Confluence of hypergeometric functions and integrable hydrodynamic type systems

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.