Dynamic states of a continuum traffic equation with on-ramp

We study the phase diagram of the continuum traffic flow model of a highway with an on-ramp. Using an open boundary condition, traffic states and metastabilities are investigated numerically for several representative values of the upstream boundary flux $f_{\mathrm{up}}$ and for the whole range of the on-ramp flux $f_{\mathrm{rmp}}.$ An inhomogeneous but time-independent traffic state (standing localized cluster state) is found and related to a recently measured traffic state. Due to the density gradient near the on-ramp, a traffic jam can occur even when the downstream density is below the critical density of the usual traffic jam formation in homogeneous highways, and its structure varies qualitatively with $f_{\mathrm{rmp}}.$ The free flow, the recurring hump (RH) state, and the traffic jam can all coexist in a certain metastable region where the free flow can undergo phase transitions either to the RH state or to the traffic jam state. We also find two nontrivial analytic solutions. These solutions correspond to the standing localized cluster state and the homogeneous congested traffic state (one form of the traffic jam), which are observed in numerical simulations.