Gelfand hypergeometric function as a solution to the 2-dimensional Toda-Hirota equation

TL;DR

This paper demonstrates that the Gelfand hypergeometric function provides explicit solutions to the 2-dimensional Toda-Hirota equation by linking it to Euler-Poisson-Darboux equations and root vector relations.

Contribution

It establishes a novel connection between Gelfand hypergeometric functions and solutions to the 2d Toda-Hirota equation using Euler-Poisson-Darboux equations.

Findings

Gelfand HGF solves the 2d Toda-Hirota equation.

Solutions are expressed via Euler-Poisson-Darboux equations.

Contiguity relations for Gelfand HGF are utilized.

Abstract

We construct solutions of the 2-dimensional Toda-Hirota equation (2dTHE) expressed by the solutions of the system of so-called Euler-Poisson-Darboux equations (EPD) in N complex variables. The system of EPD arises naturally from the differential equations which form a main body of the system characterizing the Gelfand hypergeometric function (Gelfand HGF) on the Grassmannian GM$(2,N)$. Using this link and the contiguity relations for the Gelfand HGF, which are constructed from root vectors for the root $\epsilon_i-\epsilon_j$ for $\mathfrak{gl}(N)$, we show that the Gelfand HGF gives solutions of the 2dTHE.