On an integrable hierarchy derived from the isentropic gas dynamics

Abstract

In this paper we study a new hierarchy of equations derived from the system of isentropic gas dynamics equations where the pressure is a nonlocal function of the density. We show that the hierarchy of equations is integrable. We construct the two compatible Hamiltonian structures and show that the first structure has three distinct Casimirs while the second has one. The existence of Casimirs allows us to extend the flows to local ones. We construct an infinite series of commuting local Hamiltonians as well as three infinite series (related to the three Casimirs) of nonlocal charges. We discuss the zero curvature formulation of the system where we obtain a simple expression for the nonlocal conserved charges, which also clarifies the existence of the three series from a Lie algebraic point of view. We point out that the nonlocal hierarchy of Hunter–Zheng equations can be obtained from our nonlocal flows when the dynamical variables are properly constrained.