Integrable Dispersionless KdV Hierarchy With Self-consistent Sources

Both soliton equations with self-consistent sources and dispersionless integrable systems have wide and deep applications in mathematics and physics, but the dispersionless integrable systems with self-consistent sources have not been studied, neither have the dispersionless KdV hierarchy with self-consistent sources been studied. Via the quasiclassical limit method, this article deduces the dispersionless KdV hierarchy with self-consistent sources as well as their conservation equations for the first time. Then the bi-Hamiltonian formulation for dKdVHWS is formulated and hodograph solution for the dispersionless KdV equation with self-consistent sources (dKdVWS) is obtained via the hodograph transformation. As the generalized case, the dispersionless Gelfand-Dickey hierarchy with self-consistent sources (dGDHWS) as well as its conservation equations is presented by taking the dispersionless limit of the Gelfand-Dickey hierarchy with self-consistent sources and its Lax pair. Finally, the dispersionless modified Gelfand-Dickey hierarchy is deduced and the dispersionless Miura link between the dispersionless Gelfand-Dickey hierarchy and the dispersionless modified Gelfand-Dickey hierarchy is constructed.