Gelfand hypergeometric functions on GM(2,N), for any partition of N, give explicit solution families of the 2-dimensional Toda-Hirota equation via Laplace sequences and Bäcklund transformations.
Gelfand hypergeometric function as a solution to the 2-dimensional Toda-Hirota equation
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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.
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Gelfand hypergeometric functions as solutions to the 2-dimensional Toda-Hirota equations II
Gelfand hypergeometric functions on GM(2,N), for any partition of N, give explicit solution families of the 2-dimensional Toda-Hirota equation via Laplace sequences and Bäcklund transformations.