Gelfand hypergeometric functions as solutions to the 2-dimensional Toda-Hirota equations II

TL;DR

This paper constructs explicit solutions to the 2-dimensional Toda-Hirota equation using Gelfand hypergeometric functions on Grassmannians, linking hypergeometric systems, Radon transforms, and Bäcklund transformations.

Contribution

It introduces a novel method of solving the 2d Toda-Hirota equation with Gelfand hypergeometric functions, utilizing Laplace sequences and contiguity relations.

Findings

Solutions expressed via Gelfand hypergeometric functions on Grassmannians.

Established a connection between hypergeometric systems and the Radon transform.

Demonstrated the use of Bäcklund transformations to generate higher solutions.

Abstract

We construct solutions of the $2$-dimensional Toda-Hirota equation (2dTHE) expressed by the Gelfand hypergeometric function (Gelfand HGF) on the Grassmannian $\mathrm{GM}(2,N)$ of confluent or non-confluent type, which is labeled by a partition of $N$. A system of hyperbolic equations in $N$ complex variables is obtained from the differential equations which form a main body of the Gelfand hypergeometric system and characterize the image of Radon transform. We use the Laplace sequence of the system of hyperbolic operators to find an elementary seed solution of the 2dTHE and then use the B\"acklund transformation to obtain higher solutions expressed in terms of Gelfand HGF. In constructing the Laplace sequence, the contiguity relations (operators) for the Gelfand HGF play an important role.