Confluence of hypergeometric functions and integrable hydrodynamic type systems

TL;DR

This paper introduces a new class of integrable hydrodynamic systems derived from confluent Lauricella functions on Grassmannians, expanding the understanding of their mathematical structure and solutions.

Contribution

It constructs novel integrable hydrodynamic systems based on confluent Lauricella functions on Grassmannians, revealing their Jordan matrix form and detailed cases.

Findings

Hydrodynamic systems linked to confluent Lauricella functions are integrable and non-diagonalizable.

Explicit analysis of systems on Gr(2,5) and Gr(2,6) provides detailed case studies.

The systems satisfy degenerate Euler-Poisson-Darboux equations.

Abstract

It is known that a large class of integrable hydrodynamic type systems can be constructed through the Lauricella function, a generalization of the classical Gauss hypergeometric function. In this paper, we construct novel class of integrable hydrodynamic type systems which govern the dynamics of critical points of confluent Lauricella type functions defined on finite dimensional Grassmannian Gr(2,n), the set of 2xn matrices of rank two. Those confluent functions satisfy certain degenerate Euler-Poisson-Darboux equations. It is also shown that in general, hydrodynamic type system associated to the confluent Lauricella function is given by an integrable and non-diagonalizable quasi-linear system of a Jordan matrix form. The cases of Grassmannian Gr(2,5) for two component systems and Gr(2,6) for three component systems are considered in details.