilinear identity for the τ-functions and construct explicit solutions expressed by Wronski determinants. For the second formulation, we obtain the slightly different bilinear identity and clarify the relation to the hierarchy arising from the homogeneous vertex representation of the 2-toroidal Lie algebra sl2tor, obtained by Kakei et al.