Confluence of hypergeometric functions and integrable hydrodynamic type systems

摘要

It is known that a large class of integrable hydrodynamic type systems can beconstructed through the Lauricella function, a generalization of the classicalGauss hypergeometric function. In this paper, we construct novel class ofintegrable hydrodynamic type systems which govern the dynamics of criticalpoints of confluent Lauricella type functions defined on finite dimensionalGrassmannian Gr(2,n), the set of 2xn matrices of rank two. Those confluentfunctions satisfy certain degenerate Euler-Poisson-Darboux equations. It isalso shown that in general, hydrodynamic type system associated to theconfluent Lauricella function is given by an integrable and non-diagonalizablequasi-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 areconsidered in details.